Multiplicative Thinking

Multiplicative Thinking Workshop 1 Properties of Multiplication and Division

Learning intentions for this workshop • Recognise the properties of multiplication and division (commutative, distributive, associative, identity, and inverse) • Use these properties to analyse strategies that students are using.

factors product Let’s start with some mathematics language. • Numbers that we multiply together are called factors. • The answer in a multiplication is called the product. • For example: • 3 x 7 = 21

Two students are solving this problem: • A family has $96.00 to spend at the adventure park. Each ride costs $4.00. • How many rides can the family go on?

They think They record I need to find out how many times I can take $4.00 away from $96.00 • 96 • -4 • 92 • -4 • 88 • -4 • 84 • etc. • … Frank There are 25 lots of $4.00 in $100, $96.00 is $4.00 less, so… • 25 x 4 = 100 • 100 – 4 = 96 • So • 24 x 4 = 100 • … Awhina Discuss each student’s strategy. Are they both thinking multiplicatively?

There was a qualitative difference in the way these two students solved the problem. • Frank’s strategy was additive (rather strange when he was subtracting). • Awhina’s strategy was multiplicative (rather strange since the problem was about division).

Why encourage multiplicative strategies if additive strategies can be used? • Multiplicative strategies are often more efficient than the repeated use of an additive strategy. • Imagine if the problem was: • “A family had $396.00 to spend at the adventure park. Rides cost $4.00. How many rides can the family take?” • Awhina would take as long to do that as she did for • $96 ÷ $4 = □. • Frank would need the whole mathematics lesson!

A strategy is multiplicative is it uses any of the multiplicative properties. • In the next few slides we will look at those properties. • If 5 x 6 = 30 then 6 x 5 = 30, right? • Explain to the person next to you why that is true.

Your explanation might have involved an array of rows and columns. 6 x 5 might be seen as six sets of five. The columns give the number being multiplied (multiplicand) and the number of columns are the multiplier. 5 x 6 might be seen as five sets of six. The rows give the number being multiplied (multiplicand) and the number of rows are the multiplier. 5

6 6 6 6 6 If you had thought of 5 x 6 as five bags with six lollies in each you might have thought differently. How many bags of five lollies can be made? Why? 5 5 5 How does 5 x 6 get transformed into 6 x 5 in this scenario?

How does this look in the array model? Here is 5 x 6 with the number in a row as the multiplicand and the number of rows as the multiplier. Transforming it into 6 x 5 involves taking one object from each row to form a column of five.

This may seem a bit fussy but understanding the transformation and talking about what the numbers refer to has implications for further understanding. • The commutative property is about the order of the factors leaving the product unchanged. • For example, 99 x 3 = □ has the same number answer as 3 x 99 = □.

Which of the following students understands the commutative property? • All seventy-eight students in your syndicate each bring $5.00 for the bus. • How much money is that altogether?

5 x $78 might be easier to work out… • Notice that the top student recognises that the units are different but the result invariant, i.e. 78 lots of $5 is not the same as 5 lots of $78 but the answer will be the same! 78 x $5.00? Well, 8 x 5 = 40…

Does the commutative property hold for division? • For example, if 24 ÷ 8 = 3, does 8 ÷ 24 = 3? • No, the commutative property doesn’t hold but how are the answers to 24 ÷ 8 and • 8 ÷ 24 related? • The answers are reciprocals. They multiply to give one. 3 x ⅓ = 1

Discuss what strategies you use to solve this problem: • Each carton holds • 36 cans of spaghetti. • There are 5 cartons. • How many cans • of spaghetti is that? From: GloSS Form C

Units are an important way to analyse strategies. To solve the problem you had to create a unit-of-units. • The unit-of-units you created was • 5(36-units) which means 5 x 36. • If you applied the commutative property you might have changed this to 36(5-units).

ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten 5 x 30 = 150 150 30 180 5 x 36 = + 5 x 6 + 30 = 180 • Here is a strategy that you might have used along with an equipment representation:

The first strategy involved the distributive property. This meant that one of the factors was split additively. • 5 x 36 = 5 x 30 + 5 x 6 • The 36 was split (distributed) into 30 + 6.

7 x 8 = 4 x 8 + 3 x 8 • Here’s another example of the distributive property. What factor was split additively this time? Which factor is split additively?

ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten ten 180 5 x 36 = 36 x 5 = 18 x 10 = 180 • Another strategy you may have used on the spaghetti can problem used the commutative property in conjunction with the associative property. 180

The associative property is about regrouping the factors. • So in 36 x 5 the 36 was split multiplicatively: • 36 x 5 = (18 x 2) x 5 • = 18 x (2 x 5) • = 18 x 10 • = 180

Here’s another example of the associative property: • How many small cubes make up this model?

You could have worked that out as: • 4 x (3 x 5) (4 layers of 3 x 5) • (4 x 3) x 5 (5 layers of 4 x 3) • (4 x 5) x 3 (3 layers of 4 x 5) • Any way in which you group the factors multiplicatively leaves the product invariant (unchanged).

Another important property of multiplication and division is inverse operations. • The inverse of an operation is the operation that undoes it. That is, returns you to where you started.

Here are some inverse operations: 5 x 4 = 20 20 ÷ 5 = 4 What will undo duplicating a set five times? Distributing the total equally five ways (sharing).

If two operations are inverses the relationship is two way. 24 ÷ 8 = 3 = 24 8 x 3 This also means that dividing into eight equal sets undoes duplicating a set eight times: 8 x 3 = 24 so 24 ÷ 8 = 3.

Discuss this statement.Is it true? • Multiplication always makes the answer bigger and division always makes the answer smaller. • For example, in 4 x 8 = 32, the answer 32 is bigger than 8.

There are some obvious exceptions, like: • That’s not right. • 1 x 6 = 6. • The answer is the same as the factor. And, 0 x 6 = 0. The answer is zero, that’s less than six.

What can you say about the multiplier when multiplication makes bigger? • What can you say about the multiplier when multiplication makes smaller? • Does division ever make the answer bigger? Provide some examples.

Here are two students’ strategies. • Describe what multiplicative properties they are using.

The builder has 280 posts. • She needs 8 posts to build a pen. • How many pens can she build? That’s 280 ÷ 8 = ⁪ or ⁪ x 8 = 280. 30 x 8 = 240, that leaves 40. 5 x 8 = 40 so 35 x 8 = 240. From: GloSS Form E

Mani knows that there are 8 teams in the rugby tournament. • Each team has 15 players. • What is the total number of players at the tournament? 8 x 15 =⁪, 2 x 15 = 30, 2 x 30 = 60, 2 x 60 = 120. From: GloSS Form D

In conclusion… • It is not the vocabulary of commutative, distributive, etc. we want students to know. • It’s putting the properties to work in solving problems that is the important thing.

Multiplicative Thinking