1. Teaching Multiplication & Division
Centre for Excellence cluster of schools
Combined staff meetings
The script for the presenter is in Italics.
Information for the presenter is not in Italics.
Print copy of presentation ‘Notes Pages’ to use during presentation.
 Indicates handouts
NSW Department of Education & Training
NSW Public Schools – Leading the Way

2. Multiplication & Division in the classroom
In groups/pairs discuss
What does teaching Multiplication & Division look like in my classroom?
What do I already know about how students learn to multiply and divide?
Share
Reflecting on current classroom practice connects the participants to the content of the presentation.
Reveal ‘what does teaching…………’
Give participants time to discuss in pairs or groups.
Reveal ‘ share’
Take contributions from participants

3. Counting and grouping Coordinating Grouping Counting
Learning to multiply and divide involves the coordination of keeping count of a number of groups as well as ensuring that the count of items within a group remains equal. Many students can count and make groups, however coordinating requires counting and grouping at the same time. Many students have difficulty with coordinating groups and seeing groups or ‘composite units’ as single entities. 3 3

4. Terminology Composite units
a set of items are treated spatially (visually) or numerically (abstractly)
as a unit
We need to clarify ‘composite units’ .
Seeing both the parts and the whole in a group of things contributes to the development of composite units. A composite unit is formed when a student takes a set of unit items and treats it spatially or numerically as a unit.
Simply recognising units is not sufficient to develop the operation of multiplication and division. It is necessary to develop ways of coordinating the groups that are formed.

5. Thinking groups - not items
groups of 3
counting grouping
4 threes
coordinating 3
Some students do not appear to make the transition from repeated addition to multiplication as a coordination of units of units. Repeated addition, such as , is successive and involves thinking on one level. Multiplicative thinking involves thinking on two levels simultaneously. To think of four 3s simultaneously is a coordination of units of units rather than a progression of additions.

6. Background notes
Some students persist with counting by ones and have difficulty in progressing to grouping strategies
By focusing on groups, rather than individual items, students learn to treat the groups as composite units.
Participants read slide.
Many students continue to count by ones. This is evident in the classroom by students who make lots of tally marks in order to work out multiplication problems. 6

7. Key concepts in Multiplication and Division
What are the big ideas students need to understand in order to effectively multiply and divide? 7

8. Teaching Key concepts
In our Explicit teaching part of the lesson how do we ensure deep understanding of the concept?
How do we teach concepts?
What are the key ideas?
There are four key concepts in learning to multiply and divide but before we look at them let’s consider how we teach maths concepts.

9. Key concepts What are the key concepts in multiplication & division?
Repeated addition
Commutative property
Associative property
Inverse relationship mult/division
These are the four key concepts in learning to multiply and divide. We will take a look at them in detail.

10. Key concepts – Repeated addition
Repeated addition - Multiplication is a fast way of adding a series of numbers
= 12
(Division – repeated subtraction)
The first key concept is repeated addition.
Repeated addition involves students understanding that multiplication involves finding the total given a number of groups of items. Many students have an understanding of repeated addition although this is often the limit of their understanding.

11. Key concepts – Repeated addition
Concrete  Visual 
Abstract = 12
4 groups of 3 = 12
Let’s look at how we teach this concept in the whole class explicit teaching part of our lesson.
Concrete - Students and teachers manipulate concrete material forming equal groups and counting the total by rhythmic (1,2,3..4,5,6…) or skip counting (3,6,9…). Students will move from using informal equal groupings to using arrays.
Visual – the next step is for students to draw what they have done with the concrete material. This is an important step where students are creating their own drawing of the activity in order to construct their own understanding. As the teacher moves around the room to observe drawings, students’ understandings are evident. Following the students’ own drawings of their concrete experience the teacher should model one or more visual representations to the class.
Abstract – moving to the abstract is now a natural progression for students. Students who experience difficulty with the abstract need to draw images again and be reminded of the concrete experience.
In catering for the wide range of abilities in our classrooms, all students can access the concrete activity, all students can draw a visual represnetation of the concrete and students will need a range of levels of support in the abstract learning. x

12. Key concepts – Commutative Property
Commutative Property of Multiplication - it doesn’t matter which number is first.
3 x 4 = 12
4 x 3 = 12
3 x 4 = 4 x 3
The second key concept is the Commutative Property
The commutative property applies to multiplication and also to addition. Putting your shoes on resembles a commutative operation since it doesn't matter if you put the left or right shoe on first, the end result (having both shoes on), is the same.
The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving things around. This concept is critical for students. Students with a good understanding of multiplication ‘move things around’ in order to use known facts to solve unknowns.

13. Key concepts – Repeated addition
Key concepts – Commutative Property
Key concepts – Repeated addition
Concrete  Visual 
Abstract x 4 = 4 x 3
Let’s look at how we teach this concept in the whole class explicit teaching part of our lesson.
Concrete - Students and teachers manipulate concrete material forming 3 groups of 4 and also 4 groups of 3. Students can begin by making both sets of groups in order to compare. Then students can make only 3 groups of 4 and move items to produce 4 groups of 3. This is an important step as it replicates what we do in the abstract ie. Change 3 x 4 to 4 x 3.
Visual – the next step is for students to draw what they have done with the concrete material. Arrays are very useful to demonstrate the commutative property. Students can colour a 3 x 4 grid and a 4 x 3 grid and place them on each other. Following the students’ own drawings of their concrete experience the teacher should model one or more visual representations to the class.
Abstract – showing 3 x 4 = 4 x 3 also confirms for students what the equal sign means ie. The value of both sides is the same. Many students think the equal sign means that the answer comes next. Students may have misconceptions about the equal sign and this can lead to problem in stage 4 when formal algebra is introduced. Drawing a see-saw assists students in their understanding of what the = sign means. 

14. Key concepts – Distributive Property
2 x 14 = 2 x (10 + 4)
= (2 x 10) + (2 x 4) =
= 28
The third key concept is the distributive property. The distributive property relates to splitting up groups in order to make them more manageable. In this case we have 14 groups of 2. The 14 groups are more easily managed by considering 10 groups and a further 4 groups. This
Many students use the distributive property as a strategy for working out solutions to times tables questions before they know their times tables by heart.

15. Key concepts – Distributive Property
Concrete  Visual 
Abstract 12 x 2 = 24 12
2 x
Let’s look at how we teach this concept in the whole class explicit teaching part of our lesson. Working with ’12’ times tables is a good model for this concept and students only need to be familiar with the 10 times and the 2 times tables. Students also are motivated by being able to work out 12 times tables.
Concrete – As the teacher models, students make 12 groups of twos and then split the array up into 10 x 2 and 2 x 2. Students need to see where the two sets of 2s have come from so it is critical to start with the 12 groups of 2s.
Visual – Provide students with blank paper or grid paper. The next step is for students to draw what they have done with the concrete material. This is an important step where students are creating their own drawing of the activity in order to construct their own understanding. Following the students’ own drawings of their concrete experience the teacher should model one or more visual representations to the class.
Abstract – Students need to see both form of the abstract. This concept is particularly useful in discussing how the algorithm works. Consider providing the algorithm and asking students to write an explanation as how their visual relates to the algorithm.
Written explanations can be challenging for students (and adults) however students will develop skills in expressing their understanding and this provides excellent assessment. The blank piece of paper is often the best assessment tool a teacher can have in mathematics. This form of assessment is open and allows higher achieving students to demonstrate the full range of their understanding. A number of these types of assessments are necessary in order to be able to award an A.

16. Key concepts – Distributive Property
Splitting the product into known parts.
8 x 7 = x 7 = 49
8 x 7 =
8 x 7 = 56
Splitting the product is an important strategy which relies on the distributive property. In this way students can move to unknown products by manipulating known products. Many students rely heavily on this strategy while working towards knowing their times tables.

17. Key concepts – Inverse relationships 
3 x 4 = 12
12 ÷ 3 = 4
Sometimes as teachers we delay work in division due to students experiencing difficulty with multiplication. Consider teaching both at the same time. Sometimes seeing both multiplication and division in the concrete helps students understand how groups work. Seeing both can give students a deeper understanding of working with groups.
Importantly, students need to understand that division is when you start with the total and split/divide the items into equal groups.
Handout Key Concepts in Multiplication & Division
Division start with the total items.

18. Key concepts Repeated addition Commutative property
Distributive property
Inverse relationship mult/division
These are the four key concepts in learning to multiply and divide. We will take a look at them in detail.

19. Deep Understanding
Students who understand how to coordinate composite units are able to make efficient use of known facts
What is the answer to 8 x 4?
8 x 4 is the same as 4 x 8
If 5 x 8 = 40
Then 4 x 8 must equal 32
What is the answer to 9 x 3?
Double 9 is 18,
is 20
is 27
Participants read slide.
With a good understanding of the key concepts students can manipulate questions in order to use the most efficient strategy. 19

20. Concrete /Manipulatives
Counters
Base 10 material
Pencils
Blocks
Egg cartons
Many items can be used to model concrete representations of concepts. Concrete materials are necessary for all students of all ages in order to develop deep understanding of mathematical concepts. Following the explicit teaching concrete materials should remain available to students. When students are accustomed to working with concrete materials they will use pencils or counters themselves when required.

21. Visuals Arrays Numberline 100 chart
There are many useful visuals to deepen understanding of concepts. We will look more closely at arrays shortly. Number lines are useful to represent rhythmic and skip counting. Number lines are more effective if the students have constructed them themselves. Students are required to think more when placing a range of numbers on the line.
As we have seen earlier in the explicit teaching part of the lesson where we teach the concept it is important for students to draw their own visual representation of their concrete experience before the teacher provides a visual. We need to see what understanding students have rather than them using the teacher’s visual in the first instance. Following this the teacher can lead them towards a particular visual which has been selected by the teacher. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

22. Visuals - Arrays Classroom tools – arrays
Structures such as arrays can provide students with opportunities to access spatial support through the rows and columns when coordinating groups.
As students develop the concept of multiplication, students focus on groups of items and learn to treat the groups as items themselves. Students can place counters on a blank page in an array and draw groups around the counters.
Just as students’ counting strategies are limited by their knowledge of the sequence of number words, so too their early multiplication and division strategies are often limited by their knowledge of the sequence of multiples. Students need to be able to say the multiples but saying them in order does not ensure students are thinking in groups.

23. Learning Object – The Array
There are a number of learning objects which assist in developing understanding of multiplication and division.
Handout Accessing Learning Objects

24. Learning object - Arrays: factor families
This is another learning object

25. Counting-On website Handout Counting-On website
The Counting-On website has detailed information relating to teaching the number strand in Stages 2,3 and 4.
Hyperlink click on world icon to link to site if desired

26. Count Me In Too website Handout – Count Me In Too website
The Count Me In Too website also has many resources. In particular there are many online activities for student use.
Hyperlink click on world icon to link to site if desired