Lesson 2.2.2

HW: Day 1: 2-59 to 2-62

DAY 2: 2-63 to 2-67

Learning Target: Scholars will understand speed as a rate. Scholars will apply contextual meaning to m and b.

Today you will focus on the meaning of “rate of change” in various situations. What does a rate of change represent? How can you use it? As you graph the results of a competitive tricycle race today, think about how the participants’ rates of change compare to each other.

2-53. THE BIG RACE − HEAT 1

Before a big race, participants often compete in heats, which are preliminary races that determine who competes in the final race.

In the first heat, Leslie, Kristin, and Evie rode tricycles toward the finish line. Leslie began at the starting line and rode at a constant rate of 2meters every second. Kristin got an 8-meter head start and rode 2 meters every 5seconds. Evie rode 5 meters every 4 seconds and got a 6-meter head start.

1.  On neatly scaled axes, graph and then write an equation in terms of x and y for the distance Leslie travels. Let x represent time in seconds and y represent distance in meters. Then do the same for Kristin and Evie using the same set of axes.

2.  After how many seconds did Leslie catch up to Evie? How far were they from the starting line when Leslie caught up to Evie? Confirm your answer algebraically and explain how to use your graph to justify your answer.

3.  The winner of this heat will race in the final Big Race. If the race is 20 meters long, who won? Use both the graph and the equations to justify your answer.

4.  How long did it take each participant to finish the race?

5.  The school newspaper wants to report Kristin’s speed. How fast was Kristin riding? Write your answer as a unit rate.

2-54.THE BIG RACE − HEAT 2

In the second heat, Elizabeth, Kaye, and Hannah raced down the track. They knew the winner would compete against the other heat winners in the final race.

6.  When the line representing Kaye’s race is graphed, the equation is. What was her speed (in meters per second)? Did she get a head start?

7.  Elizabeth’s race is given by the equation. Who is riding faster, Elizabeth or Kaye? How do you know?

8.  Just as she started pedaling, Hannah’s shoelace came untied! Being careful not to get her shoelace tangled in the pedal, she rode slowly. Hannah’s race is represented by the table to the right. At what unit rate was she riding? Write your answer as a unit rate.

9.  To entertain the crowd, a clown rode a tricycle in the race described by the equationf(x) = 20 − x. Withoutgraphing or making a table, fully describe the clown’s ride.

2-55.OTHER RATES OF CHANGE

The slope of a graph can represent many things. In this lesson you concentrated on situations where the rate of change of a line (the slope) represented speed. However, the rate of change can represent many other things besides speed, depending on the situation.

1.  For each graph below,

o  Explain what real-world quantities the slope and y-intercept represent.

o  Find the rate of change for each situation.

2.  In each of the situations, would it make sense to draw a different line with a negative y-intercept?

2-56. TAKE A WALK

The president of the Line Factory is so impressed with your work that you have been given a special assignment: to analyze the graph below, which was created when a customer walked in front of a motion detector. The motion detector recorded the distance between it and the customer.

Obtain the Lesson 2.2.2 Resource Page from your teacher. The graph is a piecewise graph. A piecewise graph is a graph that has a different equation for different intervals along the x-axis. Working with your team, explain the motion that the graph describes.

Make sure you describe:

·  If the customer was walking toward or away from the motion detector.

·  Where the customer began walking when the motion detector started collecting data.

·  Any time the customer changed direction or stopped.

·  When the customer walked slowly and when he or she walked quickly by calculating the rate of change. Find the speed in feet per second.

·  An equation representing each piece of the graph.

·  The domain (the interval along the xaxis) for which each of the equations isvalid.

2-59.Find the rule for the following tile pattern. 2-59 HW eTool (Desmos).

2-60.Copy and complete each of the Diamond Problems below. The pattern used in the Diamond Problems is shown at right.

2-61. THE BIG RACE − HEAT 3

Barbara, Mark, and Carlos participated in the third heat of “The Big Race.” Barbara thought she could win with a 3meter head start even though she only pedaled 3 meters every 2 seconds. Mark began at the starting line and finished the 20meter race in 5seconds. Meanwhile, Carlos rode his tricycle so that his distance (y) from the starting line in meters could be represented by the equationy = + 1, where x represents time in seconds. 2-61 HW eTool (Desmos).

1.  What is the dependent variable? What is the independent variable?

2.  Using the given information, graph lines for Barbara, Mark, and Carlos on the same set of axes. Who won the 20 meter race and will advance to the final race?

3.  Find equations that describe Barbara’s and Mark’s motion.

4.  How fast did Carlos pedal? Write your answer as a unit rate.

5.  When did Carlos pass Barbara? Confirm your answer algebraically.

2-62.Create a table and a graph for the liney= 5x− 10. Find thex-intercept andy-intercept in the table and on the graph. 2-62 HW eTool (Desmos).

2-63. Find the slope of the line containing the points in the table below. 2-63 HW eTool (Desmos).

IN (x) / 2 / 4 / 6 / 8 / 10
OUT (y) / 4 / 10 / 16 / 22 / 28

2-64.Use what you know abouty=mx+bto graph each of the following equations quickly on the same set of axes.

1.  y= 3x+ 5

2.  y= −2x+ 10

3.  y= 1.5x

2-65.Review what you know about graphs by answering the following questions. 2-65 HW eTool (Desmos).

4.  Find the equation of the line graphed at right.

5.  What are itsx- andy-intercepts?

2-66.Use the idea of cube root from problem 1-35 to evaluate the following expressions.

6. 

7. 

8. 

9. 

2-67.Each part (a) through (d) below represents a different tile pattern. For each one, determine how the pattern is growing and the number of tiles in Figure 0.

10. 

11. 

12.  y= 3x− 14

13. 

x / −3 / −2 / −1 / 0 / 1 / 2 / 3
y / 18 / 13 / 8 / 3 / −2 / −7 / −12

Lesson 2.2.2

·  2-53. See below:

  1. See graph below. Leslie: y = 2x, Kristin: y = x + 8, Evie: y = x + 6
  2. After8 seconds, both racers were 16 meters from the starting line.
  3. Leslie won the race.
  4. Leslie finished the race in 10 seconds, while Kristin took 30 seconds and Evie took 11.2seconds.
  5. Her speed was 2 meters every 5seconds, so her unit rate of change is meters per second.

·  2-54. See below:

  1. She traveled meters per second, and got a 1meter head start.
  2. Elizabeth is riding faster. Her speed is meters per second, which is faster than Kaye.
  3. Hannah rides 4 meters every 14 seconds, so she rides meters per second.
  4. The clown started at the finish line and rode toward the starting line with a speed of 1meter per second.

·  2-55. See below:

  1. i: The slope represents the number of feet a tree grows per year, and the y-intercept represents the height when the measuring began (perhaps when it was planted), rate of change is 2 feet per year.
    ii: The slope represents the amount of money being spent from a bank account each month, while the y-intercept represents the beginning balance, rate of change is –10 dollars per month.
    iii: The slope represents the distance that can be traveled on a gallon of gas, while the y-intercept indicates that you cannot go any distance with zero gallons of gas, rate of change is 22 miles per gallon.
  2. Possible answer: The yintercepts in parts (i) and (iii) cannot be negative because it is does not make sense to have negative height or distance, but the bank balance in part (ii) could be negative if money was owed.

·  2-56. The customer starts 5 feet away from the detector and walks extremely slowly toward the detector at foot per second. The equation for the first section is y = −x + 5, where y is the distance to the detector (feet) and x is the time (seconds), and the domain is 0 to 4 seconds. After 4 seconds (when the customer has travelled only one foot), the customer stops and stands still for two seconds. The equation for the second interval is y = 4, the domain is 4 to 6 seconds. Then at six seconds, the customer walks much more quickly away from the detector at (1.667) feet per second. The equation for the third interval is y = x − 6, the domain is 6 to 9 seconds. The detector stops collecting data after 9 seconds.

· 

·  2-59. y = 2x + 3

·  2-60. Find solutions in the diamonds below:

·  2-61. See below:

1.  The dependent variable is distance in meters and the independent variable is time in seconds.

2.  See graph below. Mark won the race, finishing in 5 seconds.

3.  Barbara: y = x + 3, Mark: y = 4x

4.  5 meters every 2 seconds, or meters per second.

5.  2seconds after the start of the race, when each is 6 meters from the starting line.

·  2-62. See graph and table below. x-intercept: (2, 0), y-intercept: (0, –10)

·  2-63. m = 3

·  2-64. See graph below:

·  2-65. See below:

1.  y = −2x + 1

2.  x: (0.5, 0), y: (0, 1)

·  2-66. See below:

1.  1

2.  0

3.  2

4.  7

·  2-67. See below:

1.  G =5, Fig 0=3

2.  G =–2, Fig 0=3

3.  G =3, Fig 0=–14

4.  G =–5, Fig 0=3