lesson 4 homework practice proportional and nonproportional relationships

Chapter 6, Lesson 4: Proportional and Nonproportional Relationships

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Curriculum  /  Math  /  7th Grade  /  Unit 1: Proportional Relationships  /  Lesson 4

Proportional Relationships

Lesson 4 of 18

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Write equations for proportional relationships presented in tables.

Common Core Standards

Core standards.

The core standards covered in this lesson

Ratios and Proportional Relationships

7.RP.A.2.B — Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

7.RP.A.2.C — Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

Foundational Standards

The foundational standards covered in this lesson

Expressions and Equations

6.EE.B.7 — Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

6.RP.A.3 — Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Determine the constant of proportionality from a table. 
• Write an equation for a proportional relationship in the form $$y=kx$$  where $$k$$  represents the constant of proportionality.
• Explain what the constant of proportionality means in context of a situation.
• Explain the role of the constant of proportionality in an equation.
• Use an equation to solve problems and determine additional values for the situation.
• Decontextualize situations to represent them as equations and re-contextualize equations to explain their meanings as they relate to situations (MP.2).

Suggestions for teachers to help them teach this lesson

Lessons 4 and 5 focus on representing proportional relationships as equations. Equations are abstract and can be challenging for some students to grasp. Encourage students to return to the table to show the relationship between the two quantities, either adding a column to show the constant of proportionality or drawing an arrow across rows and indicating the multiplication. Ensure that students know what the variables in the equation represent to keep the context connected to the abstract form.

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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Felix worked at a music shop during the summer. The table below shows some of Felix’s hours and earnings. The amount of money he earned is proportional to the number of hours he worked.

a.   Fill in the rest of the table and then write an equation that represents the relationship.

b.   If Felix worked 35 hours, then how much money did he earn?

c.   If Felix earned $198, then how many hours did he work?

Guiding Questions

The table below shows measurement conversions between cups and ounces.

Let  $$x$$ represent the number of cups and $$y$$  represent the number of ounces. Write an equation that represents this relationship.

The students in Ms. Baca’s art class were mixing yellow and blue paint. She told them that two mixtures will be the same shade of green if the blue and yellow paint are in the same ratio.

The table below shows the different mixtures of paint that the students made.

a.   How many different shades of paint did the students make?

b.   Write an equation that relates $$y$$ , the number of parts of yellow paint, and $$b$$ , the number of parts of blue paint for each of the different shades of paint the students made.

Art Class, Variation 2 , accessed on Aug. 1, 2017, 3:07 p.m., is licensed by Illustrative Mathematics under either the  CC BY 4.0  or  CC BY-NC-SA 4.0 . For further information, contact Illustrative Mathematics .

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

A lemonade is made by mixing flavored powder, $$p$$ , with water, $$w$$ . The chart below shows some measurements that can be used to make different amounts of lemonade.

a.    Which equation represents this relationship?

a.    $$p=4w$$

b.    $$w=4p$$

c.    $$w=4+p$$

d.    $$p=4\div w$$

b.   What is the constant of proportionality, and what does it mean in this example?

Student Response

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Batches of recipe ingredients
• Distance, time, speed
• Miles per gallon
• Unit prices
• Unit conversions
• Paint or other mixtures
• Include problems where students explain the relationship between the table, constant of proportionality, and equation in context.
• EngageNY Mathematics Grade 7 Mathematics > Module 1 > Topic B > Lesson 8 — Example 1 only
• EngageNY Mathematics Grade 7 Mathematics > Module 1 > Topic B > Lesson 9 — Example 2, Problem Set 2 and 4

Topic A: Representing Proportional Relationships in Tables, Equations, and Graphs

Solve ratio and rate problems using double number lines, tables, and unit rate.

7.RP.A.1 7.RP.A.2

Represent proportional relationships in tables, and define the constant of proportionality.

7.RP.A.2 7.RP.A.2.B

Determine the constant of proportionality in tables, and use it to find missing values.

7.RP.A.2.A 7.RP.A.2.B

7.RP.A.2.B 7.RP.A.2.C

Write equations for proportional relationships from word problems.

7.RP.A.2 7.RP.A.2.C

Represent proportional relationships in graphs.

7.RP.A.2 7.RP.A.2.A 7.RP.A.2.D

Interpret proportional relationships represented in graphs.

7.RP.A.2 7.RP.A.2.D

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Topic B: Non-Proportional Relationships

Compare proportional and non-proportional relationships.

Determine if relationships are proportional or non-proportional.

Topic C: Connecting Everything Together

Make connections between the four representations of proportional relationships (Part 1).

7.RP.A.2 7.RP.A.2.A 7.RP.A.2.B 7.RP.A.2.C 7.RP.A.2.D

Make connections between the four representations of proportional relationships (Part 2).

Use different strategies to represent and recognize proportional relationships.

Topic D: Solving Ratio & Rate Problems with Fractions

Find the unit rate of ratios involving fractions.

Find the unit rate and use it to solve problems.

7.RP.A.1 7.RP.A.3

Solve ratio and rate problems by setting up a proportion.

Solve ratio and rate problems by setting up a proportion, including part-part-whole problems.

Solve multi-step ratio and rate problems using proportional reasoning, including fractional price increase and decrease, commissions, and fees.

Use proportional reasoning to solve real-world, multi-step problems.

7.RP.A.1 7.RP.A.2 7.RP.A.3

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Chapter 3, Lesson 6: Proportional and Nonproportional Relationships

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Identifying Proportional And Non-Proportional Relationships In Tables (continued)

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Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships

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Lesson 1: Representing Linear Nonproportional Relationships

• Representing Linear Nonproportional Relationships – Page No. 98
• Representing Linear Nonproportional Relationships – Page No. 99
• Representing Linear Nonproportional Relationships – Page No. 100

Lesson 2: Determining Slope and y-intercept

• Determining Slope and y-intercept – Page No. 104
• Determining Slope and y-intercept – Page No. 105
• Determining Slope and y-intercept – Page No. 106

Lesson 3: Graphing Linear Nonproportional Relationships Using Slope and y-intercept

• Graphing Linear Nonproportional Relationships Using Slope and y-intercept – Page No. 110
• Graphing Linear Nonproportional Relationships Using Slope and y-intercept – Page No. 111
• Graphing Linear Nonproportional Relationships Using Slope and y-intercept – Page No. 112

Lesson 4: Proportional and Nonproportional Situations 

• Proportional and Nonproportional Situations – Page No. 117
• Proportional and Nonproportional Situations – Page No. 118
• Proportional and Nonproportional Situations – Page No. 119
• Proportional and Nonproportional Situations – Page No. 120

Lesson 5: Representing Linear Nonproportional Relationships – Model Quiz

• Representing Linear Nonproportional Relationships – Model Quiz – Page No. 121

Mixed Review 

• Mixed Review – Page No. 122

Guided Practice – Representing Linear Nonproportional Relationships – Page No. 98

Make a table of values for each equation.

Explanation: y = 2x + 5 Choose several values for x and substitute in the equation to find y. x = 2(-2) + 5 = 1 x = 2(-1) + 5 = 3 x = 2(0) + 5 = 5 x = 2(1) + 5 = 7 x = 2(2) + 5 = 9

Explanation: y = \(\frac{3}{8}\)x − 5 Choose several values for x and substitute in the equation to find y. x = 3/8(-8) – 5 = -8 x = 3/8(0) – 5 = -5 x = 3/8(8) – 5 = -2 x = 3/8(16) – 5 = 1 x = 3/8(24) – 5 = 4

Explain why each relationship is not proportional.

Answer: The relationship is not proportional

Explanation: Find y/x 3/0 = undefined 7/2 = 3.5 11/4 = 2.75 15/6 = 2.5 19/8 = 2.375 The ratio is not constant, hence relationship is not proportional.

Answer: The graph is a straight line but does not pass through the origin. So, the relationship is not proportional.

Complete the table for the equation. Then use the table to graph the equation.

Explanation: y = x – 1 Choose several values of x and substitute in the equation to find y to draw a table. x = -2; y = -2 – 1 = -2 x = -1; y = -1 -1 = -2 x = 0; y = 0 -1 = -1 x = 1; y = 1 – 1 = 0 x = 2; y = 2 -1 = 1 Also, Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions

Essential Question Check-In

Question 6. How can you choose values for x when making a table of values representing a real-world situation? Type below: ____________

Answer: When choosing values for x in a real-world situation, you choose positive values with an appropriate interval to represent the array of data.

Independent Practice – Representing Linear Nonproportional Relationships – Page No. 99

State whether the graph of each linear relationship is a solid line or a set of unconnected points. Explain your reasoning.

Question 7. The relationship between the number of $4 lunches you buy with a $100 school lunch card and the money remaining on the card ____________

Answer: Set of unconnected points.

Explanation: You cannot buy a fractional part of a lunch. Set of unconnected points.

Question 8. The relationship between time and the distance remaining on a 3-mile walk for someone walking at a steady rate of 2 miles per hour. ____________

Answer: A solid line

Explanation: The relationship between time and the distance remaining on a 3-mile walk for someone walking at a steady rate of 2 miles per hour. The distance remaining can be a fraction. The time can be in a fraction as well. A solid line

Explanation: y = 8x + 12 Choose several values for x and substitute in the equation to find y.

Question 9. b. Draw a graph to represent the situation. Include a title and axis labels. Type below: ____________

Explanation: Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions

Answer: It is not proportional as the graph does not pass through the origin

Explanation: When a relationship is proportional, the graph of the equation passes through the origin. It is not proportional as the graph does not pass through the origin

Question 9. d. Does it make sense to connect the points on the graph with a solid line? Explain. Type below: ____________

Explanation: No; The subscription is renewed for the entire year and cannot be done for a fraction of the year. The number of years must be a whole numb, so the total cost goes up in $8 increments.

Representing Linear Nonproportional Relationships – Page No. 100

Question 10. Analyze Relationships A proportional relationship is a linear relationship because the rate of change is constant (and equal to the constant of proportionality). What is required of a proportional relationship that is not required of a general linear relationship? Type below: ____________

Answer: The ratio between one quantity to the other quantity should be constant for a proportional linear relationship. The graph should be a straight line that passes through the origin.

Lesson 4.1 Representing Linear Nonproportional Relationships Answer Key Question 11. Communicate Mathematical Ideas Explain how you can identify a linear non-proportional relationship from a table, a graph, and an equation. Type below: ____________

Answer: In a table, the ratios y/x will not be equal. A graph will not pass through the origin. An equation will be in the form y = mx + b where b is not equal to 0.

Focus on Higher Order Thinking

Answer: The ratio is not constant, hence the relationship cannot be proportional.

Explanation: Find y/x 90/1 = 90 150/2 = 75 210/3 = 70 270/4 = 67.5 330/5 = 66 The ratio is not constant, hence the relationship cannot be proportional.

Question 13. Make a Conjecture Two parallel lines are graphed on a coordinate plane. How many of the lines could represent proportional relationships? Explain. Type below: ____________

Answer: Maximum one

Explanation: When there are two parallel lines, only one can pass through the origin and a line representing a proportional relationship must pass through the origin. Maximum one

Guided Practice – Determining Slope and y-intercept – Page No. 104

Find the slope and y-intercept of the line in each graph.

Answer: slope m = -2 y-intercept b = 1 m = -2 b = 1

Explanation: Find the slope using two points from the grapgh by Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 1) and (x2, y2) = (2, -3) Slope m = (y2 -y1)/(x2 – x1) = (-3 – 1)/(2 – 0) = -4/2 = -2 From the graph when x = 0 y-intercept (b) = 1

Answer: slope m = 5 y-intercept b = -15 m = 5 b = -15

Explanation: Find the slope using two points from the graph by Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (3, 0) and (x2, y2) = (0, -15) Slope m = (y2 -y1)/(x2 – x1) = (-15 – 0)/(0 – 3) = 15/3 = 5 From the graph when x = 0 y-intercept (b) = -15

Answer: slope m = 3/2 y-intercept b = -2 m = 3/2 b = -2

Explanation: Find the slope using two points from the graph by Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, -2) and (x2, y2) = (2, 1) Slope m = (y2 -y1)/(x2 – x1) = (1 – (-2))/(2 – 0) = 3/2 From the graph when x = 0 y-intercept (b) = -2

Answer: slope m = -3 y-intercept b = 9 m = -3 b = 9

Explanation: Find the slope using two points from the grapgh by Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (3, 0) and (x2, y2) = (0, 9) Slope m = (y2 -y1)/(x2 – x1) = (9 – 0))/(0 – 3) = -9/3 = -3 From the graph when x = 0 y-intercept (b) = 9

Find the slope and y-intercept of the line represented by each table.

Answer: slope m = 3 y-intercept b = 1 m = 3 b = 1

Explanation: Find the slope using two points from the grapgh by Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (8, 25) and (x2, y2) = (6, 19) Slope m = (y2 -y1)/(x2 – x1) = (19 – 25)/(6 – 8) = 6/2 = 3 From the graph when x = 0 y-intercept (b) = 1

Answer: slope m = -4 y-intercept b = 140 m = -4 b = 140

Explanation: Find the slope using two points from the grapgh by Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (5, 120) and (x2, y2) = (15, 80) Slope m = (y2 -y1)/(x2 – x1) = (80 – 120)/(15 – 5) = -40/10 = -4 From the graph when x = 0 y-intercept (b) = 140

Question 7. How can you determine the slope and the y-intercept of a line from a graph? Type below: ____________

Answer: Choose any two points on the line from the graph and use it to find the slope. Determine the point where the line crosses the y-axis to find the y-intercept.

Independent Practice – Determining Slope and y-intercept – Page No. 105

Answer: Find the slope using two points Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 125) and (x2, y2) = (3, 225) Slope m = (y2 -y1)/(x2 – x1) = (225 – 125)/(3 – 1) = 100/2 = 50 Find the initial value when the value of x is 0 Work backward from x = 1 to x = 0 (175 – 125)/(2 – 1) = 50/1 = 50 Subtract the difference of x and y from the first point. x = 1 – 1 = 0 y = 125 – 50 = 75 y-intercept (b) = 75 The slope/rate of change represents the increase in the cost of cleaning the rooms for a unit increase in the number of rooms. The y-intercept shows the initial cost of carpet cleaning.

Explanation: Find the slope using two points Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 17) and (x2, y2) = (2, 29) Slope m = (y2 -y1)/(x2 – x1) = (29 – 17)/(2 – 1) = 12/1 = 12 Find the initial value when the value of x is 0 Work backward from x = 1 to x = 0 (29 – 17)/(2 – 1) = 12/1 = 12 Subtract the difference of x and y from the first point. x = 1 – 1 = 0 y = 17 – 12 = 5 The cost to park for a day is $5.

Question 9. b. What will Lin pay if she rents a paddleboat for 3.5 hours and splits the total cost with a friend? Explain. $ _____________

Answer: $23.5

Explanation: When Lin paddles for 3.5hr Total Cost = 3.5(12) + 5 = 47 Lin’s cost = 47/2 = 23.5

Answer: $25

Explanation: Find the slope using two points Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 55) and (x2, y2) = (2, 85) Slope m = (y2 -y1)/(x2 – x1) = (85 – 55)/(2 – 1) = 30/1 = 30 Rate of change is $30 for per lesson Find the initial value when the value of x is 0 Work backward from x = 1 to x = 0 (85 – 55)/(2 – 1) = 30/1 = 30 Subtract the difference of x and y from the first point. x = 1 – 1 = 0 y = 55 – 30 = 25 The initial value of the group lesson is $25.

Question 10. b. Find the rate of change and the initial value for the private lessons. Type below: _____________

Explanation: Find the slope using two points Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 75) and (x2, y2) = (2, 125) Slope m = (y2 -y1)/(x2 – x1) = (125 – 75)/(2 – 1) = 50/1 = 50 The rate of change is $50 per lesson Find the initial value when the value of x is 0 Work backward from x = 1 to x = 0 (125 – 75)/(2 – 1) = 50/1 = 50 Subtract the difference of x and y from the first point. x = 1 – 1 = 0 y = 75 – 50 = 25 The initial value of the private lesson is $25.

Question 10. c. Compare and contrast the rates of change and the initial values. Type below: _____________

Answer: The initial value for both types of lessons is the same. The rate of change is higher for private lessons than group lesson

Explanation: Compare the results of a and b The initial value for both types of lessons is the same. The rate of change is higher for private lessons than group lesson

Vocabulary – Determining Slope and y-intercept – Page No. 106

Explain why each relationship is not linear.

Answer: The rate of change is not constant, hence the relationship is not linear

Explanation: Find the rate of change using two points Slope m = (y2 -y1)/(x2 – x1) (6.5 – 4.5)/(2 – 1) = 2 (8.5 – 6.5)/(3 – 2) = 2 (11.5 – 8.5)/(4 – 3) = 3 The rate of change is not constant, hence the relationship is not linear

Explanation: Find the rate of change using two points Slope m = (y2 -y1)/(x2 – x1) (126 – 140)/(5 – 3) = -7 (110 – 126)/(7 – 5) = -8 (92 – 110)/(9 – 7) = -9 The rate of change is not constant, hence the relationship is not linear

Question 13. Communicate Mathematical Ideas Describe the procedure you performed to derive the slope-intercept form of a linear equation. Type below: _____________

Answer: Express the slope m between a random point (x, y) on the line and the point (0, b) where the line crosses the y-axis. Then solve the equation for y.

Question 14. Critique Reasoning Your teacher asked your class to describe a real-world situation in which a y-intercept is 100 and the slope is 5. Your partner gave the following description: My younger brother originally had 100 small building blocks, but he has lost 5 of them every month since. a. What mistake did your partner make? Type below: _____________

Answer: If the brother loses 5 blocks every month, the slope would be -5 and not 5.

Explanation: When the initial value is decreasing, the slope is negative. If the brother loses 5 blocks every month, the slope would be -5 and not 5.

Question 14. b. Describe a real-world situation that does match the situation. Type below: _____________

Answer: I bought a 100-card pack and buy 5 additional cards every month.

Explanation: Real-world situation I bought a 100-card pack and buy 5 additional cards every month.

Answer: After parking 60 cars, John’s earnings become $600 double his initial base salary of $300. Hence, after parking 61 cars, his earning from the fee becomes more than his fixed salary.

Explanation: He earns the same in fees as his fixed salary for parking 300/5 = 60 After parking 60 cars, John’s earnings became $600 double his initial base salary of $300. Hence, after parking 61 cars, his earning from the fee becomes more than his fixed salary.

Guided Practice – Graphing Linear Nonproportional Relationships Using Slope and Y-intercept – Page No. 110

Graph each equation using the slope and the y-intercept.

Explanation: y = 1/2 x – 3 The y-intercept is b = -3. Plot the point that contains the y-intercept (0, -3) The slope m = 1/2. Use the slope to find a second point. From (0, -3) count 1 unit up and 2 unit right. The new point is (2, -2) Draw a line through the points

Explanation: y = -3x + 2 The y-intercept is b = 2. Plot the point that contains the y-intercept (0, 2) The slope m = -3/1. Use the slope to find a second point. From (0, 2) count 3 unit down and 1 unit right. The new point is (1, -1) Draw a line through the points

Explanation: y = 4x + 2 The y-intercept is b = 2. Plot the point that contains the y-intercept (0, 2) The slope m = 4. Use the slope to find a second point. From (0, 2) count 4 unit up and 1 unit right. The new point is (1, 6) Draw a line through the points

Question 3. b. Discuss which points on the line do not make sense in this situation. Then plot three more points on the line that do make sense. Type below: _____________

Explanation: The points with a negative value of x or y do not make sense as the number of cards or weeks cannot be negative.

Question 4. Why might someone choose to use the y-intercept and the slope to graph a line? Type below: _____________

Answer: When the relationship is given in the form y = mx + b, the y-intercept (b) and the slope (m) are easily accessible and easily calculable. Therefore, it is a good practice to use them to graph the line.

Independent Practice – Graphing Linear Nonproportional Relationships Using Slope and Y-intercept – Page No. 111

Question 5. Science A spring stretches in relation to the weight hanging from it according to the equation y = 0.75x + 0.25 where x is the weight in pounds and y is the length of the spring in inches. a. Graph the equation. Include axis labels. Type below: _____________

Explanation: y = 0.75x + 0.25 Slope m = 0.75 and y-intercept = 0.25 Plot the point that contains the y-intercept (0, 0.25) The slope is m = 0.75/1. Use the slope to find a second point. From (0,0.25) count 0.75 unit up and 1 unit right. The new point is (1, 1)

Question 5. b. Interpret the slope and the y-intercept of the line. Type below: _____________

Answer: The slope represents the increase in the length of spring in inches for each increase of pound of weight. y-intercept represents the unstretched length of the spring When there is no weight attached.

Question 5. c. How long will the spring be if a 2-pound weight is hung on it? Will the length double if you double the weight? Explain Type below: _____________

Answer: When there is a 2-pound weight hung, the length of the spring would be 1.75 inches. No, When there is a 4-pound weight hung, the length of the spring would be 3.25 inches and not 3.5 inches.

Look for a Pattern

Identify the coordinates of four points on the line with each given slope and y-intercept.

Question 6. slope = 5, y-intercept = -1 Type below: _____________

Answer: (2, 9) (3, 14)

Explanation: slope = 5, y-intercept = -1 Plot the point that contains the y-intercept (0, -1) The slope is m = 5/1. Use the slope to find a second point. From (0, -1) count 5 unit up and 1 unit right. The new point is (1, 4) Follow the same procedure to find the remaining three points. (2, 9) (3, 14)

Question 7. slope = -1, y-intercept = 8 Type below: _____________

Answer: (2, 6) (3, 5)

Explanation: slope = -1, y-intercept = 8 Plot the point that contains the y-intercept (0, 8) The slope is m = -1/1. Use the slope to find a second point. From (0, 8) count 1 unit down and 1 unit right. The new point is (1, 7) Follow the same procedure to find the remaining three points. (2, 6) (3, 5)

Question 8. slope = 0.2, y-intercept = 0.3 Type below: _____________

Answer: (2, 0.7) (3, 0.9)

Explanation: slope = 0.2, y-intercept = 0.3 Plot the point that contains the y-intercept (0, 0.3) The slope is m = 0.2/1. Use the slope to find a second point. From (0, 0.3) count 0.2 unit up and 1 unit right. The new point is (1, 0.5) Follow the same procedure to find the remaining three points. (2, 0.7) (3, 0.9)

Question 9. slope = 1.5, y-intercept = -3 Type below: _____________

Answer: (2, 0) (3, 1.5)

Explanation: slope = 1.5, y-intercept = -3 Plot the point that contains the y-intercept (0, -3) The slope is m = 1.5/1. Use the slope to find a second point. From (0, -3) count 1.5 unit up and 1 unit right. The new point is (1, -1.5) Follow the same procedure to find the remaining three points. (2, 0) (3, 1.5)

Question 10. slope = −\(\frac{1}{2}\), y-intercept = 4 Type below: _____________

Answer: (4, 2) (6, 1)

Explanation: slope = −\(\frac{1}{2}\), y-intercept = 4 Plot the point that contains the y-intercept (0, 4) The slope is m = −\(\frac{1}{2}\)/1. Use the slope to find a second point. From (0, 4) count 1 unit down and 2 unit right. The new point is (2, 3) Follow the same procedure to find the remaining three points. (4, 2) (6, 1)

Question 11. slope = \(\frac{2}{3}\), y-intercept = -5 Type below: _____________

Answer: (6, -1) (9, 1)

Explanation: slope = \(\frac{2}{3}\), y-intercept = -5 Plot the point that contains the y-intercept (0, -5) The slope is m = \(\frac{2}{3}\). Use the slope to find a second point. From (0, -5) count 2 unit up and 3 unit right. The new point is (3, -3) Follow the same procedure to find the remaining three points. (6, -1) (9, 1)

Question 12. A music school charges a registration fee in addition to a fee per lesson. Music lessons last 0.5 hour. The equation y = 40x + 30 represents the total cost y of x lessons. Find and interpret the slope and y-intercept of the line that represents this situation. Then find four points on the line. Type below: _____________

Answer: y = 40x + 30 Slope = 40 y-intercept = 30 The slope represents the fee of the classes per lesson and the y-intercept represents the registration fee. Plot the point that contains the y-intercept (0, 30) The slope is m = 40/1. Use the slope to find a second point. From (0, 30) count 40 units up and 1 unit right. The new point is (1, 70) Follow the same procedure to find the remaining three points. (2, 110) (3, 150)

Graphing Linear Nonproportional Relationships Using Slope and Y-intercept – Page No. 112

Answer: Yes

Explanation: Yes; Since the horizontal and vertical gridlines each represent 25 units, hence moving up 3 gridlines and right 1 gridline represents a slope of 75/25 or 3

Question 13. b. Graph the equation y = 3x + 50. Include axis labels. Then interpret the slope and y-intercept. Type below: _____________

Explanation: Slope = 3 y-intercept = 50 The slope represents the fee of the classes per lesson and the y-intercept represents the registration fee. Plot the point that contains the y-intercept (0, 50) The slope is m = 3/1. Use the slope to find a second point. From (0, 50) count 3 units up and 1 unit right. The new point is (1, 53)

Question 13. c. How many visits to the pool can a member get for $200? ______ visits

Answer: 50 visits

Question 14. Explain the Error A student says that the slope of the line for the equation y = 20 − 15x is 20 and the y-intercept is 15. Find and correct the error. Type below: _____________

Answer: The slope is -15 as it represents the change in y per unit change in x. The y-intercept is 20 when x = 0.

Explanation: y = 20 − 15x The slope is -15 as it represents the change in y per unit change in x. The y-intercept is 20 when x = 0.

Question 15. Critical Thinking Suppose you know the slope of a linear relationship and a point that its graph passes through. Can you graph the line even if the point provided does not represent the y-intercept? Explain. Type below: _____________

Answer: Yes. You can plot the given point and use the slope to find a second point. Connect the points by drawing a line.

Explanation: let’s tale the example y = 3x y = 3x – 3 y = 3x + 3 We notice that the lines are parallel to each other: the slopes of the lines are equal but the y-intersection point differs.

Guided Practice – Proportional and Nonproportional Situations – Page No. 117

Determine if each relationship is a proportional or nonproportional situation. Explain your reasoning.

Answer: Proportional relationship

Explanation: Proportional relationship The graph passes through the origin. Graph of a proportional relationship must pass through the origin

Answer: Non-proportional relationship

Explanation: The graph does not pass through the origin. The graph of a proportional relationship must pass through the origin Non-proportional relationship

Lesson 4 Skills Practice Proportional and Nonproportional Relationships Answer Key Question 3. q = 2p + \(\frac{1}{2}\) Compare the equation with y = mx + b. _____________

Answer: q = 2p + \(\frac{1}{2}\) The equation is in the form y = mx + b, with p being used es the variable instead of x and q instead of y. The value of m is 2, and the value b is 1/2. Since b is not 0, the relationship presented through the above equation is non-proportional.

Question 4. v = \(\frac{1}{10}\)u _____________

Explanation: v = \(\frac{1}{10}\)u Compare with the form of equation y = mx + b. The equation represents a proportional relationship if b = 0 Proportional relationship

Proportional and Nonproportional Situations – Page No. 118

The tables represent linear relationships. Determine if each relationship is a proportional or nonproportional situation.

Answer: proportional relationship

Explanation: Find the ratio y/x 12/3 = 4 36/9 = 4 84/21 = 4 Since the ratio is constant, the relationship is proportional.

Answer: non-proportional

Explanation: Find the ratio y/x 4/22 = 2/11 8/46 = 4/23 10/58 = 5/29 Since the ratio is not constant, the relationship is non-proportional.

Explanation: Find the ratio y/x 12/15,000,000 = 0.0000008 16/20,000,000 = 0.0000008 20/25,000,000 = 0.0000008 Since the ratio is constant, the relationship is proportional.

Question 8. How are using graphs, equations, and tables similar when distinguishing between proportional and nonproportional linear relationships? Type below: _____________

Answer: The ratio between y to x is constant when the relationship is proportional. Graphs, tables, and equations all can be used to find the ratio. The ratio is not constant when the relationship is non-proportional.

Independent Practice – Proportional and Nonproportional Situations – Page No. 119

Answer: Non-proportional

Explanation: The graph does not pass through the origin. Graph of a proportional relationship must pass through the origin Non-proportional

Question 9. b. Identify and interpret the slope and the y-intercept. Type below: _____________

Answer: Slope m = (y2 -y1)/(x2 – x1) = (12 – 10)/(4 – 0) = 0.5 y-intercept is the weight of the empty cooler, which is 10 lbs.

Explanation: Find the slope using two points from the grapgh by Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 10) and (x2, y2) = (4, 12) Slope m = (y2 -y1)/(x2 – x1) = (12 – 10)/(4 – 0) = 0.5 From the graph when x = 0 y-intercept (b) = 10 y-intercept is the weight of the empty cooler, which is 10 lbs.

In 10–11, tell if the relationship between a rider’s height above the first floor and the time since the rider stepped on the elevator or escalator is proportional or nonproportional. Explain your reasoning.

Explanation: As there is a pause of 10 seconds, it would be the y-intercept of the graph (when x = 0) Non-proportional

Representing Linear Nonproportional Relationships Lesson 4.1 Answer Key Question 11. Your height, h, in feet above the first floor on the escalator is given by h = 0.75t, where t is the time in seconds. _____________

Answer: Proportional

Explanation: Comparing with y = mx + b, where b = 0 Proportional

Analyze Relationships

Compare and contrast the two graphs.

Answer: Graph A represents a linear relationship while Graph B represents an exponential relationship. They both pass through the origin and the value of y increases with an increase in x.

Proportional and Nonproportional Situations – Page No. 120

Question 13. Represent Real-World Problems Describe a real-world situation where the relationship is linear and nonproportional. Type below: _____________

Answer: The entrance fee to the amusement park is $8 and there is a fee of $2 per ride.

Question 14. Mathematical Reasoning Suppose you know the slope of a linear relationship and one of the points that its graph passes through. How can you determine if the relationship is proportional or nonproportional? Type below: _____________

Answer: Use the graph and the given point to determine the second point. Connect the two points by a straight line. If the graph passes through the origin, the relationship is proportional and if the graph does not pass through the origin, the relationship is non-proportional.

Answer: No, the relationship is not proportional.

Explanation: Compare the equation B to the form: y = mx + b. Since b is not equal to 0, the relationship is non-proportional. Find the ratio between the Kelvin and Degrees Celsius. Since the ratio is not constant, the relationship is non-proportional. 281.15/8 = 35.14 288.15/15 = 19.21 309.15/36 = 8.59 No, the relationship is not proportional.

Question 15. b. Is the relationship between degrees Celsius and degrees Fahrenheit proportional? Why or why not? _____________

Explanation: Compare the equation A to the form: y = mx + b. Since b is not equal to 0, the relationship is non-proportional. No, the relationship is not proportional.

4.1 Representing Linear Nonproportional Relationships – Model Quiz – Page No. 121

4.2 Determining Slope and Y-intercept

Answer: Slope = 3 y-intercept (b) = 1

Explanation: Find the slope using two points from the grapgh by Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 1) and (x2, y2) = (1, 4) Slope m = (y2 -y1)/(x2 – x1) = (4 – 1)/(1 – 0) = 3/1 From the graph when x = 0 y-intercept (b) = 1

4.3 Graphing Linear Nonproportional Relationships

Explanation: Slope = 2 y-intercept = -3 Plot the point that contains the y-intercept (0, -3) The slope is m = 2/1. Use the slope to find a second point. From (0, -3) count 2 unit up and 1 unit right. The new point is (1, -1) Draw a line through the points

4.4 Proportional and Nonproportional Situations

Answer: Since the ratio is constant, the table represents a proportional linear relationship.

Explanation: Find the ratio y/x 4/1 = 4 8/2 = 4 12/3 = 4 16/4 = 4 20/5 = 4 Since the ratio is constant, the table represents a proportional linear relationship.

Question 5. Does the graph in Exercise 2 represent a proportional or a nonproportional linear relationship? _____________

Answer: It represents a non-proportional linear relationship

Explanation: The line of the graph does not pass through the origin. The graph of a proportional relationship must pass through the origin. It represents a non-proportional linear relationship

Question 6. Does the graph in Exercise 3 represent a proportional or a nonproportional relationship? _____________

Explanation: The line of the graph does not pass through the origin. The graph of a proportional relationship must pass through the origin It represents a non-proportional linear relationship

Essential Question

Question 7. How can you identify a linear nonproportional relationship from a table, a graph, and an equation? Type below: _____________

Answer: In a table, the ratio of y/x is not constant for a non-proportional relationship. In a graph, the line of the graph does not pass through the origin for a non-proportional relationship. In an equation, the b is not equal to y = mx +b for a non-proportional relationship.

Selected Response – Mixed Review – Page No. 122

Answer: c. y = −4x − 6

Explanation: From the table, you can see that the y-intercept (when x = 0) is b = -6. Comparable to y = mx + b The table is represented by Option C y = -4x – 6

Answer: a. y = −2x + 3

Explanation: From the table, you can see that the y-intercept (when x = 0) is b = 3. Comparable to y = mx + b The Option B and D are rejected. Since the graph is slanting downwards, the slope is negative. Option C is rejected The graph represents y = -2x + 3

Answer: b. -2

Explanation: Find the rate of change (7 – 4)/(3 – 2) = (10 – 7)/(4 – 3) = 3 Find the value of y for x = 0 Works backward from x = 2 to x = 1 x = 2 – 1 = 1 y = 4 – 3 = 1 x = 1 – 1 = 0 y = 1 – 3 = -2 y-intercept = -2

Question 4. Which equation represents a nonproportional relationship? Options: a. y = 3x + 0 b. y = −3x c. y = 3x + 5 d. y = \(\frac{1}{3}\)x

Answer: c. y = 3x + 5

Explanation: For a non-proportional relationship, the equation is y = mx + b and b is not equal to 0. Option C represents a non-proportional relationship y = 3x + 5

Answer: c. 18

Explanation: Find the ratio y/x 6/4 = 3/2 Since the relationship is proportional, the ratio is constant. Using the ratio to find missing y 3/2 = y/12 y = 3/2 × 12 = 18

Question 6. What is 0.00000598 written in scientific notation? Options: a. 5.98 × 10 -6 b. 5.98 × 10 -5 c. 59.8 × 10 -6 d. 59.8 × 10 -7

Answer: c. 59.8 × 10 -6

Explanation: 0.00000598 Move the decimal 6 points 59.8 × 10 -6

Question 7. b. What is the slope of the line? _______

Answer: Slope m = -2

Explanation: Find the slope using two points from the graph by Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, -2) and (x2, y2) = (2, 1) Slope m = (y2 -y1)/(x2 – x1) = (-3 -1)/(0 + 2) = -4/2 = -2

Question 7. c. What is the y-intercept of the line? _______

Answer: y-intercept (b) = -3

Explanation: From the graph when x = 0 y-intercept (b) = -3

Question 7. d. What is the equation of the line? Type below: ____________

Answer: y = -2x – 3

Explanation: Substitute m and b in the form: y = mx + b y = -2x – 3

Conclusion:

Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships for Download. All the beginners can easily start their practice and learn the maths in an easy way. Quickly start your practice with Go Math Grade 8 Answer Key.

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• Grade 8 HMH Go Math - Answer Keys

Explanation:

\(\text{q} = 2\text{p}+\large\frac{1}{2}\)

Compare the equation with \(\text{y} = \text{mx + b}\).

\(\text{v} = \large\frac{1}{10}\small\text{u}\)

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