Planting Flowers [Day 2]

RATIOS AND RATES

Introduction to ratios and rates. Students will explore both types of ratios; composed units and multiplicative comparison. They will use partitive division to reveal a rate and quotative division to determine a scale factor.

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PRESENTATION MODE

Intentionality & Unit Overview

Length of Unit: 5 Days

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Students will explore equivalent ratios and reveal rates through partitive division.

Intentionality…

The purpose of the Day 2 activities is to reinforce key concepts from Day 1. Students will engage in a string of related problems through a math talk and will have an opportunity to complete independent purposeful practice. The math talk and purposeful practice serve to develop a deeper understanding of the following big ideas.

There are two types of ratios; composed unit and multiplicative comparison;
A composed unit is often (not always) a ratio with two distinct units;
A composed unit can be scaled in tandem;
Equivalent ratios are derived from the same rate;
When you divide a composed unit through partitive division, you reveal a rate.

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.

$Planting Flowers [Day 2] - Purposeful Practice - 01 - MATH TALK Ratio 8 to 4 image$

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.

$Planting Flowers [Day 2] - Purposeful Practice - 02 - MATH TALK Ratio MMM Ration 8 by 4 image2$

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.

$Planting Flowers [Day 2] - Purposeful Practice - 03 - MATH TALK Ratio 8 by 4 image003$

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.

$Planting Flowers [Day 2] - Purposeful Practice - 04 - MATH TALK Ratio 6 to 4 image001$

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:

$Planting Flowers [Day 2] - Purposeful Practice - 05 - MATH TALK Ratio 6 to 4 image002$

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

$Planting Flowers [Day 2] - Purposeful Practice - 06 - MATH TALK Ratio 6 to 4 image003$

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.