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

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

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

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

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 137.5 minutes? What information are we given in this problem? CFU

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 137.5 minutes? a. We will organize our work using a table for time distance: Time (in minutes) Distance (in miles) 25 2 CFU

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 137.5 minutes? b. How many miles could Paul walk in 50 minutes? Explain. Time (in minutes) Distance (in miles) 25 2 50 4 CFU

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 137.5 minutes? How many miles could Paul walk in 75 minutes? Explain. Time (in minutes) Distance (in miles) 25 2 50 4 75 6 CFU

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 137.5 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

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 137.5 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

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 137.5 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

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 137.5 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 25 25 𝑦= 2 25 𝑥 CFU

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 137.5 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

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

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

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

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 38 38 𝑦= 4 38 𝑥 𝑦= 2 19 𝑥 → CFU

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