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PrimaryMaths

Teach Your Primary Pupils Mathematical Equivalences With ‘True or False’ Mini Lessons

Need to improve your children᾿s reasoning skills? Then introduce some cognitive conflict with these 10-minute routines, says Mike Askew…

Mike Askew
by Mike Askew
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PrimaryMaths

Is this mathematical equivalence true or false?

64 ÷ 14 = 32 ÷ 28

What about this one?

64 ÷ 14 = 32 ÷ 7

Many upper Key Stage 2 children, and, in my experience, quite a lot of adults, will say that these are both true. A moment’s reflection may reveal this cannot be the case: 64 divided by 14 cannot be equal to both 32 divided by 28 and 32 divided by 7, as these clearly have different answers. So which is true and which is false? Putting each calculation into a simple context helps sort out what is going on. Imagine, say, putting apples into bags. So 64 ÷ 14 is the mathematical model for ‘I have 64 apples that I want to put into bags of 14’ and similarly, 32 ÷ 28 is a model for ‘I have 32 apples that I want to put into bags of 28’.

It should be clear that the second situation is going to require far fewer bags than the first. On the other hand, putting 32 apples into bags of 7 has both halved the number of apples, and halved the number of apples going into each bag. Everything is kept in proportion.

Reasoning revisited

I recently wrote about ‘reasoning mini-lessons᾿ – short, 10-minute routines that focus on answers in terms of reasoning about what is going on, rather than only calculating.

I looked at ‘reasoning chains’, a type of mini-lesson where a series of calculations are constructed to be linked and that build towards a big idea – like the distributive or associative rules. Here, I want to look at another type of reasoning mini-lesson – true or false? These mini-lessons are based around a small number of statements of equivalence, each of which is either true or false. The routine for working on these is, as with reasoning chains, putting up a statement, giving children a few moments to think (on their own) about their answer and then ‘turn-and-talk’, where they share their thoughts with the person next to them.

Following this, the whole class indicates whether they think the statement is true or false – a thumb up for true, down for false. At this point, three things can happen – as shown in this scenario, where pupils are deciding about 64 ÷ 14 = 32 ÷ 7.

• The class is generally in agreement about whether the statement is true or false, and are correct in that agreement. For example, all agreeing that 64 ÷ 14 = 32 ÷ 7. In that case, I’d invite one or two children to share why they have arrived at that conclusion, and keep the dialogue fairly short.

• The class is split over whether they think the statement is true or false. When this happens, I recommend throwing it back to the learners – ‘OK, it cannot be both true and false. Turn and talk to your neighbour and agree on which you think it is, and whether you can convince everyone else.’

While those private conversations are going on, listen in and choose the two or three that you think will provoke a good whole-class conversation.

• The class is generally in agreement about whether the statement is true or false, and are incorrect in that agreement. For example, all agreeing that the first statement, 64 ÷ 14 = 32 ÷ 7, is false. If this is the case, I don’t recommend trying to ‘fix’ the learners’ reasoning at this juncture.

Why? Because if the variation between the examples has been carefully thought through (as in the pair here) then, when the second problem goes up, there is a moment of ‘cognitive conflict’ that confronts the learners with having to think about what is going on.

(Even without a linked statement, if you have near universal misunderstanding then you are unlikely to change everyone’s thinking in a few minutes. Better to plan to come back and address this at length later.)

To help put this in context, here are some more examples of the sorts of statements that provoke productive reasoning.

Key Stage 1 – Working with images

Young learners can be introduced to the idea through deciding whether or not an image accurately represents a situation. Since the metaphor of a balance is at the heart of equivalence, this gives us a suitable visual to discuss.

Additive equivalences 20 + 8 = 10 + 18 20 + 8 = 19 + 9 80 + 8 = 60 + 18 8 + 90 = 40 + 50 + 8

Again, in discussing these, the image of a balance helps. For example, 20 + 8 = 19 + 9 could be set up with two boxes on one side of a balance – a box of 20 and a box of 8, as shown below.

If, on the other side of the balance, there is a box of 19 and a box containing an unknown quantity, then what has to happen to the boxes of 20 and 8 in order to keep everything in balance? In the case of a false statement, such as 80 + 8 = 60 + 18, a valuable follow-up question is ‘What could we change to make the statement true?’ A number of possibilities emerge, such as changing the 80 to 70, or the 60 to 70, or the + 8 to – 2, and so forth.

Multiplicative equivalences 10 + 4 = 4 x 10 4 x 10 = 4 + 4 + 4 + 4 10 + 10 + 10 + 10 = 4 x 10

Regular readers will know I’m a fan of the array when it comes to showing what is going on in multiplicative situations. For example, 4 x 10 is a four by ten array, while 4 + 4 + 4 + 4 can be represented as a four by four array. If either of these sets of examples look familiar, it᾿s because I created them by adapting questions taken from the reasoning paper of the 2016 KS1 national mathematics test!

Lower key stage 2 – reasoning, not calculating

Additive equivalences 37 + 56 = 56 + 37 37 + 56 = 38 + 59 37 + 56 = 36 + 57

Exactly as in KS1, using images of a balance helps children, literally and figuratively, see what is going on.

Multiplicative equivalences 3 x 5 = 5 x 3 3 x 5 = 3 x 4 + 5 3 x 5 = 3 x 4 + 3

Again, drawing arrays helps make clear the reasoning. For example, comparing a 3 by 5 array with a 3 by 4 array and then adding 5 to the latter shows that these two cannot be equal.

26 x 40 = 40 x 26 26 x 40 = 26 x 39 + 40 26 x 40 = 26 x 39 + 26

At first glance, this sequence of examples may appear much harder than the ones above – but in fact, the underlying structure is exactly the same. All I have done is switch 3 for 26 and 5 for 40. But while the first equations can be checked by arithmetic, increasing the size of the numbers encourages a move to reasoning. And this pair of examples illustrates an important point about mathematical reasoning – it is not dependent on the ability to calculate. A child who may not yet know how to calculate the answer to 26 x 39 may, nevertheless, be able to reason why 26 x 39 + 26 must be equal to 26 x 40 – provided, of course, that she has done lots of work on arrays to help her think about what these calculations might represent.

Upper Key Stage 2 – drawing conclusions

Additive equivalences 56 – 38 = 56 – 37 – 1 56 – 38 = 56 – 39 + 1 56 – 38 = 56 – 36 – n

The final statement here introduces a new idea: if … then. If n is equal to 2, then it is true.

Multiplicative equivalences The example below provides another tweaking of the structure, by giving learners a statement they are told is true, and then asking them to figure out whether other conclusions can be drawn from this. As with the earlier examples, the following are based on a question from one of the 2016 reasoning questions from the national tests.

TRUE: 5,542 ÷ 17 = 326 True or false? 326 ÷ 17 = 5,542 5,542 ÷ 326 = 17 17 x 326 = 5,542 326 x 16 = 5,542 – 16 18 x 326 = 5,542 + 326

Variety or variation?

Behind the design of all these examples is the idea of providing learners with experience of variation, as opposed to variety. Although they sound similar, variety and variation are very different. Variety is what most textbook pages, or worksheets, provide. For example, a page of calculations providing practice in, say, multiplication will be written in such a way that, although the numbers might get larger or more cumbersome in the later examples in essence, the examples could be shuffled into a different order without a great deal of damage to the experience of working through them.

In working through a variety of problems, it is most likely that the learner, having completed the first question, will go onto question two without much further thought about question one, and so on. Variation is different. It comes from a theory of learning that has, at its core, the view that we learn as much from the differences between things as we do from the similarities. Variation means that the order of the examples, and the changes from one example to the next, has been very carefully considered.

That is done so that the learner’s attention is not simply on each individual example, but drawn to the connections, similarities and differences between them – which brings me back to the opening pair of statements in this article. Looking at each, on its own, would be okay, but the experience of thinking about what᾿s going on across the two examples is far more powerful.

Mike Askew is adjunct professor of education at Monash University, Melbourne and a freelance primary maths consultant; for more information, visit www.mikeaskew.net or follow @mikeaskew26

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