Project Web Access quick reference guide for managers Web viewInstructional Strategies. Inferential statistics based on Normal probability models is a topic for Advanced Placement

Ohio’s Learning Standards - Clear Learning TargetsIntegrated Math III(listing of included sources attached)

Understand and evaluate random processes underlying statistical experiments.

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. would a result of 5 tails in a row cause you to question the model?

Essential UnderstandingStudents need to know that data collection and the analysis of the data influence most areas of our lives; these analyses are what we call statistics and are important to our health, wealth, and happiness, when applied appropriately.

Students need to be able to identify whether a particular statistical model is effective in a particular context.

Students need to know that data can be distorted in several ways; bad samples result from the use of inappropriate methods to collect data and will bias the results.

Extended UnderstandingStudents should be provided opportunities to use technology to make it possible to generate plots, regression functions, and correlation coefficients, and to simulate many possible outcomes in a short amount of time.

Students should be provided opportunities to make connections to functions and modeling. Functions may be used to describe data; if the data suggest a linear relationship, the relationship can be modeled with a regression line, and its strength and direction can be expressed through a correlation coefficient.

Academic Vocabulary/ Languagebiased, binomial distribution, bivariate, categorical, cluster, confidence interval, convenience, data, empirical rule, independence test, inference, judgment, margin of error, mean, measure of center, measures of spread, median, mode, nonrandom samples, normal distribution, null hypothesis, outlier, paired t-test, population, population mean, population proportion, purposive, p-value, qualitative, quantitative, quota, random, sample, sample mean, sample proportion, sample survey, significance, simulation, simple, systematic, stratified, skew, snowball, spread, standard deviation, treatment, t-test, uniform, univariate, variance

Tier 2 Vocabularyconsistent, experiment, evaluate, process, understand

CCSSM Description Decisions or predictions are often based on data—numbers in context. These decisions or predictions would be easy if the data always sent a clear message, but the message is often obscured by variability. Statistics provides tools for describing variability in data and for making informed decisions that take it into account. This section will require students to determine the correct mathematical model and to use the model appropriately to solve problems.

Columbus City Schools Clear Learning Targets Integrated Math III 2016-2017

S.IC.1, IC.2

1

I Can Statements I can choose and use appropriate mathematics to analyze situations.

Instructional StrategiesInferential statistics based on Normal probability models is a topic for Advanced Placement Statistics (e.g., t-tests). The idea here is that all students understand that statistical decisions are made about populations (parameters in particular) based on a random sample taken from the population and the observed value of a sample statistic (note that both words start with the letter “s”). A population parameter (note that both words start with the letter “p”) is a measure of some characteristic in the population such as the population proportion of American voters who are in favor of some issue, or the population mean time it takes an Alka Seltzer tablet to dissolve. As the statistical process is being mastered by students, it is instructive for them to investigate questions such as “If a coin spun five times produces five tails in a row, could one conclude that the coin is biased toward tails?” One way a student might answer this is by building a model of 100 trials by experimentation or simulation of the number of times a truly fair coin produces five tails in a row in five spins. If a truly fair coin produces five tails in five tosses 15 times out of 100 trials, then there is no reason to doubt the fairness of the coin. If, however, getting five tails in five spins occurred only once in 100 trials, then one could conclude that the coin is biased toward tails (if the coin in question actually landed five tails in five spins). A powerful tool for developing statistical models is the use of simulations. This allows the students to visualize the model and apply their understanding of the statistical process. Provide opportunities for students to clearly distinguish between a population parameter which is a constant, and a sample statistic which is a variable.

Columbus City Schools Clear Learning Targets Integrated Math III 2016-2017 2

Common Misconceptions/ChallengesStudents may believe: That population parameters and sample statistics are one in the same, e.g., that there is no difference between the population mean which is a constant and the sample mean which is a variable. Making decisions is simply comparing the value of one observation of a sample statistic to the value of a population parameter, not realizing that a distribution of the sample statistic needs to be created.

Common Core SupportInstitute for Mathematics and Education Learning Progressions NarrativesHigh School Statistics and Probabilityhttp://commoncoretools.me/wp-content/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf

Ohio Department of Education Model Curriculum http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Statistics-and-Probability_Model-Curriculum_March2015.pdf.aspx

Illustrative Mathematicshttps://www.illustrativemathematics.org/blueprints/M3/5


https://www.illustrativemathematics.org/blueprints/M3/5

http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Statistics-and-Probability_Model-Curriculum_March2015.pdf.aspx

http://commoncoretools.me/wp-content/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf

Textbook and Curriculum ResourcesIntegrated Math III, McGraw HillChapter 10

Adventures with Mathematics: Statistics and Probability: Aligned With the Common Core State Standards, MCTM, 2012Ants on the Loose, p. 14Forecasting the Weather, p.20When Bambi Hits the Blacktop, p. 22Scrabble Express Vowel Bias, p. 28The Magical Number 7, p. 36The Definite Activity, p. 38Which Gum Lasts Longer, p. 40An A-MAZE-ING Comparison, p. 42The Spelling Bee, p. 44Archaeological Sampling, p. 46

Prior KnowledgeThe four-step statistical process was introduced in Grade 6, with the recognition of statistical questions. At the high school level, students need to become proficient in all the steps of the statistical process. Using simulation to estimate probabilities is a part of the Grade 7 curriculum as is initial understanding of using random sampling to draw inferences about a population.

Future LearningNext lessons will include making inference and justifying conclusions from sample surveys, experiments, and observational studies.


Performance Assessments/TasksClick on the links below to access performance tasks.NRICH http://nrich.maths.org/public/search.php?search=statistics

Illustrative MathematicsInterpreting Data: Muddying the Watershttp://map.mathshell.org/download.php?fileid=1774

Career ConnectionsStudents can explore the concepts of direct marketing, a marketing database, and a sales promotion as described in the High School Operations Research Modules (http://hsor.org/modules.cfm?name=Gamz_Inc). Use the provided case studies to lead a discussion on how this content is critical to tasks performed across various career fields (e.g., business, marketing, finance). Students will use the discussion to guide their research of related careers for developing future career goals.


http://hsor.org/modules.cfm?name=Gamz_Inc

http://map.mathshell.org/download.php?fileid=1774

http://nrich.maths.org/public/search.php?search=statistics


Make inferences and justify conclusions from sample surveys, experiments, and

observational studies. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each other.

Use data from a sample survey to estimate a population mean or proportion: develop a margin of error through the use of simulation models for random sampling.

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Evaluate reports based on data.

Essential Understanding Students should know and understand the different methods of data collection, specifically the difference between an observational study and a controlled experiment, and know the appropriate use for each.

Students should be able to choose and use appropriate mathematics to analyze situations; students should be able to determine the correct mathematical model and use the model to solve problems.

Extended UnderstandingWhile virtually all aspects of our human experience have benefited from a responsible use of statistics, data can be presented in ways that are misleading. At times, this occurs through carelessness or ignorance but other times it is designed to be deceptive for the purpose of obscuring unfavorable data or accentuating data, which supports a certain point of view. Provide an opportunity for students to use their skills and identify information found in magazines, newspapers, on television, and via the Internet that consumers should all be cautious of due to potential misuses and abuses of statistical data.

Academic Vocabulary/ Languagebiased, binomial distribution, bivariate, categorical, cluster, confidence interval, convenience, data, empirical rule, independence test, inference, judgment, margin of error, mean, measure of center, measures of spread, median, mode, nonrandom samples, normal distribution, null hypothesis, outlier, paired t-test, population, population mean, population proportion, purposive, p-value, qualitative, quantitative, quota, random, sample, sample mean, sample proportion, sample survey, significance, simulation, simple, systematic, stratified, skew, snowball, spread, standard deviation, treatment, t-test, uniform, univariate, variance


CCSSM DescriptionStatistical inference refers simply to the process of drawing conclusions from statistical data. Students need to be able to identify whether a particular model is effective in a particular context. Data are gathered, displayed, summarized, examined, and interpreted to discover patterns and deviations from patterns. Which statistics to compare, which plots to use, and what the results of a comparison might mean, depend on the question to be investigated and the real-life actions to be taken. The conditions under which data are collected are important in drawing conclusions from the data. In critically reviewing uses of statistics in public media and other reports, it is important to consider the study design, how the data were gathered, and the analyses employed as well as the data summaries and the conclusions drawn.

I Can StatementsI can choose and use appropriate mathematics to analyze situations.

I can estimate a sample mean or sample proportion given data from a sample survey; I can estimate the population value.

I can determine the correct mathematical model and use the model to solve problems.


S.IC.3, S.IC.4, S.IC.5, S.IC.6

6

Instructional StrategiesThis cluster is designed to bring the four-step statistical process (GAISE model) to life and help students understand how statistical decisions are made. The mastery of this cluster is fundamental to the goal of creating a statistically literate citizenry. Students will need to use all of the data analysis, statistics, and probability concepts covered to date to develop a deeper understanding of inferential reasoning. Students learn to devise plans for collecting data through the three primary methods of data production: surveys, observational studies, and experiments. Randomization plays various key roles in these methods. Emphasize that randomization is not a haphazard procedure, and that it requires careful implementation to avoid biasing the analysis. In surveys, the sample selected from a population needs to be representative; taking a random sample is generally what is done to satisfy this requirement. In observational studies, the sample needs to be representative of the population as a whole to enable generalization from sample to population. The best way to satisfy this is to use random selection in choosing the sample. In comparative experiments between two groups, random assignment of the treatments to the subjects is essential to avoid damaging problems when separating the effects of the treatments from the effects of some other variable, called confounding. In many cases, it takes a lot of thought to be sure that the method of randomization correctly produces data that will reflect that which is being analyzed. For example, in a two-treatment randomized experiment in which it is desired to have the same number of subjects in each treatment group, having each subject toss a coin where Heads assigns the subject to treatment A and Tails assigned the subject to treatment B will not produce the desired random assignment of equal-size groups. The advantage that experiments have over surveys and observational studies is that one can establish causality with experiments.

Also addressed with these standards estimation of the population proportion parameter and the population mean parameter. Data need not come from just a survey to cover this topic. A margin-of-error formula cannot be developed through simulation, but students can discover that as the sample size is increased, the empirical distribution of the sample proportion and the sample mean tend toward a certain shape (the Normal distribution), and the standard error of the statistics decreases (i.e. the variation) in the models becomes smaller. The actual formulas will need to be stated.

Finally, this cluster of standards addresses testing whether some characteristic of two paired or independent groups is the same or different by the use of resampling techniques. Conclusions are based on the concept of p-value. Resampling procedures can begin by hand but typically will require technology to gather enough observations for which a p-value calculation will be meaningful. Use a variety of devices as appropriate to carry out simulations: number cubes, cards, random digit tables, graphing calculators, computer programs.

Common Misconceptions/ChallengesStudents may believe:That collecting data is easy; asking friends for their opinions is fine in determining what everyone thinks.That causal effect can be drawn in surveys and observational studies, instead of understanding that causality is in fact a property of experiments.That inference from sample to population can be done only in experiments. They should see that inference can be done in sampling and observational studies if data are collected through a random process

Common Core SupportInstitute for Mathematics and Education Learning Progressions NarrativesHigh School Statistics and Probabilityhttp://commoncoretools.me/wp-content/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf







Textbook and Curriculum Resourceshttp://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17Integrated Math III, McGraw HillChapter 10

Adventures with Mathematics: Statistics and Probability: Aligned With the Common Core State Standards, MCTM, 2012 Forecasting the Weather, p. 20When Bambi Hits the Blacktop, p. 22The Blob, p. 25Scrabble Express Vowel Bias, p. 28The State of Drunk Driving, p. 30The Magical Number 7, p. 36The Definite Activity, p. 38Which Gum Lasts Longer?, p. 40An A-MAZE_ING Comparison, p. 42The Spelling Bee, p. 44Archaeological Sampling, p. 46

Prior KnowledgeThe four-step statistical process was introduced in middle school, with the first step likely more often generated by teachers than students. At the high school level, students need to become proficient in the first step of generating meaningful questions, as well as designing a plan to collect their data using the three primary methods: surveys, observational studies, and experiments. Using simulation to estimate probabilities is a part of the Grade 7 curriculum, as is introductory understanding of using random sampling to draw inferences about a population

Future LearningNext lessons will include interpreting categorical and quantitative data.


http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17


Illustrative MathematicsInterpreting Data: Muddying the Watershttp://map.mathshell.org/download.php?fileid=1774

Interpreting and Using Data: Testing a New Producthttp://map.mathshell.org/download.php?fileid=1704

Career ConnectionsStatistics is the study of data organization to provide specific information and for measuring and determining uncertainty and probability. The discipline can apply to problems in economics, engineering, education, biology and sports. Some of its uses include sports information for baseball players, calculations for car insurance premiums and analyses of business efficiency.






Summarize, represent, and interpret data on a

single count or measurement variable.Use the mean and standard deviation of a data set to fit it to the normal distribution and to estimate population percentages. Recognize that there are data sets which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Essential UnderstandingStudents should be able to recognize that there are data sets for which such a procedure is not appropriate.

Student should know that when data is notably skewed or when meaningful outliers are present, the median should be used to describe the distribution.

Students should know that the mean and standard deviation should be used to describe unimodal and symmetric data.

Students should use summary statistics and/or graphical representations to write critical analyses of a situation within the context of the given data.

Extended Understanding Opportunities should be provided for students to work through the statistical process. Teachers and students should make extensive use of resources in order to perfect; make use global web resources for projects.

Academic Vocabulary/ Language biased, binomial distribution, bivariate, categorical, cluster, confidence interval, convenience, empirical rule, independence test, inference, judgment, margin of error, mean, measure of center, measures of spread, median, mode, nonrandom samples, normal distribution, null hypothesis, outlier, paired t-test, population, population mean, population proportion, purposive, p-value, qualitative, quantitative, quota, random, sample, sample mean, sample proportion, sample survey, significance, simulation, simple, systematic, stratified, skew, snowball, spread, standard deviation, treatment, t-test, uniform, unimodal distribution, univariate, variance


CCSSM DescriptionStudents will calculate and use summary statistics such as mean, median, range, lower and upper quartile, interquartile range and standard deviation to help describe the shape of data. The processes by which mean and median are calculated have been previously taught; however, students have not been introduced to standard deviation, and must understand the process behind the calculation. Technology should be used to calculate the standard deviation. Students will build on their understanding of these calculations to comment on possible outliers in a data set and to make well-informed decisions about the best summary statistics to represent given data.

I Can Statements

I can use the normal distribution to make estimates of frequencies (which can be expressed as probabilities.

I can recognize that only some data are well described by a normal distribution.

I can describe the characteristics of a normal distribution.

I can use a calculator, spreadsheet, and table to estimate areas under the normal curve

I can use the mean and standard deviation of a data set to fit it to a normal distribution.

I can use normal distribution to estimate population percentages.


S.ID.4

10

Instructional Strategies It is helpful for students to understand that a statistical process is a problem-solving process consisting of four steps: formulating a question that can be answered by data; designing and implementing a plan that collects appropriate data; analyzing the data by graphical and/or numerical methods; and interpreting the analysis in the context of the original question. Opportunities should be provided for students to work through the statistical process. The richer the question formulated, the more interesting is the process. Teachers and students should make extensive use of resources to perfect this very important first step. Global web resources can inspire projects. Although this domain addresses both categorical and quantitative data, there is no reference to categorical data. This would be a good place to discuss graphs for one categorical variable (bar graph, pie graph) and measure of center (mode). Have students practice their understanding of the different types of graphs for categorical and numerical variables by constructing statistical posters. Note that a bar graph for categorical data may have frequency on the vertical (student’s pizza preferences) or measurement on the vertical (radish root growth over time - days). Measures of center and spread for data sets without outliers are the mean and standard deviation, whereas median and interquartile ranges are better measures for data sets with outliers. Introduce the formula of standard deviation by reviewing the previously learned MAD (mean absolute deviation). The MAD is very intuitive and gives a solid foundation for developing the more complicated standard deviation measure. Informally observing the extent to which two boxplots or two dotplots overlap begins the discussion of drawing inferential conclusions. Don’t shortcut this observation in comparing two data sets. As histograms for various data sets are drawn, common shapes appear. To characterize the shapes, curves are sketched through the midpoints of the tops of the histogram’s rectangles. Of particular importance is a symmetric unimodal curve that has specific areas within one, two, and three standard deviations of its mean. It is called the Normal distribution and students need to be able to find areas (probabilities) for various events using tables or a graphing calculator.

Common Misconceptions/ChallengesStudents may believe:That a bar graph and a histogram are the same. A bar graph is appropriate when the horizontal axis has categories and the vertical axis is labeled by either frequency (e.g., book titles on the horizontal and number of students who like the respective books on the vertical) or measurement of some numerical variable (e.g., days of the week on the horizontal and median length of root growth of radish seeds on the vertical). A histogram has units of measurement of a numerical variable on the horizontal (e.g., ages with intervals of equal length).

That the lengths of the intervals of a boxplot (min, Q1), (Q1, Q2), (Q2, Q3), (Q3, max) are related to the number of subjects in each interval. Students should understand that each interval theoretically contains one-fourth of the total number of subjects. Sketching an accompanying histogram and constructing a live boxplot may help in alleviating this misconception.

That all bell-shaped curves are normal distributions. For a bell-shaped curve to be Normal, there needs to be 68% of the distribution within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations.Common Core SupportInstitute for Mathematics and Education Learning Progressions NarrativesHigh School Statistics and Probabilityhttp://commoncoretools.me/wp-content/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf






Textbook and Curriculum Resourceshttp://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17Integrated Math III, McGraw HillChapter 10

Adventures with Mathematics: Statistics and Probability: Aligned With the Common Core State Standards, MCTM, 2012The Definite Activity, p. 38

The Spelling Bee, p. 44

Prior KnowledgeThe four-step statistical process was introduced in Grade 6, with the recognition of statistical questions. At the high school level, students need to become proficient in the first step of generating meaningful questions.

Future LearningThe next focus of study includes interpreting the structure of expressions in a variety of functions.




Illustrative MathematicsRepresenting Data with Frequency Graphshttp://map.mathshell.org/download.php?fileid=1780

Representing Data with Box Plotshttp://map.mathshell.org/download.php?fileid=1782

Career ConnectionsStatistics is the study of data organization to provide specific information and for measuring and determining uncertainty and probability. The discipline can apply to problems in economics, engineering, education, biology and sports. Some of its uses include sports information for baseball players, calculations for car insurance premiums and analyses of business efficiency. Other: Program Analyst, Data Specialist.





Ohio’s Learning Standards – Clear Learning TargetsIntegrated Mathematics III

6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Essential Understanding

- Students will understand the concept of fairness as it applies to probability.

Extended Understanding

- Students will be able to analyze decisions and strategies using concepts of probability.

Academic Vocabulary/Language

- theoretical probability- experimental probability- random

Tier 2 Vocabulary

- use- analyze- fair

I Can Statements

I can compute Theoretical and Experimental Probabilities.

I can use probabilities to make fair decisions (e.g. drawing by lots, using random number generator).

I can recall prior understandings of probability. I can analyze decision and strategies using probability

concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Common Misconceptions and Challenges

Students may believe that probabilities and expected values are not useful in making decisions that affect one’s life. Students need to see that these are not merely textbook exercises.


S.MD.6-7

14

Instructional Strategies


Textbook and Curriculum ResourcesMcGraw-Hill: Integrated Math II

Chapter 13-3, 13-4

S-CP, S-MD But mango is my favorite… S-MD Fred's Fun Factory

http://ccssmath.org/?s=md.6


Career Connections

Computer and mathematical occupationsActuariesComputer programmersComputer software engineersMathematiciansStatisticians

Architects, surveyors, and cartographersSurveyors, cartographers, photogrammetrists, and surveying technicians

Food preparation and serving related occupationsChefs, cooks, and food preparation workers

Personal care and service occupationsAnimal care and service workers


http://www.xpmath.com/careers/jobsresult.php?groupID=3&jobID=18










https://www.illustrativemathematics.org/content-standards/HSS/MD/B/7/tasks/1197

https://www.illustrativemathematics.org/content-standards/HSS/MD/B/7/tasks/1333



Interpret the structure of expressions.

Use the structure of an expression to identify ways to rewrite it. For example, see x4 –y4 as (x2 )2 - (y2 )2, thus recognizing it as difference of squares than be be factored as (x 2 – y2) (x2 + y2).

Essential UnderstandingStudents will need to be able to rewrite algebraic expressions in different equivalent forms such as factoring or combining like terms.

Students will need to be able to use factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to factor completely.

Students will need to be able to simplify expressions including combining like terms, using the distributive property and other operations with polynomials.

Extended UnderstandingStudents can use spreadsheets or a computer algebra system (CAS) to experiment with algebraic expressions, perform complicated algebraic manipulations, to better understand how algebraic manipulations behave.

Academic Vocabulary/ Languagecombining like terms, common factors, difference of squares, difference of two cubes, equivalent, factoring, factor completely, grouping, sum of two cubes

Tier 2 Vocabularyanalysis, manipulations, properties, rewrite, structure

CCSSM DescriptionReading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation.

I Can Statements

I can identify ways to rewrite expressions, such as difference of squares, factoring out a common monomial, regrouping, etc.

I can identify ways to rewrite expressions based on the structure of the expression.

I can use the structure of an expression to identify ways to rewrite it.

I can classify expression by structure and develop strategies to assist in classification.

Instructional Strategies Extending beyond simplifying an expression, this cluster addresses interpretation of the components in an algebraic expression. A student should recognize that in the expression 2x + 1, “2” is the coefficient, “2” and “x” are factors, and “1” is a constant, as well as “2x” and “1” being terms of the binomial expression. Development and proper use of mathematical language is an important building block for future content. Using real-world context examples, the nature of algebraic expressions can be explored. For example, suppose the cost of cell phone service for a month is represented by the expression 0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one minute (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the number of cell phone minutes used in the month. Similar real-world examples, such as tax rates, can also be used to explore the meaning of expressions. Factoring by grouping is another example of how students might analyze the structure of an expression. To factor 3x(x – 5) + 2(x – 5), students should recognize that the “x – 5” is common to both expressions being added, so it simplifies to (3x + 2)(x – 5). Students

A.SSE.2



http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx

http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Algebra_Model_Curriculum_March2015.pdf.aspx

http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf



Create equations that

describe numbers or relationships.Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratics functions, and simple rational and exponential functions.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Essential UnderstandingStudents are expected to know how to solve all available types of equations and inequalities, including root equations and inequalities, in one variable.

Students are expected to know how: to describe the relationships between the quantities in the problem (for example, how the quantities are changing or growing with respect to each other); express these relationships using mathematical operations to create an appropriate equation or inequality to solve; compare and contrast problems that can be solved by different types of equations; Students are expected to know how to identify the quantities in a mathematical problem or real world situation that should be represented by distinct variables and describe what quantities the variables represent; students will be expected to know how to graph one or more created equations on a coordinate axes with appropriate labels and scales.

Extended UnderstandingProvide examples of real-world problems that can be solved by writing an equation, and have students explore the graphs of the equations on a graphing calculator to determine which parts of the graph are relevant to the problem context.

Academic Vocabulary/ Language

absolute value function

function quadratic inequality

coordinate axes

graph rational function

cube root function

linear function

relationship

equation piecewise function

square root function

exponential function

quadratic function

variable

Tier 2 Vocabulary

appropriate describe justify

create distinct label

CCSSM DescriptionAn equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.


A.CED.1, A.CED.2

19

https://www.engageny.org/sites/default/files/resource/attachments/algebra-i-m1-teacher-materials.pdf


I Can Statements

I can solve quadratic equations in one variable; I can solve quadratic inequalities in one variable.

I can create quadratic equations and inequalities in one variable and use them to solve problems; I can create quadratic equations and inequalities in one variable to model real-world situations.

I can identify the quantities in a mathematical problem or real-world situation that should be represented by distinct variables and describe what quantities the variables represent.

I can graph one or more created equation on a coordinate axes with appropriate labels and scales; I can determine appropriate units for the labels and scale of a graph depicting the relationship between equations created in two or more variables between equations created in two or more variables.

I can create at least two equations in two or more variables to represent relationships between quantities.

I can justify which quantities in a mathematical problem or real-world situation are dependent and independent of one another and which operations represent those relationships.

Instructional StrategiesProvide examples of real-world problems that can be modeled by writing an equation or inequality. Begin with simple equations and inequalities and build up to more complex equations in two or more variables that may involve quadratic, exponential or rational functions. Discuss the importance of using appropriate labels and scales on the axes when representing functions with graphs. Examine real-world graphs in terms of constraints that are necessary to balance a mathematical model with the real world context. For example, a student writing an equation to model the maximum area when the perimeter of a rectangle is 12 inches should recognize that y = x(6 – x) only makes sense when 0 < x < 6. This restriction on the domain is necessary because the side of a rectangle under these conditions cannot be less than or equal to 0, but must be less than 6. Students can discuss the difference between the parabola that models the problem and the portion of the parabola that applies to the context. Explore examples illustrating when it is useful to rewrite a formula by solving for one of the variables in the formula. For example, the formula for the area of a trapezoid (A = 1 2 h(b 1 + b2) ) can be solved for h if the area and lengths of the bases are known but the height needs to be calculated.

Use a graphing calculator to demonstrate how dramatically the shape of a curve can change when the scale of the graph is altered for one or both variables. Give students formulas, such as area and volume (or from science or business), and have students solve the equations for each of the different variables in the formula.


Common Misconceptions/ChallengesStudents may believe that equations of linear, quadratic and other functions are abstract and exist only “in a math book,” without seeing the usefulness of these functions as modeling real-world phenomena. Additionally, they believe that the labels and scales on a graph are not important and can be assumed by a reader, and that it is always necessary to use the entire graph of a function when solving a problem that uses that function as its model.

Common Core SupportProgressions for the Common Core State Standards in Mathematics (draft) http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf

The Common Core in Ohiohttps://www.ixl.com/standards/ohio/math

Ohio’s New Learning Standardshttps://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx




https://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx

https://www.ixl.com/standards/ohio/math


Textbook and Curriculum Resourceshttp://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17Integrated Math III, McGraw HillChapter 0Chapter 1Chapter 2

Common Core State Standards Station Activities for Mathematics I, Walch Education, 2014Solving Inequalities, pp. 14-24Solving Equations, pp. 25-50

Algebra 1 Station Activities for Common Core State Standards, Walch Education, 2013Solving Linear Equations, pp. 54-79Graphing Linear Equations/Solving Using Graphs, pp.28-45Writing Linear Equations, pp. 46-53

Problem-Based Tasks for Mathematics I, Walch Education, 2013Phone Card Fine Print, pp. 1-4Investing Money, pp. 5-9Rafting and Hiking Trip, pp. 10-13Free Checking Accounts, pp. 22-24Population Change, pp. 25-28

Problem-Based Tasks for Mathematics II, Walch Education, ,2013Dancing for Charity, pp. 86-90

Prior KnowledgeWorking with expressions and equations, including formulas, is an integral part of the curriculum in Grades 7 and 8. In high school, students explore in more depth the use of equations and inequalities to model real-world problems, including restricting domains and ranges to fit the problem’s context, as well as rewriting formulas for a variable of interest.

Future Learning A.CED.1 and A.CED2 will be studied again when rational functions become the topic in Grading Period 2.



Performance Assessments/TasksClick on the links below to access performance tasks.NRICHhttp://nrich.maths.org/public/search.php?search=statistics http://nrich.maths.org/public/search.php?search=creating equations&filters[ks4]=1

Illustrative MathematicsMaximizing Profits: Selling Boomerangs http://map.mathshell.org/download.php?fileid=1718

Triangular Frameworkshttp://map.mathshell.org/download.php?fileid=814

Fearless Frameshttp://map.mathshell.org/download.php?fileid=806

Pythagorean Tripleshttp://map.mathshell.org/download.php?fileid=812

Best Buy Ticketshttp://map.mathshell.org/download.php?fileid=824

Skeleton Towerhttp://map.mathshell.org/download.php?fileid=810

Printing Ticketshttp://map.mathshell.org/download.php?fileid=772

Functionshttp://map.mathshell.org/download.php?fileid=762

Career Connections Occupations in Management: Computer, Engineering; Farmers Funeral; Industrial production managers; Medical and health services managers; Property, real estate, and community association managers ; Purchasing managers, buyers, and purchasing agents--Business and financial operations occupations: Insurance Computer and mathematical occupations: Actuaries, Computer programmers, Computer software engineers, Computer systems analysts, Mathematicians, StatisticiansEngineers: Aerospace engineers , Computer hardware engineers, Environmental engineers , Industrial engineers, Nuclear engineers

http://www.xpmath.com/careers/topicsresult.php?subjectID=2&topicID=3





























Create equations that describe numbers or relationships.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods

Essential Understanding Students should be able to create and solve

equations in one variable to answer questions.

Students should be able to interpret word problems and form expressions, equations and inequalities in order to solve a problem. They must be able to translate a word problem into an algebraic equation.

Students need to be able to identify when a common formula is needed for the given context.

Extended UnderstandingStudents can start learning quadratic, rational, and exponential functions to address all aspects of this standard. Once students are familiar with these operations individually, they should be asked to distinguish them from each other.


coefficient inequality rational

equation Linear system of equation

exponential literal variable

function polynomial

Tier 2 Vocabulary

describe solve

greater than

solution

interpret translate

reasoning unknown

CCSSM DescriptionStudents need to be able to interpret results after translating words into expressions, equations, and inequalities. They must be able to analyze an equation and problem to see if they have followed all procedures correctly; students must come up with the correct answer and determine if the answer makes sense.. Finally, students must be able to manipulate equations, following all the rules of Algebra, in order to solve for a given variable (literal equation).


A.CED.3

24

I Can Statements

I can recognize when a modeling context involves constraints I can interpret solutions as viable or nonviable options in a modeling context I can determine when a problem should be represented by equations,

inequalities, systems of equations and/or inequalities I can represent constraints by equations or inequalities, and by systems of

equations and/or inequalities

Instructional Strategies Provide examples of real-world problems that can be modeled by writing an equation or inequality. Begin with simple equations and inequalities and build up to more

complex equations in two or more variables that may involve quadratic, exponential or rational functions.

Discuss the importance of using appropriate labels and scales on the axes when representing functions with graphs. Examine real-world graphs in terms of constraints that are necessary to balance a mathematical model with the real world context. For example, a student writing an equation to model the maximum area when the perimeterof a rectangle is 12 inches should recognize that y=x (6 – x) only makes sense when 0 < x < 6. This restriction on the domain is necessary because the side of a rectangle under these conditions cannot be less than or equal to 0, but must be less than 6. Students can discuss the difference between the parabola that models the problem and the portion of the parabola that applies to the context’

Provide examples of real-world problems that can be solved by writing an equation, and have students explore the graphs of the equations on a graphing calculator to determine which parts of the graph are relevant to the problem context.

Use a graphing calculator to demonstrate how dramatically the shape of a curve can change when the scale of the graph is altered for one or both variables.

Give students formulas, such as area and volume (or from science or business), and have students solve the equations for each of the different variables in the formula


Common Misconceptions/Challenges Students may believe that equations of linear, quadratic and other functions are abstract and exist only “in a math book,” without seeing the usefulness of these functions as modeling real-world phenomena.

Students believe that the labels and scales on a graph are not important and can be assumed by a reader, and that it is always necessary to use the entire graph of a function when solving a problem that uses that function as its model.

Common Core SupportIllustrative Mathematicshttps://www.illustrativemathematics.org/content-standards/HSA/CED/A/3

Hunt Institute Video exampleshttp://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htm


http://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htm

https://www.illustrativemathematics.org/content-standards/HSA/CED/A/3

Textbook and Curriculum ResourcesIntegrated Math III McGraw Hill

Chapter 3

CCSS Mathhttp://ccssmath.org/?page_id=2121

Shmoophttp://www.shmoop.com/common-core-standards/ccss-hs-a-ced-3.html

Engage NYhttps://www.engageny.org/ccls-math/aced3

Sophiahttps://www.sophia.org/ccss-math-standard-9-12aced3-pathway

LearnZillionhttps://learnzillion.com/resources/72824-represent-constraints-by-equations-or-inequalities-and-by-systems-of-equations-and-or-inequalities

Prior KnowledgeWorking with expressions and equations, including formulas, is an integral part of the curriculum in Grades 7 and 8. In high school, students explore in more depth the use of equations and inequalities to model real-world problems,including restricting domains and ranges to fit the problem’s context, as well as rewriting formulas for a variable of interest.

Future LearningFuture learning will include the study of linear equations and inequalities in two variables.


https://learnzillion.com/resources/72824-represent-constraints-by-equations-or-inequalities-and-by-systems-of-equations-and-or-inequalities

https://www.sophia.org/ccss-math-standard-9-12aced3-pathway

https://www.engageny.org/ccls-math/aced3

http://www.shmoop.com/common-core-standards/ccss-hs-a-ced-3.html

http://ccssmath.org/?page_id=2121

Performance Assessments/TasksClick on the links below to access performance tasks.Math Assessment Project Maximizing Profits: Selling Boomerangs http://map.mathshell.org/download.php?fileid=1718

Modeling Motion: Rolling Cupshttp://map.mathshell.org/download.php?fileid=1746

Sorting Equations of Circle 1http://map.mathshell.org/download.php?fileid=1766

Sorting Equations of Circle 2http://map.mathshell.org/download.php?fileid=1768

Proving the Pythagorean Theorem http://map.mathshell.org/download.php?fileid=1756

Inside MathematicsNumber Towers http://www.insidemathematics.org/assets/common-core-math-tasks/number%20towers.pdf

Expressionshttp://www.insidemathematics.org/assets/common-core-math-tasks/expressions.pdf

Sorting the Mix http://www.insidemathematics.org/assets/problems-of-the-month/sorting%20the%20mix.pdf

Career/Everyday ConnectionsThere are many practical connections to creating equations describing numbers or relationships: deciding metered cab fares, mailing packages based upon weight, chemistry (preparing solutions mixing two given solutions; you will need to find how much of each given solution should be used to make your new solution).business (determining inventory), etc.


http://www.insidemathematics.org/assets/problems-of-the-month/sorting%20the%20mix.pdf

http://www.insidemathematics.org/assets/common-core-math-tasks/expressions.pdf

http://www.insidemathematics.org/assets/common-core-math-tasks/number%20towers.pdf







Reasoning with equations and inequalities.

Solve simple rational and radical equations in one variable and give examples showing how extraneous solutions may arise.

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Essential UnderstandingStudents are expected to be able to give examples showing how extraneous solutions mayarise when solving rational and radical equations.

Students are expected to be able to determine the domain of a rational function; students are expected to know how to determine the domain of a radical function.

Students are expected to know how to solve radical equations in one variable; students are expected to know how to solve rational equations in one variable; students are expected to be able to recognize and use function notation to represent linear, polynomial, rational, absolute value, exponential, and radical equations.

Extended Understanding Provide visual examples of radical and rational equations with technology so that students can see the solution as the intersection of two functions and further understand how extraneous solutions do not fit the model. It is very important that students are able to reason how and why extraneous solutions arise.


absolute value function

function notation

radical equation

domain linear function

rational equation

exponential function

logarithmic function

rational functions

function polynomial function

variable

Tier 2 Vocabulary

approximate

solve

recognize successive

CCSSM DescriptionRational equations mean that fractions are involved. Radical equations mean that square roots are involved. Students should know how to deal with both separately and together. Also, students should understand that an equation and its graph are just two different representations of the same thing. The graph of the line or curve of a two-variable equation shows in visual form all of the solutions (infinite as they may be) to our equation in written form. When two equations are set to equal one another, their solution is the point at which graphically they intersect one another. Depending on the equations (and the alignment of the planets), there might be one solution, or more, or none at all.


A.REI.2, A.REI.11

29

I Can Statements

I can determine the domain of a rational function.

I can determine the domain of a radical function.

I can solve radical equations in one variable.

I can solve rational equations in one variable.

I can give examples showing how extraneous solutions may arise when solving rational and radical equations.

I can approximate or find the solutions to a system.

I can explain why the solution to a system will occur at the point(s) of intersection.

Instructional StrategiesChallenge students to justify each step of solving an equation. Transforming 2x - 5 = 7 to 2x = 12 is possible because 5 = 5, so adding the same quantity to both sides of an equation makes the resulting equation true as well. Each step of solving an equation can be defended, much like providing evidence for steps of a geometric proof. Provide examples for how the same equation might be solved in a variety of ways as long as equivalent quantities are added or subtracted to both sides of the equation, the order of steps taken will not matter. Connect the idea of adding two equations together as a means of justifying steps of solving a simple equation to theprocess of solving a system of equations. A system consisting of two linear functions such as 2x + 3y = 8 and x - 3y = 1equation 2x - 4 = 5 can begin by adding the equation 4 = 4. Begin with simple, one-step equations and require students to write out a justification for each step used to solve the equation. Ensure that students are proficient with solving simple rational and radical equations that have no extraneous solutions before moving on to equations that result in quadratics and possible solutions that need to be eliminated.

Using technology, have students graph a function and use the trace function to move the cursor along the curve. Discuss the meaning of the ordered pairs that appear at the bottom of the calculator, emphasizing that every point on the curve represents a solution to the equation. Begin with simple linear equations and how to solve them using the graphs and tables on a graphing calculator. Then, advance students to nonlinear situations so they can see that even complex equations that might involve quadratics, absolute value, or rational functions can be solved fairly easily using this same strategy. While a standard graphing calculator does not simply solve an equation for the user, it can be used as a tool to approximate solutions. Use the table function on a graphing calculator to solve equations. For example, to solve the equation x2 = x + 12, students can examine the equations y = x2 and y = x + 12 and determine that they intersect when x = 4 and when x = -3 by examining the table to find where the y-values are the same.


Common Misconceptions/ChallengesStudents may believe that the graph of a function is simply a line or curve “connecting the dots,” without recognizing that the graph represents all solutions to the equation. Students may also believe that graphing linear and other functions is an isolated skill, not realizing that multiple graphs can be drawn to solve equations involving those functions. Additionally, students may believe that two-variable inequalities have no application in the real world. Teachers can consider business related problems (e.g., linear programming applications) to engage students in discussions of how the inequalities are derived and how the feasible set includes all the points that satisfy the conditions stated in the inequalities.

Common Core SupportProgressions for the Common Core State Standards in Mathematics (draft) http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf

The Common Core in Ohiohttps://www.ixl.com/standards/ohio/math

Ohio’s New Learning Standardshttps://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx

Illustrative Mathematicshttps://www.illustrativemathematics.org/blueprints/M3/5 https://www.illustrativemathematics.org/blueprints/M3 Textbook and Curriculum Resourceshttp://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17Integrated Math III, McGraw HillChapter 1Chapter 4Chapter 6Chapter 7

Common Core State Standards Station Activities for Mathematics I, Walch Education, 2014Comparing Linear Models, pp. 37-50Using Systems in Applications, pp. 51-63

Problem-Based Tasks for Mathematics I, Walch Education,2013 Senior Trip, pp. 67-70




https://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx



Prior KnowledgeSolving linear equations in one variable and analyzing pairs of simultaneous linear equations is part of the Grade 8 curriculum. These ideas are extended in high school, as students explore paper-and-pencil and graphical ways to solve equations, as well as how to graph two variable inequalities and solve systems of inequalities

Future Learning These standards will be revisited when studying rational functions.


Performance Assessments/TasksClick on the links below to access performance tasks.NRICHhttp://nrich.maths.org/public/search.php?search=creating%20equations%20and%20inequalities&filters[ks3]=1

INISIDE MathematicsGraphs 2006http://www.insidemathematics.org/assets/common-core-math-tasks/graphs%20(2006).pdf

Hexagonshttp://www.insidemathematics.org/assets/common-core-math-tasks/hexagons.pdf

Magic Squareshttp://www.insidemathematics.org/assets/common-core-math-tasks/magic%20squares.pdf

Math Assessment ProjectEvaluating Statements about Radicals http://map.mathshell.org/download.php?fileid=1714

Building and Solving Complex Equationshttp://map.mathshell.org/download.php?fileid=1722

Maximizing Profit: Selling Boomerangshttp://map.mathshell.org/download.php?fileid=1718

Representing Inequalities Graphicallyhttp://map.mathshell.org/download.php?fileid=1742

Career ConnectionsOccupations using equations and inequalities: Management: Computer, Engineering; Farmers Funeral; Industrial production managers; Medical and health services managers; Property, real estate, and community association managers ; Purchasing managers, buyers, and purchasing agents--Business and financial operations occupations: Insurance Computer and mathematical occupations: Actuaries, Computer programmers, Computer software engineers, Computer systems analysts, Mathematicians, StatisticiansEngineers: Aerospace engineers , Computer hardware engineers, Environmental engineers , Industrial engineers, Nuclear engineers
























http://www.insidemathematics.org/assets/common-core-math-tasks/magic%20squares.pdf

http://www.insidemathematics.org/assets/common-core-math-tasks/hexagons.pdf

http://www.insidemathematics.org/assets/common-core-math-tasks/graphs%20(2006).pdf

http://nrich.maths.org/public/search.php?search=creating%20equations%20and%20inequalities&filters%5Bks3%5D=1

Ohio’s Learning Standards - Clear Learning TargetsIntegrated Math III

(listing of included sources attached)

Interpret funtions that arise in applications in terms of the context.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Essential Understanding- Students must be able to interpret functions that arise in applications in terms of a specific context.

-Functions can be represented numerically, graphically, algebraically (symbolically), and/or verbally.

Extended Understanding-Students will be expected to move flexibly between the different representations of the same function for comparison.

-Students will learn about independent and dependent variables; an understanding of these concepts provides the basis for later work with functions


average rate of change

growth periodicity

continuous increasing range

decreasing intercept rate of change

domain interval representation

end behavior

linear slope

exponential maximum symmetry

function minimum

Tier 2 Vocabulary

algebraically

features range

application graphically sketch

CCSSM DescriptionBecause we continually make theories about dependencies between quantities in nature and society, functions are important tools in the construction of mathematical models. Students will use multiple representations to represent functions in different contexts.


F.IF.4, F.IF.6

34

I Can Statements

I can, given a function, identify key features in graphs and tables including: intercepts, intervals (increasing, decreasing, positive, negative), relative maximums and minimums, symmetries, end behavior, and periodicity; I can, given the key features of a function, sketch the graph.

I can, given the graph of a function, determine the practical domain of the function as it relates to the numerical relationship it describes.

I can calculate the average rate of change over a specified interval of a function presented symbolically or in a table; I can estimate the average rate of change over a specified interval of a function from the function’s graph; I can interpret,in context, the average rate of change; I can demonstrate that the rate of change of a non-linear function is different for different intervals.

Instructional StrategiesPractice moving from examining a graph and describing its characteristics (e.g., intercepts, relative maximums, etc.) to using a set of given characteristics to sketch the graph of a function.

Examine a table of related quantities and identify features in the table, such as intervals on which the function increases, decreases, or exhibits periodic behavior.

Begin with simple, linear functions to describe features and representations, and then move to more advanced functions, including non-linear situations.

Use various representations of the same function to emphasize different characteristics of that function.

Graphing utilities on a calculator and/or computer can be used to demonstrate the changes in behavior of a function as various parameters are varied.


Common Misconceptions/ChallengesStudents may believe that all relationships having an input and an output are functions, and therefore, misuse the function terminology.

Students may experience challenges moving between the different representations.

Students may believe that it is reasonable to input any x-value into a function, so they will need to examine multiple situations in which there are various limitations to the domains.

.

Common Core SupportCommon Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm

Institute for Mathematics and Education Learning Progressions Narrativeshttp://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf

Ohio Department of Education Model Curriculumhttp://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Functions_Model_Curriculum_March2015.pdf.aspx

Illustrative Mathematics: https://www.illustrativemathematics.org/blueprints/M1


https://www.illustrativemathematics.org/blueprints/M1

http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Functions_Model_Curriculum_March2015.pdf.aspx

http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf

http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm

Textbook and Curriculum ResourcesProblem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013Identifying Key Features of Linear and Exponential Graphs, pp. 95-98Proving Average Rate of Change, pp. 99-103Recognizing Average Rate of Change, pp. 104-108.

Learnzillionhttps://learnzillion.com/lessonsets/470

NCTM Illuminations:Domain Representations: http://illuminations.nctm.org/Lesson.aspx?id=2071Growth Rate: http://illuminations.nctm.org/Lesson.aspx?id=2265

Khan Academyhttps://www.khanacademy.org/math/algebra/algebra-functions

Integrated Math I, McGraw HillChapters 1,2,3

Prior KnowledgeStudents have learned about correspondences between equations, verbal descriptions, tables, and graphs and have studied regularity or trends.

Future LearningStudents will be expected to increase flexibility with moving between the multiple representations. The Rule of 4 representing mathematical functions--- visually (graphs, tables, charts), symbolically (algebraically), numerically (concrete examples), and verbally (natural language) --- will become increasingly prominent throughout students’ studies of mathematics. Students will be expected to be proficient with modeling and interpreting functions in terms of a context.


https://www.khanacademy.org/math/algebra/algebra-functions

http://illuminations.nctm.org/Lesson.aspx?id=2265

http://illuminations.nctm.org/Lesson.aspx?id=2071

https://learnzillion.com/lessonsets/470

Performance Assessments/TasksClick on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.Math Assessment ProjectLinear Graphshttp://map.mathshell.org/download.php?fileid=1106

Interpreting Functionshttp://map.mathshell.org/download.php?fileid=840

Inside MathematicsGraphshttp://www.insidemathematics.org/assets/common-core-math-tasks/graphs%20(2004).pdf

Sorting Functionshttp://www.insidemathematics.org/assets/common-core-math-tasks/sorting%20functions.pdf

Career ConnectionsEconomics, Investment Brokers, Insurance, Actuarial Science, Architectshttp://www.educationworld.com/a_curr/mathchat/mathchat010.shtml


http://www.educationworld.com/a_curr/mathchat/mathchat010.shtml

http://www.insidemathematics.org/assets/common-core-math-tasks/sorting%20functions.pdf





Analyze functions using different representations. Graph square root, cube root, and

piecewise functions, including step functions and absolute value functions.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Essential UnderstandingStudents will be expected to master flexible movement between the multiple representations.

Students are expected to increase comfort level in understanding other representations mentally even when only one representation is given.

Extended UnderstandingStudents can better understand the characteristics of representations by providing opportunities to study the eight major families of functions.


constant rate of change

x- intercept

exponential sequence

y-intercept

linear slope

parent function

standard form

Tier 2 Vocabulary

algebraically

domain sketch

CCSSM DescriptionStudents learn in different ways. The Rule of 4 ---visually (graphs, tables, charts), symbolically (algebraically), numerically (concrete examples), and verbally (natural language) facilitates and deepens understanding by presenting the same concept in different modes.


F.IF.7B, F.IF.9

39

I Can Statements

I can graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions, by hand in simple cases or using technology for more complicated cases, and show/label key features of the graph.

I can determine the difference between simple and complicated linear, quadratic, square root, cube root, and piecewise-defined functions, including step functions and absolute value functions and know when the use of technology is appropriate.

I can compare and contrast the domain and range of absolute vale, step and piece-wise defined functions with linear, quadratic, and exponential.

I can write a function in equivalent forms to show different properties of the function.

I can explain the different properties of a function that are revealed by writing a function in equivalent forms.

I can identify types of functions based on verbal, numerical, algebraic, and graphical descriptions and state key properties (e.g. intercepts, growth rates, average rates of change, and end behaviors).

I can differentiate between exponential and linear functions using a variety of descriptors (graphically, verbally, numerically, and algebraically); I can use a variety of function representations algebraically, graphically, numerically in tables, or by verbal descriptions) to compare and contrast properties of two functions.

Instructional StrategiesGraphing utilities on a calculator and/or computer can be used to demonstrate the changes in behavior of a function as various parameters are varied.

Add families of functions, one at a time, to the students’ knowledge base so they can see connections among behaviors of the various functions.

Provide numerous examples of real-world contexts such as exponential growth and decay situations (e.g., a population that declines by 10% per year) to help students apply an understanding of functions in context.


Common Misconceptions/ChallengesStudents may believe that each family of functions (e.g., quadratic, square root, etc.) is independent of the others, so they may not recognize commonalities among all functions and their graphs.

Students may believe that the process of rewriting equations into various forms is simply an algebra symbol manipulation exercise rather than serving a purpose of allowing different features of the function to be exhibited.




Illustrative Mathematicshttps://www.illustrativemathematics.org/blueprints/M1






Textbook and Curriculum ResourcesProblem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013Fund-raising Concert, pp. 109113Trout Pond, pp. 114-117Supply and Demand, pp. 118-122Analyzing Kidney Function, pp. 123-126

Station Activities for Mathematics I, Walch Education, 2014:Interpreting Functions, pp. 94-117

Algebra I Station Activities for Common Core State StandardsInterpreting Functions, pp. 231-245

ORC (Ohio Resource Center: The Ohio State University)http://www.ohiorc.org/search/results/?txtSearchText=functions

https://learnzillion.com/lessonsets/470Virtual Nerdhttp://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17

Integrated Math I, McGraw HillChapter 1Extend Lesson 1-7: Graphing Technology Lab: Representing FunctionsLesson 1-8: Interpreting Graphs of Functions

Prior KnowledgeStudents have been exposed to the idea that rewriting an expression can provide more information on the expression. This idea is expanded upon as students explore functions in high school and recognize how the form of theequation can provide clues about zeros, asymptotes, etc.

Future LearningLearning features of parent functions, the simplest form of a family of function, and features of family functions can increase understanding of functions.




http://www.ohiorc.org/search/results/?txtSearchText=functions

Performance Assessments/TasksClick on the links below to access performance tasks where students are asked to demonstrate an understanding of the concept of functions.Inside MathematicsGraphs 2004http://www.insidemathematics.org/assets/common-core-math-tasks/graphs%20(2004).pdf

Graphs 2007http://www.insidemathematics.org/assets/common-core-math-tasks/graphs%20(2007).pdf

NRICHhttp://nrich.maths.org/773http://nrich.maths.org/5872

Illustrative MathematicsThrowing Baseballshttps://www.illustrativemathematics.org/content-standards/HSF/IF/C/9/tasks/1279

Modeling London's Populationhttps://www.illustrativemathematics.org/content-standards/HSF/IF/C/7/tasks/1595

Running Timehttps://www.illustrativemathematics.org/content-standards/HSF/IF/C/7/tasks/1539

Career ConnectionsAny career that involves the need to articulate verbally the relationships between variables arising in everyday contexts can utilize the study of functions. This include health care area, science, and any career involving sales.

Students can complete the following concept development activities (Representing Functions of Everyday Situations) where they are asked to :• Translate between everyday situations and sketch graphs of relationships between variables.• Interpret algebraic functions in terms of the contexts in which they arise.• Reflect on the domains of everyday functions and in particular whether they should be discrete or Continuous




https://www.illustrativemathematics.org/content-standards/HSF/IF/C/7/tasks/1539



http://nrich.maths.org/5872





Build a function that models. Write a linear function that

describes a relationship between two quantities.

Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Essential UnderstandingExamination of functions is extended to include recursive and explicit representations and sequences of numbers that may not have a linear relationship.

Extended Understanding Using a variety of functions (e.g., linear, exponential, constant, students can increase understanding of the different representations by representing functions as a set of ordered pairs, a table, a graph, and an equation.


arithmetic sequence

explicit formula

inverse relationship

function recursive

correspondence geometric sequence

quantities

direct variation inverse function

Tier 2 Vocabulary

compare model prove

construct observe

CCSSM DescriptionFunctions can be used to make predictions about future behaviors when modeling real life situations. For students to recognize a functional relationship, they need to recognize there is a correspondence and see/understand the correspondence matches each element of the first set with an element of the second set. Once it is known that the relationship is a function, students can determine the rule for the function.

I Can Statements

I can, from context, either write an explicit expression, define a recursive process, or describe the calculations meeded to model a function between two quantities.

I can compose functioins; I can build standard functions to represent relevant relationships/quantities.

I can determine which arithmetic operation should be performed to build the appropriate combined function; I can combine two functions using the operations of addition, subtraction, multiplication, and division.

I can relate the combined function to the context of the problem; I can evaluate the domain of the combined function.


F.BF.1, F.BF.1B

44

Instructional StrategiesProvide a real-world example (e.g., a table showing how far a car has driven after a given number of minutes, traveling at a uniform speed), and examine the table by looking “down” the table to describe a recursive relationship, as well as “across” the table to determine an explicit formula to find the distance traveled if the number of minutes is known.

Write out terms in a table in an expanded form to help students see what is happening. For example, if the y-values are 2, 4, 8, 16, they could be written as 2, 2(2), 2(2)(2), 2(2)(2)(2), etc., so that students recognize that 2 is being used multiple times as a factor.

Focus on one representation and its related language – recursive or explicit – at a time so that students are not confusing the formats.

Provide examples of when functions can be combined, such as determining a function describing the monthly cost for owning two vehicles when a function for the cost of each (given the number of miles driven) is known.

Common Misconceptions/Challenges Students may believe that the best (or only) way to generalize a table of data is by using a recursive formula. Students naturally tend to look “down” a table to find the

pattern but need to realize that finding the 100th term requires knowing the 99thterm unless an explicit formula is developed.

Students may also believe that arithmetic and geometric sequences are the same. Students need experiences with both types of sequences to be able to recognize the difference and more readily develop formulas to describe them.

Advanced students who study composition of functions may misunderstand function notation to represent multiplication (e.g., f(g(x)) means to multiply the f and g function values).

When studying functions, students sometimes interchange the input and output values. This will lead to confusion about domain and range, and determining if a relation is a function. This can also interfere with a student being able to find the appropriate inverse function, or the correct equation to model a relationship between two quantities.Common Core SupportCommon Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm


Illustrative Mathematics: Practice and Content Standards: Toward Greater Focus and Coherencehttps://www.illustrativemathematics.org/standards


https://www.illustrativemathematics.org/standards



Textbook and Curriculum ResourcesProblem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013Texting for the Win, pp. 127-131Jai’s Jeans, pp. 132-134New Tablet, pp. 135-138Glass Recycling, pp. 139-142

Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013To Drill or Not to Drill?, pp. 138-Pushing Envelopes, pp. 142-145

Common Core State Standards: Station Activities for Mathematics IRelations Versus Functions/Domain and Range, pp. 85-93Sequences, pp. 118-130Real-World Situation Graphs pp. 194-208-

High School CCSS Mathematics I Curriculum Guide-Quarter 1 Curriculum Guide, 2013, pp. 161-204

Prior KnowledgeIn Grade 8, students learn to compare functions by looking at equations, tables and graphs, and focus primarily on linear relationships.

Future LearningFuture learning will include working with inverse functions.


Performance Assessments/TasksClick on the links below to access performance tasks.Math Assessment ProjectGeneralizing Patterns: Table Tiles http://map.mathshell.org/download.php?fileid=1716

Representing Linear and Exponential Growthhttp://map.mathshell.org/download.php?fileid=1732


Inside MathematicsInfinite Windowshttp://www.insidemathematics.org/assets/problems-of-the-month/infinite%20windows.pdf

Slice and Dicehttp://www.insidemathematics.org/assets/problems-of-the-month/slice%20and%20dice.pdf

Calculating Palindromeshttp://www.insidemathematics.org/assets/problems-of-the-month/calculating palindromes.pdf

First Ratehttp://www.insidemathematics.org/assets/problems-of-the-month/first%20rate.pdf

Cut It Outhttp://www.insidemathematics.org/assets/problems-of-the-month/cut%20it%20out.pdf

Illutrative MathematicsSummer Internhttps://www.illustrativemathematics.org/content-standards/HSF/BF/A/1/tasks/72


https://www.illustrativemathematics.org/content-standards/HSF/BF/A/1/tasks/72

http://www.insidemathematics.org/assets/problems-of-the-month/cut%20it%20out.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/first%20rate.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/calculating%20palindromes.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/slice%20and%20dice.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/infinite%20windows.pdf




Career ConnectionsStudents can research and evaluate several options when purchasing a vehicle (e.g., new versus used, lease versus own, down payment, and interest rate). They will examine the differences in gas mileage consumption by selecting two vehicles to evaluate (e.g., SUV versus compact hybrid). Once they choose a vehicle, they will use their evaluations to show why they chose the vehicle. Their research will include interviewing automotive professionals, visiting dealerships, and navigating company websites.

Applicable careers include business, finance, insurance and any career focused on making scholarly predictions.



Build a function that models. Identify the effect on

the graph of replacing f(x) by f(x) + k, k(f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Find the inverse functions.

Essential UnderstandingStudents should be able to identify appropriate types of functions to model specific contexts.

Students should be able to find inverse functions.

Students should recognize that not all functions have inverses.

Extended UnderstandingUse real-world examples of functions and their inverses. For example, students might determine that folding a piece of paper in half 5 times results in 32 layers of paper, but that if they are given that there are 32 layers of paper, they can solve to find how many times the paper would have been folded in half. Provide applied examples of exponential and logarithmic functions, such as the use of a logarithm to determine pH or the strength of an earthquake on the Richter Scale. Both pH and Richter Scale values are powers of 10 and are, therefore, logarithms. For example, the magnitude of an earthquake, M, on the Richter Scale can be calculated using the formula M = log10A, where A represents the amplitude of measured seismic waves


algebraic expression

functions logarithmic function

arithmetic sequence

function odd functions


parameters


quantities

even function inverse relationship

value

Tier 2 Vocabulary

analyze identify model

appropriate illustrate observe

compare judgement prove

CCSSM DescriptionStudents identify appropriate types of functions to model a situation. They adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit.


F.BF.3, F.BF.4

49

I Can Statements

Identify, through experimenting with technology, the effect on the graph of a function by replacing f(x) with f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative).

Given the graphs of the original function and a transformation, determine the value of (k).

Recognize even and odd functions from their graphs and equations.

Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠1; I can verify by composition that one function is the inverse of another.

Read values of an inverse function from a graph or a table, given that the function has an inverse.

Produce an invertible function from a non-invertible f.

Instructional StrategiesUse graphing calculators or computers to explore the effects of a constant in the graph of a function. For example, students should be able to distinguish between the graphs of y = x 2 , y = 2x2 , y = x2 + 2, y = (2x) 2 , and y = (x + 2)2 . This can be accomplished by allowing students to work with a single parent function and examine numerous parameter changes to make generalizations. Distinguish between even and odd functions by providing several examples and helping students to recognize that a function is even if f(-x) = f(x) and is odd if f(-x) = -f(x). Visual approaches to identifying the graphs of even and odd functions can be used as well. Provide examples of inverses that are not purely mathematical to introduce the idea.

Students should also recognize that not all functions have inverses. Again using a nonmathematical example, a function could assign a continent to a given country’s input, such as g(Singapore) = Asia. However, g-1 (Asia) does not have to be Singapore (e.g., it could be China). Exchange the x and y values in a symbolic functional equation and solve for y to determine the inverse function. Recognize that putting the output from the original function into the input of the inverse results in the original input value. Also, students need to recognize that exponential and logarithmic functions are inverses of one another and use these functions to solve real-world problems. Nonmathematical examples of functions and their inverses can help students to understand the concept of an inverse and determining whether a function is invertible.

Introduce finding the inverse of a function with the activity “Introduction to Inverse Functions” (included in the CCS Curriculum Guide, 2013, Math III, p. 112). In this activity, students will intuitively attempt to find the inverse of functions, and then look at the actual inverse graphically and algebraically. Students will connect the domain and range of the original function to the domain and range of the inverse relations. Students can do this in groups or individually. Have students work on the handout “Conversions – Applications of Inverses” (included in CCS Curriculum Guide, 2013, Math III, p. 122) in order to see how inverse functions could be used in real life situations. They will use real life formulas to connect functions with their inverses


Common Misconceptions/Challenges Students may believe that the graph of y = (x – 4)3 is the graph of y = x3 shifted 4 units to the left (due to the subtraction symbol). Examples should be explored by hand and

on a graphing calculator to overcome this misconception. Students may also believe that even and odd functions refer to the exponent of the variable, rather than the sketch of the graph and the behavior of the function. Additionally, students may believe that all functions have inverses and need to see counter examples, as well as examples in which a non-invertible function can be made into an invertible function by restricting the domain. For example, f(x) = x 2 has an inverse (f -1 (x) = square root of x ) provided that the domain is restricted to x ≥ 0.

Common Core SupportCommon Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources http://www.ct4me.net/Common-Core/hsfunctions/hsf-building-functions.htm






http://www.ct4me.net/Common-Core/hsfunctions/hsf-building-functions.htm

Textbook and Curriculum Resourceshttp://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17 Integrated Math III, McGraw Hill,Chapter 5Chapter 6Chapter 7

Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013 Gym Fees, pp. 143-145

Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013The Catch, pp. 146-149Fewer Parabolas, Please, pp. 150-155Falling Keys, pp. 156-159

Common Core State Standards: Station Activities for Mathematics II, Walch Education, 2010Quadratics Transformations in Vertex Form, pp. 17-34

High School CCSS Mathematics IIII Curriculum Guide-Quarter 1 Curriculum Guide, 2013

Prior KnowledgeUnderstanding functional relationships as input and output values that have an associated graph is introduced in Grade 8. In high school, changes in graphs is explored in more depth, and the idea of functions having inverses is introduced. Advanced students also expand their catalog of functions to include exponential and logarithmic cases.

Future Learning These standards will be revisited when studying rational functions.



Performance Assessments/TasksClick on the links below to access performance tasks.Math Assessment ProjectRepresenting Trigonometric Functionshttp://map.mathshell.org/download.php?fileid=1738

Representing Polynomials Graphicallyhttp://map.mathshell.org/download.php?fileid=1744

NYS COMMON CORE MATHEMATICS CURRICULUMChoosing a Modelhttps://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0ahUKEwjx483vxs3NAhUI6yYKHU3ODbkQFggrMAE&url=https%3A%2F%2Fwww.engageny.org%2Ffile%2F109501%2Fdownload%2Falgebra-ii-m3-topic-c-lesson-22-teacher.pdf%3Ftoken%3DyFz_dJHn&usg=AFQjCNFk8Swf8dKGbZzhWu2MVVInJcrTsA&cad=rja

ILLUSTRATIVE MATHEMATICSTransforming the Graph of a Functionhttps://www.illustrativemathematics.org/content-standards/tasks/742https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758

INSIDE MATHEMATICSTri-Triangleshttp://www.insidemathematics.org/assets/problems-of-the-month/tri-triangles.pdf

Career Connections /Real World ApplicationsEngineers, Scientists


http://www.insidemathematics.org/assets/problems-of-the-month/tri-triangles.pdf

https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758

https://www.illustrativemathematics.org/content-standards/tasks/742

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0ahUKEwjx483vxs3NAhUI6yYKHU3ODbkQFggrMAE&url=https%3A%2F%2Fwww.engageny.org%2Ffile%2F109501%2Fdownload%2Falgebra-ii-m3-topic-c-lesson-22-teacher.pdf%3Ftoken%3DyFz_dJHn&usg=AFQjCNFk8Swf8dKGbZzhWu2MVVInJcrTsA&cad=rja







Interpret the structure of expressions.Interpret expressions that represent a

quantity in terms of its context.

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expression by viewing one or more of their parts as a single entity. For example, see x4-y4 as (x2)2 – (y2)2, thus recognizing it as difference of squares that can be factored as (x2-y2) (x2+y2).

Essential UnderstandingStudents should know that variable expressions are used to communicate and model authentic problems.

Students should know that math is a language and has structures to ensure effective communication.

Students should know: how to write expressions from descriptive words and from patterns in data.

Students should be able to describe in words an expression in a given context.

Students should know how to explain the difference between a variable and a constant.

Extended UnderstandingHands-on materials, such as algebra tiles, can be used to establish a visual understanding of algebraic expressions and the meaning of terms, factors and coefficients.

Academic Vocabulary/ Languageaverage rate of change, binomial, coefficient, constant, degree, difference of squares, divisor, expression, factor, end behavior, maximum, minimum, monomial, polynomial, power, quotient, rational, remainder, roots, terms, trinomial, x-intercepts,y-intercepts, zero product property, zeros

Tier 2 Vocabularycomplex, context, identify, interpret, recognizing, represent

CCSSM DescriptionReading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure.

I Can Statements

I can identify the different parts of the expression and explain their meaning within the context of a problem.

I can decompose expressions and make sense of the multiple factors and terms by explaining the meaning of the individual parts: terms, factors, and coefficients

I can interpret complex expressions by examining their variables

I can, for expressions that represent a contextual quantity, interpret complicated expressions, in terms of the context, by viewing one or more of their parts as a single entity.


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54

Instructional StrategiesExtending beyond simplifying an expression, this cluster addresses interpretation of the components in an algebraic expression. A student should recognize that in the expression 2x + 1, “2” is the coefficient, “2” and “x” are factors, and “1” is a constant, as well as “2x” and “1” being terms of the binomial expression. Development and proper use of mathematical language is an important building block for future content. Using real-world context examples, the nature of algebraic expressions can be explored. For example, suppose the cost of cell phone service for a month is represented by the expression 0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one minute (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the number of cell phone minutes used in the month. Similar real-world examples, such as tax rates, can also be used to explore the meaning of expressions. Factoring by grouping is another example of how students might analyze the structure of an expression. To factor 3x(x – 5) + 2(x – 5), students should recognize that the “x – 5” is common to both expressions being added, so it simplifies to (3x + 2)(x – 5). Students should become comfortable with rewriting expressions in a variety of ways until a structure emerges. Have students create their own expressions that meet specific criteria (e.g., number of terms factorable, difference of two squares, etc.) and verbalize how they can be written and rewritten in different forms. Additionally, pair/group students to share their expressions and rewrite one another’s expressions.

Common Misconceptions/ChallengesStudents may believe that the use of algebraic expressions is merely the abstract manipulation of symbols. Use of real world context examples to demonstrate the meaning of the parts of algebraic expressions is needed to counter this misconception. Students may also believe that an expression cannot be factored because it does not fit into a form they recognize. They need help with reorganizing the terms until structures become evident.

.

Common Core SupportInstitute for Mathematics and Education Learning Progressions Narrativeshttp://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf

Ohio Learning Standards http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx

Illustrative Mathematics: https://www.illustrativemathematics.org/blueprints/M1






Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013Identifying Parts of an Expression in Context, pp. 14-16Searching for a Greater Savings, pp. 17-21

Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013Deck the Deck, pp. 27-29Puppy Pen, pp. 30-33

Columbus City Schools Curriculum Guide, 2013, Math III, Quarter 2


Prior KnowledgeAn introduction to the use of variable expressions and their meaning, as well as the use of variables and expressions in real-life situations is included in the Expressions and Equations Domain of Grade 7.

Future Learning Future learning will include arithmetic with polynomials and rational expressions.



Performance Assessments/TasksClick on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.The Physics Professor https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/23

Radius of a Cylinderhttps://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/1366

Mixing Fertilizerhttps://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/88

Increasing or Decreasing? Variation 1https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/89

Increasing or Decreasing? Variation 2https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/167

The Bank Accounthttps://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/390

Mixing Candieshttps://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/389

Delivery Truckshttps://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/1343

Animal Populationshttps://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/436

Modeling London’s Populationhttps://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/1595

Throwing Horseshoeshttps://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/90

Seeing Dotshttps://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/21

Career ConnectionsStudents will evaluate cell phone plans across multiple providers to identify one that is the most cost effective for their expected use. They will consider the fixed and variable costs to support their decision (e.g., unlimited plans, cost per unit, insurance protection, activation and cancellation fees). In collecting data related to the cost of service, students will research the employment opportunities available across the telecommunication companies via website, phone, and email.


https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/21














Understand the relationship between zeros and factors of polynomials.

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on a division by x-a is p(a), so p(a)=0 if and only if (x-a) is a factor of p(x).

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial

Essential UnderstandingStudents should be able to understand and apply the Remainder Theorem.

Students should know and understand that a is a root of a polynomial function if and only ifx-a is a factor of the function.

Students should be able to find the zeros of a polynomial when the polynomial is factored.Students should be able to use the zeros of a function to sketch a graph of the function.

Academic Vocabulary/ Languageaverage rate of change, binomial, coefficient, constant, degree, divisor, factor, end behavior, maximum, minimum, monomial, polynomial, power, quotient, rational, remainder, roots, terms, trinomial, x-intercepts, y-intercepts, zero product property, zeros

Tier 2 Vocabularyapply, construct, identify, know, understand


A.APR.2, A.APR.3

58

Extended UnderstandingMathematical Practice 3:Students can build a logical response, providing examples, for the following essential questions:How are zeros and factors of a polynomial related? How can a graph of a function be estimated based on the zeros and factors of a polynomial?

CCSSM DescriptionA quick review…If we divide two integers, sometimes they make another integer (6 ÷ 2 = 3), and other times they have remainders (13 ÷ 4 = 3 remainder 1). A remainder of 0 means that the second number is a factor of first number. Polynomials are the same way; if dividing polynomial p(x) by x – a has a remainder of 0, we’ll know that x – a is a factor of p(x). In other words, p(x) = q(x) × (x – a) where q(x) is a polynomial or an integer. Essentially, any polynomial p(x) can be written as a product of (x – a) and some quotient q(x), plus the remainder p(a). The zeros of a polynomial are the x values when we set the polynomial itself to equal zero. In other words, when we plug in any of the zeros of a polynomial in for x, our answer should be 0. So the zeros of x3 – 10x2 – 2x + 24 are the x values that make the equation x3 – 10x2– x+ 24= 0 true. Zero values are imporant because on the coordinate plane, zeros are the places where the function crosses the x-axis. The zeros of the polynomial (also called the solutions or “roots”) are the x-intercepts of the graph.

I Can Statements

I can understand, define and apply the Remainder Theorem.

I can use the Remainder Theorem to show the relationship between a factor and a zero.

I can understand that a is a root of a polynomial function if and only if x-a is a factor of the function.

I can find the zeros of a polynomial when the polynomial is factored.

I can use the zeros of a function to sketch a graph of the function.

I can determine the domain of a rational function.

I can factor polynomials using any method.

I can sketch graphs of polynomials using zeroes and a sign chart.


Instructional StrategiesBy using technology to explore the graphs of many polynomial functions, and describing the shape, end behavior and number of zeros, students can begin to make the following informal observations: The graphs of polynomial functions are continuous; an nth degree polynomial has at most n roots and at most n - 1 “changes of direction” (i.e., from increasing to decreasing or vice versa); an even-degree polynomial has the same end-behavior in both the positive and negative directions: both heading to positive infinity, or both heading to negative infinity, depending upon the sign of the leading coefficient; an odd-degree polynomial has opposite end-behavior in the positive versus the negative directions, depending upon the sign of the leading coefficient; an odd-degree polynomial function must have at least one real root. Students can benefit from exploring the rational root theorem, which can be used to find all of the possible rational roots (i.e., zeros) of a polynomial with integer coefficients. When the goal is to identify all roots of a polynomial, including irrational or complex roots, it is useful to graph the polynomial function to determine the most likely candidates for the roots of the polynomial that are the x-intercepts of the graph.

Common Misconceptions/ChallengesDifference between roots and zeros: the solution for a polynomial equation is called a root. The words root and zero are often used interchangeably, but technically, you find the zero of a function and the root of an equation.



Illustrative Mathematics: https://www.illustrativemathematics.org/blueprints/A2/3


https://www.illustrativemathematics.org/blueprints/A2/3



Textbook and Curriculum ResourcesIntegrated Math III, McGraw Hill Chapter 4

Common Core State Standards Station Activities for Mathematics II, Walch Education, 2014Operations with Polynomials, pp. 44-56



Prior KnowledgeStudents have had exposure to expressions in equations in middle school which should provide some comfort level with the understanding of roots and zeros.

Future LearningStudy of polynomial functions will continue with using polynomial identities to solve problems.



Performance Assessments/TasksClick on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.Math Assessment ProjectRepresenting Polynomials Graphicallyhttp://map.mathshell.org/download.php?fileid=1744

New York State Common Core Mathematics Curriculum: Engage New York: Math Ihttps://www.engageny.org/resource/algebra-ii-module-1-topic-b-lesson-12

Polynomials and Factoringhttps://www.sophia.org/topics/polynomials-and-factoring

Career Connections/Everyday ApplicationsPolynomials can have real-world uses. Some careers require you to use complex math, including polynomials, to solve problems, draw conclusions and make predictions. Economists: Economists use data and mathematical models and statistical techniques to conduct research, prepare reports, formulate plans and interpret and forecast market trends. Their jobs often involve addressing economic problems related to the production and distribution of goods and services and monetary and fiscal policies; Statisticians: Statisticians use mathematical techniques to analyze and interpret data and draw conclusions. Their work often influences economic, social, political and military decisions, according to the BLS. Statisticians may work in government, education, health care and manufacturing. Because the job requires the use of polynomials and other complex math, statisticians generally need at least a bachelor’s degree in statistics or math with coursework in differential and integral equations, mathematical modeling and probability theory; Engineering Careers: Aerospace engineers, chemical engineers, civil engineers, electrical engineers, environmental engineers, mechanical engineers and industrial engineers all need strong math skills. Their jobs require them to make calculations using polynomial expressions and operations. For example, aerospace engineers may use polynomials to determine acceleration of a rocket or jet, and mechanical engineers use polynomials to research and design engines and machines, according to WeUseMath.org.; Science Careers: Physical and social scientists, including archaeologists, astronomers, meteorologists, chemists and physicists, need to use polynomials in their jobs. Key scientific formulas, including gravity equations, feature polynomial expressions. These algebraic equations help scientists to measure relationships between characteristics such as force, mass and acceleration. Astronomers use polynomials to help in finding new stars and planets and calculating their distance from Earth, their temperature and other features, according to school-for-champions.com.


https://www.sophia.org/topics/polynomials-and-factoring

https://www.engageny.org/resource/algebra-ii-module-1-topic-b-lesson-12




Use polynomial identities to solve problems.

Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 – y2)2=(x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.

Essential UnderstandingStudents should be able to explain that an identity shows a relationship between two quantities or expressions, which is true for all values of the variables, over a specified set.

Students should be able to prove polynomial identities and use polynomial identities to describe numerical relationships

Extended UnderstandingSome information below includes additional mathematics that students should learn in order

Academic Vocabulary/ Language average rate of change, binomial, coefficient, constant, degree, divisor, factor, end behavior, identity, maximum, minimum, monomial, polynomial, power, quotient, rational, remainder, roots, terms, trinomial, x-intercepts, y-intercepts, zero product property, zeros

Tier 2 Vocabulary generate, prove, triples


A.APR.4

63

to take advanced courses such as calculus, advanced statistics, or discrete mathematics and goes beyond the mathematics that all students should study in order to be college- and career-ready: Ask students to use the vertical multiplication to write out term-by-term multiplication to generate (x + y) 3 from the expanded form of (x + y) 2. Then use that expanded result to expand (x + y) 4, use that result to expand (x + y) 5 , and so on. Students should begin to see the arithmetic that generates the entries in Pascal’s triangle.

CCSSM DescriptionThe operations of addition, subtraction and multiplication can be applied to polynomials. A polynomial identity is just a true equation, often generalized so that it can apply to more than one situation. Identities are proven by showing that one side of an equation is equal to the other. This takes the same skills used to organize equations and expressions.

I Can Statements

I can prove polynomial identities


Instructional StrategiesThis cluster is an opportunity to highlight polynomial identities that are commonly used in solving problems. To learn these identities, students need experience using them to solve problems. Students should develop familiarity with the following special products. Students should be able to prove any of these identities. Furthermore, they should develop sufficient fluency with the first four of these so that they can recognize expressions of the form on either side of these identities in order to replace that expression with an equivalent expression in the form of the other side of the identity: (x + y) 2 = x 2 + 2xy + y 2

(x - y) 2 = x 2 - 2xy + y 2

(x + y)(x - y) = x 2 - y 2

(x + a)(x + b) = x 2 + (a + b)x + ab (x + y) 3 = x 3 + 3 x 2 y + 3xy 2 + y 3

(x - y) 3 = x 3 - 3x 2 y + 3xy 2 – yWith identities such as these, students can discover and explain facts about the number system. For example, in the multiplication table, the perfect squares appear on the diagonal. Diagonally, next to the perfect squares are “near squares,” which are one less than the perfect square. Why? • Why is the sum of consecutive odd numbers beginning with 1 always a perfect square? • Numbers that can be expressed as the sum of the counting numbers from 1 to n are called triangular numbers. What do you notice about the sum of two consecutive triangular numbers? Explain why it works. • The sum and difference of cubes are also reasonable for students to prove. The focus of this proof should be on generalizing the difference of cubes formula with an emphasis toward finite geometric series.

Common Misconceptions/ChallengesStudents often look at a polynomial in a standard window on a grapher and do not investigate further properties that may be seen using different window settings. It is important for students to use their knowledge of polynomials to predict what its graph may look like, then check their predictions on their grapher.



Illustrative Mathematics: https://www.illustrativemathematics.org/blueprints/M3/2


https://www.illustrativemathematics.org/blueprints/M3/2%20%20%20%20%20%20%20%20%20%20%20







Prior KnowledgeIn Grade 6, students began using the properties of operations to rewrite expressions in equivalent forms.

Future LearningThe study of polynomial functions will continue with students learning to rewrite rational expressions.



Performance Assessments/TasksClick on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.New York State Common Core Mathematics Curriculum: Engage New York: Math Ihttps://www.engageny.org/resource/algebra-ii-module-1-topic-b-lesson-12

POLYNOMIALShttps://www.sophia.org/search/tutorials?q=polynomials

ILLUSTRATIVE MATHEMATICShttps://www.illustrativemathematics.org/blueprints/M3/2

Non Negative Polynomialshttps://www.illustrativemathematics.org/content-standards/HSA/APR/A/1/tasks/1656

Powers of 11https://www.illustrativemathematics.org/content-standards/HSA/APR/A/1/tasks/1654



https://www.illustrativemathematics.org/content-standards/HSA/APR/A/1/tasks/1654



https://www.sophia.org/search/tutorials?q=polynomials

https://www.sophia.org/




Rewrite rational expressions.

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) +r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspections, long division, or for more complicated examples, a computer algebra system.

Essential Understanding Students should be able to use inspection to rewrite simple rational expressions in different form.

Students should be able to use long division to rewrite simple rational expressions in different forms.

Academic Vocabulary/ Languagecoefficient, CAS, constant, degree, divisor, expression, factor, long division, monomial, polynomial, power, quotient, radical, rational, remainder, roots, terms, synthetic division, trinomial, x-intercepts, y-intercepts, zero product property, zeros


A.APR.6

68

Students should be able to use a computer algebra system to rewrite complicated rational expressions in different form.

Extended UnderstandingThe use of synthetic division may be introduced as a method but students should recognize its limitations (division by a linear term). When students use methods that have not been developed conceptually, they often create misconceptions and make procedural mistakes due to a lack of understanding as to why the method is valid. They also lack the understanding to modify or adapt the method when faced with new and unfamiliar situations. Suggested viewing Synthetic Division: How to understand It by not doing it. http://www.youtube.com/watch?v=V-Q6jBYn3Oc

Tier 2 Vocabulary inspection, rewrite, simple, strategy

CCSSM DescriptionStudents will learn strategies for rewriting rational expressions in different forms. In order to rewrite simple rational expressions in different forms, students need to understand that the rules governing the arithmetic of rational expressions are the same rules that govern the arithmetic of rational numbers (i.e., fractions). To add fractions, we use a common denominator: The operations of addition, subtraction and multiplication can be applied to polynomials. This cluster is the logical extension of the earlier standards on polynomials and the connection to the integers. This takes the same skills used to organize equations and expressions. Some students may need a review with the arithmetic of simple rational expressions.

I Can StatementsI can rewrite rational expressions using different strategies: inspection, long or synthetic division, computer algebra systems.

Instructional StrategiesThis cluster is the logical extension of the earlier standards on polynomials and the connection to the integers. Now, the arithmetic of rational functions is governed by the same rules as the arithmetic of fractions, based first on division. This cluster is the logical extension of the earlier standards on polynomials and the connection to the integers. Now, the arithmetic of rational functions is governed by the same rules as the arithmetic of fractions, based first on division. In order to rewrite simple rational expressions in different forms, students need to understand that the rules governing the arithmetic of rational expressions are the same rules that govern the arithmetic of rational numbers (i.e., fractions). To add fractions, we use a common denominator. Suggested resources/tools include: graphing calculators, graphing software (including dynamic geometry software), Computer Algebra Systems.


http://www.youtube.com/watch?v=V-Q6jBYn3Oc

http://www.youtube.com/watch?v=V-Q6jBYn3Oc

Common Misconceptions/ChallengesStudents with only procedural understanding of fractions are likely to cancel terms (rather than factors of) in the numerator and denominator of a fraction. Emphasize the structure of the rational expression: that the whole numerator is divided by the whole denominator. In fact, the word “cancel” likely promotes this misconception. It would be more accurate to talk about dividing the numerator and denominator by a common factor.





https://www.illustrativemathematics.org/blueprints/M3/2%20%20%20%20%20%20%20%20%20%20%20







Prior KnowledgeIn Grade 6, students began using the properties of operations to rewrite expressions in equivalent forms.

Future LearningThe study of polynomial functions will continue with students learning to representing and solving equations and inequalities graphically.



Performance Assessments/TasksClick on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.New York State Common Core Mathematics Curriculum: Engage New York: Math Ihttps://www.engageny.org/resource/algebra-ii-module-1-topic-b-lesson-12

POLYNOMIALShttps://www.sophia.org/search/tutorials?q=polynomials

ILLUSTRATIVE MATHEMATICShttps://www.illustrativemathematics.org/blueprints/M3/2

Non Negative Polynomialshttps://www.illustrativemathematics.org/content-standards/HSA/APR/A/1/tasks/1656

Powers of 11https://www.illustrativemathematics.org/content-standards/HSA/APR/A/1/tasks/1654










Graph polynomial functions, identifying zeros when suitable

factorizations are avaialble and showing end behavior.

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Essential UnderstandingStudents should be able to accurately graph polynomial functions.

Extended UnderstandingAdd families of functions, one at a time, to the students’ knowledge base so they can see connections among behaviors of the various functions. Provide numerous examples of real-world contexts, such as exponential growth and decay situations (e.g., a population that declines by 10% per year) to help students apply an understanding of functions in context. Examine rational functions on a graphing calculator and discuss why, for example, the tabular representation shows an “Error” message for some values of y. Students need to be able to verbalize why a function has asymptotes and distinguish between asymptotes and holes.


constant parent function

standard form

end behavior

polynomial function

x- intercept

exponential rate of change

y-intercept

factorization sequence zeros

linear slope

Tier 2 Vocabulary

algebraically

domain sketch

analyze graph symbolically

compare identify terms

contrast numerically variable

CCSSM DescriptionA function can be described in various ways, such as by a graph. The graph of a function is often a useful way of visualizing the relationship of the function models. Manipulating a mathematical expression for a function can throw light on the function’s properties. A graphing utility or a computer algebra system can be used to experiment with properties of these functions and their graphs .

I Can Statements

I can graph polynomial functions accurately.

I can graph functions expressed symbolically and show key features of the graph.

I can graph simple cases by hand and use technology to show more complicated cases

I can identify zeros when factorable and show end behavior.


F.IF.7C

73

Instructional StrategiesGraphing utilities on a calculator and/or computer can be used to demonstrate the changes in behavior of a function as various parameters are varied. Real-world problems, such as maximizing the area of a region bound by a fixed perimeter fence, can help to illustrate applied uses of families of functions

Common Misconceptions/ChallengesStudents may believe that each family of functions (e.g., quadratic, square root, etc.) is independent of the others, so they may not recognize commonalities among all functions and their graphs.

Students may believe that each family of functions (e.g., quadratic, square root, etc.) is independent of the others, so they may not recognize commonalities among all functions and their graphs. Students may also believe that skills such as factoring a trinomial or completing the square are isolated within a unit on polynomials, and that they will come to understand the usefulness of these skills in the context of examining characteristics of functions.




Illustrative Mathematicshttps://www.illustrativemathematics.org/blueprints/M1






Textbook and Curriculum ResourcesORC (Ohio Resource Center: The Ohio State University)http://www.ohiorc.org/search/results/?txtSearchText=functions

https://learnzillion.com/lessonsets/470Virtual NerdGraphing Polynomialshttp://www.virtualnerd.com/search/search.php?query=graphing+polynomials&search=1

Integrated Math I, McGraw HillChapter 4

Prior KnowledgeIn Grade 7, students are exposed to the idea that rewriting an expression can shed light on the meaning of the expression. This idea is expanded upon as students explore functions in high school and recognize how the form of the equation can provide clues about zeros, asymptotes, etc.

Future LearningLearning features of parent functions, the simplest form of a family of function, and features of family functions can increase understanding of functions. These skills will be needed when students study trigonometric functions later in the year.


http://www.virtualnerd.com/search/search.php?query=graphing+polynomials&search=1


http://www.ohiorc.org/search/results/?txtSearchText=functions

Performance Assessments/TasksClick on the links below to access performance tasks where students are asked to demonstrate an understanding of the concept of functions.Inside MathematicsSorting Functionshttp://www.insidemathematics.org/assets/common-core-math-tasks/sorting%20functions.pdf

NRICHhttp://nrich.maths.org/773

Illustrative MathematicsModeling London's Populationhttps://www.illustrativemathematics.org/content-standards/HSF/IF/C/7/tasks/1595

Career ConnectionsAny career that involves the need to articulate verbally the relationships between variables arising in everyday contexts can utilize the study of functions. This include health care area, science, and any career involving sales.

Students can complete the following concept development activities (Representing Functions of Everyday Situations) where they are asked to :• Translate between everyday situations and sketch graphs of relationships between variables.• Interpret algebraic functions in terms of the contexts in which they arise.• Reflect on the domains of everyday functions and in particular whether they should be discrete or Continuous






http://www.insidemathematics.org/assets/common-core-math-tasks/sorting%20functions.pdf


Write expressions in equivalent forms to solve problems.

Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t to reveal thee approximate equivalent monthly interest rate if the annual rate is 15%.

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

Essential Understanding Students should be able to use the properties of exponents to transform simple expressions for exponential functions.

Students should know the difference between an arithmetic sequence and a geometric sequence.

Students should be able to use a formula to solve real world problems.


arithmetic expression numerical expression

coefficient finite sequence

completing the square

geometric sequence

derive linear

Tier 2 Vocabulary

equivalent phenomena properties

CCSSM Description

The ability to interpret and create expressions to model mathematical phenomena is one of the most important skills an education in mathematics can offer. The different expressions can tell us about the quantities they represent; being able to rewrite expressions in another form leads to efficiency when solving a problems. Changing the forms of expressions, such as factoring or completing the square, or transforming expressions from one exponential form to another, are processes that are guided by goals (e.g., investigating properties of families of functions and solving contextual problems).

Extended Understanding Provide opportunities for students to use graphing utilities to explore the effects of parameter changes on a graph.


A.SSE.3C, ASSE.4

77

I Can Statements

I can identify the different parts of the expression and explain their meaning within the context of a problem.

I can use properties of exponents (such as power of a power, product of powers, power of a product, and rational exponents, etc.) to write an equivalent form of an exponential function to reveal and explain specific information about its approximate rate of growth or decay.

I can develop the formula for the sum of a finite geometric series when the ratio is not 1.

I can use the formula to solve real world problems such as calculating the height of a tree after n years given the initial height of the tree and the rate the tree grows each year.

I can calculate mortgage payments


This cluster focuses on linking expressions and functions, i.e., creating connections between multiple representations of functional relations – the dependence between a quadratic expression and a graph of the quadratic function it defines, and the dependence between different symbolic representations of exponential functions. Teachers need to foster the idea that changing the forms of expressions, such as factoring or completing the square, or transforming expressions from one exponential form to another, are not independent algorithms that are learned for the sake of symbol manipulations. They are processes that are guided by goals (e.g., investigating properties of families of functions and solving contextual problems).

Factoring methods that are typically introduced in elementary algebra and the method of completing the square reveals attributes of the graphs of quadratic functions, represented by quadratic equations.• The solutions of quadratic equations solved by factoring are the x – intercepts of the parabola or zeros of quadratic functions. • A pair of coordinates (h, k) from the general form f(x) = a(x – h) 2 +k represents the vertex of the parabola, where h represents a horizontal shift and k represents a vertical shift of the parabola y = x2 from its original position at the origin.• A vertex (h, k) is the minimum point of the graph of the quadratic function if a › 0 and is the maximum point of the graph of the quadratic function if a ‹ 0. Understanding an algorithm of completing the square provides a solid foundation for deriving a quadratic formula.

Translating among different forms of expressions, equations and graphs helps students to understand some key connections among arithmetic, algebra and geometry. The reverse thinking technique (a process that allows working backwards from the answer to the starting point) can be very effective. Have students derive information about a function’s equation, represented in standard, factored or general form, by investigating its graph. Offer multiple real-world examples of exponential functions.


Common Misconceptions/ChallengesSome students may believe that factoring and completing the square are isolated techniques within a unit of quadratic equations. Teachers should help students to see the value of these skills in the context of solving higher degree equations and examining different families of functions. Students may think that the minimum (the vertex) of the graph of y = (x + 5)2 is shifted to the right of the minimum (the vertex) of the graph y = x 2 due to the addition sign. Students should explore examples both analytically and graphically to overcome this misconception. Some students may believe that the minimum of the graph of a quadratic function always occur at the y-intercept. Some students cannot distinguish between arithmetic and geometric sequences, or between sequences and series. To avoid this confusion, students need to experience both types of sequences and series

Common Core Support

Illustrative Mathematics: Learning Progressions http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf

Ohio Department of Education Model Curriculum https://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Algebra_Model_Curriculum_March2015.pdf.aspx


https://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Algebra_Model_Curriculum_March2015.pdf.aspx


Textbook and Curriculum ResourcesProblem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013 Options of Interest, pp. 45-48

Columbus City Schools Curriculum Guide, Math III Quarter 3, 2013

https://learnzillion.com/lessonsets/470 http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17 Integrated Math I, McGraw Hill Chapter 6

Prior KnowledgeIn Grade 8, students compare tables, graphs, expressions and equations of linear relationships.

Future Learning Future learning will continue study of exponential functions with creating equations that describe numbers or relationships.




Performance Assessments/TasksClick on the links below to access performance tasks.Illustrative MathematicsForms of exponential expressionshttps://www.illustrativemathematics.org/content-standards/HSA/SSE/B/3/tasks/1305

A Lifetime of Savingshttps://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/1283

Triangle Serieshttps://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/442

Cantor Sethttps://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/929

Course of Antibioticshttps://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/805

YouTube Explosionhttps://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/1797

Career/ Everyday ConnectionsInsurance, Real Estate, Sales, Science & Engineering


https://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/805




Ohio’s Learning Standards – Clear Learning TargetsIntegrated Mathematics III

Understand that polynomials form a system analogous to theintegers, namely, they are

closed under the operations of addition,subtraction, and multiplication; add, subtract, and multiply polynomials.

Essential Understanding

- Students can add, subtract, and multiply polynomials.


- Students can divide polynomials.- Students can factor polynomials.

Academic Vocabulary/Language

- Polynomial- Monomial- Binomial- Trinomial- distribute- like terms

Tier 2 Vocabulary

- understand- analogous

I Can Statements

I can identify that the sum, difference, or product of two polynomials will always be a polynomial, which means that polynomials are closed under the operations of addition, subtraction, and multiplication.

I can define “closure”. I can apply arithmetic operations of addition, subtraction, and multiplication to polynomials.


A.APR.1

82



Common Misconceptions and Challenges

Textbook and Curriculum ResourcesMcGraw-Hill: Integrated Math II

Chapter 1-1, 1-2, 1-3, 1-3 (explore), 1-4

http://ccssmath.org/?s=apr.1

A-APR Non-Negative Polynomials A-APR Powers of 11

Polynomial Addition and Subtraction- APR.1

Polynomial Multiplication- APR.1

Adding, Multiplying, and Subtracting Monomials- APR.1

https://www.sophia.org/ccss-math-standard-9-12aapr1-pathway

https://sites.google.com/site/commoncorewarwick/home/unit-of-studies/algebra-2/a-apr-1

Career Connections

Social scientists and related occupationsEconomists

Education, training, library, and museum occupationsTeachers-adult literacy and remedial and self-enrichment educationTeachers-postsecondaryTeachers-preschool, kindergarten, elementary, middle, and secondaryTeachers-special education

Health diagnosing and treating occupationsRegistered nurses

Aerospace engineers Chemical engineers Civil engineers Electrical engineersEnvironmental engineers Industrial engineersMaterials engineers Mechanical engineersNuclear engineers Petroleum engineers




















https://www.sophia.org/ccss-math-standard-9-12aapr1-pathway

http://www.mathworksheetsland.com/algebra/8mono.html

http://www.mathworksheetsland.com/algebra/9polymul.html

http://www.mathworksheetsland.com/algebra/8poly.html



http://ccssmath.org/?s=apr.1



Analyze functions using different representations.

Graph exponential and logarithmic functions, showing intercepts and end behavior.

Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change functions such as y-(1.02)t , y=(0.97)t , y=(1.01)12t , y=(1.2)t/10 , and classify them as representing exponential growth or decay.

Essential Understanding Students will be expected to be able to graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*(Modeling standard).

Students will be expected to be able to graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Students will be expected to write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function

Extended Understanding Involve students in activities that include collection and analysis of data to generalize function behaviors. For example, they can take a cup filled with pennies, spill them onto a table, count how many came up “heads,” put only those pennies back in the cup, and repeat this process several times. In the end, they will generate a table of values that will model an exponential decay function with a base of ½.

Academic Vocabulary/ Languagearithmetic, axes, base, constant, decay, differences, equation, explicit, exponential, expression, factors, formula, function, geometric sequence, graph, growth, input, intervals, inverse, linear, logarithm, model, ordered pair, output, parameters, percent, polynomial, quadratic, quantity, rate, recursive, relation, scale, sequence, table, unit

Tier 2 Vocabulary graph, identify, interpret,

CCSSM Description A function can be described in various ways, such as by a graph (e.g., the trace of a seismograph); by a verbal rule, as in, “I’ll give you a state, you give me the capital city;” by an algebraic expression like f(x) = a + bx; or by a recursive rule. The graph of a function is often a useful way of visualizing the relationship of the function models, and manipulating a mathematical expression for a function and can throw light on the function’s properties. A graphing utility or a computer algebra system can be used to experiment with properties of these functions and their graphs and to build computational models of functions.

I Can Statements

I can graph exponential, logarithmic, and trigonometric functions.

I can describe key features of exponential, logarithmic, and trigonometric functions.

I can classify the exponential function as exponential growth or decay by examining the base.

I can use the properties of exponents to interpret expressions for exponential functions in a real-world context.


F.IF.7E, F.IF.8B

85

Instructional StrategiesExplore various families of functions and help students to make connections in terms of general features. Use various representations of the same function to emphasize different characteristics of that function. For example, the y-intercept of the function y = x2 -4x – 12 is easy to recognize as (0, -12). However, rewriting the function as y = (x – 6)(x + 2) reveals zeros at (6, 0) and at ( -2, 0). Furthermore, completing the square allows the equation to be written as y = (x – 2)2 – 16, which shows that the vertex (and minimum point) of the parabola is at (2, -16).

Hands-on materials (e.g., paper folding, building progressively larger shapes using pattern blocks, etc.) can be used as a visual source to build numerical tables for examination.

Common Misconceptions/ChallengesStudents oversimplify rules of exponents. For example, a student might think/claim 𝑒 𝑎+𝑏 = 𝑒 𝑎 + 𝑒 𝑏. This may be the result of students failing to attribute meaning to exponential symbols. Students interpret negative exponents incorrectly or fail to connect the negative symbol back to the idea of inverses. Students make the assumption that a correctly followed algorithm will only ever give correct answers. For example, in solving 𝑙𝑜𝑔2 (𝑥 − 4) = 3 − 𝑙𝑜𝑔2(𝑥 + 3) a student might correctly follow the solution algorithm and claim the answer is 𝑥 = −4, 5 without noting the fact that -4 is an invalid solution since plugging it in to 𝑙𝑜𝑔2 (𝑥 − 4) results in an input value outside the domain of the logarithmic function.

Common Core SupportInstitute for Mathematics and Education Learning Progressions Narrativeshttp://commoncoretools.me/wp-content/uploads/2012/12/ccss_progression_functions_2012_12_04.pdf


Illustrative Mathematics:https://www.illustrativemathematics.org/blueprints/M3





Common Core State Standards Station Activities for Mathematics I, Walch Education, 2014Interpreting Exponential Functions, pp. 110-117



Prior Knowledge In Grade 8, students compare functions by looking at equations, tables and graphs, and focus primarily on linear relationships. In high school, examination of functions is extended to include recursive and explicit representations and sequences of numbers that may not have a linear relationship.

Future LearningFuture learning will include analyzing functions—comparing properties of two functions each represented in a different way.



Performance Assessments/TasksClick on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.POLYNOMIALShttps://www.sophia.org/search/tutorials?q=polynomials

ILLUSTRATIVE MATHEMATICSClassifying Equations of Parallel and Perpendicular Lineshttps://www.illustrativemathematics.org/blueprints/M3/2 http://map.mathshell.org/download.php?fileid=1724

Representing Quadratic Functions Graphicallyhttp://map.mathshell.org/download.php?fileid=1734

Representing Functions of Everyday Situationshttp://map.mathshell.org/download.php?fileid=1740

Representing Polynomials Graphicallyhttp://map.mathshell.org/download.php?fileid=1744

Career Connections/Everyday ApplicationsLogarithms (graphing/analyzing), the inverses of exponential functions, are used in many occupations. Perhaps the most well-known use of logarithms is in the Richter scale, which determines the intensity and magnitude of earthquakes. Yet, there are many other professionals who use logarithms in their careers. Anyone who calculates the quantity of things that increase or decrease exponentially uses logarithms. This includes engineers, coroners, financiers, computer programmers, mathematicians, medical researchers, farmers, physicists and archaeologists. Because there is no definitive list of careers that require the use of logarithms, below is a brief sampling of how some careers employ these log

Read more : http://www.ehow.com/info_8649362_careers-use-logarithms.html


http://www.ehow.com/info_8649362_careers-use-logarithms.html








Ohio’s Learning Standards - Clear Learning TargetsIntegrated Math I


Build a function that models. Write a linear function that

describes a relationship between two quantities.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Write arithmetic sequences both recursively and with an explicit formula use them to model situations, and translate between the two forms.

Essential UnderstandingExamination of functions is extended to include recursive and explicit representations and sequences of numbers that may not have a linear relationship.

Extended Understanding Using a variety of functions (e.g., linear, exponential, constant, students can increase understanding of the different representations by representing functions as a set of ordered pairs, a table, a graph, and an equation.


arithmetic sequence

explicit formula

inverse relationship

function recursive


quantities


Tier 2 Vocabulary

compare model prove

construct observe

CCSSM DescriptionFunctions can be used to make predictions about future behaviors when modeling real life situations. For students to recognize a functional relationship, they need to recognize there is a correspondence and see/understand the correspondence matches each element of the first set with an element of the second set. Once it is known that the relationship is a function, students can determine the rule for the function.


F.BF.1, F.BF.1A, F.BF.2

89

I Can Statements

I can, from context, either write an explicit expression, define a recursive process, or describe the calculations meeded to model a function between two quantities.

I can combine standard function types, such as linear and exponential, using arithmetic operations. I can compose functioins.

I can write arithmetic sequences recursively and explicityly, use the two forms to model a sitation and translate between the two forms.

I can write geometric sequences recursively and expliciitly, use the tow forms to model a sistuation, and translate between the two forms.

I can understand that linear functions are the explicity form of recursievely-defined arithmetic sequences and that exponential functions are the explicit form of recursively-defined geometric sequences.

Instructional StrategiesProvide a real-world example (e.g., a table showing how far a car has driven after a given number of minutes, traveling at a uniform speed), and examine the table by looking “down” the table to describe a recursive relationship, as well as “across” the table to determine an explicit formula to find the distance traveled if the number of minutes is known.

Write out terms in a table in an expanded form to help students see what is happening. For example, if the y-values are 2, 4, 8, 16, they could be written as 2, 2(2), 2(2)(2), 2(2)(2)(2), etc., so that students recognize that 2 is being used multiple times as a factor.

Focus on one representation and its related language – recursive or explicit – at a time so that students are not confusing the formats.

Provide examples of when functions can be combined, such as determining a function describing the monthly cost for owning two vehicles when a function for the cost of each (given the number of miles driven) is known.

Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual models to generate sequences of numbers that can be explored and described with both recursive and explicit formulas. Emphasize that there are times when one form to describe the function is preferred over the other.


Common Misconceptions/Challenges Students may believe that the best (or only) way to generalize a table of data is by using a recursive formula. Students naturally tend to look “down” a table to find the

pattern but need to realize that finding the 100th term requires knowing the 99thterm unless an explicit formula is developed.

Students may also believe that arithmetic and geometric sequences are the same. Students need experiences with both types of sequences to be able to recognize the difference and more readily develop formulas to describe them. Advanced students who study composition of functions may misunderstand function notation to represent multiplication (e.g., f(g(x)) means to multiply the f and g function values).

When studying functions, students sometimes interchange the input and output values. This will lead to confusion about domain and range, and determining if a relation is a function. This can also interfere with a student being able to find the appropriate inverse function, or the correct equation to model a relationship between two quantities.








Textbook and Curriculum ResourcesProblem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013Texting for the Win, pp. 127-131Jai’s Jeans, pp. 132-134New Tablet, pp. 135-138Glass Recycling, pp. 139-142

Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013To Drill or Not to Drill?, pp. 138-Pushing Envelopes, pp. 142-145

Common Core State Standards: Station Activities for Mathematics IRelations Versus Functions/Domain and Range, pp. 85-93Sequences, pp. 118-130Real-World Situation Graphs pp. 194-208-

High School CCSS Mathematics I Curriculum Guide-Quarter 1 Curriculum Guide, 2013, pp. 161-204

Prior KnowledgeIn Grade 8, students learn to compare functions by looking at equations, tables and graphs, and focus primarily on linear relationships.

Future LearningFuture learning will include working with inverse functions.


Performance Assessments/TasksClick on the links below to access performance tasks.Math Assessment ProjectGeneralizing Patterns: Table Tiles http://map.mathshell.org/download.php?fileid=1716

Representing Linear and Exponential Growthhttp://map.mathshell.org/download.php?fileid=1732


Inside MathematicsInfinite Windowshttp://www.insidemathematics.org/assets/problems-of-the-month/infinite%20windows.pdf

Slice and Dicehttp://www.insidemathematics.org/assets/problems-of-the-month/slice%20and%20dice.pdf

First Ratehttp://www.insidemathematics.org/assets/problems-of-the-month/calculating palindromes.pdf http://www.insidemathematics.org/assets/problems-of-the-month/first%20rate.pdf

Cut It Outhttp://www.insidemathematics.org/assets/problems-of-the-month/cut%20it%20out.pdf

Illustrative MathematicsSummer InternCareer ConnectionsStudents can research and evaluate several options when purchasing a vehicle (e.g., new versus used, lease versus own, down payment, and interest rate). They will examine the differences in gas mileage consumption by selecting two vehicles to evaluate (e.g., SUV versus compact hybrid). Once they choose a vehicle, they will use their evaluations to show why they chose the vehicle. Their research will include interviewing automotive professionals, visiting dealerships, and navigating company websites.

Applicable careers include business, finance, insurance and any career focused on making scholarly predictions.


http://www.insidemathematics.org/assets/problems-of-the-month/cut%20it%20out.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/first%20rate.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/calculating%20palindromes.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/slice%20and%20dice.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/infinite%20windows.pdf






Build new functions from existing functions. Find inverse functions.

Solve a linear equation of the form f(x)=c for a simple function f that has an inverse and write an expression for the inverse.

Essential UnderstandingStudents should understand that an inverse function does the reverse of a given function. Square and square root functions are examples of an inverse function within the domain of nonnegative numbers.

Extended UnderstandingAdvanced students can expand their catalog of functions to include exponential and logarithmic cases. Students can learn to contrast an invertible and non-invertible function which is mentioned in the Functions Progressions document as a reasonable extension of the standard.


dependent variable

inverse


function invertible

independent variable

non-invertible

Tier 2 Vocabulary

build

interchanging

CCSSM DescriptionIn simple terms, an inverse function undoes what the original function does. Continued studies with parent functions can facilitate deeper understanding of functions.


F.BF.4, F.BF.4A

94

I Can StatementsI can solve a function for the dependent variable and write the inverse of a function by interchanging the values of th dependent and independent variables.

Instructional StrategiesProvide examples of inverses that are not purely mathematical to introduce the idea. For example, given a function that names the capital of a state, f(Ohio) = Columbus, the inverse would be to input the capital city and have the state be the output such that f—1 (Denver) = Colorado.

Allow students to initially make tables of values by hand for some simple examples, such as y = x + 3 to examine the effects of changing the constant, including the existence of inverses. Students can then examine additional effects and more complicated functions with technology.

Use real-world examples of functions and their inverses. For example, students might determine that folding a piece of paper in half 5 times results in 32 layers of paper, but that if they are given that there are 32 layers of paper, they can solve to find how many times the paper would have been folded in half.


Common Misconceptions/Challenges Students may believe that the graph of y = (x – 4)3 is the graph of y = x 3 shifted 4 units to the left (due to the subtraction symbol). Examples should be explored by hand and

on a graphing calculator to overcome this misconception.

Students may also believe that even and odd functions refer to the exponent of the variable, rather than the sketch of the graph and the behavior of the function.In f -1 (x) =3x+3, students may think -1 is an exponent.


Illustrative MathematicsHigh School: Functions https://www.illustrativemathematics.org/content-standards/HSF/BF

Integrated Math Ihttps://www.illustrativemathematics.org/blueprints/M1



https://www.illustrativemathematics.org/content-standards/HSF/BF


Textbook and Curriculum ResourcesProblem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013 Finding Inverse Functions, pp. 156-159

LearnZillion https://learnzillion.com/resources/31835#fndtn-resource__content

YouTubeKhan Academy: Finding Inverse Functionshttps://www.youtube.com/watch?v=W84lObmOp8M

Inverse Functionshttps://www.youtube.com/watch?v=Y-wxZdMMcYc

Inverse Functions- The Basics!:patrickjmthttps://www.youtube.com/watch?v=nSmFzOpxhbY

Integrated Math I, McGraw HillChapter 4

Prior Knowledge Understanding functional relationships as input and output values that have an associated graph is introduced in Grade 8. In high school, changes in graphs is explored in more depth, and the idea of functions having inverses is introduced.

Future Learning Students will begin studies of linear equations and inequalities in one variable and exponentials in future studies.


https://www.youtube.com/watch?v=nSmFzOpxhbY

https://www.youtube.com/watch?v=Y-wxZdMMcYc

https://www.youtube.com/watch?v=W84lObmOp8M

https://learnzillion.com/resources/31835#fndtn-resource__content

Performance/Assessment TasksClick on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.Inside Mathematics Digging Dinosaurshttp://www.insidemathematics.org/assets/problems-of-the-month/digging%20dinosaurs.pdf

Illustrative Mathematicshttps://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579 Invertible or Not?https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758 https://www.illustrativemathematics.org/content-standards/HSF/BF/B/4/tasks/1374

Householdshttps://www.illustrativemathematics.org/content-standards/HSF/BF/B/4/tasks/234

Temperature Conversionshttps://www.illustrativemathematics.org/content-standards/HSF/BF/B/4/tasks/364

Temperature in Degrees Fahrenheit and Celsiushttps://www.illustrativemathematics.org/content-standards/HSF/BF/B/4/tasks/501

Career ConnectionsPopulation Studies: choosing a linear function to model the given data, and then use the inverse function to interpolate a data point (see Households task above).


https://www.illustrativemathematics.org/content-standards/HSF/BF/B/4/tasks/501






http://www.insidemathematics.org/assets/problems-of-the-month/digging%20dinosaurs.pdf



Construct and compare linear, quadratics, and exponential models and

solve problems.

For exponential models, express as logarithm the solution to abct =d where a,c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Essential UnderstandingStudents will be expected to know how to express logarithms as solutions to exponential functions using bases 2, 10, and e.

Students will be expected to know how to use technology to evaluate a logarithm.


Academic Vocabulary/ Languagearithmetic, axes, base, constant, decay, differences, equation, explicit, exponential, expression, factors, formula, function, geometric sequence, graph, growth, input, intervals, inverse, linear, logarithm, model, ordered pair, output,


F.LE.4

99

Provide opportunities where students care given examples of real-world situations that apply linear and exponential functions to compare their behaviors.

parameters, percent, polynomial, quadratic, quantity, recursive, relation, scale, sequence, solution, table, unit

Tier 2 Vocabulary express, properties, technology

CCSSM DescriptionGiven sufficient information, e.g., a table of values together with information about the type of relationship represented, students will learn how to construct/evaluate the appropriate model. Technology will be used.

I Can Statements

I can use the properties of logs.

I can describe the key features of logs.

I can use logarithmic form to solve exponential models.


Instructional StrategiesUse technology to solve exponential equations such as 3*10x = 450. (In this case, students can determine the approximate power of 10 that would generate a value of 150.) Students can also take the logarithm of both sides of the equation to solve for the variable, making use of the inverse operation to solve.

Instructional Resource Tools: Examples of real-world situations that apply linear and exponential functions to compare their behaviors; graphing calculators or computer software that generates graphs and tables of functions; a graphing tool such as the one found at nlvm.usu.edu is one option.

Common Misconceptions/ChallengesStudents may believe that all functions have a first common difference and need to explore to realize that, for example, a quadratic function will have equal second common differences in a table. Students may also believe that the end behavior of all functions depends on the situation and not the fact that exponential function values will eventually get larger than those of any other polynomial functions.








Textbook and Curriculum ResourcesIntegrated Math III, McGraw Hill Chapters 2-7



Prior Knowledge While students in Grade 8 examine some nonlinear situations, most of the functions explored are linear. Students will build on the understanding of exponents that began in Grade 8.8.EE.1.

Future Learning Next studies will include trigonometric functions---interpreting functions.



Performance Assessments/TasksClick on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.

Carbon 14 Datinghttps://www.illustrativemathematics.org/content-standards/HSF/LE/A/4/tasks/369

Bacteria Populationshttps://www.illustrativemathematics.org/content-standards/HSF/LE/A/4/tasks/370

Comparing Exponentialshttps://www.illustrativemathematics.org/content-standards/HSF/LE/A/tasks/213

Newton’s Law of Coolinghttps://www.illustrativemathematics.org/content-standards/HSF/LE/A/4/tasks/382

Exponential Kisshttps://www.illustrativemathematics.org/content-standards/HSF/LE/A/4/tasks/1824

Other Taskshttps://www.illustrativemathematics.org/content-standards/HSF/LE/A/4

Career Connections/Everyday ApplicationsJobs using quadratics/exponential models include: Military and Law Enforcement; Engineering , Science, Management and Clerical Work, Agriculture:

http://www.ehow.com/info_8711999_careers-use-quadratic-equations.html?%20%20%20utm_source=eHowMobileShare%26utm_medium=email


http://www.ehow.com/info_8711999_careers-use-quadratic-equations.html?%20%20%20utm_source=eHowMobileShare%26utm_medium=email

https://www.illustrativemathematics.org/content-standards/HSF/LE/A/4

https://www.illustrativemathematics.org/content-standards/HSF/LE/A/4/tasks/1824


https://www.illustrativemathematics.org/content-standards/HSF/LE/A/tasks/213





Extend domain of trigonometric function using the

unit circle.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Essential UnderstandingStudents are expected to know that the radian measure of an angle is the length of the arc on the unit circle subtended by the angle.

Students are expected to be able to explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers interpreted as radian measures of angles traversed counterclockwise around the unit circle

Academic Vocabulary/ Languageamplitude, angle, arc, arccosine(arccos), arcsine (arcsin), arctangent (arctan), axes, circle, clockwise, constant, coordinate, cosine (cos), counterclockwise, degree, differences, equation, expression, formula, Frequency, function, graph, identity, input, intervals, inverse, midline, model, ordered pair, output, period, quadrant, quantity, radian, relation, sine (sin), subtend, table, tangent (tan), trigonometric, unit, unit circle

Tier 2 Vocabularyexplain, understand

CCSM DescriptionA unit circle is a circle with a radius of one. In trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. Because the radius of the unit circle is one, the trigonometric functions sine and cosine have special relevance for the unit circle.

I Can Statements

I can understand and explain that if the length of an arc subtended by an angle is the same length as the radius of the circle, then the measure of the angle is 1 radian.

I can understand and explain that the graph of the function, f, is the graph of the equation y=f(x).

I can explain how radian measures of angles rotated counterclockwise in a unit circle are in a one-to-one correspondence with the nonnegative real numbers, and that angles rotated clockwise in a unit circle are in a on-to-one correspondence with the non-positive real numbers.


F.TF.1, F.TF.2

104

Instructional StrategiesUse a compass and straightedge to explore a unit circle with a fixed radius of 1. Help students to recognize that the circumference of the circle is 2π, which represents the number of radians for one complete revolution around the circle. Students can determine that, for example, π/4 radians would represent a revolution of 1/8 of the circle or 45°. Having a circle of radius 1, the cosine, for example, is simply the x-value for any ordered pair on the circle (adjacent/hypotenuse where adjacent is the x-length and hypotenuse is 1). Students can examine how a counterclockwise rotation determines a coordinate of a particular point in the unit circle from which sine, cosine, and tangent can be determined. Some information below includes additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics and goes beyond the mathematics that all students should study in order to be college- and career-ready: Some students can use what they know about 30-60-90 triangles and right isosceles triangles to determine the values for sine, cosine, and tangent for π/3, π/4, and π/6. In turn, they can determine the relationships between, for example, the sine of π/6, 7π/6, and 11π/6, as all of these use the same reference angle and knowledge of a 30-60-90 triangle. Provide students with real-world examples of periodic functions. One good example is the average high (or low) temperature in a city in Ohio for each of the 12 months. These values are easily located at weather.com and can be graphed to show a periodic change that provides a context for exploration of these functions. Allow plenty of time for students to draw – by hand and with technology – graphs of the three trigonometric functions to examine the curves and gain a graphical understanding of why, for example, cos (π/2) = 0 and whether the function is even (e.g., cos(-x) = cos(x)) or odd (e.g., sin(-x) = -sin(x)). Similarly, students can generalize how function values repeat one another, as illustrated by the behavior of the curves.Common Misconceptions/ChallengesStudents may believe that there is no need for radians if one already knows how to use degrees. Students need to be shown a rationale for how radians are unique in terms of finding function values in trigonometry since the radius of the unit circle is 1. Students may also believe that all angles having the same reference values have identical sine, cosine and tangent values. They will need to explore in which quadrants these values are positive and negative.

Other challenges include: failure to identify the advantages of radian measurements over degree measurement.; overgeneralization; e.g. assuming all trigonometry functions have a range of -1 to 1; confusion regarding domain restrictions when defining inverses.; confusion over inverse notation; failure to connect the Pythagorean Theorem to other aspects of trigonometry; failure to identify the relationship between various trig functions such as sine and cosine; algorithmic oversimplification; e.g. assuming sin(𝑎 + 𝑏) = sin(𝑎) + sin(𝑏).






Textbook and Curriculum ResourcesIntegrated Math III, McGraw HillChapters 11-12

High School CCSS Mathematics I Curriculum Guide-Quarter 3 Curriculum Guide, 2013

Prior KnowledgeStudents begin studying right triangles using the Pythagorean Theorem in Grade 8.

Future LearningFuture learning will include modeling periodic phenomena with trigonometric functions.


Performance Assessments/TasksClick on the links below to access performance tasks.Illustrative MathematicsBicycle Wheel https://www.illustrativemathematics.org/content-standards/HSF/TF/A/1/tasks/1873

What Exactly Is A Radian?https://www.illustrativemathematics.org/content-standards/HSF/TF/A/1/tasks/1874

Trigonometric functions for arbitrary angleshttps://www.illustrativemathematics.org/content-standards/HSF/TF/A/2/tasks/1692

Trigonometric Identities and Rigid Motionshttps://www.illustrativemathematics.org/content-standards/HSF/TF/A/2/tasks/1698

Trig Functions and the Unit Circlehttps://www.illustrativemathematics.org/content-standards/HSF/TF/A/2/tasks/1820

Properties of Trigonometric Functionshttps://www.illustrativemathematics.org/content-standards/HSF/TF/A/2/tasks/1704

https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579Career Connections Management Occupations, Administrative Support, Construction, Production, Professional, Farming, Installationwww.xpmath.com/careers


http://www.xpmath.com/careers


https://www.illustrativemathematics.org/content-standards/HSF/TF/A/2/tasks/1704







Model periodic phenomena with trigonometric

functions.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Essential Understanding Students are expected to be able to define the parameters of trigonometric functions.

Students are expected to be able to interpret trigonometric functions in context.

Students are expected to identify and model periodic phenomena in real world situations.

Extended UnderstandingProvide students with a list of real-world applications of periodic situations that can be modeled by using trigonometric functions for students to explore. Utilize graphing calculators or computer software to generate the graphs of trigonometric functions.


Tier 2 Vocabularyexplain, understand

CCSM DescriptionThe study of trigonometry is reserved for high school students. In the Geometry conceptual category, students explore right triangle trigonometry, with advanced students working with laws of sines and cosines. In the conceptual category of Functions, students connect the idea of functions with trigonometry and explore the effects of parameter changes on the amplitude, frequency and midline of trigonometric graphs.

I Can Statements

I can define and recognize the parameters of trigonometric functions.

I can interpret trig functions in real-world situations.

I can identify and model periodic phenomena in real-world situations.


F.TF.5

108

Instructional Strategies Allow students to explore real-world examples of periodic functions. Examples include average high (or low) temperatures throughout the year, the height of ocean tides as they advance and recede, and the fractional part of the moon that one can see on each day of the month. Graphing some real-world examples can allow students to express the amplitude, frequency, and midline of each. Help students to understand what the value of the sine (cosine, or tangent) means (e.g., that the number represents the ratio of two sides of a right triangle having that angle measure). Using graphing calculators or computer software, as well as graphing simple examples by hand, have students graph a variety of trigonometric functions in which the amplitude, frequency, and/or midline is changed. Students should be able to generalize about parameter changes, such as what happens to the graph of y = cos(x) when the equation is changed to y = 3cos(x) + 5. Some information below includes additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics and goes beyond the mathematics that all students should study in order to be college- and career-ready: Some students can explore the inverse trigonometric functions, recognizing that the periodic nature of the functions depends on restricting the domain. These inverse functions can then be used to solve real-world problems involving trigonometry with the assistance of technology

Common Misconceptions/Challenges Students may believe that all trigonometric functions have a range of 1 to -1. Students need to see examples of how coefficients can change the range and the look of the graphs. Students may also believe that restrictions to the domain of trigonometric functions are not necessary for defining inverse functions. Students may also believe that sin-1 A = 1/sin A, thus confusing the ideas of inverse and reciprocal functions. Additionally, students may not understand that when sin A = 0.4, the value of A represents an angle measure and that the function sin-1 (0.4) can be used to find the angle measure.






Textbook and Curriculum ResourcesMath III, McGraw HillChapters 11-12


Prior KnowledgeStudents begin studying right triangles using the Pythagorean Theorem in Grade 8.

Future Learning Future learning will include proving and applying trigonometric identities.


Performance Assessments/TasksClick on the links below to access performance tasks.Illustrative MathematicsAs the Wheel Turnshttps://www.illustrativemathematics.org/content-standards/HSF/TF/B/5/tasks/595

Foxes and Rabbits 2https://www.illustrativemathematics.org/content-standards/HSF/TF/B/5/tasks/816

Foxes and Rabbits 3https://www.illustrativemathematics.org/content-standards/HSF/TF/B/5/tasks/817

Hours of Daylight 1https://www.illustrativemathematics.org/content-standards/HSF/TF/B/5/tasks/1832

https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758

Career ConnectionsManagement Occupations, Administrative Support, Construction, Production, Professional, Farming, Installation http://www.xpmath.com/careers/math_jobs.php




http://www.xpmath.com/careers/math_jobs.php



https://www.illustrativemathematics.org/content-standards/HSF/TF/B/5/tasks/1832





Prove and apply trigonometric identities.

Prove the Pythagorean identity sin2 (Ө) + cos2 (Ө) =1 and use it to find sin(Ө), cos(Ө), or tan(Ө) and the quadrant of the angle.

Essential Understanding Students are expected to be able to define the trigonometric ratios.

Students are expected to prove the Pythagorean identity.

Students are expected to be able to use the Pythagorean identity to find sin (Ө), cos (Ө), or tan (Ө), given sin(Ө), cos(Ө), or tan(Ө), and the quadrant of the angle.

Extended UnderstandingProvide students opportunity to draw the unit circle; drawings can be useful in showing why the Pythagorean relationship must be true. Dynamic geometry software, such as Geometer’s Sketchpad or Geogebra, can be used to demonstrate that, regardless of the angle measure, the Pythagorean relationship always holds in the unit circle.


Tier 2 Vocabularydefine, explain, prove, understand, use

I Can Statements

I can define trigonometric ratios as related to the unit circle.

I can prove the Pythagorean identity sin2 (Ө) + cos2 (Ө) =1.

I can use the Pythagorean identity, sin2 (Ө) + cos2 (Ө) =1, to find sin (Ө), cos (Ө), or tan (Ө), given sin(Ө), cos(Ө), or tan(Ө), and the quadrant of the angle.


F.TF.8

112

Instructional Strategies In the unit circle, the cosine is the x-value, while the sine is the y-value. Since the hypotenuse is always 1, the Pythagorean relationship sin 2 (θ) + cos2 (θ) = 1 is always true. Students can make a connection between the Pythagorean Theorem in geometry and the study of trigonometry by proving this relationship. In turn, the relationship can be used to find the cosine when the sine is known, and vice-versa. Provide a context in which students can practice and apply skills of simplifying radicals. Some information below includes additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics and goes beyond the mathematics that all students should study in order to be college- and career-ready: Some students can explore other trigonometric identities, such as the half-angle, double-angle, and addition/subtraction formulas to extend on the Pythagorean relationship. Formulas should be proven and then used to determine exact values when given an angle measure, to prove identities, and to solve trigonometric equations. For example, by dividing the formula sin2 (θ) + cos2 (θ) = 1 by cos2 (θ), a new formula is generated ( tan2 (θ) +1= sec 2 (θ) ).

Common Misconceptions/ChallengesStudents may believe that there is no connection between the Pythagorean Theorem and the study of trigonometry. Students may also believe that there is no relationship between the sine and cosine values for a particular angle. The fact that the sum of the squares of these values always equals 1 provides a unique way to view trigonometry through the lens of geometry. Additionally, students may believe that sin(A +B) = sinA + sinB and need specific examples to disprove this assumption.






Textbook and Curriculum ResourcesMath III, McGraw HillChapters 11-12


Prior KnowledgeStudents in Grade 8 grade learn to use the Pythagorean Theorem, while high school students in a geometry unit study right triangle trigonometry. This cluster allows high school students to connect these ideas as they derive a Pythagorean relationship for the trigonometric functions.

Future Learning Future learning will include study of coordinated geometry.


Performance Assessments/TasksClick on the links below to access performance tasks.Illustrative MathematicsTrigonometric Ratios and the Pythagorean Theoremhttps://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579 https://www.illustrativemathematics.org/content-standards/HSF/TF/C/8/tasks/1693https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758Finding Trig Valueshttps://www.illustrativemathematics.org/content-standards/HSF/TF/C/8/tasks/1835

Calculations with sine and cosinehttps://www.illustrativemathematics.org/content-standards/HSF/TF/C/8/tasks/1868

Career ConnectionsManagement Occupations, Administrative Support, Construction, Production, Professional, Farming, Installation http://www.xpmath.com/careers/math_jobs.php




http://www.xpmath.com/careers/math_jobs.php

https://www.illustrativemathematics.org/content-standards/HSF/TF/C/8/tasks/1868






Use coordinates to prove simple geometric theorems

algebraicallyUse coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given point in the coordinate plane is a rectangle; prove or disprove that the point (1, √3 ) lies on the circle centered at the origin and containing the point (0, 2).

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Essential UnderstandingStudents are expected to prove simple geometric theorems algebraically; students are expected to prove the slope criteria for parallel and perpendicular lines and sue them to solve problems; students are expected to know how to use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

Extended UnderstandingProvide students opportunities to: find pictures of real world examples of parallel lines. They can use magazines, clip art, internet pictures or take pictures themselves. Overlaying graph paper on their picture, instruct them to prove the lines are parallel. ; use Google Earth to find a real-world shape (i.e, a metro park, their yard, the stadium at OSU). Ask the students to determine the perimeter and area of their diagram using coordinate geometry. Discuss scale factor with students, reminding them to use a realistic scale to determine the perimeter and area. (You might also have several students use the same picture so they can compare their perimeters and areas)

Academic Vocabulary/ Languagealtitude, area, centroid, diagonal, directed segment, distance formula, intersecting lines, line segment, median, midpoint, ordered pair, parallel, parallelogram, partitioning a segment, perimeter, perpendicular, perpendicular bisector, polygon, Pythagorean Theorem, quadrilateral, ratio, reciprocal, segment bisector, segment partition, slope

Tier 2 Vocabularydefine, explain, find, prove, understand, use

CCSM Description The correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and understanding algebra. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof.


G.GPE.4, G.GPE.5, G.GPE.6, G.GPE.7

116

I Can Statements

I can use coordinate geometry to prove geometric theorems algebraically; I can, using slope, prove lines are parallel or perpendicular ; I can find equations of lines based on certain slope criteria such as; finding the equation of a line parallel or perpendicular to a given line that passes through a given point;I can, given two points, find the point on the line segment between the two points that divides the segment into a given ratio; I can use coordinate geometry and the distance formula to find the perimeters of polygons and the areas of triangles and rectangles.

Instructional StrategiesReview the concept of slope as the rate of change of the y-coordinate with respect to the x-coordinate for a point moving along a line, and derive the slope formula. Use similar triangles to show that every nonvertical line has a constant slope. Review the point-slope, slope-intercept and standard forms for equations of lines. Investigate pairs of lines that are known to be parallel or perpendicular to each other and discover that their slopes are either equal or have a product of –1, respectively. Pay special attention to the slope of a line and its applications in analyzing properties of lines. Allow adequate time for students to become familiar with slopes and equations of lines and methods of computing them. Use slopes and the Euclidean distance formula to solve problems about figures in the coordinate plane such as: Given three points, are they vertices of an isosceles, equilateral, or right triangle? Given four points, are they vertices of a parallelogram, a rectangle, a rhombus, or a square? Given the equation of a circle and a point, does the point lie outside, inside, or on the circle? Given the equation of a circle and a point on it, find an equation of the line tangent to the circle at that point. Given a line and a point not on it, find an equation of the line through the point that is parallel to the given line. Given a line and a point not on it, find an equation of the line through the point that is perpendicular to the given line. Given the equations of two non-parallel lines, find their point of intersection. Given two points, use the distance formula to find the coordinates of the point halfway between them. Generalize this for two arbitrary points to derive the midpoint formula. Use linear interpolation to generalize the midpoint formula and find the point that partitions a line segment in any specified ratio. Use the distance formula to find the length of each side of a polygon whose vertices are known, and compute the perimeter of that figure.


Common Misconceptions/ChallengesStudents may claim that a vertical line has infinite slopes. This suggests that infinity is a number. Since applying the slope formula to a vertical line leads to division by zero, we say that the slope of a vertical line is undefined. Also, the slope of a horizontal line is 0. Students often say that the slope of vertical and/or horizontal lines is “no slope,” which is incorrect.

Common Core SupportCommon Core State Standards: Geometryhttp://www.corestandards.org/Math/Content/HSG/introduction/




http://www.corestandards.org/Math/Content/HSG/introduction/

Textbook and Curriculum ResourcesMath III, McGraw HillChapters 14, 15

Problem Based Tasks for Mathematics I, Walch Education, 2013Field of Dreams, pp. 228-236 Building Fences, pp. 242-246

Problem Based Tasks for Mathematics II, Walch Education, 2013A Circle Graph for Lunch, pp. 308-311Points of Shade, pp. 312-316

Geometry Station Activities for Common Core State Standards, Walch Education, 2013Similarity, Right Triangles, and Trigonometry, pp. 109-135

Common Core State Standards Station Activities for Mathematics I, Walch Education, 2014Parallel Lines, Slopes, and Equations, pp. 159-168Perpendicular Lines, pp. 169-180Coordinate Proof with Quadrilaterals, pp. 181-190

Prior KnowledgeRates of change and graphs of linear equations were studied in Grade 8 and generalized in the Functions and Geometry Conceptual Categories in high school. Therefore, an alternative way to define the slope of a line is to call it the tangent of an angle of inclination of the line. In calculus, the concept of slope will be extended again to the slope of a curve at a particular point

Future Learning The next area of study will be geometric constructions and measurement.


Performance Assessments/TasksClick on the links below to access performance tasks.Illustrative MathematicsMidpoint Miraclehttps://www.illustrativemathematics.org/content-standards/HSG/GPE/B/4/tasks/605

SRT Unit Squares and Triangleshttps://www.illustrativemathematics.org/content-standards/HSG/GPE/B/4/tasks/918

Parallel Lines in the Coordinate Planehttps://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1880

SRT Slope Criterion for Perpendicular Lineshttps://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1876

Triangles inscribed in a Circlehttps://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1332

Equal Area Triangles on the Same Base Ihttps://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1347

Equal Area Triangles on the Same Base IIhttps://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1348

Career ConnectionsManagement Occupations, Administrative Support, Construction, Production, Professional, Farming, Installation, Computer and mathematical Occupations, Architects/Surveyors/Cartographers, Engineering, Business/Finance, Scientists, Pilots,






https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1348









Make geometric constructions.Make formal geometric

constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment: copying an angle; bisecting a segment; constructing perpendicular lines, including the perpendicular bisector of a line segment and constructing a line parallel to a given line through a point not on the line.

Construct an equilateral triangle, a square, and a regular hexagon inscribe in a circle.

Essential Understanding Students should be able to apply definitions,

properties, theorems about line segments, rays, and angles to support geometric constructions.

Student should be able to apply properties, theorems about parallel and perpendicular lines to support geometric constructions.

Students should be able to construct a square equilateral triangle, regular hexagon inscribed in a circle.

Extended Understanding Students can create drawings using nothing more than a compass and straightedge: e.g., stars inside of a circle, dodecagons; students can then calculate the each inscribed image.


arc equilateral regular hexagon

bisector triangle regular polygon

circle inscribe square

circumscribe parallel straightedge

congruent perpendicular triangle

diameter radius

CCSSM DescriptionStudents should be able to formalize and explain the construction of geometric figures using a variety of tools and methods.

compass draw sketch

construct explain

Tier 2 Vocabulary


G.CO.12, G.CO.13

121

I Can Statements

I can copy: a segment, an angle.

I can bisect: a segment, an angle.

I can construct perpendicular lines, including the perpendicular bisector of a line segment.

I can construct a line parallel to a given line to a point not on the line.

I can consturct an equilateral triangle so that each vertex of the triangel is on the circle.

I can construct a square so that each vertex of the squate is on the circle.

I can construct a regular hexagon so that each vertex of the regular hexagon is on the circle.

Instructional Strategies Students should analyze each listed construction in terms of what simpler constructions are involved (e.g., constructing parallel lines can be done with two

different construction of perpendicular lines).

Challenge students to perform the same construction using a compass and a string. Use paper folding to produce a reflection; use bisections to produce reflections.

Ask students to produce “how to” manuals, giving verbal instructions for particular constructions.

Provide meaningful opportunities (constructing the centroid or the incenter of a triangle) to offer students practice in executing basic constructions.

Compare dynamic geometry commands to sequences of compass-and- straightedge steps. Utilize technology in construction activities.

To ensure that students are correctly making instructions and not just estimating a parallel line or the bisector of an angle, remind students that you will be looking for the marks made by the sharp points of th3e compass and that there should be arcs made of the drawing; it should be clear where the arcs cross each other.


Common Misconceptions/ChallengesSome students believe that construction is the same as sketching or drawing.

Teachers should emphasize the need for precision and accuracy when doing constructions. Stress the ideas that a compass and straightedge are identical to a protractor and ruler. Explain the definition of measurement and construction.

If not using safety compasses, make certain that students know to use tool in a cautious, safe manner.

Remind students to keep compass opened at the same setting throughout the entire construction unless they are told to readjust the tool.

Common Core SupportIllustrative Mathematics Constructions and Rigid Motions https://www.illustrativemathematics.org/blueprints/G/1

Achieve the Core Modules, Resourceshttp://achievethecore.org/category/416/mathematics-tasks?&g%5B%5D=9&g%5B%5D=10&g%5B%5D=11&g%5B%5D=12&sort=name The Common Core in Ohiohttp://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htm





http://achievethecore.org/category/416/mathematics-tasks?&g%5B%5D=9&g%5B%5D=10&g%5B%5D=11&g%5B%5D=12&sort=name

https://www.illustrativemathematics.org/blueprints/G/1

Textbook and Curriculum ResourcesIntegrated Math I, McGraw Hill Chapters 10,11,12,13, 14

Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013Copying Segments and Angles, pp. 199-204Bisecting Segments and Angles, pp. 205-208Constructing Perpendicular and Parallel Lines, pp. 209-212

Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013Sensing Distance, pp. 177-180Calibrating Consoles, pp. 181-184Life-Size Support, pp. 185-188Sailing Centroid, pp. 189-195

Common Core Standards Station Activities for Mathematics II, Welch Education, 2014 Circumcenter, Incenter, Orthocenter, and Centroid, pp. 94-107

Geometry Station Activities Common Core State Standards: Welch Education, 2013 Classifying Triangles and Angle Theorems, pp. 13-27 Bisectors, Medians, and Altitudes, pp. 50-63 Triangle Inequalities, pp. 64-75 Ratio Segments, pp. 123-135

Prior KnowledgeDrawing geometric shapes with rulers, protractors, and technology is developed in Grade 7. In high school, students perform formal geometric constructions using a variety of tools. Students will utilize proofs to justify validity of their constructions.

Future LearningFuture learning will include study of basic geometric definitions and rigid motions, geometric relationships and properties.


http://connected.mcgraw-hill.com/connected/

Performance Assessments/TasksClick on the links below to access performance tasks.Math Assessment ProjectInscribing and Circumscribing Right Triangles http://map.mathshell.org/download.php?fileid=1758

Transforming 2D Figureshttp://map.mathshell.org/download.php?fileid=1772

Evaluating Statements About Length and Areahttp://map.mathshell.org/download.php?fileid=1750

Inside Mathematics Circles in Triangleshttp://www.insidemathematics.org/assets/common-core-math-tasks/circles%20in%20triangles.pdf

What’s My Angle?http://www.insidemathematics.org/assets/problems-of-the-month/what's%20your%20angle.pdf

The Shape of Thingshttp://www.insidemathematics.org/assets/problems-of-the-month/the%20shape%20of%20things.pdf

Polly Gonehttp://www.insidemathematics.org/assets/problems-of-the-month/polly%20gone.pdf

Once Upon a Timehttp://www.insidemathematics.org/assets/problems-of-the-month/once%20upon%20a%20time.pdf

Career/Everyday ConnectionsWith regard to constructing perpendicular lines and bisectors, engage students in a discussion where students will identify the applications of concepts/skills in career areas such as: landscaping, agriculture, construction, architecture, logistics, and engineering.


http://www.insidemathematics.org/assets/problems-of-the-month/once%20upon%20a%20time.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/polly%20gone.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/the%20shape%20of%20things.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/what's%20your%20angle.pdf

http://www.insidemathematics.org/assets/common-core-math-tasks/circles%20in%20triangles.pdf





Visualize relationships between two-

dimensional and three-dimensional objects.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Essential Understanding When given a three-dimensional object, students will be

expected to identify the shape made when the object is cut into cross sections.

Students are expected to know the three-dimensional figure that is generated when a two dimensional figure is rotating.

Students are expected to know that a cross section of a solid is an intersection of a plane (two dimensional) and a solid (three-dimensional).

Extended Understanding Provide opportunities such as the following, for students

to engage in experiences using skills learned in this sections:

Tennis Balls in a Can http://www.illustrativemathematics.org/illustrations/512

- a real life situation using a can of tennis balls and an x-ray machine at the airport to see the cross sections of the can, and to determine what the cross section would look like in different circumstances.

Academic Vocabulary/ Languagearea, base, bisect, circle, circumference, construct, coplanar, cone, cross section, cutting plane, cube, cylinder, diameter, dimension, equilateral, line, parallel, perpendicular, pi, plane, radius, regular, rotation, slid, solid of revolution, volume

CCSSM Description

Students will learn to analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Tier 2 Vocabularyidentify


G.GMD.4

126

http://www.illustrativemathematics.org/illustrations/512

I Can Statements

I I can, given a three- dimensional object, identify the shape made when the object is cut into cross-sections.

I can, when rotating a two- dimensional figure, such as a square, know the three-dimensional figure that is generated, such as a cylinder. Understand that a cross section of a solid is an intersection of a plane (twodimensional) and a solid (three-dimensional).

Instructional StrategiesReview vocabulary for names of solids (e.g., right prism, cylinder, cone, sphere, etc.). Slice various solids to illustrate their cross sections. For example, cross sections of a cube can be triangles, quadrilaterals or hexagons. Rubber bands may also be stretched around a solid to show a cross section. Cut a half-inch slit in the end of a drinking straw, and insert a cardboard cutout shape. Rotate the straw and observe the three-dimensional solid of revolution generated by the two-dimensional cutout. Java applets on some web sites can also be used to illustrate cross sections or solids of revolution. Encourage students to create three-dimensional models to be sliced and cardboard cutouts to be rotated. Students can also make three-dimensional models out of modeling clay and slice through them with a plastic knife.


Common Misconceptions/Challenges Some cross sections are more difficult to visualize than others. For example, it is often easier to visualize a rectangular cross section of a cube than a hexagonal cross section. Generating solids of revolution involves motion and is difficult to visualize by merely looking at drawings.

Common Core SupportIllustrative Mathematics Constructions and Rigid Motionshttps://www.illustrativemathematics.org/blueprints/G/1

The Common Core in Ohiohttp://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htm






Textbook and Curriculum ResourcesIntegrated Math I McGraw HillChapter 15

High School CCSS Mathematics III Curriculum Guide-Quarter4 Curriculum Guide, 2013,

Prior KnowledgeStudents have had experiences with visualizing two and three dimensional figures in middle school: 7.G.3.( Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids).

Future LearningFuture learning will include study of circles and conics.


http://connected.mcgraw-hill.com/connected/

Performance Assessments/TasksClick on the links below to access performance tasks.Math Assessment Project Modeling Motion: Rolling Cupshttp://map.mathshell.org/download.php?fileid=1746

Representing 3D Objects in 2Dhttp://map.mathshell.org/download.php?fileid=1762

Calculating Volumes of Compound Objectshttp://map.mathshell.org/download.php?fileid=1764

Inside MathematicsPiece It Togetherhttp://www.insidemathematics.org/assets/problems-of-the-month/piece%20it%20together.pdf

Cutting a Cubehttp://www.insidemathematics.org/assets/problems-of-the-month/cutting%20a%20cube.pdf

Global Positioning System Ihttps://www.illustrativemathematics.org/content-standards/HSG/GMD/B/4/tasks/1215

Global Positioning System II https://www.illustrativemathematics.org/content-standards/HSG/GMD/B

Career/Everyday ConnectionsTransportation, Art, Architecture, Medicine, Engineering, Event Planner

Careers using geometry: http://work.chron.com/careers-require-geometry-10361.html


http://work.chron.com/careers-require-geometry-10361.html

https://www.illustrativemathematics.org/content-standards/HSG/GMD/B

https://www.illustrativemathematics.org/content-standards/HSG/GMD/B/4/tasks/1215

http://www.insidemathematics.org/assets/problems-of-the-month/cutting%20a%20cube.pdf

http://www.insidemathematics.org/assets/problems-of-the-month/piece%20it%20together.pdf





Understand and apply theorems

about circles.

Prove that all circles are similar

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Essential Understanding Students should know that unlike polygons that have dimensions

independent of one another (base and height, for instance), a circle's size depends only on one measurement: the radius r.

Students should know that since all aspects of a circle's size depend on r; the size can be changed of any circle simply by dilating the radius by a constant scale factor.

Students should already know that dilations, whether they're expansions or contractions, are similarity transformations; the size off the circle is changing but not its shape.

Extended UnderstandingProvide opportunities for students to engage in activities that will allow them to enhance understanding such as:

ttp://learnzillion.com/lessonsets/427-prove-that-all-circles-are-similar

This is an all in one unit to prove all circles are similar. It includes talk about using translations and dilations as well as triangles to prove that

all circles are similar.

Academic Vocabulary/ Languagecenter, central angle, centroid, chord, circle, circumcenter, circumference, , circumscribed angle, cyclic, diameter, dilations, equidistant, focus , incenter, inscribed angle, latus rectum, proportions, quadrilateral, Radian, radius, scalersimilar, translations

Tier 2 Languageconstructdrawsketch

CCSSM Description Learning to recall, understand, apply, prove and extend theorems about circles is useful because it leads to being able to find angles in and around circles; it becomes a functional (real-life) application skill used in occupations such as engineering and design and, this leads to developing skills at geometric proof and geometric


G.C.1, G.C.2

131

I Can Statements

I can, using the fact that the ratio of diameter to circumference is the same for circles, prove that all circles are similar.

I can, using definitions, properties, theorems, identiry and describe relationships among inscribed angles, radii, and chords. Include central, inscribed, and circumscribed angles.

I can understand that inscribed angles on a diameter are right angles.

I can understand that the radius of a circle is perpendicular to the tangent where the radius intersects the circle

Instructional Strategies Given any two circles in a plane, show that they are related by dilation. Guide students to discover the center and scale factor of this dilation and make a

conjecture about all dilations of circles. Starting with the special case of an angle inscribed in a semicircle, use the fact that the angle sum of a triangle is 180° to show that this angle is a right angle. Using dynamic geometry, students can grab a point on a circle and move it to see that the measure of the inscribed angle passing through the endpoints of a diameter is always 90°. Then extend the result to any inscribed angles. For inscribed angles, proofs can be based on the fact that the measure of an exterior angle of a triangle equals the sum of the measures of the nonadjacent angles. Consider cases of acute or obtuse inscribed angles. Use properties of congruent triangles and perpendicular lines to prove theorems about diameters, radii, chords, and tangent lines. Use formal geometric constructions to construct perpendicular bisectors of the sides and angle bisectors of a given triangle. Their intersections are the centers of the circumscribed and inscribed circles, respectively.


Common Misconceptions/ChallengesStudents sometimes confuse inscribed angles and central angles. Students may think they can tell by inspection whether a line intersects a circle in exactly one point. It may be beneficial to formally define a tangent line as the line perpendicular to a radius at the point where the radius intersects the circle

Common Core SupportIllustrative MathematicsCircleshttps://www.illustrativemathematics.org/blueprints/G/6


The Common Core in Ohiohttp://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htmhttps://www.ixl.com/standards/ohio/math






Textbook and Curriculum Resources Integrated Math III, McGraw Hill

High School CCSS Mathematics I Curriculum Guide -Quarter 4- Columbus City Schools, 2013, pp. 83-207

Geometry Station Activities for Common Core State Standards, Walch Education, 2013Circumference, Angles, Arcs, Chord, and Inscribed Angles, pp. pp. 147-160Special Segments, Angle Measurements, and Equations of Circles, pp. 161-172

Common Core State Standards: Problem-Based Tasks for Mathematics II, Walch Education, 2013Following in Arhimedes’ Footsteps, pp. 265-267Masking the Problem, pp. 268-270The Circus Is In Town, Is It Safe?, pp. pp. 271-274

Prior Knowledge Middle school experiences with circles in Geometry in 7th grade is when they are expected to draw, construct and describe geometrical figures and describe the relationships between them and solve real-life and mathematical problems involving angle measure, area, surface area, and volume. In 8th grade students begin work with volume of cylinders, cones and spheres.

Future Learning Future learning includes constructing inscribed and circumscribed circles of a triangle, and proving properties of angles for a quadrilateral inscribed in a circle.


Performance Assessments/TasksClick on the links below to access performance tasks.Math Assessment ProjectInscribing and Circumscribing Right Triangles http://map.mathshell.org/download.php?fileid=1758

Solving Problems with Circles and Triangleshttp://map.mathshell.org/download.php?fileid=1760http://schools.nyc.gov/NR/rdonlyres/49162FEC-37E2-4A96-93C1-6671664FACD5/0/NYCDOEHSMathCompanyLogo_Final.pdfINSIDE MathematicsCircles In Squareshttp://www.insidemathematics.org/assets/common-core-math-tasks/circle%20and%20squares.pdf

Circles In Triangleshttp://www.insidemathematics.org/assets/common-core-math-tasks/circles%20in%20triangles.pdf

Similar Circleshttps://www.illustrativemathematics.org/content-standards/HSG/C/A/1/tasks/1368

Right triangles inscribed in Circles Ihttps://www.illustrativemathematics.org/content-standards/HSG/C/A/2/tasks/1091

Right triangles inscribed in Circles IIhttps://www.illustrativemathematics.org/content-standards/HSG/C/A/2/tasks/1093

Career/Everyday Connections Architectural Engineering Construction Engineering Forensics Landscaping Engineering

Crop circles are an interesting and controversial phenomenon that can best be described as a pattern in a field where the crop (usually wheat) has been flattened. Many believe that the circles are made using a string and a piece of wood to flatten the crops.

The geometry of a basketball court, food engineering and efficiency, and landscaping a great are great studies of circle applications.

http://algebralab.org/practice/practice.aspx?file=Word_WP-CircleApplications.xml



https://www.illustrativemathematics.org/content-standards/HSG/C/A/2/tasks/1093




http://www.insidemathematics.org/assets/common-core-math-tasks/circle%20and%20squares.pdf

http://schools.nyc.gov/NR/rdonlyres/49162FEC-37E2-4A96-93C1-6671664FACD5/0/NYCDOEHSMathCompanyLogo_Final.pdf




Understand and apply theorems about circles.Construct the inscribed and

circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle,

Essential Understanding Students need to understand that when

a circle is inscribed in a polygon, then the polygon is circumscribed about the circle and when a circle is circumscribed about a polygon, then the polygon is inscribed in the circle.

Students need to understand that when a circle is inscribed in a polygon, then the polygon is circumscribed about the circle and when a circle is circumscribed about a polygon, then the polygon is inscribed in the circle.

Extended UnderstandingChallenge students to generalize the results about angle sums of triangles and quadrilaterals to a corresponding result for n-gons.


acute triangle, angles, alternate interior angles, alternate exterior angles, base, base angles, bisect, bisector, centroid, circumcenter, circumscribe, concurrent, consecutive interior angles, corresponding angles, diagonal, equiangular triangle, equidistant, equilateral triangle, exterior angle, hypotenuse, incenter, inscribe, inscribed arc, inscribed angle, inscribed quadrilateral, interior angle, isosceles triangle, leg, linear pair, lines, midsegment, obtuse triangle, orthocenter, parallel lines, parallelogram, perpendicular, perpendicular bisector, quadrilateral, rectangle, remote angle, rhombus, right angles, scalene triangle, square, , transversal line, vertex angle, vertical angles

CCSSM Description A circle is inscribed in a polygon if each side of the polygon is tangent to the circle, so an inscribed circle touches each side of the polygon at exactly one point.

A circle is circumscribed about a polygon if each vertex of the polygon lies on the circle. A circumscribed circle passes through each vertex of the polygon.

Tier 2 Language

constructdrawsketch


G.C.3

136

I Can Statements

I can construct inscribed circles of a triangle.

I can construct circumscribed circles of a triangle.

I can, using definitions, properties, and theorems, prove properties of angles for a quadrilateral inscribed in a circle

Instructional Strategies Given any two circles in a plane, show that they are related by dilation. Guide students to discover the center and scale factor of this dilation and make a

conjecture about all dilations of circles.

Starting with the special case of an angle inscribed in a semicircle, use the fact that the angle sum of a triangle is 180° to show that this angle is a right angle. Using dynamic geometry, students can grab a point on a circle and move it to see that the measure of the inscribed angle passing through the endpoints of a diameter is always 90°. Then extend the result to any inscribed angles. For inscribed angles, proofs can be based on the fact that the measure of an exterior angle of a triangle equals the sum of the measures of the nonadjacent angles. Consider cases of acute or obtuse inscribed angles.

Use properties of congruent triangles and perpendicular lines to prove theorems about diameters, radii, chords, and tangent lines. Use formal geometric constructions to construct perpendicular bisectors of the sides and angle bisectors of a given triangle. Their intersections are the centers of the circumscribed and inscribed circles, respectively.

Dissect an inscribed quadrilateral into triangles, and use theorems about triangles to prove properties of these quadrilaterals and their angles.


Common Misconceptions/ChallengesStudents sometimes confuse inscribed angles and central angles.

Students may think they can tell by inspection whether a line intersects a circle in exactly one point. It may be beneficial to formally define a tangent line as the line perpendicular to a radius at the point where the radius intersects the circle.

Remembering which point of concurrency is created by the four special triangle segments. The medians make the centroid, the perpendicular bisectors make the circumcenter, the angle bisectors make the incenter, and the altitudes make the orthocenter.

Common Core SupportIllustrative MathematicsCircleshttps://www.illustrativemathematics.org/blueprints/G/6








Textbook and Curriculum Resources Integrated Math I, McGraw Hill Chapter 15

Integrated Math II, McGraw Hill Chapter 11

High School CCSS Mathematics I Curriculum Guide -Quarter 3- Columbus City Schools, 2013, pp. 83-207

Geometry Station Activities for Common Core State Standards, Walch Education, 2013Circumcenter, Incenter, Orthocenter, and Centroid, pp. 173-186

Common Core State Standards: Problem-Based Tasks for Mathematics II, Walch Education, 2013First Aid Station, pp. 275-278Building a New Radio Station, pp. 279-282King Arthur and His Round Table, pp. 283-285

Prior KnowledgeConstructing inscribed and circumscribed circles of a triangle is an application of the formal constructions studied in G – CO.12

Future LearningStatistics will be topic covered in next lessons.


Performance Assessments/TasksClick on the links below to access performance tasks.NYC Department of Education G.C.3: Circles: Understand And Apply Theorems About Circles: Understand And Apply Theorems About Circleshttps://www.engageny.org/ccls-math/gc3

Math Assessment ProjectInscribing and Circumscribing Right Triangles http://map.mathshell.org/download.php?fileid=1758

Solving Problems with Circles and Triangleshttp://map.mathshell.org/download.php?fileid=1760

http://schools.nyc.gov/NR/rdonlyres/49162FEC-37E2-4A96-93C1-6671664FACD5/0/NYCDOEHSMathCompanyLogo_Final.pdfINSIDE MathematicsCircles In Squareshttp://www.insidemathematics.org/assets/common-core-math-tasks/circle%20and%20squares.pdf

Circles In Triangleshttp://www.insidemathematics.org/assets/common-core-math-tasks/circles%20in%20triangles.pdf

What’s My Anglehttp://www.insidemathematics.org/assets/problems-of-the-month/what's%20your%20angle.pdf





29.30



http://www.insidemathematics.org/assets/problems-of-the-month/what's%20your%20angle.pdf


http://www.insidemathematics.org/assets/common-core-math-tasks/circle%20and%20squares.pdf




https://www.engageny.org/ccls-math/gc3


Apply geometric concepts in modeling situations in modeling.

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Essential Understanding Students are expected to apply and model

geometric concepts.

Extended Understanding Encourage students to engage in a project(s) using real-world applications of geometry. Resources: A Sourcebook of Applications of School Mathematics, compiled by a Joint Committee of the Mathematical Association of America and the National Council of Teachers of Mathematics (1980); Mathematics: Modeling our World, Course 1 and Course 2, by the Consortium for Mathematics and its Applications (COMAP); Geometry & its Applications (GeoMAP) -- an exciting National Science Foundation project to introduce new discoveries and real-world applications of geometry to high school students. Produced by COMAP; Measurement in School Mathematics, NCTM 1976 Yearbook.

Academic Vocabulary/ Languagegometric concepts, geometric methods, properties

Tier 2 Vocabularyanalyze, describe, design, model, solve

CCSSM Description An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts—interpreting a schematic drawing, estimating the amount of wood needed to frame a sloping roof, rendering computer graphics, or designing a sewing pattern for the most efficient use of material. Modeling activities are a good way to show connections among various branches of mathematics and science. Dynamic geometry environments provide students with experimental and modeling tools that allow them to investigate geometric phenomena.


G.MG.1, G.MG.2, G.MG.3

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I Can Statements

I can use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

I can use the concept of density when referring to situations involving area and volume models, such as persons per square mile.

I can solve design problems by designing an object or structure that satisfies certain constraints, such as minimizing cost or working with a grid system based on ratios (i.e., The enlargement of a picture using a grid and ratios and proportions)

Instructional Strategies Genuine mathematical modeling typically involves more than one conceptual category. For example, modeling a herd of wild animals may involve

geometry, measurement, proportional reasoning, estimation, probability and statistics, functions, and algebra. It would be somewhat misleading to try to teach a unit with the title of “modeling with geometry.” Instead, these standards can be woven into other content clusters. A challenge for teaching modeling is finding problems that are interesting and relevant to high school students and, at the same time, solvable with the mathematical tools at the students’ disposal. The resources listed below are a beginning for addressing this difficulty.


Common Misconceptions/Challenges When students ask to see “useful” mathematics, what they often mean is, “Show me how to use this mathematical concept or skill to solve the homework problems.” Mathematical modeling, on the other hand, involves solving problems in which the path to the solution is not obvious. Geometry may be one of several tools that can be used.

Common Core Supporthttps://www.illustrativemathematics.org/blueprints/G/3






Textbook and Curriculum Resources Integrated Math III, McGraw Hill

High School CCSS Mathematics I Curriculum Guide –Quarter - Columbus City Schools, 2013

Common Core State Standards: Problem-Based Tasks for Mathematics II, Walch Education, 2013Designing a Tablecloth, pp. pp. 317-320Cylinders of Sand, pp. 321-324

Prior KnowledgeStudents have acquired knowledge to empower them to experience modeling geometric concepts/skills.

Future Learning Precalculus


Performance Assessments/Tasks

Click on the links below to access performance tasks.

Math Assessment Project Solving Quadratic Equations http://map.mathshell.org/download.php?fileid=1736

Modeling Motion: Rolling Cupshttp://map.mathshell.org/lessons.php?unit=9300&collection=8&redir=1http://schools.nyc.gov/NR/rdonlyres/49162FEC-37E2-4A96-93C1-6671664FACD5/0/NYCDOEHSMathCompanyLogo_Final.pdf








http://map.mathshell.org/lessons.php?unit=9300&collection=8&redir=1


Documents

Project Web Access quick reference guide for managers Web viewInstructional Strategies. Inferential statistics based on Normal probability models is a topic for Advanced Placement