The following sequence of problems provides an opportunity for students to apply their ratio reasoning skills to scale in tandem given a proportional relationship involving a composed unit ratio between distance in kilometres and time in minutes.
This context was intentionally selected to promote the use of a linear model like a double bar or double number line model to emerge scaling in tandem. Since this context involves a composed unit ratio – a ratio that can be thought of as a single unit and lends itself to scaling up and down – scaling in tandem is a helpful tool to not only determine unknown quantities, but to also promote student flexibility and fluency with multiplicative thinking.
Throughout this series of visual number talk prompts, be sure to avoid rushing to setting up and solving a proportion as that will defeat the intent of this string of related problems.
If the terms ratio reasoning, composed unit ratios, scaling in tandem and multiplicative thinking are foreign to you, consider deepening your understanding of proportional reasoning by taking our course, The Concept Holding Your Students Back: Unpacking Key Understandings In Proportional Reasoning So You Can Reach Every Student.
Show students the following visual math talk prompt and be prepared to pause the video where indicated:
Then ask students:
A car travels 24 kilometres every 15 minutes.
How far would the car travel in 5 minutes?
Craft a convincing argument without the use of a calculator.
With this visual up for all to see, give students a reasonable amount of time to reflect on what this question is asking them and to leverage their understanding of magnitude of number to spatially approximate where 5 minutes and the corresponding number of kilometres travelled should be placed on the double number line.
The goal here is to give students a low floor opportunity to leverage their understanding of magnitude as well as to emerge the fractional thinking necessary to determine that 5 minutes is 1 third of 15 minutes.
Since we are working with a composed unit ratio, 24 kilometres to 15 minutes, there are an infinite number of equivalent ratios that exist in this proportional relationship. The ratio x to 5 minutes is only one of those equivalent ratios.
Without much prompting, students should come to the conclusion that since 5 minutes should be placed 1 third of the distance between 0 minutes and 15 minutes, then the number of kilometres travelled 1 third of the distance between 0 kilometres and 24 kilometres should be 8 kilometres.
Note that while some students may be using multiplicative thinking (i.e.: knowing 5 is 1 third of 15), others may be using more of a guess and check approach with additive thinking (i.e.: skip counting by 4s, then 5s to realize that adding 5 repeatedly will get me to 15). Be supportive of any and all strategies students bring to the table, but think of the purposeful questioning you might use to nudge each student forward towards more efficient or “clever” approaches without sounding negative or discouraging.
Something that the facilitator should be able to notice and name here is that the missing quantity that students have solved for here is the same quantity that they would determine when solving this proportion:
\(\frac{24 km}{15 min} = \frac{x km}{5 min}\)
There should be no rush at this point to show students that they have just solved a proportion of equivalent ratios unless students are arriving in your class having already been exposed and/or rushed to an algorithm like cross multiplying, “magic circle” or any other set of steps that they do not understand conceptually. While we will not shut students down from using a trick they have learned previously, we do want to encourage them to work towards crafting a convincing argument as to why their answer is correct. If they cannot convince someone of why their method works, then we can deem their approach a “trick” rather than a justifiable strategy.
Show students the following visual math talk prompt and be prepared to pause the video where indicated:
Then ask students:
A car travels 24 kilometres every 15 minutes.
How many minutes will it take the car to travel 32 kilometres?
Craft a convincing argument without the use of a calculator.
With this second prompt, you might even consider explicitly asking students to leverage the use of the double number line to support them in solving this problem. While we want to welcome all student generated approaches in our math classroom, we also want to push students outside of their comfort zone. Whether you have a student who is using mental math strategies in their head or a student who has confidence setting up and solving a proportion, we want to nudge these students to bring another mathematical model into their repertoire – not to restrict them from using their tried and true “go to” strategy, but to arm them with more tools to build increased flexibility and fluency.
This prompt is yet another situation where the facilitator should be able to notice and name some of the different proportions of equivalent ratios that students might be leveraging in their thinking.
For example, one such proportion is:
\(\frac{24 km}{15 min} = \frac{32 km}{x min}\)
However, some other students might recognize that this is also an equivalent proportion based on our first visual number talk prompt:
\(\frac{8 km}{5 min} = \frac{32 km}{x min}\)
As well as any other number of equivalent ratios can be used to solve this problem. This is an excellent opportunity to highlight to all students why the double number line is a helpful mathematical model to easily scale in tandem multiplicatively to reveal any number of equivalent ratios in the proportional relationship that might help us solve for the unknown quantity we are seeking to find.
Whether students skip count by 8 kilometres and 5 minutes (additive thinking) or extend their thinking from the first visual number talk prompt to double 8 kilometres and 5 minutes to 16 kilometres and 10 minutes before doubling again (multiplicative thinking), the double number line acts as both a tool for all learners to think and represent their thinking as a means to convince others.
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