## PLANTING FLOWERS [DAY 5]

### 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.

## Intentionality & Unit Overview

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Students will demonstrate the concepts and skills related to ratios and rates that they have developed through this contextual learning experience.

### Intentionalityâ€¦

The purpose of this assessment is for students to demonstrate their 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.
• A multiplicative comparison is a ratio with the same unit;
• The scale factor between a multiplicative comparison can be revealed through quotative division.

The independent assessment will allow students to demonstrate concepts and skills related to the following learning goals:

• I will compare and contrast composed units and multiplicative comparisons.
• I will communicate examples of a composed unit.
• I will communicate examples of a multiplicative comparison.
• I will use partitive division to reveal a rate.
• I will use quotative division to reveal a scale factor.

Students might write and/or model their answers independently. Students can also be interviewed by their teacher.

An alternative assessment opportunity should be offered if this tool will not successfully capture what the student knows, understands or can do.

## Math Talk

In todayâ€™s math talk, present the following ratios. For each ratio, ask students to share an example of a composed unit and a multiplicative comparison. Prompt students to add the appropriate units for each context.

For the composed unit examples, ask students to divide the quantities using partitive division to reveal a rate.

For the multiplicative comparison examples, ask students to divide the quantities using quotative division to reveal a scale factor.

Then, compare and contrast the two scenarios for each set of ratios.

75 : 15

9 : 6

An example of a context that might be considered for thinking of these ratios as a composed unit might be a gardener who has 75 seeds to plant and 15 pots. How many seeds per pot can the gardener plant?

An example of a context that might be considered for thinking of these ratios as a multiplicative comparison might be a gardener who has 75 seeds to plant and wants to plant 15 seeds per pot. How many pots will the gardener need? In other words, how many groups of 15 seeds are in 75 seeds?

This string of related problems should be led by the facilitator on a chalkboard/whiteboard/tablet to model student thinking â€œin the moment”, however you might consider to also leverage the visual math talk clips included below for each problem in the string.

## Visual Math Talk Prompt #1

In the first problem from the string, students are asked:

Create an example of a composed unit for the ratio 75:15.

Then, use a model to reveal a rate.

As always, value the voice of your students and do your best to model their thinking. While the context that students create for this composed unit ratio of 75:15, similar strategies may emerge such as using a fair sharing strategy to distribute 75 of a quantity into 15 groups (seeds to planter pots, goals to games played, etc.).

If students do not leverage any linear models, make an attempt to take their thinking and represent it using stacked bar models and/or a double number line then ask them if they can see their thinking in that model.

Helping students to see value in a linear model – like the double number line for example – can be helpful for them to build their multiplicative thinking skills.

Some students may quickly realize that there are 3 groups of 5 in 15, and therefore we could partition the number line into thirds to help reveal how many seeds 5 planter pots would receive.

By scaling the composed unit in tandem like we see above, students are now working with a more manageable equivalent ratio of 25:5. Partitioning that portion of the number line into fifths will allow us to reveal the ratio of number of seeds to 1 pot or a rate of 5 seeds per pot.

Alternatively, some students may jump straight to using partitive division on 75 seeds and 15 pots to reveal the rate of 5 seeds per pot (or 5 seeds/pot).

## Visual Math Talk Prompt #2

In the second problem from the string, students are asked to think of the ratio 75:15 as a multiplicative comparison:

Create an example of a composed unit for the ratio 75:15.

Then, use a model to reveal a scale factor.

Again, value the voice of your students and do your best to model their thinking. By this point in our problem based unit on ratios and rates, we hope that students will know and understand that the context they used for a composed unit in the previous problem will not suffice as a multiplicative comparison. When thinking of 75:15 as a multiplicative comparison, we are thinking about this ratio quotatively. In other words, we are wondering â€œhow many 15s are in 75″ or visa versa â€œwhat fraction of 75 is 15?”

One possible strategy and model might be to stack bars in order to visually compare how much larger (or smaller) one quantity is than the other.

Observe and listen for students thinking additively (i.e.: 75 seeds is 60 seeds greater than 15 seeds) vs. multiplicatively (i.e.: 75 seeds is 5 times greater than 15 seeds).

Note that while students will likely prefer to reveal a scale factor of 5 (i.e.: 75 seeds is 5 times larger than 15 seeds), be sure to highlight the reciprocal scale factor (i.e.: 15 seeds is 1 fifth times as great as 75 seeds).

It is worthwhile to leverage student thinking from stacking bars to connect to a double number line. If student thinking is shared in a vertical fashion, you might consider creating a vertical number line. The same is true if students are sharing thinking in a horizontal fashion.

By emerging a double number line, the idea of equivalent ratios can be quickly revealed. For example, from the double number line, we can see that 75:15 is equivalent to x:1.

Avoid rushing to an algorithm and rather promote reasoning for students to determine what the missing scale factor would be in this case. This means we are not setting up a proportion and solving using cross multiplication.

While it might seem easier to see that 75 seeds is 5 times larger than 15 seeds, it is also worth noting that if our unit is 75 seeds, that 15 seeds is 1 fifth of that amount.

Encourage students to explore the behaviours of the two types of ratios (composed units and multiplicative comparisons) on the double number line and you might be surprised what procedures they highlight.

## Visual Math Talk Prompt #3

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## Visual Math Talk Prompt #4

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## Assessment Considerations…

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## Assessment: Ratios & Rates

### Question 1:

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### Question 2:

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### Question 3:

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### Question 4:

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### Question 5:

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## Considerations to Promote Student Growth

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