D22: Fraction of a fraction - switching numerators

This theme was inspired by the photo below, which appears on page 108 of an excellent book by Catherine Fosnot and Maarten Dolk (2002): Young Mathematicians at Work: Constructing Fractions, Decimals and Percents. 

[My copy is an ex-library book from the University of Winchester. The book is in excellent condition, so did no one ever read it? Perhaps they replaced it with something on Mastery....]

TASK 22A: This is based on the fractions shown in the photo. The book contains several, similar tasks.

A basic way of finding the desired fraction is to count squares: the yellow rectangle covers 12 grid squares, while the blue rectangle covers 72 grid squares. So the fraction is 12/72 = ⅙. 

Or: each grid square represents ⅟₇₂ (Why?). So the yellow rectangle represents 12 × ⅟₇₂ = ¹²⁄₇₂ = ⅙.

In the Fosnot and Dolk book, the focus is on multiplying fractions. We can think of the yellow region as ⅜ of ⁴⁄₉ of the blue square (or ⁴⁄₉ of ⅜), so the fraction is ⅜ × ⁴⁄₉ = ¹²⁄₇₂ or ⅙. Fosnot and Dolk make the point that we can also find the fraction by rotating the yellow rectangle (shown here by the pink rectangle), which gives us a new pair of fractions and a new product ⁴⁄₈ × ³⁄₉. Notice that the numerators and denominators are the same as before, but we happen to get 'nicer' fractions leading to this simpler product: ½ × ⅓ = ⅙. Notice too, that the existence of these simpler fractions means it is also easier to see that the pink rectangle covers one 6th of the blue rectangle, as in the diagram below.

Rotating a rectangle as we have done here won't always lead to fractions that can be simplified. However, where it does, as in the current task, it helps to explain how we could get directly from our first product, ⅜ × ⁴⁄₉, to the simpler ½ × ⅓. In effect, this shows that we can extend our rule for simplifying a single fraction (divide the numerator and denominator by a common factor) to simplifying a product of fractions (divide a numerator and the denominator of the same or another fraction by a common factor). 

Of course, there are other ways of deriving this rule, for example by using the associative and commutative laws, as here:
⅜ × ⁴⁄₉ = 3×⅛×⅓×⁴⁄₃ = 3×⅓×⅛×⁴⁄₃ = ⅛×⁴⁄₃, and so on.

TASK 22B: Here is a similar task (though not directly from the book).

Again, we can count squares, or multiply the given fractions. Or we can rotate the yellow square, as below, where we can fairly easily see that it covers ⅓ of the blue rectangle, and where we get the very simple product 1×⁴⁄₁₂ = 1×⅓ = ⅓.

TASK 22C: Another variant.

This time the fraction is one quarter:
15/60 unit squares, or  ³⁄₁₀ × ⅚  or  ⁵⁄₁₀ × ³⁄₆.
By rotating the rectangle we can see the quarter easily.

TASK 22D: A chance to meet an improper fraction....

We get one quarter again, as can be seen below.

TASK 22E: From here on, we go on a bit of a tangent - there are no fractions as such but the maths (which one could think of as involving proportion and scaling) is intriguing. 

The previous tasks could be said to use different horizontal and vertical scales - the length of a horizontal unit is different from the length of a vertical unit. The same thing is happening here: while the grid appears to be composed of small squares, they are actually 5 units wide and 7 units high.

The tinted rectangles both cover 10 grid 'squares' so their areas are clearly the same. Each grid square has an area of 5×7 unit squares, so each rectangle has an area of 10×5×7 = 350 unit squares.

Because the axes have different scales, the rectangles are not congruent despite looking as though they are. Their dimensions are 10 by 35 units (yellow) and 25 by 14 units (pink). Of course, 10×35 does not just happen to be the same as 25×14. They are equal because 10×35 = 2×5 × 5×7 has the same factors as 25×14 = 5×5 × 2×7. This is similar to the property of the earlier tasks, where the different pairs of fractions had the same two numerators and the same two denominators.

TASK 22F: These are the same two rectangles as in the previous task, but this time the axes have the same scale. We can see that the rectangles are clearly different. However, as we noted before, their areas are the same because 10×35 and 25×14 are composed of the same factors.

TASK 22G: This combines the previous two tasks. It shows how the diagram where the axes are drawn to the same scale (Task 22F) has been 'compressed' to make the rectangles look the same (Task 22E).

TASK 22H: This explores in greater depth the effect that 'compressing' the plane has on the way a shape looks.

This task is challenging! It might help to draw the pink rectangle in different orientations on a 'normal' grid, where the horizontal and vertical scales are the same. It turns out the the angles of rotation b and c are 45 and 90 respectively.

The diagram below-left is the same as the diagram above. The diagram below-right is the 'decompressed' version.

 

Comments welcome

mietmau@gmail.com
@ProfSmudge

D21: Fractions as division (sharing)

 In Theme 13 I looked briefly at how a fraction, in this case ¾, can be thought of as (or as the result of) the division 3÷4. I used the scenario of 3 pancakes shared between 4 people, with this diagram showing some of the ways it could be done:

 

In each of these examples, the objects (pancakes), or parts thereof, are distributed until there is nothing left. In this theme, I consider a slightly different approach, whereby some individuals are given a whole object (be it a pancake, as above, or a bar, as below). Parts of each object are then taken away and given to the individuals who had nothing. Here is the first example:

TASK 21A: Here we have the scenario of 5 bars (of some sort!) being shared equally between 7 people. Initially, the 5 bars are given to 5 of the 7 people, with two people having nothing; small, equal parts are now removed from each bar and distributed among the two people with nothing. The critical challenge is to determine how large these pieces should be so that everyone ends up with the same amount.

In the diagram (above) the size of the green and the yellow pieces is such that everyone ends up with the same amount - as can be seen in the third column of the diagram. Everyone gets the equivalent of 5 of the small pieces, and we can see, from the second and third columns, that 7 small pieces make up one bar.
So each piece is ⅟₇ of a bar and everyone gets ⁵⁄₇ of a bar.

I would argue that the diagram can be used to demonstrate this outcome very nicely. Of course, this begs the question of how the certain size of the small pieces was determined in the first place. The diagram below shows what happens if the chosen size of the pieces is slightly too small (left) o slightly too large (right).

TASK 21B: Here we consider a slightly simpler case - only one person is without a bar initially. Again, the diagram should allow pupils to deconstruct what has happened very nicely - this time each small piece is ⅙ of a bar and everyone gets ⅚. We can write the statement below.

5 bars shared between 6 people   =   5 ÷ 6   =   ⅚.

[The statement might seem almost too obvious to bother writing down. However, for younger pupils each of the two steps represents a significant achievement.]

The strength of the task, in particular its diagram, could be said to lie in the ease with which it allows one to analyse the result of a successful sharing process. If you want to encourage students to predict what will happen, as well as analyse the result, you could present them with one of these incomplete or less detailed versions of the diagram:

 or

TASK 21C: Here most pupils will know from the start that each person gets one half of a bar. So the interest in the task is not in finding the answer but in seeing how it emerges as this particular method of sharing plays out. 

The three pieces make ½ because each is ⅙.

TASK 21D: Here we are almost back to the first task, which showed 5 ÷ 7. The current task is perhaps slightly more complex as there are more people who start with nothing.

TASK 21E: This involves the same situation as in the previous task but shows a simpler, more orthodox way of sharing the bars. This time, the bars are cut into pieces before any are distributed and the bars are shared equally one by one among the given number of people. The method involves the same principle as Method 4 at the top of the page. The outcome is not very 'interesting' but the method always works, and shows very clearly how we can get from B ÷ P to B/P:  

P people get 1/P of each bar, and so if there are B bars they get B × 1/P = B/P of a bar altogether.

 

D20: Dividing by three-sevenths

In Theme D18 we looked at how the two interpretations of division (measurement and sharing, or quotition and partition) apply when dividing by a fraction. In particular, we considered whether they threw light on the formal division rule of 'multiply by the reciprocal of the divisor' (flip-and-multiply). I raised the possibility that it might sometimes (or for some students) be more illuminating to think in terms of general fractions rather than using specific fractions.

In this theme I stick with specific fraction divisions, namely 10÷³⁄₇, ⅘÷³⁄₇ and ⅖÷³⁄₇. How well might the specific numbers/fractions work? In other words, how easy is it to see them as generic? Does their effectiveness differ for the two interpretations of division and for the three chosen dividends (10, ⅘ and ⅖)?

Division as measurement (quotitive division)

For the division 10÷³⁄₇, the measurement interpretation can be thought of as asking, 'How many three-sevenths are there in 10?'. We can solve this in stages. 

First we can ask, 'How many 7ths are there in 10?'. This is fairly easy to construe: there are seven 7ths in 1, so there will be 10 times as many in 10, which gives us 10×7.

Next we can ask, 'How many three-sevenths are there in 10?'. Three-sevenths is 3 times as large as one-seventh, so the number of times it fits into 10 will be one-third of the number of times one-seventh fits. So it fits 10×7÷3 times. 

Overall, it is fairly easy to see, but also to see why, we get the desired answer by multiplying 10 by the denominator of the divisor ³⁄₇, and by then dividing by the numerator. It follows, though this might be a step too far for some pupils, that we can rewrite our division 10÷³⁄₇ as 10×⁷⁄₃, where ⁷⁄₃ is the reciprocal (or multiplicative inverse) of ³⁄₇. 

[There is a further step one could take, for this and the other divisions, of scrutinising the actual answer. For this division we have 10×⁷⁄₃ = 23⅓. We can think of 23 as telling us that there are 23 lots of ³⁄₇ in 10. But what about the ⅓? Is that referring to 10, to 1, or to ³⁄₇? This is an important issue, but one that I am not going to focus on here.]

Let's now consider the division ⅘÷³⁄₇. This time the measurement interpretation can be thought of as asking, 'How many three-sevenths are there in ⅘?'. We can apply the same steps as before. However, one of these is less easy to construe this time. 

We can again start with the simple and powerful idea that there are seven 7ths in 1; however, some pupils may well balk at the next step, namely that there will be ⅘ as many 7ths in ⅘, giving us ⅘×7. Previously we multiplied 10 by 7 rather than ⅘ by 7, which fits the common sense notion that 'multiplication makes bigger'; this time we can see that there will be fewer 7ths in our dividend than in 1 - can this be achieved by multiplying?

Our third case is the division ⅖÷³⁄₇. For the measurement interpretation we can apply the same steps as before. However, the dividend and divisor are much closer together than in the previous two examples, so pupils might struggle at first to get a good sense of what is going on. How does ³⁄₇ go into something almost the same size, indeed that is slightly smaller? This situation is eased somewhat by the first step, if pupils manage to take it, since we start with single sevenths rather than three sevenths. From then on, it is likely that the steps will be as meaningful (or not) as in the previous example.

In sum, I would argue that the use of specific numerical examples to illustrate the measurement interpretation of division, can enable pupils to get a sense of why the 'flip-and-multiply' rule works. However, it is likely that it is easier to see what is going on when the dividend is substantially larger than the divisor, and larger than 1.

Division as sharing (partitive division)

Consider a division such as 10÷4, where the divisor is a whole number. We can think of this as fitting a story like '10 kg of potatoes are shared fairly between 4 people. How much does each person get?". Let us rephrase this, a little awkwardly, as "10 is 4 shares, how much is 1 share?". We can then interpret 10÷³⁄₇ as "10 is ³⁄₇ of a share, how much is 1 share?".

Again, we can solve this in stages, as in the diagram below. As with the measurement interpretation, a powerful first step is to consider one seventh. If 10 provides ³⁄₇ of a share, then ¹⁄₇ of a share will be one third of 10, which we can write as 10÷3. In turn, seven sevenths, or a whole share, will be 7 times that: 10÷3×7. 

Overall, it is fairly straightforward to see, and probably easier than for the measurement interpretation of division, that we arrive at the answer by dividing the dividend 10 by the numerator of ³⁄₇ and then multiply by the denominator. So we can rewrite our division 10÷³⁄₇ as 10×⁷⁄₃, where ⁷⁄₃ is the reciprocal (or multiplicative inverse) of ³⁄₇. But again, this might be a step too far for some pupils.

We can interpret ⅘÷³⁄₇ as "⅘ is ³⁄₇ of a share, how much is 1 share?". Here it is less easy to get a sense of what the answer might be, compare to the previous example. This might prevent some pupils from knowing what to do - even though we can carry out the same steps as before and, once started, they are probably just as easy to visualise and to appreciate how the numerator and denominator of ³⁄₇ come into play.

We can interpret our final division, ⅖÷³⁄₇, as "⅖ is ³⁄₇ of a share, how much is 1 share?". This is probably no more difficult to make sense of and solve than the previous division, in contrast to when the divisions are interpreted as measurement.

In sum, it is probably easier to interpret and solve a division by a fraction task using a sharing interpretation when the dividend is larger, or at least, markedly different from the divisor. However, overall, it is probably easier to see what effect the numerator and denominator of the divisor have when division is interpreted as sharing than when it is interpreted as measurement - as long as pupils are willing to accept, for the division u÷v, the reformulation of sharing as meaning "u is v of a share, how much is 1 share?".

D19: Fraction of a fraction of a fraction....

This theme has similarities with several other themes where we create a fraction by subdividing a square or rectangle several times - for example, D03, D10 and D11. The theme is distinctive in that we subdivide numerous times and often by repeating an operation, such as halving, to produce a regular pattern, as in the first task below.

TASK 19A: Here we make use of the classic 'diamond' method of halving the area of a square - by rotating a given square through 45˚ and shrinking it until it fits tightly inside the original square. The orange region is formed by halving the blue outer square and then halving and halving again. So its area is one eighth of the outer blue square. The lemon region is formed by taking one third of the blue outer square, then taking a third of the resulting strip. So its area is one ninth of the blue outer square.

TASK 19B: For the green outer square on the left, we apply the halving operation 4 times. This means that successive squares have areas that are ½, ¼, ⅛, and ¹⁄₁₆ of the area of the green outer square. The fact that the halving operation occurs so many times might nudge some students away from determining this string of individual fractions to using a more formal approach of operating on the operations; so they might construct an expression like ½×½×½×½ or (½)⁴ and calculate its value. The area of the lemon region is ¼ of ¼ of its green outer square, and so has the same area as the orange region.

TASK 19C: The area of the orange region is ½×½×⅓×⅓ of its blue outer square. The area of the lemon region is ⅓×⅓×½×½ of its blue outer square. So they are both ¹⁄₃₆ of the blue outer square.

It will be interesting to see whether students form the above expressions to solve the task, or whether they solve the task in a step by step manner, by finding the area of successive smaller squares.

TASK 19D: Here we take a slight detour by considering the 'diamond' shape that we met in earlier tasks, and exploring what happens when the square that envelops the shape changes into a rectangle with a smaller and smaller height.

It turns out that the area of the orange region is ½, ⁵⁄₁₂, ⅓ and ¼ of the area of the square and the three successive rectangles respectively. Beyond the fact that the fractions get smaller, pupils will probably feel that there is no simple pattern here. However, there must be a pattern, surely?!

It transpires that the fraction is equal to h/12, where h is the height of the rectangle. Some pupils might spot this, if given enough time and encouragement! We can derive it algebraically, of course:
the area of the 'diamond' is ½h², and so the fraction is ½h²/6h = h/12.

TASK 19E: A further detour. Here we can think of the 'diamond' shape as being formed by overlapping flaps of an 'envelope' rather than being enveloped by a rectangle!

The lemon region turns out to cover 0, ⅟₆₀, ⅟₁₂ and ¼ of the area of the square and the three successive rectangles respectively. This time a pattern in the fractions is even harder to spot.
The fraction is given by the expression (36 – 12h + h²)/12h, where h is the height of the rectangle.

TASK 19F: We return to the earlier form of task.

The orange region covers ⅟₈₀ of its outer green square. Pupils might find the fraction in a step by step way (such as ⅟₂₀ → ⅟₄₀ → ⅟₈₀) or by deriving and evaluating expressions such as ⅟₂₀ × ¼ or ⅕ × ¼ × ½ × ½.

The lemon region covers ⅟₈₁ of its outer green square. Pupils might find the fraction in a step by step way (though this could get quite complicated) or by deriving and evaluating an expressions such as ⅟₉ × ⅟₉.

TASK 19G: This is more straightforward. We can express the fractions as, say, ⅟₂₅ × ½ and ⅟₇ × ⅟₇ respectively. So the lemon region is slightly larger.

It might well appear that the orange region looks larger than the lemon region. Here we have rotated the region and placed it close to the lemon region. It is a tiny bit smaller!

TASK 19H: A nice simple one to finish with. We have ⅟₂₅ × ¼, say, and ⅟₁₀ × ⅟₁₀, say. So they are both ⅟₁₀₀.

 

D18: Dividing a fraction by a fraction

Another interlude. Here I look at a paper by Gregg and Gregg on the division of a fraction by a fraction. This is not as fundamental as the issue of connecting fractions with division, as in the discussion about statement like 3÷4 = ¾, in Theme D13 (Parts of a whole and sharing).

The paper is concerned with the rule that allows us to change a division into a multiplication, as in the statement below. Gregg&Gregg consider two aspects of division (measurement, or quotitive division, and sharing, or partitive division) and present contexts to help students (in their case student teachers) make sense of these forms of division when they involve fractions. Their aim is to help the students see why we can transform a division involving fractions into a multiplication.

The paper is interesting on several counts. One concerns problem strings: they present some nicely designed ones, but how effective might they be? Second, and more fundamental, how easy is it to devise situations that help students see why a formal transformation like the one above is valid? And third, which aspect of division might be more salient when it comes to this particular transformation?

In the introduction to the paper we are presented with this task:

The authors claim that their students will often solve this by writing ½ × ⁴⁄₃ = ⁴⁄₆ = ⅔, on the basis that "...it's really ½ ÷ ¾, but I flipped the second fraction and multiplied". They go on to say that students struggle to explain why this 'flip-and-multiply' procedure works.

It is perhaps not surprising that their students generally don't have a good explanation for the 'flip-and-multiply' rule. On the other hand, I am surprised that they seem so readily able to construe the structure of the task in terms of a formal expression: ½ ÷ ¾. This, it seems to me, is quite impressive. I would have expected a more grounded approach to have been more common, such as this:
Half the floors in three-quarters of an hour means ⅙ of the floor in one quarter of an hour, which means ⁴⁄₆ or ⅔ of the floors in an hour.
This of course raises the issue, how readily does a grounded approach lead to a more formal understanding?

Division as measurement (quotitive division)

The paper continues with a look at tasks that tease-out division as measurement (though this is left implicit for quite a while). This is one of the problem strings they use with their students:

This is nicely sequenced, but I do wonder, as students work their way through so many tasks, do they become aware of the underlying structure, or are their senses dulled by so much information, rather like the frog in the slowly heating saucepan of water? Notice, too, that the string doesn't take us very far. One of the fractions is always ½, when the eventual goal is something more generic, like ¾ ÷ ⅔. Even if students can't yet resolve such a more challenging version, would it not give them a sense of where they need to go?

An alternative to presenting a fairly long problem string, like the one above, might be to start with the last problem in the string (the place you want students to get to) and then, if students struggle, ask them, individually or as a class, to come up with a more intuitable variant, by keeping the wording but changing the numbers.

[We can represent ¾ ÷ ⅔ as in the diagram below. Division as measurement amounts to asking how many times does ⅔ go into ¾? We can see that it fits once, with a light-green-bit left over. So the question becomes, what fraction of the dark green region is the light green region? It might not be immediately obvious but this turns out to be ⅛ - the light green region is one 12th of the original strip and the dark green region is eight 12ths of the original strip. So ¾ ÷ ⅔ = 1⅛. But how much light does this throw light on the 'flip-and-multiply' rule?]

[A careful scrutiny of the above diagram, in all its detail, might allow us to be confident about the answer, and to see how we could construct a similar diagram for another pair of fractions. As such, the diagram can be empowering. However, to get more insight into the actual 'flip-and-multiply' rule, in the case where division is interpreted as measurement, it might be helpful to consider a less grounded, more general approach:

• Imagine dividing something by a fraction 1/n. From a measurement point of view, the larger n is, the smaller 1/n will be and the more times it will fit into the dividend, D (where D is any rational number, whole or fractional). To be precise, 1/n will fit into 1 n times, and so it will fit into D, Dn times. So dividing by 1/n is the same as multiplying by n.
• Now, if we were to divide by m/n, which is m times as large as 1/n, our new fraction will fit into the dividend, D, 1/m times as often. So D ÷ m/n = (D × n) ÷ m = D × n/m.

The general form of the above argument might make it is inaccessible to some pupils. On the other hand some (older?) pupils might find it to be illuminating through its very lack of detail. Would the argument work as well if we replaced m/n by ⅔ and D by ¾?]

Returning to the above problem string used by Gregg&Gregg, the notion that these tasks can be thought of as division is not raised with their students until they have worked through two more strings of tasks. One of these involves fractions with the same denominator and that is where the work is heading: students experience that a (still rather simple) division like ¾ ÷ ⅓ can be solved by converting the fractions into 12ths, giving us ⁹⁄₁₂ ÷ ⁴⁄₁₂ which is 9 ÷ 4, which is 2¼. This is a nice grounded approach, but does it throw light on the 'flip-and-multiply' rule? My sense is, not much, since the fractions in their original form are no longer there!

Division as sharing (partitive division)

Towards the end of the paper, Gregg&Gregg look at division as sharing. At first glance, this might seem rather odd - how does one share something into a fractional number of groups?! They use a rather nice scenario of cakes and containers to do this. 

They start by sharing their (fractional amount of) cake into a whole number of containers, as in the examples below. This is easy enough to understand and to construe as involving division. [Strangely, perhaps, sharing a fractional amount in this way is probably easier than sharing a whole amount, as in the classic 3 cakes for 4 people.]

Eventually they go from whole containers to fractions of a container, as in the examples below, where they end up with a quite general case involving the non-unit fractions ¾ and ⅔.

 

The crucial question here is, do we still see this as a division? I think the scenario, and its formulation, works well, though some people might be reluctant to accept it. And I do wonder whether we couldn't have modified a more orthodox sharing story, as here:

Three-quarters of a cake is shared equally between 2 people. What do they get each?

Three-quarters of a cake is the amount given to 2 people. What does one person get?

Three-quarters of a cake is the amount given to ⅔ of a person. What does one person get?

Three-quarters of a cake is ⅔ of the amount given to a person. What does the person get?

Van de Walle et al (2010) argue that partitive division includes not just sharing problems but also rate problems. As well as a problem like "24 apples to be shared with 4 friends", it includes a problem like "If you walk 12 miles in 3 hours, how many miles do you walk per hour?". In essence, partitive division problems ask, How much is one? (ibid, p321)

Let's return to the Gregg&Gregg formulation, in particular their Q3 (above):
I have ¾ of a cake. It fills up exactly ⅔ of a container.

If we now consider ⅓ of a container, rather than ⅔, and then consider a whole container, it is fairly easy to see that we can get to the answer by dividing by 2 and then multiplying by 3. Gregg&Gregg express this as follows:

The steps in this sequence of equations are not trivial, but, by relating them to the cake scenario, I think they show in a fairly grounded way that dividing by a fraction is equivalent to multiplying by its (multiplicative) inverse.

This is an interesting outcome: though the sharing view of division by a fraction is perhaps less 'natural' than the measurement view, in that we have to accept a somewhat contrived formulation of sharing, it provides us with a quite well grounded representation of the 'flip-and-multiply' rule.

PS: We can of course justify the 'flip-and-multiply' rule in a formal way. In the case of our division expression ¾ ÷ ⅔, we could, for example, multiply both fractions by ³⁄₂. This doesn't change the value of the expression but transforms it into (¾ × ³⁄₂)÷1, or just ¾ × ³⁄₂.

 

 

D17: Naming neighbours to one-third on a number line

These tasks complement the tasks from the previous theme. This time, rather than finding the position of a fraction relative to the position of ⅓, we try to name the fraction, when we are given its position relative to ⅓ (and to 0). 

It is likely that pupils will find these tasks to be easier than the corresponding tasks in themes 16 and 15, as they are slightly more structured.

TASK 17A: Here the most plausible, simple answer is ½, as in Task 16A (though of course we can't be absolutely sure, even if we measure the intervals).

TASK 17B: Here the 'intended' fraction is ¼ as in Task 16B.

Again, one can't be sure that the arrow is pointing exactly to ¼, but it is the simplest 'plausible' answer. You might want to challenge pupils by asking,

If the fraction is not exactly ¼, what fraction might it be?

An interesting variation on this (and all the other tasks) is to provide some extra, equally spaced, marks. This won't necessarily make the task easier!

TASK 17C: The intended fraction here is ⅙, as in Task 16C. Easy!

 TASK 17D: Here we were thinking of ²⁄₉. It will interesting to see whether pupils come up with other, plausible fractions; for example, ⅕ would be a good estimate!

 TASK 17E: Here we were thinking of ¹¹⁄₃₀, as in Task 16E. However, one wouldn't expect pupils to come up with this precise fraction, unless they were asked (or they decided) to make very careful measurements.

The task could lead to an interesting class discussion: what fractions do pupils come up with and how do they try to evaluate the various suggestions?