Theory
Haylock (2014) discusses three division structures:
Equal-sharing between – is the equal distribution of a quantity to the required portions.
Inverse-of-multiplication – the question changes so you are trying to find the quantity within the portion.
Ratio – comparing two quantities.
In order for students to be efficient with multiplication and division mental strategies they must have a good understanding of the multiplication facts.
If a student struggles with this they will need to understand other strategies for working out what they do not know (Haylock, 2014).
Knowledge of commutative, associative and distributive law assist students in the early stages to deal more easily with complicated equations (Haylock, 2014).
Commutative law – a x b = b x a
It is not applicable to division. It is important for students to understand they can alternate the quantities to assist problem solving.
Example: 5 x 2 = 10
2 x 5 = 10
Associative law – (a + b) + c = a + (b + c)
Example: (2 + 5) + 3 = 10
2 + (5 + 3) = 10
Distributive law - (a + b) x c = (a x c) + (b x c) or (a + b) ÷ c = (a ÷ c) + (b ÷ c)
Example: (2 + 5) x 3 = 21
(2 x 3) + (5 x 3) = 21
Demonstrate to students that a number can be 'distributed' across addition to simplify the multiplication or division.
For example: 36 x 4 could be (30 x 4) + (6 x 4)
(Haylock, 2014)
Other Student-invented strategies for multiplication (Van de Walle, Karp & Bay- Williams, 2014)
Students may also use compensation strategies, but this is only suitable for certain computations.
For example: 45 x 5
45 x 10 = 450 Double the 5
450 ÷ 2 = 225 Halve the answer
So doubling the 5 made it much easier to compute the multiplication mentally, then halving the number was also simpler.
Students may also use repeated addition but this is less efficient than other strategies and requires attention for developing new strategies.
The cluster problems approach allows the student to use a combination of facts they already know.
For example: 93 x 5
40 x 5 = 200
50 x 5 = 250
3 x 5 = 15
200 + 250 + 25 = 475
Then there are also area models which are explained under multiplicative thinking.
Student-invented strategies for division (Van de Walle et al., 2014)
The trial-and-error strategy looks at the quotient to find what number multiplied by that is equal to the dividend.
Cluster problems approach also looks at the quotient to find what number multiplied by that is equal to the dividend. In this case you are building up the divisors in a logical method.
For example: 512 ÷ 8
400 ÷ 8 = 50 That leaves 112
80 ÷ 8 = 10 That leaves 32
32 ÷ 8 = 4 Add the dividends 50 + 10 + 4 = 64 so 512 ÷ 8 = 64
Equal-sharing between – is the equal distribution of a quantity to the required portions.
Inverse-of-multiplication – the question changes so you are trying to find the quantity within the portion.
Ratio – comparing two quantities.
In order for students to be efficient with multiplication and division mental strategies they must have a good understanding of the multiplication facts.
If a student struggles with this they will need to understand other strategies for working out what they do not know (Haylock, 2014).
Knowledge of commutative, associative and distributive law assist students in the early stages to deal more easily with complicated equations (Haylock, 2014).
Commutative law – a x b = b x a
It is not applicable to division. It is important for students to understand they can alternate the quantities to assist problem solving.
Example: 5 x 2 = 10
2 x 5 = 10
Associative law – (a + b) + c = a + (b + c)
Example: (2 + 5) + 3 = 10
2 + (5 + 3) = 10
Distributive law - (a + b) x c = (a x c) + (b x c) or (a + b) ÷ c = (a ÷ c) + (b ÷ c)
Example: (2 + 5) x 3 = 21
(2 x 3) + (5 x 3) = 21
Demonstrate to students that a number can be 'distributed' across addition to simplify the multiplication or division.
For example: 36 x 4 could be (30 x 4) + (6 x 4)
(Haylock, 2014)
Other Student-invented strategies for multiplication (Van de Walle, Karp & Bay- Williams, 2014)
Students may also use compensation strategies, but this is only suitable for certain computations.
For example: 45 x 5
45 x 10 = 450 Double the 5
450 ÷ 2 = 225 Halve the answer
So doubling the 5 made it much easier to compute the multiplication mentally, then halving the number was also simpler.
Students may also use repeated addition but this is less efficient than other strategies and requires attention for developing new strategies.
The cluster problems approach allows the student to use a combination of facts they already know.
For example: 93 x 5
40 x 5 = 200
50 x 5 = 250
3 x 5 = 15
200 + 250 + 25 = 475
Then there are also area models which are explained under multiplicative thinking.
Student-invented strategies for division (Van de Walle et al., 2014)
The trial-and-error strategy looks at the quotient to find what number multiplied by that is equal to the dividend.
Cluster problems approach also looks at the quotient to find what number multiplied by that is equal to the dividend. In this case you are building up the divisors in a logical method.
For example: 512 ÷ 8
400 ÷ 8 = 50 That leaves 112
80 ÷ 8 = 10 That leaves 32
32 ÷ 8 = 4 Add the dividends 50 + 10 + 4 = 64 so 512 ÷ 8 = 64
Common Student Misconceptions
- Students require a good understanding of multiples and factors for mental computation. This can assist students with finding missing quotients in division and using the distributive law.
- Students may think that partitioning is only an additive composition but they can also use subtractions. Eg. 19 could be partitioned to 20 - 1
Mathematics Language for Multiplication and Division
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Activities
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Word Problems |
Year 2
Equal sets of apples Recognise and represent division as grouping into equal sets and solve simple problems using these representations (ACMNA032) Objective: student will display their knowledge of known number facts to create equal sets of apples. (Adapted from Australian Mathematical Science Institute, 2011)
Year 3 Solving Word problems Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies (ACMNA057) Objective: Students demonstrate their ability to understand word problems using manipulatives, representing it through drawing and writing in numerical form. They will demonstrate different mental strategies in order to the solve problem. (Adapted from Victoria Department of Education and Training, 2014)
Year 4 Number patterns to help recall of multiplication facts Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075) Objective: Students develop fluency in multiplication recall. (Adapted from Victoria Department of Education and Training, 2014)
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Division Partition model Michael has 45 lollies. If Lesley shares the lollies among 5 bags, how many lollies will each bag contain? Measurement model Michael paid $10 for lolly bags. Each bag cost $2. How many bags did Michael get? Comparison model Libby pencil case held 48 pencils. Libby’s fit 4 times as many as Charlotte’s pencil case. How many pencils did Charlotte’s pencil case hold? Measurement model Libby’s pencil case held 48 pencils, whereas Charlotte’s only held 12 pencils. How many times as many pencils does Libby’s pencil case hold? (Adapted from Van de Walle et al., 2014) |
Resources
Curriculum Map
In the Early Years Learning Framework (Document 16, outcome 5, p. 6) children will learn the key learning concept of patterns including the recognition of repeating designs in the environment; and ability to create, copy and extend repeating designs using colours, sounds, shapes, objects, stamps, pictures and actions.
(Charles Sturt University Early Years Learning Framework Consortium, 2009)
Multiplication and division is found in the Australian Curriculum from Foundation through to Year 6.
FOUNDATION
YEAR 1
YEAR 2
YEAR 3
YEAR 4
YEAR 5
YEAR 6
(Charles Sturt University Early Years Learning Framework Consortium, 2009)
Multiplication and division is found in the Australian Curriculum from Foundation through to Year 6.
FOUNDATION
- Represent practical situations to model addition and sharing (ACMNA004)
YEAR 1
- Develop confidence with number sequences to and from 100 by ones from any starting point. Skip count by twos, fives and tens starting from zero (ACMNA012)
YEAR 2
- Investigate number sequences, initially those increasing and decreasing by twos, threes, fives and ten from any starting point, then moving to other sequences. (ACMNA026)
- Recognise and represent multiplication as repeated addition, groups and arrays (ACMNA031)
- Recognise and represent division as grouping into equal sets and solve simple problems using these representations (ACMNA032)
YEAR 3
- Recall multiplication facts of two, three, five and ten and related division facts (ACMNA056)
- Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies (ACMNA057)
YEAR 4
- Investigate number sequences involving multiples of 3, 4, 6, 7, 8, and 9 (ACMNA074)
- Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)
- Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder (ACMNA076)
YEAR 5
- Identify and describe factors and multiples of whole numbers and use them to solve problems (ACMNA098)
- Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies (ACMNA100)
- Solve problems involving division by a one digit number, including those that result in a remainder (ACMNA101)
- Use efficient mental and written strategies and apply appropriate digital technologies to solve problems (ACMNA291)
YEAR 6
- Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123)
- Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies (ACMNA129)
- Multiply and divide decimals by powers of 10 (ACMNA130)
References
Australian Mathematical Science Institute. (2011). Multiplication and division: A guide for teachers - years F to 4. Retrieved from
http://amsi.org.au/teacher_modules/multiplication_and_division.html#MODELLING_DIVISION
Charles Sturt University Early Years Learning Framework Consortium. (2009). Document 16 outcome 5: Children are effective communicators. Retrieved
from http://www.earlyyears.sa.edu.au/files/links/16_Outcome_5.pdf
Education Services Australia. (n.d.) Curriculum. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1
Haylock, D. (2014). Mathematics explained for primary teachers (5th ed.). London, England: Sage Publications.
Thompson, I. (2011). Mental calculation: Why don't they get it? Mathematics Teaching, 221, 30-31. Retrieved from https://web-b-
ebscohost-com.ezproxy2.acu.edu.au/ehost/pdfviewer/pdfviewer?vid=1&sid=c0b6dc84-4f40-4d10-abd3-b26f4febdf4a
%40sessionmgr112&hid=118
Van de Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Victoria Department of Education and Training. (2014). Recall of multiplication facts: Level 4. Retrieved from
http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/continuum/Pages/multifacts30.aspx
http://amsi.org.au/teacher_modules/multiplication_and_division.html#MODELLING_DIVISION
Charles Sturt University Early Years Learning Framework Consortium. (2009). Document 16 outcome 5: Children are effective communicators. Retrieved
from http://www.earlyyears.sa.edu.au/files/links/16_Outcome_5.pdf
Education Services Australia. (n.d.) Curriculum. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1
Haylock, D. (2014). Mathematics explained for primary teachers (5th ed.). London, England: Sage Publications.
Thompson, I. (2011). Mental calculation: Why don't they get it? Mathematics Teaching, 221, 30-31. Retrieved from https://web-b-
ebscohost-com.ezproxy2.acu.edu.au/ehost/pdfviewer/pdfviewer?vid=1&sid=c0b6dc84-4f40-4d10-abd3-b26f4febdf4a
%40sessionmgr112&hid=118
Van de Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Victoria Department of Education and Training. (2014). Recall of multiplication facts: Level 4. Retrieved from
http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/continuum/Pages/multifacts30.aspx