Using Numbers to Highlight Connections

This year I had the pleasure of contributing a chapter to a book edited by Ed Southall and published by The Mathematical Association called ‘If I Could Tell You One Thing‘. The chapter discusses a collection of situations where the numbers that are used in a worked example or practice question can obscure the reasoning behind the calculations or even fuel a misconception. However, numbers don’t just cause problems; they can also create opportunities! This blog post discusses some of the positive power that number choices can have in examples and questions! In particular, how carefully chosen numbers can help draw attention towards some wonderful connections that can be found within mathematics.

As a mathematics teacher, I am fascinated by deep structures that underpin mathematical ideas and the connections between different ideas. When looking at topics through this lens, numbers sometimes seem irrelevant and calculations can just be a distraction from what is going on at a conceptual level. It is arguably much easier for teachers than students to see mathematics in this way because we have already experienced the whole mathematics curriculum as a student and then several times again as a teacher. However, I tend to find that when students are looking at mathematics a lot of their attention to is focused on the numbers and calculations. So how can we use numbers and calculations to steer their attentions towards these deeper structures?

I’ve previously written a blog post about situations where questions from different topics can be solved using the same method – occasions where it’s almost like we have the same question dressed up in different disguises. An example of this is with equivalent fractions, equivalent ratios and other problems with proportion. The way that the questions and methods are laid out below helps highlight the similarities between them (even if they are are not necessarily the methods we might normally use). But the fact that the numbers are all the same in each question makes the connections even more obvious to anyone who is just focusing on getting answers – not only are the answers all the same, so are the calculations!

There are also occasions in mathematics where we are not quite doing the same thing but still find ourselves using similar processes in different topics. For example, we can simplify a ratio by finding for the highest common factor and dividing each part by it. We also find ourselves doing this when factorising an algebraic expression into a single bracket. The highest common factor plays a slightly different role in these two problems, both conceptually and in the final solution, but the mental methods are certainly similar.

Students could be prompted to notice the connections between these two problem types by solving one of each consecutively or seeing an example of each side-by-side on the board. If this was done using the two examples above then students might spot that they are doing similar calculations with each example. However, it’s still fairly subtle. Both the questions and solutions appear quite different in appearance and the main similarities are in the mental calculations, which they can not necessarily see on the page. Also, if students are used to completing exercises with unrelated questions (such as in mixed topic starters) then they might not think to look out for connections between them. However, if the two questions use the same numbers then it could help shine a brighter spotlight over the connections.

For students who are focusing mostly on the numbers and calculations they’re using, they might now find themselves not doing similar calculations but doing the exact same calculations in each. This might be enough to spark intrigue about why they’re getting a feeling of déjà vu. But if not then it leaves a nice opportunity for a teacher address it explicitly by asking questions such as, “Look at these two questions that have the same numbers. Why do their solutions also have the same numbers? What did we do in each case?” This connection could be exaggerated even further by using two questions like the one below.

However, it seems important in these cases for a teacher to also draw attention to the differences between the two concepts and processes. In particular highlight the differences in the role that the HCF plays in each question, just in case students then start to make mistakes like the one below.

There are some situations in mathematics where different topics appear completely unrelated on the surface but have some hidden connections when we look under the bonnet. Take histograms and density, which I have previously written about here. The two questions below could not appear more different if they tried…

However, these questions do share something common: at some point we find ourselves dividing the amount of “stuff” there is by the amount of space available for it. On the left we do this when calculating density by dividing mass by volume; on the right we do then when calculating frequency density by dividing frequency by class width. This might not be obvious to students, even if it’s explicitly explained to them. However, by presenting two questions together that are in a similar format and use the same numbers (such as below), it might help similarities to stand out a little more clearly. It’s arguably a little contrived, but its purpose is to shine a bright spotlight on the connections between these two contrasting topics.

But even when we have questions, answers and calculations that all appear completely different, there are still cases where our methods are driven by similar principles. Take for example calculating with fractions and calculating with standard form.

These questions look quite different and we’re not really doing similar mental calculations either. However, by looking at each of calculations from the first set side by side there are some connections to be found. In both topics, multiplication seems to have the most straight forward process: in each case we multiply the parts that correspond with each other and adjust/simply our product at the end if we need to.

Addition is quite straight forward when the “things” that we have are the same (i.e. the powers of 10 or the denominators). In these situations, we can just add together the other “things” (i.e. the decimals or numerators) and adjust/simply our solution at the end if necessary. Students also see this when simplifying simple algebraic expressions.

However, for both standard form and fractions, addition becomes trickier when the critical parts are not the same. In both cases, these problems can be solved by finding an equivalent addition where the parts do match, and then adding as before.

So students can think about how even though the methods and calculations are different, the ideas behind those methods are the same. They might conclude that the reason why these two types of addition problems are similar is because “that’s just how addition works: you can add when the two numbers are amounts of the same thing” which would be a helpful takeaway. However, we can shine a brighter light on why these two problem types are so similar by exploring pairs of questions like the ones below. In this case it’s potentially clearer for students to see why the methods are linked, because they are seeing two different ways of writing the same addition.

For one final example, I’d like to look at the connection between multiplying together a pair of two digit numbers and expanding a pair of binomials. This too can be highlighted through a careful choice of matching numbers.

To be honest, I’m not a particular fan of using the grid method as a final method for solving either of the questions above but more so as a stepping-stone to help students understand the reasoning behind more efficient methods. In this case, solving these two questions side-by-side using the same method demonstrates that despite the questions appearing different, they are structurally similar. It can also lead to some other interesting questions to reason with, such as “What value of x would make these two questions exactly the same?”

Also, it may also encourage creative thinking about other potential methods we could use (even if they are not the best). For example, if we can answer both of the questions above using the grid method, then what would it look like if we used the FOIL method that we normally use for expanding brackets to perform the calculation?

And what would it look like if we used the column method that we normally use to calculation long multiplication to expand the brackets?

I haven’t particularly warmed to the idea of using FOIL to calculate multiplications or column method to expand brackets (although I’d be interested if anyone does). But exploring methods that are normally associated with one problem-type to another can sometimes draw attention to aspects of each method that may have become automated or unnoticed. For example, expanding brackets using the column method emphasises the role that the ‘place holder zero’ plays in lining digits up correctly in long multiplication.

When I was an early career teacher, I used to regard the numbers I chose for worked examples and practice questions as fairly unimportant (so long as they didn’t make the calculations distractingly complicated or give decimals at unwanted times). However, as discussed in Chapter 18 of ‘If I Could Tell You One Thing’ and this post, careless number choices can lead to problems and careful number choices can lead to opportunities!

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Using Numbers to Highlight Connections

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