1. Activating Prior Knowledge – Notes
M4:LSN10
A Critical Look at Proportional Relationships
Activating Prior Knowledge – Notes
Solve.
1. 𝟑 𝟕 = 𝒙 𝟏𝟒
2. 𝒙 𝟖 = 𝟗 𝟏𝟐
3. 𝟐 𝒙−𝟏 = 𝟔 𝟐𝒙
4. 𝟑 𝒙+𝟐 = 𝟒 𝒙
Tie to LO

2. Learning Objective
Today, we will look critically at and solve proportional relationships.
CFU

3. M4:LSN10
A Critical Look at Proportional Relationships
Concept Development - Pair Share
What does a variable represent?
x + 3
4 - 2x
7 + 8x = 25
Why would we have two variables in a linear equation?
3y + 5x
y = 4x + 2
CFU

4. Concept Development – Notes #1 & 2
M4:LSN10
A Critical Look at Proportional Relationships
Concept Development – Notes #1 & 2
1. The formula connecting distance to rate and time is d = rt.
2. D = distance, r = rate, and t = time.
CFU

5. What information are we given in this problem?
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Notes #3
3. Paul walks 2 miles in 25 minutes. How many miles can Paul walk in minutes?
What information are we given in this problem?
CFU

6. Skill Development/Guided Practice – Cont. Notes #3
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Cont. Notes #3
3. Paul walks 2 miles in 25 minutes. How many miles can Paul walk in minutes?
a. We will organize our work using a table for time distance:
Time (in minutes)
Distance (in miles) 25 2
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7. Skill Development/Guided Practice – Cont. Notes #3
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Cont. Notes #3
3. Paul walks 2 miles in 25 minutes. How many miles can Paul walk in minutes?
b. How many miles could Paul walk in 50 minutes? Explain.
Time (in minutes)
Distance (in miles) 25 2 50 4
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8. Skill Development/Guided Practice – Cont. Notes #3
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Cont. Notes #3
3. Paul walks 2 miles in 25 minutes. How many miles can Paul walk in minutes?
How many miles could Paul walk in 75 minutes? Explain.
Time (in minutes)
Distance (in miles) 25 2 50 4 75 6
CFU

9. Skill Development/Guided Practice – Cont. Notes #3
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Cont. Notes #3
3. Paul walks 2 miles in 25 minutes. How many miles can Paul walk in minutes?
How many miles could Paul walk in 100 minutes? Explain.
Time (in minutes)
Distance (in miles) 25 2 50 4 75 6
100 8
CFU

10. Skill Development/Guided Practice – Cont. Notes #3
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Cont. Notes #3
3. Paul walks 2 miles in 25 minutes. How many miles can Paul walk in minutes?
How many miles could Paul walk in 125 minutes? Explain.
Time (in minutes)
Distance (in miles) 25 2 50 4 75 6
100 8
125 10
CFU

11. Skill Development/Guided Practice – Cont. Notes #3
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Cont. Notes #3
3c. Paul walks 2 miles in 25 minutes. How many miles can Paul walk in minutes?
How many miles y can Paul run in x minutes?
Time
(in mins)
Distance
(in miles)
We can answer this question with a proportion. 25 2
We know for a fact that Paul can walk 2 miles in 25 minutes, so we can write the ratio 𝟐𝟓 𝟐 . We can also write another ratio for the number of miles 𝒚 Paul walk in 𝒙 minutes. It is 𝒙 𝒚 . 50 4 75 6
100
Now we can write the two ratios as a proportion, 𝟐𝟓 𝟐 = 𝒙 𝒚 . We can use cross multiplication to solve this proportion for the variable 𝒚. 8
125 10
CFU

12. Skill Development/Guided Practice – Cont. Notes #3
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Cont. Notes #3
3c. Paul walks 2 miles in 25 minutes. How many miles can Paul walk in minutes?
How many miles y can Paul run in x minutes?
d. What does the equation mean?
25 2 = 𝑥 𝑦
Paul can walk 𝟐 𝟐𝟓 miles every minute.
25𝑦=2𝑥
𝑦= 2 25 𝑥
CFU

13. Skill Development/Guided Practice – Cont. Notes #3
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Cont. Notes #3
3e. How many miles could Paul walk in minutes?
The relationship between the distance Paul walks and the time it takes him to walk that distance is proportional.
𝐖𝐡𝐚𝐭 𝐝𝐨𝐞𝐬
𝐲 𝐫𝐞𝐩𝐫𝐞𝐬𝐞𝐧𝐭?
𝟐𝟓 𝟐 = 𝟏𝟑𝟕.𝟓 𝒚
Distance
(in miles)
Time
(in mins) 25 2
𝟐𝟓𝒚=𝟏𝟑𝟕.𝟓(𝟐) 50 4
𝟐𝟓𝒚=𝟐𝟕𝟓 75 6
÷𝟐𝟓 ÷𝟐𝟓
100 8
𝒚=𝟏𝟏
125 10
CFU

14. Skill Development/Guided Practice – Notes # 4 & 5
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Notes # 4 & 5
4. Suppose a person walks a distance d (miles) in a given time interval t (minutes).
Then the average speed in the given time interval is 𝑑 𝑡 in miles per minute.
5. If we assume that someone can actually walk at the same average speed over any time interval, then we say that the person is walking a constant speed C.
CFU

15. Skill Development/Guided Practice – Notes #6
M4:LSN10
A Critical Look at Proportional Relationships
Skill Development/Guided Practice – Notes #6
How many miles y can be traveled in any number of hours x?
𝐖𝐡𝐚𝐭
𝐝𝐨𝐞𝐬 𝐱
𝐫𝐞𝐩𝐫𝐞𝐬𝐞𝐧𝐭?
𝟏𝟐𝟑 𝟑 = 𝒚 𝒙
Time (in hours) 3
123 6
246 9
369 12
492 y
Distance (in miles)
𝐖𝐡𝐚𝐭
𝐝𝐨𝐞𝐬 𝐲
𝐫𝐞𝐩𝐫𝐞𝐬𝐞𝐧𝐭?
𝟏𝟐𝟑𝒙=𝟑𝒚
÷𝟑 ÷𝟑
𝟒𝟏𝒙=𝒚
𝐖𝐡𝐚𝐭 𝐝𝐨𝐞𝐬 𝐭𝐡𝐞
𝐞𝐪𝐮𝐚𝐭𝐢𝐨𝐧
𝐲=𝟒𝟏𝐱 𝐦𝐞𝐚𝐧?
It means that the distance traveled y is equal to the rate of 41 multiplied by number of hours x traveled at that rate.
CFU

16. Independent Practice– Notes #7
M4:LSN10
A Critical Look at Proportional Relationships
Independent Practice– Notes #7
Amanda runs 4 miles in 38 minutes. How many miles can Amanda run in 95 minutes?
Time(in minutes) 19 2 38 4 57 6 76 8 95 10
Distance (in miles)
Amanda can run 10 miles in 95 minutes.
CFU

17. M4:LSN10
A Critical Look at Proportional Relationships
Exit Ticket – #4
How many miles y can Amanda run in x minutes? Write the equation using the given information.
Time (in hours) 19 2 38 4 57 6 76 8
Distance (in miles)
38 4 = 𝑥 𝑦
38𝑦=4𝑥
𝑦= 4 38 𝑥
𝑦= 2 19 𝑥 →
CFU

18. Homework Closure – End of notes CFU 1. What did we learn today?
2. Why is this important to you?
3. How is a proportion a “linear equation in disguise”?
Homework
Problem Set 1 – 5. Complete all parts on a separate sheet of paper for credit.
CFU