Exploring Linear Relations Math B College Prep

Module #2 Homework 2016-2017

Created in collaboration with Utah Middle School Math Project A University of Utah Partnership Project

San Dieguito Union High School District

2.1A Homework: Proportional Relationships* Name:

Period:

1. Fill in the table below. Graph

Table of Values

Proportional Constant

Unit Rate

Linear Proportional Relationship

2. Use the table of values to complete the following. a. Create a story that represents the table of values.

b. What equation represents this situation?

Hours

Cost ($)

3

36

6

72

9

108

12

144

c. Is this situation proportional? Explain why or why not using the table of values. 3

d. What is the cost for 4 of an hour?

3. Use the graph to complete the following. a. Create a story that represents the graph.

b. What equation represents this situation?

c. Is this situation proportional? Explain why or why not using the graph.

d. $30 relates to how many hours?

4. Captain Cable charges $50.00 per hour to install internet service at your friend’s house. a. State the independent and dependent variables for this situation.

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b. Complete the table of values.

𝒚:

𝒙:

c. What is the unit rate? Describe what it represents.

1 2 1

d. What equation represents this situation?

75 e. If you only have $20 to spend, how many minutes does the cable company have to finish your installation?

f. If Super Cable charges $40 per hour to install internet service, how would the graph representing this situation differ from Captain Cable’s graph?

5. Miguel is taking a road trip and driving at a constant speed of 65 mph. He is trying to determine how many miles he can drive based on how many hours he drives. a. What is the independent variable for this situation? What is the dependent variable?

b. Complete the graph and table below for this relationship. Label the columns in your table and the axes in your graph. Should the points be connected on your graph? Explain why or why not.

x:

y: 300

0 240

1 180

130 120

195 60

4 0

1 2 3 4 5 6 7 8 9 10

c. What is the unit rate? What does the unit rate represent in the context?

d. Write an equation that represents this situation.

e. Is this relationship proportional? Explain why or why not using the table of values and graph. f. If Josh takes a road trip and drives at a constant speed of 40 mph, how would the graph representing this situation differ from Miguel’s graph? SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

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6. DJ is pouring cement for his backyard patio that has an area of 100 square feet. The cement comes out of the truck at a constant rate. It is very important that he gets all the cement poured before 12:00 noon when it gets too hot for the cement to be mixed properly. It is currently 11:00 AM and he has poured 75 square feet of concrete in the last 3 hours. a. What is the independent variable? What is the dependent variable?

b. Complete the graph and table below for this relationship. Label the columns in your table and the axes in your graph. Should the points be connected on your graph? Explain why or why not.

x:

100

y:

90

0

80 70

1

60

50

50 40

3 (11:00 am)

30 20

100

10 0

1

2

3

4

5

c. Write an equation that represents this relationship.

d. Does this relationship have a unit rate? Describe what it represents.

e. Will DJ finish the job by noon? Justify your answer.

f. If Kevin pours 35 square feet of concrete per hour, how would the graph representing this situation differ from DJ’s graph?

7. Barry and Felicity are making super smoothies to re-energize them after a long workout. Felicity follows the recipe which calls for 2 cups of strawberries for every 3 bananas. Barry wants twice as much as Felicity, so he makes a smoothie with 4 cups of strawberries and 5 bananas. Barry tastes his smoothie and says, “This tastes too tart; there are too many strawberries!” a. Explain why Barry’s smoothie is too tart.

b. Find and describe the unit rate for Felicity’s smoothie.

c. Find and describe the unit rate for Barry’s smoothie.

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d. Write an equation that relates the cups of strawberries, 𝑥, to the number of bananas, 𝑦, for Barry’s and Felicity’s smoothies. Felicity:

Barry:

e. Using your equations above, complete the tables of values and graph both lines on the same coordinate plane. Label your axes and lines on the graph. Felicity’s Smoothie x

y

x

0

0

0

2 6 9

7

Barry’s Smoothie

6

y

5 2.5

4

5

3 2

6

1 0

f. Which line is steeper? Explain why this occurs.

1

2

3

4

5

6

7

8. Ben, Boston, and Bryton have each designed a remote control monster truck. They lined them up to race in the driveway. The lines on the graph below show the distance in inches that each monster truck travels over time in seconds. a. Bryton states that each truck is traveling at a constant rate. Is his statement correct? Explain why or why not.

Ben

Bryton

b. What do the values in the ordered pairs given on the graph represent?

Distance (inches)

(6,8)

Boston (1,4)

c. Find the unit rate for each truck.

(4,2)

Ben: Bryton:

0

Time (seconds)

Boston: d. Which truck is moving the fastest? Explain why.

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9. The graph shows the distance two snowboarders have traveled down a hill for several seconds. Hannah is traveling 18 meters per second. a. Which equation below is the best choice to describe the distance Torah travels after 𝑥 seconds?

y  29 x

y  17 x

y  10 x

y  18x

b. Explain your reasoning for your choice above.

c. Christina travels at a unit rate of 10 meters per second, going down the same hill. Draw a line on the graph that represents her speed.

Spiral Review: Directions: Solve each equation. Show all of your work. 10.

3 8

8 − 𝑛 = −4

3

1

11. − 4 − 𝑥 = 2

12. 4(𝑥 + 3) − 5 = 7(𝑥 − 1) + 9

13.

14. 1.4𝑎 + 1.1 = 8.3 − 𝑎

15.

16. (5𝑥 + 9) − (3𝑥 − 13) = 2(11 + 𝑥)

17.

2(1 + 4𝑡) = 8 − (3 − 8𝑡)

7𝑛−1 8

𝑛 3

3

=8−𝑛

5

+ 4 = 6𝑛 − 1

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2.1B Homework: Proportional Relationships as Linear Relationships* Name:

Period:

1. Describe in words, the characteristics of a table of values that represents a proportional linear relationship and a table of values that represents a non-proportional linear relationship. Provide examples to support your reasoning. Proportional 𝑥

𝑦

Non-Proportional 𝑦 𝑥

𝑥

𝑦

𝑦 𝑥

2. Describe in words, the characteristics of an equation that represents a proportional linear relationship and an equation that represents a non-proportional linear relationship. Justify your description by writing an equation for each of your table of values in problem #1.

Directions: Do the table of values below have a first difference. Explain why or why not. 3.

4.

x

y

-8

0

3

1

-3

1

6

2

2

2

12

3

7

3

24

4

12

4

48

x

y

0

5. Sunset Wave charges an entrance fee of $5.00 and then charges $2.00 per hour to swim. Paradise Pool charges $3.00 per hour to swim. Complete the table of values that represent each situation. Sunset Wave t (hours)

C ( dollars)

Paradise Pool C t

t (hours)

0

0

1

1

2

2

3

3

4 5

C (dollars)

C t

4 5

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a. Graph both situations on the same coordinate plane and label each line and each axis. Should the points on your graph be connected? Explain why or why not. 20

b. State the unit rate for each situation. What does the unit rate represent in the context?

18 16 14 12

c. Write an equation that represents the cost, C, for t hours for each situation.

10 8 6

Sunset Wave: Paradise Pool:

4 2 0

1

2

3

4

5

6

7

8

9

10

d. Which situation, Sunset Wave or Paradise Pool, represents a proportional relationship? Justify your answer using your tables, graphs and equation.

6. During her Tuesday shift at Soccer World, Fiona sells the same amount of cleats per hour. Two hours into her shift, Fiona sold 8 pairs. a. What is the independent variable? What is the dependent variable?

b. Complete the table and label the columns. Complete the graph, label your axes and name your line Tuesday. Should the points be connected on your graph? Explain why or why not.

x:

20

y:

18 16

0

14

1

12 10

8

8 6

3

4 2 0

1

2

3

4

5

6

7

8

9

10

c. Write an equation that represents this relationship.

d. State the unit rate and constant of proportionality.

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7.

On Saturday, Fiona gets to work 15 minutes early and sells three pairs of cleats before her shift even begins. She then sells 4 pairs every hour for the rest of her shift. a. Graph this relationship on the same coordinate plane that shows Tuesday’s information. Label your line Saturday. b. Write an equation that represents this relationship.

c. State the unit rate and constant of proportionality.

d. Which situation, Tuesday or Saturday, represents a non-proportional relationship? Justify using your graphs and equations.

e. Does Fiona sell cleats at a faster rate on Tuesday or Saturday? Explain your reasoning.

Directions: State if the given relationship is proportional or non-proportional. Explain your reasoning. 8. A candle is 10 inches tall when lit. The candle burns 2 inches in the first hour. After 2 more hours, the candle is 4 inches tall.

10.

x

y

3

14

1

8

8

-19

10

-25

9.

x

y

-2

-5

4

10

10

25

16

40

11.

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2.1C Homework: Connect the Rule to the Pattern* Name:

Period:

1. Use the pattern below to complete the following.

Stage 1

Stage 2

Stage 3

Stage 4

a. Draw the figure at stage 4 in the space above. How did you draw your figure in stage 4? Explain or show on the picture how you see the pattern growing from one step to the next.

b. How many blocks are in stage 4? Stage 10?

c. Write a rule that gives the number of blocks, b, for any stage, s. Show how your rule relates to the pattern (geometric model). Simplify your rule.

Stage 1

Stage 2

Stage 3

d. Write a different rule that gives the total number of blocks b for any stage, s. Show how your rule relates to the pattern (geometric model). Simplify your rule.

Stage 1

Stage 2

Stage 3

e. Use your rule to determine the number of blocks in stage 100.

f.

Use your rule to determine which stage has 28 blocks.

g. How many blocks are in stage 0 of the pattern? How does the number of blocks in stage 0 relate to the simplified form of your rule?

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2. Use the pattern below to complete the following.

Stage 1

Stage 2

Stage 3

Stage 4

a. Draw the figure at stage 4 in the space above. How did you draw your figure in stage 4? Explain or show on the picture how you see the pattern growing from one step to the next.

b. How many blocks are in stage 4? Stage 10?

c. Write a rule that gives the total number of blocks, b, for any stage, s. Show how your rule relates to the pattern (geometric model). Simplify your rule.

Stage 1

Stage 2

Stage 3

d. Write a different rule that gives the total number of blocks, b, for any stage, s. Show how your rule relates to the pattern (geometric model). Simplify your rule.

Stage 1

Stage 2

Stage 3

e. Use your rule to determine the number of blocks in stage 100.

f.

Use your rule to determine which stage has 37 blocks.

g. How many blocks are in stage 0 of the pattern? How does the number of blocks in stage 0 relate to the simplified form of your rule?

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Spiral Review: Directions: Solve each equation or inequality. Show all of your work. 3.

5𝑥+1 8

= −3

5. −12 ≥

4−𝑛 4

7. 0.65 = −0.5𝑥 + 2.45

9.

2 𝑥 5

+1>4+𝑥

1 3

4. 𝑎 − 7 = − (6 − 3𝑎)

4

1

6. 3𝑛 − 5 = 3 𝑛

8. 3(𝑥 + 1) − 5 = 3𝑥 − 2

10.

1 4

1

−𝑛 = 3+𝑛

Directions: Solve each inequality. Show all of your work. Graph your answer on a number line. 11. −4𝑛 + 9 ≤ 𝑛 − 21

13.

2 𝑎 3

−5< 4+𝑎

12.

4𝑥−2 5

≥ −4

14. 5(𝑥 + 4) − 2(𝑥 + 5) > 6(𝑥 + 1) − 1

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2.1D Homework: Representations of a Linear Pattern* Name:

Period:

1. Use the pattern to complete the following.

a. How many new blocks are added to the pattern from one stage to the next?

Stage 1

Stage 2

b. Complete the table and state the first difference and unit rate.

Stage 3

Stage (s)

# of Blocks (b)

1

c. Show where you see the unit rate in your table.

2 3 4 d. Create a graph of this data. Show the unit rate on your graph.

5 y 14 12

e. What is the equation that gives the total number of blocks, 𝑏, for any stage, 𝑠? Where do you see the different parts of the equation in the geometric model, table, and graph?

10 8 6 4

Equation:

2 0

Model

Table

2

4

6

8

10 12 14

x

Graph

f. Is this pattern a linear pattern? Use supporting evidence from each of the representations to justify your answer.

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2. Use the pattern to complete the following.

a. How many new blocks are added to the pattern from one stage to the next? Stage 1 b. Complete the table and state the first difference and unit rate.

Stage 2

Stage (𝑠)

Stage 3

# of Blocks (𝑏)

1 2 c. Show where you see the unit rate in your table.

3 4 5

d. Create a graph of this data. Show the unit rate on your graph.

y 14 12 10

e. What is the equation that gives the total number of blocks, 𝑏, for any stage, 𝑠? Where do you see the different parts of the equation in the geometric model, table, and graph?

8 6 4 2 0

Equation:

Model

2

4

6

Table

8

10 12 14

x

Graph

f. Explain in words how the graph would differ if the equation changed to 𝑏 = 2𝑠.

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3. Jan is using tiles to make a pattern for her new patio garden. Here are the first three stages in her pattern:

Stage 1

Stage 2

Stage 3

a. Complete the table: Stage

1

2

3

4

5

Number of tiles

b. Graph this situation and label each axis. Should your points be connected? Explain why or why not.

c. Use your graph to predict how many tiles are needed at stage 10.

d. Write an equation that gives the number of tiles, 𝑦, for any given stage, 𝑥. Use your equation to find the exact amount of tiles needed at stage 10.

e. How many tiles are needed at stage 25?

f. If Jan used 137 tiles in her patio garden, how many stages did she complete?

g. For your equation, where is the constant value shown on your graph?

Spiral Review: 4. The first side of a triangle is 3 inches shorter than the second side. The third side is 4 times as long as the first side. The perimeter of the triangle is 27 inches. Find the length of each side. Show all of your work.

5. Adam is 20 years younger than Brian. In two years, Brian will be twice as old as Adam will be. How old are they now? Show all of your work.

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2.1E Homework: Representations of a Linear Context* Name:

Period:

1. A landscape company charges a certain rate per hour and also an initial fee of $100 for equipment. a. State the independent and dependent variables.

b. Complete the table of values and graph for this situation. Label table columns and each axis on the graph. Should your points be connected on your graph? Explain why or why not.

1400

y:

x:

1200

500 1000

4 1300

800 600

c. What is the unit rate? Describe what it represents. Show your unit rate on the graph.

400 200

d. What is the y-intercept of your graph? What does the y-intercept represent in the context?

0

2

4

6

8

10

12

14

e. Write an equation for the cost, 𝑐, of landscaping for ℎ hours.

f. Is this context linear? Explain why or why not.

g. How much will it cost if the crew worked for 8 hours?

h. If you have $5,000 to spend on landscaping, how many hours can the crew work?

i. How would the context and equation change if the unit rate was 150 and there was no equipment fee?

j. How would the graph change to reflect this new context?

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2. Serena is always losing her tennis balls. At the beginning of tennis season, she has 20 tennis balls. a. State the independent and dependent variables.

b. Complete the table of values and graph for this situation. Label the table columns and each axis on the graph. Should your points be connected on your graph? Explain why or why not. 20

y:

x:

18 16

4

14

6

12

4

10 8

c. On average, how many tennis balls does Serena lose each week?

6 4 2 0

1

2

3

4

5

6

7

8

9

10

d. What is the y-intercept of your graph? What does the y-intercept represent in the context?

e. Write an equation that represents this relationship.

f. Is this context linear? Explain why or why not.

g. In how many weeks will Serena run out of tennis balls? How is this information shown on the graph?

h. How would the context and equation change if the unit rate was -3?

i. How would the graph change?

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Directions: In each problem, complete the remaining representations that are not given. When needed, label the columns in the table and the axes on the graph. 3. A Community Garden Context Gavin is buying tomato plants to plant in his local community garden. Tomato plants are $9 per flat (a flat contains 36 plants). No partial flats can be purchased. Consider the relationship between total cost and number of flats purchased.

Table independent variable: dependent variable: x:

y:

2 4 6

Graph Should the points on your graph be connected? Explain why or why not.

Equation

Is this situation proportional? Explain why or why not using the context. 126 108 90

How many flats can Gavin purchase if he has $150 to spend? Show your work.

72 54 36 18 0

2

4

6

8

10 12 14

a. What is the unit rate? Describe what it represents

b. What is the y-intercept of your graph? Where do you see the y-intercept in the graph? Where do you see the y-intercept in the equation?

c. What does the y-intercept represent in the context?

d. How would the context and equation change if the unit rate was 10 and there was a $5 entrance fee to the garden?

e. Would the graph of the new line be steeper or less steep than the original?

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4. Context

Table x:

Graph

y:

Equation

y

𝑦 = 5𝑥 + 30

70 60

What is the unit rate? Describe what it represents.

50 40

What is the y-intercept of your graph? What does the y-intercept represent in your context?

30 20 10 0

2

4

6

8

10 12 14

x

Directions: Determine if the statement is true or false. If false, re-write the statement to make it true. 5. In a non-proportional relationship, the y-intercept is (0,0)

6. All proportional and non-proportional relationships are linear if they show a constant unit rate.

7. A situation with unit rate 5 has a graph that is steeper than a situation with unit rate 7.

Spiral Review: 8. A mistake was made when solving the equation below. Circle and explain the mistake made. Solve the equation correctly. Equation: −3(𝑥 − 2) − 5𝑥 = 10

Mistake made:

−3𝑥 − 6 − 5𝑥 = 10 −8𝑥 − 6 = 10

Equation solved correctly:

−8𝑥 = 16

𝑥 = −2 SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

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Section 2.1: Review Name:

Period:

1. A proportional constant of being planted.

1 3

relates the number of inches a flower grows to the number of weeks since

a. State the independent and dependent variables. b. Complete the table.

x:

1

3

9

30

2

y:

c. Write an equation that represents this relationship and use the equation to predict how tall the flower will be after 8 weeks. d. Can the flower continue to grow in this manner forever?

e. Is this situation proportional? Explain why or why not using your table of values and equation.

2. Laura has a job delivering newspapers. Laura gets paid $100 dollars for delivering 200 papers. a. State the independent and dependent variables.

b. Find the unit rate. State what it describes.

c. Complete the table of values and graph for this situation. Label each axis and label your line Laura. Should your points be connected? Explain why or why not. x:

180

y:

160

100 100

140 120

150

100 80

d. Write an equation for this situation. 3. Kali also has a job delivering newspapers. She gets paid $20 for expenses and then $140 for delivering 350 papers. a. Find the unit rate for this situation. State what it describes.

60 40 20 0

40

80

120

160

200

240

280

b. Graph this situation on the same coordinate plane used above. Label your line Kali. c. Write an equation for this situation. d. Which situation, Laura or Kali, represents a non-proportional relationship? Justify using your graph and equations. SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

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4. John took 16 pounds of aluminum cans to the recycling center and received $12.00. a. What rate did the center pay for the aluminum cans?

b. Is the unit rate a constant of proportionality? Explain your reasoning.

5. Colby has $30 in his piggy bank. Each week, he takes out the same amount of money. After 5 weeks, he has no money left in his piggy bank. a. How much money does Colby take out of his piggy bank each week? Explain your reasoning.

b. Is the unit rate a constant of proportionality? Explain why or why not.

6. Nayala bought 5 pounds of mangos for $6.25. a. What is the price per pound for the mangos that she bought?

b. Which line on the graph, A, B, or C, represents Nayala’s situation?

7. Emma is putting together an order for sugar, flour, and salt for her restaurant pantry. The graph shows the cost 𝑦 to buy 𝑥 pounds of sugar and flour. One line shows the cost of buying 𝑥 pounds of flour and the other line shows the cost of buying 𝑥 pounds of sugar.

Sugar

a. From the graph, which ingredient costs more to buy per pound? Justify your answer. Flour b. The cost to buy salt by the pound is less than sugar and flour. Draw a possible line that could represent the cost to buy x pounds of salt.

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8. Write two different rules that describe the pattern where 𝑠 is the stage number and 𝑏 is the total number of blocks. Explain how your rules connect to the pattern.

Stage 1 Rule 1:

Stage 2

Stage 3

Rule 2:

a. Simplify both rules. What do you notice?

b. How many blocks are in stage 0? How is this value represented in the simplified rule?

c. Use your rule to find the number of blocks in the 50th stage.

d. Which stage has 582 blocks?

9. Use the pattern to complete the following. a. Draw stage 4. b. How many new blocks are added to the pattern from one stage to the next?

c. Complete the table. State the first difference and unit rate.

Stage 1

Stage 2

Stage (s)

Stage 3

Stage 4

# of Blocks (b)

1 2

d. Show the unit rate in the table.

3 4

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e. Create a graph of this data. Show the unit rate on your graph.

y 14

# of Blocks

12

f. What is the simplified form of the equation that gives the number of blocks, 𝑏, for any stage 𝑠? Where do you see the different parts of the equation in the geometric model, table, and graph?

10 8 6 4 2

Equation: 0

2

4

6

8

10 12 14

x

Stage # Model

Table

Graph

10. For each of the representations given below, identify the unit rate and initial value or y-intercept. a. The local community center charges a monthly fee of $15 to use their facilities plus $2 per visit.

unit rate: initial value:

b. unit rate: initial value: Stage 1

c.

d.

Stage 2

x 2 3

y 10 5

4

0

1

𝑦 = 2𝑥 −3

Stage 3

unit rate: y-intercept:

unit rate: y-intercept:

e.

unit rate: y-intercept:

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Directions: Complete the remaining representations that are not given. When needed, label the columns in the table and axes on the graph. 11. Context The number of students currently enrolled at Discovery Place Preschool is 24. Enrollment is increasing by 6 students each year. Consider the relationship between the number of years from now and the number of students enrolled.

Table independent variable: dependent variable: x:

y: 2 4 6

Graph Should your points be connected on your graph? Explain why or why not.

84

Equation

Is this situation proportional? Explain why or why not using the context.

72 60 48 36

What is the unit rate? Describe what it represents.

24 12 0

2

4

6

8

10

12

14

a. What is the y-intercept of your graph? How is the y-intercept shown in the equation?

b. What does the y-intercept represent in the context?

c. How would you change the context so that the relationship between number of years and number of students enrolled is 𝑦 = 6𝑥 + 40?

d. What would happen to the graph if the maximum enrollment at the school was 72?

e. How would the graph of the line change if enrollment increased to 10 students each year?

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12. Context

Table x:

Graph

y:

Equation

What is the unit rate? Describe what it represents.

What is the y-intercept of the graph? Where do you see the y-intercept in the equation?

What does the y-intercept represent in your context?

a. How would your context, graph, and equation change if the y-intercept of the graph was changed to 75? Context Graph Equation

b. How would your context, graph and equation change if the unit rate was changed to −2? Context

Graph

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

Equation

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2.2A Homework: Rate of Change and Building Stairs* Name:

Period:

1. Design a set of stairs. Your stairs should begin at ground level and end at 5 feet high. Answer the following questions as you develop your design. a. How many steps do you want or need?

b. How deep should each step be? Why do you want this run depth?

c. How tall will each step be? Why do you want this rise height?

d. What is the total distance (total depth for all steps) you will need (at the base) for all of the stairs? This is the measurement at ground level from the stair start point to the stair end point.

2. Sketch your staircase design (as viewed from the side) on the graph below.

Height (feet)

Stair Base (feet)

3. From your staircase design above, find and record the following measurements. Rise height (total height you’ve climbed at this point)

Run depth (total distance covered from stair-base beginning)

Ratio

rise run

Reduced ratio

At the first step At the third step At the last step

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

26

a. What do you notice about the reduced ratios for your first step, third step and last step?

b. On your stair drawing, draw a line from the origin (0,0) to the point representing the final height of your staircase. What is the slope of this line?

c. What would happen to the slope of the line for your stairs if the rise of your stairs was higher, but the run depth remained the same?

d. What would happen to the slope of the line for your stairs if the rise of your stairs was lower, but the run depth remained the same?

Directions: Determine if the table of values has a constant rate of change, and whether or not it represents a linear relationship. 4.

6.

Number

2

4

7

9

Cost

38

76

133

171

x

8

12

16

24

y

8

5

3

-2

5.

7.

Time

1

2

8

11

Distance

2

5

65

122

x

-7

-1

2

5

y

5

3

2

1

8. Use the stairs below to answer the following questions. a. What is the slope of the stairs? Write your ratio in simplest form. b. If you’d prefer to climb a set of stairs that were steeper, what are possible values of the rise height and run depth of your preferred set of stairs?

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

27

5

9. If the slope of the line is 9 What is the value of 𝑦? Explain your reasoning.

27 𝑦

10. Roadway signs are used to warn drivers of an upcoming steep down grade that could lead to a dangerous situation. If the grade shown in the sign is 8%, what is the slope?

1

Directions: Each staircase below has been drawn to the scale of 8 inch = 4 inches. Use a ruler to find the actual rise height, run depth and slope for each staircase. (this problem is adapted from Illuminations) 11.

12.

13.

Actual rise =

Actual rise =

Actual rise =

Actual run =

Actual run =

Actual run =

Slope =

Slope =

Slope =

14. Which staircase is the steepest? Explain why.

15. Which staircase is built closest to building code? Justify your answer using research from the internet.

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

28

2.2B Homework: Dilations and Proportionality* Name:

Period:

1. Apply the given instructions to the image below. a. Connect point B to C. Label triangle ABC as the pre-image. b. Halve the length of ̅̅̅̅ 𝐴𝐵 on ⃡𝐴𝐸 Label the new segment ̅̅̅̅̅ 𝐴𝐵ˈ

D

c. Halve the length of ̅̅̅̅ 𝐴𝐶 on ⃡𝐴𝐷 ̅̅̅̅ Label the new segment 𝐴𝐶ˈ

C

d. Connect B’ to C’. Label triangle AB’C’ as the image. e. What do you notice about B ' C ' in relationship to BC ?

A

B

E

f. What observations can you make about the segments of the pre-image and image?

g. What kind of figures are these triangles in relationship to one another? Explain your reasoning.

2. What type of figure does a dilation produce? Which images from the drawing above are dilations of each other?

Directions: In the figures below, B is a dilation of A. Find the values of the variables. Write a proportion and show all of your work. 3.

4.

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

29

5.

6.

7. Which figures are similar to ∆𝐴𝐵𝐶? Select all that apply. A.

B.

C.

D.

8. Assuming the two triangles are similar, find the tower’s height from the given measurements. Show all of your work.

9. If an 18 ft. tall tree casts a 9 ft. long shadow, how tall is an adult giraffe that casts a 7 ft. shadow. Write a proportion and solve. Show all of your work.

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

30

Spiral Review: Directions: Write an expression for each unknown quantity and an equation that represents the word problem. Solve your equation and write your solution in a complete sentence. Show all of your work. 10. Lucy has $1.15 in nickels and dimes. She has 16 coins total. How many nickels and how many dimes does Lucy have?

11. The sum of three consecutive integers is 72. What is the largest integer?

12. Kerri is 25 years younger than her mother Pam. In 6 years, the sum of their ages will be 75. What is Pam’s current age?

13. Write a word problem that represents the given information and equation. Solve your word problem and write your solution in a complete sentence. Show all of your work. The Cost of Dinner

Word Problem:

Cost of drink: 𝑥 Cost of side order of mac and cheese: 2.5𝑥 Cost of calzone: 2𝑥 + 4 2(𝑥) + 2.5𝑥 + (2𝑥 + 4) = 17.00

Solution:

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

31

2.2C Homework: Similar Triangles and Slope* Name:

Period:

1. Describe in words how to use a right triangle to calculate slope.

Directions: Draw a right triangle on each line to calculate the slope. State the slope and simplify your ratio. y

2.

3.

-5

-5

5 x

𝑚=

𝑚=

10

5

5

5

-10

10 x

-5

-5

-10

-10

=

𝑚=

𝑟𝑖𝑠𝑒 𝑟𝑢𝑛

-5

10 x

-10

-5

5

-10

=

𝑚=

𝑟𝑖𝑠𝑒 𝑟𝑢𝑛

= y

10.

10

10

5

5

5

10 x

-10

-5

5

10 x

-10

-5

5

-5

-5

-5

-10

-10

-10

𝑟𝑖𝑠𝑒 = 𝑟𝑢𝑛

𝑚=

10 x

-5

y

9.

5

𝑚=

5

-5

10

-10

y

7.

y

y

8.

𝑟𝑖𝑠𝑒 = 𝑟𝑢𝑛

10

5

𝑟𝑖𝑠𝑒 𝑟𝑢𝑛

𝑚=

10

-5

x

-5

𝑟𝑖𝑠𝑒 = 𝑟𝑢𝑛

6.

y

-10

5

-5

=

5.

5

-5

5 x

-5 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛

4.

5

5

𝑚=

y

y

𝑟𝑖𝑠𝑒 = 𝑟𝑢𝑛

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

𝑚=

10 x

𝑟𝑖𝑠𝑒 = 𝑟𝑢𝑛 32

3

11. If the slope of the line is 4, what is the value of x?

x 7.5

12. The ratio of the vertical side length to the horizontal side length of each triangle formed by the slope of a 2 line is 5. Find two possible slopes for the line. Justify your reasoning.

13. Describe, using similar triangles, why a line going down from left to right has a negative slope.

14. Describe the steepness of a horizontal line using the slope ratio,

𝑟𝑖𝑠𝑒 . 𝑟𝑢𝑛

Directions: Determine if the given statement is true or false. If false, explain why. 15. All lines have slope.

1

5

16. A line with slope 2 is less steep than a line with slope 10.

17. A line with slope 4 has the same steepness as a line with slope -4.

18. A line with slope -2 is steeper than a line with slope -3 because -2 is a greater value than -3.

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

33

Directions: For the statements below, fill in the blank with “Always”, “Sometimes” or “Never.” If “Sometimes,” justify your answer. 19. Similar triangles have corresponding sides that are

20. A dilation is

proportional.

a reduction of the pre-image.

21. Lines

have zero or undefined slope.

22. For all linear relationships, the rate of change is

23. A linear relationship

the same value as slope.

has a y-intercept.

Spiral Review: 24. A mistake was made when solving the equation below. Circle and explain the mistake made. Solve the equations correctly. Equation:

1 2

+𝑥 =4−𝑥

1 2 ( + 𝑥) = 4 − 𝑥 2

Mistake made:

Equation solved correctly:

1 + 2𝑥 = 4 − 𝑥 3𝑥 = 3 𝑥=1 25. Create an equation with variables on both sides of the equal sign that has no solution. Solve your equation and explain why your equation has no solution using the structure of your equation.

26. Create an equation with variables on both sides of the equal sign that has infinitely many solutions. Solve your equation and explain why your equation has infinitely many solutions using the structure of your equation.

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

34

2.2D Homework: Finding Slope from Two Points* Name:

Period:

Directions: Find the slope of each line. 1.

2.

3.

4.

5. Graph the points (-1,4) and (0,1) and use the graph to justify that the slope is -3.

6. Graph the line that passes through the point (4,-2) 2 and has a slope of .

y

3

10

5

-10

-5

5

10

x

-5

-10

7. Circle the sets of ordered pairs that represent a vertical line and have undefined slope. (-1, 3) and (2,5)

(-9,3) and (-9,20)

(7,4) and (7,-1)

(-15,8) and (-15,6)

(0,2) and (0,5)

(-4,11) and (9,13)

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

35

8. Circle the sets of ordered pairs that represent a horizontal line and have zero slope. (4, 3) and (-10,3)

(-6,8) and (14,10)

(0,7) and (-11,7)

(-1,8) and (-1,9)

(18,2) and (-5,2)

(3,20) and (-8,20)

9. Without graphing, how can you determine if a line has zero or undefined slope by examining the ordered pairs.

Directions: Find the slope of the line that passes though the given points using the slope formula. Show all of your work. 10. (-4, -1) and (0, 2)

11. (14, -3) and (14, -7)

12. (1, 42) and (4, 40)

13. (-10, 3) and (0, -7)

14. (-9, 7) and (18, 7)

15. (7, 3) and (-3, 0)

16. (32, -23) and (-6, -2)

17. (-5, 36) and (-4, 3)

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

36

18. (18, -20) and (4, 8)

19.

(-5, 1) and (-5,-2)

20. (-6, -5) and (4,0)

21. (2,-3) and (5, -3)

Directions: Calculate the slope from each table. Show all of your work. 22. Slope =

24. Slope =

26. Slope =

x

y

3

23. Slope =

x

y

-9

8

1

5

-15

6

3

9

-27

11

-33

2 -4

7 13

x 0 1

y 4 -5

x 10

y 1

8

1

2

-14

12

1

3

-23

-4

1

x

y

x

y

-5

2

-3

5

5

6

-3

10

10

8

-3

15

15

10

-3

20

25. Slope =

27. Slope =

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

37

Directions: Find the value of 𝑥 or 𝑦 so that the line passing through the points has the given slope. Show all of your work. 5

29. (6, 𝑦) and (1, -5) with slope 2

28. (𝑥, 5) and (2, 15) with slope 2

2

4 3

30. (𝑥, 3) and (-1, 5) with slope − 3

31. (5, 𝑦) and (2, -3) with slope

32. (2, 6) and (𝑥, −3) with slope -1

33. (8, 3) and (1, 𝑦) with slope − 7

6

Spiral Review: 34. Solve:

−3𝑎−4 2

=8

35. Solve: 3(𝑥 + 1) − 5 = 8𝑥 − 2 − 5𝑥

36. Solve:

𝑥 2

1

𝑥

1

−3 = 3−2

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

38

2.2E Homework: Finding Slope in Context* Name:

Period:

1. Jackson’s soccer team is going out for hot dogs after Saturday’s tournament. Frank’s Grill is having a special on hot dogs, four hot dogs for three dollars. a. State the independent and dependent variables. Complete the table below. x:

4

y:

$3

b. Graph the information from the table and label each axis. Should your points be connected? Explain why or why not.

30 28 26 24 22 20 18 16 14 12 10 8 6 4 2

c. What is the unit rate? What does it represent in the context?

d. What is the slope of the line?

0

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

e. What is the y-intercept? What does it represent in the context?

f. Write an equation to find the cost for any amount of hot dogs. How many hot dogs can the team purchase for $20?

g. Re-write the context to represent a non-proportional relationship. Write an equation to represent your new context.

2. The San Diego County Fair costs $2 admission, plus $0.50 per ticket for rides. Complete the following table, showing the cost for getting into the fair, plus additional tickets for rides. # of tickets for rides

0

total expense

$2

1

10 $4.50

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

30 $12

$20

39

a. Graph the information from the table and label each axis. Should your points be connected? Explain why or why not.

20 18 16 14

b. What is the unit rate? What does it represent in the context?

12 10 8 6 4

c. What is the slope of the line?

2

0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

d. What is the y-intercept? What does it represent in the context?

e. Write an equation to find the total expense at the fair with any amount of ride tickets purchased.

f. If a family of 6 want to go to the fair on Saturday, and each of them plan to ride 10 rides, what will be the total cost for that family?

g. Re-write the context to represent a proportional relationship. Write an equation to represent your new context.

3. BeBop Bakery and Moonlight Deli offer catering options for local businesses. The table below shows how much money each bakery charges for their catered meals. a. Use the information in the table to graph BeBop Bakery and Moonlight Deli’s meal prices. Graph both lines on the same coordinate plane below. Label each line and label each axis.

# of catered Meals 1 2 3 4 5

BeBop Bakery $20 $40 $60 $80 $100

Moonlight Deli $30 $45 $60 $75 $90

b. What is the slope of the line representing BeBop Bakery? What does the slope represent in the context?

100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

1

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

2

3

4

5

40

c. What is the slope of the line representing Moonlight Deli? What does the slope represent in the context?

d. What does the slope tell you about these two situations?

e. Explain the difference in y-intercepts.

f. Write an equation for each situation. BeBop Bakery:

Moonlight Deli:

g. Which situation is a better deal? Explain your reasoning.

Directions: Find the slope of the line represented in the graph, table of values or ordered pairs. 4. (3, -4) and (-2, 8)

5.

6.

7. (-5, 3) and (2, 7)

8.

# of Raffle Tickets

Total Cost

2

1

4

2

8

4

10

5 9. # of Lawns

Total Earned

3

25.50

5

42.50

7

59.50

9

76.50

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

41

2.2F Homework: Deriving the equations y = mx and y = mx + b * Name:

Period:

1. Fill in the blanks using the words proportional or linear: “All proportional relationships are

, but not all linear relationships are

.

2. True or False, Equations of the form 𝑦 = 𝑚𝑥 + 𝑏 sometimes show a proportional relationship. Directions: For each of the equations given below, make a table of values and graph the line. Circle the slope in your equation and put a star next to the y-intercept in your equation. On the graphs, draw a right triangle to represent slope and put a star next to the y-intercept. 1

3. 𝑦 = − 𝑥

x

y

x

y

x

y

4

Slope (m): y-intercept (b):

1 2

4. 𝑦 = 𝑥 − 2 Slope (m): y-intercept (b):

5. 𝑦 = −3𝑥 + 1 Slope (m): y-intercept (b):

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

42

6. Use dilations and proportionality to show that the equation of a line that goes through the origin and has a 1 1 slope of 3 is 𝑦 = 3 𝑥. Explain your process. Explanation:

7. Use dilations and proportionality to show that the equation of a line that goes through the point (0, −2) 3 3 and has a slope of 2 is 𝑦 = 2 𝑥 − 2. Explanation:

Directions: Write a proportion to solve for 𝑥. 8. Slope of the line is

3 4

9. Slope of the line is −4

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

10. Slope of the line is 6.

43

Directions: Solve each proportional equation. Show all of your work. 11.

2𝑥 5

13.

𝑥+8 𝑥−1

15.

=

𝑥−2 8

7

=4

5𝑥−2 5𝑥−1

4

=5

𝑦−3

12.

𝑥−3 3

14.

3𝑥+1 6

16.

−4𝑥+1 2

=

𝑥+2 6

=

𝑥−5 10

=

−𝑥+7 7

1

17. Olivia used the proportion 𝑥 = 3 when deriving the equation of a line with similar triangles. Which statement below is correct? Select two that apply. 1

1

A. The line has a slope of − 3

B. The line has a slope of 3

C. The line has a slope of 3

D. The line has a y-intercept of −3 1

F. The line has a y-intercept of 3

E. The line has a y-intercept of 3

18. Zach began deriving the equation of line 𝐴𝐶 by defining the lengths of line segments 𝐴𝐵 and 𝐵𝐸 as 𝑦 − 4 and −𝑥, respectively. What proportion should he set up next using similar triangles 𝐴𝐵𝐸 and 𝐶𝐷𝐸? A.

3 5

= 𝑦−4

−𝑥

C.

3 5

=

𝑦−4 −𝑥

B.

3 5

= 𝑦+4

𝑥

D.

3 5

=

𝑦+4 𝑥

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

44

Section 2.2: Review Name:

Period:

1. How are rate of change and slope of a line related?

2. How is slope of a line, unit rate, and constant of proportionality related?

3. Compare the equations below using a Venn Diagram. 𝒚 = −𝒙 + 𝟔

𝒚 = −𝒙

4. Which of the proportions below can be used when deriving the equation of the line 𝑦 = 4𝑥 − 5 with similar triangles? Select three that apply. A.

𝑦 𝑥+5

=

24 6

B.

8 2

D.

𝑦+5 𝑥

=

20 5

E.

3 12

=

𝑦+5 𝑥

𝑥

= 𝑦+5

C.

4 16

F.

𝑦−5 𝑥

=

𝑦+5 𝑥

=

28 7

Directions: For each line graphed below, draw a right triangle to calculate slope. Label the rise and run of each triangle and simplify the ratio. Then, find any two points on the line and verify the slope using the slope formula. 5.

6.

𝑟𝑖𝑠𝑒 𝑟𝑢𝑛

=

Slope formula =

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

𝑟𝑖𝑠𝑒 𝑟𝑢𝑛

=

Slope formula =

45

7.

8. 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛

𝑟𝑖𝑠𝑒 𝑟𝑢𝑛

=

Slope formula =

=

Slope formula =

y 5

9. Graph the line that passes through the point 2

(1, -1) and has a slope of 3. -5

5

x

-5

10. Write a proportion and solve for the missing values. a. a. Find the value of x if the 1 slope of the line is 4.

b. Find the value of y if the slope of the line is -4.

c. Find the value of x if the slope of the line is 6.

Directions: Find the slope from each table of values. Show all of your work. 11. Slope =

13. Slope =

x 8

y 1

6 2 -4

3 7 13

x

y

-5

2

5

6

10 25

12. Slope =

x

y

10

1

8

1

12

1

-4

1

x

y

-3

5

-3

10

8

-3

15

14

-3

20

14. Slope =

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

46

Directions: Determine if the given ordered pairs represent a line with zero or undefined slope. If neither, find the slope of the line. 15. (2,-1) (2,4) (2,-15)

16. (0,1) (5,4) (-10,-5)

17. (12,3) (0,3) (-3,3)

Directions: Calculate the slope of the line that passes though the given points using the slope formula. Show all of your work. 18. (-2, 7) and (14, 4)

19. (15, -7) and (1, -7)

20. (-4, -1) and (0, 2)

21. (-19, 5) and (-11, 12)

22. (8,3) and (8,-2)

23. (-6, -5) and (4, 0)

24. (5, 5) and (1, -5)

25. (6, 9) and (18, 7)

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

47

26. In the coordinate plane below, ∆𝐴𝐵𝐶 is similar to ∆𝐷𝐸𝐹. What is the value of y? Show all of your work.

27. Molly and Sam began deriving the equation of line 𝐶𝐹 by calculating the length of both line segment 𝐴𝐵 and line segment 𝐵𝐶 as 1. Then, they defined the length of line segment 𝐷𝐸 as 𝑥 and the length of line segment 𝐸𝐹 as 𝑦 − 3. What should they get for the equation of the line once they set up a proportion and simplify?

28. Ethan can solve 10 equations in 8 minutes. Complete the table below. x:

2

8 30

y:

5

30

28 26 24 22

a. Graph the information from the table and label each axis. Should your points be connected? Explain why or why not.

20 18 16 14 12 10

b. What is the slope of the line? What does the slope represent in the context?

8 6 4 2 0

2

4

6

8 10 12 14 16 18 20 22 24

c. What is the y-intercept? What does it represent in the context?

d. Write an equation to find the number of equations solved for any number of minutes.

e. Is this relationship proportional? Explain why using the equation.

f. Re-write the context to represent a non-proportional relationship. Write an equation to represent your new context.

SDUHSD Math B College Prep Module #2 – HOMEWORK 2016-2017

48