Presented by John Santos

What is tonight all about?• The aim of tonight’s Maths session is to clear up some

of the confusion that surrounds the way we teach some maths concepts in today’s classrooms.

• As parents, we have all experienced the situation where our child has asked for help with their maths homework and when we showed them ‘the way we did it at school’, they stared at us with a look of total confusion, or said “That’s not right!”

• After tonight ‘s presentation, I hope you will feel better equipped and have greater confidence when you next assist your child with their Maths. • The areas we will focus on are:Addition and Subtraction mental strategies. tradingMultiplication and division mental strategies. Multiplication and division algorithms

The Language of Addition and Subtraction+ -

plus minus

add take away

addition subtraction

and difference

total remaining

altogether what’s left

added to more than

sum of change from $

how much remove

join groups give away

combine groups

difference

Addition and Subtraction Developmental Levels

There are 5 main developmental levels that children move through on their way to becoming independent and efficient addition and subtraction problem solvers.

Addition and Subtraction Developmental Levels

1 3 7 4 5

1 2 3 4 5 6 7 8 9 10

A child at this level may or may not be able to count from 1 to 10. The child cannot count objects correctly.

Emergent

A child at this level needs to see or touch the groups of objects and counts each object one at a time.

1 2 3 4

1 2 31 2 3 4 5 6 7

Perceptual

A child at this level can build a picture of objects in his/her head and will count each pictured object one at a time, starting from one

1 2 3 4

1 2 3 1 2 3 4 5 6

7

Figurative

Counting OnAt this level, a child can add two

numbers by holding the larger number in their head and counting on by ones e.g. 10 + 5 =

10 11, 12, 13, 14, 15

A child at this level counts by numbers other than one, and uses strategies such as the Jump, Split and Compensation.

Using mental strategies

Friends of TenTo develop their understanding

of our number system it is very important that children learn their addition and subtraction combinations to 10.

This means the numbers that add together to total ten. We call these the ‘Friends of Ten’, for example 3 and 7, 2 and 8, 5 and 5 etc.

Bridging to Ten 6 + 8 = 14

6

+4

10

+4

14

6 + 4 = 10, 10 + 4 = 14I know my ‘friends of 10’ so I know that 6+ 4 = 10. That still leaves 4 to add so I add that to 10 and get 14.

+ 20

63 + 20 = 83, 83 + 9 = 92I kept the 83 whole and split the 29 into 20 and 9. Then I added 20 to 63 and got 83. Then I added the 9 and got 92.

Jump Strategy

63 + 29 =

+ 9

63 83 92

Bridging to Ten / Jump Strategy

6 + 14 = 20

6 + 4 = 10, 10 + 10 = 20First I added 4 to the 6 to get 10, then I added another 10 and got 20.

+4

6

+10

10 20

Jump Strategy / Bridging to Ten

63 + 20 = 83, 83 + 7 = 90, 90 + 2 = 92I kept the 83 whole and split the 29 into 20 and 9. Then I added 20 to 63 and got 83. Then I added 7 because 3 and 7 make a ten and got 90. Then I added the other 2 and got 92.

63 + 29 =

+20 +7 +2

63 83 9092

+20

60 + 20 = 80, 3 + 9 = 12, 80 + 12 = 92I split the 63 into 60 and 3, and the 29 into 20 and 9. Then I added the 60 and the 20 and got 80. Then I added the 3 and the 9 and got 12. Then I added the 80 and the 12 and got 92.

Split Strategy

63 + 29 =

+3 +9

60 8083 92

63 + 30 = 93, 93 – 1 = 92First I added 30 to 63 because 29 is nearly 30 and it’s easier to add tens. I got 93. Then I had to take one away because 30 is one more than 29 and I got 92.

63 + 29 =

+ 30 -1

63 92 93

Compensation Strategy

Addition Algorithm Procedure

+ 29

2We say:3 plus 9 equals 12, write down the 2 and trading one 10 to the tens column. 6 plus 2 equals 8, plus the 1 equals 9.

63¹

9

•The older a person gets the more they will use mental strategies to solve mathematical problems. •Accordingly, it is important that children develop a conceptual understanding of mental strategies and place value rather than rely solely on algorithms.

Decomposition4

1 8

We say:2 minus 8 you can’t do so we trade a ten from the tens column. Now my 2 is 12. 12 minus 8 you can do. It leaves 4. Write down the 4.4 minus 1 equals 3. Write down the 3.

3-

Subtraction Algorithm Procedures

2¹5

4

Decomposition with Zeros

8 0 0 06

Subtraction Algorithm Procedures:

- 37

9 917

10-3=7, 9-7=2, 9-6=3, 7-0=7 ….

Oh forget it! Let’s just use the compensation strategy …….

We say:0 minus 3 you can’t do. So I need to get a ten from the tens column but there aren’t any. So I need to get a hundred from the hundreds column to give to the tens column but there aren’t any. So I can get a thousand from the thousands column to give to the hundreds column. That leaves 7 in thousands column and 10 in the hundreds column. I give one hundred to the tens column. That leaves 9 in the hundreds column and 10 in the tens column. NOW I can give a ten from the tens column to the ones column …..

1 1

7 723

7 9 9 9 - 6 7 3

7 3 2 6

Change the 8000 into 7999 + 1.Subtraction Algorithm Procedures: Compensation

7326 + 1 = 7327

51 8

3We say:2 minus 8 you can’t do so we add a ten to the ones column in the top number and a ten to the tens column in the bottom number.Now my 2 is 12. 12 minus 8 you can do. It leaves 4. Write down the 4.5 minus 2 equals 3. Write down the 3.

-

Subtraction Algorithm Procedures

Equal Addends

2¹

4¹

The Language of Multiplication and Division

X ÷multiply divide

equal groups equal share

times equal groups

multiples equal parts

factors quotient

equal rows remainder

array equal rows

double, triple array

product fraction

percentage

Multiplication and Division Levels

Forming Equal Groups

A child at this level will be able to form the objects into equal groups but will then count each object by ones, to find the total number of objects.

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

3 6 9

Multiplication and Division Levels

PerceptualA child at this level will be able to form the objects into equal groups and will skip count or rhythmically count while looking at or touching the counters, to find the total.

A child at this level will not need to see the individual objects but will need a marker for each group. They will be able to skip count or rhythmic count to find the total number of objects represented by the group markers.

1 2 3 4 5 6 7 8 9

3 6 9

Multiplication and Division LevelsFigurative Units

A child who is at this level will use known facts or doubles.

2 x 7 = 14, 2 x 14 = 28, 2 x 28 = 56

4 x 7 = 28, so 8 x 7 = 56

I know 7 x 7 = 49, so 7 x 8 = 56

Multiplication and Division LevelsRepeated Abstract Composites

8 x 7 =

Multiplication and Division Levels

Multiplication & Division as Operations

7 x 8 = 56

Arrays3 rows of 4

makes 12

4 + 4 + 4 = 12

4 x 3 = 12

3 x 4 = 12

12 ÷ 4 = 3

12 ÷ 3 = 4

x 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

Mental Strategies: Multiplication and Division

26 x 4 =

26 + 26 = 52, 52 + 26 = 78, 78 + 26 = 104

Repeated Addition

I added 26 and 26 and got 52. Then I added another 26 and got 78. Then I added the 4th 26 and got 104.

0 26 52 78 104

+26 +26 +26 +26

Doubling

Mental Strategies: Multiplication and Division

26 x 4 =

Double 26 = 52. Double 52 = 104

I doubled 26 and got 52. Then I knew I needed another 2 26s which I knew was another 52 so I doubled 52 and got 104.

0

Double 52Double 26

26 52 104

Mental Strategies: Multiplication and Division

26 x 4 =Compensation Strategy

4 x 25 = 100, 100 + 4 = 104

I knew that 25 times 4 is 100. Then I needed 1 more 4 to make 26 4s. So 100 plus 4 made 104.

0

4 x 25

104100

Split Strategy

Mental Strategies: Multiplication and Division

26 x 4 =

4 x 20 = 80, 4 x 6 = 24 then 80 + 24 = 104

I knew that 26 was made of 20 plus 6 20 times 4 is 80 6 times 4 is 24.

80 plus 24 is 104

0

4 x 64 x 20

10480

The Multiplication Algorithm:Extended Form

We say:4 times 6 equals 24, write down the 24. We write a zero in the ones column. Then we say 4 times 2 equals 8, and write it in the tens column. We then add 4 and 0 to equal 4 and 2 and 8 to equal 10.

26

X 424

80

104

The Multiplication Algorithm:Contracted Form

104

² 26 X 4

We say:4 times 6 equals 24, write down the 4. and carry the 2. 4 times 2 equals 8, plus the 2 equals 10. Write down the 10.

When solving an algorithm, we treat each digit as a ‘one’, even

the ‘tens’ and ‘hundreds’!

The Division Algorithm

6

4 1 0 4 )

0 2

- 8

4

- 2 4

0

2

4 into 1 goes 0 times, write down the 0 4 into 10 goes 2. Write down the 2. Check that division fact using multiplication: 2 x 4 = 8. Write down the 8 below the 10. Subtract the 8 to find the remainder:10 – 8 = 2. Write it below the 8. Bring down the next number which is 4. 4 into 24 goes 6. Write 6 above the 4. Check that division fact using multiplication: 6 x 4 = 24. Write it below the other 24. Subtract the 24 to find the remainder: 24 – 24 = 0.

Extended Form

The Division Algorithm

4 1 0 4 )

4 into 1 goes 0 times, write down the 0. 4 into 10 goes 2. Write down the 2 above the 10. 2 x 4 = 8 so there are 2 left over, write it in front of the 4. 4 into 24 goes 6, write 6 above the 4.

2

When solving an algorithm, we treat each digit as a ‘one’, even the ‘tens’ and ‘hundreds’! A reliance on the algorithm limits children’s conceptual understanding of division and place value.

0 2 6

Contracted Form

When we have taken students through this exhaustive process and tried our very best to teach them Maths, what do we turn to

when all else fails?????

What Happens If this Doesn’t Work?

Mr O’Connor