lesson 5: graphing stories – writing...

ALGEBRA I

M1 Hart Interactive – Algebra 1 Lesson 5

Lesson 5: Graphing Stories – Writing Equations

Exploratory Activity

Consider the story:

Maya and Earl live at opposite ends of the hallway in their apartment building. Their doors are 50 ft. apart.

Each starts at his or her own door and walks at a steady pace toward each other and stops when they

meet.

1. A. What two aspects of their walk should we measure?

B. What would their graphing stories look like if we put them on the same graph?

2. When the two people meet in the hallway, what would be happening on the graph?

3. Sketch a graph that shows their distance from Maya’s door.


S.31

This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org

This file derived from ALG I-M1-TE-1.3.0-07.2015

This work is licensed under a

Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

ALGEBRA I


Consider the story:

Duke starts at the base of a ramp and walks up it at a constant rate. His elevation increases by 3 ft. every

second. Just as Duke starts walking up the ramp, Shirley starts at the top of the same 25 ft. high ramp and

begins walking down the ramp at a constant rate. Her elevation decreases 2 ft. every second.

Understanding the Problem

4. Fill in the chart below to help you understand how quickly Duke and Shirley move and where they start.

Duke’s Path Shirley’s Path

Time in

Seconds

Elevation

in Feet

Time in

Seconds

Elevation

in Feet

0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9

10

11

12

5. Use the extra row in each table to write an

expression that describes the elevation pattern.

6. Use the grid above to graph the data for Duke and Shirley. Be sure to finish the legend to help the reader

understand your graph.

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ele

va

tio

n i

n F

ee

t

Time in Seconds

The Story of Duke and Shirley

Duke:

Shirley:


S.32





ALGEBRA I


Writing an Equation of a Line

The equation of a line can be in the form y mx b= + , where m represents the slope of the line and b

represents the y-intercept.

7. Determine the slope of Duke’s line and Shirley’s line. Why is one of the slopes negative?

Duke’s slope: ___________________ Shirley’s slope: ___________________

8. What is the y-intercept for each line?

Duke’s y-intercept: _______________ Shirley’s y-intercept: ______________

9. Write the equation of Duke’s line and Shirley’s line.

Duke’s line: ___________________ Shirley’s line: ___________________

10. Where do the two lines intersect? _____________ This is the point of intersection. What time do Duke

and Shirley pass each other?

11. How could you use the equations to find the point of intersection?

The equation you wrote in Exercise 9 are in slope-intercept form or y =mx + b. Point-slope is another very

useful form of a linear equation. For point-slope we need any point on the line, not just the y-intercept and

the slope of the line. The general form looks like y – y1 = m(x – x1) where (x1, y1) is any point on the line and m

is the slope of the line.

12. A. Use the points when t = 3 seconds and slope from Exercise 7 with the point-slope equation to get an

equation for Duke and Shirley.

B. How do these equations compare to the ones you wrote in Exercise 9?


S.33





ALGEBRA I


Winter-Wonderland Party!

Nancy and Jane want to throw a Winter-Wonderland party. They want to split the cost evenly between them.

Nancy owes her parents $30 for the dress she wanted to have for the first day back to school. She must pay

them back before she can contribute to the party expenses. Jane has been putting money away all summer

and had $66 before she spent $6 on some new songs by her favorite band. Nancy earns $15 each week for

keeping up on her chores. Jane earns $18 every week for allowance, but pays her brother half for doing her

pooper-scooper duty and cleaning up after their two dogs. It is mid-August and the girls promise to not

spend their earnings and plan to save for the party. They each try to figure out how long it will take to have

the same amount of savings in order to maximize the party budget.

13. Create a table to help you organize the information you have so far. Then graph your data and include a

legend. Does it make sense to connect the points? Finally, write an equation for each girl.

Time in Weeks

Since Mid-August

Nancy’s

Savings

Jane’s

Savings

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

n

-30

-20

-10

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

210

220

230

0 2 4 6 8 10 12 14 16 18

Mo

ne

y S

ave

d i

n D

oll

ars

Time in Weeks

Winter-Wonderland Party

This is where you can write the expression for each girl’s saving.


S.34





ALGEBRA I


14. How many weeks does it take for the girls to have the same amount of money? Check that this answer

makes sense on the graph and in each equation.

15. If the girls determine that they need $150 each for the party, which week would they have enough

money? Remember that they each have to put in the same amount. Justify your answer using the tables,

graph or equations.


S.35





ALGEBRA I


Lesson Summary

Linear Equations: If you know the slope and y-intercept you can use the equation y mx b= + to

write an equation for the line. If you have the slope and any point on line you can use the point-

slope equation, y – y1 = m(x – x1) where (x1, y1) is any point on the line and m is the slope of the

line.

Example 1: The slope of a line is -3 and the y-intercept is 5. An equation of this line is

___________________.

Example 2: The slope of a line is ½ and the point (2, 4) is on the line. The equation of this line

is ___________________.

Point of Intersection: The point of intersection is an ordered pair that is a solution to both

equations. On a graph this is where the two graphs cross each other.

Example: Determine the point of intersection for the graph below.

Point of intersection: _________


S.36





ALGEBRA I


Homework Problem Set

1. Maya and Earl live at opposite ends of the hallway in their apartment building. Their doors are 50 ft. apart. Each starts at his or her own door and walks at a steady pace toward each other and stops when

they meet.

Suppose that Maya walks at a constant rate of 2 ft. every second and Earl walks at a constant rate of 4 ft. every second starting from 50 ft. away. Create equations for each person’s distance from Maya’s door

and determine exactly when they meet in the hallway. How far are they from Maya’s door at this time?

Let y = distance from Maya’s door in feet

Let x = time in seconds

2. Suppose two cars are travelling north along a road.

Car 1 travels at a constant speed of 50 mph for two hours, then speeds up and drives at a constant speed

of 100 mph for the next hour. The car breaks down and the driver has to stop and work on it for two

hours. When he gets it running again, he continues driving recklessly at a constant speed of 100 mph.

Car 2 starts at the same time that Car 1 starts, but Car 2

starts 100 mi. farther north than Car 1 and travels at a

constant speed of 25 mph throughout the trip.

A. Sketch the distance-versus-time graphs for Car 1 and

Car 2 on a coordinate plane at the right. Be sure to

include a legend.

B. Approximately when do the cars pass each other?

C. Tell the entire story of the graph from the point of

view of Car 2. (What does the driver of Car 2 see

along the way and when?)


S.37





ALGEBRA I


3. Challenge Problem

A. Write linear equations representing each car’s distance in terms of time (in hours). Note that you will

need four equations for Car 1 and only one for Car 2.

B. Use these equations to find the exact coordinates of when the cars meet.

4. Suppose that in Problem 2 above, Car 1 travels at the constant speed of 25 mph the entire time.

A. Sketch the distance-versus-time graphs for the two

cars on a graph below.

B. Do the cars ever pass each other? Explain.

C. What is the linear equation for Car 1 in this case?


S.38





ALGEBRA I


5. The graph at the right shows the revenue (or

income) a company makes from designer

coffee mugs and the total cost (including

overhead, maintenance of machines, etc.)

that the company spends to make the coffee

mugs.

A. How are revenue and total cost related to

the number of units of coffee mugs

produced?

B. What is the meaning of the point (0, 4000) on the total cost line?

C. What are the coordinates of the intersection point? What is the meaning of this point in this

situation?

0

2000

4000

6000

8000

10000

12000

14000

0

10

0

20

0

30

0

40

0

50

0

60

0

70

0

80

0

90

0

10

00

Do

lla

rs

Units Produced and Sold

Total Cost

Revenue


S.39





ALGEBRA I


6. Challenge Problem

Consider the story:

May, June, and July were running at the track. May started first and ran at a steady pace of 1 mi. every

11 min. June started 5 min. later than May and ran at a steady pace of 1 mi. every 9 min. July started

2 min. after June and ran at a steady pace, running the first lap �14 mi. � in 1.5 min. She maintained this

steady pace for 3 more laps and then slowed down to 1 lap every 3 min.

A. Sketch May, June, and July’s distance-versus-time graphs on a coordinate plane below . Be sure to label

your axes, put a title on your graph, and include a legend.

B. Write linear equations that represent each girl’s mileage in terms of time in minutes. You will need two

equations for July since her pace changes after 4 laps (1 mi.).

C. Who was the first person to run 3 mi.?

D. Did June and July pass May on the track? If they did, when and at what mileage?

E. Did July pass June on the track? If she did, when and at what mileage?


S.40





lesson 5: graphing stories – writing...

Documents