Number Bonds

Students who struggle with addition and subtraction often cope by counting on by ones - a safe, reliable, but very inefficient strategy and one that must be replaced in order for them to be successful when adding and subtracting numbers.

Moving On From Counting On

It is important that students move on from thinking about numbers as only collections of single units. For example, helping them see 5 as being a 3 and a 2 will allow them to add 5 to 7 add by first adding the 3 to make 10 and then adding the 2 to get the answer of 12. This is a more efficient strategy than starting at 7 and counting on 5 by ones. The activities and worksheets below will help move students beyond just counting by ones. There is more on making 10 here.

The worksheets below are based on the concept of Cuisenaire Rods and they provide practice activities as a transition from the concrete through pictorial to the abstract. e.g. 3 + 2 = 5

Cuisenaire Rods

white (1) cuisenaire rod 1
red (2) cuisenaire rod 2
light green (3) cuisenaire rod 3
magenta (4) cuisenaire rod 4
yellow (5) cuisenaire rod 5
dark green (6) cuisenaire rod 6
black (7) cuisenaire rod 7
brown (8) cuisenaire rod 8
blue (9) cuisenaire rod 9
orange (10) cuisenaire rod 10

If you are able to buy a set of Cuisenaire Rods then using them with your children will help them see numbers as more than just a group of ones. You will find more about Cuisenaire Rods hereexternal link icon. As the graphics below show, these rods can be arranged to support the idea of numbers being made up from two components.

What are Number Bonds?

"Number bonds" is a term for components of numbers or number pairs shown pictorially. Students should be able to recall number pairs for all numbers up to 10 as this will greatly help with mental calculations. The number bonds for 3, 4, 5, 6, 7, 8, and 9 are shown below using cuisenaire rods.

3
light green (3) cuisenaire rod
red (2) cuisenaire rodwhite (1) cuisenaire rod
white (1) cuisenaire rodred (2) cuisenaire rod
light green (3) cuisenaire rod

 

4
magenta (4) cuisenaire rod
light green (3) cuisenaire rodwhite (1) cuisenaire rod
red (2) cuisenaire rodred (2) cuisenaire rod
white (1) cuisenaire rodlight green (3) cuisenaire rod
magenta (4) cuisenaire rod
5
yellow (5) cuisenaire rod
magenta (4) cuisenaire rodwhite (1) cuisenaire rod
light green (3) cuisenaire rodred (2) cuisenaire rod
red (2) cuisenaire rodlight green (3) cuisenaire rod
white (1) cuisenaire rodmagenta (4) cuisenaire rod
yellow (5) cuisenaire rod

 

6
dark green (6) cuisenaire rod
yellow (5) cuisenaire rodwhite (1) cuisenaire rod
magenta (4) cuisenaire rodred (2) cuisenaire rod
light green (3) cuisenaire rodlight green (3) cuisenaire rod
red (2) cuisenaire rodmagenta (4) cuisenaire rod
white (1) cuisenaire rodyellow (5) cuisenaire rod
dark green (6) cuisenaire rod
7
black (7) cuisenaire rod
dark green (6) cuisenaire rodwhite (1) cuisenaire rod
yellow (5) cuisenaire rodred (2) cuisenaire rod
magenta (4) cuisenaire rodlight green (3) cuisenaire rod
light green (3) cuisenaire rodmagenta (4) cuisenaire rod
red (2) cuisenaire rodyellow (5) cuisenaire rod
white (1) cuisenaire roddark green (6) cuisenaire rod
black (7) cuisenaire rod
8
brown (8) cuisenaire rod
black (7) cuisenaire rodwhite (1) cuisenaire rod
dark green (6) cuisenaire rodred (2) cuisenaire rod
yellow (5) cuisenaire rodlight green (3) cuisenaire rod
magenta (4) cuisenaire rodmagenta (4) cuisenaire rod
light green (3) cuisenaire rodyellow (5) cuisenaire rod
red (2) cuisenaire roddark green (6) cuisenaire rod
white (1) cuisenaire rodblack (7) cuisenaire rod
brown (8) cuisenaire rod
9
blue (9) cuisenaire rod
brown (8) cuisenaire rodwhite (1) cuisenaire rod
black (7) cuisenaire rodred (2) cuisenaire rod
dark green (6) cuisenaire rodlight green (3) cuisenaire rod
yellow (5) cuisenaire rodmagenta (4) cuisenaire rod
magenta (4) cuisenaire rodyellow (5) cuisenaire rod
light green (3) cuisenaire roddark green (6) cuisenaire rod
red (2) cuisenaire rodblack (7) cuisenaire rod
white (1) cuisenaire rodbrown (8) cuisenaire rod
blue (9) cuisenaire rod

 

3
2 + 1 = 3
1 + 2 = 3
 
 

 

4
3 + 1 = 4
2 + 2 = 4
1 + 3 = 4
 
5
4 + 1 = 5
3 + 2 = 5
2 + 3 = 5
1 + 4 = 5

 

6
5 + 1 = 6
4 + 2 = 6
3 + 3 = 6
2 + 4 = 6
1 + 5 = 6
7
6 + 1 = 7
5 + 2 = 7
4 + 3 = 7
3 + 4 = 7
2 + 5 = 7
1 + 6 = 7
8
7 + 1 = 8
6 + 2 = 8
5 + 3 = 8
4 + 4 = 8
3 + 5 = 8
2 + 6 = 8
1 + 7 = 8
9
8 + 1 = 9
7 + 2 = 9
6 + 3 = 9
5 + 4 = 9
4 + 5 = 9
3 + 6 = 9
2 + 7 = 9
1 + 8 = 9

Commutative Law

Ensure your children are familiar with the commutative law for addition. They do not need to know actual name of this law but they must be able to use it. In other words, they if they find that 4 + 2 = 6 then they should know that 2 + 4 = 6. This law allows the 35 number pairs above to be reduced to only 19 that need to be quickly recalled.

The worksheets below are based on the concept of Cuisenaire Rods and they provide practice activities as a transition from the concrete through pictorial to the abstract. e.g. 3 + 2 = 5

Number Bonds Worksheets

Complements of 10

Quickly recalling the complements of 10 will greatly assist your children with addition and subtraction. Thanks to the Commutative Law, there are only five facts (six if you include 10 + 0 = 10) that need to be memorized; 9 + 1, 8 + 2, 7 + 3, 6 + 4, and 5 + 5. Knowing these facts will really help when students move on to add and subtract with numbers above 10 when they will "make" 10 and multiples of 10.

Complements of 10
orange (10) cuisenaire rod 10 + 0 = 10
blue (9) cuisenaire rodwhite (1) cuisenaire rod 9 + 1 = 10
brown (8) cuisenaire rodred (2) cuisenaire rod 8 + 2 = 10
black (7) cuisenaire rodlight green (3) cuisenaire rod 7 + 3 = 10
dark green (6) cuisenaire rodmagenta (4) cuisenaire rod 6 + 4 = 10
yellow (5) cuisenaire rodyellow (5) cuisenaire rod 5 + 5 = 10
magenta (4) cuisenaire roddark green (6) cuisenaire rod 4 + 6 = 10
light green (3) cuisenaire rodblack (7) cuisenaire rod 3 + 7 = 10
red (2) cuisenaire rodbrown (8) cuisenaire rod 2 + 8 = 10
white (1) cuisenaire rodblue (9) cuisenaire rod 1 + 9 = 10
orange (10) cuisenaire rod 0 + 10 = 10

The worksheets and games below provide practice with the complements of 10.

Worksheets

Games

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