A lemonade recipe calls for the juice of 5 lemons, 2 cups of water, and 2 tablespoons of honey.
Invent four new versions of this lemonade recipe:
One that would make more lemonade but taste the same as the original recipe.
One that would make less lemonade but taste the same as the original recipe.
One that would have a stronger lemon taste than the original recipe.
One that would have a weaker lemon taste than the original recipe.
4.2
Activity
Visiting the State Park
Standards Alignment
Building On
Addressing
7.RP.2.a
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
Entrance to a state park costs $6 per vehicle, plus $2 per person in the vehicle.
How much would it cost for a car with 2 people to enter the park? 4 people? 10 people? Record your answers in the table.
number of
people in vehicle
total entrance cost
in dollars
2
4
10
For each row in the table, if each person in the vehicle splits the entrance cost equally, how much will each person pay?
How might you determine the entrance cost for a bus with 50 people?
Is the relationship between the number of people and the total entrance cost a proportional relationship? Explain how you know.
4.3
Activity
Running Laps
Standards Alignment
Building On
Addressing
7.RP.2.a
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
Han and Clare were running laps around the track. The coach recorded their times at the end of laps 2, 4, 6, and 8.
Han's run:
distance (laps)
time (minutes)
pace (minutes per lap)
2
4
4
9
6
15
8
23
Clare's run:
distance (laps)
time (minutes)
pace (minutes per lap)
2
5
4
10
6
15
8
20
Is Han running at a constant pace? Is Clare? How do you know?
Write an equation for the relationship between distance and time for anyone who is running at a constant pace.
Student Lesson Summary
Here are the prices for some smoothies at two different smoothie shops:
Smoothie Shop A
smoothie
size (fl oz)
price
($)
dollars
per ounce
8
6
0.75
12
9
0.75
16
12
0.75
0.75
Smoothie Shop B
smoothie
size (fl oz)
price
($)
dollars
per ounce
8
6
0.75
12
8
0.67
16
10
0.625
???
???
For Smoothie Shop A, smoothies cost $0.75 per ounce no matter which size we buy. There could be a proportional relationship between smoothie size and the price of the smoothie. An equation representing this relationship is where represents size in ounces and represents price in dollars. (The relationship could still not be proportional, if there were a different size on the menu that did not have the same price per ounce.)
For Smoothie Shop B, the cost per ounce is different for each size. Here the relationship between smoothie size and price is definitely not proportional.
In general, two quantities in a proportional relationship will always have the same quotient. When we see some values for two related quantities in a table and we get the same quotient when we divide them, that means they might be in a proportional relationship—but if we can't see all of the possible pairs, we can't be completely sure. However, if we know the relationship can be represented by an equation of the form , then we are sure it is proportional.
Glossary
None
Have feedback on the curriculum?
Help us improve by sharing suggestions or reporting issues.
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
