Week 16: Here we revisit the idea that we met in Week 8, of determining what a minor change to the numbers in an expression does to its value, without simply calculating the modified expression.
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Monday: This might look daunting at first, but after a brief pause for thought it should become apparent that the change made to the first expression is fairly straightforward. What does it do to the expression's value?
We have increased the value of the expression by 100. Its new value is 7336.
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Tuesday: This is a lot more demanding. A first glance might suggest the expression has just been increased by 1.
It can help if pupils try to give some kind of meaning to 4×38 (and similarly to 4×39), for example by applying a context such as '4 boxes of 38 pencils' or '38 bunches of 4 flowers'. Or, more abstractly, reading the expression as '4 lots of 36' or '4, 38 times'. Pupils are then likely to see that changing 38 to 39 increases the expression by 4. Of course, some pupils will simply calculate 4×38 and 4×39, and thus determine that the value has increaed by 4. This is perfectly valid, but encourage these pupils to try to make sense of our neater, 'incremental' approach.
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Wednesday: It is likely that many pupils will find this more challenging than Tuesday's task.
Formally, this can be said to rest on the distributive property of division. We can think of 2421
45÷5 as 2421
35÷5 +
10÷5. So we have increased the value of the expression by 2.
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Thursday: We are not asking for the actual value of the second expression, only whether it is smaller or larger than the first. Nonetheless, this is challenging!
Here again, it can help pupils to make sense of the expressions if they give them some kind of meaning. For example, we can imagine sharing £46,385 between 3 people and an amount that is £1 more between 4 people. Or we can ask, how many packs of 3 apples can we get from 46385 apples, and what would happen if each pack were to contain an extra apple and we had 1 more apple altogether.
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Friday: The simplest way to solve this task is to 'add a naught' to some of the numbers. But which ones?
There is quite a lot going on here! Multiplication is distributive over addition, so both '26×435' and '18' have to be multiplied by 10. However, multiplication is
not distributive over multiplication, so when we multiply '26×435' by 10 we don't multiply both numbers in the expression by 10!
Here are two ways of making the whole expression '10 times as large':
260×435 + 180 and 26×4350 + 180.
It is likely that some pupils will 'add a naught' to just one of the three given numbers, while others might do so to all of the numbers.
