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: = 5

Concrete Stage: Subtraction Concrete stage: with real objects taking 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: = 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 Subtraction Place value addition using tens and units: = 23

Place Value Addition and Subtraction Place 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: =

Subtraction By Counting Back Using A Blank Number Line: 37 – 24 =

Expanded Vertical Layout For addition: For subtraction: – (7+6) ( ) = 211

Subtraction Using The Expanded Vertical Layout With Decomposition: 53 – 28 = = 25

The Compact Written Method For addition:

The Compact Written Method For subtraction: 53 – 28: –

Multiplication and Division Part 2

Multiplication Concrete stage putting together equal sets, with counting: / /

Multiplication Drawing stage representing the concrete stage in pictures, with repeated addition: = 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: = 9

Multiplication Introducing the multiplication symbol as a shorthand form of recording: = 6 or 3 x 2 = = 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 = 50 5 x 100 = 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 =744

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

Multiplication Multiplication using the compact written method: 38 X 124 X

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 : sharing representing 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 -: – 60 (10 X 6) 12 – 12 (2 X 6) 0 So, 72 -:- 6 = 12

Division Short formal written method for division: and with remainders: 20 r 4 or

Division Long formal written method for division: 78 remainder