03x 13 T7 SAB 2012 13 · Copyright!©!2012!Charles!A.!Dana!Center!at!the!University!of!Texas!at!Austin,!Learning!Sciences!ResearchInstitute!at!the!University!of!Illino isatChicago

Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc.

Unit 3 Rate of change

This unit introduces an important mathematical concept: rate of change. You will draw on your previous work with multiple representations as you develop and apply your understanding of rate of change.

You will explore how learning feels and learn that, by better recognizing the feelings associated with the learning process, you can be more comfortable with these feelings as you learn new things.

OUTLINE

Topic 7: Exploring rate of change in motion problems

Understanding and representing how quantities change is an important idea in algebra. In this topic, you explore the concept of rate by analyzing motion over time. You investigate the rate at which distance changes numerically and graphically.

In this topic, you will:

• Understand the concepts of speed and rate • Create motion graphs (distance vs. time) and investigate how changes in motion affect the graph • Demonstrate what effect a change in speed or rate has on a motion graph • Find rates for data in tables • Compare different constant rates • Find rates from graphs

Topic 8: Algebraic thinking and how learning feels

You further develop your problem-‐solving capabilities and algebraic thinking by working on non-‐routine problems involving balance scales. You analyze the feelings you experience as you work on the problem, and learn that confusion is a natural state of learning and that learning often involves backsliding and breakthrough moments.


• Think in a reflective way about how it feels to be engaged in a challenging learning task • Understand the learning process to empower you to take on more challenging learning tasks • Extend the study of rates to unit rates • Solve non-‐routine problems involving balance scales, work rates and job completion

Topic 9: Exploring rate of change in other situations

You may have heard of growth rates, pay rates, pulse rates, interest rates, birth rates, postal rates, passing rates, conversion rates, and many other kinds of rates. These are all rates because they are comparisons of two quantities with units that are not the same. In this topic, you continue your investigation of important ideas related to rates, but in contexts that go beyond the context of motion.


• Find rates of change for linear functions from tables, graphs, or verbal descriptions • Contrast proportional linear relationships with non-‐proportional linear relationships • Identify relationships as linear or non-‐linear from tables, graphs, or descriptions • Identify from tables whether a relationships is linear, exponential, quadratic, or inverse variation


Topic 7: Exploring rate of change in motion problems 255


EXPLORING RATE OF CHANGE IN MOTION PROBLEMS Lesson 7.1 Rates and motion graphs

7.1 OPENER

Here are three situations that suggest different speeds. Situation A: Running a foot race Situation B: Riding a bike Situation C: Racing a car

1. Here are three numerical speeds. Write each situation next to the speed that makes sense.

• 15 miles per hour ______________________ • 110 miles per hour ______________________ • 6 miles per hour ______________________

2. List three other situations that deal with speeds. Then write a numerical speed that makes sense for each situation.

7.1 CORE ACTIVITY

1. Complete the math journal.

Term My understanding of this term… An example that illustrates this term…

Speed

Rate

2. How is speed like a rate? How is speed different from a rate?

256 Unit 3 – Rate of change


3. A motion detector is used to detect a skateboarder’s movement. Different situations for the skateboarder are described. For each situation, sketch a graph and then describe how your sketch matches the description provided.

a. The skateboarder moves at a moderate, steady rate away from the motion detector.

Sketch a graph Describe in words

b. The skateboarder moves at a faster, steady rate away from the motion detector.


c. The skateboarder moves away from the motion detector for about a second and then stands still.


d. The skateboarder starts far from the motion detector and skates toward the motion detector.




e. The skateboarder skates slowly and then speeds up.


f. The skateboarder skates quickly and then slows down.


4. Based on the information in the graph, what happened 7 seconds into Terrence’s skate?

7.1 CONSOLIDATION ACTIVITY

1. Sketch a graph for each of the following situations. Then check your graph using the animation. a. Sketch a graph of Terrence starting 4 feet from the motion

detector and skating at a constant, steady pace away from the motion detector.



b. Sketch a graph of Terrence starting 4 feet from the motion detector and skating at a constant, steady pace toward the motion detector.

c. Start Terrence 1 foot away from the motion detector. Move him away slowly at first, then speed up.

d. Start Terrence 1 foot away from the motion detector. Move him away quickly at first, then slow down.

e. Start Terrence 7 feet away from the motion detector. Move him toward the motion detector slowly at first, then speed up.

f. Start Terrence 7 feet away from the motion detector. Move him toward the motion detector quickly at first, then slow down.



2. Now look back over your work and answer the following questions.

a. How is Terrence skateboarding when his graphs most look like lines? How is he skateboarding when the graphs look like curves?

b. How is Terrence skateboarding when his graphs rise from left to right? How is he skateboarding when the graphs fall from left to right?

3. Use the graph provided to answer the following questions.

a. Where is Terrence at 4 seconds? How do you know?

b. What is happening between 7 and 10 seconds? How do you know?



HOMEWORK 7.1 Notes or additional instructions based on whole-‐class discussion of homework assignment:

Part I: Match each description in questions 1 and 2 with the corresponding graph. Then answer question 3.

1. For each description of the skateboarder’s motion, write the letter of the corresponding graph.

a. Terrence started at 2 feet and skated quickly away from the motion detector.

____

b. Terrence started at 2 feet and skated slowly away from the motion detector.

____

c. Terrence started at 5 feet and skated quickly away from the motion detector.

____

d. Terrence started at 2 feet and paused to catch his breath. ____

e. Terrence started at 5 feet and skated slowly away from the motion detector.

____

2. For each description of the skateboarder’s motion, write the letter of the corresponding graph.

a. Terrence started at 6 feet and skated slowly toward the motion detector.

____

b. Terrence started at 6 feet and skated quickly toward the motion detector.

____

c. Terrence started at 3 feet and skated quickly toward the motion detector.

____

d. Terrence started at 3 feet and skated slowly toward the motion detector.

____

3. Describe your strategy for matching the descriptions and graphs in Part I.

Part II. Match each description with the corresponding graph. Write the letter in the blank.

4. Terrence skated quickly toward the motion detector for 3 seconds and then slowly away from it for 3 seconds.

____

5. Terrence skated slowly away from the motion detector for 3 seconds and then quickly toward it for 3 seconds.

____

6. Terrence skated quickly toward the motion detector for 2 seconds and then slowly away from it for 4 seconds.

____

7. Terrence skated slowly away from the motion detector for 2 seconds and quickly toward it for 4 seconds.

____



Part III. Explain in words:

8. José says the graph represents Terrence skating up a hill and then down a hill. What do you say to José?

9. What would a graph look like if Terrence skateboarded quickly?

10. What would a graph look like if Terrence skateboarded slowly?

11. What would a graph look like if Terrence skateboarded away from the motion detector?

12. What would a graph look like if Terrence skateboarded toward the motion detector?



STAYING SHARP 7.1 Practic

ing algebra skills & con

cepts

1. Complete the following table.

Directions Example For any number, n

Choose a number. n

Multiply by 4.

Add 3. 23

2. Simplify:

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9− 26− 4

Answer:

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3. Teri is downloading a file and the computer shows this image after taking 2 minutes to download. About how long will it take Teri to download the whole file? Answer with supporting work:

4. If you popped 3 bags of popcorn in 15 minutes, how many bags did you pop per minute?

Answer with supporting work:

Review

ing pre-‐algebra ideas

5. a. Find the average of 27 and 45. b. Find the average of 10, 18, and 23.

6. Label the quadrants.

In which quadrant is each of the points?

a. (-‐2,55) _______

b. (55,-‐2) _______

c. (-‐2,-‐55) _______

Time: 2:00 minutes

10% downloaded...



Lesson 7.2 Match the graph

7.2 OPENER 1. Which line could model each situation? Write a letter of a line in each blank. Some lines may fit more than one situation.

a. Terrence skated quickly away from the motion

detector.

b. Terrence skated slowly away from the motion detector.

c. Terrence skated quickly toward the motion detector.

d. Terrence took a break, standing still in front of the motion detector.

e. Terrence skated slowly toward the motion detector.

2. Describe the strategy you used to complete this activity.

7.2 CORE ACTIVITY

1. Using the terms provided, fill in the blanks to complete the statements.

rate time

distance speed

a. The skateboarder begins at a of 7 feet from the motion detector.

b. How many seconds the skateboarder’s ride lasts is the .

c. The skateboarder’s is negative.

2. The graph of Terrence’s skateboarding is shown here, with each segment labeled. Use the graph to answer the questions.

a. For each of the segments, describe how you would move the skateboarder to match that portion of the graph.

Segment A

Segment B

Segment C

b. What is different about the ways you would move the skateboarder to match the three different graph segments?



3. Where was the skateboarder's starting position this time? How do you know?

4. Complete this puzzle to test your understanding of how the shape of a graph relates to the rate of change. Fill in the blanks to complete each statement.

a. Segment ____ is horizontal.

b. Segment ____ is steeper than segment ____.

c. Segment ____ rises, but is flatter than segment ____.

d. Segment ____ shows a steady, fast rate.

e. Segment ____ shows a steady, slow rate.

f. Segment ____ shows no movement.

5. Describe how you would move the skateboard to generate each of the following graphs.

a.

b.



c.

6. Complete this puzzle to demonstrate your understanding of the relationship between the steepness of the graph and the skateboarder's speed. Match each description of the skateboarder's speed with a line segment.

a. ____ He moved quickly and at a constant rate.

b. ____ He stopped moving.

c. ____ He moved slowly at a constant rate.

d. ____ He moved most quickly here.

7.2 REVIEW END-OF-UNIT ASSESSMENT

Today you will review the end-‐of-‐unit assessment.




1. Sketch a graph (Elapsed time, Distance) for each description. Keep in mind that you must walk for the full 10 seconds and cannot travel farther than 10 feet from the motion detector.

a. You start 3 feet from the motion detector and walk slowly away from the motion detector.

b. You start 2 feet from the motion detector and walk quickly away from the motion detector.

c. You start 7 feet from the motion detector and walk slowly toward the motion detector.

d. You start 9 feet from the motion detector and walk quickly toward the motion detector.

e. You walk toward the motion detector, stand still, and then walk away from the motion detector.

f. You stand still, then walk away from the motion detector, and then back toward the motion detector.

Elapsed time (sec)

Distan

ce from

motion de

tector (ft)

Elapsed time (sec)

Distan

ce from

motion de

tector (ft)

Elapsed time (sec)

Distan

ce from

motion de

tector (ft)

Elapsed time (sec)

Distan

ce from

motion de

tector (ft)

Elapsed time (sec)

Distan

ce from

motion de

tector (ft)

Elapsed time (sec)

Distan

ce from

motion de

tector (ft)



2. Mark is running in a foot race. The graph below shows Mark’s distance from the finish line as related to time. Use the graph to

answer the following questions.

a. How long was the race? Explain.

b. How long did it take Mark to finish the race? Explain.

c. Write a story about Mark’s race.





cepts

1.

a. Calculate the following products. Look for a pattern.

b. Can you use the pattern to help you find 9 ⋅8 ?

Explain:

2. Simplify:

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Answer:

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3. If Jonah sees this image on his computer while he is downloading a file, about how long will the entire download take?

Explain your answer:

4. If an elevator traveled 10 floors in 5 seconds, how many floors did it travel in 1 second?

Explain your reasoning:

Review


5. a. José made these grades on tests: 78, 88, 98. Without doing any computing, what is his average? ______

b. If he makes a _____ on his next test, his average will stay the same.

c. If he makes a _____ on his next test, his average will go up.

d. If he makes a _____ on his next test, his average will go down.

6. Plot the following points on the grid provided.

A (1,7); B (5,2); C (8,0); D (9,9)

10 ⋅91 ⋅99 ⋅910 ⋅ 71 ⋅ 79 ⋅ 7

8:00 minutes remaining

40% remaining...



Lesson 7.3 Motion with non-constant rates

7.3 OPENER

The graph shows a skateboarder’s distance in front of a motion detector.

After looking at the graph, Stephen says that the skateboarder started 8 feet from the motion detector and skated away from it at a constant rate.

Is Stephen’s analysis correct? If so, explain how you know. If not, correct his analysis and explain how you know you are right.

7.3 CORE ACTIVITY


a.

b.


a.



b.

c.

d.


1. Sketch a graph to model the verbal description provided.

a. Start 8 feet away from the motion detector. Move toward the motion detector quickly and then gradually slow down.



b. Start 2 feet away from the motion detector. Move away from the motion detector quickly and then gradually slow down.

c. Start 8 feet away from the motion detector. Move toward the motion detector slowly and then gradually speed up.

d. Start 2 feet away from the motion detector. Move away from the motion detector slowly and gradually speed up.

e. Start 8 feet away from the motion detector. Move toward the motion detector quickly and then stop.

2. Explain how to move to correctly match the graph.

a. Explain graph:



b. Explain graph:

c. Explain graph:

d. Explain graph:

3. Study the graphs and then answer the questions that follow.

A. B. C.



D.

E. F.

a. Which of the graphs cannot be created by walking in front of a motion detector? Explain why.

b. Which of the graphs can be created by walking in front of a motion detector? Explain how you would walk to produce the graphs.

4. Looking back over your work, answer the following questions.

a. How are you walking when your graphs look most like straight lines?

b. How are you walking when your graphs look most like curves?

c. How are you walking when your graphs look the steepest?

d. How are you walking when your graphs look the shallowest?




1. Match each description with the corresponding graph. Write a letter in each blank.

a. You started walking quickly away from the motion detector and gradually slowed down.

____

b. You started walking away slowly from the motion detector and gradually sped up.

____

c. You walked toward the motion detector at a fast, constant rate and then you walked away at a slower constant rate.

____

d. You walked away from and then toward and then away from and then toward the motion detector.

____

e. You started walking quickly toward the motion detector and gradually slowed down.

____

f. You started walking slowly toward the motion detector and gradually sped up.

____

2. Complete the math journal.

Scenario: Describe in words what the graph would look like.

Sketch the graph.

a. You start walking slowly and speed up.

b. You start walking quickly and slow down.

c. You walk at a constant rate.





cepts

1. Do the following calculations. Look for a pattern. 98 + 47100 + 45

Can you use the pattern to help you figure 97 + 74 ? Explain:

2. Simplify:

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1− 96− (−2)

Answer:

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3. Twelve minutes after he starts downloading a file, Mario looks at his computer and sees this image. About how long will it take to download the entire file?

30% downloaded...

Time: 12:00 minutes


4. a. Tali jogs a mile in 12 minutes. How many miles per hour is Tali running?

b. Sandra runs twice as fast as Tali. How many miles per hour is Sandra running? How long does it take for Sandra to run a mile?

Review


5. Two numbers have an average of 17. If one of the numbers is 11, what is the other number? Answer with supporting work:

6. Label each point in the graph with its coordinates written as an ordered pair.





Lesson 7.4 Elevator graphs

7.4 OPENER

Here is the graph of an elevator ride, shown in (Elapsed time in seconds, Distance from ground in floor numbers). Interpret the graph by answering the following questions.

1. Where did the elevator start?

2. What happened between 0 and 4 seconds?



7.4 CORE ACTIVITY

1. Sketch the graph of each elevator ride described.

a. Start the elevator on floor 4 and make the elevator rise at a rate of 1 floor per second.

b. Start the elevator on floor -‐3 and make the elevator rise at a rate of 2 floors per second.



c. Start the elevator on floor 11 and make the elevator descend at a rate of 1 floor per second.

d. Start the elevator on floor 12 and make the elevator descend at a rate of 2 floors per second.

2. Why do some of the graphs end with a horizontal segment?

3. What is the meaning of the point (7,-‐2) on graph 1d?

4. Suppose the hotel has two elevators: elevator A and elevator B. The elevators’ rides are described. Sketch the graphs of elevator A and elevator B on the same grid. Use two colors, one for each elevator.

a. Elevator A: Start at floor 5 at rate 1 floor per second Elevator B: Start at floor 2 at rate 4 floors per second

b. Elevator A: Start at floor 11 at rate 2 floors per second Elevator B: Start at floor -‐3 at rate 2 floors per second

c. Elevator A: Start at floor 3 at rate -‐2 floors per second

Elevator B: Start at floor 3 at rate 2 floors per second d. Elevator A: Start at floor 11 at rate -‐1 floor per second

Elevator B: Start at floor 11 at rate -‐3 floors per second



e. Elevator A: Start at floor 1 at rate -‐1 floor per second Elevator B: Start at floor 12 at rate -‐3 floors per second


1. Use the graph to determine starting floor and rate for each elevator. a. Elevator A: Start: ____ Rate: ____

Elevator B: Start: ____ Rate: ____ b. Elevator A: Start: ____ Rate: ____

Elevator B: Start: ____ Rate: ____



c. Elevator A: Start: ____ Rate: ____ Elevator B: Start: ____ Rate: ____

d. Elevator A: Start: ____ Rate: ____ Elevator B: Start: ____ Rate: ____

e. Elevator A: Start: ____ Rate: ____

Elevator B: Start: ____ Rate: ____ f. Elevator A: Start: ____ Rate: ____

Elevator B: Start: ____ Rate: ____

2 a. Look back over your work. How did you decide what the starting floor was?

b. How did you decide what the rate was?

3. Sketch a prediction for each of the following descriptions of elevator motion.

a. Elevator A: Start at floor 7 at a rate of 1 floor per second Elevator B: Start at floor 7 at a rate of 2 floors per second

b. Elevator A: Start at floor 7 at a rate of -‐1 floor per second Elevator B: Start at floor 7 at a rate of -‐2 floors per second



c. Elevator A: Start at floor 4 at a rate of -‐4 floors per second Elevator B: Start at floor 1 at a rate of 4 floors per second

d. Elevator A: Start at floor 11 at a rate of 2 floors per second Elevator B: Start at floor 11 at a rate of -‐2 floors per second

4 a. Look back over your work. How does the rate of the elevator relate to the steepness of the graph?

b. What is the connection between the starting floor for the elevator and the starting point for the graph? 5. Five descriptions of graphs are provided here. Use these descriptions to fill in the blanks in the statements about elevator

motion. the graph falls and is steeper

the graph rises and is steeper

the graph is a horizontal line

the graph rises and is less steep

the graph falls and is less steep

a. When the elevator pauses on a floor, _________________________________________________________.

b. When the elevator rises rapidly, ______________________________________________________________.

c. When the elevator descends rapidly, _________________________________________________________.

d. When the elevator rises slowly, _______________________________________________________________.

e. When the elevator descends slowly, __________________________________________________________.




1. Match each description with the corresponding graph. Write a letter in each blank.

a. The elevator starts on floor 3 and rises at 4 floors per second. ____

b. The elevator starts on floor 8 and descends at 1 floor per second.

____

c. The elevator starts on floor 8 and descends 4 floors per second. ____

d. The elevator pauses on floor 2. ____

e. The elevator starts on floor 5 and descends at 4 floors per second.

____

f. The elevator starts on floor 5 and descends at 1 floor per second.

____

g. The elevator starts on floor 2 and rises at 4 floors per second. ____

h. The elevator starts on floor 3 and rises at 0.5 floors per second. ____

i. The elevator starts on floor 2 and rises at 0.5 floors per second. ____

2. For each description of an elevator’s motion, sketch a graph. Label each graph.

a. Rate of -‐2 floors per second, starting at floor 3 b. Rate of 2 floors per second, starting at floor 3



c. Starting at floor 8, rate of

12 (1 floor per 2 seconds)

d. Starting at floor -‐1, rate of − 1

3 floors per second (-‐1

floor per 3 seconds)

3. Given the following elevator graphs, find the starting floor and the rate.

a. Starting floor: _____

Rate: _____

b. Starting floor: _____

Rate: _____





cepts

1. Simplify each of the following:

34+ 14

34+ 24

34+ 12

2. Simplify:

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Answer:

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3. If the computer screen shows this image as Calinda is downloading a file, how big is the file she is downloading?

80% downloaded

16 KB downloaded...

Explain your reasoning:

4. Terrence’s friend, Tony, skates on inline skates. Tony skated the 2 miles to school in 14 minutes. Terrence skateboarded 4 miles to school in 30 minutes. Who skated faster? Justify your answer. Answer:

Review


5. Three numbers have an average of 15. If two of the numbers are 10 and 12, then what is the other number?


6. The tick marks are evenly spaced. What number belongs in the blank?

23

13



Lesson 7.5 What’s my rate?

7.5 OPENER

Use what you have learned about rate so far to answer the following questions.

1. Terrence skated 8 feet in 4 seconds at a constant rate. How fast did he skate in feet per second?

2. An elevator traveled from the 10th floor to the 2nd floor in 6 seconds. What was the rate of the elevator in floors per second?

3. A car traveled 300 miles in 6 hours. What was the car’s rate in miles per hour?

7.5 CORE ACTIVITY 1. This graph models Terrence's movement while skateboarding. How

can you use the graph to find his rate during a specific time period?

2. Fill in the table with the information from Terrence’s skateboarding.

3. Notice that the graph is horizontal from 4 seconds to 10 seconds. What was Terrence's rate then?



4. What is the rate for the skate modeled by this graph? Show your work.

5. Here is another graph that shows Terrence's movement while skateboarding. Fill in the blanks to complete the following statements about the different rates of change you see on the graph.

a. Segment _ shows a rate of change of 0 ft/sec.

b.

Segment _ shows a negative rate of change.

c. Segment _ shows a positive rate of change.

d. Looking at segment A, Terrence’s rate of change for

the first 2 seconds was ___ ft/sec.

e Looking at segment C, Terrence’s rate of change for

the last 5 seconds was _ ft/sec.

6. The table shows the information for one of Terrence’s skates.

Time (sec) Distance (ft) 0 2 1 4 2 6 3 8

a. Use the table to create a graph

for this skate. b. How fast was Terrence skating?



7. According to the graph, how fast were the elevators moving? Elevator A: ____________ Elevator B: ____________

8. According to the graph, how fast were the elevators moving? Elevator A: ____________ Elevator B: ____________

9. Describe how you found the rates from the graphs.

7.5 CONSOLIDATION ACTIVITY Hare challenges Tortoise to three more races. Each race will be 100 meters.

Race 1: Tortoise leaves the starting line, "sprints" at the rate of 4 meters per minute, and runs a steady race. Hare leaves the starting line running at a rate of 20 meters per minute and runs steadily for 2 minutes. He then decides to take a break, as he knows he is far ahead of Tortoise. Hare's break lasts 8 minutes. Then Hare runs back to the starting line (at 20 meters per minute again) to grab a towel that he left behind. He does not stop but grabs the towel on the run and heads down the racecourse again. He then runs at a steady 20 meters per minute to the finish line. 1. Draw a graph that shows the progress of the race. Use the same

grid for both Tortoise and Hare. Show “Time from start of race in minutes” on the horizontal axis. Show “Distance from the starting line in meters” on the vertical axis. Use one color for the tortoise’s graph and one color for the hare’s graph.

2. Who wins the first race?

3. Are Tortoise and Hare ever side by side in the race? How do you know?



Race 2: This time Hare leaves the starting line 15 minutes after Tortoise. Hare runs at the rate of 20 meters per minute for 3 minutes, then rests for 5 minutes. Hare then runs for 1 minute at the same rate. He stops to tie his shoe (which takes 2 minutes) and then runs to the finish line at 20 meters per minute. Tortoise, unaffected by the results from the first race, sprints at a steady 4 meters per minute throughout this second race. 4. Draw a graph that shows the progress of the race. Use the same

grid for both Tortoise and Hare. Show “Time from start of race in minutes” on the horizontal axis. Show “Distance from the starting line in meters” on the vertical axis.

5. Who wins the second race? How do you know?

Race 3: Tortoise decides he needs to pick up his speed, so he sprints at 5 meters per minute and runs another steady race. Hare leaves 5 minutes after Tortoise, because he changes his running shoes for a newer pair. Hare runs for 3 minutes at a rate of 20 meters per minute. Then Hare stops for a 10-‐minute rest, knowing he is far in the lead. After Hare’s 10-‐minute nap, he resumes the race, again running at 20 meters per minute. 6. Make another graph to show the progress of each racer. Show

“Time from start of race in minutes” on the horizontal axis. Show “Distance from the starting line in meters” on the vertical axis.

7. Who wins the third race? How do you know?




Part I: Find the rate for each graph or table.

1. Rate = ______ 2. Rate = ______

Time (sec) Dist (ft)

0 1

1 6

2 11

Time (sec) Dist (ft)

0 1

3 7

6 13

3. Rate = ______ 4. Rate = ______

Part II: Match each rate with a table and graph.

TABLE (A-‐D) GRAPH (W-‐Z)

5. Rate of 2 feet per second

6. Rate of -‐2 feet per second

7. Rate of 1 foot per second

8. Rate of -‐1 foot per second

A. Time (sec) Dist (ft) B. Time (sec) Dist (ft)

0 2 0 2 4 6 2 6 8 10 4 10 12 14 6 14

C. Time (sec) Dist (ft) D. Time (sec) Dist (ft) 0 12 0 12 2 8 4 8 4 4 8 4 6 0 12 0



9. Describe the strategy you used to match the rates with the tables and graphs.

Part III: Sketch a graph of distance versus time for each description. Label each graph.

10. Elevator: rate of 2, starting at floor 5 11. Terrence started 2 feet from the motion detector and skated away at 5 feet per second.

12. Elevator: rate of -‐1, starting at floor 10 13. Terrence started 10 feet from the motion detector and skated toward the motion detector at 3 feet per second





cepts

1. Which value is the closest to!!

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?

a. 0 b. 0.5 c. 1

d. 2120

Explain your answer:

2. If a = 7, b = 3, c = 9, and d = 1, find the value of the following expression:

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a − bc − d

Answer:

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3. Adriana is downloading a file and notices this image. How much of the file has been downloaded?

40% downloaded

Total 32 KB

4. Sketch a graph of an elevator starting at the first floor and rising by 1 floor per second.

Review


5. Invent a six-‐value data set for which the median is 5. Answer:

6. If you were plotting the following points on the grid provided, what would you choose as your scales for the x-‐ and y-‐axes? (3,20), (-‐14,11), (0,13), (19,2), (1, -‐17)

One unit on the x-‐axis equals: One unit on the y-‐axis equals:





Lesson 7.6 More about rates of change

7.6 OPENER

1. Sketch two different graphs on the same grid of distance versus elapsed time for an elevator rising at a rate of 1 foot per second.

2. Sketch two different graphs on the same grid of distance versus elapsed time for an elevator descending at a rate of 2 feet per second.

3. Suppose everyone in the class added their lines to the same grid on which you sketched your graphs. What patterns, if any, would you see?

7.6 CORE ACTIVITY

1. A graph of Terrence’s skateboarding is shown. Use the graph to answer the following questions.

a. Build a table from the graph. You may have to estimate distances.

Time (sec) Distance (ft)

b. Now use the table to find Terrence’s average rate from 0 seconds to 1 second.

c. Now find Terrence's average rate between 1 second and 2 seconds. Is this rate the same as or different from the average rate from 0 seconds to 1 second? What does a comparison of the rates tell you?

Elapsed time in seconds

Distan

ce in fe

et


Distan

ce in fe

et



2. Here is another graph that models Terrence's movement while skateboarding. Using the answer choices provided, fill in the blanks to complete the statements about the rates shown on the graph. a. Section A indicates a rate of change that is

_________________.

not constant more quickly

positive 0

constant more slowly negative 5

b. Section B indicates a rate of change that is constant and

________________.

c. Section C indicates a rate of change that is

________________ and ________________.

d. Section D indicates a rate of change that is

________________ and ________________.

e. Section E indicates a rate of change that is

________________ and ________________.

f. According to the graph, Terrence was moving

________________ during the time represented by

section B than during the time represented by section C.

7.6 CONSOLIDATION ACTIVITY A student recorded three different skates, each with a constant rate. The student recorded some of the information for each skate in tables. For each skate, fill in the table, create a graph of the data, and find the rate. 1.

Time (sec) Distance (ft) 0 5

1 2 7

3

2.


0 3

1 2

3 9



3.

Time (sec) Distance (ft) 0 8

2 6 4

6

4. Here is data from another skate.

Time (sec) Distance (ft) 0

3 5

6 9 8

a. Assume the skater was moving at a constant rate. Create a graph of the skater’s distance over time and label it “a.”

b. Assume the skater was not moving at a constant rate. Sketch two possible graphs for the skate on the same grid as your graph for part a.

c. For each of the graphs you sketched in part b, describe the story of the skate. Be sure each story includes information about specific average rates over different intervals of time during the skate.




1. Fill in the missing information about these skates recorded by a student. (Fill in the table, create the graph and/or find the rate.) Assume that these graphs are linear.

a. How fast was he skating? ______

Time (sec)

Distance (ft)

0 6 1 5 2 3

b. How fast was he skating? ______

Time (sec)

Distance (ft)

c. How fast was he skating? ______

Time (sec)

Distance (ft)

1 1 2 3 5 4

2. What are the differences between a positive rate and a negative rate? Use a graph and a table to support your answer.

3. What does it mean for an elevator to have a rate of 12 floors per second? Use a graph and table to support your answer.



4. The graph shows data collected from a skateboarder’s skate. Fill in the missing data from the table. Then use the table to answer the following questions.

Time (sec)

Distance (ft)

0

1

3

6

10

a. Find the skateboarder’s average rate between 0 and 1 second.

b. Find the skateboarder’s average rate between 1 and 3 seconds.

c. Find the skateboarder’s average rate between 3 and 6 seconds.

d. Find the skateboarder’s average rate between 6 and 10 seconds.

e. Look over your answers for parts a-‐d. Compare the rates for parts a and b to the rates for parts c and d. What do you notice? What does this mean?

5. Consider the graph in question 4. Write a story to describe the skateboarder’s skate.





cepts

1. Which value is the closest to !!

�

0.9 ⋅997? a. 1,000

b. 81

c. 10,000

d. 10


2. If (e,f) = (7,2) and (g,h) = (11,5), find the value of the following expression:

!!

�

h− fg− e

Answer:

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3. Ravi glanced at the computer as he was downloading a file. What percent had downloaded?


4. Fill in the blanks for the area model:

( )( )16 12 10 __ __ 2• = + +

Review


5. Invent a five-‐value data set for which the mean is 6 and the mode is not 6.

6. Choose an appropriate scale for each axis and plot the following points on the grid provided: (20,0) (-‐8,4) (12,-‐12) (0,-‐16).

One unit on the x-‐axis equals:

One unit on the y-‐axis equals:

_______% downloaded

36 KB of 40 KB downloaded...2

10 100

12



Lesson 7.7 Rates in context

7.7 OPENER 1. Use the table provided to sketch a graph. Then use the table and your graph to answer the following questions.


0 0 1 2 2 4 3 6 4 6 5 6 6 6 7 7 8 8 9 9 10 10

2. What is the rate between 0 and 3 seconds? What does the graph look like between these times?

3. What is the rate between 3 and 6 seconds? What does the graph look like between these times?

4. What is the rate between 6 and 10 seconds? How does this piece of the graph compare to the piece of the graph you analyzed in question 2?

7.7 CORE ACTIVITY On Memorial Day, Tamira and her family go on a day trip to an outlet mall. They leave their house at 9 a.m. and travel at a constant speed of 70 miles per hour for one hour, arriving at the mall just as the stores open. They shop for two hours and then drive farther away from home for 15 minutes, at a speed of 40 miles per hour. Then they stop at a local café for one hour to enjoy some lunch. After their lunch break, they return home, still at a constant rate of speed, but traffic is very heavy, so they have to travel much more slowly than they did earlier in the day. The trip takes a total of 10 hours.

1. Tamira’s trip can be divided into 5 segments. Complete the table by filling in the length of time and the speed of the car for each of the segments for the trip

Segment Length of time Speed of car A B C D E


Distance in fe

et



2. Use the verbal description of Tamira’s trip and the information in your table to sketch a graph of the data.

3. Fill in the blank in the following statement. Then explain why the statement is true.

For the segments representing the times during which Tamira and her family stopped to shop and eat lunch, the graph is a _________________ line.

4. Tamira’s classmate Robert graphs the trip he took on Memorial Day. Use the graph to complete the table for Robert’s trip.

Time Distance from

home Change in distance from last time

Time since last change

0

1

2

3.5

4

4.5

6

5. Write a story, based on the graph, describing Robert’s trip.

6. Janie made several walks in front of a motion detector to explore rate of change. This is a graph of one of Janie’s walks, where T(S) represents time in seconds and D(M) represents Janie’s distance from the motion detector in meters. Use the graph to answer the following questions.

a. Fill in the table below by estimating the distance to two decimal places.

Time (seconds) Distance (meters) 0

2

3

5

b. Estimate Janie’s rate between 0 and 2 seconds. Estimate her rate between 3 and 5 seconds. How do these rates

compare?



7.7 CONSOLIDATION ACTIVITY 1. Tamira and Robert want to match this motion graph. They decide Tamira will hold the motion detector while Robert walks in

front of it to try to match the graph. Answer the following questions about how Robert should move to match the graph.

a. Where should Robert begin his movement?

b. How should Robert move for the first 2 seconds?

c. What should Robert do between 2 and 3 seconds?

d. What should Robert do between 3 and 4 seconds?

e. How fast should Robert walk between 4 and 6 seconds?

f. How fast should Robert walk between 6 and 7 seconds?

g. When should Robert move most quickly? Why? What is the fastest rate?

2. By yourself, create a graph modeling a motion detector walk. When you are done, exchange graphs with your partner and try to create a story to go along with your partners graph.




1. Match each rate with a table and graph.

TABLE (A-‐D) GRAPH (V-‐Z)

a. Rate of 2.5 feet per second

b. Rate of -‐1.5 feet per second

c. Rate of -‐3 feet per second

d. Rate of 0 feet per second

e. Rate of 0.5 foot per second

A. Time (sec) Distance

(ft) B. Time (sec) Distance

(ft)

0 21 0 39 4 23 2 33 8 25 4 27 12 27 6 21

C. Time (sec) Distance (ft)

D. Time (sec) Distance (ft)

0 21 0 21 2 26 4 15 4 31 8 9 6 36 12 3

E.


7 21 11 21 13 21

2. Describe the strategy you used to match the rates with the tables and graphs.





cepts

1. Rewrite each expression without parentheses:

4 − 3− x( )

−4 −3+ x( )

2. If (m,n) = (3,7) and (o,p) = (8,-‐2), find the value of the following expression:

!!

�

n− pm− o

Answer:

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3. Create your own download problem. Specify the following:

• The total size of the file, in KB: ___________

• The amount downloaded so far, in KB:__________

• The amount left to download, in KB: __________

• The percentage downloaded: ____________

• The percentage remaining: ______________

Make sure your answers agree with one another. Then draw a download bar to illustrate your problem.

4. Fill in the blanks in the area model:

8

10

100

40

( )( )( ) ( ) ( ) ( )

18 15 10 8 10 5

10 10 10 8 8 ___ ___ ___

• = + +

= • + • + • + •

Review


5. Identify the following values for the box-‐and-‐whisker plot. (You may need to select a scale for the graph first.)

Median: ______

Minimum: ______

Maximum: ______

6. The two points plotted on the grid have the coordinates (3,6) and (15,12).

a. Label each point with its coordinates. b. Label the tick marks on each axis. c. Plot the point (12,9) on the grid and label it. d. Plot the point (5,10) on the grid and label it.





Lesson 7.8 Putting it all together

7.8 OPENER

1. Sketch two different graphs of distance versus time for two elevators, both starting at floor 2, but rising or descending at different rates.

2. Sketch two different graphs of distance versus time for two elevators, both starting on floor 5, but rising or descending at different rates.

3. If everyone in the class added their lines to your grids above, what patterns, if any, would you see?

7.8 CORE ACTIVITY

Use the graph of three athletes (Jaime, Brad, and Chris) running a race to answer the following questions. 1. At 50 seconds into the race, who is leading?

2. What can you observe about how each runner ran the race?

3. When is each athlete in the lead?

4. What happens about 75 seconds into the race, according to the graph?

5. Who wins the race?

6. What can you say about each of the runners’ final times?



7. Now that you have looked at some specifics about the race, write a detailed interpretation of each athlete's race experience, including estimates of time and distance but also mentioning sprints, slowdowns, and speed throughout the race.

7.8 ONLINE ASSESSMENT

Today you will take an online assessment.




Part I: Fill in the missing information about these skates recorded by a student. (Fill in the table and/or reconstruct the graph, then find the rate.) Assume the skater is traveling at a constant rate.

1. How fast was he skating?

Time (sec)

Distance (ft)


Time (sec)

Distance (ft)

0 3 6 3


Time (sec)

Distance (ft)

3 3 5 7 7 9


Time (sec)

Distance (ft)

2 0 4 6 8 3




Time (sec)

Distance (ft)

1 7 -‐ -‐ 5 2


Time (sec)

Distance (ft)

2 3 5 4 7

7. Choose two table and graph pairs in questions 1-‐6, then write stories about them to match the representations.

a. Make one set of representations a skateboard-‐motion detector story.

b. Make one set of representations an elevator story.

Part II: Reflect on what you have learned so far in this topic. Complete the following math journal.

Big idea from the unit Describe how the idea challenged you. Describe the skills you used to overcome this challenge.

1.

2.

3.

4.

5.





cepts

1. Calculate the following products. Look for a pattern.

3 ·∙ 20 =

3 ·∙ 20 =

3 ·∙ 18 =

9 ·∙ 20 =

9 ·∙ 20 =

9 ·∙ 18 =

Use the pattern to help you find the value of 6 ·∙ 18.

Explain:

2. If (m, n) = (3, 7) and (o, p) = (8, -‐2), find the value of the expression:

!!

�

n− pm− o


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3. Nina is trying to download a file of 520 MB in total size. So far, it has taken 117 seconds and 130 MB has completed.

Enter the facts given into the correct places in this table:

Time Part completed 130 MB

Whole 520 MB Then, find the time for Nina to download the whole file, if the file continues to download at a constant rate. Answer:

4. Michael did 45 pushups in 3 minutes. Michelle did 36 pushups in 2 minutes. Who did pushups at a faster rate? How much faster was that person?


Review


5. Consider the data set 3, 4, 5, 6, 7, 7, 10.

Find each measure of central tendency for the data set: median mode average

Circle the three true statements in the list below: median < mode mode < average median < average

median = mode mode = average median = average

median > mode mode > average median > average

6. Choose an appropriate scale for each axis and plot these points on the grid: (100, 5) (10, -‐80) (-‐40, -‐10) (15, 60)

One unit on the x-‐axis equals: One unit on the y-‐axis equals:



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03x 13 T7 SAB 2012 13 · Copyright!©!2012!Charles!A.!Dana!Center!at!the!University!of!Texas!at!Austin,!Learning!Sciences!ResearchInstitute!at!the!University!of!Illino isatChicago