Maths Without Limits
Opening Young Minds to Endless Possibilities

 Why do they do sums this way instead of this way? INTERESTING FACT:  Children who are held back from learning the written methods for addition and subtraction, and are first encouraged to work things out mentally, become much more confident with numbers! With the written method you can make silly mistakes if you don’t think carefully.. Adding up the columns is fine with a sum like this… but doesn’t work so well with a sum like this one… where you will often get mistakes like this! What??  How do children get that?  Simple!  4 and 3 makes 7.           5 and seven makes 12.         Answer 712!  And you get even more problems with subtraction. Because of this, many schools now put a strong emphasis in the early Primary School years on mental calculation skills and delay the teaching of traditional written methods until the children have a firm understanding of the number system.   Written Recording of Mental Calculations Even when you are learning to work things out in your head, it is still important to get into the habit of showing your working.  So you need a way of writing things down. The horizontal recording is designed to encourage a child to think about the calculation process: you can’t just add up the columns, because there aren’t any! So if your child’s school is teaching horizontal recording, they will be very happy if you do NOT try to show your child the traditional vertical method.  It will come eventually. You just have to be patient!   How is the horizontal recording taught? Isn’t it much more difficult? The secret to all good teaching is building up from easy skills to harder ones step by step. Before learning to do calculations like the one above, your child will first have to become secure with simpler calculations. It works something like this:Step 1: Learn to add single-digit numbers. (5 + 7 = 12)Step 2: Learn to add tens. (40 + 30 = 70)Step 3: Learn to add a single-digit to a two-digit number (45 + 7 = 52)Step 4: Learn to add a tens number to a two-digit number (45 + 30 = 75)Step 5: Learn to add two two-digit numbers together. (45 + 37 = 82) It sounds complicated at first, doesn’t it? But children who learn this way, and become secure at each step, become SO much more confident with manipulating numbers later on. The process works exactly the same way with subtraction, so many schools will teach the addition and subtraction process together, emphasising the links between the two. ( 12 – 5 = 7 because 7 + 5 = 12 etc) If you want to support the learning process at home, the games that follow will help your child at each of the different stages.

Step 1: Adding and subtracting single-digit numbers

Did you know that children use different strategies for adding and subtracting, depending on their level of skill and understanding of numbers?  And you can bet that the strategies they choose will determine how quickly they can work things out.

The games in this section will help your child to become skilled in the most effective methods.

Counting on and Counting Back

When children first learn to add and subtract they will learn to count two groups of objects together, counting all the objects beginning at 1. A quicker method is to count on or count back from the first number. For example, if you are adding 6 and 5 then you THINK 6 and BEGIN COUNTING from 7: To do this accurately a child needs to be able to begin counting from any number (without starting from 1) and know not to count the number that you start on (in this case 6) but to begin counting from the next number (in this case 7).

The Envelope Game will help with this.

Number Bonds: Using Known Facts

Children who find it easy to learn number facts will discover that a much more efficient way to add and subtract is to learn the number bonds.

4 + 7 = 11
5 + 6 = 11
6 + 5 = 11

etc

Your child’s school will spend time teaching these during the early years in Primary School.

Some children are lucky in that they find it easy to commit facts to memory. Others find this more of a challenge. But even the most talented of us cannot learn all the possible number facts off by heart!  The secret of being good at maths is to learn certain key facts and to know how to use them to work out others.

6 + 6 = 12

so

6 + 7 = 13

etc

Number Stories

The most important addition and subtraction facts to learn by heart are the number pairs that add together to make 10.

 10 + 0 = 109 + 1 = 108 + 2 = 107 + 3 = 106 + 4 = 105 + 5 = 104 + 6 = 103 + 7 = 102 + 8 = 101 + 9 = 100 + 10 = 10 10 – 0 = 1010 – 1 = 910 – 2 = 810 – 3 = 710 – 4 = 610 – 5 = 510 – 6 = 410 – 7 = 310 – 8 = 210 – 9 = 110 – 10 = 0

The complete set of facts for 10 is know as the ‘Story of Ten’.

If you know these really well then you can use them to help you with other calculations:

2 + 8 = 10    so   20 + 80 = 100

4 + 6 = 10    so   10 – 6 = 4

Your child will also learn the stories of other numbers (story of 6, story of 11 etc).  These are useful, but the most important is the Story of 10.

Also very important are the ‘doubles’ and the ‘teens’ number facts.

 Doubles 0 + 0 = 01 + 1 = 22 + 2 = 43 + 3 = 64 + 4 = 85 + 5 = 106 + 6 = 127 + 7 = 148 + 8 = 169 + 9 = 1810 + 10 = 20 Teens Number Facts 10 + 0 = 1010 + 1 = 1110 + 2 = 1210 + 3 = 1310 + 4 = 1410 + 5 = 1510 + 6 = 1610 + 7 = 1710 + 8 = 1810 + 9 = 1910 + 10 = 20

Tap, Say, Turn Number Bonds is a good game for consolidating any facts that your child is finding difficult.

Missing Numbers

OK, so now we know how to add and subtract small numbers, we think!

And then we meet those nasty questions like

15  –  ?  =  7

or, worse still,

?  –  8  =  7

Number tracks, which show the numbers in order, can be really useful here.

Find the Treasure is an excellent game using number tracks which will REALLY help with these kinds of problems.  Do give it a go!

Practise, Practise, Practise!

Once learned, any skill needs practice. Sums and Differences is a great little game for practising all the addition and subtraction skills within 20.

 Step 2: Adding and subtracting tens Once you are secure with the number bonds to ten, you can use these facts to help add and subtract larger numbers. 3 + 7 = 10 so  30 + 70 = 100  etc But first you need to be confident with the number sequence up to 100.   A lot of early mathematical activities in school will involve counting or ordering the numbers. You can support these at home using equipment which is very easy to make. The Counting Stick is a wonderfully versatile tool for making children comfortable with the order of any size of numbers.  It is brilliant for learning the positions of all the numbers to 100 and for estimating.  It is definitely worth making one of these. All you need is a strip of wood marked off into ten equal sections. A trip to your local hardware store for some 2 by 1 will do the trick nicely. If you get the piece cut to exactly 1m long, you can use it for measuring activities as well. There are several excellent activities you can play using a couple of metre sticks. Schools purchase these through catalogues, but you can easily make your own in the same way as a counting stick. Divide the stick into ten equal sections again and mark in the tens numbers 0, 10, 20 etc. Then put smaller marks to show the individual centimetres, but only write in the 5s numbers (ie write in 5, 15, 25, etc but not 1, 2, 3, 4, 6, 7, etc. Two Hands One Piano is really good for consolidating the positions of all the numbers from 0 to 100.  Back to Back Metre Sticks is excellent for learning the pairs of numbers that add to make 100.  Metre Stick Jumps practises addition and subtraction of tens within 100.   40 + 30 = 70 80 – 20 = 60 etc When you are ready to venture beyond 100, Wake Up Chum is a hilarious game which teaches the crucial skill of counting over from one hundred into the next and back again. 80, 90, 100, 110, 120  etc Again it is very easy to make.
 Step 3: Two-digit numbers plus or minus single digits  When we are confident with these: 9 – 3 = 6 5 + 7 = 12 then alongside learning these: 90 – 30 = 60 50 + 70 = 120 we can learn how to do these. 47 + 6 = 53 53 – 6 = 47 The Maths programmes used in many schools support children’s mental addition and subtraction skills in three main ways: counting on and counting back exploring number patterns using the blank number line   Counting On and Counting Back Counting on and counting back work the same as they do with single-digit numbers. Example:  47 + 6 Think 47 and then count: 48, 49, 50, 51, 52, 53. Example: 53 – 6 Think 53 and then count backwards: 52, 51, 50, 49, 48, 47   Number Patterns Children’s understanding of how addition and subtraction work can be built up very effectively by exploring patterns.   Example: 7 + 6 = 13 17 + 6 = 23 27 + 6 = 33 37 + 6 = 43 etc The exploration of number patterns is fully developed in the online patternbuilders.   The Blank Number Line An important part of becoming confident with Maths is learning to visualise the number line in your head.  As a step on the way to this, you can draw a blank number line and use this to help with addition and subtraction.  Here’s how it works. Example:  47 + 6 Example:  53 – 6 All of these ideas can be explored at home using the metre stick.  No specific instructions are given here for this, since it is expected that you will want to leave the school to do this bit, but if you want to explore the ideas with your child at home, then read through the guidance for parents on Tips for Maths at Home.
 Step 4: Two-digit numbers plus or minus tens numbers When we are confident with these: 90 – 30 = 60 50 + 70 = 120 and these: 47 + 6 = 53 53 – 6 = 47 then we can learn how to do these. 47 + 30 = 77 77 – 30 = 47 We can use exactly the same techniques as with the last step. Explore Number Patterns Example: 47 + 10 = 57 47 + 20 = 67 47 + 30 = 77 47 + 40 = 87 etc   The Blank Number Line Example:  47 + 30 Example:  77 – 30 Mental Counting On and Counting Back Example:  47 + 30 Think 47 and then count: 57, 67, 77. Example: 77 – 30 Think 77 and then count backwards: 67, 57, 47
 Step 5: Two-digit numbers plus or minus two-digit numbers Finally, when we are confident with these: 47 + 6 = 53 53 – 6 = 47 and these: 47 + 30 = 77 77 – 30 = 47 then we can learn how to do these 47 + 36 = 83 83 – 36 = 47 Example:  47 + 36 Start with 47.  First add the tens, then add the units: 47 + 30 = 77 77 + 6 = 83   Example:  83 – 36 Start with 83.  First subtract the tens, then subtract the units: 83 – 30 = 53 53 – 6 = 47   Notice that when we do things mentally, the most natural way is to deal with the tens first (since they are the biggest thing) and then deal with the units.  This is the opposite to when we do written ‘sums’ where we always do the units first.  That’s it! Hopefully, having read this far you will now be much more confident in helping your child with adding and subtracting!