Developmental Stages in Calculation

Colehill First School

1st March 2013

Addition and Subtraction

Part 1

Concrete Stage: Addition

Concrete stage: with real objects

putting together two sets of objects, with a number sentence:

3 + 2 = 5

Concrete Stage: Subtraction

Concrete stage: with real objectstaking a set of objects away from a larger

set, with number sentence

4 – 2 = 2

Counting Forwards (Addition) and Backwards (Subtraction)

Counting forwards along labelled number tracks or lines, with number sentences:

3 + 4 = 7

Counting Forwards (Addition) and Backwards (Subtraction)

Counting backwards along labelled number tracks or lines, with number sentences:

5 – 2 = 3

Place Value Addition and SubtractionPlace value addition using tens and

units:

20 + 3 = 23

Place Value Addition and SubtractionPlace value subtraction using tens and

units:

15 – 5 = 10

Addition and Subtraction using the Hundred Square:

Addition By Counting On Using A Blank Number Line:

47 + 36 = 83

47 50 80 83

47 77 83

Subtraction By Counting Back Using A Blank Number Line:

37 – 24 = 13

13 20 30 37

13 17 37

Expanded Vertical Layout

For addition: For subtraction:

47 + 327 – 116

76 300 20 7 -

13 (7+6) 100 10 6

110 (40 + 70) 200 10 1

123 = 211

Subtraction Using The Expanded Vertical Layout With Decomposition:

53 – 28 =

50 3 - 40 13

20 8 20 8

20 5 = 25

The Compact Written Method

For addition:

47+

76

123

1

The Compact Written Method

For subtraction: 53 – 28:

4

513 – 2 8 2 5

Multiplication and Division

Part 2

Multiplication

Concrete stageputting together equal sets, with counting:

1 2 3 4 / 5 6 7 8 / 9 10 11 12

Multiplication

Drawing stagerepresenting the concrete stage in pictures,

with repeated addition:

5 + 5 = 10

Multiplication

Counting forwards in jumps of greater than 1, both mentally and along labelled number tracks or lines, or on the hundred square, with number sentences written as repeated addition:

3 + 3 + 3 = 9

Multiplication

Introducing the multiplication symbol as a shorthand form of recording:

2 + 2 + 2 = 6

or 3 x 2 = 6

5 + 5 + 5 + 5 + 5 + 5 + 5 = 35

or 7 x 5 = 35

Multiplication

Recognising equivalent multiplication, e.g. using arrays:

2 x 4 = 8 * * * *

* * * *

4 x 2 = 8 * * * *

* * * *

Multiplication

Place Value: Multiplying by 10 and multiples of 10, using 0 as a place holder:

5 x 10 = 505 x 100 = 5005 x 1000 = 5000

Multiplication

Multiplication using partitioning:

e.g. 38 x 7 = (30 x 7) + (8 x 7)

124 x 6 = (100 x 6) + (20 x 6) + (4 x 6)

Multiplication

Grid Layout (expanded method): 124 x 6

X 100 20 4

6 600 + 120 + 24

=744

Multiplication

Vertical method for multiplication: 38 x 124 x 7 6 210 (30 x 7) 600 (100 x 6) 56 (8 x 7) 120 (20 x 6) 24 ( 4 x 6) 266 744

Multiplication

Multiplication using the compact written method:

38 X 124 X

7 6

266 744

5 1 2

Division

Concrete stage : sharing

sharing a set of objects between a group of people:

4 shared between 2 people gives 2 each.

Division

Drawing stage : sharingrepresenting the concrete stage in

pictures, recording using the division symbol.

e.g Sharing 6 sweets between 3 friends:

Joe Lucy Lee

* * * * * * 6 : 3 = 2

Division

Concrete stage: grouping

making groups or sets of a certain number:

e.g. 15 grouped into 5s gives 3 groups:

# # # # # # # # # # # # # # #

Division

Division as repeated subtraction:Counting backwards in jumps of greater than

1 along labelled number tracks or lines, or on hundred squares, with number sentences:

8 -:- 4 = 2

Division

Division as repeated subtraction without the number line, continuing until 0 (or a remainder) is reached, e.g. for 18 -:- 6:

18 – 6 = 12 – 6 = 6 – 6 = 0 I took 6 away three times, and so

18 -:- 6 = 3

And for 20 -:- 6: 20 – 6 = 14 – 6 = 8 – 6 = 2 I took 6 away 3 times, leaving 2 at the end,

so: 20 -:- 6 = 3 remainder 2

Division

Division using ‘chunking’ Chunking means subtracting larger groups, or

chunks, of the divisor number. This saves time and reduces the length of the repeated subtraction:

e.g. for 72 -:- 6 72 – 60 (10 X 6) 12 – 12 (2 X 6) 0 So, 72 -:- 6 = 12

Division

Short formal written method for division:

12

6 72

and with remainders:

20 r 4 or 20.5 8 164

Division

Long formal written method for division:

78 remainder 3 7 549 49 59 56 3