Same and Different

Task

For all of these questions, it may be helpful to draw a diagram.

A steady speed

1. A snail went 6 inches in 2 minutes at a steady speed. How far did it go in one minute?
2. A caterpillar went 36 inches in 2 minutes at a steady speed. How far did it go in one minute?
3. A baby turtle went 36 inches in 12 minutes at a steady speed. How far did it go in one minute?
4. What is the same and what is different about the snail and the caterpillar? The caterpillar and the baby turtle? The baby turtle and the snail?

Buying in bulk

1. 30 kilograms of beans costs \$5. How many kilograms of beans can you buy for \$1?
2. 15 kilograms of rice costs \$5. How many kilograms of rice can you buy for \$1?
3. 30 kilograms of cornmeal costs \$10. How many kilograms of cornmeal can you buy for \$1?
4. What is the same and what is different about the beans and the rice? The rice and the cornmeal? The cornmeal and the beans?

IM Commentary

The purpose of this task is to analyze some very common contexts that can be represented by ratios and to motivate the idea of equivalent ratios for different kinds of contexts.

It can also be used to introduce students to double number line diagrams. If that is the case, we suggest providing students with an example of one along with the task statement for part (a):

The first set of questions connects the idea of constant speed with the idea of a ratio. Students should see that when you have equivalent ratios involving time and distance where the objects are each moving at a constant speed, that implies the objects are moving at the same constant speed. While students have thought about speed before in their own lives, this task is meant to help them tie that prior experience to the formal language of ratios. The numbers are chosen so that each pair-wise comparison has either the time, the distance, or the speed the same. 

The second set of questions, parallel to the first, intends to help students make the connection that equivalent ratios in the case of buying items is characterized by the same unit price. Again there is always one quantity the same for each comparison, and two quantities that are different.

Solution

A steady speed

1. The snail went 3 inches in 1 minute.
2. The caterpillar went 18 inches in 1 minute. 
3. The baby turtle went 3 inches in 1 minute. 
4. The snail and the caterpillar both moved for the same amount of time: two minutes. The caterpillar moved a greater distance in two minutes, so the caterpillar was moving faster than the snail. The caterpillar and the baby turtle moved the same distance: 36 inches. The baby turtle took more time than the caterpillar to move 36 inches, so the baby turtle was moving at a slower speed than the caterpillar. The baby turtle and the snail both moved 3 inches per minute, so the baby turtle and the snail were moving at the same speed. However they moved different distances and they took different amounts of time.

Buying in bulk

1. You can buy 6 kilograms of beans for \$1. 
2. You can buy 3 kilograms of rice for \$1. 
3. You can buy 3 kilograms of cornmeal for \$1.
4. For the beans and the rice, you are spending the same amount of money, \$5, but buying different amounts -- you get more kilograms of beans than rice for the same amount of money. The rice and the cornmeal both cost \$1 for 3 kilograms, but you are buying different amounts and paying a different amount of money. You are buying the same amount of cornmeal and beans, but paying a different amount of money -- cornmeal costs more than beans when buying the same amount of each.