Amazing Video Proof – Practice Makes Perfect Even For Mental Maths

Practice makes perfect so they say.

Here is a video showing how regular practice at using an abacus can build to an amazing level of skill over a few years.



Note how eventually some pupils are so familiar with the abacus they don’t even need a real one. Using their imagination is sufficient.

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Geometry Can Help Make Clear What Algebra and Multiplication Mean

Many people who have learned arithmetic successfully have some difficulty moving to algebra which is more abstract and symbolic.

In fact algebra is just arithmetic but using letters to be more general.

They say a picture is worth a thousand words, fortunately geometry can make the abstract ideas of algebra crystal clear and even obvious.

Multiplication can also be viewed as arranging things such as stones or pennies in a grid. Or if you have square paper just color the squares so

looks like

How about

this must also be 48 as

eqn2Add4And3Add5Is there a way for us to do the sum by doing the multiplication first rather than the addition?  The answer must still be 48.

Perhaps geometry can give us a clue.  How about if we cut up 6×8 square by drawing a line 2 squares from the left side and another 3 squares from the bottom side.

We end up with 4 rectangles whose areas add up to 48 –  the area of the whole rectangle.

This tells us that

eqnExpand2Add4Times3Add5And the rule is

  • one at a time take each number in the left hand brackets
  • then also one at a time take each number in the right hand bracket
  • and multiply each of these pairs

With any four numbers (a,b,c,d) we can draw a rectangle with sides of length (a+b) and (c+d) and divide up the rectangle as we just did

The area of the whole rectangle will be the sum of the areas of the individual rectangles so


This is exactly the same rule for multiplying brackets that was given in words above, only this time its written more succinctly in letters.

One of the  first things that is met in algebra is how to work out

Now this is similar to eqnaAddbTimescAdddand it would be possible to just rewrite this replacing

  • c with a
  • d with b

But let’s give geometry a go first

The square with sides of length a+b has been split up into 2 squares and 2 rectangles

  • A square with sides of length a
  • A square with sides of length b
  • Two rectangles with sides of length a and b

So An explanation of how we get from second to third line

In algebra it is normal to

  • miss out  multiplication signs as algebra uses single letters to represent different numbers when 2 letters are written next to each it is assumed that they are multiplied (i.e. ab means a x b).
  • as the result of multiplying 2 numbers is the same whichever way they are written (i.e. a x b = b x a).  When 2 or more letters are multiplied they are written alphabetic order (i.e.  ba  is reordered to ab).
  • multiple occurrences of similar groups are collected together (i.e. ab + ab is written 2ab).
  • letters that are multiplied by themselves are written as squares, cubes etc.

This result may not seem like much but it takes us about half the way to proving Pythagoras’ theorem.  I am not going show how to do this in this post.

Instead I’m going to head back to Arithmetic and Multiplication and note that the rule

eqnExpandaAddbTimescAdddcan be used to multiply numbers such as 15 x 23 if we take the numbers in units and tens columns separately.

This is just what is now taught in primary schools as the ‘grid method’ though laid out differently.  Something like this would be more usual

It is important to understand what is happening when multiplying numbers, this is illustrated by the large rectangle divided into 4 smaller rectangles and the algebra

eqnExpandaAddbTimescAdddThe grid method, or any other method, is just a tool for performing the calculation. Once the idea of multiplying larger numbers by

  • splitting the numbers into parts
  • multiplying the parts
  • adding the results

All that matters is that the method helps calculating the answer

  • simply
  • accurately
  • reliably

Posted in Algebra, Geometry, Multiplication, Numbers | Leave a comment

Maths Is Different – Thats Not How We Do It – Number Bonds

It is commonly said maths is different nowadays.  Up and down the country when parents try and help their children with maths homework often

  • either it is incomprehensible to the parent
  • or when they show their children something the children cry out

“That’s not how we do it”

Is it really true?  One thing I know for certain is that

Another thing I know is that in most things there are fashions and fads.  Different approaches to teaching the same material goes in and out of fashion.  New names are invented for the same ideas.

[update ]

See these posts for number bonds worksheets

Number bonds of 9 worksheets

Number bonds of 10 worksheets

Number bonds of 20 worksheets

All of the posts have a link to an Excel spreadsheet which will generate new number bonds questions each time it is recalculated.

[end update]

Number Bonds,  what are they. Well according to Wikipedia they are another name for “addition facts”.  More helpfully for any number, 5 say, the number bonds of 5 are just the pairs of numbers that add up to that 5.

numberBonds_5It’s a bit like saying if we’ve got 5 cans of beans and 2 carrier bags we can

  • Put them all the cans in  one bag and leave the other empty.
  • Put 4 cans in one bag and just 1 in the other bag, perhaps because we were worried the bag with 4 cans in would split.
  • Or put 2 cans in one bag and 3 in the other, if we were trying to share the cans as equally as possible between the two bags.

For 3 it is even simpler

numberBonds_3It is not absolutely necessary to learn number bonds. You can add using your fingers and toes, or step along a ruler.  But doing this takes longer than if you have learned some ‘number bonds’ or ‘addition facts’ so you can do the sum in your head.

If you think of starting to learn to drive a car, there seems to a mass of different things that have to be done all at the same time.  After a while you can do these tasks automatically which frees your mind to concentrate on what is happening on the road.

In a similar way learning some basic facts, such as number bonds, so they can be recalled automatically,  frees your children’s minds to think about more complicated problems.

Whether or not you call them number bonds, the number bonds for 3 and 5 will be learned as part of learning to count up to 10 and to add the numbers o to 9.

Number bonds of 9 are worth learning as it just so happens if you add the digits (the numbers in units and tens columns) for the numbers in the 9 times tables they add up to 9.  As you can see from the following table.

Actually it is not usual to list the number bonds both ways, so I’ll stop doing it!

Number bonds of 10 (without writing out reversed pairs) are well worth learning, especially the pairs of digits (1 through 9) that add up to 10 as these introduce the idea adding a pair of single digit numbers can sometimes result in a two digit number.

numberBonds_10Some pairs of digits will add up to more than 10, though usually the next number for which number bonds are learned is 20.


Comparing adding

3Add7and the corresponding teens


13Add7helps introduce the idea of adding in columns.

In all three cases adding 3 and 7 result in 0 (in the units column)

  • When adding two single digits,  a 1 is put to the left to of the 0 in the answer.
  • If one of the numbers has a digit in ‘the tens column’  that digit  is made one bigger.
  • Note we’re not coping with adding two 2 digit numbers yet.

So all in all a new name for an old idea, but something that is useful to do.

And, just like learning to ride a bike or drive a car, this is something you learn by doing.

As a parent you can help by asking your child questions which make them think about number bonds.  Or you can just get them to put cans of baked beans into 2 bags!

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A Great Free Online Tool For Drawing Graphs Of Functions

This is a short plug for free online graphing calculator which is a great tool for exploring graphs of various functions.

You start with this form.


Add one or more lines by clicking on buttons to the right

onlineGraphics_formulaClick the Graph button at the bottom of form (not shown here)

And you get a graph like this.


This has only scratched the surface of what this tool can do.

Go here free online graphing calculator and have a play. If you have children them let them have a play too.

When I were a lad we had to do all this with graph paper and pencils!

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Tips For Helping Your Children Practice Learning Times Tables

A previous post Learning Times Tables talked about how to set about helping your children learn times tables. This post gives some simple tips for how you can help your children practice learning times tables.  These tips will be especially relevant when they are just starting  out.

Learning Times Tables gave a link which allows you to download your free copy of the appendices from my report Starting Arithmetic which contain worksheets I used for my children.

If you don’t wish to leave your name and email I’ve copied the spreadsheets to Google Docs and linked to them here.

You can cut and paste into Excel or OpenOffice Calc and print them out.

And if you don’t want to do that either you can download pdf files with the templates in here.

0 To 149 Table

Times Table Template

TimesTable Grid

I recommend you let your children use these to work out the times tables.

Then they have their own version of the tables which they have written out themselves.

When you first start you can ask them tables questions when they have their written sheets in front of them.  Gradually they will begin to learn the answers and be able to answer without the sheets.

People tend to think that when you answer a times table question, the answer just pops into your mind.  So you either know the answer or your don’t.

I believe this is completely wrong.  For a start what do you do if the wrong answer pops into your mind?  How do you know it’s the wrong answer?  What do you do if nothing pops into your mind?  Just wait?

I believe that one should learn a set of methods for working out the answer.  So either

  • You know the answer.
  • You are working on a method to work out the answer and can explain what you are doing.

If an answer does just pop into your head, great.  But use the methods to check its right before saying what it is.

The methods I am going to suggest here are extremely simple, so simple that you might think I’m joking, or that these are not really “methods” at all.

  1. Read the answer from written copy of the tables.
  2. Use a blank copy of the 0 to 149 table, or a ruler, to count forwards 1 at a time.
  3. Use a blank copy of the 0 to 149 table, or a ruler, to step forwards 3,5, 10 or whatever is the right number of steps for the table.
  4. Either count forward or step forward without using any table or ruler, once the numbers have been learned.

The first suggestion may seem like cheating, but if you were going to learn a poem or a speech you would start  by looking at the words in the poem or speech.

Counting forward is more useful when beginning (or when stuck) and when answering questions from 3 or 7 times table which are harder to step through.

Stepping is more natural for 5x, 10x, 11x.

When multiplying any pair of numbers the answer is the same which ever way you multiply them.  So


Or more generally, for any two numbers i and j

i_times_jSo firstly it makes sense to explain this.

Then ask questions both ways around and check you get the same answer each time.  In other words if you ask what is

2times3also ask what is

3times2It is a good idea to ask the same question more than once just to check that you get the same answer each time.

You can use this rule to switch the question to which ever way around seems easiest to answer. So if

3times5 seems hard see if

5times3is easier.

It will take some time to learn all the tables, but some children leave primary school after 6 or 7 years of teaching not knowing all their tables.  With you help I would be surprised if your children were not fully confident within 2 or 3 years, maybe less.

Remember it doesn’t take long just 5 minutes or so a few days a week.  If the sessions are regular they can be short, which is great as it leaves more time to play.

When I was at primary school there were 30+ children each year and one teacher taught 2 years at the same time (i.e. a single teacher was teaching 60+ children).  Everyday was started with 5 minutes of tables drill when he (and the teachers were largely he in those days). Would point to different people and fire off questions such as


Even if a question was answered every 10 seconds that was only 6 a minute, 30 in 5 minutes.  So not every one answered a question every day.  Yet those that didn’t answer, were in the room and heard the questions and answers so something sunk in.

I told this story recently to a current primary school teacher and she replied

Yes I know, it’s child abuse isn’t it

Oh dear.

 So my question to you is are you prepared to spend 5 minutes a day, a few days a week helping your children learn tables?

Children learn to speak from hearing their parents and others around them talk.  Suppose you never spoke to your child.  How well do you think they would learn to speak?

I’m just suggesting you talk tables for a few minutes some days each week.


Posted in Multiplication, Numbers, Times Tables, Times Tables Grid, Times Tables Templates | Tagged | Leave a comment

Learning Times Tables

We learn tables at school because we have to, it’s part of the curriculum.

Our success or lack of it may affect our esteem and how we are judged by teachers and others.  Ultimately it may affect our lives by contributing to our choice of career, whether we go to university or not, whether we succeed in business or investment.

Curiously, considering how much importance is put on learning tables, it’s not that difficult.

It is also something that people are much happier to do in everyday life, perhaps they just get put off by the school setting.

When was the last time you saw a darts match held up because no one could work out the score.  Or snooker, pool or scrabble for that matter.

James Martin, the TV chef, says from time to time that he never passed an exam in his life but he doesn’t seem to have trouble with his recipes whens scaling them up or down for more or less people.

Plumbers, builders, decorators, carpenters all get by.  Have you ever had any say

“I’d have like to give you a quote but I couldn’t do the multiplication”

Anyway getting back to tables.  When I was at primary school we learned from 1 x 1 to 12 x 12.  Nowadays the tendency is to stop at 10 x 10 as it’s easier.  But I prefer to stick with 12 x 12, if only to get practice with answers over 100.

In addition I like to include the 0x table, as it makes for some easy answers!

So we have a table like this, 169 different answers.
Let’s discount the 0x and 1x tables as they are so easy.

Also when multiplying any pair of numbers the answer is the same which ever way you multiply them.  So


Or more generally, for any two numbers i and j

i_times_jThis cuts the number of answers down it is necessary to learn to 66.   So in the table below we are cutting out the numbers shaded in blue and ignoring the 0x and 1x tables.


I printed 11 copies of this for my children, one for each of 2 through 12.
and got them to step through in steps of 2 to 12 circling each number they step on. Then write the step number by the circle.

Here is an easy example, stepping forward in 10s
10TimesFor each of the page they circled I gave them one of the following templates for them to copy the results to,  and they wrote out the times tables for themselves.


Download your free copy of the appendices from my report Starting Arithmetic
which contain these templates and much more.

I recommend you give your children blank templates and let them circle as they step through, then copy the results to the tables template. However Starting Arithmetic appendices contain the results of stepping forward in steps of size 2 to 12. It is instructive to page through the pdf file watching how the patterns change as you do so.

It is common to start with the 2 times table as it is ‘easiest’.  But in many ways 10, 11 and 5 times tables are easier.  You have seen the circle template for 10s here are 11s. From 11 x 1 to 11 x 9 is easy, 11 x 10 is in the 10 times table.

11Timesand 5s

5TimesThere is a simple pattern to the 9 times table from 9 x 1 to 9 x 9.  The units get one smaller and the tens get one bigger.  9 x 10 is in the 10s and 9 x 11 is in the 11s.


Just these 4 tables contain 35 of the 66 answers.  Over 1/2 way already!

You may have seen you children playing “Dizzy Dinosaurs”, where they put there arms out and spin round and round, often chanting as they do so.  5s and 10s are great for this as they are so regular that they can just go on and on, until they fall down!

In the following table I’ve broken down learning tables into groups.  It seems easiest to me to start with 10x, 11x, 5x and then 9x which covers over 1/2 the 66 answers.

At school children will probably be learning 2x 3x and 4x tables and there is no reason why you can’t or shouldn’t help with these too.

Table To Learn At This Stage Number To Learn Total So Far Number Left
10 x2 to x12 11 11 55
11 x2 to x9 8 19 47
5 x2 to x9, x12 9 28 38
9 x2 to x4
x6 to x9
7 35 31
2 x2, x3, x4 3 38 28
3 x3, x4 2 40 26
4 x4 1 41 25
2 x6 x7 x8 3 44 22
3 x6 x7 x8 3 47 19
4 x6 x7 x8 3 50 16
9 x12 1 51 15
11 x11 1 52 14
12 x11 x12 2 54 12
12 x2 x3 x4
x6 x7 x8
6 60  6
6 x6 x7 x8 3 63  3
7 x7 x8 2 65  1
8 x8 1 66  0

Once your children have learned 2x 3x 4x by 6,7,8 they’re nearly 2/3 done.

There are the 4 numbers above 100


It helps if you have a little story to tell, for example

  • 12 x 12 is 144 which is the largest number in the times tables
  •   9 x 12 is 108 which is like 18 only with a 0 in the middle

11×11 and 11×12 can be worked out by stepping forward in steps of 11 from 11×10.

Next the parts of the 12x table not yet learned.  One way of working out the 12x tables is to find the answer for 11x  then add on, for example

11 x 3 = 33
33 + 3 = 36


Finally the 6 answers some children find the hardest of all.


These can be worked out by counting on from x5, once 5x table has been learned.

So that’s it then – it’s all easy?

There has to be a catch.

Well yes there is, in fact there are two.

The first is we tend to forget what we have learned, anyone who has ever crammed for an exam knows that.  To really learn facts you have to revise over and over again.   But the good news is it doesn’t take much.   Just a few minutes regularly is enough.

You can help you children learn by asking them questions about tables.  In this way you relieve them of the burden of what to learn and revise.  The downside is you take up this load.  But the good news is you only need a few minutes a day.

Think of it this way, how good would your children’s speaking be if you never spoke with them at home, as that was something schools did?

It’s ridiculous isn’t it.

Why should maths and tables be any different.

To start with choose one table, say the 10x, let your children have the sheets they circled. Then ask them some questions. After a bit ask them questions without the sheets and if they can’t answer immediately encourage them to step through the tables (10 20 30 40 …).

Answering questions helps build and strengthen the associations (between questions and answers), which are the basis of learning. Short sessions means it should never get too much. Regular sessions help ensure what has been learned is not forgotten, but rather is consolidated.

Once your children have learned one table move on to the next, but keep asking questions about the tables they have already learned in case they become  forgotten.

Posted in Counting, Numbers, Times Tables, Times Tables Grid, Times Tables Templates | Tagged , | 1 Comment

Some Simple Mistakes Children Make With Fractions

I have read that when adding fractions some children make the mistake of adding both the top and the bottom, for example

frac{1}{6} + frac{1}{6} = frac{2}{12}

Now it may be that these children were not taught very well.

Or maybe they have not had enough practice with adding fractions.

Probably the most useful piece of advice with maths is to think of a physical example.

I like to think of cake.

If a cake is cut into 6 parts and you eat 1 part and then another part how many have you eaten?

Most children can get that right, although they might think the answer is so obvious that you’re a bit silly for asking.

A slightly different questions is

If a cake is cut into 6 parts and you eat 1 part and then another part how many parts of the whole cake have you eaten?

2 parts of the 6 parts that made up the cake.

When your children are learning fractions (or any part of maths) encourage them to think of physical examples for the questions they have to answer.

When talking about fractions explain that we start with a whole of something (cake) then split it into parts.  When we add fractions we are counting the number of parts, but we need to remember how many parts the whole was split into. So when we say

one sixth and one sixth

this is short hand for

one sixth part of the whole and one sixth part of the whole

it should be more obvious that the answer is two sixth parts of the whole.

To make it more obvious why not get a real cake and cut it up!

Once your children understand what is meant they should find it easier to understand

frac{1}{6} + frac{1}{6} = frac{2}{6}

is just a short hand way of writing this down.  The number on the bottom is the number of equal parts the whole (e.g cake) was divided into, the number on the top is the number of these parts.


In English there are two sets of names for numbers,  Cardinals and Ordinals. Cardinals are just ordinary numbers.  Ordinals are the names we use for the position of something, say in a race.  Most of these names come from Anglo Saxon but a few from Latin.

Cardinal Ordinal Fraction
one first
two second half
three third third
four fourth quarter
five fifth fifth
six sixth sixth

The first few numbers are a bit irregular but from 6 onwards there is a pattern

  • The cardinal is just the ordinal with “th” at the end.
  • The fraction is the same as the cardinal.

Second and quarter come from Latin, the rest from Anglo Saxon.  Half was special even in Anglo-Saxon.

It is not hard to imagine that fraction started out as something like

One equal part of six

which over time changed to

One part of six, or one sixth part

and finally was abbreviated to just

One sixth

Even if this was not how the names of fractions developed it is a story, and stories are great ways to convey ideas.  The idea we want to  convey is that fractions are about parts of a whole.  When we add fractions we just count the number of parts, the size of the parts never changes.

Maths, at least at school level, is a means of calculating answers by manipulating symbols rather than stuff.  The rules have been designed to give the same answer as you would get if you actually did manipulate stuff.  It’s worth explaining this to your children so they understand it’s not a collection of squiggles but a description of what would happen in the real world.  For example, cutting a piece of cake into 6 equally sized pieces and then taking two of them.



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A Simple Easy Way To Help Your Children Learn To Count – Snakes And Ladders

I love snakes and ladders (or chutes and ladders) as it’s a great way for children to learn numbers up to 100 and to count forwards and backwards which introduces adding and subtraction.

It also introduces the idea that things can go down as well as up, and the fastest doesn’t always win.

To start it’s not necessary to know or be able to say the numbers.
You just throw the dice and take the number of steps that there are dots on the top side of the dice.

If you are playing with your children you can count as they take each step

1, 2, 3, 4…

Once they get the hang of that you can say the number of the squares as they step through. So if they are on square 20 and throw a 4.

21,  22,  23,  24

The knowledge just seeps in.

You may have heard of the elusive obvious.  Something right out in the open staring you in the face, something you and everyone else have seen all your life, something so obvious that it gets ignored.

When speaking english  if  you see 10, you say the word ten, you don’t say ‘one’ ‘zero’.  For all the numbers on the board, 1 to 100,  everyone says the names of the numbers in words one, two, three … ninety nine, one hundred.

This means that people get used to seeing a single digit (one of 0,1,2,3,4,5,6,7,8,9)  for the numbers zero through nine.  Two digits for ten through ninety nine, and three digits for one hundred.  It is not necessary to explain about the ‘place value system’ or units, tens and hundreds columns.

There are several versions of the ending the game.

  1. If you throw enough to get to, or past 100, you finish.
  2. If you throw exactly enough to get to 100 you finish, if not stay still.
  3. Count forward to 100.  Stop if your move ends on 100, if you threw a larger number then you step backwards to complete your move.

I prefer the 3rd version as

  • It prolongs the game
  • It doesn’t involve not moving for a turn
  • It involves counting backwards

In UK  you can buy Snakes & Ladders here

Why not make your own?

  1. You only need a piece of paper or card board.
  2. Draw 10 x 10 squares
  3. Mark 1-10 going left to right, 11 to 20 going right to left, and so on
  4. Put as many or as few snakes and ladders as you like

My board looks like this


The snakes and ladders go from

Ladders Snakes
2 to 38 98 to 78
4 to 14 95 to 75
8 to 31 93 to 73
21 to 42 91 to 71
28 to 84 87 to 24
51 to 67 64 to 60
71 to 91 62 to 18
80 to 100 56 to 53
49 to 11
47 to 26
16 to 6

I don’t know why there are more snakes than ladders, but it will help to make the game last longer.

If you haven’t got dice make one.

Or make a hexagonal top. Learn how to fold a piece of paper into to hexagon here
Then push a stick or used match through the middle of the hexagon so it can spin.

Or get empty bottle and write numbers 1 to 6 (or dots 1 to 6)  around the circumference.
Roll the bottle and see which number is on top.

Recently Unicef and IpsosMori published a report saying children need time with parents not more stuff (report summary).

OECD PISA studies publish league tables of how well children in different countries are performing at school. These show children from the UK (and US) dropping down the league tables in maths.

Why not kill two birds with one stone. Spend a little time playing games with your children it might be fun and their maths will improve too.




Or download a Snakes and Ladders printable template for free from here.

Posted in Counting, Numbers | Tagged , , , , , , , | 1 Comment

You Can’t Answer the Question If You Can’t Say The Answer

In order to answer a question you have to be able to say the answer. Sometimes when helping our children learn something new, like fractions, we may forget that not only is the idea new to them but so are the words. Children need to become familiar with these new words in order to answer questions.

It takes time to become accustomed and familiar with new words and phrases. It would be nice if people could be told something once, then be able to recall and use this information perfectly forever more. But it doesn’t often happen.

The names of fractions in English are somewhat irregular, at least to start with because English, as it developed, absorbed words from more than one other language (usually Anglo Saxon or Old English and Latin).

For example half and third are derived from the Old English words ‘healf’ and ‘pridda’. But quarter comes from the Latin ‘quartus’. Next comes fifth, not ‘fiveth’, which comes from the Old English ‘fifta’. After this, things settle down with sixth seventh and so on. But it is still necessary to learn the exceptions.

Another common difficulty is saying remainders and improper fractions. For example five divided by four is one remainder one or one and one quarter. It is very common for children to be confused at the two ones in one remainder one and in one and one quarter. After all these are not phrases that they are likely to have used before. In contrast by the time children learn to count at school they are likely to have used the words one, two and three many times. (How many ice creams would you like?). But one and one quarter is more of a tongue twister.

Another difficulty comes with converting kilograms to grams, or kilometers to meters. This is often thought of as a math problem. But it is in largely a speaking problem. There are two parts.

Firstly just repeat exactly what is said but replace kilo with thousand and thousand with kilo.
So two kilograms becomes two thousand kilo grams.
And three thousand grams becomes three kilo grams.

It really is that simple!

Secondly, learn the how to say fractions of a thousand. So 1/2 a kilogram is five hundred grams which means five hundred grams is half a kilogram.

Children seem to be comfortable with saying two thousand five hundred grams for 2 1/2 kg and three thousand five hundred grams for 3 1/2 kg.

But ten thousand five hundred grams for 10 1/2kg
or one hundred thousand five hundred grams for 100 1/2kg
seem to be more difficult to say, until they become familiar.

For some reason it seems that saying one thousand five hundred grams for 1 1/2 kg is takes longer to become familiar with.

By the time they are comfortable with saying and one hundred and one thousand five hundred grams for 101 1/2 kilograms you can be confident that they have got to grips with this particular type of problem.

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