Week 12: Hidden fractions

This is essentially about fractions, though we don't mention them by name! We use the number line to focus on the measurement and ratio aspect of fraction rather than part of a whole.

Task 12A: This gives us an opportunity to see what strategies children use to estimate parts of a line. We use a line with a large number of divisions to discourage children from simply counting (though they can do so to check their estimates) and to convey the idea that we don't expect the estimates to be spot on. We have chosen numbers where the use of benchmarks can be helpful. For example, the 18 mark will be just under halfway along the line segment, 12 will be just over a quarter (or under one third), 25 is over one half but less than three quarters, say. Some children might spot that 25 is 5/8 of the way along and, similarly, that 30 is three quarters along the 0 to 40 segment.

Task 12B: We don't require anything precise here, beyond the fact that the 13 mark is more than halfway along its line segment (actually quite close to 2/3) and that 23 is slightly less than halfway along its segment - so 13 is nearer the right-hand end of its segment. Item 2 attaches a story to the same numbers - do children appreciate that it makes sense to use ratios to compare the two types of bean rather than the differences in the number of germinating and non-germinating beans? In other words, do they appreciate that it is more appropriate to use multiplicative rather than additive reasoning?

Some children might express the relations using fractions or, more likely perhaps, decimals or percentages. If it seems appropriate, you might want to discuss the idea that comparing 13-out-of-20 and 23-out-of-50 is essentially about division: 13÷20 is greater than 23÷50.

Task 12C: Again, we don't expect a very precise answer here (the red arrow is actually pointing at 35.2), but do children at least appreciate that the red arrow is pointing more than half way along the line segment?

Some children might argue that since 11 is 1 more than half of 20, the red arrow will be pointing at 1 more than half of 64 (that is, 33). This is quite a sophisticated answer, though it involves additive rather than multiplicative reasoning. The gap between the red arrow and the halfway mark, 32, will actually be just over 3 times 1 (since 64 is just over 3 times 20).

Task 12D: The 3 mark is further along the 0 to 19 line segment than along the 0 to 20 line segment: 3 out of 19 is greater than 3 out of 20. However, some children will think the opposite, especially if they are focussed on 'parts of a whole': 3 out of 19 involves fewer parts so will be smaller.

 

Task 12E: The red arrow is pointing at the 15 mark. How close do children get to this? Do they at least chose a number below 20 (half of 40)?

Some children will recognise that we are in effect dealing with the equivalent fractions 3/8 and 15/40. And/or they might see that numbers on the bottom line are 5 times the corresponding numbers on the top line. Some children might think of splitting the top line into 8 equal parts, perhaps by halving, halving and halving again. Doing the same on the bottom line reveals the numbers 20, 10 and then 15.