Week 03: Squares (and other shapes) on a grid

Here we present children with parts of a square (or other simple shape). The shapes' vertices are grid-points and the aim is for children to analyse the grid-points and how they are related with sufficient care to be able to tell whether a desired shape 'looks right'? [Some children might take the analysis further and turn an observation such as this into a general rule: "If a line goes 4 to the right and 3 up, when I turn it through 90˚ it will go 4 up and 3 to the left". However, that is not the central aim of this week's tasks.]

Children can use a ruler if they wish, though you might want to encourage them to sketch the shapes (once the vertices have been carefully placed) as this is an important skill.

Task 03A: We can think of the line from A to B as going 4 units to the right and 2 up (or 2 to the right and 1 up, twice). Similarly the line from B to C goes 4 up and 2 to the left (or twice 2 up and 1 to the left).

We can locate D by going 4 up and 2 to the left from A (the same as going from B to C). Or we can start at C and go 4 to the left and 2 down (the same as going from B to A).

Task 03B: This is more demanding than Task 03A in that we are only given a line segment in one direction, not also in the perpendicular direction. There are two possible positions for vertex C (and D) but one of these leads to a square that is only partly on the given grid.

We can get from A to B by going 3 to the right and one up. This means C lies 3 up and 1 to the left from B (or 3 down and 1 to the right, though this would mean that D is not on the given portion of the grid).

Task 03C: One way to solve this task is by a form of trial and improvement, by, for example, trying to place B such that AB = BC and ∠ABC is a right angle. This has its merits as it encourages children to visualise.

An alternative, more analytic approach, is first to locate the other diagonal by using the properties of a square that its two diagonals are equal, bisect each other and are at right angles.

Task 03D: As children play around with this, they might discover that it is quite easy to generate more parallelograms by using some of the parallelogram properties. For example, the fact that parallelograms have rotational symmetry means that if, say, point B is moved 1 unit up the page, D has to move one unit down.

It is not easy to find the desired rectangles (at least not if their vertices are grid-points). Here are two (I don't think there are any more (apart from the square from Task 03C) where the vertices are grid-points).

Task 03E: This can be solved in the manner of Task 03C by first drawing the diagonal AC (which passes through X). Alternatively, as children play around with this, they might discover that whatever the position of point A, we can generate the vertices B, C and D by moving point A through successive rotations of 90˚ with X as the centre of rotation. Here, it might help to first draw the line segment XA.

 

Week 02: Expressions as numbers

Here we look at a series of structural arithmetic tasks that consist of simple expressions involving addition or subtraction.

Task 02A: This task can be solved by simply adding 10 to 213. Surprisingly perhaps, many children don't seem to spot this: they ignore the information about 165+48 and perform the calculation 175 + 48 from scratch. If this happens, encourage children to compare the expressions 165+48 and 175+48. How do they differ? How can we make use of this difference?

[It is 'natural' to see an expression like 165+48 as an instruction to perform an operation. However, it can be argued that to solve Task 02A structurally, one needs simultaneously to see the expression as a number in its own right, which changes as its components change. In an interesting paper by Gray and Tall, this 'amalgam of process and concept' is given the name procept. A similar phenomenon can be said to occur with algebraic expressions like 8 + g (see Task 02E, below); Kevin Collis refers to 'acceptance of lack of closure'; Anna Sfard to 'reification'.]

Gray, E. and Tall, D. (1994). Duality, ambiguity and flexibility: a proceptual view of simple arithmetic.  Journal for Research in Mathematics Education, 26(2), 115-141.

In this task we have chosen fairly 'nasty' numbers to discourage children from performing the calculation 175 + 48. However, the very presence of such numbers might put some children off, in which case you might want to try a simpler version such as the one below. Of course, here it is very much easier to calculate the second expression from scratch, but if children do so you can still ask whether they could find its value a different way.

Here is another variant which is somewhat simpler than the original task, in that the change in the component 165 occurs in the ones rather than the tens column.

Task 02B: The first part of this task provides an opportunity to discuss the variety of ways one can get a 'derived fact' like '53 – 27 = 26'. For example, one could using 'adding on':
27+3+10+13 = 53, 3+10+13 = 26; or 27+30–4=53, 30–4 = 26; or 27+27=54, so 27+26 = 53.

The second part of the task is quite demanding. You might want to try a simpler version:

This is more demanding than the original task, as it might lead some children to give the answer 162 rather than 1062:

And here's a variant where the underlying structure is rather different - and probably much simpler, once children spot it!

Task 02C: The structure in this task is different again. Some of the digits in the first expression have been switched from one term to the other, but keeping the same place value and so keeping the overall value of the expression. If one so wished (!), one could write both expressions as this:
800 + 900 + 10 + 40 + 5 + 7.

This is a simpler variant: only the ones digits have been switched. So again, the two expressions are equal.

This time the tens digits have been switched, but the numbers themselves are simpler. Again, the two expressions are equal.

Finally, two of the digits have been switched within one of the terms (947) of the expression, so their place values have changed.

Overall, the second expression is equal 1762 – 200 + 2 = 1564.

Task 02D: As with all this week's tasks, this task can be solved more easily if one pauses to take stock of the structure. If children attempt to solve this by simply working from left to right, the process is quite cumbersome and involves a negative number. On the other hand, if children look carefully they might notice that subtracting 53 (at some stage) and adding 63 is equivalent to adding 10, so the expression represents 19+10, or 29.

In this variant it is perhaps more tempting this time to simply work from left to right as it doesn't introduce a negative number. However, it is still more efficient to combine the later terms first.

Here the '53' terms cancel out, as long as children pause long enough to notice*; however, they also have to be comfortable about changing the order of the operations or accepting the relatively abstract idea that -53 will be cancelled by +53, even if the latter operation doesn't happen 'immediately'!

It might take a while to spot the contrivance that makes this variant easy to solve! It turns out that the sum of the first and third term is the same as that of the second and fourth term...

Task 02E: The items in this task don't quite match any of the other Week 2 items; their structure is slightly different and the expressions involve 'unknowns'. However, we have some interesting data about the items. Item a) is from the CSMS Algebra test that was given to large representative samples of secondary school children in Years 8, 9 and 10 in England in 1976 and again in 2008/9. Item b) was used in an informal study in 2019, involving an ad hoc sample of children mostly from Year 6.

Item a) turned out to be extremely easy. For example 97% of the 1976 Year 9 sample answered the item correctly, reduced slightly to 87% for the Year 9 sample in 2008/9. The younger children were equally successful on item b), with 87% giving the answer 40.

However, before we rush to the conclusion that these successful children can 'deal' with algebra, we should consider their performance on the items below, which also featured on the CSMS Algebra test and on the informal test with the younger children respectively. Here item a) was answered successfully by only 41% of the 1976 Year 9 sample, and by 37% of the corresponding sample in 2008/9. For many children, the correct answer, of 8+g, doesn't look like a proper answer - it is not closed. It looks like an instruction, not a number in its own right, just as for many children the expression 165+48 in Task 02A only looks like an instruction . As a consequence, some children gave the answer 8g instead of 8+g. Others came up with plausible numerical answers such as 9 (it's more than 8) or 12 (perhaps on the basis that e, f and g all equal 4) or 17 (perhaps on the basis that e, f and g equal 3, 5 and 7 respectively, or because g is the 7th letter of the alphabet). As for item b), only 1 child from the younger sample gave a correct open answer.

*Put another way, "Don't just do something, sit there!" Thanks to Tom Francome for alerting me to this phrase.

 

Week 01: Bang in the middle

Here we look at ways of finding the 'midpoint' of two numbers. The tasks are fairly straightforward but also slightly out of the ordinary, which might encourage children to try to visualise what is going on before embarking on some form of arithmetic.

It turns out that there are some nice methods for finding the middle number. The tasks can also be solved quite nicely by guessing a solution and then testing whether it is exactly in the middle. This in turn might lead to the discovery of a systematic method, but even if it doesn't, it can provide children with useful experience of estimating as well as practising simple arithmetic and working with known and derived facts.

Task 01A: This task, the first of five in this section, might seem rather abstract as it involves pure numbers. The next three tasks are more grounded in that we use a line to represent the given numbers, which also provides a nice 'frame' for analysing the situation. We finish with a context where the given the numbers represent numbers of plastic ducks....

The abstract nature of this task, and the use of relatively large (although quite 'nice') numbers might make this task quite challenging for some children. However, this is not necessarily a bad thing in that it sends the message that there is something here to think about and discover. If one started with, say, 30 and 32, there would be nothing to think about at all.

However, if you think the task is too challenging, you should change the numbers or simply go on to the next task. You should always feel free to modify these tasks and to use them in whatever order you like.

If children are stuck, encourage them to guess a number between 30 and 54, for example 38. This then brings a new challenge: how can we test whether the guess is right or how might we might improve it? A grounded way, which the children might hit upon, is to find the difference between the guess and each of the given numbers. In turn, this provides an opportunity for the children to practise a taught procedure (eg column subtraction) or to adopt an informal method such as 'adding on', as here:
30 + 8 = 38, 38 + 2 + 10 + 4 = 54;
so the differences are 8 and 2+10+4=16: so we need a larger number!

Further, this might prompt children to wonder how they might find the number where these differences are equal. This leads to a nice systematic method: find the difference between the given numbers and halve it; add this half-difference to the smaller given number or subtract it from the larger. However, don't push children towards this if they don't see it at this stage - it can come (or not!) on a later task. Most important here is that children feel that they can make progress by using their own resources, perhaps with nudges from you but not through having to be told a method.

[Note: Another systematic method is to add the two given numbers and halve the result. This, in essence, is the standard way of finding the mean: sum all the values and divide by the number of values. At a much later stage, it can be a nice challenge, and one that demonstrates the power of algebra, to express this method and the previous one sybolically: for given numbers A and B, the expressions for the middle number are (A+B)/2 and A + (B–A)/2. Can we show that these are equivalent?!]

Task 01B: This is similar to the previous task, except the numbers are perhaps slightly more accessible and we use a number line to model the numbers.

The number line might help children to visualise the position and value of the middle number that we are looking for: 

The number line might also help children realise that the two 'gaps' should be the same between the given numbers and the middle number - and that these gaps will be half of 18, the gap between 42 and 60:

The way we have drawn the number line probably favours this 'halve the difference method' over 'add the numbers and halve' (the mean method discussed above). For the latter it would help to have a longer line, as here:

It is interesting to consider what would happen if we wanted the 'middle' of three numbers, as in the diagram below. If we wanted to stick with the 'halve the difference' method, one approach would be to halve the difference between 42 and 60 again, and then consider the difference between the resulting 51 and 30, or rather between 51 and another 51 and 30! The answer is not 40½! Here, the mean method is much simpler and less error-prone: 30+42+60 = 132; 132÷3 = 44.  

Returning to the original task, some children might notice that 42 is close to 40. The number exactly halfway between 40 and 60 is relatively easy to spot: it's 50. This means our middle number will be slightly larger than 50, as 42 is slightly more than 40. But how much larger? 2 more? Or 1?! We look at this question in the next task. 

 

Task 01C: Here the numbers are again quite 'nice' but further apart than in the previous task, so the first part of the task might be slightly more demanding.

After point P has been found, we are asked to imagine that A moves 10 units to the right. Some children might initially think that the midpoint would now also move 10 units to the right, from the position 51 on the line to the position 61. If they suggest this, it is worth asking, 'Why can't that be right?' The gaps between the midpoint and the given numbers have to remain equal, so they have to change by the same amount, namely by 10÷2 = 5 units. So the midpoint moves to the position 51+5 = 56.

The final part of the task, where we imagine that A move 2 units to the left, to the position 10 on the line, might help children see things more clearly. It is fairly obvious that the midpoint of points at 10 and 90 is at 50, which means that P has moved 1 unit to the left while A has moved 2. A nice follow-up to this is to ask,

Find two more points whose midpoint is at 50. Now find another two....

When children get to the second part of the original task, some will start afresh, ie by applying the method they used for 12 and 90 to the new pair, 22 and 90. The method discussed above, where we add 5 to the previous answer, 51, is arithmetically much simpler. However, many children struggle with this kind of approach, where one examines how things change rather than simply starting anew. In part this might be due to the fact that this focus on 'increments' is unfamiliar to some children. However, I think it is also conceptually more demanding in that it seems to involve a more 'abstracted' view of the task along with a better sense of its structure. This kind of structural thinking is powerful, and we will meet many more tasks where it can be applied in the course of this blog. 

[Here is a simple example of a structural task:
37 + 96 = 133. Use this information to find 38 + 96.
The task can be solved by simply adding 1 to 133. However, not all children will adopt this structural approach but will perform the calculation 38 + 96 instead. The structural approach involves seeing the expression 37+96 not just as an instruction but also as an entity in its own right - the number 133. Further, it involves seeing how this number is affected when one of its components, 37, is changed into 38.]

Task 01D: This task uses the same numbers as in the very first task, but with the numbers represented by points on the number line.

The tasks asks for a series of midpoints, rather than just one, which should allow children to consolidate the approaches they have developed. An interesting difference from earlier tasks is that point S does not represent a whole number. Some children might hesitate at this juncture, perhaps saying that the point is 'impossible'*. After point S, the midpoints get more challenging. It will be interesting to see whether children work in decimals or with common fractions. Whichever they prefer, it is worth asking them to work with the other form as well.

The positions of points P, Q, R and S are shown below. The next point, T is at 30.75 or 30¾. Beyond this, if children are working in decimals, a calculator can help! The next points are at 30.375, 30.1875, 30.09375, and so on. And if they are working in mixed numbers, all they need do is to keep halving the denominator of the fractional part: 30³⁄₈ , 30³⁄₁₆, 30³⁄₃₂, and so on.  Will we ever get to 30?!

*It might be worth exploring this issue in its own right, for example by starting with points A and B again, but moving one point's position to an adjacent odd number, as here:

Task 01E: To round things off, we present a word problem that has the same underlying structure as the 'midpoints' tasks, set in a context where the numbers represent numbers of plastic ducks.

You might want to change the numbers in the task, to make them more challenging (and perhaps change the names if you've developed an antipathy to Piagetian stage theory - Barbie and Ken?!).

The given numbers match those used in a study by Jean Piaget. Individual young children (aged about 5, 6 or 7) were presented with a set of 14 counters and a set of 8 counters and asked to 'equalise' the sets. Piaget describes** how some children, as they move counters from the larger to the smaller set, focus only on the smaller set and so fail to notice that as this gets larger the other gets smaller. Children who succeeded on the task commonly used 1-to-1 correspondence. They would arrange the counters in the set of 8 in a row, and construct a matching row from the set of 14 counters. They would then distribute the remaining 6 counters to the two rows, perhaps first adding 2 counters to each and then 1 more to each, as in my diagram below:

 Returning to the 14 and 8 plastic ducks, once children have worked on the task, they might find the diagrams below useful, as a way of helping them see the structure and a method of solution, or as a way of helping them describe the structure and a method of solution.

[Note: The task is reminiscent of this famous, but perhaps more challenging, puzzle:
A bat and a ball cost £1.10. The bat costs £1 more than the ball. How much does the ball cost?]

**Piaget, J. (1965). The Child's Conception of Number. Norton & Co, New York, pages 190-195.
First published in French in 1941.

 

Introduction

 

 

 

This blog is aimed at people (be they parents, tutors or teachers) working with children in late-primary, early-secondary school. It contains tasks to help children develop a better sense of number and other aspects of mathematics.

The blog is aimed particularly at those children who have developed the view that mathematics is essentially about procedures and who have got out of the habit of making sense of the mathematical elements (eg numbers) that they are working with. For example, I have in mind a Year 7 child who, when asked to find 20+20, instead of coming up with 40 in their head, felt compelled to write the numbers in a column: "0 plus 0 is 0, 2 plus 2 is 4, the answer is 40".

This is not an argument against procedures, but I do wonder whether we give them too much emphasis, and at too early a stage. Schools, teachers and pupils are put under considerable pressure these days by the high-stakes tests that pupils are required to take. In the UK, this applies especially to the Key stage 2 national curriculum tests (SATs or Standardised Assessment Tests) that pupils take towards the end of Year 6 (age 10-11 years). In maths, pupils take three such tests, one on 'arithmetic', two on 'reasoning', and there is a strong focus on procedures in each of them.

Some of the tasks in this blog will use items from these tests to throw light on these procedures and to explore other ways in which the items can be solved. The blog will also contain non-standard tasks so that children who would normally reach for a procedure are given the opportunity to use their 'natural' intelligence and some of the many skills that they have developed as they grow up.

For convenience, I am arranging the blog into sets of 'weekly' tasks, usually with 5 tasks per week. However, this format should not be taken too literally. For example, it doesn't mean that you should use one task every day, and only one task, though I would urge that when you use a task, allow plenty of time; don't rush through it and on to the next. It also doesn't mean that the weekly sets, or the tasks within them, have to be used in order. The order is up to you.

My hope is that the tasks in this blog will help children get a better feel for what mathematics and mathematical thinking is primarily about and that the children will become more confident about maths and get more joy from doing it.

Dietmar Küchemann