TEN (Targeted Early Numeracy) is a program that explicitly teaches students in K-2 the fundamental skills of addition and subtraction for problem solving.

It targets students identified at risk in 5 week cycles by the classroom teacher to improve their ability to add, subtract and recognise numbers in many different learning situations.

Focus groups have 3-4 students only in them. Skills and strategies are targeted through

intense ten minute sessions every day, using games and strategies that promote success.

Students are assessed constantly to ensure appropriate intervention happens.

The program focuses on Aspect 2 of the Numeracy Continuum, Counting As a Problem Solving Process.

TENS

Addition and Subtraction LevelsEmergent

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.

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

TENS

Addition and Subtraction LevelsPerceptual

1 2 3 4 1 2 3

1 2 3 4 5 6 7

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

TENS

Addition and Subtraction LevelsFigurative

A child at this level will keep the greater number in his/her head and count on or back the lesser number.

TENS

Addition and Subtraction LevelsCounting on and back

34 + 7 …..34 35 36 37 38 39 40 41

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

TENS

Addition and Subtraction LevelsFacile

34 + 7 …..I know 4 and 6 makes 10 so that’s 40 and 1 which makes

41

Once a child gets to Year 3 they should be able to or are starting to count by numbers other than one, and use strategies such as the Jump, Split and Compensation.

We like to get the students to share their strategies with each other and say how they solved the problem.

Working Mathematically

Addition and subtraction Strategies

Mental Strategies for Addition and Subtraction

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

Mental Strategies for Addition and Subtraction

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 =Jump

+20 +7 +2

63 83 90 92

+ 20

63 + 20 = 83, 83 + 9 = 92I kept the 63 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.

Mental Strategies for Addition and Subtraction63 + 29 =Jump

+ 9

63 83 92

+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.

Mental Strategies for Addition and Subtraction63 + 29 =Split

+3 +9

60 80 83 92

Compensation

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.

Mental Strategies for Addition and Subtraction63 + 29 =

+ 30 -1

63 92 93

Addition Algorithm Procedure

+ 29

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

63¹

9

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

A reliance on the algorithm limits children’s conceptual understanding of mental strategies and place value.

52 – 18 =Number Line:

Numbers:

Mental Strategies for Addition and Subtraction

Words:

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

Decomposition

4

1 8

We say:2 minus 8 you can’t do so we get 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

What is Multiplication?joining equal groups together to see

how many altogether.repeated addition

What is Division?splitting / sharing a group into smaller

equal groups.repeated subtraction

Multiplication and Division are inverse operations.

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

In the early years students will use objects to make groups or arrays to understand what multiplication and division really is and the process.

4 groups of 3 is….. Or 4 X 3 =

A child may 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 Strategies

8 x 7 =

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

26

X 4

8

10

2 4

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.

4

0

The Multiplication Algorithm:Contracted Form

4 10

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 Multiplication Algorithm:Extended Form - 2 digits by 2 digits.

911

1

24 times 6 equals 24,

write down the 4 and carry the 2. 4 times 4 equals 16, plus the 2 equals 18.

Write down the 18.We write a zero in the ones column.

Then we say 2 times 6 equals 12, write down the 2 and carry the 1.

2 times 4 equals 8, plus the 1 equals 9.

Write down the 9.We then add 4 plus 0 equals 4.

8 plus 2 equals 10, write down the 0 and carry the 1.

1 plus 9 equals 10, plus the 1 we carried equals 11.

46X 24

418

1

0240

Mental Strategies for Multiplication and Division

104 ÷ 4 =Repeated Subtraction

0 26 52 78 104

-26 -26 -26 -26

104 – 26 = 78, 78 – 26 = 52, 52 – 26 = 26, 26 – 26 = 0.

I knew 25 x 4 was 100 so 26 x 4 was going to be 104. I demonstrated on a number line by keeping on subtracting 26.

Repeatedly subtracting 4 from 104 is inefficient – students mix with other strategies such as:

Halving

Halve 104 = 52. Halve 52 = 26. I halved 104 and got 52. Then I knew I needed to halve again because dividing by 4 is like finding a quarter and a quarter is half of a half. So I halved 52 and got 26.

Mental Strategies for Multiplication and Division

104 ÷ 4 =

0 10452

Halve 104Halve 52

26

Compensation Strategy

Mental Strategies for Multiplication and Division

104 ÷ 4 =

100 - 4 = 100, 100 ÷ 4 = 25, 4 ÷ 4 = 1, 25 + 1 = 26I took 4 away from 104 and got 100. Then I did 100 ÷ 4 = 25. I still had the 4 that I took away and 4 ÷ 4 is 1, I added the 1 to the 25 and it was 26.Mental strategies increases children’s conceptual understanding of multiplication, division and place value.

0

100 ÷ 4 = 25 - 4

100 10425 50 75

The Division Algorithm

6

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

0 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.

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.

Students start with informal units and comparison before moving on to formal units.

Students don’t start with formal units until the end of year 2 or into year 3.

Students should not use informal units if they have not understood the concepts for measuring with informal units.

A rectangle shape is not a good shape to measure area because it may be arranged in more than one way eg vertically or horizontally. This makes consistent measurement difficult.

The pattern made when measuring area with squares is an array. This is the same as the array children use in multiplication and division and fractions. The array structure provides the understanding for rectangular area to be calculated using multiplication.

1 row of 1212cm x 1cm

3 rows of 43 cm x 4 cm

2 rows of 62 cm x 6 cm

6 rows of 26 cm x 2 cm

4 rows of 34 cm x 3 cm

12 rows of 112 cm x 1 cm½ cm x 24 cm

Rectangular Prism

6 sides, opposite faces equal,

8 vertices, all vertices equal

Cube

6 sides, all faces equal,8 vertices, all vertices equal

The volume of the rectangular prism with 1 layer is ____ cubic centimetres.

The volume of the rectangular prism with 2 layers is ____ cubic centimetres.

The volume of the rectangular prism with 3 layers is ____ cubic centimetres.

The volume of the rectangular prism with 4 layers is ____ cubic centimetres.

Are there any questions?