In today’s visual math talk, we will be scaling a composed unit ratio in tandem through ratio reasoning. This strategy is useful when trying to compare the price of goods.
Through the context of the price and 12-egg cartons, students will be encouraged to leverage the ratio table to scale this composed unit ratio to reveal other values. Throughout the process, encourage students to share and model their mental math strategies and do your best to model them in the ratio table.
Show students the following visual math talk prompt and be prepared to pause the video where indicated:
Then ask students:
If the cost for 2 cartons of eggs is $7.20, how much would it be for 4 cartons of eggs?
The first prompt in this string of related visual number talk prompts involves a pretty low floor giving students two quantities that can easily be doubled by decomposing $7.20 into $7 + $0.20.
2($7 + $0.20)
= $14 + $0.40
= $14.40
Next, the visual number talk prompt video asks students to determine the cost for 1, 6, 10 and 12 egg cartons. The goal here is to promote students leveraging their understanding of scaling up and down the ratio table to reveal additional equivalent composed unit ratios.
While some may choose to add an additional $7,20 to $14.40 to arrive at $21.60 for 6 egg cartons, others may recognize that they can triple the cost of 2 egg cartons since 6 egg cartons is triple that of 2 egg cartons.
Some students may add $14.40 to $21.60 to reveal the cost for 10 egg cartons since those are the costs of 4 egg cartons and 6 egg cartons (additive thinking).
Others may multiply the cost of 2 egg cartons by 5 since 5 groups of 2 egg cartons is equivalent to 10 egg cartons.
Others still may determine the cost of a single egg carton by halving $7.20 and then scale this by a factor of 10.
While we will celebrate all approaches used by students via additive and multiplicative thinking, we do want to explicitly highlight the use of scaling in tandem as this is a multiplicative strategy that is very helpful when working with composed unit ratios.
It is worth noting here that while we can see the cost of a single carton of eggs, the rate (or “unit rate” as some call it) is not explicitly shown here.
Only through partitive division of any of these equivalent ratios (i.e.: 3.60:1, 7.20:2, 14.40:4, etc.) can we reveal the rate of $3.60 per carton of eggs.
Interested in learning more about ratios, rates and proportional relationships? Dig into our course, The Concept Holding Your Students Back.
Show students the following visual math talk prompt and be prepared to pause the video where indicated:
Then ask students:
If the cost for 3 boxes of cereal is $9.60, how much would it be for 6 boxes of cereal?
Again, a very low entry point here promoting students to use the multiplicative thinking strategy of doubling, but still offering access to those students who choose to leverage additive thinking by adding the cost of 3 boxes of cereal.
We then present students with the challenge of determining the cost of various numbers of boxes assuming a constant rate of cost per box.
The values selected here are intentional to promote tripling, thirding and other scale factors as students attempt revealing the prices for various quantities of boxes.
The values selected here are intentional to promote tripling, thirding and other multiplicative strategies.
While different students may use different scale factors to determine various prices in the ratio table, one realization they may make is that determining the price for 1 box of cereal can be helpful to easily determine other scale factors in the table.
While not likely, the following screenshot is shared to intentionally let educators in on a behaviour of scaling that they may not have noticed in the past.
Since 12 boxes is 4 thirds as large as 9 boxes, we can also multiply the cost of 9 boxes by 4 thirds.
While this isn’t extremely helpful for mental math, thinking of 4 thirds as first taking 1 third of 9 to get 3 boxes for $9.60 and then quadrupling will provide an opportunity to arrive at 12 boxes for $38.40.
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