Unit 7-1
Lesson 1-6
CW
Ratios and Proportional
Relationships: Lessons 1-6
Name
Date
Classwork Book
Math 7: Mr. Sanford
Lessons 1-6:
Proportional Relationship
Lesson 1-1
Mathematics 7: Mr. Sanford
CW
Lesson 1: An Experience in Relationships as Measuring Rate
Example 1: How fast is our class?
Record the results from the paper passing exercise in the table below.
Number of Papers
Time
Ratio of Number of Papers
Trial
Passed
(in seconds)
Passed to Time
Rate
Unit Rate
1
2
3
4
Key Terms from Grade 6 Ratios and Unit Rates
A ratio is an ordered pair of non-negative numbers, which are not both zero. The ratio is denoted 𝐴: 𝐵 to indicate the
order of the numbers: the number 𝐴 is first and the number 𝐵 is second.
Two ratios 𝐴: 𝐵 and 𝐶: 𝐷 are equivalent ratios if there is a positive number, 𝑐, such that 𝐶 = 𝑐𝐴 and 𝐷 = 𝑐𝐵.
A ratio of two quantities, such as 5 miles per 2 hours, can be written as another quantity called a rate.
The numerical part of the rate is called the unit rate and is simply the value of the ratio, in this case 2.5. This means
that in 1 hour the car travels 2.5 miles. The unit for the rate is miles/hour, read miles per hour.
Example 2: Our Class by Gender
Number of boys
Number of girls
Ratio of boys to girls
Class 1
Class 2
th
Whole 7 grade
Create a pair of equivalent ratios by making a comparison of quantities discussed in this Example.
Grade 7 Lesson 1-1:
An Experience in Relations as Measuring Rate
3
Mathematics 7: Mr. Sanford
Lesson 1-1
CW
Exercise 1: Which is the better buy?
Value-Mart is advertising a Back-to-School sale on pencils. A pack of 30 sells for $7.97 whereas a 12-pack of the same
brand cost for $4.77. Which is the better buy? How do you know?
Exercise 2: Problem Solving
Each year Lizzie’s school purchases student agenda books, which are sold in the school store. This year, the school
purchased 350 books at a cost of $1,137.50. If the school would like to make a profit of $1,500 to help pay for field trips
and school activities, what is the least amount they can charge for each agenda book? Explain how you found your
answer.
Grade 7 Lesson 1-1:
An Experience in Relations as Measuring Rate
4
Lesson 1-1
Mathematics 7: Mr. Sanford
CW
Lesson 2: Proportional Relationships
Example 1: Pay by the Ounce Frozen Yogurt!
A new self-serve frozen yogurt store opened this summer that sells its yogurt at a price based upon the total weight of the yogurt
and its toppings in a dish. Each member of Isabelle’s family weighed their dish and this is what they found.
Weight (ounces)
12.5
10
5
8
Cost ($)
5
4
2
3.20
Cost = ________ Weight
Example 2: A Cooking Cheat Sheet!
In the back of a recipe book, a diagram provides easy conversions to use while cooking.
0
½
1
1½
2
0
4
8
12
16
Cups
Ounces
Ounces = ________ Cups
Exercise 1
During Jose’s physical education class today, students visited activity stations. Next to each station was a chart depicting how many
Calories (on average) would be burned by completing the activity.
Calories burned while Jumping Rope
0
1
2
3
4
0
11
22
33
44
Time (minutes)
Calories Burned
a.
Is the number of calories burned proportional to time? How do you know?
Grade 7 Lesson 1-2:
Proportional Relationships
NOTE:
Proportional means the
relationship is made up of
equivalent ratios
5
Lesson 1-1
Mathematics 7: Mr. Sanford
b.
CW
If Jose jumped rope for 6.5 minutes, how many calories would he expect to burn?
Example 3: Summer Job
Alex spent the summer helping out at his family’s business. He was hoping to earn enough money to buy a new $220 gaming system
by the end of the summer. Halfway through the summer, after working for 4 weeks, he had earned $112. Alex made the same
amount of money each week that he worked. Alex wonders, “If I continue to work and earn money at this rate, will I have enough
money to buy the gaming system by the end of the summer?”
To check his assumption, he decided to make a table. He entered his total money earned at the end of week 4.
Week
Total
Earnings
0
1
2
3
4
5
6
7
8
$112
a.
Work with a partner to complete the table and answer Alex’s question.
b.
Are Alex’s total earnings proportional to the number of weeks he worked? How do you know?
c.
Try to write a multiplication problem using a number of weeks and the total earnings.
Lesson Summary:
Measures in one quantity are proportional to measures of a second quantity if the relationship is made up of
equivalent ratios.
We can tell if a relationship is proportional by comparing the unit rates to see if they are the same. If each set of
values can be divided to get the same unit rate we can say that each measure multiplied by the unit rate gives you
the other measure.
In the example given below, the distance is proportional to time since each measure of distance, 𝑦, can be
calculated by multiplying each corresponding time, 𝑡, by the same value, 10. This table illustrates a proportional
relationship between time, 𝑡, and distance, 𝑦.
Time (hrs), t
Distance (km), y
Grade 7 Lesson 1-2:
Proportional Relationships
0
0
1
10
2
20
3
30
6
Lesson 1-3
Mathematics 7: Mr. Sanford
CW
Lesson 3: Identifying Proportional Relationships in Tables
Opening Exercise
You have been hired by your neighbors to babysit their children on Friday night. You are paid $8 per hour. Complete the table
relating your pay to the number of hours you worked.
Hours Worked
Pay
1
Based on the table above, is pay proportional to hours
worked? How do you know?
2
3
4
4½
5
6
6.5
Examples 1–4
For Examples 1–3, determine if 𝑦 is proportional to 𝑥. Justify your answer.
1. The table below represents the amount of snow fall in 5 counties (in inches) to hours of a recent winter storm.
𝑥
Time (hrs)
𝑦
Snowfall
(In)
2
10
6
12
8
16
2.5
5
7
14
Grade 7 Lesson 1-3:
Unit Rate
(in/hr)
Identifying Proportional Relationships in Tables
7
Lesson 1-3
Mathematics 7: Mr. Sanford
2.
3.
4.
CW
The table below shows the relationship between cost of renting a movie to the number of days on rent.
𝑥
Number of
Days
𝑦
Cost
6
2
9
3
24
8
3
1
Unit Rate
($/day)
The table below shows the relationship between the amount of candy (pounds) bought and the total cost.
𝑥
Pounds
𝑦
Cost
5
10
4
8
6
12
8
16
10
20
Unit Rate
($/lb)
Randy is planning to drive from New Jersey to Florida. Randy recorded the distance traveled and the total number of gallons
used every time he stopped for gas.
Complete the table assuming miles driven is proportional to gallons consumed.
Gallons
Consumed
2
Miles Driven
54
4
8
189
10
12
216
In real life are miles driven proportional to gallons consumed?
Grade 7 Lesson 1-3:
Identifying Proportional Relationships in Tables
8
Lesson 1-4
Mathematics 7: Mr. Sanford
CW
Lesson 4: Identifying Proportional Relationships in Tables
Example: Which Team Will Win the Race?
You have decided to run in a long distance race. There are two teams that you can join. Team A runs at a constant rate of 2.5 miles
per hour. Team B runs 4 miles the first hour and then 2 miles per hour after that.
Task: Create a table for each team showing the distances that would be run for times of 1, 2, 3, 4, ,5 and 6 hours.
Time (hrs)
Team A
Distance
(miles)
Unit Rate
(m/hr)
Time (hrs)
Team B
Distance
(miles)
Unit Rate
(m/hr)
1
2
3
4
5
6
a.
For which team is distance proportional to time? Explain your reasoning.
b.
Explain how you know distance for the other team is not proportional to time.
c.
If the race were 2.5 miles long, which team would win? Explain.
If the race were 3.5 miles long, which team would win? Explain.
Grade 7 Lesson 1-5:
Identifying Proportional Relationships in Graphs
9
Mathematics 7: Mr. Sanford
Lesson 1-4
CW
If the race were 4.5 miles long, which team would win? Explain.
d.
For what length race would it be better to be on Team B than Team A? Explain.
e.
Using this relationship, if the members on the team ran for 10 hours, how far would each member run on each team?
f.
Will there always be a winning team, no matter what the length of the course? Why or why not?
g.
If the race is 12 miles long, which team should you choose to be on if you wish to win? Why would you choose this team?
Challenge: How much sooner would you finish on that team compared to the other team?
Grade 7 Lesson 1-5:
Identifying Proportional Relationships in Graphs
10
Lesson 1-5
Mathematics 7: Mr. Sanford
CW
Lesson 5: Identifying Proportional Relationships in Graphs
Opening Exercise
Isaiah sold candy bars to help raise money for his scouting troop. The table compares the amount of candy he sold to the money he
received.
Is the amount of candy bars sold proportional to the money Isaiah received? How do you know?
_____________________________________________________________________________
_____________________________________________________________________________
𝑥
Candy Bars
Sold
𝑦
Money
Received ($)
2
3
4
5
8
9
12
12
Example 1: From a Table to Graph
Complete the table below if Isaiah earned $3.00 for every 2 candy bars.
𝑥
Candy Bars
Sold
𝑦
Money
Received ($)
2
3
Important Note:
Characteristics of graphs of proportional relationships:
Grade 7 Lesson 1-5:
Identifying Proportional Relationships in Graphs
11
Mathematics 7: Mr. Sanford
Lesson 1-5
CW
Example 2
𝑥
2
𝑦
4
6
8
10
12
14
3
Similarities with Example 1:
Differences from Example 1:
Example 3
𝑥
0
𝑦
3
8
6
10
9
12
12
14
6
Similarities with Example 1:
Differences from Example 1:
Lesson Summary:
When two proportional quantities are graphed on a coordinate plane, the points lie on a straight line that passes
through the origin.
Grade 7 Lesson 1-5:
Identifying Proportional Relationships in Graphs
12
Lesson 1-6
Mathematics 7: Mr. Sanford
CW
Lesson 6: Identifying Proportional Relationships in Graphs and Tables
Today’s activity is an extension of Lesson 5. You will be working in groups to table, graph and identify whether or not the two
quantities are proportional to each other.
Classwork
Poster Layout
Use for notes
Problem
Table
Graph
Proportional or not? Explain.
Grade 7 Lesson 1-5:
Identifying Proportional Relationships in Graphs
13
Mathematics 7: Mr. Sanford
Lesson 1-6
CW
Gallery Walk: Take notes and answer the following questions:



Were there any differences found in groups that had the same problem.
Did you notice any common mistakes? How could they be fixed?
Was there any group that stood out by representing their problems and findings exceptionally clearly?
Problem 1:
Problem 2:
Problem 3:
Problem 4:
Grade 7 Lesson 1-5:
Identifying Proportional Relationships in Graphs
14
Extra Notes Page