Algebra 3-4 Curriculum Guide-2014 · research comparing the U.S. curriculum to high performing countries, surveys of college faculty and teachers, the National Math Panel, the Early

Algebra 3-4 DVUSD Curriculum 2014

Algebra 3-4

Curriculum Guide-2014


Page 2 Major Content Supporting Content Additional Content

Mathematics Curriculum The Intent and Design of the Common Core State Standards

Toward greater focus and coherence

The single most important flaw in United States mathematics instruction is that the curriculum is “a mile wide and an inch deep.” This finding comes from research comparing the U.S. curriculum to high performing countries, surveys of college faculty and teachers, the National Math Panel, the Early Childhood Learning Report, and all the testimony the CCSS writers heard. The standards are meant to be a blueprint for math instruction that is more focused and coherent. The focus and coherence in this blueprint is largely in the way the standards progress from each other, coordinate with each other and most importantly cluster together into coherent bodies of knowledge.

—Daro, McCallum, & Zimba, 2012

There are many ways to organize curricula. The challenge, now rarely met, is to avoid those that distort mathematics and turn off students. — Steen, 2007 Assessing the coherence of a set of standards is more difficult than assessing their focus. William Schmidt and Richard Houang (2002) have said that content standards and curricula are coherent if they are:

articulated over time as a sequence of topics and performances that are logical and reflect, where appropriate, the sequential or hierarchical nature of the disciplinary content from which the subject matter derives. That is, what and how students are taught should reflect not only the topics that fall within a certain academic discipline, but also the key ideas that determine how knowledge is organized and generated within that discipline. This implies that to be coherent, a set of content standards must evolve from particulars (e.g., the meaning and operations of whole numbers, including simple math facts and routine computational procedures associated with whole numbers and fractions) to deeper structures inherent in the discipline. These deeper structures then serve as a means for connecting the particulars (such as an understanding of the rational number system and its properties). (emphasis added) These Standards endeavor to follow such a design, not only by stressing conceptual understanding of key ideas, but also by continually returning to organizing principles such as place value or the properties of operations to structure those ideas.

In addition, the “sequence of topics and performances” that is outlined in a body of mathematics standards must also respect what is known about how students learn. As Confrey (2007) points out, developing “sequenced obstacles and challenges for students…absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise.” In recognition of this, the development of these Standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.



Example of how to read the grade level standards

Standards define what students should understand and be able to do. Clusters are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject. Domains are larger groups of related standards. Standards from different domains may sometimes be closely related. Conceptual Categories are unique to high school; high school standards are organized into conceptual categories, showing the body of knowledge students should learn to be college and career ready.

Algebra: Creating Equations A-CED

Create equations that describe numbers or relationships

1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional cost constraints on combinations of different foods. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law IRV = to highlight R .

These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that standards are not just promises to our children, but promises we intend to keep.

**Note: Arizona has added a letter to indicate the cluster in the coding of the standard. For example, the code for the first standard underlined above would be A-CED.A.1 rather than A-CED.1 to indicate that this is the first cluster under this domain. For more information about the state changes to coding, click here.

Standard

Cluster

http://www.azed.gov/azccrs/files/2013/10/revisionazccrs_mathcoding_explanation_10142013.pdf

http://www.azed.gov/azccrs/files/2013/10/revisionazccrs_mathcoding_explanation_10142013.pdf



Mathematics Curriculum Team Members

Adrienne Wooten

Patty Tyler

Jim Addabbo

Cindy Knoll

Nicole Donogher

Doug Evans

Lindsey Fleegle

Roberta Marshall-Bassik

Denis Ohlwiler

Jeff Williams

Ellen Sizemore

Jaymie Irwin



Introduction This document is meant to guide teachers through the adoption of the Common Core Standards in the math classroom. It is also meant to be a catalyst for grade level discussions and backwards design. It will assist in providing a district-wide guaranteed and viable curriculum.

How the Standards Are Divided There are six conceptual categories for high school in the Common Core Standards: number and quantity, algebra, functions, geometry, probability and statistics, and modeling. Within each conceptual category there are domains, which are further delineated by clusters which organize similar skill concepts together. In an effort to further connect the Common Core Standards to college and career readiness, the conceptual category modeling is not broken down into domains but woven throughout the other five conceptual categories. Modeling standards are defined by a “ “ in this document. In addition, there are 18 standards that are assessed on both the Algebra 1-2 and the Algebra 3-4 end of course PARCC exams. These standards will be identified with a “ “. For a full explanation of the limits for each course please see the table at the end of this document.

Enduring Understandings For each cluster of the Common Core Standards enduring understandings have been provided. Enduring understandings are statements summarizing important ideas and core processes that are central to a discipline and have a lasting value beyond the classroom. It synthesizes not just what a student should know or do, but what a student should understand as a result of studying a particular content area. Teachers should not be confined or obligated to use the sample enduring understandings, but instead should use them as a guide.

Essential Questions For each cluster of the Common Core Standards essential questions have been provided. According to Grant Wiggins, co-author of Understanding by Design, essential questions are important questions that recur throughout one’s life. They can also be key inquiries within a discipline. A question can be considered essential when it helps students make sense of important but complicated ideas. Teachers should not be confined or obligated to use the sample essential questions, but instead should use them as a guide.

Key Concepts For each cluster, key concepts have been provided. Some concepts are clearly of more importance than others. The key concepts provide us with the power to explore a variety of situations and events and to make significant connections. Other concepts may be meaningful in more limited situations but play a part in connecting unrelated facts. Student Friendly Objectives For each cluster of the Common Core Standards, student objectives have been provided. Teachers should not be confined or obligated to use the sample student objectives, but instead they should use them as a guide.

Academic Vocabulary For each cluster of the Common Core Standards, academic vocabulary has been provided. It is essential that students practice using the appropriate vocabulary in written work and discussions.



Depth of Knowledge (DOK) Dr. Norman Webb’s Depth of Knowledge (DOK) measures the degree to which the knowledge elicited from students on assessments is as complex as what students are expected to know and do as stated in the Common Core Standards. It accomplishes a cognitive process going across four levels of depth of knowledge:

• DOK 1 (recall): the most basic skills or definition • DOK 2 (skill concept): using the information or conceptual knowledge • DOK 3 (strategic thinking): reasoning or developing a plan; the task may have more than one answer • DOK 4 (extended thinking): requires an investigation, collection of data and analysis of results; requires time to think and process

While similar in appearance to Bloom’s Taxonomy, DOK is focused on what comes after the verb and not on the verb itself. Each level is based upon the process of the task, not on the level of the verb involved in the task. It is designed to measure depth of learning, not the category of the task.

DOK is not about verbs. Verbs are not always used appropriately. DOK is not about "difficulty" - It is not about the student or level of difficulty for the student - it requires looking at the assessment item not student work in order to determine the level. DOK is about the item/standard - not the student.

DOK is about what FOLLOWS the verb. What comes after the verb is more important than the verb itself. DOK is about the complexity of mental processing that must occur to answer a question.

Remember DOK is descriptive and not a taxonomy. It focuses on how deeply the student has to know the content in order to respond.

Assessment Rubric For each cluster a sample rubric has been provided. The rubric is based on the idea that the standard meets the qualifications on what a student requires to be labeled proficient; therefore, proficient is the standard itself. The rubric is not intended to align with grading; however, it can be used to help assess students. For example, a student could place in developing on the rubric and still receive high marks in the grade book as it is developmentally appropriate for the class. This is to say that a student that is entering the grade band should be developing and be, at the minimum, proficient as they exit the grade band.

There are four rubric categories:

• Developing-Approaching the standard; still needs scaffolding. • Proficient-This is the standard achieved independently by the student. • Advancing-Moving past the standard with some scaffolding. • Mastery-Moving past the standard independently and in a cross-curricular manner.

Resources For each cluster of the Common Core Standards, District and teacher recommended resources have been provided to support instruction.



NUMBER & QUANTITY The Real Number System (N-RN)

• Extend the properties of exponents to rational exponents Quantities (N-Q)

• Reason quantitatively and use units to solve problems The Complex Number System (N-CN)

• Perform arithmetic operations with complex numbers • Use complex numbers in polynomial identities and equations

ALGEBRA Seeing Structure in Expressions (A-SSE)

• Interpret the structure of expressions • Write expressions equivalent forms to solve problems.

Arithmetic with Polynomials and Rational Expressions (A-APR) • Understand the relationship between zeroes and factors of polynomials • Use polynomial identities to solve problems • Rewrite rational expressions

Creating Equations (A-CED) • Create equations that describe numbers or relationships

Reasoning with Equations and Inequalities (A-REI) • Understand solving equations as a process of reasoning and explain the reasoning • Solve equations and inequalities in one variable • Solve systems of equations. • Represent and solve equations and inequalities graphically

FUNCTIONS Interpreting Functions (F-IF)

• Understand the concept of a function and use function notation • Interpret functions that arise in applications in terms of the context • Analyze functions using different representations

Building Functions (F-BF) • Build a function that models a relationship between two quantities • Build new functions from existing functions

Linear, Quadratic, and Exponential Models (F-LE) • Construct and compare linear, quadratic, and exponential models and solve

problems • Interpret expressions for functions in terms of the situation they model

Trigonometric Functions (F-TF) • Extend the domain of trigonometric functions using the unit circle • Model periodic phenomena with trigonometric functions • Prove and apply trigonometric identities

GEOMETRY Expressing Geometric Properties with Equations (G-GPE)

• Translate between the geometric description and the equation for a conic section

STATISTICS Interpreting Categorical and Quantitative Data (S-ID)

• Summarize, represent, and interpret data on a single count or measurement variable

• Summarize, represent, and interpret data on two categorical and quantitative variables

Making Inferences and Justifying Conclusions (S-IC) • Understand and evaluate random processes underlying statistical

experiments • Make inferences and justify conclusions from sample surveys,

experiments and observational studies Conditional Probability and the Rules of Probability (S-CP)

• Understand independence and conditional probability and use them to interpret data

• Use the rules of probability to compute probabilities of compound events in a uniform probability model

Mathematical Practices (MP) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. For more information regarding Common Core, including Key Advances from Grades K-8, please see the Content Model Frameworks.

http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf

http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf



Algebra 3-4 SCOPE AND SEQUENCE 1st Semester (85 days) 2nd Semester (95 days)

UNIT 1: Linear Functions, Equations, & Inequalities (10 days) N-Q.A.2 -define appropriate quantities in modeling A-CED.A.1 -create linear equations & inequalities (1 var) and use to solve problems A-REI.C.6 -solve systems (exact & approx.) A-REI.D.11 -understand relationship between equations, graphs, and solutions (tech) F-IF.A.3 –sequences are functions whose domain is a subset of the integers F-IF.B.6 -rate of change F-IF.C.7(a) -graph linear functions/key features (tech) F-BF.A.2 -model with arithmetic sequences F-BF.B.3 –transformations/parent functions (tech) F-LE.A.2 -construct linear functions F-LE.B.5 -interpret the parameters in context S-ID.B.6(a) -fit a function to data (scatterplot); use function to solve problems UNIT 2: Quadratic Functions and Equations (16 days) N-Q.A.2 -define appropriate quantities in modeling N-CN.A.1-complex numbers (𝑎 + 𝑏𝑖) N-CN.A.2-add, subtract, multiply complex numbers N-CN.C.7-quadratic equations/complex solutions A-CED.A.1 -create quadratic equations (1 var) and use to solve problems A-REI.B.4(b) -solve quadratics using appropriate method; recognize when quadratic formula gives complex solutions A-REI.C.7-system of linear & quadratic equations F-IF.B.6 -rate of change F-IF.7.C.(a) -graph quadratic functions showing intercepts, max and min (tech) F-IF.8.C.(a) – factor/complete the square to show zeroes, extremes, symmetry F-IF.C.9 –compare functions F-BF.B.3 –transformations/parent functions (tech) F-BF.B.4(a)-find inverse functions S-ID.B.6(a) -fit a function to data (scatterplot); use function to solve problems

UNIT 3: Polynomial Functions (16 days) N-Q.A.2 -define appropriate quantities in modeling A-SSE.A.2 -use structure to rewrite expressions A-APR.B.2-Remainder Theorem A-APR.B.3 -identify zeroes when factored & use to construct a rough graph A-APR.C.4- prove polynomial identities & use to describe numerical relationships A-APR.D.6-rewrite rational expressions A-REI.D.11 -understand relationship between equations, graphs, and solutions (tech) F-IF.B.4 -interpret features of graphs/tables & sketch graph F-IF.B.6 -rate of change F-IF.C.7(c) -graph; ID zeroes; show end behavior F-IF.C.9 –compare functions F-BF.A.1(b) -combine functions w/operations F-BF.A.1(c) + -combine functions w/operations F-BF.B.3 –transformations/parent functions (tech) UNIT 4: Rational Functions (16 days) A-SSE.A.2 -use structure to rewrite expressions A-APR.D.7+-operations on rational expressions A-CED.A.1 -create equations (1 var) and use to solve problems A-REI.A.1 –explain steps to solve an equation or justify solution method A-REI.A.2-solve rational equations (1 var); extraneous solutions A-REI.D.11 -understand relationship between equations, graphs, and solutions (tech) F-IF.C.7(d)+ -graph rational functions UNIT 5: Radical Functions (10 days) N-RN.A.1-properties of rational exponents N-RN.A.2-rewrite expressions using props of exponents A-REI.A.1 –explain steps to solve an equation or justify solution method A-REI.A.2-solve radical equations (1 var); extraneous solutions F-BF.B.4(a)-find inverse functions

UNIT 6: Exponential Functions (10 days) N-Q.A.2 -define appropriate quantities in modeling A-SSE.B.3(c) -use properties of exponents to transform expressions for exponential functions. A-SSE.B.4 -finite geometric series A-CED.A.1 -create equations (1 var) and use to solve problems A-REI.D.11 -understand relationship between equations, graphs, and solutions (tech) F-IF.B.4 -interpret features of graphs/tables & sketch graph F-IF.B.6 -rate of change F-IF.C.7(e) -graph exponential; intercepts & end behavior F-IF.C.8(b)-properties of exponents interpret expressions F-IF.C.9 –compare functions F-BF.A.1(a) -explicit or recursive expression context F-BF.A.2 -model with geometric sequences F-BF.B.3 –transformations/parent functions (tech) F-LE.A.2 -construct exponential functions F-LE.B.5 -interpret the parameters in context S-ID.B.6(a) -fit a function to data (scatterplot); use function to solve problems UNIT 7: Logarithmic Functions (10 days) N-Q.A.2 -define appropriate quantities in modeling A-REI.D.11 -understand relationship between equations, graphs, and solutions (tech) F-IF.B.4 -intrpt features graphs/tables & sketch graph F-IF.C.7(e) -graph log functions; intercepts & end behavior F-BF.B.3 –transformations/parent functions (tech) F-BF.B.4(a)-find inverse functions F-LE.A.4 -use logs to solve exponential equations; evaluate logs using technology UNIT 8: Trigonometric Functions (20 days) N-Q.A.2 -define appropriate quantities in modeling F-IF.B.4 -interpret features of graphs/tables & sketch graph F-IF.C.7(e) -graph trig functions; show period, midline, and amplitude F-IF.C.9 –compare functions F-BF.B.3 –transformations/parent functions (tech) F-TF.A.1-conceptual understanding of radian measure F-TF.A.2-extend domain of trig functions using the unit circle (radian measure) F-TF.B.5 -choose trig functions to model periodic phenomena (amplitude, frequency, midline) F-TF.C.8-prove Pythagorean Identity & use to find other ratios G-SRT.C.8 -use trig ratios & Pythagorean Theorem

UNIT 9: Statistics (15 days) S-IC.A.1 -understand stats is a process to make inferences about populations based on random samples S-IC.B.3 -purpose & differences between sample surveys, experiments, & observational studies (randomization) S-IC.A.2 -analyze model with results of data (theoretical/empirical results) S-ID.A.4 -use the mean & standard deviation to fit to normal distribution & estimate population percentages; recognize when not appropriate; estimate areas under the normal curve S-IC.B.4 -use sample data to estimate population mean or proportion; develop a margin of error through simulation S-IC.B.5 -compare treatments; analyze differences in parameters for significance S-IC.B.6 -evaluate reports based on data UNIT 10: Probability (15 days) S-CP.A.1 -describe events as subsets of sample space using characteristics or categories or as unions, intersections, or complements of other events S-CP.B.7 -apply & interpret the Addition Rule S-CP.A.2 -define/ID independent events S-CP.A.4 -construct & interpret two-way frequency tables; use to determine independence and approximate conditional probabilities S-CP.A.3 -define/ID dependent events S-CP.A.5 -recognize & explain conditional probability & independence in context S-CP.B.6 -calculate & interpret conditional probabilities





Mathematical Practices (MP) Mathematical

Practice Description of Student Behavior

1. Make sense of problems and persevere in solving them.

High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. By high school, students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They check their answers to problems using different methods and continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively.

High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given situation and represent it symbolically, manipulate the representing symbols, and pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Students use quantitative reasoning to create coherent representations of the problem at hand; consider the units involved; attend to the meaning of quantities, not just how to compute them; and know and flexibly use different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others.

High school students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. High school students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. High school students learn to determine domains to which an argument applies, listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

4. Model with mathematics.

High school students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. High school students making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

5. Use appropriate tools strategically.

High school students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. High school students should be sufficiently familiar with tools appropriate for their grade or course to make



sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. They are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

6. Attend to precision.

High school students try to communicate precisely to others by using clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

7. Look for and make use of structure.

By high school, students look closely to discern a pattern or structure. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. High school students use these patterns to create equivalent expressions, factor and solve equations, and compose functions, and transform figures.

8. Look for and express regularity in repeated reasoning.

High school students notice if calculations are repeated, and look both for general methods and for shortcuts. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, derive formulas or make generalizations, high school students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.



Unit 1: Linear Functions, Equations, and Inequalities 2010 Standards Comments Resources

Cluster: Reason quantitatively and use units to solve problems.

N-Q.A.2. Define appropriate quantities for the purpose of descriptive modeling. For example, if you want to describe how dangerous the roads are, you may choose to report the number of accidents per year on a particular stretch of interstate. Generally speaking, it would not be appropriate to report the number of exits on that stretch to describe the level of danger. Source: NC Number & Quantity

This standard should be involved in modeling. Students need to be able to grapple with the appropriate quantities given the situation.

Glencoe: throughout

Cluster: Create equations that describe numbers or relationships.

A-CED.A.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Example Given that the following trapezoid has area 54 cm2, set up an equation to find the length of the base, and solve the equation.

source: ADE CCSS

When the standard refers to an equation in one variable, this would be an example:

24354 += x

Notice there is only one variable.

LearnZillion – A-CED.1 part 1 Paying the Rent Planes and Wheat Glencoe: 1-1 & 1-2 (review); 1-3 (like #s 29, 34, 43, 44, 59 in homework), 1-5

http://maccss.ncdpi.wikispaces.net/file/view/Number%20and%20Quantity%20Theme%20Unpacked%2004102012.pdf/359510167/Number%20and%20Quantity%20Theme%20Unpacked%2004102012.pdf

http://www.azed.gov/azcommoncore/mathstandards/hsmath/

http://learnzillion.com/lessons/763-create-linear-equations-to-solve-problems

http://www.illustrativemathematics.org/illustrations/582




Cluster: Solve systems of equations. A-REI.C.6. Solve systems of linear equations exactly and approximately

(e.g., with graphs), focusing on pairs of linear equations in two variables.

Example The space shuttle uses up fuel at a cost of y = 600x + 1500, where x is the number of gallons of fuel and y is our total cost. Boeing has decided to design a much more fuel-efficient rocket engine where the cost is calculated at y = 500x + 2700. Is this new engine actually more fuel efficient in the long run than the old one or not? a. Yes, it is more fuel-efficient when x < 12 b. Yes, it is more fuel-efficient when x > 12 c. No, it isn't more fuel-efficient when x < 12 d. No, it isn't more fuel-efficient when x > 12 Source: Shmoop

The system solution methods can include but are not limited to graphical, elimination/linear combination, substitution, and modeling (although you don’t see a star, you ought to provide systems in context). Systems can be written algebraically or can be represented in context. Students may use graphing calculators, programs, or applets to model and find approximate solutions for systems of equations (matrices would be appropriate, particularly with tech). 3x3 systems will be covered in Honors (you are welcome to extend for your regular students if you like)

Learnzillion Learnist – A-REI.6 Shmoop – A-REI.6 Glencoe: 3-1, 3-2, 3-3

Cluster: Represent and solve equations and inequalities graphically.

A-REI.D.11. 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.

Example In yesterday’s basketball game, Cheryl scored 17 points with a combination of 2-point and 3-point baskets. The number of 2-point shots she made was one greater than the number of 3-point shots she made. Use a graph to determine the amount of each type of basket she scored. source: Monterey Institute

Students need to understand that numerical solution methods (data in a table used to approximate an algebraic function) and graphical solution methods may produce approximate solutions, and algebraic solution methods produce precise solutions that can be represented graphically or numerically. Students may use graphing calculators or programs to generate tables of values, graph, or solve a variety of functions. Cases are limited to linear functions in this unit.

Learnist – A-REI.11 – linear only Solving Systems in Context with Graphing Glencoe: 3-1, supplement (see example)

http://www.shmoop.com/common-core-standards/ccss-hs-a-rei-6.html

http://learnzillion.com/lessonsets/247-solve-systems-of-linear-equations-exactly-and-approximately

http://learni.st/users/171/boards/2559-pairs-of-linear-equations-in-two-variables-common-core-standard-a-rei-6#/users/171/boards/2559-pairs-of-linear-equations-in-two-variables-common-core-standard-a-rei-6

http://www.shmoop.com/common-core-standards/ccss-hs-a-rei-6.html

http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U06_L1_T1_text_final.html

http://learni.st/users/171/boards/2765-the-intersection-of-two-equations-common-core-standard-a-rei-11#/users/171/boards/2765-the-intersection-of-two-equations-common-core-standard-a-rei-11

http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U06_L1_T1_text_final.html



Cluster: Understand the concept of a function and use of function notation.

F-IF.A.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Example Consider the sequence that follows a “plus 3” pattern: 4, 7, 10, 13, 16…

a) Write a formula for the sequence using both the na notation and

)(nf notation.

b) Graph the terms of the sequence as ordered pairs ))(,( nfn on the coordinate plane. What do you notice about the graph?

c) Describe the domain for this sequence. Source: EngageNYMod3

In an arithmetic sequence, each term is obtained from the previous term by adding the same number each time. This number is called the common difference. The common difference corresponds to the slope in the explicit form of a linear function, bmxy += and the initial value of the sequence corresponds to the y-intercept, or b. One of the main points to this standard is that sequences are functions whose domain is a subset of the integers. Students need to understand this means the graph of the function will be discrete. You can add to Extend 2-1 in Glencoe to meet this standard.

Engage NY Module 3 Topic A LearnZillion – F-IF.3 Glencoe: 2-1 (review), Extend 2-1 + supplement (see example); parts of 11.1 (particularly word problems)

Cluster: Interpret functions that arise in applications in terms of context.

F-IF.B.6. 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. Example Use the following table to find the average rate of change of g over the intervals [-2, -1] and [0,2]:

x g(x)

-2 2

-1 -1

0 -4

2 -10

source: ADE CCSS

The average rate of change of a function y = f(x) over an interval [a, b] is

abafbf

xy

−−

=∆∆ )()(

In addition to finding average rates of change from functions given symbolically, graphically, or in a table, Students may collect data from experiments or simulations (ex. falling ball, velocity of a car, etc.) and find average rates of change for the function modeling the situation. Focus on linear functions.

Learnist – F-IF.6 - linear The High School Gym Glencoe: 2-3

http://www.engageny.org/resource/high-school-algebra-i


http://learnzillion.com/lessons/3364-understand-sequences-as-functions


http://learni.st/users/S33572/boards/2378-average-rate-of-change-common-core-standard-9-12-f-if-6#/users/S33572/boards/2378-average-rate-of-change-common-core-standard-9-12-f-if-6




𝑔(𝑥) = |𝑥 − 1|

Cluster: Analyze functions using different representation. F-IF.C.7. Graph functions expressed symbolically and show key features of

the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and

minima. b. Graph piecewise-defined functions, including step functions and absolute

value functions. Example Graph the following equations.

531)( += xxf

Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Intercepts should be expressed as ordered pairs. Note this standard is here for review from Algebra 1-2 and is not an explicit 3-4 standard.

Learnist – F-IF.7 - linear Glencoe: 2-2 (graphing), Extend 2-4, 2-6

Cluster: Build a function that models a relationship between two quantities.

F-BF.A.2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Examples • Given 𝑎1 = 4 and 𝑎𝑛 = 𝑎𝑛−1 + 3, write the explicit formula. • A company purchases $24,500 of new computer equipment. For tax

purposes, the company estimates that the equipment decreases in value by the same amount each year. After 3 years, the estimated value is $9800. Write an explicit function that gives the estimated value of the computer equipment 𝑛 years after purchase.

source: Flipbook

Recursive expressions require the previous term to calculate the next term. For example, in the sequence 2, 4, 6, 8, the recursive expression would be n+2 (or more specifically 2;2 11 +== −nn aaa ).

An explicit expression is one that will give you any term without the previous term. For the example 2, 4, 6, 8, the explicit expression would be 2n. The explicit process is generally more efficient. Students should connect arithmetic sequences to linear functions and understand that linear functions are the explicit form of a recursively defined arithmetic sequence. In addition, the recursive formula for an arithmetic sequence uses addition and the explicit formula uses multiplication.

Glencoe: 11-1, 11-2, 11-5 (arithmetic sequences only)

http://learni.st/users/S33572/boards/2380-graphing-and-analyzing-functions-pt-1-common-core-standards-9-12-f-if-a-c#/users/S33572/boards/2380-graphing-and-analyzing-functions-pt-1-common-core-standards-9-12-f-if-a-c

http://www.azed.gov/azcommoncore/files/2012/11/high-school-ccss-flip-book-usd-259-2012.pdf



Cluster: Build new functions from existing functions. F-BF.B.3. 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.

Example Given the following functions describe the transformation from the table to the second function.

time (hours) 4 12 20 28 distance (miles) 4 8 12 16

521)( += xxf

Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Stick with transformations of linear functions in this unit.

Transformations of linear functions Transformations Glencoe: Explore 2-7, 2-7

Cluster: Construct and compare linear, quadratic, and exponential models to solve problems.

F-LE.A.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Example The following table shows the height (cm) of a sunflower in terms of its age in days.

Age (days)

Height (cm)

60 208 62 213 64 218 66 223 68 228

Construct a function that represents the height )(df the sunflower will be

after d days.

Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to construct linear functions.

Learnist – linear only Sandia Aerial Tram Do two points always determine a linear function? Do two points always determine a linear function II? Glencoe: 2-2, 2-4

http://my.hrw.com/math06_07/nsmedia/homework_help/alg1/alg1_ch05_09_homeworkhelp.html

http://cms.cerritos.edu/uploads/pmata/Math%20140%20Materials/Chapter%201/1.7_Transformations.pdf

http://learni.st/users/171/boards/2038-constructing-linear-and-exponential-functions-common-core-standard-f-le-2#/users/171/boards/2038-constructing-linear-and-exponential-functions-common-core-standard-f-le-2








Cluster: Interpret expressions for functions in terms of the situation they model.

F-LE.B.5. Interpret the parameters in a linear or exponential function in terms of a context. Example You're caught in the middle of Life or Death III, a brutal first person shooter, when suddenly you realize your in-game health is currently being modeled as y = 25 – 3x, where x is in minutes. What does this mean? (A)Your health increases steadily with every passing minute (B)Your health remains constant (C)Your health decreases with every passing minute slowly at first and then more rapidly (D)Your health decreases steadily with every passing minute source: Shmoop

Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to model and interpret parameters in linear functions.

Focus on parameters of linear functions. For a brief explanation of a parameter click here.

Shmoop – F-LE.5 (linear only) Learnist – Linear only US Population 1982-1988 Glencoe: supplement/throughout

Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables.

S-ID.B.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve

problems in the context of the data. Use given functions or chooses a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Example Measure the wrist and neck size of each person in your class and make a scatterplot. Find the least squares regression line. source: ADE CCSS

Shmoop – S-ID.6 Learnist – S-ID.6 Coffee and Crime – illustrates all 4 standards in this unit Linear Regression and Correlation Laptop Battery Charge 2 TI-84 – Linear Regression Glencoe: 2-5

http://www.shmoop.com/common-core-standards/ccss-hs-f-le-5.html

http://www.mathsisfun.com/definitions/parameter.html


http://learni.st/users/171/boards/2041-interpreting-the-parameters-of-linear-and-exponential-functions-common-core-standard-f-le-5#/users/171/boards/2041-interpreting-the-parameters-of-linear-and-exponential-functions-common-core-standard-f-le-5



http://www.shmoop.com/common-core-standards/ccss-hs-s-id-6.html

http://learni.st/users/S33572/boards/3238-representing-quantitative-variables-using-scatter-plots-common-core-standard-9-12-s-id-6#/users/S33572/boards/3238-representing-quantitative-variables-using-scatter-plots-common-core-standard-9-12-s-id-6


http://www.shodor.org/interactivate/lessons/LinearRegressionCorrelation/

https://www.illustrativemathematics.org/illustrations/1559

https://www.youtube.com/watch?v=484Sy437Dhw



Unit 1: Linear Functions, Equations, and Inequalities ENDURING UNDERSTANDINGS

• Linear functions and inequalities can be used to model and solve real world problems. • Graphing a relationship allows us to model real world situations and visualize the relationship between two variables.

ESSENTIAL QUESTIONS KEY CONCEPTS • What is a function? • What is the relationship between the graph, the equation and

table of values? • When is it useful to use a particular form of linear equation

(standard, slope-intercept, or point-slope)? • What does each key feature of a function mean in a given context? • What is the purpose of a line of best fit? • How are piecewise functions represented algebraically and

graphically? • How can systems and matrices be used to solve problems? • What is the difference between an explicit and a recursive

formula?

• Different forms of linear equations can be useful depending on the context. • Linear functions can represented algebraically, graphically, numerically in

tables, or by verbal descriptions. • The line of best fit can be used to predict data values. • Key features of functions can be described in terms of the context of the

function. • Linear functions grow by equal differences over equal intervals (constant

rate of change). • A sequence is a function whose domain is the positive integers. • Arithmetic sequences are represented by linear functions which have

constant differences. • An explicit formula specifies the nth term of a sequence as an expression in

n. • A recursive formula specifies the nth term of a sequence as an expression in

the previous term (or previous couple of terms).

STUDENT FRIENDLY OBJECTIVES ACADEMIC VOCABULARY • TSW determine if a relation is a function, use function notation,

and evaluate functions. • TSW manipulate constructs to create different forms of a linear

equation. • TSW determine the line of best fit to predict data values. • TSW describe key features of the graphs of linear functions

(including piecewise) and inequalities. • TSW use systems and matrices (tech) to solve problems. • TSW create recursive and explicit formulas for arithmetic

sequences and make the connection between arithmetic sequences and linear functions.

variable linear equation solution algebraic model domain range ordered pair linear inequality recursive formula arithmetic sequence

relation function slope rate of change slope-intercept form standard form point-slope formula matrices (matrix) explicit formula linear regression

piecewise-defined function absolute value function parent function transformation linear regression system of equations system of inequalities correlation coefficient



Unit 1: Linear Functions, Equations, and Inequalities DEPTH OF KNOWLEDGE

(Students should be assessed on DOK level 1-3 questions; DOK 4 should be given as an activity or project) **General Guidelines for any assessment: 35% DOK 1 40% DOK 2 25% DOK 3

DOK1 – Recall *demonstrate a rote response or perform a well-known algorithm

DOK2 – Skill Concept *make some decisions as to how to approach the problem; usually more than one step is involved

DOK 3 – Strategic Thinking *requires reasoning, planning, using evidence, and explaining their thinking; may require evidence of multiple strategies or more than one possible answer

DOK 4 – Extended Thinking *requires reasoning, planning, developing and thinking over an extended period of time; cognitive demand of task is high and work is complex; problem is non-routine

STUDENT PERFORMANCE TASKS

Task: Determine if the following sequence is arithmetic. If so, state the common difference.

5,−6,−17,−28, …

Task: Consider the sequence that follows a “plus 3” pattern: 4, 7, 10, 13, 16… Write a formula for the sequence using both the na notation and

)(nf notation.

Task: Consider the sequence that follows a “plus 3” pattern: 4, 7, 10, 13, 16… Graph the terms of the sequence as ordered pairs ))(,( nfn on the coordinate plane. What do you notice about the graph? Explain your reasoning in terms of the domain.

Task: Part 1: Create a problem in which the recursive formula for the sequence would be more efficient in answering. Give your problem, the answer, and the recursive formula used. Part 2: Create a problem in which the explicit formula for the sequence would be more efficient in answering. Give your problem, the answer, and the explicit formula used. Part 3: Compare and contrast when one formula is more efficient than the other using mathematical evidence.



Unit 1: Linear Functions, Equations, and Inequalities ASSESSMENT RUBRIC

(DVMA will include questions based on the developing and proficient levels only.) (The levels of proficiency do not correspond to DOK levels, but instead reflect a student’s progress in relation to a standard) Developing

(below standard) Proficient

(basic understanding of standard) Advancing

(greater understanding of standard)

Mastery (exceptional understanding of

standard) A student can identify an arithmetic sequence and give the common difference.

A student can write a formula for a sequence using both sequence and function notation.

A student demonstrates an understanding that a sequence is a function whose domain is a subset of the integers.

A student can compare and contrast the efficiency of explicit and recursive formulas.

DVUSD ADOPTED RESOURCES ADDITIONAL RESOURCES Glencoe Illustrative Mathematics

Mathematics Assessment Project Mathematics Vision Project North Carolina Wiki Shmoop Problem-Based Curriculum Maps Flipbook CCSS Math Tuscon Resources Howard County Algebra II Engage NY Algebra II

Technology Tools Geogebra Core Math Tools Online Graphing Calculator Function Visualizer Create A Graph Foo Plot Print Free Graph Paper

TEACHER NOTES

http://www.illustrativemathematics.org/

http://map.mathshell.org/materials/index.php

http://www.mathematicsvisionproject.org/


http://maccss.ncdpi.wikispaces.net/Unpacking+Documents

http://www.shmoop.com/

http://emergentmath.com/my-problem-based-curriculum-maps/


http://ccssmath.org/

http://www.tusd1.org/resources/iw/math_hs.asp

https://commoncorealgebra2.wikispaces.hcpss.org/

http://www.engageny.org/resource/high-school-algebra-ii

http://www.geogebra.org/cms/en/

http://www.nctm.org/standards/content.aspx?id=32706

http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

http://www.abhortsoft.hu/functionvisualizer/functionvisualizer.html

http://nces.ed.gov/nceskids/createagraph/default.aspx

http://fooplot.com/#W3sidHlwZSI6MCwiZXEiOiJ4XjIiLCJjb2xvciI6IiMwMDAwMDAifSx7InR5cGUiOjEwMDB9XQ--

http://www.printfreegraphpaper.com/



Unit 2: Quadratic Functions and Equations 2010 Standards Comments Resources


N-Q.A.2. Define appropriate quantities for the purpose of descriptive modeling. For example, what quantities could you use to describe a safe bungee apparatus? Source: NC Number & Quantity


Glencoe: throughout

Cluster: Perform arithmetic operations with complex numbers.

N-CN.A.1. Know there is a complex number i such that 12 −=i , and every complex number has the form bia + with a and b real. Example Is it possible to simplify √−16 using the Real Number System? Justify your reasoning. Source: NC Number & Quantity

Every number is a complex number of the form 𝒂 + 𝒃𝒊 where 𝒂 and 𝒃 are the elements of the Real Numbers and 𝒃𝒊 is an element of the Pure Imaginary Numbers. Students should know the sets and subsets of the Complex Number System. The identity 𝒊 = √−𝟏 is not only used to identify non-real solutions for particular functions, but is also used to find the identity 𝒊𝟐 = −𝟏 which is used to simplify expressions.

Howard County Unit 3 Shmoop – N-CN.1 Glencoe: 5-4, 5-6



https://commoncorealgebra2.wikispaces.hcpss.org/Unit+3

http://www.shmoop.com/common-core-standards/ccss-hs-n-cn-1.html



N-CN.A.2. Use the relation 12 −=i , and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Example Simplify the following expression. Justify each step using the commutative, associative and distributive properties.

( )( )ii 4723 +−−

source: ADE CCSS

When adding, subtracting, or multiplying Complex Numbers, and 𝒊𝟐 remains in the expression, use the identity 𝒊𝟐 = −𝟏, which is used to simplify expressions.

Howard County Unit 3 Glencoe: 5.4, 5.6

Cluster: Use complex numbers in polynomial identities and equations.

N-CN.C.7. Solve quadratic equations with real coefficients that have complex solutions. Examples

• Within which number system can x2 = – 2 be solved? Explain how you know.

• Solve x2+ 2x + 2 = 0 over the complex numbers. • Find all solutions of 2x2 + 5 = 2x and express them in the form a + bi.

source: ADE CCSS

Extend strategies for solving quadratics such as: taking the square root and applying the quadratic formula, to find solutions of the form 𝒂 + 𝒃𝒊 for quadratic equations. This extension is made when the identity 𝒊 = √−𝟏 is used to simplify radicals having negative numbers under the radical.

Howard County Unit 3 Glencoe: 5-5, 5-5 Extend









Example A ball thrown vertically upward at a speed of v ft/sec rises d feet in t seconds can be represented with the equation 2166 tvtd −+= . Find the time it takes a ball thrown at a speed of 88 ft/sec to rise 20 feet.

The key to this standard is equations and inequalities in one variable. In order to solve the equation in the example to the left, students will be solving the following equation for t.

21688620 tt −+=

Engage NY Module 4 Topic A & B Throwing a Ball LearnZillion – A-CED.1 LearnZillion – A-CED.1 Glencoe: throughout

Cluster: Solve equations and inequalities in one variable. A-REI.B.4. Solve quadratic equations in one variable.

b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Value of Discriminant

Nature of Roots

Nature of Graph

b2 – 4ac = 0 1 real roots intersects x-axis once b2 – 4ac > 0 2 real roots intersects x-axis twice b2 – 4ac < 0 2 complex

roots does not intersect x-axis

Examples

• Are the roots of 2x2 + 5 = 2x real or complex? How many roots does it have? Find all solutions of the equation.

• What is the nature of the roots of x2 + 6x + 10 = 0? Solve the equation using the quadratic formula and completing the square. How are the two methods related?

source: ADE CCSS

Students should solve by factoring, completing the square, and using the quadratic formula. The zero product property is used to explain why the factors are set equal to zero. Students should relate the value of the discriminant to the type of root to expect. A natural extension would be to relate the type of solutions to ax2 + bx + c = 0 to the behavior of the graph of y = ax2 + bx + c.

Complete the square with and without leading coefficients.

Howard County Unit 3 Engage NY Module 4 LearnZillion – A-REI.4(b) Glencoe: 5-2, CCSS Lab 2

http://www.engageny.org/resource/algebra-i-module-4


http://learnzillion.com/lessons/3244-write-and-solve-a-quadratic-equation

http://learnzillion.com/lessons/656-create-and-solve-quadratic-equations




https://learnzillion.com/lessonsets/98-solve-quadratic-equations-by-inspection-taking-square-roots-completing-the-square-the-quadratic-formula-and-factoring



Cluster: Solve systems of equations. A-REI.C.7. Solve a simple system consisting of a linear equation and a

quadratic equation in two variables algebraically and graphically.

Example Two friends are driving to the Grand Canyon in separate cars. Suzette has been there before and knows the way but Andrea does not. During the trip Andrea gets ahead of Suzette and pulls over to wait for her. Suzette is traveling at a constant rate of 65 miles per hour. Andrea sees Suzette drive past. To catch up, Andrea accelerates at a constant rate. The distance in miles (d) that her car travels as a function of time in hours (t) since Suzette’s car passed is given by d = 3500t2. Write and solve a system of equations to determine how long it takes for Andrea to catch up with Suzette.

source: ADE CCSS

Howard County Unit 3 LearnZillion - A-REI.11 Glencoe: supplement


F-IF.B.6. 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. Example The table below represents the value of Andrew’s stock portfolio, with 𝑉 representing the value of the portfolio, in hundreds of dollars, and 𝑡 is the time, in months, since he started investing.

How fast is Andrew’s stock value decreasing between [10, 12]? Find another two-month interval where the average rate of change is faster than [10, 12] and explain why. Source: Engage NY


abafbf

xy

−−

=∆∆ )()(

In addition to finding average rates of change from functions given symbolically, graphically, or in a table, Students may collect data from experiments or simulations (ex. falling ball, velocity of a car, etc.) and find average rates of change for the function modeling the situation. Focus on quadratic functions.

Engage NY Module 4 Topic A, B, & C Glencoe: Extend 5-7



https://learnzillion.com/lessons/3562-find-two-solutions-for-two-equations






the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Example The all-star kicker kicks a field goal for the team and the path of the ball is modeled by 𝑓(𝑥) = −4.9𝑡2 + 20𝑡. Graph the function and label the maximum or minimum. What does the maximum or minimum mean in context? adapted from: NC Functions

Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Intercepts should be expressed as ordered pairs. Students should be able to graph with T-charts, 𝒇(𝒙) form, standard form, factored form, as well as vertex form. Students should also be able to define domain and range in interval, set notation, and description forms. Note this standard is here for review from Algebra 1-2 and is not an explicit 2-4 standard.

Engage NY Module 4 Topic A, B, & C Graphs of Quadratic Functions LearnZillion – F-IF.7(a) LearnZillion – F-IF.7(a) Glencoe: 5-1

F-IF.C.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeroes, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Example In the cartoon, Coyote was chasing the Road Runner, seeing no easy escape, Road Runner jumped off a cliff towering above the roaring river below. Molly mathematician was observing the chase and obtained a digital picture of this fall. Using her mathematical knowledge, Molly modeled the Road Runner’s fall using several different quadratic functions.

64)1(16)()3)(1(16)(483216)(

2

2

+−−=

−+−=++−=

tthttthttth

a) Does Molly have three unique equations that model the same situation? Explain.

b) Explain what it means for expressions or equations to be

Students should use different forms to highlight properties and features of the function. In addition, students should be able to select which form is more efficient given the context of the situation.

Howard County Unit 3 Engage NY Module 4 Topic B & C Forming Quadratics Which Function? Profit of a Company Glencoe: CCSS Lesson 3

http://maccss.ncdpi.wikispaces.net/file/view/Functions%20Theme%20Unpacked%2004152012.pdf/359510119/Functions%20Theme%20Unpacked%2004152012.pdf



https://learnzillion.com/lessonsets/477-graph-quadratic-functions-and-show-intercepts-maxima-and-minima

https://learnzillion.com/lessonsets/405-graph-quadratic-functions-and-show-intercepts-maxima-and-minima



http://map.mathshell.org/materials/lessons.php?taskid=224





mathematically equivalent. c) Which of the equivalent equations would be most helpful in

answering each of these questions? Explain. 1. What is the maximum height the Road Runner reaches and when will it occur? 2. When would the Road Runner splash into the river? 3. At what height was the Road Runner when he jumped off the cliff?

source: NC Functions

F-IF.C.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Example Examine the functions below. Which function has the larger maximum? How do you know?

2082)( 2 +−−= xxxf

source: ADE CCSS

Engage NY Module 4 Topic C Forming Quadratics Throwing Baseballs Glencoe: supplement

http://learnzillion.com/lessonsets/477-graph-quadratic-functions-and-show-intercepts-maxima-and-minima



http://map.mathshell.org/materials/lessons.php?taskid=224






Examples

• Compare the shape and position of the graphs of f (x) = x2 and

g(x ) = 2x 2 , and explain the differences in terms of the algebraic expressions for the functions.

• Describe effect of varying the parameters a, h, and k have on the shape and position of the graph of f(x) = a(x-h)2 + k.

source: ADE CCSS

Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Students should be able to describe transformation in relation to a, h, and k in vertex form.

Engage NY Module 4 Topic C LearnZillion – F-BF.3 Medieval Archer Building a Quadratic Function from 𝑓(𝑥) = 𝑥2 Building a General Quadratic Function Glencoe: Explore 5-7, 5-7



https://learnzillion.com/lessonsets/766-manipulate-quadratic-functions






F-BF.B.4. Find inverse functions. a. 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. Example Given the following relation: (−3, 2) (−2, 2) (−1, 2) (0, 2)

a) Is the relation a function? Justify your answer. b) Find the inverse of the relation. c) Is the inverse a function? Justify your answer.

**For this unit, students should be working with tables and graphs – no Algebra at this time. This topic will progress in second semester.

Virtual Nerd Videos Glencoe: supplement


S-ID.B.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or chooses a function suggested by the context. Emphasize linear, quadratic, and exponential models. Example An independent study was done linking advertising to the purchase of an object. 400 households were used in the survey and the commercial exposure was over a one week period. See the data set below.

# of times commercial was shown, x

1 7 14 21 28 35 42 49

# of households bought item, y

2 25 96 138 88 37 8 6

a) Find a model to fit the data. b) Why do you think the amount of homes that purchased the item

went down after more exposure to the commercial?

Focus on quadratic regression. If you run Quadratic Regression on your calculator, do note that the correlation coefficient, or 𝒓, has no meaning for a curve as it measures the linear relationship between two variables. The coefficient of determination, or 𝒓𝟐, can be used to assess how closely the curve fits the data.

LearnZillion – S-ID.6 - Quadratic HotMath Purple Math TI-84 –Quadratic Regression

Glencoe: Extend 5-1, supplement

http://www.virtualnerd.com/algebra-2/radical-functions/inverse-relations/inverse-relation

https://learnzillion.com/lessons/3174-select-a-statistical-regression-model-interpolate-and-extrapolate

http://hotmath.com/hotmath_help/topics/quadratic-regression.html

http://www.purplemath.com/modules/scattreg4.htm

https://www.youtube.com/watch?v=74bdyEF6Ugw



Unit 2: Quadratic Functions and Equations ENDURING UNDERSTANDINGS

• Real world problems can be represented and solved with quadratic functions. ESSENTIAL QUESTIONS KEY CONCEPTS

• What are the algebraic structures within quadratic expressions? • How is algebraic structure used to create and solve quadratic

equations? • When is one form of a quadratic more appropriate than another? • What is the purpose of a complex number in a quadratic equation? • How are different representations of quadratic functions used to

identify key features of a quadratic function? • What contextual situations can be modeled by quadratic equation?

• The graph of a quadratic function is a symmetrical curve with a maximum or minimum point corresponding to its vertex and an axis of symmetry passing through it.

• Key features of functions can be described in terms of the context of the function.

• Solutions should be examined to determine if they are viable given the context.

• If a simplified solution includes both rational and irrational components (without a perfect square under the radical), it can’t be rewritten equivalently as a single rational or irrational number.

• Roots, solutions, and zeroes correspond to the points where the graph of the quadratic equation crosses the x-axis.

• The sign of the discriminant can be used to determine the number of solutions to a quadratic equation.

• A complex number is written in the form 𝑎 + 𝑏𝑖 and such numbers can be added, subtracted, or multiplied using the principle 𝑖2 = −1.

STUDENT FRIENDLY OBJECTIVES ACADEMIC VOCABULARY • TSW solve quadratic equations using different methods. • TSW will choose the appropriate method to solve a quadratic and

explain their reasoning. • TSW graph quadratic functions and identify the intercepts,

min/max, axis of symmetry, domain and range, and roots. • TSW solve equations with complex solutions, and add, subtract,

and multiply complex numbers. • TSW compare two quadratic functions shown in two different

ways. • TSW solve a system consisting of a linear and quadratic equation. • TSW will use quadratic regression to fit a curve to data.

degree binomial trinomial zero product property solution x-intercept zero(s) solution(s) roots imaginary number quadratic regression

quadratic expression quadratic equation vertex end behavior maximum minimum leading coefficient completing the square axis of symmetry complex number coefficient of determination

rational irrational discriminant y-intercept standard form vertex form factored form radical parabola complex solution



Unit 2: Quadratic Functions and Equations DEPTH OF KNOWLEDGE







Task: Solve the equation using the quadratic equation. 𝑥2 + 12𝑥 − 9 = 0

Task: For the following quadratic equation find the value of the discriminant and describe the number and type of roots. 3𝑥2 + 8𝑥 + 2

Task: Describe three different ways to solve 𝑥2 − 2𝑥 − 15 = 0. Which method do you prefer and why?

Task: Building a General Quadratic Function




Unit 2: Quadratic Functions and Equations ASSESSMENT RUBRIC

(The levels of proficiency do not correspond to DOK levels, but instead reflect a student’s progress in relation to a standard) Developing





standard) A student can solve a quadratic equation.

A student can solve a quadratic equation in context and describe the solution(s) in terms of the context.

A student can make the connection between the solutions, the discriminant, and the graphs of a quadratic equation.

A student can describe when which representation of a quadratic will reveal important information in context.




TEACHER NOTES























Unit 3: Polynomial Functions 2010 Standards Comments Resources




Glencoe: throughout

Cluster: Interpret the structure of expressions. A-SSE.A.2. Use the structure of an expression to identify ways to rewrite

it. Example Factor xxx 352 23 −− source: TUSD

i) Tasks are limited to polynomial, rational, or exponential expressions. ii) Examples: see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

LearnZillion – A-SSE.2 Glencoe: 6-5

Cluster: Understand the relationship between zeroes and factors of polynomials.

A-APR.B.2. Know and apply the Remainder Theorem: For a polynomial 𝑝(𝑥)and a number𝑎, the remainder on division by 𝑥 − 𝑎 is 𝑝(𝑎), so 𝑝(𝑎) = 0 if and only if (𝑥 − 𝑎)is a factor of 𝑝(𝑥). Example Let p(x )= x5 −3x 4+8x 2 − 9x +30 . Evaluate 𝑝(−2). What does your answer tell you about the factors of 𝑝(𝑥)? [Answer: 𝑝(−2) = 0 so 𝑥 + 2 is a factor.] source: ADE CCSS

The Remainder theorem says that if a polynomial 𝒑(𝒙) is divided by 𝒙 − 𝒂, then the remainder is the constant 𝒑(𝒂). That is, p(x )=q(x)(x − a)+ p(a). So if

𝒑(𝒂) = 𝟎 then 𝒑(𝒙) = 𝒒(𝒙)(𝒙 − 𝒂).

Howard County Unit 4 Shmoop – A-APR.2 LearnZillion – A-APR.2 Glencoe: 6-6


http://www.tusd1.org/contents/depart/highschool/gr9_10.asp

https://learnzillion.com/lessons/3674-factor-using-multiple-strategies



http://www.shmoop.com/common-core-standards/ccss-hs-a-apr-2.html

https://learnzillion.com/lessonsets/605-know-and-apply-the-remainder-theorem



A-APR.B.3. Identify zeroes of polynomials when suitable factorizations are available, and use the zeroes to construct a rough graph of the function defined by the polynomial. Example Factor the expression x3 + 4x2 − 59x −126 and explain how your answer can be used to solve the equation x3 + 4x2 − 59x −126 = 0 . Explain why the solutions to this equation are the same as the x-intercepts of the graph of the function f (x) = x3 + 4x2 − 59x −126 . source: ADE CCSS

Graphing calculators or programs can be used to generate graphs of polynomial functions. i) Tasks include quadratic, cubic, and quartic polynomials and polynomials for which factors are not provided. For example, find the zeroes of (𝒙𝟐 −𝟏)(𝒙𝟐 + 𝟏).

Howard County Unit 4 Graphing from Roots Shmoop – A-APR.3 LearnZillion – multiple videos A-APR.3 Glencoe: 6-5, 6-7, CCSS Lab 7

A-APR.C.4. Prove polynomial identities and use them to describe numerical relationships.

Examples Use the distributive law to explain why x2 – y2 = (x – y)(x + y) for any two numbers x and y.

Derive the identity (x – y)2 = x2 – 2xy + y2 from (x + y)2 = x2 + 2xy + y2 by replacing y by –y.

source: ADE CCSS

For example, the polynomial identity (x2+y2)2 = (x2– y2)2 + (2xy)2 can be used to generate Pythagorean triples.

Howard County Unit 4 Shmoop – A-APR.4 LearnZillion – A-APR.4 Glencoe: CCSS Lab 6

Cluster: Rewrite rational expressions. A-APR.D.6. Rewrite simple rational expressions in different forms; write

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 inspection, long division, or, for the more complicated examples, a computer algebra system. Examples

Find the quotient and remainder for the rational expression 𝑥3−3𝑥2+𝑥−6

𝑥2+2 and

use them to write the expression in a different form.

Express 𝑓(𝑥) = 2𝑥+1𝑥−1

in a form that reveals the horizontal asymptote of its graph.

[Answer: , so the horizontal asymptote is y = 2.] source: ADE CCSS

The polynomial q(x) is called the quotient and the polynomial r(x) is called the remainder. Expressing a rational expression in this form allows one to see different properties of the graph, such as horizontal asymptotes. At the regular level division is limited to monomials and binomials with no leading coefficient. Stick with synthetic rather than long division for regular.

Howard County Unit 4 Shmoop – A-APR.6 LearnZillion – multiple videos A-APR.6 Glencoe: 6-2





https://learnzillion.com/common_core/math/algebra




https://learnzillion.com/lessonsets/498-prove-and-use-polynomial-identities



http://www.shmoop.com/common-core-standards/handouts/a-arp_worksheet_6.pdf

https://learnzillion.com/common_core/math/algebra





Example The functions 𝑓(𝑥) = 𝑥 and 𝑔(𝑥) = 𝑥3 are graphed below.

Part A: Explain how you can use the graph to find the solution(s) of the equation𝑓(𝑥) = 𝑔(𝑥). In your answer, provide the approximate value(s) of the solution(s). Part B: Write the value(s) of 𝑓(𝑥) when 𝑥 equals the solution(s) from Part A. Part C: Let the function ℎ(𝑥) be defined asℎ(𝑥) = 𝑓(𝑥) − 𝑔(𝑥). What are the coordinates of the point(s) on the graph of ℎ(𝑥) when 𝑥 equals the solution(s) from Part A? Explain you reasoning.

Students need to understand that numerical solution methods (data in a table used to approximate an algebraic function) and graphical solution methods may produce approximate solutions, and algebraic solution methods produce precise solutions that can be represented graphically or numerically. Students may use graphing calculators or programs to generate tables of values, graph, or solve a variety of functions.

Howard County Unit 4 Introduction to Polynomials – College Fund Two Squares are Equal Glencoe: supplement







F-IF.B.4. 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.

Example Let 155)( 23 +−−= xxxxf . Graph the function and identify end behavior and any intervals of constancy, increase, and decrease.

source: ADE CCSS

Students may be given graphs to interpret or produce graphs given an expression or table for the function, by hand or using technology.

Howard County Unit 4 Glencoe: 6-3, 6-4, Extend 6-4

F-IF.B.6. 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. Example The volume of a cubic crystal, grown in a laboratory, can be modeled by 𝑉(𝑥) = 𝑥3, where 𝑉 is the volume measured in cubic centimeters and 𝑥 is the side length in centimeters. Determine the average rate of change in the volume of the crystal with respect to the side length as each side grows from 4cm to 5cm. Source: click here


abafbf

xy

−−

=∆∆ )()(

In addition to finding average rates of change from functions given symbolically, graphically, or in a table, Students may collect data from experiments or simulations (ex. falling ball, velocity of a car, etc.) and find average rates of change for the function modeling the situation.

Average rate of change (instantaneous not necessary) Glencoe: supplement



http://scc.scdsb.edu.on.ca/Students/onlinecourses/Sacchetto/AFIC%20web%20page/AFIC%20textbook/AFIC%20text%20pages/Ch%203/172.pdf






the graph, by hand in simple cases and using technology for more complicated cases. c. Graph polynomial functions, identifying zeroes when suitable factorizations are available, and showing end behavior. Example A standard piece of notebook paper measuring 8.5 in. by 11 in. is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let 𝑥 represent the length of a side of each such square in inches. Use the table feature of your graphing calculator to do the following.

a) Find the maximum volume of the box. b) Determine when the volume of the box will be greater than 40 in3. c) Sketch a graph of the function.

source: Pearson College Algebra & Trig text (p. 356 #102)

Key characteristics include but are not limited to maxima, minima, intercepts, symmetry, end behavior, and asymptotes. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions.

Howard County Unit 4 Graphs of Power Functions Running Time LearnZillion – F-IF.7 Glencoe: 6-3, 6-4, Extend 6-4, 6-7, CCSS Lab 7


Example If 𝑓(𝑥) is a polynomial function with an order of 7 and 𝑔(𝑥) is a linear function with a slope of 2, which function will cross the 𝑦-axis more? Source: Shmoop

Howard County Unit 4 Shmoop – F-IF.9 (polynomial) Glencoe: CCSS Lab 5, supplement


F-BF.A.1. Write a function that decribes a relationship between two quantities. b. Combine standard function types using arithmetic operations. c.+ Compose functions. Example The radius of a circular oil slick after 𝑡 hours is given in feet by 𝑟 = 10𝑡2 −0.5𝑡, for 0 ≤ 𝑡 ≤ 10. Find the area of the oil slick as a function of time. source: Flipbook

Note that composing functions is not explicitly tested at this level but a brief introduction is necessary to prepare students for inverse functions.

Howard County Unit 4 LearnZillion – F-BF.1(b) Glencoe: 6-1, 6-2, 7-1




https://learnzillion.com/lessonsets/518-graph-polynomial-functions-identifying-zeros-and-showing-end-behavior

http://www.shmoop.com/common-core-standards/handouts/f-if-worksheet_9.pdf


http://www.shmoop.com/common-core-standards/handouts/f-if-worksheet_9.pdf



https://learnzillion.com/lessonsets/771-combine-standard-function-types-using-arithmetic-operations





Example Is f(x) = x3 - 3x2 + 2x + 1 even, odd, or neither? Explain your answer orally or in written format. source: ADE CCSS

Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions.

Howard County Unit 4 LearnZillion – F-BF.3 Glencoe: CCSS Lab 5



https://learnzillion.com/lessons/3984-distinguish-between-even-and-odd-functions



Unit 3: Polynomial Functions ENDURING UNDERSTANDINGS

• Real world problems can be represented and solved with polynomial functions. ESSENTIAL QUESTIONS KEY CONCEPTS

• What is the relation between synthetic substitution and synthetic division?

• How are the factors of a polynomial equation related to the graph of the function?

• What is the difference between and among: factors, zeroes, x-intercepts, roots, and solutions?

• How can the end behavior related to the leading term of the polynomial?

• If there is no remainder after synthetic division, then the divisor is a factor.

• The solutions for a function are the points where the graph crosses the x-axis.

• The degree of the polynomial determines the number of solutions both real and complex.

• End behaviors are determined by the leading coefficient and the degree of the polynomial.

STUDENT FRIENDLY OBJECTIVES ACADEMIC VOCABULARY • TSW add, subtract, and multiply polynomials. • TSW compose functions. • TSW rewrite polynomials by factoring. • TSW use the Fundamental Theorem of Algebra to determine the number

of zeroes in a polynomial function. • TSW apply the Remainder Theorem to determine if (x – a) is a factor of a

polynomial. • TSW factor a polynomial to identify the zeroes of the function. • TSW find the rational zeroes of a polynomial function by graphing. • TSW use the rational zeroes (found by using a graphing calculator) and

synthetic division to reduce the degree of a polynomial function so it can be factored.

• TSW sketch a graph of the polynomial using the zeroes and end behavior. • TSW use zeroes from a graph or given to write a polynomial function. • TSW use synthetic division to divide polynomials. • TSW will find the average rate of change of a polynomial over a

specified interval. • TSW write a polynomial equation that models a real-world problem and

use the equation to solve the problem.

factoring polynomial function degree of polynomial function end behavior intercept symmetry polynomial identity quartic function extrema depressed polynomial

synthetic substitution synthetic division remainder theorem prime polynomials fundamental theorem of algebra relative maximum relative minimum quadratic function leading coefficient turning points

rational root complex root multiplicity possible solutions real root rate of change asymptote cubic function quantic function power function



Unit 3: Polynomial Functions DEPTH OF KNOWLEDGE







Task: Sketch the graph of an odd polynomial function with zeroes at −5,−3, 0, 2, 𝑎𝑛𝑑 4.

Task: Jin’s vending machines currently sell an average of 3500 beverages per week at a rate of $0.75 per can. She is considering increasing the price. Her weekly earnings can be represented by 𝑓(𝑥) =−5𝑥2 + 100𝑥 + 2625, where 𝑥 is the number of $0.05 increases. Graph the function and determine the most profitable price for Jin.

Task: Determine whether the following statement is sometimes, always or never true. Explain your reasoning. For any continuous polynomial function, the y-coordinate of a turning point is also a relative maximum or relative minimum.

Task: Running Time After completing the task above, research an algorithm used for running time for computer applications. How does it compare to these models?




Unit 3: Polynomial Functions ASSESSMENT RUBRIC






standard) A student can identify the degree and leading coefficient of a polynomial.

A student can describe the end behavior, determine whether it represents and odd or even degree polynomial, and state the number of real zeroes from a graph.

A student can graph a polynomial using technology, determine a reasonable viewing window, and approximate zeroes.

A student can graph a polynomial using technology, determine a reasonable viewing window, approximate zeroes, and use the information to answer questions in context.




TEACHER NOTES






















Unit 4: Rational Functions 2010 Standards Comments Resources

Cluster: Interpret the structure of expressions. A-SSE.A.2. Use the structure of an expression to identify ways to rewrite

it. Example Which of the following rational expressions equals −1? (In parts A, B, and D, 𝑥 ≠ −4, and in part C, 𝑥 ≠ 4). Select all that apply.

a) 𝑥−4𝑥+4

b) −𝑥−4𝑥+4

c) 𝑥−44−𝑥

d) 𝑥−4−𝑥−4

source: Pearson College Algebra & Trig text (p. 356 #102)

i) Tasks are limited to polynomial, rational, or exponential expressions. ii) Examples: See (x2 + 4)/(x2 + 3) as ( (x2+3) + 1 )/(x2+3), thus recognizing an opportunity to write it as 1 + 1/(x2 + 3). Students should be able to add rational expressions, such as finding the perimeter of a figure. Students should also be able to subtract, multiply, and divide rational expressions. Ensure students are able to factor trinomials, group to factor, GCF’s, and cubes.

A Cubic Identity Glencoe: 9-1, 9-2

Cluster: Rewrite rational expressions. A-APR.D.7.+Understand that rational expressions form a system

analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Examples

Students at this level should be able to add, subtract, multiply, and divide rational expressions.

LearnZillion – A-APR.7 Shmoop – A-APR.7 Glencoe: 9-1, 9-2


https://learnzillion.com/lessonsets/154-add-subtract-multiply-and-divide-rational-expressions






Example On the day of the class field trip, the chocolate factory produced three times as many plain chocolate bars as crispy bars. They produced 50 more nutty bars that crispy bars. The ratio of plain chocolate bars produced to nutty bars produced was 2 to 1. Which of the equations below could be utilized to solve for the number of crispy bars produced on the day of the field trip?

a) 3𝑐 + 2𝑐 = 50 b) 3𝑐

𝑐+50= 2

c) 2 × 3𝑐 = 1 × (𝑐 + 50) d) 𝑐+50

3𝑐= 2

Source: Shmoop

LearnZillion – A-CED.1 – video 1 Shmoop – A-CED.1 (rational) Glencoe: 9-6, Extend 9-6

4𝑥 + 3

= 5

Cluster: Understand solving equations as a process of reasoning and explain the reasoning.

A-REI.A.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Example Solve the following equation for 𝑥. Use mathematical properties to justify each step in the process. Justify your solution.

Properties of operations can be used to change expressions on either side of the equation to equivalent expressions. In addition, adding the same term to both sides of an equation or multiplying both sides by a non-zero constant produces an equation with the same solutions.

Glencoe: 9-6, Extend 9-6, supplement to meet standard

http://www.shmoop.com/common-core-standards/ccss-hs-a-ced-1.html

https://learnzillion.com/lessons/3328-write-and-solve-a-simple-rational-equation

http://www.shmoop.com/common-core-standards/ccss-hs-a-ced-1.html



1𝑥 − 5

=𝑥

𝑥2 − 25

A-REI.A.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Example Solve the equation and check your solution.

LearnZillion – A-REI.2 – several videos LearnZillion – A-REI.2 LearnZillion – A-REI.2 Howard County Unit 6 Glencoe: 9-6, Extend 9-6



Example The functions 𝑓(𝑥) = 1 − 𝑥 and 𝑔(𝑥) = 0.11

𝑥2 are defined for all values of

𝑥 > 0. The graphs are shown below.

Part A: Explain how you can use the graph to find the solution(s) of the equation𝑓(𝑥) = 𝑔(𝑥). In your answer, provide the approximate value(s) of the solution(s). Part B: Write the value(s) of 𝑓(𝑥) when 𝑥 equals the solution(s) from Part A. Part C: Let the function ℎ(𝑥) be defined asℎ(𝑥) = 𝑓(𝑥) − 𝑔(𝑥). What are


Howard County Unit 6 PARCC Sample Item Glencoe: 9-3, 9-4, supplement

https://learnzillion.com/lessonsets/501-solve-simple-rational-radical-equations-in-one-variable

https://learnzillion.com/lessons/1456-solve-rational-expressions-using-additive-and-multiplicative-inverses

https://learnzillion.com/lessons/1457-finding-extraneous-solutions-in-rational-algebraic-equations



http://www.parcconline.org/sites/parcc/files/HighSchoolAlg2Math3-GraphsofFunctions.pdf



the coordinates of the point(s) on the graph of ℎ(𝑥) when 𝑥 equals the solution(s) from Part A? Explain you reasoning. Source: PARCC Sample Items

𝑔(𝑥) =1

𝑥 − 3+ 5


the graph, by hand in simple cases and using technology for more complicated cases. d. + Graph rational functions, identifying zeroes and asymptotes when suitable factorizations are available, and showing end behavior. Example Graph the following function. Identify the asymptotes, domain and range.

Key characteristics include but are not limited to intercepts, symmetry, end behavior, horizontal & vertical asymptotes, and domain and range. Oblique asymptotes and point discontinuity not necessary in regular. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions.

Howard County Unit 6 Glencoe: 9-3, 9-4

http://www.parcconline.org/sites/parcc/files/HighSchoolAlg2Math3-GraphsofFunctions.pdf




Unit 4: Rational Functions ENDURING UNDERSTANDINGS

• Real world problems can be represented and solved with rational functions. ESSENTIAL QUESTIONS KEY CONCEPTS

• What causes a rational equation to have an extraneous solution? • What kinds of asymptotes are possible for a rational function? • What is the difference between and among: factors, zeroes, x-

intercepts, roots, solutions and asymptotes? • Are a rational expression and its simplified form equivalent?

• A rational equation, expression or function has a variable in the denominator.

• To operate with rational expressions, you can use much of what you know about operating with fractions.

• A rational function is a ratio of polynomial functions. If a rational function is in simplified form and the polynomial in the denominator is not a constant, the graph of the rational function features asymptotic behavior.

• When solving an equation involving rational expressions multiplying by the common denominator can result in extraneous solutions.

• An extraneous solution is an algebraically found number that causes the rational expression to be undefined.

• The zeroes of a function are the points where the graph crosses the x-axis.

STUDENT FRIENDLY OBJECTIVES ACADEMIC VOCABULARY • TSW determine the best method to simplify a rational expression. • TSW rewrite a rational expression using factoring or synthetic division. • TSW add, subtract, multiply, and divide rational expressions. • TSW solve a rational equation, and remember to check for extraneous

solutions. • TSW justify each step used to solve a rational equation. • TSW solve a rational equation by graphing. • TSW write a rational equation that models a real-world problem and use

the equation to solve the problem. • TSW find the zeroes of a rational function and use the zeroes in creating

its graph. • TSW will identify key features on a graph of a rational function and

relate them to the context. • TSW find the average rate of change of a rational function presented

symbolically or as a table over a specified interval.

rational expression complex fraction vertical asymptote horizontal asymptote

rational equation extraneous solution zeroes of a function Degree of a polynomial

division by zero end behavior rational function



Unit 4: Rational Functions DEPTH OF KNOWLEDGE







Task: Solve the equation and check your solution. 47

+3

𝑥 − 3=

5356

Task: How many milliliters of a 20% acid solution must be added to 40 milliliters of a 75% acid solution to create a 30% acid solution?

Task: Consider 2

𝑥−3+ 1

𝑥= 𝑥−1

𝑥−3

a) Solve the equation. Were there any extraneous solutions?

b) Graph 𝑦1 = 2𝑥−3

+ 1𝑥 and 𝑦2 = 𝑥−1

𝑥−3

on the same graph for 0 < 𝑥 < 5. c) For what value(s) of x do they

intersect? Do they intersect where x is extraneous for the original equation?

d) Use this knowledge to describe how you can use a graph to determine whether an apparent solution of a rational equation is extraneous.

Task: A Cubic Identity




Unit 4: Rational Functions ASSESSMENT RUBRIC

(DVMA will include questions based on the developing and proficient levels only.) (The levels of proficiency do not correspond to DOK levels, but instead reflect a student’s progress in relation to a standard) Developing





standard) A student can identify the LCM of a set of polynomials.

A student can simplify a monomial expression by finding a common denominator.

A student can simplify and solve a rational equation in context.

A student can create and solve a rational equation in context.




TEACHER NOTES






















Unit 5: Radical Functions 2010 Standards Comments Resources

Cluster: Extend the properties of exponents to rational exponents.

N-RN.A.1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. Examples

a) Use an example to show why 𝑥𝑚

𝑥𝑛= 𝑥𝑚−𝑛 holds true for

expressions involving rational exponents like 12 or

15.

b) Using what you know about properties of exponents, simplify the

following: �1614�3

source: NC Number & Quantity

Understand that the denominator of the rational exponent is the root index and the numerator is the exponent of the

radicand. For example, 𝟓𝟏𝟐 = √𝟓.

In order to understand the meaning of rational exponents, students can initially investigate them by consider a pattern such as:

LearnZillion – N-RN.1 Shmoop – N-RN.1 Howard County Unit 6 Glencoe: 7-6

N-RN.A.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Examples

• 2

3 2 35 5= ; 2

3 235 5=

• Rewrite using fractional exponents: 4

5 45 516 2 2= =

• Rewrite 2

xx

in at least three alternate forms.

Solution: 32

3 32

1 1 1xx xxx

−= = =

• Rewrite 4 42− .using only rational exponents.

• Rewrite 3 3 23 3 1x x x+ + + in simplest form.

source: ADE CCSS

Note that rationalizing the denominator and complex conjugates is not necessary. Leaving a radical in the denominator is acceptable (see third example to the left from ADE).

LearnZillion – N-RN.2 Shmoop – N-RN.2 Howard County Unit 6 Glencoe: 7-4, 7-5


https://learnzillion.com/lessonsets/551-relate-rational-exponents-to-integer-exponents-use-radical-notation

http://www.shmoop.com/common-core-standards/ccss-hs-n-rn-1.html



https://learnzillion.com/lessonsets/646-rewrite-expressions-involving-radicals-and-rational-exponents

http://www.shmoop.com/common-core-standards/ccss-hs-n-rn-2.html




√𝑥 + 3 = 5

Cluster: Understand solving equations as a process of reasoning and explain the reasoning.

A-REI.A.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Example Solve the following equation for 𝑥. Use mathematical properties to justify each step in the process. Justify your solution.

Properties of operations can be used to change expressions on either side of the equation to equivalent expressions. In addition, adding the same term to both sides of an equation or multiplying both sides by a non-zero constant produces an equation with the same solutions.

Glencoe: 7-7, supplement to meet standard

√𝑥 + 2 = 𝑥

A-REI.A.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Example Solve the equation and check your solution(s).

LearnZillion – A-REI.2 – video 1 LearnZillion – A-REI.2 – video 2 Howard County Unit 6 Glencoe: 7-7

𝑓(𝑥) = 𝑥2

F-BF.B.4. Find inverse functions. a. 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. Example Find the inverse of the function. The graph the function and its inverse.

**Limit to inverses of simple quadratic and cubic functions. Students should start to use composition of functions here (Unit 3).

Shmoop – F-BF.4 Glencoe: 7-2 (simple quadratic and cubic functions only)

https://learnzillion.com/lessons/1458-solve-radical-algebraic-equations

https://learnzillion.com/lessons/1459-fina-extraneous-solutions-in-algebraic-radical-expressions


http://www.shmoop.com/common-core-standards/ccss-hs-f-bf-4.html



Unit 5: Radical Functions ENDURING UNDERSTANDINGS

• Real world problems can be represented and solved with radical functions. ESSENTIAL QUESTIONS KEY CONCEPTS

• To simplify the nth root of an expression, what must be true about the expression?

• When you square each side of an equation, is the resulting equation equivalent to the original?

• How are a function and its inverse related? • What is the difference between real solutions and extraneous

solutions? • How are the solutions of a radical equation related to the graph of

the function?

• The properties of exponents can be extended to rational exponents. • Corresponding to every power there is a root. • You can write a radical expression in an equivalent form using a

fractional (rational) exponent instead of a radical sign. • You can combine like radicals using properties of real numbers. • Solving a square root equation may require that you square each side

of the equation which may result in extraneous solutions. • The solutions for a radical function are the points where the graph

crosses the x-axis.

STUDENT FRIENDLY OBJECTIVES ACADEMIC VOCABULARY • TSW model the properties of integer exponents to derive the rules for

rational exponents. • TSW change a radical expression to an exponential expression and

apply the properties of exponents. • TSW solve a simple radical equation in one variable and check for

extraneous solutions. • TSW explain why solutions are extraneous. • TSW find the inverse of a radical equation and identify the domain. • TSW sketch a graph of the radical function using the key features

of the graph to include: zeroes, end behavior, and asymptotes. • TSW justify each step used to solve an equation.

rational root complex root multiplicity possible solution radical sign radicand

extraneous solutions domain real root inverse function principal root index

zeroes asymptotes square root function nth root radical function



Unit 5: Radical Functions DEPTH OF KNOWLEDGE







Task: Simplify. �25𝑥24

Task: If the area A of a square is known, then the lengths of its sides can be

computed using ℓ = 𝐴12. You have

purchased a 169 𝑓𝑡2share in a community garden for the season. What is the length of one side of your square garden?

Task: Determine whether −𝑥−2 = (−𝑥)−2 is always, sometimes, or never true. Explain your reasoning.

Task: Rational or Irrational?




Unit 5: Radical Functions ASSESSMENT RUBRIC






standard) A student can write a simple expression with a rational exponent in radical form.

A student can simplify an expression with rational exponents or radicals.

A student can model the properties of integer exponents to derive the rules for rational exponents.

A student can explain how it might be easier to simplify an expression using rational exponents rather than using radicals.




TEACHER NOTES
























Unit 6: Exponential Functions 2010 Standards Comments Resources




Glencoe: throughout

Cluster: Interpret the structure of expressions. A-SSE.B.3. Choose and produce an equivalent form of an expression to

reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for

exponential functions.

Example The equation 𝑦 = 14000(0.8)𝑥represents the value of an automobile x years after purchase. Find the yearly and the monthly rate of depreciation of the car. source: NC Algebra

i) Tasks have a real-world context. As described in the standard, there is interplay between the mathematical structure of the expression and the structure of the situation such that choosing and producing an equivalent form of the expression reveals something about the situation. ii) Tasks are limited to exponential expressions with rational or real exponents.

Forms of Exponential Expressions LearnZillion – A-SSE.3(c) – particularly starting at 3:22 LearnZillion – A-SSE.3(c) Glencoe: 8-2 (supplement to meet standard)


http://maccss.ncdpi.wikispaces.net/file/view/Algebra%20Theme%20Unpacked%20Content%2003162012.pdf/359510039/Algebra%20Theme%20Unpacked%20Content%2003162012.pdf


http://learnzillion.com/lessons/3389-manipulate-exponential-models-to-reveal-information

http://learnzillion.com/lessons/3498-compare-exponential-models-using-properties-of-exponents



A-SSE.B.4.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. Example In February, the Bezanson family starts saving for a trip to Australia in September. The Bezanson’s expect their vacation to cost $5375. They start with $525. Each month they plan to deposit 20% more than the previous month. Will they have enough money for their trip? source: ADE CCSS

The standard specifies finite series but you are welcome to extend to infinite series.

Triangle Series Course of Antibiotics A Lifetime of Savings YouTube Explosion Shmoop – A-SSE.4 Glencoe: 11-3



Example A river has an initial minnow population of 40,000 that is growing at 5% per year. Due to environmental conditions, the amount of algae that minnows use for food is decreasing, supporting 1000 fewer minnows each year. Currently, there are enough algae to support 50,000 minnows. In what year will the minnow population exceed the amount of algae available? source: EngageNYMod3

The key to this standard is equations and inequalities in one variable. Another example would be to solve the following equation by equating exponents (click here for examples):

80)2(25

=x

Although students should experience problems in context (like the example to the left). Please note these should be fairly basic exponential equations and students should not be using logarithms until the next unit. Include population, bacteria, money, etc. Students can also employ guess and check using a table (see LearnZillion to the right), or graphing to functions to solve.

LearnZillion – A-CED.1 LearnZillion – A-CED.1 Glencoe: 8-2






http://www.shmoop.com/common-core-standards/ccss-hs-a-sse-4.html


http://www.purplemath.com/modules/solvexpo.htm

http://learnzillion.com/lessons/659-create-and-solve-exponentials-using-functions

http://learnzillion.com/lessons/3487-solve-exponential-equations-using-properties-of-exponents





Example A city has 152,817 residents. The population is increasing at a rate of 7% per year. The city council is running a contest to see who can predict how long it will take the city to reach a population of 200,000. Use a graph to make a prediction.


Population and Food Supply Glencoe: 8-1, Explore 8-2, supplement



Example The consumption of soda has increased each year since 2000. The function

tC )029.1(179= models the amount consumed in billions of liters and t

is the number of years since 2000. (adapted from: Glencoe Algebra) a) Graph the function. b) What is the y-intercept? What does it mean in the context of the

problem? c) Discuss where the function is increasing and/or decreasing in the

context of the problem.


Influenza Epidemic Howard County Unit 2 Glencoe: 8-1






F-IF.B.6. 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. Example Given the function 𝑓(𝑥) = 2𝑥+2 − 2, what is the average rate of change from [0, 2]?


abafbf

xy

−−

=∆∆ )()(

In addition to finding average rates of change from functions given symbolically, graphically, or in a table, Students may collect data from experiments or simulations (ex. falling ball, velocity of a car, etc.) and find average rates of change for the function modeling the situation.

Howard County Unit 2 Glencoe: supplement


the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Example Suppose you invest $1000 in a savings account that pays 2.5% annual interest. No additional deposits or withdrawals are made.

a) Draw a graph to represent the amount in the account as a function of time.

b) What is the y-intercept and what does it mean in context? c) Describe the end behavior and its meaning in context.

Key characteristics include but are not limited to intercepts, end behavior, asymptotes, and domain and range. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Include base e**

Identifying Graphs of Functions LearnZillion – Exponential Growth LearnZillion – Exponential Decay Glencoe: 8-1

F-IF.C.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. Example The equation 𝑦 = 14000(0.8)𝑥represents the value of an automobile x years after purchase. Find the yearly and the monthly rate of depreciation of the car. source: NC Algebra

Students can work with populations, the age of an object, compound interest, depreciation, etc.

LearnZillion – several videos – F-IF.8b Glencoe: supplement



https://learnzillion.com/lessons/3162-graph-exponential-growth-functions

https://learnzillion.com/lessons/3176-graph-exponential-decay-functions

http://maccss.ncdpi.wikispaces.net/file/view/Algebra%20Theme%20Unpacked%20Content%2003162012.pdf/359510039/Algebra%20Theme%20Unpacked%20Content%2003162012.pdf

https://learnzillion.com/lessonsets/601-use-the-properties-of-exponents-to-interpret-expressions-and-functions




Example Compare the y-intercepts and end behavior of the functions below: Function 1: 𝑓(𝑥) = 2(4)𝑥 Function 2:

x g(x) -2 4 -1 2 1 1

2

2 14

Howard County Unit 2 Glencoe: supplement


F-BF.A.1. Write a function that decribes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Example Ten bacteria are placed in a test tube and each one splits in two after one minute. After 1 minute, the resulting bacteria each split in two, creating 20 bacteria. This process continues for one hour until the test tube is full. Write an equation that will determine the number of bacteria after any number of minutes. source: NC Functions

Recursive expressions require the previous term to calculate the next term. For example, in the sequence 2, 4, 6, 8, the recursive expression would be n+2 (or more specifically 2;2 11 +== −nn aaa ).

An explicit expression is one that will give you any term without the previous term. For the example 2, 4, 6, 8, the explicit expression would be 2n. The explicit process is generally more efficient.

Compounding with a 5% Interest Rate Compounding with a 100% Interest Rate Lake Algae LearnZillion – several videos – F-BF.1a Glencoe: 11-3






https://learnzillion.com/lessonsets/770-write-a-function-that-describes-an-exponential-relationship-between-two-quantities



F-BF.A.2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Example Ten bacteria are placed in a test tube and each one splits in two after one minute. After 1 minute, the resulting bacteria each split in two, creating 20 bacteria. This process continues for one hour until the test tube is full.

a) Write a formula that uses any current number of bacteria to find the number of bacteria at the next minute.

b) Write an equation that will determine the number of bacteria after any number of minutes.


Students should connect geometric sequences to exponential functions and understand that exponential functions are the explicit form of a recursively defined geometric sequence. In addition, the recursive formula for a geometric sequence uses multiplication and the explicit formula uses exponentiation.

Howard County Unit 2 Shmoop – F-BF.2 (geometric) Glencoe: 11-3



Example Given the graph 𝑔(𝑥) = 2𝑥+3 − 2

a) Graph 𝑔(𝑥) and the parent function 𝑓(𝑥). b) Describe the transformations from 𝑓(𝑥) to 𝑔(𝑥). c) On what interval is 𝑓(𝑥) > 𝑔(𝑥)?


LearnZillion – F-BF.3 LearnZillion – shift LearnZillion – stretch and reflect Howard County Unit 2 Glencoe: 11-3, supplement



http://www.shmoop.com/common-core-standards/ccss-hs-f-bf-2.html

http://learnzillion.com/lessons/3526-interpret-graphical-behaviors-using-properties-of-exponents

https://learnzillion.com/lessons/3335-shift-exponential-and-logarithmic-functions

https://learnzillion.com/lessons/3342-stretch-and-reflect-exponential-and-logarithmic-functions





F-LE.A.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Example The price of a movie ticket increases approximately 6% a year. The Theatre Deluxe charged $7.00 for regular admission in 2000. Develop a mathematical model to predict the ticket price. source: NC Functions

Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to construct linear functions.

Two Points Determine an Exponential Function I – exponential Two Points Determine an Exponential Function II – exponential Howard County Unit 2 Glencoe: throughout Chapter 8, supplement

Cluster: Interpret expressions for functions in terms of the situation they model. F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context. Example The consumption of soda has increased each year since 2000. The function

tC )029.1(179= models the amount consumed in billions of liters and t

is the number of years since 2000. What values of C and t are meaningful in the context of the problem?

Answer: Since t represents time, 0>t . At 0=t , the consumption is 179

billion liters. Therefore in the context of the problem, 179>C is meaningful.

Newton’s Law of Cooling Illegal Fish Saturating Exponential Carbon 14 dating in practice I Howard County Unit 2 Shmoop – F-LE.5 (exponential) Glencoe: throughout Chapter 8, supplement
















S-ID.B.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or chooses a function suggested by the context. Emphasize linear, quadratic, and exponential models. Example The table shows the number of bacteria in a culture after the given number of hours. Find a good model for the data. Based on the model, how many bacteria will be in the culture after an additional 8 hours?

Hour Bacteria 1 2205 2 2270 3 2350 4 2653 5 3052 6 3417 7 3890 8 4522 9 5107 10 5724

Focus on exponential regression. If you run Exponential Regression on your calculator, do note that the correlation coefficient, or 𝒓, has no meaning for a curve as it measures the linear relationship between two variables. The coefficient of determination, or 𝒓𝟐, can be used to assess how closely the curve fits the data.

Shmoop – S-ID.6 (exponential) Exponential Regression Model Example Online Exponential Regression Tool Exponential Regression – Regents Exams TI-84 – Exponential Regression Glencoe: supplement

http://www.shmoop.com/common-core-standards/ccss-hs-s-id-6.html

http://mathbits.com/MathBits/TISection/Statistics2/exponential.htm

http://www.xuru.org/rt/ExpR.asp

http://www.jmap.org/JMAP/RegentsExamsandQuestions/PDF/WorksheetsByPI-Topic/Algebra_2_Trigonometry/Statistics_and_Probability/A2.S.7.ExponentialRegression.pdf

https://www.youtube.com/watch?v=TkMQ5n6vWGg



Unit 6: Exponential Functions ENDURING UNDERSTANDINGS

• Real world problems can be represented and solved with exponential functions. ESSENTIAL QUESTIONS KEY CONCEPTS

• How can I use an exponential function to model and solve real world problems?

• How can the properties of exponents be used to illuminate new meaning from an exponential function?

• What are the features of the graph of an exponential function and what do they mean in the context of the problem?

• What are the similarities and differences between linear and exponential functions?

• What do the parameters of an exponential function mean in the context of the problem?

• Exponential equations can be solved in various ways, including equating bases.

• Properties of exponents can be used to manipulate exponential models to reveal information.

• Key features of the graph of an exponential function include the y-intercept and increasing/decreasing behavior.

• Exponential functions grow or decay by an equal factor (as opposed to a constant factor in a linear function).

• Exponential functions are the explicit form of a recursively defined geometric sequence.

• The recursive formula for a geometric sequence uses multiplication and the explicit formula uses exponentiation.

STUDENT FRIENDLY OBJECTIVES ACADEMIC VOCABULARY • TSW use exponential functions to model and solve real world

problems. • TSW manipulate exponential models to highlight additional

information. • TSW graph exponential functions and describe key features. • TSW explain how the change of parameters b and k of kby x +=

transform the graph. • TSW describe the difference between constant and exponential

functions. • TSW solve problems with exponential growth and decay functions. • TSW create recursive and explicit expressions and formulas for

sequences. • TSW develop the formula for the sum of a finite geometric series. • TSW will use exponential regression to fit a curve to data.

exponential function asymptote intercept exponential growth function exponential decay function domain half life finite series infinite series exponential regression

parameter exponent base intervals range base e geometric sequence geometric series exponentiation coefficient of determination



Unit 6: Exponential Functions DEPTH OF KNOWLEDGE







Task: Graph the following function:

xy 3=

Task: Graph each set of equations on the same screen. Describe any similarities and differences among the graphs.

)2(61),2(3,2 xxx yyy ===

Task: Sketch and analyze the graphs of the following two situations. What information can you conclude about the types of growth each type of interest has? Explain you reasoning.

• Lee borrows $9,000 from his mother to buy a car. His mom charges him 5% interest a year, but she does not compound the interest.

• Lee borrows $9,000 from a bank to buy a car. The bank charges 5% interest compounded annually.

Task: Interesting Interest Rates – along with this task, have students research standard banking practices where interest is compounded on a daily basis (see commentary)




Unit 6: Exponential Functions ASSESSMENT RUBRIC






standard) A student can graph an exponential function.

A student can describe the similarities and differences between the graphs of multiple exponential functions.

A student can understand that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

A student can prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.




TEACHER NOTES






















Unit 7: Logarithmic Functions 2010 Standards Comments Resources




Glencoe: throughout



Example How can you use a graph to determine the solution to log(4𝑥 − 3) = 2?


Glencoe: 8-3, Extend 8-6




5 log12�𝑃

100� = 𝑡



Example The isotope bismuth-210 has a half-life of 5 days. The following log function represents the inverse of the exponential decay function:

a) Describe the domain and range of the logarithmic function. b) Describe the end behavior of the function.


Logistic Growth Model, Abstract Version Logistic Growth Model, Explicit Version Modeling London’s Population Howard County Unit 2 Glencoe: 8-3

5 log12�𝑃

100� = 𝑡


the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Example The isotope bismuth-210 has a half-life of 5 days. The following log function represents the inverse of the exponential decay function:

a) What is the y-intercept for the function and what does it mean in context?

b) Describe the end behavior of the function in context.

Key characteristics include but are not limited to intercepts, end behavior, and domain and range. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions.

Exponential Kiss LearnZillion – logarithmic functions Glencoe: 8-3






https://learnzillion.com/lessons/3234-graph-logarithmic-functions





Example The graph of 𝑓(𝑥) = log3 𝑥 is shown below. Write the function rules for 𝑔(𝑥) and 𝑓(𝑥) based on the descriptions given. Then sketch and label the graphs of 𝑔(𝑥) and ℎ(𝑥) on the same coordinate plane.

a) The graph of 𝑔(𝑥) is the translation of the graph of 𝑓(𝑥) to the

right 4 units and up 3 units. b) The graph of ℎ(𝑥) is the reflection of the graph of 𝑓(𝑥) over the x-

axis followed by a translation down two units.


Howard County Unit 2 LearnZillion – shift LearnZillion – stretch and reflect Glencoe: 8-3, supplement

𝑓(𝑥) = 3𝑥

F-BF.B.4. Find inverse functions. a. 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. Example Find the inverse of the function below.

Howard County Unit 2 Glencoe: 8-3


https://learnzillion.com/lessons/3335-shift-exponential-and-logarithmic-functions

https://learnzillion.com/lessons/3342-stretch-and-reflect-exponential-and-logarithmic-functions





F-LE.A.4. For exponential models, express as a 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. Example Shirley has $1000 to invest for 4 years. She wants to double her money during this time. What interest rate does Shirley need for this investment, assuming the interest is compounded continuously?

You don’t need to cover inequalities. Doubling Your Money Carbon 14 Dating Bacteria Populations Algae Blooms Snail Invasions Carbon 14 Dating in Practice II Accuracy of Carbon 14 Dating II Graphene Howard County Unit 2 LearnZillion – F-LE.4 set 1 LearnZillion – F-LE.4 set 2 Glencoe: 8-4, 8-5, 8-6, 8-7, 8-8










https://learnzillion.com/lessonsets/772-express-solution-to-exponential-models-as-a-logarithm-in-base-2-10-or-e

https://learnzillion.com/lessonsets/438-express-solutions-to-exponential-models-as-logarithms-evaluate-using-technology



Unit 7: Logarithmic Functions ENDURING UNDERSTANDINGS

• Real world problems can be represented and solved with logarithmic functions. ESSENTIAL QUESTIONS KEY CONCEPTS

• How do logarithmic functions model real-world problems and their solutions?

• How are exponents and logarithms related? • How are exponential functions and logarithmic functions related? • What are the features of the graph of a logarithmic function and

what do they mean in the context of the problem?

• A logarithmic function is the inverse of an exponential function. • A reflection of the graph of an exponential function over the line

𝑦 = 𝑥 results in the graph of a logarithmic function. • A common logarithm is a logarithm with base 10. • Logarithms and exponents have corresponding properties. • The log key on a calculator evaluates logs in base 10. To evaluate a

different base, the change of base formula must be used. • Logarithms can be used to solve exponential equations and vice versa. • Logarithmic equations can be solved by graphing. • Logarithmic expressions can be condensed to solve problems. • The inverse of the function 𝑦 = 𝑒𝑥 is the natural logarithmic function.

STUDENT FRIENDLY OBJECTIVES ACADEMIC VOCABULARY

• TSW find the inverse of exponential functions • TSW graph logarithmic functions and describe key features in

context. • TSW compare logarithmic functions to the parent function. • TSW convert between exponentials and logarithms. • TSW solve real world problems using logarithms. • TSW condense logarithmic expressions. • TSW find the average rate of change over an interval of a

logarithmic function. • TSW evaluate a logarithmic function using technology.

asymptote change of base formula logarithm logarithm form exponential form common logarithm logarithmic function logarithmic equation natural logarithmic function natural log

inverse function natural base base condense expand logarithm of y with base b logistic growth function base e exponential decay logarithmic scale



Unit 7: Logarithmic Functions DEPTH OF KNOWLEDGE







Task: Write the equation in logarithmic form. 54 = 625

Task: Describe how the graph below is related to the graph of the parent function. Then find the domain, range, and asymptotes. 𝑦 = 3𝑥 + 2

Task: Show that solving the equation 32𝑥 = 4 by taking the common logarithm of each side is equivalent to solving it by taking the logarithm with base 3 of each side.

Task: Doubling Your Money




Unit 7: Logarithmic Functions ASSESSMENT RUBRIC






standard) A student can write a basic exponential equation in logarithmic form.

A student can use logarithms to solve exponential equations.

A student can determine which method of solving a logarithmic equation will be most efficient and use the method to solve.

A student can explain prove a logarithmic function is the inverse of an exponential function.




TEACHER NOTES






















Unit 8: Trigonometric Functions 2010 Standards Comments Resources




Glencoe: throughout



Example The height 𝐻 in feet above the ground of a seat on a carousel can be modeled by 𝐻(𝜃) = −4 cos𝜃 + 4.5.

a) Graph the height of the seat for 0° ≤ 𝜃 ≤ 360°. b) How high was the seat when 𝜃 = 60°? c) When is the seat at its lowest?

source: Explorations in Core Math (p. 649)

Students may be given graphs to interpret or produce graphs given an expression or table for the function, by hand or using technology. Students should be graphing sine, cosine, and tangent functions.

Howard County Unit 7 Glencoe: 13-6, 13-7, 13-8






the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Example Graph one cycle of 𝑔(𝜃) = −3 cos 𝜃

2.

a) What is the period of the function? b) A student said the amplitude of the graph of 𝑔(𝜃) is −3. Describe

and correct the student’s error. source: Explorations in Core Math (p. 615-616)

Key characteristics include but are not limited to period, midline, and amplitude. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Students should be graphing sine, cosine, and tangent functions.

Howard County Unit 7 Glencoe: 13-7, 13-8


Example Compare the period, midline and amplitude for the following functions: Function 1: 𝑔(𝑥) = −2sin 4𝜃 Function 2:

Note this is comparing two functions with different representations. Students should be graphing sine, cosine, and tangent functions.

Glencoe: supplement

𝑓(𝑥)






Example Using 𝑓(𝑥) = sin 𝑥, graph 𝑏(𝑥) = −5 sin 𝑥. Describe the sequence of transformations used to create 𝑏(𝑥).

Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Students should be graphing sine, cosine, and tangent functions.

Howard County Unit 7 Exploring Sinusoidal Functions Trigonometric Identities and Rigid Motions Glencoe: Explore 13-8

Cluster: Extend the domain of trigonometric functions using the unit circle.

F-TF.A.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Example Two students are discussing moving between degrees and radians. Critique and use their arguments in the questions that follow their discussion.

James I remember that 360° = 2𝜋 radians. In finding the radian measure of 120°, I divided 120 by

360 to get 13. Since my angle was 1

3

of the whole circle and there are 2𝜋 radians in a circle, I think that

the angle will be 13 of 2𝜋 which is

2𝜋3

.

Jim If 360° = 2𝜋 radians then if I divide both sides by 360, then 1° = 𝜋

180.

Then 120° times 𝜋180

would be

radian measure, which reduces to 2𝜋3

.

a) Using James’ method, find the radian measure of 30°. b) Using Jim’s method, find the radian measure of 30°. c) Which method do you prefer? Why?


A central angle of a circle intercepts an arc on the same circle. The length of the arc is some fraction of the circumference of the circle and is the basis for discovering the radian measure of an angle. These steps lead the students to the definition of a radian. A radian is the measure of the central angle of a circle that intercepts an arc of the same measure. It is acceptable to let students define 1 radian as about 57°. Through classroom discussion and investigation, (not direct instruction) the instruction should guide students to the more precise conversion factors of 𝟑𝟔𝟎° = 𝟐𝝅 radians and 𝟏𝟖𝟎° = 𝝅 radians. (source: NC Functions)

Howard County Unit 7 LearnZillion – several videos (set 1) LearnZillion – several videos (set 1) Bicycle Wheel What exactly is a radian? Glencoe: 13-2, 13-6







https://learnzillion.com/lessonsets/572-understand-radian-measure-of-an-angle

https://learnzillion.com/lessonsets/492-understand-radian-measure-of-an-angle





F-TF.A.2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as angles traversed counterclockwise around the unit circle.

Example Part 1: Use the coordinates of the intersection point to label the length of each side of the triangle in the diagram below.

Part 2: Use the lengths from part A to find the values of sin𝜃, cos 𝜃, and tan𝜃. Part 3: Explain why sine, cosine, and tangent, as defined in Part 2, are functions. source: Explorations in Core Math (p. 569)

Howard County Unit 7 LearnZillion - several videos (set 1) LearnZillion - several videos (set 2) Shmoop – F-TF.2 Properties of Trigonometric Functions Trigonometric Functions for Arbitrary Angles Trig Functions and the Unit Circle Glencoe: 13-6 (see above and example as well)

y

x


https://learnzillion.com/lessonsets/573-explain-how-the-unit-circle-in-the-coordinate-plane-enables-the-extension-of-trigonometric-functions-to-all-real-numbers

https://learnzillion.com/lessonsets/509-extend-trigonometric-functions-to-all-real-numbers-using-the-unit-circle

http://www.shmoop.com/common-core-standards/ccss-hs-f-tf-2.html






Cluster: Model periodic phenomena with trigonometric functions.

F-TF.B.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Example At midnight the water at a particular beach is at high tide. At the same tiem a gauge at the end of a pier reads 10 feet. Low tide is reached at 6am when the gauge reads 4 feet.

a) Choose which trig function would be the best fit for this model (assuming midnight is 𝑡 = 0). Justify your choice using specific characteristics of trigonometric function graphs.

b) Determine midline, amplitude, and frequency using the above tidal information. You must show all computations and explain why you performed each computation.

c) Write a function based on parts one and two to represent the above tidal information.

d) If the times for high and low tides are reversed what (if anything) would change in the equation from part (c)? Justify your conclusion.

e) If you were instructed to let 𝑡 = 0 represent 9pm, would your function in part (b) still be the most convenient choice? Why or why not? If not, convince your teacher what a better choice would be.


Please note that this standard should be presented and assessed in context. Students should have the ability to select the appropriate trig function to model the problem. Students should be graphing sine, cosine, and tangent functions.

Howard County Unit 7 LearnZillion – F-TF.5 As the Wheel Turns Model Air Plane Acrobatics Foxes and Rabbits 2 Foxes and Rabbits 3 Glencoe: 13-7, 13-8



https://learnzillion.com/lessonsets/574-choose-trigonometric-functions-to-model-periodic-phenomena







Cluster: Prove and apply trigonometric identities. F-TF.C.8. Prove the Pythagorean identity sin2 𝜃 + cos2 𝜃 = 1 and use it to

calculate trigonometric ratios. Example The terminal sides of an angle 𝜃 intersects the unit circle at the point (𝑎, 𝑏), as shown below.

Part 1: Write 𝑎 and 𝑏 in terms of trigonometric functions involving 𝜃. 𝑎 = _______ 𝑏 = _______ Part 2: Apply the Pythagorean Theorem to the right triangle in the diagram. When a trigonometric function is squared, it is common practice to write the exponent immediately after the name of the function. For instance, write (sin𝜃)2 as sin2𝜃.

𝑎2 + 𝑏2 = 𝑐2 Write the Pythagorean Theorem. ___2 + ___2 = ___2 Substitute your values from the figure (Part 1). ___ + ___ = ___ Square each expression. Part 3: The identity is usually written with the sine function first. Write the identity this way, and explain why it is equivalent to the one in Part 2. Part 4: Confirm the Pythagorean identity for 𝜃 = 𝜋

3.

source: Explorations in Core Math (p. 627)

Please note that for regular 3-4, students only need to prove the Pythagorean identity and use it to calculate trig ratios.

Howard County Unit 7 LearnZillion – F-TF.8 Trigonometric Ratios and the Pythagorean Theorem Finding Trig Values Calculations with Sine and Cosine Glencoe: supplement (see above)

y

x


https://learnzillion.com/lessonsets/519-prove-the-pythagorean-identity-and-use-it-to-find-trigonometric-values







Cluster: Define trigonometric ratios and solve problems involving right triangles.

G-SRT.C.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Example Joe strung a 12-yard rope form the bottom of one flagpole to the top of another. He measured the angle of elevation of the rope as 38°. What is the distance between the two flagpoles? source: Shmoop

Note this standard is here for review from Geometry 1-2 and is not an explicit 3-4 standard.

Howard County – scroll down to Right Triangle Trigonometry LearnZillion – G-SRT.8 –several videos Shmoop – G-SRT.8 Ask the Pilot Learnist – G-SRT.8 Glencoe: 13-1

http://www.shmoop.com/common-core-standards/handouts/g-srt_worksheet_8.pdf

https://commoncoregeometry.wikispaces.hcpss.org/Unit+2

https://learnzillion.com/lessonsets/769-solve-problems-using-trigonometric-ratios-and-the-pythagorean-theorem

http://www.shmoop.com/common-core-standards/ccss-hs-g-srt-8.html


http://learni.st/users/60/boards/3453-trig-ratios-and-the-pythagorean-theorem-common-core-standard-9-12-g-srt-8#/users/60/boards/3453-trig-ratios-and-the-pythagorean-theorem-common-core-standard-9-12-g-srt-8



Unit 8: Trigonometric Functions ENDURING UNDERSTANDINGS

• Real world problems can be represented and solved with trigonometric functions. ESSENTIAL QUESTIONS KEY CONCEPTS

• How can you model periodic behavior? • How is the unit circle related to trigonometric ratios? • How can you use the unit circle to prove the Pythagorean identity? • What are the features of the graph of a trigonometric function and

what do they mean in the context of the problem? • What is the relationship between degrees and radians?

• Trigonometric functions can be described in terms of period, amplitude, frequency, and midline.

• Periodic phenomena can be modeled with trigonometric functions. • The unit circle has a radius of 1 unit and its center at the origin of the

coordinate plane. • The unit circle extends trigonometric functions to all real numbers. • The Pythagorean Theorem and the unit circle aid in proving the

Pythagorean identity • Radian measure of an angle is the length of the arc on the unit circle

subtended by the angle. • Trigonometric functions can be translated in the same way as other

functions. STUDENT FRIENDLY OBJECTIVES ACADEMIC VOCABULARY

• TSW use trigonometric ratios to solve problems. • TSW graph trigonometric functions and describe key features in

context. • TSW compare the properties of trigonometric functions represented in

different ways. • TSW convert degree to radian measures and radian to degree measures. • TSW describe how a radian measure is derived. • TSW create a unit circle with all trigonometric functions. • TSW explain how trigonometric graphs are transformed. • TSW prove and apply the Pythagorean identity. • TSW model periodic phenomena with specified amplitude, frequency

and midline using the appropriate trigonometric function.

sine cosine tangent Pythagorean Theorem radian degree Unit circle Pythagorean identity amplitude

cycle period frequency midline periodic function trigonometric function standard position initial side terminal side

coterminal angles central angle arc sine function sine curve cosine function tangent function phase shift



Unit 8: Trigonometric Functions DEPTH OF KNOWLEDGE







Task: Using 𝑓(𝑥) = sin 𝑥, graph 𝑏(𝑥) = −5 sin 𝑥. Describe the sequence of transformations used to create 𝑏(𝑥).

Task: The height 𝐻 in feet above the ground of a seat on a carousel can be modeled by 𝐻(𝜃) =−4 cos 𝜃 + 4.5.

a) Graph the height of the seat for 0° ≤ 𝜃 ≤ 360°.

b) How high was the seat when 𝜃 = 60°?

c) When is the seat at its lowest?

Task: Graph one cycle of 𝑔(𝜃) = −3 cos 𝜃

2.

a) What is the period of the function?

b) A student said the amplitude of the graph of 𝑔(𝜃) is −3. Describe and correct the student’s error.

Task: Model Air Plane Acrobatics




Unit 8: Trigonometric Functions ASSESSMENT RUBRIC






standard) A student can convert a degree measure to a radian measure.

A student can find the exact values of cosine and sine of a radian measure.

A student can use radian measure and arc length to answer problems in context.

A student can describe two different methods for converting between degrees and radians and determine which is more efficient.




TEACHER NOTES
























Unit 9: Statistics 2010 Standards Comments Resources

Our current resources do not address the standards in this unit. The curriculum document is based on Howard County’s Unit 5: Inferences and Conclusions from Data. The standards have been reordered in this unit to represent the flow. Please approach them in this order. For each standard, start by looking at the Lesson attachment on Howard County’s site (see illustration below) then take a look at the other attachments for the standard.

The last two standards are just tasks (see below) and include instructions on how to proceed.

PLEASE NOTE: Students will have a TI-84 like calculator on the entire statistics section of the state assessment. Although it may aid in their understanding, it is not necessary for students to do such calculations as standard deviation by hand. In addition, students would be better served creating box plots, etc., using their graphing calculators.

start here




Cluster: Understand and evaluate random processes underlying statistical experiments.

S-IC.A.1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Example From a class containing 12 girls and 10 boys, three students are to be selected to serve on a school advisory panel. Here are four different methods of making the selection:

I. Select the first three names on the class roll. II. Select the first three students who volunteer.

III. Place the names of the 22 students in a hat, mix them thoroughly, and select three names from the mix.

IV. Select the first three students who show up for class tomorrow. Which is the best sampling method, among these four, if you want the school panel to represent a fair and representative view of the opinions of your class? Explain the weaknesses of the three you did not select as the best. source: Flipbook

Students should come to Algebra 3-4 with a basic understanding of mean, standard deviation, 5 number summaries, and shapes of distribution. For a quick review on these topics, the following LearnZillion videos would be beneficial. Describe data using measures of center and spread Identify a data set’s shape using modes and symmetry

Howard County – Random Rectangles

LearnZillion videos aligned to the lesson:

Compare dot plots using center and spread

Distinguish between population and sample

Take a simple random sample

Take a systematic sample

Take a stratified sample

Take a convenience sample

Take a voluntary sample

Choose a sampling method given a situation

Determine a population and parameter from a statistical question

Make inferences about a population from a sample

Obtain a random sample through simulation

Reduce variation by increasing sample size

**The Moneyball clip for the lesson launch will be posted on my high school website before 4th quarter.

http://www.azed.gov/azccrs/files/2013/11/high-school-ccss-flip-book-usd-259-2012.pdf

https://learnzillion.com/lessons/3886-describe-data-using-measures-of-center-and-spread

https://learnzillion.com/lessons/3886-describe-data-using-measures-of-center-and-spread

https://learnzillion.com/lessons/2402-identify-a-data-sets-shape-using-modes-and-symmetry

https://learnzillion.com/lessons/2402-identify-a-data-sets-shape-using-modes-and-symmetry


https://learnzillion.com/lessons/2588-compare-dot-plots-using-center-and-spread

https://learnzillion.com/lessons/2673-distinguish-between-population-and-sample

https://learnzillion.com/lessons/2716-take-a-simple-random-sample

https://learnzillion.com/lessons/2788-take-a-systematic-sample

https://learnzillion.com/lessons/2789-take-a-stratified-sample

https://learnzillion.com/lessons/2674-take-a-convenience-sample

https://learnzillion.com/lessons/2790-take-a-voluntary-sample

https://learnzillion.com/lessons/2791-choose-a-sampling-method-given-a-situation

https://learnzillion.com/lessons/2511-determine-population-and-parameter-from-a-statistical-question

https://learnzillion.com/lessons/2511-determine-population-and-parameter-from-a-statistical-question

https://learnzillion.com/lessons/2682-make-inferences-about-a-population-from-a-sample

https://learnzillion.com/lessons/2682-make-inferences-about-a-population-from-a-sample

https://learnzillion.com/lessons/2844-obtain-a-random-sample-through-simulation

https://learnzillion.com/lessons/2960-reduce-variation-by-increasing-sample-size

https://sites.google.com/site/dvusdhsmath/home



Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

S-IC.B.3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Example Students in a high school mathematics class decided that their term project would be a study of the strictness of the parents or guardians of students in the school. Their goal was to estimate the proportion of students in the school who thought of their parents or guardians as “strict”. They do not have time to interview all 1000 students in the school, so they plan to obtain data from a sample of students.

a) Describe the parameter of interest and a statistic the students could use to estimate the parameter.

b) Is the best design for this study a sample survey, and experiment, or an observational study? Explain your reasoning.

c) The students quickly realized that, as there is no definition of “strict”, thry could not simply ask a student, “Are your parents or guardians strict?” Write three questions that could provide objective data related to strictness.

d) Describe an appropriate method for obtaining a sample of 100 students, based on your answer in part (a) above.

source: Flipbook

Students should be able to explain techniques/applications for randomly selecting study subjects from a population and how those techniques/applications differ from those used to randomly assign existing subjects to control groups or experimental groups in a statistical experiment.

In statistics, an observational study draws inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator (for example, observing data on academic achievement and socio-economic status to see if there is a relationship between them). This is in contrast to controlled experiments, such as randomized controlled trials, where each subject is randomly assigned to a treated group or a control group before the start of the treatment.

Howard County – Experimental Design **Be sure to incorporate the Lesson Seed: Surveys, Observational Studies, and Experiments as well

Georgia also has an excellent task for this standard that is more comprehensive: click here (go to page 54: We’re Watching You Learning Task)

LearnZillion videos aligned to the standard:

Design a statistical experiment

Select a design method for a statistical experiment

Conduct a survey and choose a sampling method

Develop and conduct a statistical experiment

Develop and conduct an observational study

Select a type of study

Distinguish between observational studies, surveys, and experiments

Determine whether an investigation uses a simple random sample or a systematic sample

Determine the differences between a cluster sample and a stratified sample

Understand sources and types of bias



https://www.georgiastandards.org/Common-Core/Common%20Core%20Frameworks/CCGPS_Math_9-12_AdvancedAlgebra_Unit1SE.pdf

https://learnzillion.com/lessons/3346-design-a-statistical-experiment

https://learnzillion.com/lessons/3262-select-a-design-method-for-a-statistical-experiment

https://learnzillion.com/lessons/3262-select-a-design-method-for-a-statistical-experiment

https://learnzillion.com/lessons/2738-conduct-a-survey-and-choose-a-sampling-method

https://learnzillion.com/lessons/2738-conduct-a-survey-and-choose-a-sampling-method

https://learnzillion.com/lessons/2883-develop-and-conduct-a-statistical-experiment

https://learnzillion.com/lessons/2945-develop-and-conduct-an-observational-study

https://learnzillion.com/lessons/2884-select-a-type-of-study

https://learnzillion.com/lessons/2459-distinguish-between-observational-studies-surveys-and-experiments

https://learnzillion.com/lessons/2459-distinguish-between-observational-studies-surveys-and-experiments

https://learnzillion.com/lessons/2363-determine-whether-an-investigation-uses-a-simple-random-sample-or-a-systematic-sample

https://learnzillion.com/lessons/2363-determine-whether-an-investigation-uses-a-simple-random-sample-or-a-systematic-sample

https://learnzillion.com/lessons/2406-determine-the-differences-between-a-cluster-sample-and-a-stratified-sample

https://learnzillion.com/lessons/2406-determine-the-differences-between-a-cluster-sample-and-a-stratified-sample

https://learnzillion.com/lessons/2507-understand-sources-and-types-of-bias



Avoid bias

S-IC.A.2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. 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?

Possible data-generating processes include (but are not limited to): flipping coins, spinning spinners, rolling a number cube, and simulations using the random number generators. Students may use graphing calculators, spreadsheet programs, or applets to conduct simulations and quickly perform large numbers of trials. The law of large numbers states that as the sample size increases, the experimental probability will approach the theoretical probability. Comparison of data from repetitions of the same experiment is part of the model building verification process. confidence intervals

Howard County – Pediatrician Genders LearnZillion videos aligned to the lesson:

Test a model using samples Test a hypothesis for a population parameter Test a model using a simulation with random numbers Test a model using a simple simulation Test a model against given results Compare theoretical and empirical results Explain and use the Law of Large Numbers Evaluate the effectiveness of a treatment Design a simulation

https://learnzillion.com/lessons/2508-avoid-bias


https://learnzillion.com/lessons/2861-test-a-model-using-samples

https://learnzillion.com/lessons/2931-test-a-hypothesis-for-a-population-parameter

https://learnzillion.com/lessons/3124-test-a-model-using-a-simulation-with-random-numbers

https://learnzillion.com/lessons/3124-test-a-model-using-a-simulation-with-random-numbers

https://learnzillion.com/lessons/3128-test-a-model-using-a-simple-simulation

https://learnzillion.com/lessons/3604-test-a-model-against-given-results

https://learnzillion.com/lessons/2946-compare-theoretical-and-empirical-results

https://www.georgiastandards.org/Common-Core/Common%20Core%20Frameworks/CCGPS_Math_9-12_AdvancedAlgebra_Unit1SE.pdfhttps:/learnzillion.com/lessons/2837-explain-and-use-the-law-of-large-numbers

https://learnzillion.com/lessons/2739-evaluate-the-effectiveness-of-a-treatment

https://learnzillion.com/lessons/2885-design-a-simulation



S-ID.A.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Example The bar graph below gives the birth weight of a population of 100 chimpanzees. The line shows how the weights are normally distributed about the mean, 3250 grams. Estimate the percent of baby chimps weighing 3000-3999 grams.

source: ADE CCSS

Students will need to know the different types of distributions to complete this task (see previous standard). Students may use spreadsheets, graphing calculators, statistical software and tables to analyze the fit between a data set and normal distributions and estimate areas under the curve.

Howard County – Z Tables


Identify situations that fit to normal distributions Estimate population percentages by applying the Empirical Rule Find population percentages by extending the Empirical Rule Find z-scores Find population percentages using z-scores and tables Find population percentages using technology Check whether normal model is appropriate for a data set Apply the Empirical Rule to determine whether a distribution is normal Model a data set with a normal probability distribution Predict intervals and population percentages using the Empirical Rule Predict population percentages using a graphing calculator Solve problems around normal distributions using a graphing calculator



https://learnzillion.com/lessons/2442-identify-situations-that-fit-to-normal-distributions

https://learnzillion.com/lessons/2442-identify-situations-that-fit-to-normal-distributions

https://learnzillion.com/lessons/2636-estimate-population-percentages-by-applying-the-empirical-rule

https://learnzillion.com/lessons/2636-estimate-population-percentages-by-applying-the-empirical-rule

https://learnzillion.com/lessons/2690-find-population-percentages-by-extending-the-empirical-rule

https://learnzillion.com/lessons/2690-find-population-percentages-by-extending-the-empirical-rule

https://learnzillion.com/lessons/2729-find-zscores

https://learnzillion.com/lessons/2892-find-population-percentages-using-zscores-and-tables

https://learnzillion.com/lessons/2892-find-population-percentages-using-zscores-and-tables

https://learnzillion.com/lessons/2899-find-population-percentages-using-technology

https://learnzillion.com/lessons/2437-check-whether-normal-model-is-appropriate-for-a-data-set

https://learnzillion.com/lessons/2437-check-whether-normal-model-is-appropriate-for-a-data-set

https://learnzillion.com/lessons/2312-apply-the-empirical-rule-to-determine-whether-a-distribution-is-normal

https://learnzillion.com/lessons/2312-apply-the-empirical-rule-to-determine-whether-a-distribution-is-normal

https://learnzillion.com/lessons/2392-model-a-data-set-with-a-normal-probability-distribution

https://learnzillion.com/lessons/2392-model-a-data-set-with-a-normal-probability-distribution

https://learnzillion.com/lessons/2664-predict-intervals-and-population-percentages-using-the-empirical-rule

https://learnzillion.com/lessons/2664-predict-intervals-and-population-percentages-using-the-empirical-rule

https://learnzillion.com/lessons/2710-predict-population-percentages-using-a-graphing-calculator






S-IC.B.4. 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. Example Is normal body temperature the same for men and women? Medical researchers interested in this question collected data from a large number of men and women. Random samples from that data are recorded in the table below.

a) Use a 95% confidence interval to estimate the mean body temperature of men.

b) Use a 95% confidence interval to estimate the mean body temperature of women.

c) Find the margin of error for the men and for the women. d) Which margin of error is larger? Why is it larger?

source: Flipbook

Students may use computer generated simulation models based upon sample surveys results to estimate population statistics and margins of error.

Howard County – Confidence Intervals


Determine the likelihood that a hypothesis is reasonable Approximating a sampling distribution from a simulation Approximate a confidence interval for a population proportion Compare two different populations Decrease the size of a confidence interval Distinguish between descriptive and inferential statistics Generate survey data through simulations Interpret margins of error Confidence intervals



https://learnzillion.com/lessons/3099-determine-likelihood-that-a-hypothesis-is-reasonable

https://learnzillion.com/lessons/3099-determine-likelihood-that-a-hypothesis-is-reasonable

https://learnzillion.com/lessons/3103-approximating-a-sampling-distribution-from-a-simulation

https://learnzillion.com/lessons/3103-approximating-a-sampling-distribution-from-a-simulation

https://learnzillion.com/lessons/3193-approximate-a-confidence-interval-for-a-population-proportion

https://learnzillion.com/lessons/3193-approximate-a-confidence-interval-for-a-population-proportion

https://learnzillion.com/lessons/3327-compare-two-different-populations

https://learnzillion.com/lessons/3266-decrease-the-size-of-a-confidence-interval

https://learnzillion.com/lessons/3132-distinguish-between-descriptive-and-inferential-statistics

https://learnzillion.com/lessons/3132-distinguish-between-descriptive-and-inferential-statistics

https://learnzillion.com/lessons/3206-generate-survey-data-through-simulations

https://learnzillion.com/lessons/3207-interpret-margins-of-error

https://learnzillion.com/lessons/3168-confidence-intervals



S-IC.B.5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. Example A potential growth agent is sprayed on the leaves of 12 emerging palm trees twice a week for a month. Another 12 emerging palm trees are not sprayed with the growth agent. The mean and heights of the two groups of palm trees are compared after a month.

a) State the null hypothesis. b) Suppose that the treated palm trees had a mean height that is twice

the mean height of the untreated palm trees. Should the researcher reject the null hypothesis? Does the experimental result prove that the growth agent works? Explain.

Students may use computer generated simulation models to decide how likely it is that observed differences in a randomized experiment are due to chance.

Treatment is a term used in the context of an experimental design to refer to any prescribed combination of values of explanatory variables. For example, one wants to determine the effectiveness of weed killer. Two equal parcels of land in a neighborhood are treated; one with a placebo and one with weed killer to determine whether there is a significant difference in effectiveness in eliminating weeds.

Howard County – Cholesterol Conundrum


Compare experimental treatments by comparing box plots State the null and alternative hypotheses Compare treatments using a resampling strategy **extension on the topic

S-IC.B.6. Evaluate reports based on data. Example A reporter used the two data sets below to calculate the mean housing price in Arizona as $629,000. Why is this calculation not representative of the typical housing price in Arizona?

o King River area {1.2 million, 242000, 265500, 140000, 281000, 265000, 211000}

o Toby Ranch homes {5 million, 154000, 250000, 250000, 200000, 160000, 190000}

source: ADE CCSS

Explanations can include but are not limited to sample size, biased survey sample, interval scale, unlabeled scale, uneven scale, and outliers that distort the line-of-best-fit. In a pictogram the symbol scale used can also be a source of distortion.

As a strategy, collect reports published in the media and ask students to consider the source of the data, the design of the study, and the way the data are analyzed and displayed.

Howard County – Soda vs. Heart Disease

**The article “Soft Drinks and Heart Disease – Critiquing a Statistical Study” be posted in the portal in the Unit 9 folder before 4th quarter.


https://learnzillion.com/lessons/3413-compare-experimental-treatments-by-comparing-box-plots

https://learnzillion.com/lessons/3413-compare-experimental-treatments-by-comparing-box-plots

https://learnzillion.com/lessons/3415-state-the-null-and-alternative-hypotheses

https://learnzillion.com/lessons/3414-compare-treatments-using-a-resampling-strategy

https://learnzillion.com/lessons/3414-compare-treatments-using-a-resampling-strategy





Unit 9: Statistics ENDURING UNDERSTANDINGS

• Statistics are used to summarize and compare sets of data as well as to generalize from a sample taken from a population to the population as a whole.

• A knowledge of statistics is especially useful when evaluating reports in the media and when making and analyzing decisions. ESSENTIAL QUESTIONS KEY CONCEPTS

• Which statistics best describe a given set of data? • What type of data can be described using the normal distribution? • How can the normal distribution be used to estimate probabilities? • How can theoretical and empirical results be used to evaluate the effectiveness

of a treatment? • What are different methods of collecting data and how does the collection

method affect the conclusions that can be drawn? • How can statistics be used to deal with the variability in results from experiments

and inherent randomness? • How can the margin of error be used to find a confidence interval?

• Summary statistics are chosen based on the characteristics of the data distribution. • Statistics is a process for making for making inferences about population parameters

based on a random sample from that population. • Only some data are described well by a normal distribution. • The normal distribution uses area to make estimates of probabilities. • The comparison of theoretical and empirical results can be used to evaluate the

effectiveness of a treatment. • The way in which data is collected determines the scope and nature of the

conclusions that can be drawn from the data. • Statistics is a way to deal with (but not eliminate) variability of results from

experiments and inherent randomness. • The margin of error can be used to find a confidence interval.

STUDENT FRIENDLY OBJECTIVES ACADEMIC VOCABULARY • TSW analyze samples and recognize the importance of randomizing and

replicating to calculate more accurate values. • TSW describe how statistics can be used to make inferences about population

parameters based on a random sample. • TSW design and diagram an experiment and explain the four principles of

experimental design. • TSW compare and contrast surveys, observational studies, and experiments. • TSW determine if a specified model is consistent with results from a simulation. • TSW use sample standard deviation and confidence intervals to determine

feasibility of a sample based on a specific model. • TSW construct a frequency histogram and discover & state the empirical rule

about the normally distributed data. • TSW move between Z-scores, raw scores, and percentile scores. • TSW compare the sample proportion to a null hypothesis. • TSW will construct a bootstrap sampling distribution and use it to find an

approximate confidence interval about a sample proportion. • TSW use data from a randomized experiment to compare two treatments. • TSW evaluate reports based on data.

statistics inference population population parameter sample sample statistic random sample center spread distribution confidence interval sampling distribution inferential statistics skewness bootstrap sampling distribution

mean median 5 number summary standard deviation IQR outlier replication dot plot box plot survey margin of error variability uniformity

randomization observational study experiment control blocking blinding subjects explanatory variable response variable factors frequency histogram normal distribution descriptive statistics variance

empirical rule Z-score percentile scores raw scores normal curve null hypothesis treatment sample mean sample proportion sampling variability shape symmetry peaks



Unit 9: Statistics DEPTH OF KNOWLEDGE







Task: Boxes of Cruncho cereal have a mean mass of 323 g with a standard deviation of 20 g. You choose a random sample of 36 boxes of the cereal. Write the given information for the population and sample below: 𝜇 =_____ 𝜎 =_____ 𝑛 =_____

Task: Boxes of Cruncho cereal have a mean mass of 323 g with a standard deviation of 20 g. You choose a random sample of 36 boxes of the cereal. What interval captures 95% of the means for random samples of 36 boxes?

Task: Boxes of Cruncho cereal have a mean mass of 323 g with a standard deviation of 20 g. You choose a random sample of 36 boxes of the cereal. When you choose a sample of 36 boxes, is it possible for the sample to have a mean mass of 315 g? Is it likely? Explain.

Task: Howard County – Cholesterol Conundrum





Unit 9: Statistics ASSESSMENT RUBRIC






standard) A student can define the four principles of experimental design.

A student can determine the focus and variables of interest of a study.

A student can determine a data collection plan for an experiment.

A student can design a data collection plan for an experiment and describe how particular sampling method can be used to control for confounding variables.




TEACHER NOTES






















Unit 10: Probability 2010 Standards Comments Resources

Our current resources do not address the standards in this unit. The curriculum document is based on Howard County’s Unit 5: Applications of Probability (note this is part of their Geometry course). The standards have been reordered in this unit to represent the flow. Please approach them in this order. For each standard, start by looking at the Lesson attachment on Howard County’s site (see illustration below) then take a look at the other attachments for the standard. The Overview and Starting Points at the top of the page are quite helpful and were used to design this document.

Some of the standards are just tasks or lesson seeds (see below) and include instructions on how to proceed.

Howard County includes several Teacher Professional Development Resources (videos) for Probability. Note that this unit has several + standards that will not be a part of our curriculum for Algebra 3-4 at this time. Please focus on the standards and tasks on the curriculum document.

start here




Cluster: Understand independence and conditional probability and use them to interpret data.

S-CP.A.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). Example Describe the event that the summing two rolled dice is larger than 7 and even, and contrast it with the event that the sum is larger than 7 or even. source: Flipbook

Intersection: The intersection of two sets A and B is the set of elements that are common to both set A and set B. It is denoted by A ∩ B and is read ‘A intersection B.’

Union: The union of two sets A and B is the set of elements, which are in A or in B or in both. It is denoted by A ∪ B and is read ‘A union B.’

Complement: The complement of the set A ∪ B is the set of elements that are members of the universal set U but are not in A ∪ B. It is denoted by (A ∪ B)’

Students need to use correct set notation, with appropriate symbols, to identify sets and subsets. Students should also use Venn diagrams to show relationships between sets within a sample space. Students can also use two-way tables (work with two way tables begins in 8th grade).

Howard County – Unions, Intersections and Complements LearnZillion videos aligned to the lesson:

Organize sample space information Identify the intersection of two events Identify the union of two events Identify the complement of an event Determine quantities belonging to subsets of a sample space Determine the probability of union of several events Determine the probability of intersection of several events Compare the probabilities of the union and intersection of events




https://learnzillion.com/lessons/3957-organize-sample-space-information

https://learnzillion.com/lessons/3957-organize-sample-space-information

https://learnzillion.com/lessons/3958-identify-the-intersection-of-two-events

https://learnzillion.com/lessons/3959-identify-the-union-of-two-events

https://learnzillion.com/lessons/3960-identify-the-complement-of-an-event

https://learnzillion.com/lessons/3961-determine-quantities-belonging-to-subsets-of-a-sample-space

https://learnzillion.com/lessons/3961-determine-quantities-belonging-to-subsets-of-a-sample-space

https://learnzillion.com/lessons/2396-determine-the-probability-of-union-of-several-events

https://learnzillion.com/lessons/2396-determine-the-probability-of-union-of-several-events

https://learnzillion.com/lessons/2343-determine-the-probability-of-intersection-of-several-events

https://learnzillion.com/lessons/2343-determine-the-probability-of-intersection-of-several-events

https://learnzillion.com/lessons/2344-compare-the-probabilities-of-the-union-and-intersection-of-events

https://learnzillion.com/lessons/2344-compare-the-probabilities-of-the-union-and-intersection-of-events



Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model.

S-CP.B.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Example In a math class of 32 students, 18 are boys and 14 are girls. On a unit test, 5 boys and 7 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student? source: Flipbook

Students could use graphing calculators, simulations, or applets to model probability experiments and interpret the outcomes. Two events A and B are mutually exclusive (disjoint) if they have no outcomes in common. These are two events that cannot happen at the same time. The addition rule also applies in this case but the third term 𝑷(𝑨 𝒂𝒏𝒅 𝑩) = 𝟎. Students can also use two-way tables (work with two way tables begins in 8th grade).

Howard County – Addition Rule of Probability LearnZillion videos aligned to the lesson:

Use the Addition Rule to calculate the probability of disjoint event A or B

Nondisjoint events and the Additional Rule

Solve word problems using the addition rule

Calculate probabilities by using the complement and addition rule

Use conditional probability

Calculate different probabilities by working backward

S-CP.A.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Example When rolling two number cubes:

• What is the probability of rolling a sum that is greater than 7? • What is the probability of rolling a sum that is odd? • Are the events, rolling a sum greater than 7, and rolling a sum that

is odd, independent? Justify your response. source: Flipbook

If A and B are independent events, then 𝑷(𝑨 𝒂𝒏𝒅 𝑩) = 𝑷(𝑨) ∙ 𝑷(𝑩). This relationship can also be shown with set notation: 𝑷(𝑨 ∩ 𝑩) = 𝑷(𝑨) ∙ 𝑷(𝑩). Students can also use two-way tables (work with two way tables begins in 8th grade).

Howard County – Sample Space **Be sure to incorporate the Lesson Seed: Independent and Dependent Events


Independence in the real world Determine whether events are independent Determine independence using a formula Calculate the probability of independent events Understand selecting without replacement




https://learnzillion.com/lessons/2524-use-the-addition-rule-to-calculate-the-probability-of-disjoint-event-a-or-b

https://learnzillion.com/lessons/2524-use-the-addition-rule-to-calculate-the-probability-of-disjoint-event-a-or-b

https://learnzillion.com/lessons/2470-nondisjoint-events-and-the-addition-rule

https://learnzillion.com/lessons/2533-solve-word-problems-using-the-addition-rule

https://learnzillion.com/lessons/2537-calculate-probabilities-by-using-the-complement-and-addition-rule

https://learnzillion.com/lessons/2537-calculate-probabilities-by-using-the-complement-and-addition-rule

https://learnzillion.com/lessons/2497-use-conditional-probability

https://learnzillion.com/lessons/2552-calculate-different-probabilities-by-working-backward

https://learnzillion.com/lessons/2552-calculate-different-probabilities-by-working-backward



https://learnzillion.com/lessons/2318-independence-in-the-real-world

https://learnzillion.com/lessons/2646-determine-whether-events-are-independent

https://learnzillion.com/lessons/2597-determine-independence-using-a-formula

https://learnzillion.com/lessons/2653-calculate-the-probability-of-independent-events

https://learnzillion.com/lessons/2653-calculate-the-probability-of-independent-events

https://learnzillion.com/lessons/3039-understand-selecting-without-replacement



S-CP.A.4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example A two-way frequency table is shown below displaying the relationship between age and baldness. We took a sample of 100 male subjects and determined who is or is not bald. We also recorded the age of the male subjects by categories.

a) What is the probability that a man from the sample is bald, given

that he is under 45? b) Are the events independent? Justify your answer.

source: Flipbook

Students may use spreadsheets, graphing calculators, and simulations to create frequency tables and conduct analyses to determine if events are independent or determine approximate conditional probabilities. Students begin work with two-way frequency tables in 8th grade. This standard builds on student work in 8th grade and Algebra 1-2. Two frequency tables can be used to address many of the problems in this unit, not just those listed here. Students should be collecting data to create their own two-way tables. In addition, they should be able to pose a question appropriate for a two-table. This standard ties nicely to the previous unit with appropriate sampling techniques and the summary of results.

Howard County – All the Pets LearnZillion videos aligned to the lesson:

Determine whether events are independent with a two-way table Determine independence using the products of probabilities Identify conditional probability in a two-way table



https://learnzillion.com/lessons/2876-determine-whether-events-are-independent-within-a-twoway-table

https://learnzillion.com/lessons/2876-determine-whether-events-are-independent-within-a-twoway-table

https://learnzillion.com/lessons/3037-determine-independence-using-the-products-of-probabilities

https://learnzillion.com/lessons/3037-determine-independence-using-the-products-of-probabilities

https://learnzillion.com/lessons/3404-identify-conditional-probability-in-a-twoway-table

https://learnzillion.com/lessons/3404-identify-conditional-probability-in-a-twoway-table



S-CP.A.3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Example A die is thrown twice. Determine the probability that the sum of the rolls is less than 4 given that:

a) At least one of the rolls is a 1. b) The first roll is a 1.

source: Flipbook

𝑷(𝑩|𝑨) =𝑷(𝑨 𝒂𝒏𝒅 𝑩)

𝑷(𝑨)

When the occurrence of one event affects how a second event can occur, the events are dependent. The probability that an event, B, will occur given that another event, A, has already occurred is called a conditional probability. Conditional Probability For any two events A and B with 𝑷(𝑨) ≠ 𝟎,

Howard County – Conditional Probability LearnZillion videos aligned to the lesson:

Understand conditional probability Compute conditional probability using a formula Determine independence using conditional probability Interpret conditional probabilities Identify conditional probability in a tree diagram Interpret independent variables using conditional probabilities Determine independence or association of two events Determine independence by comparing conditional probability to simple probability Show that associated events are not independent by comparing simple and conditional probability Determine whether events A and B are independent using complements



https://learnzillion.com/lessons/2365-understand-conditional-probability

https://learnzillion.com/lessons/3427-compute-conditional-probability-using-a-formula

https://learnzillion.com/lessons/3427-compute-conditional-probability-using-a-formula

https://learnzillion.com/lessons/3360-determine-independence-using-conditional-probability

https://learnzillion.com/lessons/3360-determine-independence-using-conditional-probability

https://learnzillion.com/lessons/3255-interpret-conditional-probabilities

https://learnzillion.com/lessons/3271-identify-conditional-probability-in-a-tree-diagram

https://learnzillion.com/lessons/3271-identify-conditional-probability-in-a-tree-diagram

https://learnzillion.com/lessons/3336-interpret-independent-variables-using-conditional-probabilities

https://learnzillion.com/lessons/3336-interpret-independent-variables-using-conditional-probabilities

https://learnzillion.com/lessons/2321-determine-independence-or-association-of-two-events

https://learnzillion.com/lessons/2321-determine-independence-or-association-of-two-events

https://learnzillion.com/lessons/2320-determine-independence-by-comparing-conditional-probability-to-simple-probability

https://learnzillion.com/lessons/2320-determine-independence-by-comparing-conditional-probability-to-simple-probability

https://learnzillion.com/lessons/2397-show-that-associated-events-are-not-independent-by-comparing-simple-and-conditional-probability



https://learnzillion.com/lessons/2330-determine-whether-events-a-and-b-are-independent-using-complements

https://learnzillion.com/lessons/2330-determine-whether-events-a-and-b-are-independent-using-complements



S-CP.A.5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Example Use the data given in the table, determine if owning a smart phone is independent from grade level.

source: Flipbook

Howard County – Lesson Seed – Conditional Probability and Independence LearnZillion videos aligned to the lesson:

Understand conditional probability using scenarios

Calculate conditional probabilities

Determine dependence and independence by comparing scenarios

Prove independence numerically using the independence formula

Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model.

S-CP.B.6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Example A teacher gave her class two quizzes. 30% of the class passed both quizzes and 60% of the class passed the first quiz. What percent of those who passes the first quiz also passed the second quiz? source: Flipbook

Students could use graphing calculators, simulations, or applets to model probability experiments and interpret the outcomes. Identifying that a probability is conditional when the word “given” is not stated can be very difficult for students.

Howard County – Common Characteristics Howard County – Favorite Baseball Teams (S-CP.4) LearnZillion videos aligned to the lesson:

Understand conditional probability

Calculate conditional probabilities using a two-way table




https://learnzillion.com/lessons/2446-understand-conditional-probability-using-scenarios

https://learnzillion.com/lessons/2446-understand-conditional-probability-using-scenarios

https://learnzillion.com/lessons/2447-calculate-conditional-probabilities

https://learnzillion.com/lessons/2575-determine-dependence-and-independence-by-comparing-scenarios

https://learnzillion.com/lessons/2575-determine-dependence-and-independence-by-comparing-scenarios

https://learnzillion.com/lessons/2496-prove-independence-numerically-using-the-independence-formula







https://learnzillion.com/lessons/2553-calculate-conditional-probabilities-using-a-twoway-table

https://learnzillion.com/lessons/2553-calculate-conditional-probabilities-using-a-twoway-table



Unit 10: Probability ENDURING UNDERSTANDINGS

• Probability is indispensable for analyzing data; data are indispensable for estimating probabilities. ESSENTIAL QUESTIONS KEY CONCEPTS

• How can a sample space be designed to represent possible outcomes?

• How can Venn diagrams and two-way tables be used to determine probabilities of compound events? How can these tools be used to determine if events are independent?

• What strategies might be used to find the probability of mutually exclusive (disjoint) events? Mutually inclusive (overlapping) events?

• How can you find the probabilities of complements, unions, and intersections?

• What is conditional probability? How can you find the conditional probabilities?

• What does it mean for two events to be independent in the context of the problem?

• When should the addition rule be used? What do the results mean in the context of the problem?

• The intersection of two sets A and B is the set of elements that are common to both set A and set B. 𝐴 ∩ 𝐵

• The union of two sets A and B is the set of elements, which are in A or in B or in both. 𝐴 ∪ 𝐵

• The complement of the set 𝐴 ∪ 𝐵 is the set of elements that are members of the universal set U but are not in 𝐴 ∪ 𝐵. (𝐴 ∪ 𝐵)′

• As the number of trials increases the experimental probability will approach the theoretical probability.

• When the occurrence of one event affects how a second event can occur, the events are dependent events. Otherwise they are independent.

• Two events that cannot happen at the same time are mutually exclusive (disjoint) events.

• When two events are not mutually exclusive you need to subtract the probability of the common outcomes to find 𝑃(𝐴 𝑜𝑟 𝐵).

• The probability that an event, B, will occur given that another event, A, has already occurred is called a conditional probability.

• Two way frequency tables can be used to determine independence and calculate conditional probabilities.

STUDENT FRIENDLY OBJECTIVES ACADEMIC VOCABULARY • TSW determine the probabilities of event(s), including unions,

intersections, and complements. • TSW apply the addition rule of probability of probability to

successfully solve different scenarios. • TSW determine if events are mutually exclusive or not and us this

information to solve problems. • TSW create a probability problem that applies to the addition rule. • TSW determine if events are independent using probabilities. • TSW find conditional probabilities using Venn diagrams and two-

way tables.

event outcome sample space union intersection complement mutually exclusive independent events dependent events universal set

disjoint overlap mutually inclusive tree diagram two-way table conditional probability compound events probability element



Unit 10: Probability DEPTH OF KNOWLEDGE







Task: In a math class of 32 students, 18 are boys and 14 are girls. On a unit test, 5 boys and 7 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student?

Task: A die is thrown twice. Determine the probability that the sum of the rolls is less than 4 given that: a) At least one of the rolls is a 1. b) The first roll is a 1.

Task: A two-way frequency table is shown below displaying the relationship between age and baldness. We took a sample of 100 male subjects and determined who is or is not bald. We also recorded the age of the male subjects by categories.

a) What is the probability that a man from the

sample is bald, given that he is under 45? b) Are the events independent? Justify your

answer.

Task: But Mango is My Favorite




Unit 10: Probability ASSESSMENT RUBRIC






standard) A student can define conditional probability.

A student can use two way tables to find conditional probabilities.

A student can use conditional probability to determine if two events are independent.

A student can explain what conditional probability and independence means in the context of the problem and use the information to solve the problem.




TEACHER NOTES






















ASSESSMENT LIMITS FOR STANDARDS ASSESSED ON MORE THAN ONE END-OF-COURSE TEST: AI-G-AII PATHWAY Table 1. This draft table shows assessment limits for standards assessed on more than one end-of-course test.

ACCS-M Cluster ACCS-M Key ACCS-M Standard Algebra I Assessment Limits and

Clarifications Algebra II Assessment Limits and Clarifications

Reason quantitatively and use units to solve problems

N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.

This standard will be assessed in Algebra I by ensuring that some modeling tasks (involving Algebra I content or securely held content from grades 6-8) require the student to create a quantity of interest in the situation being described (i.e., a quantity of interest is not selected for the student by the task). For example, in a situation involving data, the student might autonomously decide that a measure of center is a key variable in a situation, and then choose to work with the mean.

This standard will be assessed in Algebra II by ensuring that some modeling tasks (involving Algebra II content or securely held content from previous grades and courses) require the student to create a quantity of interest in the situation being described (i.e., this is not provided in the task). For example, in a situation involving periodic phenomena, the student might autonomously decide that amplitude is a key variable in a situation, and then choose to work with peak amplitude.

Interpret the structure of expressions

A-SSE.2 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 a difference of squares that can be factored as (x2 – y2)(x2 + y2).

i) Tasks are limited to numerical expressions and polynomial expressions in one variable. ii) Examples: Recognize 532 − 472 as a difference of squares and see an opportunity to rewrite it in the easier-to-evaluate form (53+47)(53−47). See an opportunity to rewrite a2 + 9a + 14 as (a+7)(a+2).

i) Tasks are limited to polynomial, rational, or exponential expressions. ii) Examples: see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). In the equation x2 + 2x + 1 + y2 = 9, see an opportunity to rewrite the first three terms as (x+1)2, thus recognizing the equation of a circle with radius 3 and center (−1, 0). See (x2 + 4)/(x2 + 3) as ( (x2+3) + 1 )/(x2+3), thus recognizing an opportunity to write it as 1 + 1/(x2 + 3).

Write expressions in equivalent forms to solve problems

A-SSE.3c Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.↔ (c) Use the properties of exponents to transform expressions for exponential

i) Tasks have a real-world context. As described in the standard, there is an interplay between the mathematical structure of the expression and the structure of the situation such that choosing and producing an equivalent

i) Tasks have a real-world context. As described in the standard, there is an interplay between the mathematical structure of the expression and the structure of the situation such that choosing and producing an equivalent





functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

form of the expression reveals something about the situation. ii) Tasks are limited to exponential expressions with integer exponents.

form of the expression reveals something about the situation. ii) Tasks are limited to exponential expressions with rational or real exponents.

Understand the relationship between zeros and factors of polynomials

A-APR.3 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.

i) Tasks are limited to quadratic and cubic polynomials in which linear and quadratic factors are available. For example, find the zeros of (x - 2)(x2 - 9).

i) Tasks include quadratic, cubic, and quartic polynomials and polynomials for which factors are not provided. For example, find the zeros of (x2 - 1)( x2 + 1)

Create equations that describe numbers or relationships

A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

i) Tasks are limited to linear, quadratic, or exponential equations with integer exponents.

i) Tasks are limited to exponential equations with rational or real exponents and rational functions.

ii) Tasks have a real-world context.

Understand solving equations as a process of reasoning and explain the reasoning

A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

i) Tasks are limited to quadratic equations.

i) Tasks are limited to simple rational or radical equations.

Solve equations and inequalities in one variable

A-REI.4b Solve quadratic equations in one variable. b) Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

i) Tasks do not require students to write solutions for quadratic equations that have roots with nonzero imaginary parts. However, tasks can require the student to recognize cases in which a quadratic equation has no real solutions. Note, solving a quadratic equation by factoring relies on the connection between zeros and factors of polynomials (cluster A-APR.B). Cluster

i) In the case of equations that have roots with nonzero imaginary parts, students write the solutions as a ± bi for real numbers a and b.





A-APR.B is formally assessed in A2.

Solve systems of equations

A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

i) Tasks have a real-world context. ii) Tasks have hallmarks of modeling as a mathematical practice (less defined tasks, more of the modeling cycle, etc.).

i) Tasks are limited to 3x3 systems.

Represent and solve equations and inequalities graphically

A-REI.11 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.

i) Tasks that assess conceptual understanding of the indicated concept may involve any of the function types mentioned in the standard except exponential and logarithmic functions. ii) Finding the solutions approximately is limited to cases where f(x) and g(x) are polynomial functions.

i) Tasks may involve any of the function types mentioned in the standard.

Understand the concept of a function and use function notation

F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

i) This standard is part of the Major work in Algebra I and will be assessed accordingly.

i) This standard is Supporting work in Algebra II. This standard should support the Major work in F-BF.2 for coherence.

Interpret functions that arise in applications in terms of a context

F-IF.4 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;

i) Tasks have a real-world context. ii) Tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. Compare note (ii) with standard F-IF.7.

i) Tasks have a real-world context ii) Tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. Compare note (ii) with standard F-IF.7. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.6 and F-IF.9.





symmetries; end behavior; and periodicity.

The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6 and F-IF.9.

Interpret functions that arise in applications in terms of a context

F-IF.6 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.

i) Tasks have a real-world context. ii) Tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. The function types listed here are the same as those listed in the Algebra I column for standards F-IF.4 and F-IF.9.

i) Tasks have a real-world context. ii) Tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.4 and F-IF.9.

Analyze functions using different representations

F-IF.9 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.

i) Tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. The function types listed here are the same as those listed in the Algebra I column for standards F-IF.4 and F-IF.6.

i) Tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.4 and F-IF.6.

Build a function that models a relationship between two quantities

F-BF.1a Write a function that describes a relationship between two quantities. a) Determine an explicit expression, a recursive process, or steps for calculation from a context.

i) Tasks have a real-world context. ii) Tasks are limited to linear functions, quadratic functions, and exponential functions with domains in the integers.

i) Tasks have a real-world context ii) Tasks may involve linear functions, quadratic functions, and exponential functions.





Build new functions from existing functions

F-BF.3 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.

i) Identifying 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) is limited to linear and quadratic functions. ii) Experimenting with cases and illustrating an explanation of the effects on the graph using technology is limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. iii) Tasks do not involve recognizing even and odd functions. The function types listed in note (ii) are the same as those listed in the Algebra I column for standards F-IF.4, F-IF.6, and F-IF.9.

i) Tasks may involve polynomial, exponential, logarithmic, and trigonometric functions ii) Tasks may involve recognizing even and odd functions. The function types listed in note (i) are the same as those listed in the Algebra II column for standards F-IF.4, F-IF.6, and F-IF.9.

Construct and compare linear, quadratic, and exponential models and solve problems

F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

i) Tasks are limited to constructing linear and exponential functions in simple context (not multi-step).

i) Tasks will include solving multi-step problems by constructing linear and exponential functions.

Interpret expressions for functions in terms of the situation they model

F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.

i) Tasks have a real-world context. ii) Exponential functions are limited to those with domains in the integers.

i) Tasks have a real-world context. ii) Tasks are limited to exponential functions with domains not in the integers.

Summarize, represent, and interpret data on two categorical and quantitative variables

S-ID.6a Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a) Fit a function to the data; use functions fitted to data to solve

i) Tasks have a real-world context.

ii) Exponential functions are limited to those with domains in the integers.

i) Tasks have a real-world context.

ii) Tasks are limited to exponential functions with domains not in the integers and trigonometric functions.





problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.