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Math Talk

Present the following composed unit ratios one at a time. Create a context to define the units for each ratio. Focus on ratios with two distinct units at this time. Ask students to determine and justify one or more equivalent ratios for each composed unit. Equivalent ratios can be revealed by scaling the composed unit up or down. For each ratio, reveal the rate by leveraging partitive division.
8 : 4

14 : 2

27 : 3

32 : 8 

6 : 4

As you’ll notice with this set of animated Math Visual Prompts, we begin by introducing a composed unit ratio of 8 apples to 4 baskets which provides an extremely low floor for students to access and enter the problem.

It will almost seem obvious to many students that the 8:4 ratio they are given of apples to baskets is equivalent to 2:1.

What students (and many educators) may not realize, is that while we have scaled this composed unit ratio from 8:4 down to 2:1 by “fourthing” the ratio, we can also quickly reveal a rate of 2 apples per basket through partitive division.

The goal here is to give students an opportunity to work with ratios using their intuition. Also we aim to help them realize that while we can scale any ratio up or down to find an infinite number of equivalent ratios, scaling to a unit ratio provides access to reveal a rate through partitive division (without really having to divide at all).

We also emerge the double number line as a tool for thinking as well as a tool to represent thinking.

Although there are five (5) ratios to work through in this animated Math Visual Prompt sets where we use partitive division via fair sharing (distributing apples equally to each basket) as well as through scaling in tandem on the double number line, be sure to dive into the final ratio where we reveal a fractional rate of apples per basket.

Through partitive division and the fair sharing strategy, you can quickly see that each basket will receive 1 and 1 half apples, revealing a rate of 1 ½ apples per basket:

Scaling in tandem by halving twice (or “fourthing”) reveals an equivalent ratio of 1.5:1 apples to baskets:

Consider watching this silent solution animation to help you prepare to facilitate this math talk utilizing a double bar model.

Facilitator Note: From each of these ratios, two rates can be derived. For example 8 apples : 4 baskets, the two rates can be revealed through partitive division:
8 apples ÷ 4 baskets = 2 apples/basket 

and

4 baskets ÷ 8 apples = ½ basket/apple. 

You can learn more about rates, ratios and the roadmap to proportional relationships through the course: The Concept Holding Your Students Back.

Purposeful Practice

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