Theory
Multiplication
Multiplication involves a multiplier (a), multiplicand (b) and a product (c), and is the process of counting groups of equal size. It is important to be specific that you require equal size sets (Van de Walle, Karp & Bay-Williams, 2014). |
a x b = c |
Combination/Cartesian model – is looking at the number of combinations possible between different objects or quantities.
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Rate/allocation model – can be modelled through the use of counters, number lines and arrays.
Students should develop an understanding of commutative, associative and distributive laws so they can adjust equations to make them easier to solve.
Commutative law – a x b = b x a
This does not apply to division.
Associative Law – a x (b x c) = (a x b) x c
This does not apply to division.
Distributive law - allows you to distribute multiplication or division across an addition or subtraction.
(a + b) x c = (a x c) + (b x c)
(a - b) x c = (a x c) - (b x c)
(a + b) ÷ c = (a ÷ c) + (b ÷ c)
(a - b) ÷ c = (a ÷ c) - (b ÷ c)
Students should develop an understanding of commutative, associative and distributive laws so they can adjust equations to make them easier to solve.
Commutative law – a x b = b x a
This does not apply to division.
Associative Law – a x (b x c) = (a x b) x c
This does not apply to division.
Distributive law - allows you to distribute multiplication or division across an addition or subtraction.
(a + b) x c = (a x c) + (b x c)
(a - b) x c = (a x c) - (b x c)
(a + b) ÷ c = (a ÷ c) + (b ÷ c)
(a - b) ÷ c = (a ÷ c) - (b ÷ c)
Division
Division involves a quotient (c), dividend (a) and a divisor (b) and is an extension of multiplication knowledge so teaching should be introduced soon after multiplication. Division is the separation of a whole into parts. a ÷ b = c Division can be modeled with “pictures, diagrams or concrete materials to help students learn what the factors and their product represent in various contexts” (Van de Walle et al., 2014, p. 170). Encourage students to use any method and resources that allows them to explain their process. |
Common Student Misconceptions
- Recording multiplication calculations on one line can cause errors with regrouping and renaming.
- Students may not understand that the number of groups can be exchanged so 6 x 5 can be 6 groups of 5 or 5 groups of 6.
- Students may not be able to distinguish that a grouped set of items is also a single entity.
- Student may misunderstand the multiplication symbolism in a word problem and use repeated addition instead.
- Student may misunderstand or become confused with the mathematical language used so it is important to use appropriate terminology consistently. This can be particularly the case for English Language Learners.
- A student’s ability to obtain correct answers using standard algorithms does not guarantee conceptual understanding.
- Students may not understand what to do with remainders of division unless addressed in teaching. Teaching in context will assist.
- Zeros and ones can cause conceptual challenges so they require the use of real world problems followed by class discussions.
- When working with patterns, the colours or patterns could cause confusion with learners.
- When working with patterns, if they count on from the right they may mirror the pattern instead of continuing it.
- When a student is asked to pull down or circle the repeating pattern they may pull it all down or not enough down.
Mathematics Language in Multiplicative Thinking
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Activities
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Word Problems |
Early years
Patterns The Early Years Learning Frameworks (EYLF) key numeracy concept (Outcome 5, p. 6) that is met is: the student has the ability to recognise repeating patterns in the environment; and the ability to create, copy and extend repeating designs using colours, sounds, shapes, objects, stamps, pictures and actions. Objective: Student will demonstrate their ability to recognise a pattern, copy that pattern and extend on the pattern. Foundation
Repeated patterns Represent practical situations to model addition and sharing (ACMNA004) Objective: students will demonstrate repeated addition through their use of array models. Year 1
Skip Counting 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) Objective: students will develop skip counting by twos, fives and tens from number up to 100. |
Multiplication
Equal group model Lesley has 5 bags of lollies. Each bag contains 9 lollies. How many lollies in total? Rate model If a bag of lollies cost $2, how much did Lesley pay for 5 bags? Comparison model Charlotte has 12 pencils. Libby has 4 times as many pencils as Charlotte. How many pencils did Libby have altogether? Combination model Lilly has 6 skirts and 5 tops that can be worn together. How many possible outfits could Lilly make with the skirts and tops? (Adapted from Van de Walle et al., 2014) |
(Adapted from Victoria Department of Education and Training, 2014)
Resources
Curriculum Map
EARLY YEARS
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)
Multiplicative thinking 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
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)
Multiplicative thinking 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
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
Cox, K. (n.d.). Pre-K math: Patterns [Blog Post]. Retrieved from http://www.prekinders.com/math-patterns/
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.
Quirk, L. (n.d.). Use repeated addition to find the total number of objects in an array. Retrieved from https://learnzillion.com/lessons/3953-use-repeated-
addition-to-find-the-total-number-of-objects-in-an-array
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). Skip counting: Level 2. Retrieved from
http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/continuum/pages/skipcount20.aspx
from http://www.earlyyears.sa.edu.au/files/links/16_Outcome_5.pdf
Cox, K. (n.d.). Pre-K math: Patterns [Blog Post]. Retrieved from http://www.prekinders.com/math-patterns/
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.
Quirk, L. (n.d.). Use repeated addition to find the total number of objects in an array. Retrieved from https://learnzillion.com/lessons/3953-use-repeated-
addition-to-find-the-total-number-of-objects-in-an-array
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). Skip counting: Level 2. Retrieved from
http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/continuum/pages/skipcount20.aspx