Course: 8th Grade Math DETAIL LESSON PLAN Student Objective 8.EE.B.5-1 – Graph proportional relationships, interpreting the unit rate as the slope of the graph. 8.EE.B.5-2 – Compare two different proportional relationships represented in different ways. 8.EE.B.6 – Use similar triangles to explain why the slope “m” is the same between any two distinct points; derive the equation y=mx for a line through the origin and the equation y = mx + b for a line intercepting the y-axis at b. Lesson Lesson 7.2.5 Adjustment Teachers will have students work problem 7-77 from the CPM book, then move to the PARCC practice for objectives 8.EE.5-1, 8.EE.5-2 and 8.EE.6. Homework Teacher Selected Bellwork Teacher Selected Prior Knowledge

Review homework

Review bellwork

Anticipatory Set

Read introduction from Lesson 7.2.5. Teacher Input

Work problem 7-77

Have students work the PARCC extra practice problems on slope.

Review answers to the problems.

Assessment Question the students for understanding. Closure

Teacher selected.

In Chapter 1, you identified proportional relationships in both tables and graphs. Today you will look at the connection between slope and graphs of a proportional relationship.

7-77. POLITICAL POLL

Mr. Mears was running for mayor of Atlanta. His campaign managers were eager to determine how many citizens of Atlanta would vote for him in the upcoming election. They decided to pay a respected, impartial statistical company to survey potential voters (a process called “polling”) to find out how many people would probably vote for Mr. Mears.

One afternoon, pollsters called 100 random potential voters in Atlanta to ask them how they would vote in the election. During that survey, 68 people indicated that they would vote for Mr. Mears.

a. If the pollsters had instead called 50 randomly selected potential voters, predict how many of them would have said that they would vote for Mr. Mears. _________ People

b. Do you suppose that this relationship is proportional? Why or why not? __________________________________________________________

__________________________________________________________ c. Carina’s neighborhood has 327 potential voters. If

voting in her neighborhood is similar to that of the poll, how many neighbors will probably vote for Mr. Mears? Complete the table at right.

_________ Neighbors

d. Make a graph of the data Carina has collected, and draw a line through the points. Should your line go through the origin? How do you know? _____________________________

_____________________________ e. What is the equation of the line in y = mx + b form? ___________________ f. Use the equation to help Carina figure out how many people will probably vote for Mr. Mears if 350,125 people vote in the election. What do x and y represent in your equation? _____________________________________________________________

_____________________________________________________________ g. What information would the unit rate give Carina? _____________________________________________________________

_____________________________________________________________ What is the unit rate? __________ How does the unit rate compare to the slope? _____________________________________________________________

_____________________________________________________________

Name: _________________ Slope Review

Period: _______

Section 1: Write Slope Equation; Slope Triangles

Focus: Use similar triangles to explain why the slope “m” is the same between any two distinct points on a line. Derive the equation y = mx for a line through the origin and y = mx + b for a line intercepting the y-axis at b. (8.EE.6)

1) Identify the slope of each line as positive, negative, or zero.

_____________ _____________ _____________ _____________ _____________

_____________ _____________ _____________ _____________ _____________

2) Identify the slope and y-intercept of each. Then circle whether the slope is + or ─.

y =

m= _______ b =_______ Slope type: positive or negative

y = −8x − 2 m= _______ b =_______ Slope type: positive or negative

y =

m= _______ b =_______ Slope type: positive or negative

y = 6x − 5 m= _______ b =_______ Slope type: positive or negative y = 3x − 12 m= _______ b =_______ Slope type: positive or negative y = 4x m= _______ b =_______ Slope type: positive or negative

3) The slope of a line can be written in the format of y = mx and y = mx + b. Explain when you would use each format and draw a simple sketch of what the line might look like if graphed.

Which equation represents a proportional relationship and why? 4) Create a slope triangle using the lattice points in each graph, and then write an equation for the slope of the line.

___________________ ___________________ ___________________ 5) Use the graph to answer each question below.

Part A: Mark each lattice point with a dot. Part B: Using any two lattice points, draw a slope triangle.

Part C: What is your slope? _______

Part D: What is your y-intercept? _______

Part E: Write the slope in the form of an equation. ________________

6) Notice that two slope triangles are drawn below creating similar triangles.

a. Find the slope of the line between point A and point B

. ________

b. Find the slope of the line between point B and point D. ________ c. Is it fair to say that when finding the slope of a line, you may use any two distinct points on the line? 7) Which statements are true regarding the slope of line W. Select all that apply. (PARCC clone)

___ A. The slope of is equal to the slope

of

___ B. The slope of is equal to the slope

of

___ C. The slope of is equal to the slope of line

___ D. The slope of line W is equal to

___ E. The slope of line W is equal to

___ F. The slope of line W is equal to

Section 2: Graph Unit Rate as Slope

Focus: Graph proportional relationships, interpreting the unit rate as the slope of the graph. (8.EE.5-1) 8) Olivia sold water bottles over four days. The quantity she sold each day is listed in the table below. a. Determine if the quantities of bottles and number of days are proportional. If the

quantities are proportional, what is the unit rate? __________ b. Graph the points from the table and connect the points with a line.

c. Use any two lattice points to create a slope triangle. What is the slope? __________ d. What do you notice about the unit rate from part a and the slope of the line from part c?

e. Write an equation to represent the slope of the line. _______________

9) A candy bar is made up of 80% chocolate. (PARCC clone) a. Complete the table below showing the amount of chocolate compared to the number of chocolate bars. You may use a calculator.

b. Is the relationship in the table proportional, if so what is the unit rate? ______________ c. Next, graph the points from the table, then use a slope triangle along with the formula

to find the slope. What is the slope of the line? __________

d. How does the slope of the line compare to the unit rate?

e. Write an equation to represent the slope of the line. ____________

Section 3: Proportional Relationships Represented in Different Ways

Focus: Compare two different proportional relationships represented different ways. (8.EE.5-2) Directions: In problems 10 and 11, you have two separate proportional relationships. Analyze each proportion then answer the question that follows. (PARCC clone) 10) Proportion A Proportion B y = 7x

a. What is the rate of change (growth/unit rate) in portion A. __________ What is the rate of change (growth/unit rate/slope) in portion B? __________ b. The rate of change in Portion A is than the rate of change in Portion B.

11) Proportion A Proportion B y = 4x

The rate of change in Portion A is than the rate of change in Portion B.

Answer Key

In Chapter 1, you identified proportional relationships in both tables and graphs. Today you will look at the connection between slope and graphs of a proportional relationship.

7-77. POLITICAL POLL

Mr. Mears was running for mayor of Atlanta. His campaign managers were eager to determine how many citizens of Atlanta would vote for him in the upcoming election. They decided to pay a respected, impartial statistical company to survey potential voters (a process called “polling”) to find out how many people would probably vote for Mr. Mears.

One afternoon, pollsters called 100 random potential voters in Atlanta to ask them how they would vote in the election. During that survey, 68 people indicated that they would vote for Mr. Mears.

a. If the pollsters had instead called 50 randomly selected potential voters, predict how many of them would have said that they would vote for Mr. Mears. 34 People

b. Do you suppose that this relationship is proportional? Why or why not? Yes, it is a proportional relationship. Based on the original results of calling 100 people and having 68 say they would vote for Mr. Mears, you could assume that if you make half the calls it would result in half the number in favor of Mr. Mears. If you double the number of calls you could expect double in favor of Mr. Mears. If you graphed the numbers, the line would pass through the origin (0, 0); because if you make zero calls, zero people would say they were in favor of Mr. Mears.

c. Carina’s neighborhood has 327 potential voters. If voting in her neighborhood is similar to that of the poll, how many neighbors will probably vote for Mr. Mears? Complete the table at right.

222 Neighbors Find the unit rate using any (x, y) value in the table.

Unit Rate =

=

= .68

Multiply 327 .68 = 222.36 = 222 Some students may set up a proportion to find the answer.

100x = 22,236 x = 222.36

d. Make a graph of the data Carina has collected, and draw a line through the points. Should your line go through the origin? How do you know?

If you make zero calls, zero people would say they were in favor of Mr. Mears. e. What is the equation of the line in y = mx + b form?

Growth =

Starting Point = 0 y = .68x + 0 or y = .68x

f. Use the equation to help Carina figure out how many people will probably vote for Mr. Mears if 350,125 people vote in the election. y = .68x + 0 y = .68 • 350,125 y = 238,085 What do x and y represent in your equation? “x” represents the number of voters called. “y” is the number of voters called that said they would vote for Mr. Mears. g. What information would the unit rate give Carina? Since, UNIT means one, the unit rate would tell her how many votes to expect per one potential voter. What is the unit rate? .68 How does the unit rate compare to the slope?

The unit rate (.68) is the same as the slope of the line.

Answer Key Slope Review

Section 1: Write Slope Equation; Slope Triangles

Focus: Use similar triangles to explain why the slope “m” is the same between any two distinct points on a line. Derive the equation y = mx for a line through the origin and y = mx + b for a line intercepting the y-axis at b. (8.EE.6)

1) Identify the slope of each line as positive, negative, or zero.

2) Identify the slope and y-intercept of each. Then circle whether the slope is + or ─.

3) The slope of a line can be written in the format of y = mx and y = mx + b. Explain when you would use each format and draw a simple sketch of what the line might look like if graphed.

The equation y = mx would represent the slope of a line with a y-intercept of zero.

The equation y = mx + b would represent the slope of a line that intercepts the y-axis at “b”. Which equation represents a proportional relationship and why? y = mx because it is a straight line going through the origin (0, 0) 4) Create a slope triangle using the lattice points in each graph, and then write an equation for the slope of the line.

5) Use the graph to answer each question below.

Part A: Mark each lattice point with a dot. Part B: Using any two lattice points, draw a slope triangle. Part C: What is your slope? Part D: What is your y-intercept?

Part E: Write the slope in the form of an equation.

6) Notice that two slope triangles are drawn below creating similar triangles.

a. Find the slope of the line between point A and point B

.

b. Find the slope of the line between point B and point D. c. Is it fair to say that when finding the slope of a line, you may use any two distinct points on the line? 7) Which statements are true regarding the slope of line W. Select all that apply. (PARCC clone) A. The slope of is equal to the slope of B. The slope of is equal to the slope of C. The slope of is equal to the slope of line

___ D. The slope of line W is equal to

E. The slope of line W is equal to

___ F. The slope of line W is equal to

Section 2: Graph Unit Rate as Slope

Focus: Graph proportional relationships, interpreting the unit rate as the slope of the graph. (8.EE.5-1) 8) Olivia sold water bottles over four days. The quantity she sold each day is listed in the table below. a. Determine if the quantities of bottles and number of days are proportional. If the quantities are proportional, what is the unit rate? b. Graph the points from the table and connect the points with a line. c. Use any two lattice points to create a slope triangle. What is the slope?

Slope =

d. What do you notice about the unit rate from part a and the slope of the line from part c?

If the relationship is proportional, the slope and unit rate will be the same value. e. Write an equation to represent the slope of the line.

y = mx y = 4x

9) A candy bar is made up of 80% chocolate. (PARCC clone) a. Complete the table below showing the amount of chocolate compared to the number of chocolate bars. You may use a calculator.

b. Is the relationship in the table proportional, if so what is the unit rate? ______________ Yes, because y ÷ x for each set of coordinates in the table is .8. and the line goes through (0,0).

Unit rate

c. Next, graph the points from the table, then use a slope triangle along with the formula

to find the slope. What is the slope of the line?

Slope =

d. How does the slope of the line compare to the unit rate? When the relationship is proportional, the slope and unit rate will be the same value. e. Write an equation to represent the slope of the line.

y = mx y = .8x

Section 3: Proportional Relationships Represented in Different Ways

Focus: Compare two different proportional relationships represented in different ways. (8.EE.5-2) Directions: In problems 10 and 11, you have two separate proportional relationships. Analyze each proportion then answer the question that follows. (PARCC clone) 10) Proportion A Proportion B y = 7x a. What is the rate of change (growth/unit rate) in portion A. What is the rate of change (growth/unit rate/slope) in portion B? b. The rate of change in Portion A is than the rate of change in Portion B.

11) Proportion A Proportion B y = 4x

The rate of change in Portion A is than the rate of change in Portion B.