The purpose of this unit of five lessons is to develop understanding of the symbols for and operations of multiplication and division, of their inverse relationship, and of how to use these operations in problem solving situations.
This sequence of lessons makes the connection between repeated addition and multiplication. It introduces division and explores the relationship between the operations of multiplication and division.
There are three main understandings being developed within these lessons.
The first is to have students understand that multiplication is much more than basic facts to be rote learned and memorised, and that division ‘somehow goes with multiplication too, only it’s harder’. Rather the focus is on developing an understanding of how quantities relate to one another (they can be equal to, or bigger or smaller than one another) and how number operations, in this case multiplication and division, behave. The emphasis is on the relationship between quantities and between operations, and in particular, the inverse relationship between multiplication and division.
Secondly, students need to fully understand the symbols that they use to record their ideas and mathematical ‘actions’. Symbols are used to express a ‘mathematical event’ but are also, in themselves, tools for helping us to think about what is happening. To understand and use symbols, the associated language must be developed, both the informal student ‘sense-making language’, and the formal and precise mathematical language.
Thirdly, these understandings are best developed within a practical context (in this case using a patchwork quilt or array). Arrays are a powerful way of showing the structure and pattern of multiple groups, and, in this case, strongly linking multiplication and division to measurement. It is however recognised that there are situations other than equal groups and rectangular area, for multiplication and division. These include rates and ratios, for example, and are explored in other units and resources.
In exploring the structure and pattern of multiplication and division, the focus is also on developing an early understanding of number properties. The commutative property of multiplication is formally explored in these lessons. The distributive property, in which factors are ‘transformed’, (eg. 12 x 55 = 10 x 55 + 2 x 55), is accessible to those students with more advanced multiplicative thinking.
In exploring the behavior of the operations of multiplication and division, it is important that students make generalisations in which they can state ‘what always happens’ when certain actions are undertaken. For example, recognising that the ‘turn around’ rule (commutative) is always true for multiplication, but that is it not true for division.
This series of lessons focuses on single digit factors and divisors. It recognises that to build a sound understanding of how we use multiplication and division symbols and expressions to think mathematically, and to express relationships, students must have many opportunities to write, as equations, the mathematics found in word problems. They also need to be able to imagine or suggest a context or scenario which an equation expresses. Making the connections between language and symbols is essential to developing a sound understanding of mathematical ideas and concepts.
The activities suggested in this series of lessons can form the basis of independent practice tasks.
Links to the Number Framework
Early additive (Stage 5)
Advanced additive (Stage 6)
at least two rectangular quilts (or pictures of them)
coloured plastic square tiles (or small squares of different coloured card)
Write two equations on the class chart, one multiplication and one division.
For example: 6 x 5 = 30 28 ÷ 4 = 7. Read them together.
Have each student write a description in words of the quilt that each equation is expressing, and show each with a diagram (refer to these in Session 2).
Conclude the session by reviewing the operation and relationship symbols and their meanings.
Return to the quilts (pictures). Explain that some young children like alphabet quilts in which each patch shows a picture of something starting with a different letter of the alphabet. Talk about what some of these might be. For example: A might show an apple, B a butterfly, C a cat and so on.
Make available to the students paper, pencils and felt pens.
Pose the problem: You are going to make an alphabet quilt for a young child (in the Children’s ward or hospice.) You have till the end of the session today to plan your design and how you will arrange your ‘patches’. There might be a challenge somewhere in the problem. You decide the best way to solve this for your quilt design.
(26 will only make an even 2 x 13 array, which is undesirable for a quilt of this kind. Students will encounter ‘a remainder’ (6 x 4 + 2r, 5 x 5 + 1r) or they’ll find they are some ‘patches’ short (7 x 4). Accept realistic solutions for the context. (eg; 5 x 5 quilt: put 2 letters on one patch, 6 x 4 quilt: make it 7 x 4 and include 2 novel or blank patches.)
Show an alternative quilt array. For example:
Have four students record one of each of the related facts.
(6 x 5 = 30, 5 x 6 = 30, 30 ÷ 5 = 6, 30 ÷ 6 = 5) and explain each fact with reference to the quilt, including demonstrating the commutative (turn around) property of multiplication. Turn the quilt to demonstrate this.
Distribute Attachment 1. Emphasise the inverse operations, and the need for students to show or explain how multiplication helps to solve division problems.
Review key learning from Session 4. Have students work in pairs to share their solutions to the quilt problems from Session 4, Activity 4. Encourage them to question each other.
Display some quilt examples:
Write on the class chart.
One quilt of sixteen patches:
One quilt of nine patches:
One quilt of thirty patches:
Ask students to record multiplication equations for each of these statements.
One quilt of sixteen patches: 1 x 16 = 16
One quilt of nine patches: 1 x 9 = 9
One quilt of thirty patches: 1 x 30 = 30
Pose these: 16 ÷ 1 = ☐ , 9 ÷ 1 = ☐ , 30 ÷ 1 = ☐
Have students discuss this, then explain and justify their thinking.
Write these, discuss, highlighting that these fractions are also representations of division.
Have students complete the equations, and discuss.
|16||= ☐||9||= ☐||30||= ☐|
Record students’ summary statements about what happens when we multiply by one or divide by one. Have them suggest, and record, other examples of their own.
Pose these: 16 ÷ 16 = ☐, 9 ÷ 9 = ☐, 30 ÷ 30 = ☐
Have a student explain in words what each equation says and is asking us to find out.
Have a student write each of these division expressions in this way:
|16||= ☐||9||= ☐||30||= ☐|
and explain why each example will give us one (quilt).
Record students’ summary statements about what happens when we divide a number by itself. Have them suggest, and record, other examples of their own.
Have them play in groups of 2 to 4, Is it a fact? (Attachment 2).
(Purpose: To discriminate between correct and incorrect multiplication and division equations and expressions, and be able to explain why, justifying their decision)
Copy equations and expressions onto cardboard and cut into separate cards.
Students take turns to take a card and explain to others in the group, if and why the statement is a fact, or if and why it is incorrect (true or false).
Spare (blank cards) can be used for students to create more Is it a fact? Cards to add for others to use.
Print onto sheets for individual students. Have them decide Yes or No (true or false) and write about or draw a diagram in the blank adjacent space, to justify their decision.
Conclude this session by reviewing the learning that has happened over the five sessions.