In previous lessons, students learned that if two situations involve equivalent ratios, we can say that the situations are described by the same rate. In this lesson, students compare ratios to see if two situations in familiar contexts involve the same rate. The contexts and questions are:
- Two people run different distances in the same amount of time. Do they run at the same speed?
- Two people pay different amounts for different numbers of concert tickets. Do they pay the same cost per ticket?
- Two recipes for a drink are given. Do they taste the same?
In each case, the numbers are purposely chosen so that reasoning directly with equivalent ratios is a more appealing method than calculating how-many-per-one and then scaling. The reason for this is to reinforce the concept that equivalent ratios describe the same rate, before formally introducing the notion of unit rate and methods for calculating it. However, students can use any method. Regardless of their chosen approach, students need to be able to explain their reasoning (MP3) in the context of the problem.
- Choose and create diagrams to help compare two situations and explain whether they happen at the same rate.
- Justify that two situations do not happen at the same rate by finding a ratio to describe each situation where the two ratios share one value but not the other, i.e., $a:b$ and $a:c$, or $x:z$ and $y:z$.
- Recognize that a question asking whether two situations happen “at the same rate” is asking whether the ratios are equivalent.
Let’s use ratios to compare situations.
- I can decide whether or not two situations are happening at the same rate.
- I can explain what it means when two situations happen at the same rate.
- I know some examples of situations where things can happen at the same rate.
double number line diagram
A double number line diagram uses a pair of parallel number lines to represent equivalent ratios. The locations of the tick marks match on both number lines. The tick marks labeled 0 line up, but the other numbers are usually different.
meters per second
Meters per second is a unit for measuring speed. It tells how many meters an object goes in one second.
For example, a person walking 3 meters per second is going faster than another person walking 2 meters per second.
The word per means “for each.” For example, if the price is \$5 per ticket, that means you will pay \$5 for each ticket. Buying 4 tickets would cost \$20, because \(4 \boldcdot 5 = 20\).
We use the words same rate to describe two situations that have equivalent ratios.
For example, a sink is filling with water at a rate of 2 gallons per minute. If a tub is also filling with water at a rate of 2 gallons per minute, then the sink and the tub are filling at the same rate.
The unit price is the cost for one item or for one unit of measure. For example, if 10 feet of chain link fencing cost \$150, then the unit price is \(150 \div 10\), or \$15 per foot.