What’s going on? Properties of multiplication and division

Purpose

This unit develops students’ recognition of pattern (consistency) in equations involving multiplication and division with whole numbers.

Achievement Objectives
NA4-8: Generalise properties of multiplication and division with whole numbers.
Specific Learning Outcomes
  • Describe and represent the commutative property of multiplication by attending to multiplication as repeated addition.
  • Describe and represent the distributive property of multiplication by attending to place value, and multiplication as repeated addition.
  • Recognise that multiplication and division are inverse operations, and interpret division as either equal sharing or measuring.
  • Find relationships in the difference of perfect squares, e.g. 7 x 7 = 49 so 8 x 6 = 48.
Description of Mathematics

This unit develops students’ recognition of pattern (consistency) in equations involving multiplication and division with whole numbers. The patterns of pairs of equation embody important properties of multiplication and division, such as commutativity, distributivity, and inverse. Students learn to represent specific examples where the properties are used then provide convincing arguments about why the properties hold in all circumstances.

An important consequence is that students learn to consider variables as generalised numbers, and express relationships involving whole numbers under multiplication and division.

In this unit we build on research by Deborah Schifter, and colleagues, about the development of algebraic thinking. Shifter works for The Educational Development Centre, a non-profit research organisation in USA. Her approach follows several steps that can be linked to ‘folding back’ in the Pirie-Kieren model of conceptual development, that is commonly used in New Zealand classrooms.

The phases of the approach are as follows:

In this unit claims are developed through equation sets involving multiplication and division. The sets aim at developing students’ understanding of the properties of multiplication (commutativity, distributivity, associativity, identity and inverse). By expanding the equation sets to include division students will learn how these properties hold or do not hold when the operation is changed.

Required Resource Materials
  • Square tiles, connecting cubes
  • Place value materials
  • Calculators (optional)
  • Square grid paper
  • Copymaster 1
  • PowerPoint 1
Activity

All lessons in this unit follow the same sequence of phases as given in the diagram above. A poster of the phases is provided as Copymaster One for students to refer to. The notes suggest possible student ideas and teacher reactions to those responses. It is not feasible to anticipate all ideas students might give so you are encouraged to be flexible in how you respond to students rather than ‘teach’ the sample ideas and representations provided.

PowerPoint One contains seven equation sets that drive the unit. The sets might form the basis of a week-long unit. The phases for each equation set are described below. The sets are labelled in the top left corner of each slide for reference.

Equation Pairs Set One

Slide one has the first pattern to look at. The pattern involves the commutative property, i.e. a x b = b x a, which students should be familiar with. It is used as an example to familiarise students with the approach.

Noticing Regularity

Use ‘think, pair, share’ by inviting students to look independently at the four examples, work out the missing values, then share their ideas with a partner. In the class discussion expect students to express their observations in ways that are clear to others. Students should re-express their ideas if others do not understand what they are saying. You may need to remind students that the ‘something going on’ relates to all four examples, not just one. Expect responses like:

S: The numbers are just turned around, like 9 x 4 becomes 4 x 9.
T: Can you be more specific. Which numbers are turned around?
S: The numbers being multiplied each time.

Discussion opens the possibility of using correct mathematical terms, like factor (number being multiplied), and product (answer to multiplication).

S: The products (answers) are always the same.
T: All four patterns have the same product? What do you mean?
S: No. The products are the same when the factors are turned around.

Articulating a claim

Encourage students to state a claim about what is going on with all four examples in Pattern One. They might do so individually at first then work in a small team to refine their ideas and the way they express those ideas.

Expect ideas like:

  • If the factors are the same, and you turn them around, the product doesn’t change.
  • The first factor changes places with the second factor. The product is the same.

Your aim is for students to express their claims in clear, minimal terms, using correct mathematical language. For example, ‘turning around’ is not as clear as order of the factors.

Representing

In this phase students choose representations to show why the pattern holds consistently. Students might choose physical manipulatives, such as linking cubes or counters, draw diagrams such as number lines or arrays, and use contexts from everyday life. Encourage students to begin with the first two examples of equation pairs then consider how the same relationships might generalise to the last, and other similar, equation pairs.

Examples might be:

  • I made 7 x 5 first using cubes. Then I took one off each five to make a seven. I found out I could make five stacks of seven from the seven fives.
  • I drew 7 x 5 as an array. The fives were the columns. When I turned the array around the sevens became the columns, but the total number of cubes was the same.

Some representations are less helpful than others in terms of understanding the structure of the commutative property. For example, jumps on a number line do not show how each item in a set is used to form the new sets. It is very important for students to recognise the multiplier as the first factor and the multiplicand as the second factor. Your questioning is important to support them to connect the symbols and other representations.

  • Explain where the 7 and 5 are in your representation.
  • If you start with 7 x 5, why can you only make five sets of seven?
  • What do the 5 and 7 represent in 5 x 7?
  • How does your representation show that the product stayed the same?

Constructing an argument

In this phase students are asked to formalise their noticing by creating a statement that generalises to all cases. The discussion may start with a specific equation pair but must be amended to deal with what occurs in general.

S: With 7 x 5, one from each five makes a set of seven. Because there are five in the sets that means exactly five sevens can be made.
T: So how does that work in the same way with 9 x 4, 8 x 99, and 5 x 36?

This might lead to expressing the property in general terms.

S: The first factor multiplied by the second factor has the same product as the second factor multiplied by the first factor.
T: If we gave names to the first and second factors, like a and b, could we express the property more simply?

Some students might experiment with algebraic notation such as a x b = c so b x a = c. Note that this represents the starting equation pairs. In general, a sets of b can be remodelled. Taking one object from each set of b creates sets of size, a. This can be done b times, resulting in b x a (b sets of a).

T: Do we need to say both equations have an answer of c? Do we need c?
S: We could just write a x b = b x a.

Focus on the class of numbers that have been used, i.e. whole numbers. Encourage students to investigate if the commutative property holds for integers and rational numbers, e.g. If ½ x 36 = 18 does 36 x ½ = 18.

Equation Pairs Set Two

Ask the students to approach the second equations set more independently. From this point each equation set is discussed succinctly using the phases of the approach.

Noticing regularity

The four equations apply doubling and halving, thirding and trebling, etc. of the factors in the first equation. This strategy is sometime called proportional adjustment since it underpins the concept of equivalent fractions. The completed sets should be:

8 x 3 = 24                                                      6 x 10= 60

4 x 6 = 24 [Doubling 3, halving 8]             12 x 5 = 60 [Doubling 6, halving 10]

 

9 x 9 = 81                                                      7 x 6 = 42

27 x 3 = 81 [Trebling 9, thirding 9]            14 x 3 = 42 [Doubling 7, halving 6]

Articulating a claim

In natural language expect the students to use phrases like “one number doubles, the other halves”. Introduce important vocabulary such as factor and product to clarify what numbers are being referred to in the claims. If the claim is restricted to doubling and halving draw attention to 9 x 9 = 81 and 27 x 3 = 81. The aim is to broaden the claim to the equivalent of “one factor is divided by n, the other factor is multiplied by n. The product stays constant (the same).”

Representation

Expect both physical and diagrammatic representations to be used. A cube stack representation might look like this:

Diagrams of a ‘sets’ representation might look like this:

Arrays might also be used as a powerful representation. Initially cubes might be used as units of area leading to a more abstract use of side lengths.

Argumentation

Look for students to justify that a given quantity, say 24, can be created by multiplying two factors, say 4 x 6. Keeping our quantity constant, one factor can be divided in equal parts, say each 6 is divided into three equal parts (2s). Now there are three times more of those parts making up 24. The number of parts a factor is divided into is a variable. Some students may be comfortable with using a label, like n, to represent the number of equal parts. The factors and product are also variables and might be represented with shapes or letters. Algebraically the relationship might be expressed as:

Look for students to generalise ‘undoing’ nature of the inverse operations, i.e. divided by n, multiplied by n. At this level students are progressing towards the use of letters to represent variables. You might also introduce the quotient interpretation of fractions, e.g. a÷n can be expressed as a/n. Multiplication can also be represented without the x symbol, e.g. a×b can be represented as ab.

Equation Pairs Set Three

Noticing regularity

The four equations apply the distributive property. This property is used a lot in the multiplication of multi-digit numbers. The completed sets should be:

7 x 10 = 70                                                    5 x 20= 100

7 x 11 = 77 [Adding 7 x 1]                           5 x 22 = 110 [Adding 5 x 2]

 

9 x 50 = 450                                                  6 x 100 = 600

9 x 53 = 477 [Adding 9 x 3]                        6 x 105 = 630 [Adding 6 x 5]

Articulating a claim

In natural language expect the students to use phrases like “adding on so many lots of the number”. Expect the use of mathematical vocabulary such as factor and product to clarify what numbers are being referred to in the claims. Encourage clarity by asking questions like:

  • Can you know how much more the second product is than the first? How?
  • What does the first factor mean? What does the second factor mean?

The aim is to state the claim as something like “If a number is added to the second factor, then the product increases by the first factor multiplied by that number.”

Representation

Expect both physical and diagrammatic representations to be used. Since most the first equations involve multiples of ten or 100, place value blocks (MAB) might be useful.

Diagrams of a tens and hundreds can be made schematic to highlight important structure.

Arrays illustrate how the first factor ‘acts’ on the second factor as it is changed.

Argumentation

Look for students to justify that two factors multiply to a given product. The starting product might be expressed as a x b. Adding a number to b results in the second factor becoming b + n (n is the number being added). The product increase by a x n. It is important for students to consider what is happening with all four equation sets, in that n is a variable, and can be ‘any number.’ Students are working towards expressing relationships among variables using letters and equations. So progress can be encouraged by working with the notations that students develop themselves.

Algebraically the relationship might be expressed as:

a×b=c so a×(b+n)=(a×b)+(a×n)

Multiplication can also be represented without the x symbol, e.g. a×b can be represented as ab, and unnecessary brackets (due to order of operations) can be removed.

ab=c so a(b+n)=c+an

Discuss the removal of unnecessary variables as well. c is redundant as ab expresses the product. This reduces the property to:

a(b+n)=ab+an

Equation Pairs Set Four

Noticing regularity

The four equations apply the inverse relationship between multiplication and division. This property is used by students to solve division problems by measurement, e.g. “How many x’s go into y?  The completed sets should be:

8 x 6 = 48                                                      7 x 3 = 21

48 ÷ 6 = 8 [expressing as division]                21 ÷ 3 = 7 [expressing as division]

 

12 x 25 = 300                                                68 x 9 = 612

300 ÷ 25 = 12 [expressing as division]          612 ÷ 9 = 68 [expressing as division]

Articulating a claim

In natural language expect the students to use phrases like “the factors are being put into a division equation.” Students might indicate what they see in a diagram.

Expect the use of mathematical vocabulary such as factor and product to clarify what numbers are being referred to in the claims. You may need to introduce division terms like divisor (number being divided by), quotient (answer to division) and dividend (the quantity being divided). Encourage clarity by asking questions like:

  • What does the second factor become in the division equation? (divisor)
  • What does the first factor become in the division equation? (quotient)
  • What does the second factor mean?

The aim is to state the claim as something like “Two factors multiply to give a product. The product divided by one factor equals the other factor.”

Representation

Expect both physical and diagrammatic representations to be used. Be aware that students may interpret division in two ways, as equal sharing (most common) or as measuring. Either interpretation can be used to represent the equations. Here is a measurement interpretation since 48 ÷ 6 is seen as “How many sixes are in 48?”

A sharing view interprets 48 ÷ 6 as “48 is equally shared among six parties. How much does each party get?”

Schematic diagrams, like arrays, show the factors as side lengths, and the product as the area. The missing number in an equation can be shown as an empty measure in the diagram.

Argumentation

Look for students to justify that if a product, a x b, is the multiplication of two factors a and b, then the product can be divided into a sets of b or b sets of a. So the product in multiplication can be thought of as a dividend in division. Either factor can be the divisor, but the other factor is the quotient.

Look for students to accept that the factors and product are variables, meaning they can take up any value. The product is dependent on the factors so can always be represented as a x b, or ab.

Some students may use the repeated addition view of multiplication like this:

7 x 3 means seven sets of three, so seven sets of three can be made from 21.

a x b means a sets of b, so a sets of b can be made from ab.

Equation Pairs Set Five

Noticing regularity

The equation set applies proportional adjustment, halving, of the dividend and explores the effect on the quotient. This property can be used by students to solve division problems. The completed sets should be:

24 ÷ 4 = 6                                                                  40 ÷ 5 = 8

12 ÷ 4 = 3 [halving of dividend and quotient]      20 ÷ 5 = 4

 

264 ÷ 11 = 24                                                           72 ÷ 9 = 8

132 ÷ 11 = 12                                                           144 ÷ 9 = 16

Articulating a claim

In natural language expect the students to use phrases like “halving the number being divided halves the answer.”

Expect the use of mathematical vocabulary associated with division, like divisor (number being divided by), quotient (answer to division), and dividend (the quantity being divided). Encourage clarity by asking questions like:

  • What changes in each pair of equations?
  • What stays the same?
  • Look at equation four. It is different to the others. How?

The aim is to state the claim as something like “Multiplying or dividing the dividend by a number, while keeping the divisor the same, results in the quotient being multiplied or divided by the same number.”

Representation

Both equal sharing, or measuring interpretations of division can be used to model the relationships in the equation pairs. Sets models, like stacks of cubes, work well but encourage the progress towards schematic diagrams like arrays.

Note that the fourth equation pair shows the effect of multiplying both dividend and divisor by two.

Argumentation

Look for students to justify, using words or diagrams, that if a dividend, ab, is divided by a divisor, b, then the quotient, ab ÷ b or ab/b, determines now many b’s measure ab. If ab is divided by two, then the number of b’s that can fit into ab/equals half of ab/b, written as ab/2b.

Look for students to accept that the dividend and divisor are variables, meaning they can take up any value. The quotient is dependent on those variables, so can always be represented as ab ÷ b or just a.

Some students may transfer the repeated addition view of multiplication to division like this:

8 x 5 means eight sets of five, which totals 40. So, with half that total, 20, it is possible to make half as many fives.

If the factors are regarded as variables, then a more general finding might be ‘proven’ geometrically with arrays.

If a amounts of b equal ab, then a/amounts of b equal half of ab (ab/2).

Equation Pairs Set Six

Noticing regularity

The equation set highlights the difference of squares. This property can be used by students to solve multiplication problems. The completed sets should be:

9 x 9 = 81                                                      5 x 5 = 25

8 x 10 = 80 [One less than 81]                  4 x 6 = 24 [One less than 25]

 

20 x 20 = 400                                                12 x 12 = 144

21 x 19 = 399 [One less than 400]            13 x 11 = 143 [One less than 144]

Articulating a claim

In natural language expect the students to use phrases like “the first equation has the same factor times itself. The second equation has one more and one less and the answer is one less.” Ask students to be more specific in their description.

  • Which factor is increased by one?
  • Which factor is decreased by one?
  • Is this true for all four equations?

The aim is to state the claim as something like “If a number is chosen, one less than the number, multiplied by one more than the number, equals the square of the number less one.”

Representation

Students are likely to use specific examples to convince other about how the relationships work.

Argumentation

Arrays are a very useful representation to ‘prove’ what occurs with the difference of perfect squares.

In general, a2 can be transformed spatially into (a-1)(a+1) by removing one (1 x 1) and moving the unit of (a-1) to create rectangle with sides of (a-1) and (a+1).

Students are not expected to use algebraic notation at this level. High achieving students might like to see that the transformation can be represented as:

a2-1=a2+a-a-1

=(a-1)(a+1)

Equation Pairs Set Seven

Use set seven as an opportunity to see how well students engage in the generalisation process independently. Ask them to record their claims, representations, and arguments in ways that work for them. Some students may prefer writing their work while others may prefer to capture their ideas using a digital recording.

Noticing regularity

The equation set highlights the difference of squares. This property can be used by students to solve multiplication problems. The completed sets should be:

80 ÷ 8 = 10                                                    60 ÷ 5 = 12

72 ÷ 8 = 9 [One less set of 8]                     55 ÷ 5 = 11 [One less set of 5]

 

270 ÷ 9 = 30                                                  700 ÷ 7 = 100

261 ÷ 9 = 29 [One less set of 9]                686 ÷ 7 =   98 [Two less sets of 7]

Articulating a claim

In natural language expect the students to use phrases like “the first equation has a dividend divided by a divisor. The second equation has one or two times the divisor taken off the dividend, so the quotient is one or two less.” Ask students to be more specific in their description. Diagrams might be useful.

Ask questions like:

  • Show how the dividend is decreased by one or two times the divisor.
  • What is the effect on the quotient?
  • Is this true for all four equations?

The claim using mathematical language might be something like, “If the dividend is reduced by one times the divisor, then the quotient is reduced by one.” Note that the fourth equation pair reduces the dividend by two times the divisor.

Representation

Given the dividends are multiples of ten or 100, place value blocks (MAB) might be a suitable representation.

Schematic diagrams, like arrays, provide a clearer view of the structure found in examples.

Argumentation

Students might go back to the repeated addition view of multiplication to convince others about their claim. A specific example might look like this:

 

See if students can take the specific examples and turn them into a more generalised form, by using letters or other symbols to present the dividend as the product of two variables:

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