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Teacher’s guide

Table of ContentsVolume I

Teacher’s GuideIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TG1Pacing.Guide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TG5Standards.Correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TG29Graphic.Organizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TG31

Unit 1: Function FamiliesLesson.1:.Plotting.Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..1Lesson.2:.Domain.and.Range.and.Maximum/Minimum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..40Lesson.3:.Introduction.to.Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Lesson.4:.Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Lesson.5:.Rates.of.Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Lesson.6:.Exploring.the.Function.f(x).=.1/x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Lesson.7:.Exploring.the.Function.f(x).=.f x x( ) = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Lesson.8:.Exploring.the.Functions.f(x)3,.f(x).=.x2.and.Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Answer.Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Station.Activities:.Evaluating.and.Graphing.Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Unit 2: Algebra InvestigationsLesson.1:.Writing.Equivalent.Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Lesson.2:.Multiplying.Binomials.and.Special.Patterns .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 302Lesson.3:.Algebraic.Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353Lesson.4:.Working.with.Square.Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375Answer.Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393Station.Activities:.Algebra.Investigations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

Unit 3: Geometry GalleryLesson.1:.Exploring.Polygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Lesson.2:.Characteristics.of.Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437Lesson.3:.Congruent.Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456Lesson.4:.Bisectors,.Medians,.and.Altitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476Lesson.5:.Points.of.Concurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491Lesson.6:.Applications.of.Points.of.Concurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503Lesson.7:.Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510Answer.Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559Station.Activities:.Pythagorean.Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

Volume II

Unit 4: The Chance of WinningLesson.1:.Equally.Likely.Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

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Lesson.2:.Expected.Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604Lesson.3:.Not.Equally.Likely.Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614Lesson.4:.Creating.a.Spinner.from.Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665Lesson.5:.Applying.Expected.Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680Lesson.6:.Mutually.Exclusive.Events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684Lesson.7:.Permutations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694Lesson.8:.Sample.Statistics.and.Population.Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705Lesson.9:.Comparing.Distributions.I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737Lesson.10:.The.Binomial.Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750Lesson.11:.Comparing.Distributions.II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792Lesson.12:.Comparing.Distributions.III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802Lesson.13:.More.on.Conditional.Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817Lesson.14:.Calculating.Probabilities.from.a.Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827Lesson.15:.Applying.Conditional.Probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835Lesson.16:.Geometric.Probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848Lesson.17:.Dependent.Events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858Lesson.18:.Comparing.Samples.to.the.Population. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875Lesson.19:.Statistical.Analyses.of.Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886Answer.Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911Station.Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925

Set.1:.Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925Set.2:.More.Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935

Unit 5: Algebra in ContextLesson.1:.Solving.Quadratic.Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943Lesson.2:.Graphing.Quadratic.Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986Lesson.3:.Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000Lesson.4:.Odd.Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040Lesson.5:.Rational.Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056Lesson.6:.Solving.Quadratic.Equations.by.Taking.the.Square.Root .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 1092Answer.Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103Station.Activities:.Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117

Unit 6: Coordinate GeometryLesson.1:.The.Distance.Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129Lesson.2:.The.Midpoint.Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166Lesson.3:.Quadrilaterals.on.the.Coordinate.Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172Lesson.4:.Calculating.the.Shortest.Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233Answer.Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245Station.Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1251

Set.1:.Right.Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1251Set.2:.Using.the.Midpoint.Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1261

Teacher’s guideTable of Contents

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Lesson 1: Plotting PointsUnit 1 • Function Families

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InstructionGeorgia Performance Standards

MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques.

a. Represent functions using function notation.

b. Graph the basic functions f(x) = xn, where n = 1 to 3, f x x( ) = , f(x) = |x|, and f(x) = 1/x.

c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes.

e. Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior.

Essential Questions

1. How do you use a table to determine if a relation is a function? A graph?

2. How can a graph be used to determine unknown values of x or y?

3. What is the difference between the independent and dependent quantity? How can you see this difference on a graph?

WORDS TO KNOW

dependent quantity the quantity whose value is dependent on the input quantity; the output quantity

dependent variable the variable whose value is dependent on the input variable; the output variable

domain the set of all x-values, or input values, for which a function is defined

function a relationship in which each input value has one and only one input value

function notation a naming convention of a function, where if f is the name of the function, and x is the independent variable, the name of the function would be f (x)

independent quantity the input quantity of a function

independent variable the input variable of a function

range the set of all y-values, or output values, for which a function is defined

relation any set of ordered pairs

x-y plane the two-dimensional coordinate grid with x-values on the horizontal axis and y-values on the vertical axis

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Lesson 1.1.1: Tables and GraphsPrerequisite Skills

• Plotting points in the x-y plane

• Constructing a coordinate grid

• Setting up the graph with appropriate scales

• Determining independent vs. dependent variables

Plotting Points in the x-y PlaneInstruction

Students will need to understand how to translate a table of x- and y-values into points on a coordinate grid. Help students to identify each row of the table as a point to be plotted. It may be helpful to have students rewrite the points presented in the table in the form (x, y) to understand the connection between these two representations. Students will be provided with a labeled coordinate grid, and asked to plot each point. In the exercises, students will be asked to label each point with its number, from 1 to 10.

Walk students through the following example on the board or overhead.

Example

Plot the points listed below on the grid that follows:

x y 1 –1 0 4–3 –2–4 1 1 3

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Instruction

Identify the x-axis as the horizontal number line, and the y-axis as the vertical number line. Each row of the table corresponds to a point. The location of the point is dependent on both values. Locate each point by starting at the origin (0), and moving left or right to the x-value on the x-axis, then moving up or down to the value on the y-axis. Each point can also be written in the form (x, y), where x is the value of the coordinate in the x column, and y is the value of the coordinate in the y column.

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(–4, 1)

(0, 4)

(1, 3)

(1, –1)

(–3, –2)

Unit 1 • Function FamiliesLesson 1: Plotting Points

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Prerequisite Practice 1.1.1: Plotting Points in the x-y PlanePlot each point on the coordinate grid below. Label each point with the corresponding problem number below.

1. (3, 2) 6. (–4, 3)

2. (–2, 0) 7. (2, –2)

3. (4, 4) 8. (2, 1)

4. (–1, –3) 9. (4, 0)

5. (0, –1) 10. (–3, 1)

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Constructing a Coordinate GridInstruction

Students will build on the concept of plotting individual points and begin creating their own coordinate grids. They will need to make decisions regarding the units of each axis, as well as the scale. Reinforce the concept of independent and dependent variables by asking students to first decide which quantity is independent or dependent, and then to relate the independent and dependent quantities to the general independent and dependent variables.

Walk students through the following example on the board or overhead.

Example

Christian decides to track his texting activity in February. At the end of each week, Christian records the total number of texts sent during February. Create a coordinate grid and plot the data.

Week Total Texts Sent1 1232 2413 3874 610

Before creating a grid, the independent and dependent quantities need to be identified. Christian is tracking his total texts sent based on the total time elapsed. So, the time, in weeks, is the independent quantity. His total texts sent will be the dependent quantity. When plotting the points on the x-y coordinate grid, the x-axis will be the time in weeks, and the y-axis will be the total texts sent.

Next, determine the range of x- and y-values on both axes. Since the x-values are from 1 to 4, use a range of 0 to 5. The y-values are from 123 to 610, so use a range of 0 to 650. Decide how to label each axis by looking at the range of values. The x-values can be labeled every unit, but the y-axis would look too cluttered if each unit was labeled. Instead, try labeling every 100 units, with lines every 50 units to help plot each point. Describe the decision of the range and labeling of units as the scale of each axis.

Draw the axes, labeling the horizontal as x and the vertical as y, and label the values as determined.

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Instruction

Plot the points on the grid. The y-axis isn’t labeled every unit, so the y-value of each point can be approximated.

Unit 1 • Function FamiliesLesson 1: Plotting Points

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Instruction

Unit 1 • Function FamiliesLesson 1: Plotting Points

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Prerequisite Practice 1.1.1: Constructing a Coordinate GridUse each problem statement to complete the numbered steps below each table. Create coordinate grids on separate graph paper.

An office records the total sheets of paper used by its main printer over the course of a work week. The data is recorded in the table below.

Day Total Sheets Used1 582 1193 3264 5035 725

1. Identify the independent and dependent quantities.

2. Determine which axis on the x-y coordinate grid corresponds to each quantity.

3. Determine the scale of each axis.

4. Draw an x-y coordinate grid and plot the data.

continued

Unit 1 • Function FamiliesLesson 1: Plotting Points

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Ryan records the total number of calories consumed during one day. She records her data in the table below. She counts zero hours as midnight.

Time (in hours) Total Calories Consumed 0 0 4 0 8 31010 31014 1,02617 1,45020 1,93522 2,008

5. Identify the independent and dependent quantities.

6. Determine which axis on the x-y coordinate grid corresponds to each quantity.

7. Determine the scale of each axis.

8. Draw an x-y coordinate grid and plot the data.

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Instruction

Lesson 1.1.1 PreviewIn Exploring Functions with Fiona, question 1, emphasize that plotting data on a coordinate grid allows us to analyze trends in the data. To effectively plot data, students need to be comfortable translating information from a table to a plot on the x -y plane. Students need to establish which quantities, and corresponding variables, are independent and dependent, and then use this information to create a scale for both axes. A review of plotting information from tables and setting up a coordinate grid for the plot is found in the Prerequisite Skills.

After translating from a table to a coordinate grid, students will need to decide if the points on the plot should be connected. It is important that students understand that if points are connected, it is implied that in order to move from one y-value to the next, all y-values in between must also be attained. In this case, because Fiona is looking at data involving height over time, her mother must have been every height in between one year and the next. The line that connects two heights approximates when her mother was at each height in between the known values.

Students will also develop an understanding that a steeper line indicates a faster rate of change. In this case, a steeper line indicates that Fiona’s mother grew at a faster rate. Students will also observe that a horizontal line connecting two values indicates no change, or in Fiona’s mother’s case, no growth. Allow students to draw from outside knowledge to hypothesize that she stopped growing because she had reached her maximum height by the age of 16, which is common for girls. Guide students to understand that if Fiona’s mother continued to stay 66 inches tall, the graph of her height after age 16 would be a horizontal line at y = 66.

Unit 1 • Function FamiliesStation Activities: Evaluating and Graphing Functions

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Station 1Answer each question individually and within your group. Assign the count-off numbers 1, 2, 3, 4, etc., to each member of your group. Use separate sheets of graph paper for problems 2 and 5.

1. Write a function in the form y = ax, where a is your count-off number.

2. Graph both the function y = x and your function from problem 1.

3. Compare and contrast the two graphs. Discuss rate of change, domain, range, minimums, maximums, symmetry, and any place either graph crosses the x- or y-axis.

4. Compare your unique linear function with the functions of your group members. Describe any differences between both your formulas and your graphs.

5. Graph a function in the form yax=

1. Compare and contrast y = x, y = ax, and y

ax=

1

within your group.