KS1 and KS2 Maths – number lines

In his new series, professor Mike Askew shows how children’s understanding will come on leaps and bounds if they can make connections between important mathematical concepts...

We know that learning mathematics is more powerful, deep and lasting when children make connections between different mathematical ideas. For example, place value is commonly taught as an additive process – exchanging 10 units for one ten, and 10 tens for one hundred. But thinking of place value as a multiplicative process – scaling up the unit by a factor of 10, and then scaling up the ten by a factor of 10 again to make a hundred – is a big idea.

Why is this a big idea? Well, for one reason, decimals take on a whole new meaning. Rather than being seen as a type of fraction, decimals come to be understood as an extension of the place value system. A hundred can be scaled down (shrunk) to one tenth of its size, giving a ten. This in turn is shrunk to one tenth of its size, becoming a one (or a unit if you prefer). There is no need to stop this shrinking process at this point; the one can be shrunk to one tenth of its size, giving rise to the first decimal place, and that in turn can be shrunk to one tenth of its size, giving a hundredth, and so on.

Thinking of place value as enlarging and shrinking (as opposed to adding and separating) is what I mean by a ‘big idea’. The idea is big for two reasons. First it’s mathematically a big idea: the development of the history of mathematics would have been very different if no one had come up with the idea of place value as multiplying by ten, and scaling up or down by factors of ten. It’s also a big idea cognitively – it helps learners come to understand mathematics as a network of interconnected ideas, not a series of separate ones.

What is a big idea?

There is no agreed definition of ‘big ideas’ but there are certain criteria that, for me, define a big idea in mathematics. First of all, the idea must be big enough to connect together seemingly disparate aspects of mathematics, but not so big that it is unwieldy. For example, ‘mathematics is all about problem solving’ is a big idea of sorts, but rather too big to be helpful. An idea like ‘fractions, decimals and percentages are different ways to represent equal quantities’ is a big idea that links separate aspects of the curriculum together, yet it is small enough to be thought about in practical terms.

Second, the idea must have currency across all the years of primary schooling. This is important because it means children get to revisit big ideas across the year groups. The ideas will grow and develop, but there will be a core of similarity on which to build. It also means that all children can be engaged in thinking about the big idea at different developmental levels; working with big ideas is a means of dealing with classroom diversity and being inclusive.

Over this series of articles, I’m going to look at a number of big ideas. I make no claims for these being exhaustive of the number of big ideas that children need to meet, nor for them being the most important of the big ideas. They are the ideas, however, that my experience and the research suggest would improve learning were explicit attention paid to them.

Big idea number 1

The numbers in the primary curriculum all have a unique position on the number line.

Mathematically, this is a big idea because it links together two other ideas: that quantities can be discrete (pebbles, children, cows) or continuous (sea water, height, milk). We can count discrete quantities but we have to measure continuous quantities. And mathematicians realising that discrete counting numbers can be placed on a continuous line brings counting and measuring together.

The idea that numbers are uniquely positioned is also mathematically important. Consider these two problems:

• A pizza is cut up into three equal pieces and Joe eats two of the pieces. How much pizza does Joe get to eat?
• Joe and two friends share two pizzas equally. How much pizza does Joe get to eat?

In each case Joe gets two thirds of one pizza, but this arises from very different situations, and it is not immediately obvious that two thirds of a pizza is the same in each case.

Modelling these problems on a number line shows that the answers are the same: the point at which you arrive by taking one unit on the line and marking two thirds of that unit turns out to be the same point reached by taking two units on the line and finding one third of that length.

Cognitively the fact that numbers are uniquely positioned on the number line is a big idea because it helps children make connections between different types of numbers – whole counting numbers, fractions, and negative numbers. It introduces them to notions of infinity: the counting numbers extend indefinitely; there are an infinite number of fractions between any two numbers on the line. And it ‘ties together’ numbers that, on first encounter, may seem very disparate (are fractions really numbers anyway? The number line representation shows that they are).

The idea of positioning numbers on the number line so they are evenly spaced is also cognitively important. It may seem obvious that the gap on a number line between one and two is the same as the gap between 98 and 99, but when children construct number lines they often squeeze numbers closer together as they get larger. (There is some sense to this. A pile of three stones is perceptually much larger than a pile of two stones, whereas a pile of 98 stones is hard to distinguish from a pile of 99 stones).

Being able to accurately place numbers on a number line turns out to be a powerful predictor of children’s mathematical attainment. But this is not an ‘all or nothing’ skill. A child who can accurately position 0 − 10 on a number line may not necessarily transfer this skill to putting the multiples of 10 on a number line from 0 to 100. And marking up a line from 2.0 to 3.0 with the numbers to one decimal place cannot be assumed to be a skill children can accurately do.

Big idea activities

EARLY YEARS: PREDICT
AND PLACE

Draw sections of a number line from 0 to 100 on strips of sugar paper and give each section to a different group of children. For example, five groups could have 0 − 20, 21 − 40, 41 − 60, 61 − 80, 81 − 100. Each section of the number line should only be partially marked up. You can provide more or less structure depending on how much scaffolding you think the children may need. For example, one group might be given a number line marked with the multiples of five, another multiples of 10, while a line with just 41 at one end and 60 at the other will be sufficient for some.

Give each group a collection of six or so numeral cards (one for each member and a few more) that are in the range of the section of the number line belonging to that group.

Each child takes one card and places it on the number line where they think that number needs to be marked. The challenge is for the group to get the card positioned appropriately, in approximately the right place and in the right order. Once they are agreed over the placing of their cards, they mark that place on the line and label it with the number. They then agree on the position of the remaining cards and mark these too.

Groups come to the board in order to construct the entire number line from 0 to 100, sharing the strategies they used for deciding where to position their numbers on the line.

MIDDLE YEARS: ADD
ANOTHER NUMBER

Provide the children with (or get them to draw) a number line with the left hand end marked and one other mark to the right, but not at the end. When first introducing this activity, you might specify what this second number is, say, 100. Once the children have understood the nature of the task, you could allow them some choice over the value of this number, for example ‘mark the blank division with a number between 10 and 20’.

Ask the children to mark another division on the line (you might specify whether this has to be to the left or right of the marked number, or leave it up to them) and they have to decide what value they think is indicated by that mark. Have children come to the board to share their answers and explain their thinking behind the values they put on the markings.

UPPER YEARS:
FRACTION SQUEEZE

In pairs, children need a pack of playing cards with the picture cards removed and a number line from 0 to 5 (they can make this themselves). They take it in turns to select two cards and to make a fraction with the numerals – if they choose to make an improper fraction, they convert this to a mixed number. For example, if one player turns over three and five, they can use this to make 3/5 or 5/3, which they need to convert to 1 2/3. They mark their choice of number on the line, showing it is their number by either marking their initials against it, or using a different coloured pen. The winner is the child who marks three fractions in a row without their partner squeezing a fraction in-between two of theirs.

Why use number lines?

Imagine two test tubes. You pour three units of liquid into one (filling over half) and four units of the same size into the other. Just by looking, it would be easy to tell which test tube contained the most liquid. Now imagine two other, same sized test tubes. Using much smaller units, you pour 103 units into the first, and 104 into the second. It’s fairly obvious that it would now be difficult to tell which held the more liquid. This is the analogy that psychologist Stanislas Dehaene uses to explain how humans, and other animals, mentally store and compare quantities: not as discrete amounts but through a sort of ‘accumulator’. Drawing on a range of research he presents a convincing case that human infants have a rudimentary number sense, which is wired into the brain, and why the number line is so powerful in helping children develop a good number sense.