1. 2.5 The Man Who Ran from Marathon to Athens Graphing Direct Proportions
WARM UP
A baby elephant nurses for the first two years of its life. Shortly after birth, the baby elephant searches for its mother‘s milk. It drinks about 10 liters of milk every day.
Assume that the elephant maintains the same rate of consumption.
Let d = number of days the milk is consumed and let m = number of liters of milk consumed.
1. Write an equation representing the relationship between amount of milk consumed and time it spends consuming the milk.
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.
2. What is the constant of proportionality?
3. Use your equation to complete the table.
4. Graph the values on the coordinate plane, with time (days) on the x axis and amount of milk consumed (liters) on the y-axis.
5. What do you notice about the points on the graph?

2. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Lesson Opener as a class.
(pg. 105)
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

3. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
Do #1-3 with your group
(pg 106)
You have 4:45 min.
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

4. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon (p. 106)
The distance (d) in miles a runner runs varies directly with the amount of time (t) in hours spent running. Suppose Antonio’s constant of proportionality is 9.
1. Write an equation that represents the relationship between the distance ran, and the time spent running.
𝑑=9𝑡
Is there another way to write this equation?
2.25
How do you know if your equation is correct?
0.5
2. What does the constant of proportionality represent?
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.
Antonio’s rate: 9 miles per hour.
6.75
How many minutes is 0.25 hours? 1
How did you determine the distance when the time was 0.25 hours?
11.25
How did you determine the time, when the distance was 4.5 miles?
13.5 2

5. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

6. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
Do #4a-d with your group
(pg 107)
You have 2:15 min.
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

7. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
a. What do you notice about the points on the graph?
It forms a straight line that starts at the origin.
b. Would it make sense to connect points on the graph?
Yes – Antonio could have run any number of hours and could have run any distance.
c. Interpret the meaning of the point (0,0) for the graph.
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.
(0,0) means that for 0 hours, Antonio ran 0 miles. The ordered pair (0,0) represents the beginning of the run.
d. Interpret the meaning of point (1.5, 13.5) for the graph.
At 1.5 hours, Antonio ran 13.5 miles.

8. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
Remember:
Graph of two variables that are directly proportional is a
LINE THAT PASSES THROUGH THE ORIGIN (0,0).
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

9. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
Do #5 with your group
(pg 108)
You have 2:55 min.
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

10. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
5. For each of the points on the graph, write a ratio in the form 𝑦 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑥 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 in the table.
Thinking about the equation that generated the points 𝑑=9𝑡, why would you expect all of the ratios to be equivalent?
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.
How is the constant of proportionality related to the ratios you determined?

11. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

12. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
Do #8-11 with your group
(pg 109)
You have 2:55 min.
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

13. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
8. Locate the points (1,9) and (1.5, 13.5).
a. What is the horizontal distance (from left to right) from 1 to 1.5?
0.5 hour
b. What is the vertical distance from 9 to 13.5?
4.5 miles
c. What is the ratio of the vertical distance to the horizontal distance?
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.
4.5 𝑚𝑖𝑙𝑒𝑠 0.5 ℎ𝑜𝑢𝑟 or 9 miles per hour.

14. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
9. Locate the points (1.25, 11.25) and (2, 18).
a. What is the horizontal distance (from left to right) from 1.25 to 2?
0.75 hour
b. What is the vertical distance from to 18?
6.75 miles
c. What is the ratio of the vertical distance to the horizontal distance?
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.
6.75 𝑚𝑖𝑙𝑒𝑠 0.75 ℎ𝑜𝑢𝑟 or 9 miles per hour.

15. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 1 Running a Marathon
11. What do you notice about the ratios?
The ratios are all the same as the constant of proportionality (k), which is 9.
How is the ratio of the
𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑝𝑜𝑖𝑛𝑡𝑠 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑝𝑜𝑖𝑛𝑡𝑠
related to the constant of proportionality?
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

16. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 4 Equations, Tables, and Graphs – Oh My!
Do #1a-d with your group
(pg )
You have 2:30 min.
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

17. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 4 Equations, Tables, and Graphs – Oh My!
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.
Does not show direct variation – does not go through the origin
Does not show direct variation – graph is not a straight line

18. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 4 Equations, Tables, and Graphs – Oh My!
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.
Yes – straight line; goes through origin
𝑘=3
Yes – straight line; goes through origin
𝑘= 1 5 𝑜𝑟 0.2
How do the 𝑦 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑥 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 ratios of the points relate to each other?

19. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 4 Equations, Tables, and Graphs – Oh My!
Do #2a-d with your group
(pg )
You have 4:28 min.
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.

20. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 4 Equations, Tables, and Graphs – Oh My! 9
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1. 21 24

21. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 4 Equations, Tables, and Graphs – Oh My!
0.5
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1. 2
3.2

22. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 4 Equations, Tables, and Graphs – Oh My! 18 8
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.
2.88
1.8

23. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Problem 4 Equations, Tables, and Graphs – Oh My! 2 10
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.
11.5 15

24. 2.5 The Man Who Ran From Marathon to Athens Graphing Direct Proportions
Homework:
Due Wednesday
Lesson 2.5 Skills Practice
Weekly Math #21 (Monday and Tuesday)
Circulate and see what kids have for this. Make sure that they are able to create some sort of ratio. There is no need to go over this as a class since they will be doing a similar one in the beginning of today’s lesson. Just use it as a quick formative assessment. Address student misconceptions in Problem 1, #1.