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Curriculum Map
2014 - 2015
Mathematics Florida Standards Volusia County Curriculum Maps are revised annually and updated throughout the year.
The learning goals are a work in progress and may be modified as needed.
Mathematics Florida State Standards Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them. (MAFS.K12.MP.1)
Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process which sometimes requires
perseverance, flexibility, and a bit of ingenuity.
2. Reason abstractly and quantitatively. (MAFS.K12.MP.2)
The concrete and the abstract can complement each other in the development of mathematical understanding: representing a concrete situation
with symbols can make the solution process more efficient, while reverting to a concrete context can help make sense of abstract symbols.
3. Construct viable arguments and critique the reasoning of others. (MAFS.K12.MP.3)
A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and supporting evidence.
4. Model with mathematics. (MAFS.K12.MP.4)
Many everyday problems can be solved by modeling the situation with mathematics.
5. Use appropriate tools strategically. (MAFS.K12.MP.5)
Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen mathematical understanding.
6. Attend to precision. (MAFS.K12.MP.6)
Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical explanations.
7. Look for and make use of structure. (MAFS.K12.MP.7)
Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea.
8. Look for and express regularity in repeated reasoning. (MAFS.K12.MP.8)
Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results more quickly and
efficiently.
M/J Pre-Algebra, Advanced: Mathematics Florida State Standards
In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
(1) Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m(A). Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade, fitting the model, and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and y-intercept) in terms of the situation. Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and concept of logical equivalence, they maintain the solutions of the original equation. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems. (2) Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations. (3) Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work on volume by solving problems involving cones, cylinders, and spheres.
Fluency Recommendations 8.EE.7 Students have been working informally with one-variable linear equations since as early as kindergarten. This important line of development culminates in grade 8 with the solution of general one-variable linear equations, including cases with infinitely many solutions or no solutions as well as cases requiring algebraic manipulation using properties of operations. Coefficients and constants in these equations may be any rational numbers. 8.G.9 When students learn to solve problems involving volumes of cones, cylinders and spheres — together with their previous grade 7 work in angle measure, area, surface area and volume (7.G.4–6) — they will have acquired a well-developed set of geometric measurement skills. These skills, along with proportional reasoning (7.RP) and multistep numerical problem solving (7.EE.3), can be combined and used in flexible ways as part of modeling during high school — not to mention after high school for college and careers.
The following English Language Arts LAFS should be taught throughout the course: LAFS.68.RST.1.3: Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks. LAFS.68.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific
scientific or technical context relevant to grades 6–8 texts and topics. LAFS.68.RST.3.7: Integrate quantitative or technical information expressed in words in a text with a version of that information expressed
visually (e.g., in a flowchart, diagram, model, graph, or table). LAFS.68.WHST.1.1: Write arguments focused on discipline-specific content. LAFS.68.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. LAFS.8.SL.1.1: Engage effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on
grade 8 topics, texts, and issues, building on others’ ideas and expressing their own clearly. LAFS.8.SL.1.2: Analyze the purpose of information presented in diverse media and formats (e.g., visually, quantitatively, orally) and evaluate
the motives (e.g., social, commercial, political) behind its presentation. LAFS.8.SL.1.3: Delineate a speaker’s argument and specific claims, evaluating the soundness of the reasoning and relevance and sufficiency
of the evidence and identifying when irrelevant evidence is introduced. LAFS.8.SL.2.4: Present claims and findings, emphasizing salient points in a focused, coherent manner with relevant evidence, sound valid reasoning, and well-chosen details; use appropriate eye contact, adequate volume, and clear pronunciation.
M/J Pre-Algebra, Advanced: Mathematics Florida State Standards At A Glance
First Quarter Second Quarter Third Quarter Fourth Quarter
DSA
Unit 1- Real Number System MAFS.8.NS.1.1 MAFS.8.NS.1.2 Unit 2-Solving Equations Prior Knowledge of 1&2 Step Equations MAFS.8.EE.3.7
DIA - Unit 1 & 2 Unit 3- Linear Functions MAFS.8.F.1.1 MAFS.8.F.1.2 MAFS.8.F.1.3 MAFS.8.F.2.4 MAFS.8.F.2.5
DIA - Unit 3
Unit 4-Linear & Non-Linear Models & Tables MAFS.8.EE.2.5 MAFS.8.EE.2.6 MAFS.8.SP.1.1 MAFS.8.SP.1.2 MAFS.8.SP.1.3 MAFS.8.SP.1.4
DIA - Unit 4 Unit 5-Solving Systems of Equations MAFS.8.EE.3.8
SSA
Unit 6- Exponents MAFS.8.EE.1.1 MAFS.8.EE1.2 MAFS.8.EE1.3 MAFS.8.EE1.4 Unit 7-Properties of Exponents MAFS.912.N-RN.1.1 MAFS.912.N-RN.1.2 MAFS.912.F-IF.3.8b
DIA - Unit 6 & 7 Unit 8- Angles/Pythagorean Theorem MAFS.8.G.1.5 MAFS.8.G.2.6 MAFS.8.G.2.7 MAFS.8.G.2.8
DIA - Unit 8
Unit 9-Transformations/Volume MAFS.8.G.1.1 MAFS.8.G.1.2 MAFS.8.G.1.3 MAFS.8.G.1.4 MAFS.8.G.3.9
DIA - Unit 9 Unit 10- Polynomials MAFS.912.A-SSE.1.1 MAFS.912.A-SSE.1.2 MAFS.912.A-APR.1.1
STATE EOC
Course: Pre-Algebra Advanced
Unit One: Real Numbers
Essential Question(s): In what ways can rational numbers be useful?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.NS.1.1 Know that numbers that are not rational are call irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. SMP # 2
classify a number as a rational or irrational based on its decimal expansion.
convert a repeating decimal into a rational number.
Prior Knowledge http://www.cpalms.org/Resources/PublicPreviewResource9955.aspx http://www.cpalms.org/Resources/PublicPreviewResource35460.aspx http://www.cpalms.org/Resources/PublicPreviewResource8647.aspx http://www.riversidepublishing.com/commoncore/pdf/A2KMathCCSItemSaMPler.pdf http://www.khanacademy.org/math/arithmetic/absolute-value/abs_value_tutorial/v/absolute-value-1 http://www.khanacademy.org/math/recreational-math/vi-hart/vi-cool-stuff/v/what-was-up-with-pythagoras http://www.khanacademy.org/math/arithmetic/decimals/percent_tutorial/v/converting-decimals-to-percents--ex-1 http://www.khanacademy.org/math/arithmetic/decimals/percent_tutorial/e/converting_percents_to_decimals
MAFS.8.NS.1.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g. π2). SMP # 6
use reasoning to determine between which two consecutive whole numbers a square root will fall.
plot the estimated value of an irrational number on a number line.
estimate the value of an irrational number by rounding to a specific place value.
use estimated values to compare two or more irrational numbers.
Prior knowledge
Course: Pre-Algebra Advanced
Unit Two: Solving Equations /Intro to Functions
Essential Question(s): How can algebraic equations be used to model, analyze, and solve mathematical situations?
How are functions useful? Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.EE.3.7a Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x=a, a=a or a=b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
SMP # 7
apply the properties of real numbers to determine the solution of a linear equation.
give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.
show which of these possibilities is the case by successfully transforming the given equation into simpler forms, until an equivalent equation of the form x =a , a = a, or a = b results (where a and b are different numbers).
simplify a linear equation by using the distributive property and/or combining like terms.
solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Examples of Opportunities for In-Depth Focus This is a culminating standard for solving one-variable linear equations. Students may think that checking solutions is redundant and useless. Disregarding signs when manipulating expressions. Students often forget to verify that their answer makes sense in the original context of the problem.
Students will sometimes combine unlike terms such as 2x + 5 combines to 7x. Students will sometimes forget to multiply the outside number by both numbers in the parentheses with distributive property. While students may understand to distribute the coefficient, they sometimes will forget to distribute the negative sign.
http://map.mathshell.org/materials/lessons.php?gradeid=23 http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5022&t=9120&id=11342 http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5022&t=9120&id=11985 http://map.mathshell.org/materials/download.php?fileid=1286
http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=1008&t=9449&id=16901
Course: Pre-Algebra Advanced
Unit Three: Linear Functions
Essential Question(s): How can algebraic equations be used to model, analyze, and solve mathematical situations?
How are functions useful? Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.F.1.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is a set of ordered pairs consisting of an input and the corresponding output. SMP # 3
explain that a function represents a relationship between an input and an output where the output depends on the input; therefore, there can only be one output for each input.
show the relationship between the inputs and outputs of a function by graphing them as ordered pairs on a coordinate grid.
Examples of Opportunities for In-Depth Focus When students work toward meeting this standard, they build on what they know about two-variable linear equations, and they enlarge the varieties of real-world and mathematical problems they can solve. Students sometimes consider an equations such as x = 10 a linear function. This is a linear equation, but it is not a linear function because there are infinite output values for the input value of 10.
http://www.cpalms.org/Resources/PublicPreviewResource7034.aspx http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5022&t=9121&id=11450
http://www.youtube.com/watch?v=VUTXsPFx-qQ&safety_mode=true&persist_safety_mode=1&safe=active http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=1008&t=9449&id=16899
Course: Pre-Algebra Advanced
Unit Three: Linear Functions Continued
Essential Question(s): How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations?
How are functions useful? Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.F.1.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). SMP# 7
determine the properties of a function written in algebraic form (e.g., rate of change, meaning of y-intercept, linear, non-linear).
determine the properties of a function when given the inputs and outputs in a table.
determine the properties of a function represented as a graph.
determine the properties of a function when given the situation verbally.
compare the properties of two functions that are represented differently (e.g., as an equation, in a table, graphically, or a verbal representation).
Examples of Opportunities for In-Depth Focus
Work toward meeting this standard repositions previous work with tables and graphs in the new context of input/output rules. For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Students may not relate that all three forms are representing the same information. Students may not understand that the x and y on a table are the ordered pairs on the graph. Students may have difficulty with the vocabulary words and may need to be reminded of them.
http://education.ti.com/calculators/timathnspired/US/Activities/Detail?id=11347&sa=5022&t=5030 http://www.cpalms.org/Resources/PublicPreviewResource8617.aspx
Course: Pre-Algebra Advanced
Unit Three: Linear Functions Continued
Essential Question(s): How can algebraic equations be used to model, analyze, and solve mathematical situations?
How are functions useful? How can you describe the graph of the equation y =mx + b?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.F.1.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. SMP #3
explain why the equation y = mx + b represents a linear function and interpret the slope and y-intercept in relation to the function.
give examples of relationships that are non-linear functions.
analyze the rate of change between input and output values to determine if function is linear or non-linear.
create a table of values that can be defined as a non-linear function.
For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Some students will mistakenly think of a straight line as horizontal or vertical only. Some students will mix up x- and y-axes on the coordinate plane, or mix up the ordered pairs.
http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=8592&MICROSITE=ACTIVITYEXCHANGE http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=5488&MICROSITE=ACTIVITYEXCHANGE http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=9630&MICROSITE=ACTIVITYEXCHANGE
http://education.ti.com/calculators/downloads/US/Activities/Detail?id=4784&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fAdvanced%3fs%3d5022%26d%3d1%26sort%3dRecommendedAllTime%26size%3d15%26page%3d47
Course: Pre-Algebra Advanced
Unit Three: Linear Functions Continued
Essential Question(s): How can algebraic equations be used to model, analyze, and solve mathematical situations?
How are functions useful? How can you describe the graph of the equation y =mx + b?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.F.2.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
SMP #1
write a linear function that models a given situation given verbally, as a table of x and y values, or as a graph.
define the initial value of the function in relation to the situation.
define the rate of change in relation to the situation.
define the y-intercept in relation to the situation.
explain any constraints on the domain in relation to the situation.
Some students may not pay attention to the scale on a graph, assuming that the scale units are always “one”. When making axes for a graph, some students may not using equal intervals to create the scale. Some students graph incorrectly because they don‘t understand that x usually represents the independent variable and y represents the dependent variable.
http://departments.jordandistrict.org/curriculum/mathematics/secondary/iMPact/Algebra/Alg%205%20Linear%20patterns%20and%20functions/Alg%205.1a%204%20problems%20TGSE.pdf http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=1008&t=9450&id=17166 http://www.regentsprep.org/Regents/math/ALGEBRA/AD4/indexAD4.htm www.ixl.com http://www.mathsisfun.com/data/scatter-xy-plots.html
http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=4125&MICROSITE=ACTIVITYEXCHANGE
MAFS.8.F.2.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
SMP #4
match the graph of function to a given situation.
write a story that describes the functional relationship between two variables depicted on a graph.
create a graph of function that describes the relationship between two variables.
NOTE: The notion of a function is introduced in Grade 8. Linear functions are a major focus, but note that students are also expected to give examples of non-functions. The main focus in Grade 8 is linear functions, those of the form y = mx + b, where m and b are constants. Formal language, such as domain and range, and function notation may be postponed until High school.
Course: Pre-Algebra Advanced
Unit Four: Linear Models and Tables
Essential Question(s): How can algebraic equations be used to model, analyze, and solve mathematical situations?
How are functions useful? How can you describe the graph of the equation y = mx + b?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.EE.2.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. SMP #8
graph a proportional relationship in the coordinate plane.
interpret the unit rate of a proportional relationship as the slope of a graph.
justify that the graph of a proportional relationship will always intersect the origin (0, 0) of the graph.
use a graph, a table, or an equation to determine the unit rate of a proportional relationship and use the unit rate to make comparisons between various proportional relationships.
Examples of Opportunities for In-Depth Focus When students work toward meeting this standard, they build on grades 6–7 work with proportions and position themselves for grade 8 work with functions and the equation of a line. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Students may still have difficulty finding slope. They can still mix up the relationship between change of y and change of x. For example: dividing the change in x by the change in y subtracting the x coordinate from the y coordinate subtracting the x and y coordinates in different order.
http://zonalandeducation.com/mmts/functionInstitute/linearFunctions/new/lsif0.php http://www.ixl.com/math/grade-8 http://www.zweigmedia.com/RealWorld/tutorialsf0/framesLA.html http://www.mathopenref.com/linearexplorer.html http://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/graphing_with_intercepts/v/x--and-y-intercepts http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=5028&MICROSITE=ACTIVITYEXCHANGE
MAFS.8.EE.2.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. SMP #1
create right triangles by drawing a horizontal line segment and a vertical line segment from any two points on a non-vertical line in the coordinate plane.
justify that these right triangles are similar by comparing the ratios of the lengths of the corresponding legs.
justify that since the triangles are similar, the ratios of all corresponding hypotenuses, representing the slope of the line, will be equivalent.
justify that an equation in the form of y = mx will represent the graph of a proportional relationship with a slope of m and a y-intercept of 0.
justify that an equation in the form of y = mx represents the graph of a linear relationship with a slope of m and a y-intercept of b.
Students must be sure to use corresponding sides of triangles when the figures are turned around.
Course: Pre-Algebra Advanced
Unit Four: Linear Models and Tables Continued
Essential Question(s): How can algebraic equations be used to model, analyze, and solve mathematical situations?
How are functions useful? How can you use statistical displays to analyze, interpret and predict data?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.SP.1.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
SMP#4
plot ordered pairs on a coordinate grid representing the relationship between two data sets.
describe patterns in the plotted points such as clustering, outliers, positive or negative association, and linear or non-linear association and describe the pattern in the context of the measurement data.
interpret the patterns of association in the context of the data sample.
When trying to decide correlation of data set students will sometimes have confusion about whether there is positive, negative, or no correlation.
http://education.ti.com/calculators/downloads/US/Activities/Detail?id=5786 http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=4197&MICROSITE=ACTIVITYEXCHANGE
MAFS.8.SP.1.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
SMP#3
recognize whether or not data plotted on a scatter plot have a linear association.
draw a straight trend line to approximate the linear relationship between the plotted points of two data sets.
make inferences regarding the reliability of the trend line by noting the closeness of the data points to the line.
NOTE: In Grade 8, students plot bivariate data in the coordinate plane (by hand or electronically) and use linear functions to analyze the relationship between two paired variables
Students think the line of best fit must go through (0,0). This is sometimes true, but students need to be able to understand what the y-intercept means in the situation and determine if it makes sense given the situation. In general, students think there is only one correct answer in mathematics. Students may mistakenly think their lines of best fit for the same set of data will be exactly the same. Because students are informally drawing lines of best fit, the lines will vary slightly. Students will often times only chose data points that are plotted on the graph in order to find the line of best fit. Sometimes this will work because their line will go through two of the original data points. However, often times the line may NOT go through any of the data points and students still pick only the plotted points.
Course: Pre-Algebra Advanced
Unit Four: Linear Models and Tables Continued
Essential Question(s): How can functions be used to model, analyze, and solve mathematical situations?
How are functions useful? How can you use statistical displays to analyze, interpret and predict data?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.SP.1.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. SMP#2
determine the equation of the trend line that approximates the linear relationship between the plotted points of two data sets.
interpret the y-intercept of the equation in the context of the collected data.
interpret the slope of the equation in the context of the collected data.
use the equation of the trend line to summarize the given data and make predictions regarding additional data points.
For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Students think the line of best fit must go through (0,0). This is sometimes true, but students need to be able to understand what the y-intercept means in the situation and determine if it makes sense given the situation.
MAFS.8.SP.1.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. SMP#3
create a two-way table to record the frequencies of bivariate categorical values.
determine the relative frequencies for rows and/or columns of a two-way table.
use the relative frequencies and context of the problem to describe possible associations between the two sets of data.
Students may believe that bivariate data is only displayed in scatter plots. It is important to provide the opportunity to display bivariate, categorical data in a table.
Course: Pre-Algebra Advanced
Unit Five: Solving Systems of Equations
Essential Question(s): How can algebraic equations be used to model, analyze, and solve mathematical situations?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.EE.3.8 Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations
c. Solve real-world and mathematical problems leading to two linear equations in two variables.
SMP #2
explain how a line represents the infinite number of solutions to a linear equation with two variables.
solve systems of two linear equations in two variable algebraically.
estimate solutions by graphing equations.
Examples of Opportunities for In-Depth Focus
When students work toward meeting this standard, the build on what they know about two-variable linear equations, and they enlarge the varieties of real-world and mathematical problems they can solve. Example; given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Misconception: students stop after solving one variable.
https://wiki.eee.uci.edu/index.php/System_of_Equations#Methods_of_Teaching http://www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/index.php
http://www.purplemath.com/modules/systlin2.htm http://www.charleston.k12.il.us/cms/Teachers/math/PreAlgebra/paunit8/RV8.PDF
Course: Pre-Algebra Advanced
Unit Six: Exponents
Essential Question(s): How can algebraic expressions be used to model, analyze, and solve mathematical situations?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.EE.1.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. SMP # 7
determine the properties of integer exponents by exploring patterns and applying my understanding of properties of whole number exponents.
apply the properties of integer exponents to simplify expressions.
For example, 32 × 3
–5 = 3
–3 = 1/3
3 = 1/27.
Students may make the relationship that in scientific notation, when a number contains one nonzero digit and a positive exponent, that the number of zeros equals the exponent. This pattern may incorrectly be applied to scientific notation values with negative values or with more than one nonzero digit. Students may mix up the product of powers property and the power of a power property.
http://insidemathematics.org/problems-of-the-month/pom-pollygone.pdf http://katm.org/wp/wp-content/uploads/flipbooks/8thFlipFinaledited.pdf http://www.youtube.com/watch?v=xidvf2YwCvA http://www.cpalms.org/Standards/PublicPreviewBenchmark5492.aspx http://education.ti.com/calculators/downloads/US/Activities/Detail?id=5302&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fSubject%3fs%3d5022%26sa%3d1008%26t%3d1134%26d%3d9%26size%3d15%26page%3d2
MAFS.8.EE.1.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. SMP # 2
recognize taking a square root as the inverse of squaring a number.
recognize taking a cube root as the inverse of cubing a number.
evaluate the square root of a perfect square.
evaluate the cube root of a perfect cube.
justify that the square root of a non-perfect square will be irrational.
For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations Students locate rational and irrational numbers on the number line. Students compare and order rational and irrational numbers. Additionally, students understand that the value of a square root can be approximated between integers and that non-perfect square roots are irrational. Students also recognize that square roots may be negative and written as - 28.
Course: Pre-Algebra Advanced
Unit Six: Exponents Continued
Essential Question(s): How can algebraic expressions be used to model, analyze, and solve mathematical situations?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.EE.1.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. SMP # 7
write an estimation of a large quantity by expressing it as the product of a single-digit number and a positive power of ten.
write an estimation of a very small quantity by expressing it as the product of a single-digit number and a negative power of ten.
compare quantities written as the product of a single-digit number and a power of ten by stating their multiplicative relationships.
For example, estimate the population of the United States as 3 × 10
8 and the population
of the world as 7 × 109, and determine that
the world population is more than 20 times larger. Students express numbers in scientific notation. Students compare and interpret scientific notation quantities in the context of the situation. If the exponent increases by one, the value increases 10 times. Students understand the magnitude of the number being expressed in scientific notation and choose an appropriate corresponding unit. For example, 3 x 10
8 is equivalent to 30
million, which represents a large quantity. Therefore, this value will affect the unit chosen
http://www.cpalms.org/Resources/PublicPreviewResource42388.aspx http://www.cpalms.org/Resources/PublicPreviewResource10082.aspx http://www.khanacademy.org/math/arithmetic/exponents-radicals http://education.ti.com/calculators/downloads/US/Activities/Detail?id=1643&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fSubject%3fs%3d5022%26sa%3d5022%26t%3d5036%26d%3d9
MAFS.8.EE.1.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. SMP # 6
add and subtract two numbers written in scientific notation.
multiply and divide two numbers written in scientific notation.
select the appropriate units for measuring derived measurements when comparing quantities written in scientific notation. identify and interpret the various ways scientific notation is displayed on calculators and through computer software.
Course: Pre-Algebra Advanced
Unit Seven: Properties of Exponents
Essential Question(s): How does knowledge of integers help when working with rational and irrational numbers?
How can the relationship between quantities best be represented? Standard
The students will: Learning Goals
I can:
Remarks Resources
MAFS.912.N-RN.1.1. 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. SMP #3, #7, #8 MAFS.912.N-RN.1.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. SMP #7
evaluate and simplify expressions containing zero and integer exponents.
multiply monomials.
apply multiplication properties of exponents to evaluate and simplify expressions.
divide monomials.
apply division properties of exponents to evaluate and simplify expressions.
apply properties of rational exponents to simplify expressions.
convert between radicals and rational exponents.
Students sometimes misunderstand the meaning of exponential operations, the way powers and roots relate to one another, and the order in which they should be performed. Attention to the base is very important.
Consider examples: (
) and
. The position of a negative sign of a term with a rational exponent can mean that the rational exponent should be either applied first to the base, 81,
and then the opposite of the result is taken, (
), or the rational
exponent should be applied to a negative term
. The
answer of
will be not real if the denominator of the exponent is even. If the root is odd, the answer will be a negative number. Students should be able to make use of estimation when incorrectly using multiplication instead of exponentiation. Students may believe that the fractional exponent in the
expression
means the same as a factor of 1/3 in multiplication expression, 36 ● 1/3 and multiple the base by the exponent.
Tasks & Mini-Projects
http://insidemathem
atics.org/common-
core-math-
tasks/high-
school/HS-A-
2009%20Quadratic2
009.pdf
MAFS.912.F-IF.3.8 b 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. SMP #2, #7
distinguish between exponential functions that model exponential growth and exponential decay.
interpret the components of an exponential function in the context of a problem (e.g., y =
5
describes a quantity that was initially 5 and increases 22.5% every three years.)
Introduce the idea of classify them as representing exponential growth or decay. For example, identify percent rate of change in functions such as y
= , y = , y = , y = ,
exponential function to emphasize one of its properties (e.g.,
y = 5 ●
5 ● , which means that increasing 22.5% in three years is about the same as increasing 7% per year.
Course: Pre-Algebra Advanced
Unit Eight: Angles/Pythagorean Theorem
Essential Question(s): How can you use similar triangles to solve problems?
How are figures, angles, similarity and congruency related to Pythagorean Theorem? Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.G.1.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. SMP#4
prove that the sum of any triangle’s interior angles will have the same measure as a straight angle.
prove that the sum of any polygon’s exterior angles will be 360-degrees.
make conjectures regarding the relationships and measurements of the angles created when two parallel lines are cut by a transversal.
apply proven relationships to establish minimal properties to justify similarity.
Students will need reminders and review of the different types of angles formed by two lines cut by a transversal. Students will sometimes mistake angles that are supplementary with angles that are congruent, when two parallel lines are cut by a transversal. Students may have difficulty decomposing polygons into triangles. Students may struggle to develop the formula to find sums of interior angles of polygons.
http://www.mathwarehouse.com/geometry/triangles/index.php http://illuminations.nctm.org/LessonDetail.aspx?id=L745 http://www.sophia.org/angles-formed-by-transversal-lines-tutorial http://illuminations.nctm.org/LessonDetail.aspx?id=L765 http://learnzillion.com/lessons/1260-use-the-pythagorean-theorem-to-see-if-a-triangle-is-a-right-triangle
MAFS.8.G.2.6 Explain a proof of the Pythagorean Theorem and it converse. SMP#4, 7
demonstrate the relationship of the three side lengths of any right angle by the use of visual models.
apply algebraic reasoning to relate the visual model to Pythagorean Theorem.
determine if a given triangle is a right triangle using Pythagorean Theorem.
Students often miss the idea that this works for only RIGHT triangles. It is important for students to explore the relationships of the areas of the side length of non-right triangles to clarify that a2 + b2 = c2 is only true for right triangles. Students will sometimes substitute the hypotenuse in for a or b instead of c in the equation.
Course: Pre-Algebra Advanced
Unit Eight: Angles/Pythagorean Theorem
Essential Question(s): How can you use similar triangles to solve problems?
How are figures, angles, similarity and congruency related to Pythagorean Theorem? Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.G.2.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. SMP#4
apply the Pythagorean Theorem to find an unknown side length of a right triangle.
draw a diagram and use the Pythagorean Theorem to solve real-world problems involving right triangles.
draw a diagram to find right triangles in a three-dimensional figure and use the Pythagorean Theorem to calculate various dimensions.
Examples of Opportunities for In-Depth Focus The Pythagorean theorem is useful in practical problems, relates to grade-level work in irrational numbers and plays an important role mathematically in coordinate geometry in high school. Students will make the mistake of just adding the side lengths (a + b = c) Students will do a2 + b2 = c and forget to take the square root of c2
to find the length of the hypotenuse.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L684 http://illuminations.nctm.org/LessonDetail.aspx?ID=L683
MAFS.8.G.2.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
SMP#2
connect any two points on a coordinate grid to a third point so that the three points form a right triangle.
apply the Pythagorean Theorem with the characteristics of a right triangle to find the distance between the original two points.
When finding a length of a segment drawn on dot paper, students count the dots instead of length in units.
Course: Pre-Algebra Advanced Unit Nine: Transformations/Volume
Essential Question(s): How does transformations relate to similarity and congruency?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.G.1.1 Verify experimental properties of rotations, reflections and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
SMP #3
verify by measuring and comparing lengths, angle measures, and parallelism of a figure and its image – that after a figure has been translated, reflected, or rotated, corresponding lines and line segments remain the same length, corresponding angles have the same measure, and corresponding parallel lines remain parallel.
Students sometimes believe that a reflection over y = x or y= -x is like a rotation of it. The correct reflection will actually look wrong compared to the original drawing. One way to help with this is to have the students physically fold their paper to make sure it does reflect over the correct line
http://www.shodor.org/interactivate/lessons/TranslationsReflectionsRotations/ http://www.math-drills.com/graphpaper.shtml http://www.math-drills.com/geometry.shtml http://explore.firstinmath.com/common-core/grade-8/geometry.aspx http://www.opusmath.com/common-core-standards/8.g.3-describe-the-effect-of-dilations-translations-rotations-and-reflections
MAFS.8.G.1.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. SMP#4
explain how transformations can be used to prove that two figures are congruent.
perform a series of transformations (reflections, rotations, and/or translations) to prove or disprove that two given figures are congruent.
When labeling transformed images, students sometimes mistakenly apply the wrong label. Either they use the wrong letter, or forget the prime notation
MAFS.8.G.1.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. SMP#2
describe the changes occurring to x- and y- coordinates of a figure after a translation, reflection, rotation, or dilation.
Students may identify the incorrect line of symmetry (x-axis vs. y-axis) when reflecting. In translations, students can sometimes count to their points incorrectly or switch the order of the coordinates when writing the point down.
Course: Pre-Algebra Advanced
Unit Nine: Transformations/Volume Continued
Essential Question(s):
How does transformations relate to similarity and congruency?
Standard
The students will: Learning Target
I can:
Remarks Resources
MAFS.8.G.1.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. SMP#4
explain how transformations can be used to prove that two figures are similar.
describe a sequence of transformations to prove or disprove that two given figures are similar.
Students may have difficulty working with rotations. Students may only know terms used in earlier grades such as flip, slide, and turn.
http://education.ti.com/calculators/downloads/US/Activities/Detail?id=10278&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fSubject%3fd%3d1007%26size%3d15%26page%3d94 http://insidemathematics.org/common-core-math-tasks/8th-grade/8-2006%20Aaron's%20Designs.pdf http://education.ti.com/calculators/downloads/US/Activities/Detail?id=17252&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fSubject%3fd%3d1007%26size%3d15%26page%3d100
MAFS.8.G.3.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. SMP#4
describe the similarity between finding the volume of a cylinder and the volume of a right prism.
recall the formula to find the volume of a cylinder, a cone, and a sphere.
informally prove the relationship between the volume of a cylinder and the volume of a cone with the same base; and the volume of a sphere and the volume of a circumscribed cylinder.
use the formulas to find the volume of cylinders, cones, and spheres.
solve real-world problems involving the volume of cylinders, cones, and spheres.
A common misconception among middle grade students is that “volume” is a “number” that results from substituting other numbers into a formula. For these students there is no recognition that “volume” is a measure – related to the amount of space occupied.
It is important to provide opportunities for hands on experiences where students fill three dimensional objects.
Course: Pre-Algebra Advanced
Unit Ten: Polynomials
Essential Question(s): Why structure expressions in different ways?
How can the properties of the real number system be useful when working with polynomials and rational expressions? Standard
The students will: Learning Goals
I can:
Remarks Resources
MAFS.912.A-SSE.1.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. SMP #7
classify and write polynomials in standard form.
evaluate polynomial expressions.
define expression, term, factor, and coefficient.
interpret the real-world meaning of the terms, factors and coefficients of an expression in terms of their units.
group the parts of an expression differently in order to better interpret their meaning.
NOTE: The middle grades standards in Expression and Equations build a ramp from arithmetic expressions in elementary school to more sophisticated work with algebraic expressions in high school. As the complexity of expressions increases, students continue to see them as being built out of basic operations: they see expressions as sums of terms and products of factors.
Students may believe that an expression cannot be factored because it does not fit into a form they recognize. They need help with reorganizing the terms until structures become evident. Students will often combine terms that are not like terms. For example, 2 + 3x = 5x or 3x + 2y = 5xy.
Students sometimes forget the coefficient of 1 when adding like terms. For example, x + 2x + 3x = 5x rather than 6x.
Students will change the degree of the variable when adding/subtracting like terms. For example, 2x + 3x = 5x
2 rather than 5x. Students will forget
to distribute to all terms when multiplying. For example, 6(2x + 1) = 12x + 1 rather than 12x + 6.
Students may not follow the Order of Operations when simplifying expressions.
Students fail to use the property of exponents correctly when using the distributive property. For example, 3x(2x – 1) = 6x – 3x = 3x instead of simplifying as 3x(2x – 1) = 6x2 – 3x.
Students fail to understand the structure of expressions. For example, they will write 4x when x = 3 is 43 instead of 4x = 4•x so when x = 3, 4x = 4•3 = 12. In addition, students commonly misevaluate –32 = 9 rather than –32 = –9.
Students will incorrectly translate expressions that contain a difference of terms. For example, 8 less than 5 times a number is often incorrectly translated as 8 – 5n rather than 5n – 8.
Tasks & Mini-Projects
http://engageny.org/resource/common-core-exeMPlar-for-high-school-math
MAFS.912.A-SSE.1.2 Use the structure of an expression to identify ways to rewrite it. SMP #7
multiply and divide polynomials by monomials.
multiply two binomials using the distributive property.
apply models for factoring polynomials to rewrite expressions.
For example, see x4- y
4 as (x²)² – (y²)², thus recognizing it as a difference
of squares that can be factored as (x² – y²)(x² + y²).
Course: Pre-Algebra Advanced
Unit Ten: Polynomials Continued
Essential Question(s): Why structure expressions in different ways?
How can the properties of the real number system be useful when working with polynomials and rational expressions? Standard
The students will: Learning Goals
I can:
Remarks Resources
MAFS.912.A-APR.1.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. SMP #2, #7
add and subtract polynomials.
multiply and divide polynomials by monomials.
multiply two binomials using the distributive property.
Students often forget to distribute the subtraction to terms other than the first one. For example, students will write (4x + 3) – (2x + 1) = 4x + 3 – 2x + 1 = 2x + 4 rather than 4x + 3 – 2x – 1 = 2x + 2. Students will change the degree of the variable when adding/subtracting like terms. For example, 2x + 3x = 5x2 rather than 5x. Students may not distribute the multiplication of polynomials correctly and only multiply like terms. For example, they will write (x + 3)(x – 2) = x2 – 6 rather than x2 – 2x + 3x – 6.
