Here we adopt a particular interpretation of division - quotition or measurement. For a division like 100÷20, we ask, How many times does 20 go into 100? We can also think of division in terms of partition, or sharing. This is probably the interpretation that children meet first, but, strangely perhaps, when we perform division, for example by using chunking or the long division algorithm, we tend to use quotition. Division by a rational number is also more easily interpretative as quotition than partition. Consider, for example, 10 ÷ ⅓. It seems more natural to think of this as 'How many ⅓s are there in 10?' than 'What is 1 share if ⅓ of a share is 10?' [Or, compare 'How many ⅓ pint bottles can I fill with 10 pints of milk?' with 'If 10 pints is ⅓ of my milk supply, how much milk do I have?' We might be able to solve the latter fairly easily, but it seems an odd way to view a situation.]
Task 10A: In these items we mostly divide by 20 and provide children with a list of multiples of 20 to help then visualise the situation. However, we also try to get children to think beyond what they see in front of them by asking about multiples not in the list, or about multiples of other numbers than 20. Each of the items involves a new cognitive leap, and this might be going much too fast for many children. Thus, here and on later tasks, you might want to dwell longer on a particular idea by devising variants that are similar to a previous item (and for some children, you might want to devise variants that are even more challenging!).
Item 3 can be solved using the fact that 280 is twice 140 and will thus contain twice as many multiples of 20. This might be quite an abstract idea for some children, who might decide to list all the multiples instead.
Item 4 can be solved using the idea that 280 will contain half as many multiples of 40 than of 20. Again, some children might solve the task in a more grounded way, by listing multiples of 40.
Item 5 is intended to bring out the idea that asking 'How many...' is a way of interpreting division. Do children see the connection, or, indeed, have they already done so? Some children might solve this or subsequent items using the idea of sharing. You might want to spend some time comparing the two interpretations of division. In the case of 100÷20, for example, one could compare 'How many 20cm lengths can be cut from a 100cm rod' and 'How long is each piece if a 100cm rod is cut into 20 equal pieces?'. In one case the answer is 5; in the other it is 5cm.
Task 10B: This is similar to the previous task but involves multiples of 15 which are probable slightly less familiar to children than multiples of 20. We also push things a bit further by involving a fractional divisor (7.5 in Item 6) and a fractional answer (1½ in Item 7).
Many of these items can be solved in several ways, which can provide plenty of scope for discussion. In Item 5, children might notice that the multiples of 75 (namely 75, 150, 225 and 300) lie on a slanting line in the table - why? Some might notice that Items 4 and 6 have the same answer, since the numbers in Item 6 are both one tenth of the corresponding numbers in Item 4.
Task 10C: Here we move into the realm of decimals and you might need to decide how far to delve into the more challenging ideas and how much to linger on simpler ones by devising straight-forward variants of the earlier items.
Task 10E: This task comes from the CSMS Fractions (3 and 4) Test. It was given to a representative sample of over 200 Year 9 students in 1976/7 and to a similar sample in 2008/9. It was answered correctly by 57% of the sample in 1976/7, but by only 29% in 2008/9.
