8 Grade Algebra 1 First Quarter Module 1: Relationships ...

HIGLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL ALIGNMENT

8th Grade Algebra 1 First Quarter

Module 1: Relationships Between Quantities and Reasoning with Equations and Their Graphs (40 days) Unit 1: Representing Relationships Mathematically

By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. Now, students analyze and explain precisely the process of solving an equation. Students, through reasoning, develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and make conjectures about the form that a linear equation might take in a solution to a problem. They reason abstractly and quantitatively by choosing and interpreting units in the context of creating equations in two variables to represent relationships between quantities. They master the solution of linear equations and apply related solution techniques and the properties of exponents to the creation and solution of simple exponential equations. In this unit, students solidify their previous work with functional relationships as they begin to formalize the concept of a mathematical function. This unit provides an opportunity for students to reinforce their understanding of the various representations of a functional relationship—words, concrete elements, numbers, graphs, and algebraic expressions. Students review the distinction between independent and dependent variables in a functional relationship and connect those to the domain and range of a function. The standards listed here will be revisited multiple times throughout the course, as students encounter new function families.

Big Idea:

• Units and quantities define the parameters of a given situation and are used to solve problems. • The different parts of expressions, equations and inequalities can represent certain values in the context of a situation and help

determine a solution process. • Relationships between quantities can be represented symbolically, numerically, graphically, and verbally in the exploration of real world

situations. Essential

Questions: • When is it advantageous to represent relationships between quantities symbolically? numerically? graphically? • Why are procedures and properties necessary when manipulating numeric or algebraic expressions?

Vocabulary

Standard Common Core Standards Explanations & Examples Comments

N.Q.A.1 A.Reason qualitatively and units to solve problems

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and

N.Q.1 Interpret units in the context of the problem N.Q.1 When solving a multi-step problem, use units to evaluate the appropriateness of the solution. N.Q.1 Choose the appropriate units for a specific formula and interpret the meaning of the unit in that context.

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interpret the scale and the origin in graphs and data displays.

HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.6. Attend to precision.

N.Q.1 Choose and interpret both the scale and the origin in graphs and data displays Include word problems where quantities are given in different units, which must be converted to make sense of the problem. For example, a problem might have an object moving 12 feet per second and another at 5 miles per hour. To compare speeds, students convert 12 feet per second to miles per hour:

hr24day1

min60hr1

sec60min1sec24000 ••• which is more than 8 miles per

hour. Graphical representations and data displays include, but are not limited to: line graphs, circle graphs, histograms, multi-line graphs, scatterplots, and multi-bar graphs.


Define appropriate quantities for the purpose of descriptive modeling. HS.MP.4. Model with mathematics. HS.MP.6. Attend to precision.

N.Q.2 Determine and interpret appropriate quantities when using descriptive modeling. Examples:

• What type of measurements would one use to determine their income and expenses for one month?

• How could one express the number of accidents in Arizona?

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.


Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

HS.MP.5. Use appropriate tools strategically. HS.MP.6. Attend to precision.

N.Q.3 Determine the accuracy of values based on their limitations in the context of the situation. The margin of error and tolerance limit varies according to the measure, tool used, and context. Example: Determining price of gas by estimating to the nearest cent is appropriate because you will not pay in fractions of a cent but the cost

of gas is gallon

479.3$.

A.SSE.A.1

A.Interpret the structure of expressions

Interpret expressions that represent a quantity in terms

Students should understand the vocabulary for the parts that make up the whole expression and be able to identify those parts and interpret their meaning in terms of a context.

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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. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.MP.7. Look for and make use of structure.

A.SSE.1a Identify the different parts of the expression and explain their meaning within the context of a problem. A.SSE.1b Decompose expressions and make sense of the multiple factors and terms by explaining the meaning of the individual parts.

A.SSE.A.2 A.Interpret the structure of expressions

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

HS.MP.2. Reason abstractly and quantitatively. HS.MP.7. Look for and make use of structure.

A.SSE.2 Rewrite algebraic expressions in different equivalent forms such as factoring or combining like terms.

• Use factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to factor completely.

• Simplify expressions including combining like terms, using the distributive property and other operations with polynomials.

A.CED.A.1 A.Create equations that describe numbers or relationships

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

Equations can represent real world and mathematical problems. Include equations and inequalities that arise when comparing the values of two different functions, such as one describing linear growth and one describing exponential growth. Examples:

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

Lava coming from the eruption of a volcano follows a parabolic path. The height h in feet of a piece of lava t seconds after it is ejected from the volcano is given by ℎ(𝑡) = −𝑡2 + 16𝑡 + 936. After how many

In Algebra I, tasks are limited to linear, quadratic, or exponential equations with integer exponents. To make the strongest connection between students’ previous work and the work of this course, the focus for A.CED.A.1, A.CED.A.3 should be on linear functions and equations.

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seconds does the lava reach its maximum height of 1000 feet? A.CED.1 Create linear, quadratic, rational and exponential equations and inequalities in one variable and use them in a contextual situation to solve problems.


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

HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically.

A.CED.2 Create equations in two or more variables to represent relationships between quantities. A.CED.2 Graph equations in two variables on a coordinate plane and label the axes and scales.


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 and cost constraints on combinations of different foods.


To make the strongest connection between students’ previous work and the work of this course, the focus for A.CED.A.1, A.CED.A.3 should be on linear functions and equations.

A.REI.D.10 D. Represent and solve equations and inequalities graphically Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics.

A.REI.10 Understand that all solutions to an equation in two variables are contained on the graph of the equation. Example:

• Which of the following points is on the circle with equation

(a) (1, -2) (b) (2, 2) (c) (3, -1) (d) (3, 4)

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F.BF.A.1a

A.Build a function that models a relationship between two quantities 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.

HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.6. Attend to precision. HS.MP.7. Look for and make use of structure. HS.MP.8. Look for and express regularity in repeated reasoning.

F.BF.1a From context, either write an explicit expression, define a recursive process, or describe the calculations needed to model a function between two quantities. Students will analyze a given problem to determine the function expressed by identifying patterns in the function’s rate of change. They will specify intervals of increase, decrease, constancy, and, if possible, relate them to the function’s description in words or graphically. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to model functions. Examples:

• You buy a $10,000 car with an annual interest rate of 6 percent compounded annually and make monthly payments of $250. Express the amount remaining to be paid off as a function of the number of months, using a recursion equation.

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8th Grade Algebra 1 First Quarter

Module 1: Relationships Between Quantities and Reasoning with Equations and Their Graphs (40 days) Unit 2: Linear Equations and Inequalities

Students have written and solved linear equations and inequalities in their previous mathematics courses. The work of this unit should be on bringing students to mastery of this area of their mathematical study. This unit leverages the connection between equations and functions and explores how different representations of a function lead to techniques to solve linear equations, including tables, graphs, concrete models, algebraic operations, and "undoing" (reasoning backwards). This unit provides opportunities for students to continue to practice their ability to create and graph equations in two variables, as described in A-CED.A.2 and A-REI.D.10.

Big Idea:

• The different parts of expressions, equations and inequalities can represent certain values in the context of a situation and help determine a solution process.

• Relationships between quantities can be represented symbolically, numerically, graphically, and verbally in the exploration of real world situations.

• Rules of arithmetic and algebra can be used together with notions of equivalence to transform equations and inequalities. • Equivalent forms of an expression can be found, dependent on how the expression is used.

Essential Questions:

• How are equations and inequalities used to solve real world problems? • When is it advantageous to represent relationships between quantities symbolically? numerically? graphically? • Why are procedures and properties necessary when manipulating numeric or algebraic expressions? • How can the structure of expressions/equations/inequalities help determine a solution strategy?

Vocabulary Literal equation, absolute value equation



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.

Equations can represent real world and mathematical problems. Include equations and inequalities that arise when comparing the values of two different functions, such as one describing linear growth and one describing exponential growth. Examples:

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

The work of A.CED.A.1 should focus on linear equations and inequalities. Exponential equations will be addressed in The Exponential functions and equations unit, and quadratic equations will

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Lava coming from the eruption of a volcano follows a parabolic path. The height h in feet of a piece of lava t seconds after it is ejected from the volcano is given by ℎ(𝑡) = −𝑡2 + 16𝑡 + 936. After how many seconds does the lava reach its maximum height of 1000 feet?

A.CED.1 Create linear, quadratic, rational and exponential equations and inequalities in one variable and use them in a contextual situation to solve problems.

be addressed in the Quadratic equations unit. Rational equations should be addressed in Algebra II. In Algebra I, tasks are limited to linear, quadratic, or exponential equations with integer exponents.


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 and cost constraints on combinations of different foods.


Example: • A club is selling hats and jackets as a fundraiser. Their budget

is $1500 and they want to order at least 250 items. They must buy at least as many hats as they buy jackets. Each hat costs $5 and each jacket costs $8. o Write a system of inequalities to represent the situation. o Graph the inequalities. o If the club buys 150 hats and 100 jackets, will the

conditions be satisfied? o What is the maximum number of jackets they can buy

and still meet the conditions?


Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.


Examples: • The Pythagorean Theorem expresses the relation between the

legs a and b of a right triangle and its hypotenuse c with the equation a2 + b2 = c2. o Why might the theorem need to be solved for c? o Solve the equation for c and write a problem situation

where this form of the equation might be useful.

o Solve 343

V rπ= for radius r.

• Motion can be described by the formula below, where t =

A.CED.4 Solve multi-variable formulas or literal equations, for a specific variable.

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HS.MP.7. Look for and make use of structure. time elapsed, u=initial velocity, a = acceleration, and s = distance traveled s = ut+½at2

o Why might the equation need to be rewritten in terms of a?

o Rewrite the equation in terms of a.

A.REI.A.1 A.Understand solving equations as a process of reasoning and explain the reasoning

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.

HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.7. Look for and make use of structure.

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. Other operations, such as squaring both sides, may produce equations that have extraneous solutions. Examples:

• Explain why the equation x/2 + 7/3 = 5 has the same solutions as the equation 3x + 14 = 30. Does this mean that x/2 + 7/3 is equal to 3x + 14?

• Show that x = 2 and x = -3 are solutions to the equation 𝑥2 + 𝑥 = 6. Write the equation in a form that shows these are the only solutions, explaining each step in your reasoning.

A.REI.1 Assuming an equation has a solution, construct a convincing argument that justifies each step in the solution process. Justifications may include the associative, commutative, and division properties, combining like terms, multiplication by 1, etc.

A.REI.1 Assuming an equation has a solution, construct a convincing argument that justifies each step in the solution process. Justifications may include the associative, commutative, and division properties, combining like terms, multiplication by 1, etc. Algebra I, students should focus on and master A.REI.1 for linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Students will solve exponential equations with logarithms in Algebra II.

A.REI.B.3 B.Solve equations and inequalities in one variable

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

HS.MP.2. Reason abstractly and quantitatively. HS.MP.7. Look for and make use of structure.

A.REI.3 Solve linear equations in one variable, including coefficients represented by letters. A.REI.3 Solve linear inequalities in one variable, including coefficients represented by letters.

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HS.MP.8. Look for and express regularity in repeated reasoning.

Examples:

• 7 8 1113

y− − =

• 3x > 9 • ax + 7 = 12

• 4

97

3 −=

+ xx

• Solve for x: 2/3x + 9 < 18 8.EE.C.8 C.Analyze and solve linear equations and

pairs of simultaneous linear equations 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. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solve real-world and mathematical problems leading to two linear equations in two variables. For 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.

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

8.MP.2. Reason abstractly and quantitatively.

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

8.MP.4. Model with mathematics.

8.EE.8 Systems of linear equations can also have one solution, infinitely many solutions or no solutions. Students will discover these cases as they graph systems of linear equations and solve them algebraically. Students graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will generate the given y-value for both equations. Students recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have infinitely many solutions. By making connections between algebraic and graphical solutions and the context of the system of linear equations, students are able to make sense of their solutions. Students need opportunities to work with equations and context that include whole number and/or decimals/fractions. Students define variables and create a system of linear equations in two variables Example 1: 1. Plant A and Plant B are on different watering schedules. This affects their rate of growth. Compare the growth of the two plants to determine when their heights will be the same. Solution: Let W = number of weeks Let H = height of the plant after W weeks

Students’ perseverance in solving real---world problems with systems of equations requires that they work with various solution methods and learn to discern when each method is most appropriate (MP.1). As with the previous unit, writing and solving systems require that students make use of structure (MP.7) and attend to precision (MP.6) as students apply properties of operations to transform equations into simpler forms.

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8.MP.5. Use appropriate tools strategically.

8.MP.6. Attend to precision.

8.MP.7. Look for and make use of structure.

8.MP.8. Look for and express regularity in repeated reasoning.

2. Based on the coordinates from the table, graph lines to represent each plant. Solution:

3. Write an equation that represents the growth rate of Plant A and Plant B. Solution: Plant A H = 2W + 4 Plant B H = 4W + 2 4. At which week will the plants have the same height? Solution:

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After one week, the height of Plant A and Plant B are both 6 inches.

Given two equations in slope-intercept form (Example 1) or one equation in standard form and one equation in slope-intercept form, students use substitution to solve the system. Example 2: Solve: Victor is half as old as Maria. The sum of their ages is 54. How old is Victor?

If Maria is 36, then substitute 36 into v + m = 54 to find Victor’s age of 18. Note: Students are not expected to change linear equations written in standard form to slope-intercept form or solve systems using elimination. For many real world contexts, equations may be written in standard

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form. Students are not expected to change the standard form to slope-intercept form. However, students may generate ordered pairs recognizing that the values of the ordered pairs would be solutions for the equation. For example, in the equation above, students could make a list of the possible ages of Victor and Maria that would add to 54. The graph of these ordered pairs would be a line with all the possible ages for Victor and Maria.

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8th Grade Algebra 1 First/Second Quarter

Module 2: Descriptive Statistics (25 days) Unit 3: Statistical Models

This module builds upon students’ prior experiences with data, providing students with more formal means of assessing how a model fits data. Students display and interpret graphical representations of data, and if appropriate, choose regression techniques when building a model that approximates a linear relationship between quantities. They analyze their knowledge of the context of a situation to justify their choice of a linear model. With linear models, they plot and analyze residuals to informally assess the goodness of fit. This unit reviews the univariate data representations students studied previously and then introduces statistical models for bivariate categorical and quantitative data. Students have already addressed in previous units many of the standards in this unit, and they should now be able to apply their understandings from that previous work in the new work with the statistics standards in this unit. This unit provides opportunities to reinforce students’ work from the previous units with representing linear functions symbolically, as described in A---SSE.A.1a, A---CED.A.2. Students use their understanding of data distribution or shape to determine more precise comparisons of data sets. Students have worked with a variety of units in the past and will bring that information to this unit of study. They will recall labeling graphs and units. Students use statistics to compare center and spread of two or more different data sets, including the use of scatter plots, histograms, box plots, and standard deviation. Students will interpret outliers and recognize associations and trends in the data.

Big Idea:

• Data can be represented and interpreted in a variety of formats. • Extreme data points (outliers) can skew interpretations of a set of data. • Synthesizing information from multiple sets of data results in evidence-based interpretation. • Center and spread of a data set may be compared in multiple ways. • Data in a two –way frequency table can be summarized using relative frequencies in the context of the data.

Essential Questions:

• How is attention to units and quantities meaningful in data analysis and problem solving? • How do various representations of data lead to different interpretations of the data? • When and how can extreme data points impact interpretation of data? • Why are multiple sets of data used? • How are center and spread of data sets described and compared? • How is a data set represented in a two-way frequency table summarized?

Vocabulary Joint relative frequency, marginal relative frequency, conditional relative frequency, outlier, skewed distribution, correlation coefficient, two-way frequency table, standard deviation, interquartile range


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S.ID.A.1 A.Summarize, represent, and interpret data on a single count or measurement variable

Represent data with plots on the real number line (dot plots, histograms, and box plots).

HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically.

Construct dot plots, histograms and box plots for data on a real number line.

Algebra I students take a more sophisticated look at using a linear function to model the relationship between two numerical variables. In addition to fitting a line to data, students assess how well the model fits by analyzing residuals. The expectation in Algebra I is to build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship.


Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics.HS.MP.5. Use appropriate tools strategically.

Students may use spreadsheets, graphing calculators and statistical software for calculations, summaries, and comparisons of data sets. Examples:

• The two data sets below depict the housing prices sold in the King River area and Toby Ranch areas of Pinal County, Arizona. Based on the prices below which price range can be expected for a home purchased in Toby Ranch? In the King River area? In Pinal County? o King River area {1.2 million, 242000, 265500, 140000,

281000, 265000, 211000} o Toby Ranch homes {5million, 154000, 250000, 250000,

200000, 160000, 190000}

In grades 6 – 8, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points.

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HS.MP.7. Look for and make use of structure. • Given a set of test scores: 99, 96, 94, 93, 90, 88, 86, 77, 70, 68, find the mean, median and standard deviation. Explain how the values vary about the mean and median. What information does this give the teacher?

-Describe a distribution using center and spread. -Use the correct measure of center and spread to describe a distribution that is symmetric or skewed. -Identify outliers (extreme data points) and their effects on data sets. -Compare two or more different data sets using the center and spread of each.


Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.7. Look for and make use of structure.

Students may use spreadsheets, graphing calculators and statistical software to statistically identify outliers and analyze data sets with and without outliers as appropriate. Interpret differences in different data sets in context. Interpret differences due to possible effects of outliers.

In grades 6 – 8, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points.

8.SP.A.1 A.Investigate patterns of association in bivariate data

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.




Bivariate data refers to two-variable data, one to be graphed on the x-axis and the other on the y-axis. Students represent numerical data on a scatter plot, to examine relationships between variables. They analyze scatter plots to determine if the relationship is linear (positive, negative association or no association) or nonlinear. Students can use tools such as those at the National Center for Educational Statistics to create a graph or generate data sets. (http://nces.ed.gov/nceskids/createagraph/default.aspx) Data can be expressed in years. In these situations it is helpful for the years to be “converted” to 0, 1, 2, etc. For example, the years of 1960, 1970, and 1980 could be represented as 0 (for 1960), 10 (for 1970) and 20 (for 1980). Example 1:

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



Data for 10 students’ Math and Science scores are provided in the chart. Describe the association between the Math and Science scores.

Solution: This data has a positive association. Example 2: Data for 10 students’ Math scores and the distance they live from school are provided in the table below. Describe the association between the Math scores and the distance they live from school.

Solution: There is no association between the math score and the distance a student lives from school. Example 3: Data from a local fast food restaurant is provided showing the number of staff members and the average time for filling an order are provided in the table below. Describe the association between the number of staff and the average time for filling an order.

Solution: There is a positive association. Example 4: The chart below lists the life expectancy in years for people in the United States every five years from 1970 to 2005. What would you expect the life expectancy of a person in the United States to be in 2010, 2015, and 2020 based upon this data? Explain how you

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determined your values.

Solution: There is a positive association. Students recognize that points may be away from the other points (outliers) and have an effect on the linear model. NOTE: Use of the formula to identify outliers is not expected at this level. Students recognize that not all data will have a linear association. Some associations will be non-linear as in the example below:


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.

Students understand that a straight line can represent a scatter plot with linear association. The most appropriate linear model is the line that comes closest to most data points. The use of linear regression is not expected. If there is a linear relationship, students draw a linear model. Given a linear model, students write an equation. Example: The capacity of the fuel tank in a car is 13.5 gallons. The table below

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shows the number of miles traveled and how many gallons of gas are left in the tank. Describe the relationship between the variables. If the data is linear, determine a line of best fit. Do you think the line represents a good fit for the data set? Why or why not? What is the average fuel efficiency of the car in miles per gallon?


Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. 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.






Linear models can be represented with a linear equation. Students interpret the slope and y-intercept of the line in the context of the problem. Example 1: 1. Given data from students’ math scores and absences, make a scatterplot.

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2. Draw a linear model paying attention to the closeness of the data points on either side of the line.

3. From the linear model, determine an approximate linear equation that models the given data

4. Students should recognize that 95 represents the y-intercept and -25/3 represents the slope of the line. In the context of the problem, the y-intercept represents the math score a student with 0 absences could expect. The slope indicates that the math scores decreased 25 points for every 3 absences. 5. Students can use this linear model to solve problems. For example, through substitution, they can use the equation to determine that a student with 4 absences should expect to receive a math score of

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about 62. They can then compare this value to their line.


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. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?


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





8.SP.4 Students understand that a two-way table provides a way to organize data between two categorical variables. Data for both categories needs to be collected from each subject. Students calculate the relative frequencies to describe associations. Example 1: Twenty-five students were surveyed and asked if they received an allowance and if they did chores. The table below summarizes their responses.

Of the students who do chores, what percent do not receive an allowance? Solution: 5 of the 20 students who do chores do not receive an allowance, which is 25% Example 2: The table illustrates the results when 100 students were asked the survey questions: Do you have a curfew? and Do you have assigned chores? Is there evidence that those who have a curfew also tend to have chores?

Solution: Of the students who answered that they had a curfew, 40

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had chores and 10 did not. Of the students who answered they did not have a curfew, 10 had chores and 40 did not. From this sample, there appears to be a positive correlation between having a curfew and having chores.

S.ID.B.5 B.Summarize, represent, and interpret data on a single count or measurement variable

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.8. Look for and express regularity in repeated reasoning.

Students may use spreadsheets, graphing calculators, and statistical software to create frequency tables and determine associations or trends in the data.

Examples:

Two-way Frequency Table

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.

The total row and total column entries in the table above report the marginal frequencies, while entries in the body of the table are the joint frequencies.

Two-way Relative Frequency Table

The relative frequencies in the body of the table are called conditional relative frequencies.

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-Create a two-way table from two categorical variables and read values from two way table. Interpret joint, marginal, and relative frequencies in context. -Recognize associations and trends in data from a two-way table.

S.ID.B.6 B.Summarize, represent, and interpret data on a single count or measurement variable

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.

b. Informally assess the fit of a function by plotting and analyzing residuals.

c. Fit a linear function for a scatter plot that suggests a linear association.

HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.7. Look for and make use of structure. HS.MP.8. Look for and express regularity in repeated reasoning.

The residual in a regression model is the difference between the observed and the predicted for some ( the dependent variable and the independent variable). So if we have a model , and a data point the residual is for this point is: . Students may use spreadsheets, graphing calculators, and statistical software to represent data, describe how the variables are related, fit functions to data, perform regressions, and calculate residuals. Example: Measure the wrist and neck size of each person in your class and make a scatterplot. Find the least squares regression line. Calculate and interpret the correlation coefficient for this linear regression model. Graph the residuals and evaluate the fit of the linear equations. S.ID.6 Create a scatter plot from two quantitative variables. S.ID.6 Describe the form, strength and direction of the relationship. S.ID.6a Categorize data as linear or not. Use algebraic methods and technology to fit a linear function to the data. Use the function to predict values. S.ID.6a Explain the meaning of the slope and y-intercept in context. S.ID.6a Categorize data as exponential. Use algebraic methods and technology to fit an exponential function to the data. Use the function to predict values. S.ID.6a Explain the meaning of the growth rate and y-intercept in context. S.ID.6a Categorize data as quadratic. Use algebraic methods and technology to fit a quadratic function to the data. Use the function to predict values. S.ID.6a Explain the meaning of the constant and coefficients in context.

Tasks have a real-world context. In Algebra I, exponential functions are limited to those with domains in the integers. Algebra I students take a more sophisticated look at using a linear function to model the relationship between two numerical variables. In addition to fitting a line to data, students assess how well the model fits by analyzing residuals. S.ID.6b should be focused on linear models, but may be used to preview quadratic functions.

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S.ID.6b Calculate a residual. Create and analyze a residual plot. S.ID.6c Categorize data as linear or not. Use algebraic methods and technology to fit a linear function to the data. Use the function to predict values.

S.ID.C.7 C. Interpret linear models

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.6. Attend to precision.

Students may use spreadsheets or graphing calculators to create representations of data sets and create linear models.

Example:

• Lisa lights a candle and records its height in inches every hour. The results recorded as (time, height) are (0, 20), (1, 18.3), (2, 16.6), (3, 14.9), (4, 13.2), (5, 11.5), (7, 8.1), (9, 4.7), and (10, 3). Express the candle’s height (h) as a function of time (t) and state the meaning of the slope and the intercept in terms of the burning candle.

Solution:

h = -1.7t + 20 Slope: The candle’s height decreases by 1.7 inches for each hour it is burning. Intercept: Before the candle begins to burn, its height is 20 inches.

S.ID.7 Explain the meaning of the slope and y-intercept in context. Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship.


Compute (using technology) and interpret the correlation coefficient of a linear fit.

HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.8. Look for and express regularity in repeated reasoning.

Students may use spreadsheets, graphing calculators, and statistical software to represent data, describe how the variables are related, fit functions to data, perform regressions, and calculate residuals and correlation coefficients.

Example:

Collect height, shoe-size, and wrist circumference data for each student. Determine the best way to display the data. Answer the following questions: Is there a correlation between any two of the three indicators? Is there a correlation between all three indicators? What patterns and trends are apparent in the data? What inferences can be made from the data?

S.ID.8 Use a calculator or computer to find the correlation coefficient for a linear association. Interpret the meaning of the value in the context of the data. Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the

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computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship.


Distinguish between correlation and causation.

HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. HS.MP.6. Attend to precision.

Some data leads observers to believe that there is a cause and effect relationship when a strong relationship is observed. Students should be careful not to assume that correlation implies causation. The determination that one thing causes another requires a controlled randomized experiment. Example: Diane did a study for a health class about the effects of a student’s end-of-year math test scores on height. Based on a graph of her data, she found that there was a direct relationship between students’ math scores and height. She concluded that “doing well on your end-of-course math tests makes you tall.” Is this conclusion justified? Explain any flaws in Diane’s reasoning.

S.ID.9 Explain the difference between correlation and causation. The important distinction between a statistical relationship and a cause- and-effect relationship arises in S.ID.9.

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8 Grade Algebra 1 First Quarter Module 1: Relationships ...

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