## Multiplication and division symbols, expressions and relationships

Purpose

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.

Achievement Objectives
NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.
Specific Learning Outcomes
• Understand the words ‘number operation’.
• Read, write and understand the multiplication and division symbols, and the language associated with them.
• Write story contexts for given multiplication and division equations.
• Recognise that the operation of multiplication is commutative.
• Identify related multiplication and division facts (‘fact family’).
• Recognise the inverse relationship of the operations of multiplication and division.
• Recognise that division is not commutative.
• Use the words ‘factor’ and ‘product’ appropriately and identify factors of given amounts.
Description of Mathematics

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.

Required Resource Materials

at least two rectangular quilts (or pictures of them)

coloured plastic square tiles (or small squares of different coloured card)

unifix cubes

playing cards

Activity

Session 1

SLOs:

• Understand the words ‘number operation’.
• Read, write and understand the multiplication symbol, and the language associated with it.
• Read, write and understand the division symbol and the language associated with it.
• Write equations for story contexts, emphasising the meaning of equivalence.
• Write story contexts for given multiplication and division equations.

Activity 1

1. Begin by showing the students two different rectangular patchwork quilts (or pictures of them). For example:  Explain (real or hypothetical): a/the class is going to make a quilt for the children’s ward at the local hospital (or hospice).
Engage them in a quilt discussion, establishing their knowledge and understanding of a quilt, as appropriate.
2. Ask, “ What mathematics is there in these quilts?” (eg. the 3 x 3 quilt)
Record the students’ ideas on the class chart. (These may include number, geometry, measurement statements: for example, 3 + 3 + 3 = 9, 3 x 3 = 9, 9 squares, one big square, sides the same, divided by 3) and with reference to both quilts: 4 + 4 + 4 + 4 = 16, 16 is more than 9, 16 >9).
3. Highlight the operation and relationship symbols (or words) that have been recorded. For example: 4. Write each symbol on a separate A4 sheet of paper. Have student pairs take one sheet (one symbol) and each take turns to record, in 2 minutes, using words and pictures/diagrams, a brainstorm of all they know about this symbol (or word).
5. Ask students to return to the mat, sitting in separate two groups: a group with operation symbols and a group with relationship symbols. Have selected student pairs explain why they are sitting where they are, and what ideas they have recorded for their symbols.
In this discussion, emphasise the language used, develop the understanding of what a number operation is (a mathematical process that changes an number or amount), and review the meaning of the equals sign.
Keep the brainstorm sheets for future reference.

Activity 2

1. Prepare bags of 12, 18 and 20 plastic tiles, small coloured small squares of card, or fabric squares. Make these, pencils and paper, available to student pairs. Pose the problem. “Show, using diagrams and equations, how many different ways can you arrange these patches to make a ‘mini patchwork quilt’?”
Have students work in pairs to record their ideas.
2. Have students pair share their ideas with a pair who had the same number of tiles, and record any arrangements they had not thought of.
3. As a class, share ideas, exploring and recording key understandings on the class chart. Keep this student work for Session 2.
For example: From the bag of 18 ‘patches’ (tiles). In this discussion, build on ideas shared in Activity 1 (above), highlighting and recording in words, these ideas:
• The ‘patch’ arrangements can be recorded using different operations.
• Multiplication, using the x symbol, can show the same idea as repeated addition (of equal amounts), using the + symbol.
• The symbol for division, or sharing into equal groups, is ÷. It’s called the division symbol (not the “divided by” symbol.)
• This even arrangement with rows and columns is called an array.
4. Pose and record: “ 9 + 9 = 6 x 3. Do you agree or disagree.” Have student pairs discuss this and be prepared justify their position (explain why they agree or disagree, and how they know that they are right).
Record student justification, highlighting the relationship of equivalence (both are equal to 18, there are 18 patches altogether in both arrays.)

Activity 3

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).

Activity 4

Conclude the session by reviewing the operation and relationship symbols and their meanings.

Session 2

SLOs:

• Write equations for story contexts.
• Write equality and inequality multiplication statements.
• Draw a multiplication array, and recognise remainders.

Activity 1

1. Without referring to the equations written in Session 1, Activity 3, Step 1, begin by having at least two students share the quilt story that they wrote and illustrated. Have another student record the equation for each. Highlight the fact that real life maths can be written in both words and symbols.
2. Brainstorm on the class chart several other places in our lives where we see and use multiplication or division. As students share ideas, require them to suggest specific numbers. Record these.
For example: We see multiplication:
“When little packets of raisins are wrapped together in a bigger pack – four packets in a row and three rows.”
“When you buy two packets of gum with ten pieces in each”
“When we make four teams of six for a game in PE.”
3. Read them again together. Explain that they are to use symbols to record the equations for each of these in their books/on whiteboard/paper. Those who finish quickly can draw these or should write more of their own context stories.
Have them pair share their equations. If students have recorded using repeated addition, have them also record multiplication equations.

Activity 2

1. Review symbol information from Session 1, Activity 1 and again highlight the operation symbols, + - x ÷ , and review equals =, greater than > and less than < relationship symbols.
Have students now work in pairs using the information and equations from Session 2, Activity 1 (above), or from Session 1, Activity 2, Step 1. They should discuss and see how many equations and other relationship expressions they can each write using this information: For example:
3 x 4 = 4 x 3
3 x 4 < 2 x 10
4 x 6 > 2 x 10 > 4 x 3
They should use diagrams to show how they know that they are correct.
2. Have students pair share their work. As they do, they should take turns to read aloud what they have written.

Activity 3

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.)

Session 3

SLOs:

• Recognise that the operation of multiplication is commutative.
• Identify related multiplication and division facts (‘fact family’).
• Recognise that division is not commutative.
• Use the words ‘factor’ and ‘product’ appropriately and identify factors of given amounts.

Activity 1

1. Have students share their alphabet quilt designs from Session 2, Activity 3 (above).
Discuss the ‘remainder issue’ and commend creative solutions.
Point out that often division problems don’t work out evenly. We call what is left over, a remainder. One way we could record this problem, 26 ÷ 6 = 4 r2 (This is a point of interest rather than a key idea at this point.)
2. On the class chart quickly draw the arrays that have been decided upon. Discuss the ‘dimensions’ of the array, introducing the words factors and product. Model with an example: Have each student record below their quilt designs, what is in the box above, adjusting the numbers for their own design.

Activity 2

1. Pose: Look at the factors and product that you have written. You have written one equation using these numbers. Can you write three more equations using these numbers?
2. Share and record these. Accept any addition equations written, taking the opportunity to emphasise the words addend and sum.
Now record a set of multiplication an division equations such as:
4 x 7 = 28 7 x 4 = 28 28 ÷ 4 = 7 28 ÷ 7 = 4
3. Pose:
1. Using these numbers and these operations we can write more equations.
2. 4 x 7 = 7 x 4 and 28 ÷ 4 = 4 ÷ 28

Have student work in pairs to discuss both statements. Have them decide if they agree or disagree. The must be prepared to justify their position (explaining how they know they are right).
4. Have students pair share and appoint one person to report back to the class on behalf of the group of four.
5. Summarise findings on a class chart. For example:
• There are four related facts only (family of facts) and no more.
4 x 7 = 28 7 x 4 = 28 28 ÷ 4 = 7 28 ÷ 7 = 4
• Multiplication is a ‘turn around’ operation. You can ‘do it ‘ both ways without changing the product. (It’s like addition.)
We say that multiplication (and addition) are commutative.
4 x 7 = 7 x 4 = 28
• Division isn’t a ‘turn around’ operation.
28 ÷ 4 = 7 4 ÷ 28 does not equal 7. (It equals 1/7)
We say that division (and subtraction) are not commutative.

Activity 3

1. Have students play in pairs, Multiply, Draw and Write.
They need playing cards, 2 to 9, pencil and paper.
The winner is the person who, after six rounds, has the most pairs of products that are the same, but which are made with different factors.
For example: 6 x 4 = 8 x 3 = 24,    or 4 x 4 = 2 x 8 = 16
How to play:
The cards are shuffled and placed face down in a pile between both players.
Players take turns to turn over two cards from the pile. These are the factors. The player must draw the array for the multiplication, write in the factors, and write the four related family of facts.
For example: 2. Students conclude the session by writing word scenarios for their equation sets (family of facts). These do not have to be quilt scenarios.
For example: “There were three bags with five apples in each. So fifteen shared among three bags is five. If these fifteen apples were put into five bags, there would be three in each. That would be five lots of three.”

Session 4

SLOs:

• Recognise that the commutative property involves regrouping.
• Recognise the inverse relationship of the operations of multiplication and division.
• Recognise the associative property of multiplication.

Activity 1

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.

Activity 2

1. Make linking cubes (or coloured counters) available to the students. Have pairs of students each take 12 cubes. Ask what factors could make 12. Record 4 x 3 and 6 x 2.
Have one student of each student pair model 4 x 3, linking the cubes. Then have their partner use the same cubes to model 3 x 4. Discuss what happened. (They needed to regroup these). Repeat with 6 x 2 and 2 x 6. Emphasise that the commutative property involves the same factors and product, but that it can involve both turn around and regrouping.
2. Ask students to suggest more examples of single digit multiplication and explore these with the cubes, using the language of same factors and product, focusing on regrouping.

Activity 3

1. Record again one familiar multiplication equation on the class chart. For example, 6 x 2 = 12. Have one of students in each pair model this, by making 6 groups of 2 and linking the cubes together into one line of 12. Record 12 ÷ 6 = 2. Have the other student in the pair enact this with the cubes. Have students describe what has happened and record ideas such as: it’s the opposite, division undid multiplication, it’s the reverse, we’re back where we started.
Ask, Is this always true? How can we find out? Accept student ideas. These should include the students exploring more examples.
2. Conclude that it is impossible to try all multiplication and division this is something that is known to be true for all. It is the way these multiplication and division behave.
Ask if anyone knows what we call this.
Write inverse relationship on the class chart. Discuss words similar to inverse eg. reverse, and their meaning. Make the connection to the inverse relationship between addition and subtraction. Highlight that in each operation pair, one operation or action undoes the other.
3. Return to the quilt in Activity 1 (above) and to the equations recorded:
(6 x 6 = 30, 5 x 6 = 30, 30 ÷ 5 = 6, 30 ÷ 6 = 5)
Have a student explain the ‘undoing’ (the inverse relationship again, with reference to the quilt. (this is a little harder to see, because this array cannot physically been ‘undone.’)
4. Write on the class chart:
Knowing that multiplication and division are inverse operations is helpful because……..
Have students suggest reasons and record these, including: we can use multiplication to help us solve division problems.

Activity 4

Distribute Attachment 1. Emphasise the inverse operations, and the need for students to show or explain how multiplication helps to solve division problems.

Session 5

SLOs:

• Understand and explain the result of having a factor of one or a divisor of one.
• Understand and explain the result of dividing a number by itself.
• Discriminate between correct and incorrect multiplication and division equations and expressions.

Activity 1

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.

Activity 2

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.

Activity 3

Write these, discuss, highlighting that these fractions are also representations of division.

 16 9 30 1 1 1

Have students complete the equations, and discuss.

 16 = ☐ 9 = ☐ 30 = ☐ 1 1 1

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.

Activity 4

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 = ☐ 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.

Activity 5

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.
Or
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.

Activity 6

Conclude this session by reviewing the learning that has happened over the five sessions.

Attachments