Here we look at the mediant, the fraction formed when one adds the numerators and the denominators of two fractions. The mediant always lies between the fractions, though not necessarily exactly in the middle.
The tasks help address the common misconception that the sum of two fractions can be found by adding their numerators and their denominators - a classic case of 'natural number bias'. The result cannot be the sum, as it is smaller than one of the (positive) fractions being added. We also address another aspect of natural number bias, based on the fact that one can't fit a natural number between two consecutive natural numbers. For any two fractions, one can always fit a fraction between them.
Task 11A: A response like Cath's in this task is very common. However, if pupils are given time to reflect, it is not too difficult to see that 2/5 is smaller than the larger of the two fractions (1/2 = 2/4 > 2/5). And it is larger than 1/3 (1/3 = 2/6 < 2/5). This suggests Dean and/or Erin is right. In fact, only Dean is right. The mean of 1/3 and 1/2 is 5/12 which is not equivalent to 2/5.
Finding the mean of 1/3 and 1/2 can be challenging for pupils. It can be achieved in different ways, for example by adding the fractions and halving the result; or by finding the difference between the fractions, halving this and adding it to the smaller fraction! Some pupils might claim that 1/2½ is the mean. This might seem plausible, especially as it is equivalent to 2/5!
Task 11B: Here we reinforce the ideas visited in Task 11A, starting with a slightly more 'generic' pair of fractions (albeit ones that are relatively easy to compare). The number line is a salient way of showing that the mediant does indeed lie between the two initial fractions, but that it is not exactly in the middle. It turns out that the mediant is closer to 7/8 than 1/4 - do pupils have any sense of why this might be?
Task 11C: Here we revisit the pair of fractions, 1/3 and 1/2, used in Task 11A. We see what happens to the mediant when we replace a fraction by one that is equivalent.
The task provides a nice opportunity for pupils to further develop their 'fraction sense': it turns out that the mediant depends not just on the 'values' of the two fractions, but on the numbers used to represent them - the larger a particular number is, the greater its 'pull'.
Cath's, Dean's and Erin's statements are all true.
The mediant is (slightly) closer to 1/3 than 1/2 (because 3>2).
Dean's mediant, 5/14 is even closer to 1/3 (1/3 = 5/15).
Erin's mediant is the same as the mean because the fractions have the same denominator - how might one prove this?
Pupils might find it instructive also to compare the fractions by converting them to decimals, using a calculator or spreadsheet - although this quite challenging as many of the decimals recur.
Task 11D: This task tackles the important (revolutionary?!) idea that we can continually find fractions between fractions. It can be solved using mediants, but there are also other ways. For example, some pupils might express the fractions using the common denominator 63 (giving 35/63 and 45/63). So if a pupil chooses, say, 40/63 as their first fraction, they could continue with 39/63, 38/63, 37/63 and 36/63 - at which point they would have to change tack slightly, for example by writing the fractions as 70/126 and 72/126.
It is also quite fun to solve the task using mediants:
the mediant of 5/9 and 5/7 is 5/8 (when simplified);
the mediant of 5/9 and 5/8 is 10/17;
the mediant of 5/9 (or 10/18) and 10/17 is 20/35;
the mediant of 5/9 (or 20/36) and 20/35 is 40/71,
and so on.
Can pupils spot the pattern in the resulting fractions?
These fractions are of course getting closer and closer to 5/9. One way to see this is to turn the fractions into decimals: 5/9 is 0.5 recurring; here are the fractions 5/8, 10/17, 20/35 and 40/71 (to 5 dp):
| 0.62500 |
| 0.58824 |
| 0.57143 |
| 0.56338 |
Task 11E: This is similar to the previous task, but this time the denominators are the same. If pupils meet the task in isolation, it is quite likely that some will say that there is no fraction between 4/9 and 5/9. Others might suggest 4½/9, which can be written as 9/18 or 1/2.
As with the previous task, one way to find lots of 'inbetweeners' is to change the given fractions to equivalent fractions with a large denominator, for example 40/90 and 50/90.
One could also use mediants again:
the mediant of 4/9 and 5/9 is 9/18;
the mediant of 4/9 (or 8/18) and 9/18 is 17/36;
the mediant of 4/9 (or 16/36) and 17/36 is 33/72;
the mediant of 4/9 (or 32/72) and 33/72 is 65/144,
and so on.
