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Do we really make the most of our wonderful number system?
In this, the final of a threepart series, Barbara Carr discusses the place value of our measurement system and the powers of 10.
Just as the children need to memorise the family place value names of numbers, I believe the place value headings of the units of our measurement system need to be memorised too. In the same way young children cope with terms like digraphs and phonemes, there is no reason why they can’t master the prefixes associated with the metric system.
The Latin prefixes are as follows:

k = kilo = thousand

h = hecto = hundred

da = deca = ten

d = deci = tenth

c = centi = hundredth

m = milli = thousandth
The SI standard units of measure are the kilogram, the litre and the metre.
It is uncommon to use the terms decametre (10 metres) and hectometre (100 metre) but we do use the term kilometre (1000 metres). It is my belief that not having heading names for the multiples of 10 and 100 metres and litres adds to children’s confusion.
Decimal notation
The Arabic system used powers of ten to create new columns.
1 x 10 = 10 (Tens column) 10^{1}
10 x 10 = 100 (Hundreds column) 10^{2}
10 x 10 x 10 = 1000 (Thousands column) 10^{3}
This is easy to remember because there is one zero in 10^{1}, two for 10^{2} and three for 10^{3}.
One is 10^{0}, 1 with no zero.
It makes sense then that another column to the right could be called 10 ^{1}, which is a tenth or 0.1. So 10^{2} is a hundredth or 0.01 and 10^{3} is one thousandth or 0.001. As a visual learner, I find colours help me to see the pattern.
There is a bit of a glitch to our HTU colour system. Positive (whole) numbers follow the HTU HTU HTU pattern but we don’t start our decimals with a unitth but with a tenth. This is really confusing until we look at the pattern a different way. You may find the diagram below really helpful.
There is a line in the units/ones column followed by the decimal point. I teach the children that the units/ones column and the decimal point should be seen as one.
So 1 ÷ 10 = 0.1 and we can write 1 ÷ 10 as a fraction 1/10 (a tenth)
1 ÷ 100 = 0.01 and we can write 1 ÷ 100 as a fraction 1/100 (a hundredth)
1 ÷ 1000 = 0.001 and we can write 1 ÷ 1000 as a fraction 1/1000 (a thousandth)
The use of colour supports this idea. Yellow is used to represent the tens that are positioned one place to the left of the units/ones and the tenths are positioned one place to the right of the units/ones. It is easy to work out what comes next: hundreds and hundredths represented by green, then thousands and thousandths represented by red. Next are tens of thousands and tens of thousandths represented by yellow again and hundreds of thousands and hundreds of thousandth represented by green and so it goes on ad infinitum. This can help the children appreciate the infinity of number.
As usual, if we draw children’s attention to pattern in mathematics it makes sense. If something makes sense then it is remembered.
Please note that there is some diversity in how the decimal point is represented. It is worth making children aware of this. In the UK some people place the point above the line, others place it on the line. In the US it is placed on the line and in France and some other countries a comma is used to represent the decimal point.
Unlike many of us who were taught to move the decimal point when we multiply or divide by a power of ten, primary school children are taught that the decimal point never moves. In the same way as our place value headings do not switch position, the decimal stays firmly in place to the right of the units/ones. This is really important when children have to start converting units of measurement.
Williams and Shuard (1980) recognise the importance of representing numbers:

We can carry figures in our head as well as write them on paper.

We can see the way a number is organized

We can see what structures it contains

We can see how it is related to other numbers

We can work out what will be the effect upon it of operations we carry out mentally
Structured apparatus helps a child to understand the multiplicative reasoning required to understand our base 10 number system.
Base 10 equipment like this, is based on a cube with six square faces. A small cube represents one.
10 small cubes can be built upwards to form what is termed a ‘long’ or a ten.
10 ‘longs’ can be stacked backwards to form a ‘flat’ or a hundred.
10 ‘flats’ can be stacked sideways to form a ‘cube’ or a thousand.
It is vitally important to call each piece of equipment by its correct name. Base 10 equipment is proportional equipment that can be used to represent digits in any number based on powers of ten.
Decimal measurements
Base 10 blocks can be used flexibly.
If we take two digits 2 and 1 and sit them side by side.
We know that the digit to the left is ten times bigger than its neighbour.
We could make 21 using 2 ‘longs’ and a ‘one/unit’.
We could also make 2.1 using the same equipment with a decimal point in between.
Why is this? Will this not confuse the children?
The one (cube) is ten times smaller than the ‘long’.
The long could be cut into 10 equal smaller pieces.
The smaller pieces would represent one tenth of the ‘long’
So if the ‘long’ represent a whole number or unit, we can show tenths using a ‘one’.
Decimal notation can confuse children.
If we use Base 10 equipment to help them there should be no problem.
Consider, for example, money. Children would represent 321p using 3 ‘flats’, 2 ‘longs’ and 3 ‘ones’. The same equipment can represent decimal notation because of the relationship between the powers of ten.
£1 is ten times larger than 0.1 (tenth of a pound) and one hundred times larger than 0.01 (hundredth of a pound).
Asking children to build up decimal notation helps them to recognise that decimal place headings are just an extension of whole number headings. The multiplicative relationship becomes clear. They just need the opportunities to explore!
As children are introduced to measurement they work with numbers to 3 decimal places.
If you ask a child to build 1234 using Base 10 equipment they will use 1 large cube, 2 flats 3 longs and 4 small cubes.
When asked to build 1.234m the same equipment can be used.
Let’s take 1 234 and look at the value of each digit in terms of powers of ten:

4 is 10 times smaller than the digit 3 in this number

1 is 1000 times bigger than the digit 4 in this unit
If we represented the number 1 234 mm using Dienes we would use 1 cube, 2 ‘flats’ 3 ‘longs’ and 4 ‘ones’. 1 234 mm is equivalent to 1.234 metres.
The same equipment can represent one unit (using the large cube), 2 flats representing two tenths, 3 longs representing 3 hundredths and 4 small cubes representing 4 thousandths.
Try it out before attempting to teach this to children.
Representing SI units of measurement on a place value board.
I mentioned earlier that it is not helpful to NOT teach children the names of place value headings and that I believe we should. Having recently spoken to a onetoone tutor, she said that every pupil she works with struggles with converting units of measurement. She believes that this is because the children are currently taught by rote and expected to memorise facts through rehearsal.
If we could teach children these prefixes using the headings on a place value board, this will support understanding.
Once children have mastered heading up place value boards for measures they can then see the value of the tenth, hundredth and thousands using base 10 equipment.
Converting units of measurement
Trying to explain this is tricky… the unit of measure is expressed verbally and by a symbol.
3 kilograms is represented 3kg. 1 metre is represented as 1m. The symbol is the unit of measurement. So for 3.4cm, the centimetre is the unit of measurement.
This is helpful to know because we can now build this number using base 10 equipment on a place value board.
When asked how many millimetres are in 3.4 centimetres we know that the 3 next door to the 4 is worth ten times more ie 30. So there are 34 millimetres in 3.4 cm.
So if base 10 and powers of ten are not taught as soon as possible, children will find all of this rather overwhelming (as may you when you read my attempt to explain all of this!!!).
Here is a harder example.
How many centimetres are in 3.124m? The metre is the unit of measure, followed by the dm, cm and mm headings. Build this using Dienes.
Now underline everything to the right of the centimetre heading. Ignore the decimal point and you have 312. This may look odd, but the proportional relationship between each piece of Dienes is correct.
312 is represented by 3 large cubes and 1 ‘flat’ and 2 ‘longs’
We need 10 ‘longs’ to make a ‘flat’ and 10 ‘flats’ to make a ‘cube’.
We need 100 ‘longs’ to make a ‘cube’.
Play around with this as reading this is confusing.
Multiplying by a power of ten
The place value board is often used to model dividing or multiplying a number by 10, 100 and 1000.
1 x 10 = 10
Take a one cube and multiply this by 10 to make 10 ones.
Exchange these for a ‘long’.
Give children opportunities to explore the effect of multiplying one by ten, and ten by ten then dividing a one by ten and ten by ten, exchanging and talking about what they notice.
Now that the new curriculum mentions powers of 10 I wonder if it may also provide an opportunity to explore multiplying and dividing by powers of 10.
10^{3} ÷ 10 = 10^{2} which is the equivalent of 1000 ÷ 10 = 100
10^{2} x 10 = 10^{1} which is equivalent to 0.01 x 10 = 0.1
Summary
It is my belief that by oversimplifying the primary curriculum over the years, we have lost some key ‘tools’ that help us to understand what is going on with our base 10 number system.
The problem is that it is written in such a precise format that this will not be fully understood by many primary teachers. It has taken me all morning to write an explanation of my understanding of it all and I am lucky enough to have time to get to grips with this as a mathematics specialist during SATS week!!!
“Our number notation and the measures in daily use are so closely bound up with our history that new meaning is given to them if their origins are known. There is some considerable value in letting children read for themselves about the inventions that have gone to the making of our number system.” (Williams and Shuard ,1980:163)
I was listening to Thought for the Day on the way to work one day last year by the (then) Chief Rabbi, Lord Sacks. He was talking about a man who had lost his memory and the difficulties he faced. He finished his talk by saying “Today needs a yesterday if we are to plan for tomorrow. If we as individuals or as a humanity are to shape a better future, we need to take time to remember the past.”
Many thanks to Barbara for sharing her article with us. We hope that you found the series interesting. If you have anything you would like to share with us, please let us know.
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