Week 08: Subtracting on the number line

In this week's task we use the number line to model (and perform) subtraction, A – B. (We restrict ourselves to natural numbers, with A greater than B, and focus on the specific subtraction 44 – 28). 

We use the number line representation to model subtraction in two ways:
1. We find the difference between two positions (A and B, or 44 and 28) on the number line, in other words the displacement that takes us from one position (B, or 28) to the other (A, or 44); we can think of this as 'adding on':
B + ? = A;
2. We start at one position (A, or 44) on the number line and see where we end up after a displacement (B, or 28).

Further, we see what happens when we use the 'constant difference' principle, by, in particular, changing 44 – 28 into 46 – 30.

[Note: an interesting tension can arise when one uses a model based on a representation like the number line. Such a model can help one think more clearly about the mathematical elements being modelled; however, until one has become familiar with the model, its detailed features can get in the way of such thinking. This might well happen with some children using this week's tasks, in which case it might be best to concentrate for a while on just one of the tasks, for example 08A or 08C, and apply the particular model to a range of subtraction expressions.]

Task 08A: Here we are thinking of the subtraction 44 – 28 as finding a difference, which we model as a displacement that takes us from the location 28 on the number line to the location 44. 

Andy has found the displacement that takes us from location 28 to 44 as three skips, of 2 units, then 10, then 4. Children might think of it in other equally valid ways, for example as a skip of 2 and then 14, or of 10 and then 6, or even as a single skip of 16.

Whatever particular set of skips one might use to get from 28 to 44, it is worth emphasising that they involve number sense. If we had been given a fully marked number line as in the diagram below, we could simply have counted. To underline the point, you might want to show children this diagram.

When it comes to the last part of the task, children should be encouraged to use a semi or fully 'empty' number line as in the diagrams below.

 

Task 08B: Here we are again thinking of the subtraction 44 – 28 as finding a difference between the locations 28 and 44 on the number line. However, Bella uses the 'constant difference' principle to change 44 – 28 into an equivalent expression 46 – 30, where the difference is easier to see.

Bella has found the displacement that takes us from location 30 to 46 as two skips, of 10 units, then 6, in contrast to Andy's 3 skips in the previous task. Children might think of it in other equally valid ways, and more might see it as a single skip of 16, given the nicer numbers.

Task 08C: In this task, 44 is interpreted as a location on the number line, and 28 as a displacement that takes us to a new location.

Carl interprets the displacement of 28 units as 3 skips, of 20 units, then 4, then another 4, which lands on the location 16. Children might come up with other, equally valid skips that land on 16, for example 4, then 20, then 4.

Task 08D: Here we again interpret 44 as a location on the number line, and 28 as a displacement that takes us to a new location. However, Dee moves shifts the location 44 two units to the right and increases the displacement by the same amount. This will still take her to the desired location but using a displacement that it is easier to apply. As with Task 08B, this makes use of the 'constant difference' principle, though it might not be as easy to construe!

Dee interprets the displacement of 30 units as a single skip that lands directly on 16. Children might come up with other, equally valid skips that land on 16, for example 6, then 20, then 4.

Task 08E: Here we use some closely related stories to represent the subtractions 44 – 28 and 44 – 16. Do some of the stories seem easier to solve than others?

If we use the number line to represent the stories, then Story W fits the model in Task 08A and Story X fits the model in Task 08B. The resulting displacements are 16 and 28 units respectively, so Story W has the smaller displacement, and I think it is for this reason that Story W seems easier to me.

Similarly, Stories Y and Z fit the models in Tasks 08A and 08B respectively, with resulting displacements of 28 of 16 units, which seems to make Z easier (for me!).

[If this doesn't seem very convincing, keep the number 44 in each story, but change 28 and 16 to, say, 38 and 6.]