Algorithms for multiplication tables of 9 to 10
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Teaching algorithms for multiplication
In the primary school, children are taught multiplication using a formal written method that is based on:

the place value system

multiplication tables up to 10 by 10

the distributive property of multiplication over addition.
Understanding the formal written algorithm for multiplication depends on assembling together understanding of several separate steps. Therefore the ideas must be introduced through a number of stages. Students need to be competent and comfortable with each stage prior to moving onto the next stage. Ample experience with place value materials prior to the introduction of symbolic notation will assist children consolidate knowledge at each stage.
The sections below give the reasoning behind the steps of the formal written algorithm, the intermediate forms which teachers use before students learn the most efficient procedure and link to explanations using place value material.
Learning the standard written algorithm
Stage 1: Develop meaning(s) for multiplication
(Repeated addition is sufficient for multiplying whole numbers)
Stage 2: Multiplication by a single digit
Stage 3: Multiplication by ten
Stage 4: Multiplication by a multiple of ten
Stage 5: Multiplication by numbers with two or more digits
Click below to see how to explain multiplication using concrete materials in stages 2 and 3, Multiplication using place value material
Other ways of setting out the standard algorithm
Other algorithms for whole number multiplication
Stage 2: Multiplication by a single digit
2 3
x 4
23 is 2 tens and 3 ones.
3 ones multiplied by 4 gives 12 ones and
2 tens multiplied by 4 gives 8 tens (that is 80).
80 and 12 are added to give the final product 92.

Children should write multiplication in this form for some time, until the procedure is familiar and the concepts (especially the distributive property) is well understood. Ruling up (or using squared paper) and labeling columns for tens and ones is recommended in the early stages. Later it can be reduced to a more compact form:

The 3 ones are first multiplied by 4 giving the product 12, which is 1 ten and 2 ones. 2 is written in the ones column and the 1 is recorded in the tens column. Now the 2 tens are multiplied by 4 to give 8 tens. The 1 ten recorded before is added on, so the product has 9 tens.
In the primary school, children are taught multiplication using a formal written method that is based on:

the place value system

multiplication tables up to 10 by 10

the distributive property of multiplication over addition.
Understanding the formal written algorithm for multiplication depends on assembling together understanding of several separate steps. Therefore the ideas must be introduced through a number of stages. Students need to be competent and comfortable with each stage prior to moving onto the next stage. Ample experience with place value materials prior to the introduction of symbolic notation will assist children consolidate knowledge at each stage.
The sections below give the reasoning behind the steps of the formal written algorithm, the intermediate forms which teachers use before students learn the most efficient procedure and link to explanations using place value material.
Learning the standard written algorithm
Stage 1: Develop meaning(s) for multiplication
(Repeated addition is sufficient for multiplying whole numbers)
Stage 2: Multiplication by a single digit
Stage 3: Multiplication by ten
Stage 4: Multiplication by a multiple of ten
Stage 5: Multiplication by numbers with two or more digits
Click below to see how to explain multiplication using concrete materials in stages 2 and 3, Multiplication using place value material
Other ways of setting out the standard algorithm
Other algorithms for whole number multiplication
Stage 2: Multiplication by a single digit
2 3
x 4
23 is 2 tens and 3 ones.
3 ones multiplied by 4 gives 12 ones and
2 tens multiplied by 4 gives 8 tens (that is 80).
80 and 12 are added to give the final product 92.

Children should write multiplication in this form for some time, until the procedure is familiar and the concepts (especially the distributive property) is well understood. Ruling up (or using squared paper) and labeling columns for tens and ones is recommended in the early stages. Later it can be reduced to a more compact form:

The 3 ones are first multiplied by 4 giving the product 12, which is 1 ten and 2 ones. 2 is written in the ones column and the 1 is recorded in the tens column. Now the 2 tens are multiplied by 4 to give 8 tens. The 1 ten recorded before is added on, so the product has 9 tens.
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