Week 09: Transforming expressions

Here we take a gentle stroll through the art of transforming expressions into equivalent expressions, including the method of 'constant difference' used in the previous week's tasks.

Task 09A: Here we have the classic situation, which might be quite familiar to some children, where an addend or subtrahend is close to a 10s, 100s or 1000s boundary, etc. In the case of a simple addition, we use the 'compensation' principle: if we increase one component we have to decrease the other by the same amount to keep the expression's value unchanged. In the case of a simple subtraction, we apply the same change to both components.

There is plenty of scope here for devising similar tasks if you feel children would benefit from further exploration or practice.

Task 09B: Here we look more closely at the 'constant difference' principle. Changing the component 27 to 30 or to 20 is equally beneficial in that they result in similar, more simple subtraction expressions. However, it could be argued that the change made to the other component, 642, is (slightly?) more demanding in the latter case since it involves
• a larger number (7 rather than 3),
• subtraction rather than addition and
• bridging a tens boundary (at 40).

Task 09C: Here we are using 'compensation' to transform an addition expression and this time it can be argued that the transformation of the first expression is more demanding since it is here that the change made to the other component, 642,  involves
• a larger number (7 rather than 3),
• subtraction rather than addition and
• bridging a tens boundary (at 40).

Some children might feel that the difference in demand of the two methods is minimal. That is a perfectly valid stance to take! The aim of the task is not to suggest that the difference is so vital that children should learn to adopt one method and reject the other. The aim is more modest and diffuse, of helping children develop a greater sensitivity about the various elements that form part of a mathematical method.
Task 09D: Here the constant difference principle is enacted in stages.

When the given method is applied to 416 – 27, we get this:
416 – 27 = 419 – 30 = 489 – 100 = 389.
There are of course other methods for performing the calculation and you might want to ask children for some. One that works for these particular numbers is to 'decompose' 27 into 16+11, which gives this:
416 – 27 = 416 – 16 – 11 = 400 – 11.
And one way to continue would be this: 400 – 11 = 399 – 10 = 389.

Task 09E: Some children will attempt to solve this task by simply adding together all the terms of the expression. However, it is hoped that the length of the expression, and the clue about its approximate value, might prompt them to look for an alternative approach.

We are told the value of the expression is close to 300, which might encourage children to consider 50 + 50 + 50 + 50 + 50 + 50 as a way of checking whether this is true. How easily can they see that this is indeed equal to 300? By using 6×50 = 300, or perhaps by thinking of (50+50) + (50+50) + (50+50)? In turn, can they spot that the original expression will differ from 300 by this amount:
1 + 4 – 2 – 6 – 3 + 8 = 13 – 11 = 2.
This is greater than 0 and so the expression is greater than 300.

 

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