Week 10: A ÷ B, how many times does B go into A?

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 10D: This is similar to the previous task, but children might find it to be more challenging as there is a tendency for children to be more comfortable with simple decimals than simple vulgar fractions.

Many children will have been taught a procedure for converting mixed numbers into improper fractions. It will be interesting to see whether they see any connection between the procedure and what is being asked in Items 1 and 3. (And Item 2, for that matter!)

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.
 

It always surprises me that the item turned out to be so difficult, according to these data. Is it because children reach for a taught procedure which many of them then get wrong, and which diverts their attention from thinking about the story, of runners on a running track? My hunch is, that if children paused and paid attention to the story, rather than reaching for a procedure, then many of them would be successful at mentally marking-off 1/8 km lengths until one gets to 3/4 km. (Admittedly, this grounded approach involves some knowledge of equivalent fractions, but then eighths, quarters and halves are part of the most familiar family of equivalent fractions that there is.)

If you have any comments about the blog, or would like to share experiences of using the tasks, please email me here: mathsuntangle25@gmail.com

Week 09b: Sum decimals and fractions

Here we embark on a gentle excursion into adding fractions and compare the effect of expressing numbers as common fractions or decimals.

Task 09A: This is a fairly straightforward task, but it is interesting to see what strategy children use to choose values for the numbers A, B and C, especially at the point where they are asked for another ten examples (a request that might seem rather excessive!).

A simple way of approaching this task is to choose fairly random values for A and B (with A and B positive and A+B<10), and then to derive C by subtracting their sum from 10. However, there are many subtle features that children could employ. For example, they might limit their choice to numbers with one place of decimals; they might allow 0 in that decimal place for some or all of the three numbers; they might try to limit the amount of 'carrying' by choosing decimal parts that sum to 1 rather than to 2; they might form new sets of numbers by re-ordering an existing set, or by modifying the set in a simple and systematic way. Where children have used restrictions that massively simplify the task, you might want to challenge children to work without them.

Task 09B: This is also fairly straightforward, and is of interest mainly for how it compares to later tasks where the decimals are replaced by common fractions.

It is likely that children will initially choose values for A and B that have only one decimal place, as in 0.1 + 0.4. However, they will soon run out of such numbers when asked to 'find ten more'. How readily do they see that they could use numbers with two decimal places, and whose sum is 0.50, as in 0.01 + 0.49?

Task 09C: Some children might find this task a good deal more challenging than the previous task. A neat way to solve it is to find a fraction equivalent to ½ (for example, ⁴⁄₈) and then to split this into two (for example, ⅛ and ⅜). An alternative approach is to choose a value for A and then derive the value for B. Children might discover that this is simpler when the denominator of A is even (as with 7/20) than when it is odd (as with 4/9). Note that the notion of equivalent fractions (and, implicitly, common denominators) is not needed in Task 09B, beyond the idea that 0.5 is equivalent to 0.50, etc.
 

Some children might solve the last part of the task, where A = 4/9, by thinking of ½ as being equivalent to 4½/9, so that B = ½/9 which is 1/18.

Task 09D: This task reinforces the idea that it is easier to find B when the denominator of A is even. It is also worth pointing out that this task would not work well if the fractions were written as decimals, as in Task 09B: most unit fractions would turn out to be recurring decimals (and when A is recurring, B is too).

As children work their way down the table, they might become more fluent in finding the values of B. If so, they might enjoy continuing the table. Children might notice that for a unit fraction A with an odd denominator, the denominator is 2 more than the numerator of B.
 

Task 09E: We open things out slightly by choosing a less familiar value for A+B, namely ⅓. Where previously it was relatively easy to find B when the denominator of A was even, now this only applies when the denominator is a multiple of 3.

Again, it can be fun to continue the table. This time children might notice that for a unit fraction A with a denominator that is not a multiple of 3, the denominator is 3 more than the numerator of B.
 

Week 09: Transforming expressions

Here we take a gentle stroll through the art of transforming expressions into equivalent expressions, including the method of 'constant difference' used in the previous week's tasks.

Task 09A: Here we have the classic situation, which might be quite familiar to some children, where an addend or subtrahend is close to a 10s, 100s or 1000s boundary, etc. In the case of a simple addition, we use the 'compensation' principle: if we increase one component we have to decrease the other by the same amount to keep the expression's value unchanged. In the case of a simple subtraction, we apply the same change to both components.

There is plenty of scope here for devising similar tasks if you feel children would benefit from further exploration or practice.

Task 09B: Here we look more closely at the 'constant difference' principle. Changing the component 27 to 30 or to 20 is equally beneficial in that they result in similar, more simple subtraction expressions. However, it could be argued that the change made to the other component, 642, is (slightly?) more demanding in the latter case since it involves
• a larger number (7 rather than 3),
• subtraction rather than addition and
• bridging a tens boundary (at 40).

Task 09C: Here we are using 'compensation' to transform an addition expression and this time it can be argued that the transformation of the first expression is more demanding since it is here that the change made to the other component, 642,  involves
• a larger number (7 rather than 3),
• subtraction rather than addition and
• bridging a tens boundary (at 40).

Some children might feel that the difference in demand of the two methods is minimal. That is a perfectly valid stance to take! The aim of the task is not to suggest that the difference is so vital that children should learn to adopt one method and reject the other. The aim is more modest and diffuse, of helping children develop a greater sensitivity about the various elements that form part of a mathematical method.
Task 09D: Here the constant difference principle is enacted in stages.

When the given method is applied to 416 – 27, we get this:
416 – 27 = 419 – 30 = 489 – 100 = 389.
There are of course other methods for performing the calculation and you might want to ask children for some. One that works for these particular numbers is to 'decompose' 27 into 16+11, which gives this:
416 – 27 = 416 – 16 – 11 = 400 – 11.
And one way to continue would be this: 400 – 11 = 399 – 10 = 389.

Task 09E: Some children will attempt to solve this task by simply adding together all the terms of the expression. However, it is hoped that the length of the expression, and the clue about its approximate value, might prompt them to look for an alternative approach.

We are told the value of the expression is close to 300, which might encourage children to consider 50 + 50 + 50 + 50 + 50 + 50 as a way of checking whether this is true. How easily can they see that this is indeed equal to 300? By using 6×50 = 300, or perhaps by thinking of (50+50) + (50+50) + (50+50)? In turn, can they spot that the original expression will differ from 300 by this amount:
1 + 4 – 2 – 6 – 3 + 8 = 13 – 11 = 2.
This is greater than 0 and so the expression is greater than 300.

 

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Week 08: Subtracting on the number line

In this week's task we use the number line to model (and perform) subtraction, A – B. (We restrict ourselves to natural numbers, with A greater than B, and focus on the specific subtraction 44 – 28). 

We use the number line representation to model subtraction in two ways:
1. We find the difference between two positions (A and B, or 44 and 28) on the number line, in other words the displacement that takes us from one position (B, or 28) to the other (A, or 44); we can think of this as 'adding on':
B + ? = A;
2. We start at one position (A, or 44) on the number line and see where we end up after a displacement (B, or 28).

Further, we see what happens when we use the 'constant difference' principle, by, in particular, changing 44 – 28 into 46 – 30.

[Note: an interesting tension can arise when one uses a model based on a representation like the number line. Such a model can help one think more clearly about the mathematical elements being modelled; however, until one has become familiar with the model, its detailed features can get in the way of such thinking. This might well happen with some children using this week's tasks, in which case it might be best to concentrate for a while on just one of the tasks, for example 08A or 08C, and apply the particular model to a range of subtraction expressions.]

Task 08A: Here we are thinking of the subtraction 44 – 28 as finding a difference, which we model as a displacement that takes us from the location 28 on the number line to the location 44. 

Andy has found the displacement that takes us from location 28 to 44 as three skips, of 2 units, then 10, then 4. Children might think of it in other equally valid ways, for example as a skip of 2 and then 14, or of 10 and then 6, or even as a single skip of 16.

Whatever particular set of skips one might use to get from 28 to 44, it is worth emphasising that they involve number sense. If we had been given a fully marked number line as in the diagram below, we could simply have counted. To underline the point, you might want to show children this diagram.

When it comes to the last part of the task, children should be encouraged to use a semi or fully 'empty' number line as in the diagrams below.

 

Task 08B: Here we are again thinking of the subtraction 44 – 28 as finding a difference between the locations 28 and 44 on the number line. However, Bella uses the 'constant difference' principle to change 44 – 28 into an equivalent expression 46 – 30, where the difference is easier to see.

Bella has found the displacement that takes us from location 30 to 46 as two skips, of 10 units, then 6, in contrast to Andy's 3 skips in the previous task. Children might think of it in other equally valid ways, and more might see it as a single skip of 16, given the nicer numbers.

Task 08C: In this task, 44 is interpreted as a location on the number line, and 28 as a displacement that takes us to a new location.

Carl interprets the displacement of 28 units as 3 skips, of 20 units, then 4, then another 4, which lands on the location 16. Children might come up with other, equally valid skips that land on 16, for example 4, then 20, then 4.

Task 08D: Here we again interpret 44 as a location on the number line, and 28 as a displacement that takes us to a new location. However, Dee moves shifts the location 44 two units to the right and increases the displacement by the same amount. This will still take her to the desired location but using a displacement that it is easier to apply. As with Task 08B, this makes use of the 'constant difference' principle, though it might not be as easy to construe!

Dee interprets the displacement of 30 units as a single skip that lands directly on 16. Children might come up with other, equally valid skips that land on 16, for example 6, then 20, then 4.

Task 08E: Here we use some closely related stories to represent the subtractions 44 – 28 and 44 – 16. Do some of the stories seem easier to solve than others?

If we use the number line to represent the stories, then Story W fits the model in Task 08A and Story X fits the model in Task 08B. The resulting displacements are 16 and 28 units respectively, so Story W has the smaller displacement, and I think it is for this reason that Story W seems easier to me.

Similarly, Stories Y and Z fit the models in Tasks 08A and 08B respectively, with resulting displacements of 28 of 16 units, which seems to make Z easier (for me!).

[If this doesn't seem very convincing, keep the number 44 in each story, but change 28 and 16 to, say, 38 and 6.]

Week 07: Counting dots

Here we look at quick ways of counting a collection of dots by putting dots into groups. The collections in the first two task don't have a particularly clear, overriding pattern so there can be lots of fairly efficient ways of grouping them! The aim is to encourage children to look for (or impose) structure, even though the structure might not be general. In the later tasks there is the potential to generalise, which some children might find both challenging and exciting, but the tasks can be worked on fruitfully at a non-general level.

Task 07A: The collection of dots here is not entirely random - it has rotational symmetry. However, initially, children might group the dots in fairly random ways, in which case it would help to show them groupings that contain more of a pattern. [It would also help to provide copies of the dot-collection - or squared or dotted paper for children to draw their own copies.]

Once children get the hang of it, they might enjoy the fact that one can group the dots in many, not-entirely-random, ways. Here are some examples.

 

Task 07B: Another collection with rotational symmetry. You might also want to create you own collection, or ask children to do so.

Some possible groupings:

Task 07C: This collection is already quite structured, with two lines of symmetry (and rotational symmetry). Some children might spot that we can think of the collection as showing two copies of the third triangle number, 6. If we push the triangles together horizontally we can form a  rectangle containing 4 row of 3 dots, as hinted at in the given example.

Here are some more possible groupings. The first example hints at the idea of pushing the two triangular sub-groups together vertically, to form a rectangle containing 4 columns of 3 dots. The 4th example introduces the idea of 'phantom' dots which have to be taken away.

Task 07D: Here we have moved entirely away from the idea of a semi-random collection of dots that we had in Tasks 07A and 07B. This collection is highly structured, which makes it mathematically more interesting, including in ways that are quite challenging.

A nice, but challenging, feature of these ways of structuring the dots is that they can all be generalised. Imagine a 'square' of dots with, say, 10 dots rather than 5 along each side. Then we would get these structurings corresponding to the ones in the figure below:
4×10 – 4, 4×(10–1), 4×(10–2) + 4, and 10×10 – (10–2)×(10–2).
You might want to see whether children can generalise any of their structurings in this way.

Task 07E: Here is another well-structured pattern, which can be generalised should you wish to do so.

The collection of dots might remind children of the way 5 dots are arranged on dice. One can think of the collection as three overlapping 5s, as in the first example, below: there are three 5s but with 2 lots of 2 overlapping dots. The third example includes a phantom dot.

As in Task 07D, these structurings can be generalised, although they are probably even more challenging. Consider, for example, a collection that looks like a row of 10 overlapping 5s. Then we would get these structurings correspoding to the three in the figure below:
10×5 – (10–1)×2, 5 + (10–1)×3, and 3×(10+1) – 1.
Again, you might want to see whether children can generalise any of their structurings in this way.

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Week 06: SATuration

This week's tasks are based on some simple arithmetic items from a Key Stage 2 SAT test (Paper 1: arithmetic, 2025). We use variants of the items to help children look more closely at the structure of numbers and numerical expressions.

Task 06A: The SAT item here is the first item in the 2025 Paper 1 test and is a simple item about place value: what does the 5 represent? Our variants are sightly more complex in that they also involve seeing how a change in one component of an expression affects other components (as in the Week 02 tasks). So, for example, whereas the missing number in the original item is 500, in Variant 1 it is 1 less than 500, because 40+9 (or 49) in the original item has been changed to 50.

Children might find it quite demanding to invent their own variants for this particular SAT item, but it is worth given children the opportunity, here and in the rest of this week's tasks, if they are willing to do so.

Task 06B: The SAT item here is Q2 from the 2025 Paper 1 test. Children might well use column arithmetic to solve it as this is what they will probably have been taught to do. However, encourage children to work mentally to solve the three variants.

In Variant 1, 15 is added to each term of the given expression 456 – 385. This doesn't change its value. In Variant 2, 55 has been subtracted from each term. Variant 3 involves 'adding on' or 'the shopkeeper's method'.

Task 06C: Here we are not asking children to solve the given SAT item (Q4) but to get some sense of what the answer will look like. 

The numbers in the given expression are slightly less than 48 and 52, so their sum is slightly less than 100. 

The second number in the given expressions contains three decimal places, with the digit 3 in the third place. The first number only contains two decimal places, so we can think of it as having 0 in the third place. 0 + 3 = 3.

 

Task 06D: The SAT item here is Q5 from the 2025 Paper 1 test. Here we use the same principle as in Task 06B, that for a subtraction expression A – B, its value does not change if we add (or subtract) the same number to A and to B.

In Variants 1, 2 and 3, we transform the given expression 904–8 by, respectively, subtracting 4, adding 2, and adding 92 to each term. 

Task 06E: This SAT item is Q6 from the 2025 Paper 1 test. Here we find ways of transforming a division expression without changing its value.

In Variants 1 and 2 we transform the given expression by dividing each term by 2 and by 4 respectively. In Variant 3 we change the division into two 'partial' divisions, using the distributive law. Note that this only works in one direction. If instead of splitting 84 into two parts (60+24, say) we split 12 into two (eg 8 + 4), this wouldn't get us anywhere. 84 ÷ (8+4) is not equivalent to 84÷8 + 84÷4 !

Some children might express the view that these variants don't really make the task any easier. If so, it is worth reiterating that this is not the prime purpose of the task. Rather we are looking at the properties of division. And sometimes such transformations can be quite powerful, as in this example:

Week 05: Playing with words

Here we take an informal look at rotation and reflection, by considering words that have been written upside down or rendered as mirror writing. Intuitively one can think of mirror writing as the result of an action in three dimensions - flipping. However, mathematically, both kinds of action can be described as transformations of the plane!

Task 05A: The three given instructions are MIND THE GAP!, SLOW DOWN! and WATCH YOUR STEP! Mathematically, the phrases have been transformed, respectively, by means of
• a reflection in a horizontal mirror line
• a 180˚ rotation
• a reflection in a vertical mirror line.

However, if one thinks of this in a more concrete way, for example with the phrases written on a piece of glass, then one can think of the transformed phrases being the result, respectively, of
• flipping the glass over (through 180˚ about a 'horizontal' axis in the glass)
• rotating the glass (through 180˚ about an axis perpendicular to the glass)
• flipping the glass over (through 180˚ about a 'vertical' axis in the glass)

Some children will have difficulty reconciling the mathematical and the everyday interpretations of these transformations. However, the primary aim of this week's tasks is that children visualise the various transformations in whatever way they can. If they are more comfortable doing this in a concrete way, then you might want to leave the mathematical interpretation for now! (We refer explicitly to the mathematical transformations in Task 05C.) Also, if children find the task very challenging, you might want to go straight to Task 05B, where we apply the transformations to single words.

Children might notice that some letters are easier to transform than other, perhaps depending on the particular transformation. We look at this explicitly in Task 05D, but you might want to start discussing it here too.

This shows the transformations of HOLD TIGHT!, corresponding to the ways the original instructions were transformed.

Task 05B: Here we transform single words. The words are quite short, but they involve an interesting variety of letters - do children notice their various characteristics? You might also want to use some other words - including, for example, the children's own names.

These are the transformations of SIXTY, corresponding to those of TEN.

Task 05C: Here we describe the transformations mathematically and include images of the centre of rotation and mirror lines. However, if children prefer to stick to the everyday interpretations, then let them do so!

Here are the solutions:
 

Task 05D: Here we begin to look at the symmetry properties of the letters in a more thorough way, which we continue in Task 05E.

For the second part of the task, children might want to see what happens with the words AVA, VIV and BIX. Can they find a word that fits the third condition? 

Task 05E: Here we go through the letters of the alphabet one-by-one, noting whether they have each of the symmetries in turn.

The symmetries of the letters are shown here. Some letters have several symmetries, some have none. Note that the letters with rotational symmetry have either no lines of symmetry or two lines; none have just one line of symmetry. Conversely, the letters with two lines of symmetry all have rotational symmetry as well.
 

The letters fit nicely into five groups:
• those with rotational symmetry only
• those with rotational symmetry and 2 lines of symmetry
• those with only a horizontal line of symmetry
• those with only a vertical line of symmetry
• those with no symmetry.

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