• Students compute unit rates associated with ratios of quantities measured in different units. Students use the context of the problem to recall the meaning of value of a ratio, equivalent ratios, rate and unit rate, relating them to the context of the experience.
• Students understand that two quantities are proportional to each other when there exists a constant (number) such that each measure in the first quantity multiplied by this constant gives the corresponding measure in the second quantity.
• When students identify the measures in the first quantity with x and the measures in the second quantity with y, they will recognize that the second quantity is proportional to the first quantity if y = kx for some positive number k. They apply this same relationship when using variable choices other than x and y.
Lesson 1 Summary
Unit Rate is often a useful means for comparing ratios and their associated rates when measured in different units.
The unit rate allows us to compare varying sizes of quantities by examining the number of units of one quantity per
1 unit of the second quantity. This value of the ratio is the unit rate.
Lesson 2 Summary
Measures in one quantity are proportional to measures of a second quantity if there is a positive number so that for every measure x of the first quantity, the corresponding quantity y is given by kx. The equation y = kx models this relationship.
A proportional relationship is one in which the measures of one quantity are proportional to the measures of the second quantity.
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.
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.
Lesson 1 Example 1: How Fast is our Class?
Record the results from the paper passing exercise in the table below.
Lesson 2 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.
Does everyone pay the same cost per ounce? How do you know?
Isabelle’s brother takes an extra-long time to create his dish. When he puts it on the scale, it weighs 15 ounces. If everyone pays the same rate in this store, how much will his dish cost? How did you calculate this cost?
Lesson 2 Example 2: A Cooking Cheat Sheet!
In the back of a recipe book, a diagram provides easy conversions to use while cooking.What does the diagram tell us?
Lesson 2 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 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 1 and his total money earned at the end of Week 4.
a. Work with a partner to answer Alex’s question.
b. Are Alex’s total earning proportional to the number of weeks he worked? How do you know?