Developing Multiplicative

Thinking 2011

Alison Howard and Lisa Heap

Numeracy Facilitators

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Reflection on Numeracy Teaching:

Is there something new you have started doing since the last workshop?

• Used the teaching model for strategy teaching?• Introduced a new add/sub strategy?• Tried some new activities?• Used your modeling book?• Taught number knowledge as whole class, to groups and

individual practice?• Listening to students’ thinking?• Any other exciting ideas to share?

Where are you now? What are your next steps?

Objectives:• Understand the progressive strategy stages of

multiplication & division

• Understand where basic facts in multiplication/division “fits” with multiplicative thinking

• Know how to use Numeracy book six and other supportive resources to help teach multiplication and division

• Explore problem progressions, equipment and activities used to teach multiplication and division strategies.

Mult Game

Basic Facts:

• Independent reading: “ Call for more times table chants…”

• Discuss key messages in Thinking Groups

• Discuss how this will impact on your future teaching

• Brainstorm Stage 5, 6 and 7 Multiplication Basic Facts

• Check using your Knowledge Framework

Development of Multiplicative Thinking

There are 6 minivans outsidethe school, they are going on a school trip. There are 5 children in each minivan. How many children are going on the trip?

• How would a student at the different stages solve this problem?

Mult/Div Strategy Framework• 2/3 CA Counts all the objects

• 4 AC Uses skip counting

• 5 EA Repeated addition or using known facts

• 6 AA Derived multiplication and known facts

• 7 AM Choosing efficiently from a range of strategies

using whole numbers

• 8 AP Choosing efficiently from a range of strategies

using decimals and fractions

e.g. Roll a three and a four: 3 x 4 or 4 x 3

Multiplication Grid GR

e.g. Roll a three and a four: 3 x 4 or 4 x 3

Multiplication Grid GR

3 x 4

4 x 3

What is Multiplicative Thinking ?

Multiplicative thinking is not about the type of problem you

solve but how you solve it.

E.g. Although 3 x 18 is a multiplication problem, if it is solved by adding

18 + 18 + 18 then you are not thinking multiplicatively but are using

an additive strategy.

Similarly an addition problem e.g. 27 + 54 can be solved multiplicatively

by doing (3 x 9) + (6 x 9) = 9 x 9Refer to pages 3 - 4.

Discuss the strategies you would use to solve the following problem:

Each carton holds 36 cans of spaghetti

There are five cartons.

How many cans of spaghetti is that?

Let’s look at the possibilities…

• You may have used the distributive property. This meant that one of the factors was split additively.

• 5 x 36 = (5 x 30) + (5 x 6) = 150 + 30 = 180

• Or… 5 x 36 = 5 x 40 = 200 200 – (5 x 4) = 180

Another strategy:

• You may have used the commutative property in conjunction with the associative property.

5 x 36 = 36 x 5 (commutative) = 18 x 10 (associative) = 180

The associative property is about grouping factors:• So in 36 x 5, the 36 was split multiplicatively: 36 x 5 = (18 x 2 ) x 518 x 10 = 180

Using the Associative Property:

• There were 12 children. Each had 33 marbles. How many marbles are there altogether?

• Using the Associative Property, regroup the factors to make this an easier problem to solve!!

Proportional Adjustment:

• Transforming the factors to create a simpler problem.

• 12 x 33 becomes…• (4 x 3) x 33• 4 x (3 x 33)• 4 x 99 EASY!

A Multiplication Lesson:

• Watch the video and in your thinking groups discuss the following:– What was the key purpose of the lesson?– What stage was the lesson aimed at?– How was the key idea developed though the lesson?– What mathematical language was being developed?– When were the mathematical symbols introduces?– How did the written recording support the students

understanding?

Stage 2 - 3:

Aim: Working towards children seeing sets of numbers as a whole unit rather than by

counting one by one.• Building number knowledge: i.e. skip counting in

2’s, 5’s and 10’s. Using bead strings, flip boards, body percussion,

hundreds squares, calculator constant, number line pegs, animal strips.

• Introduce multiplication language: e.g.”groups of”, “lots of”, “sets of”, “teams of” etc..

Getting to know Book 6:In thinking groups explore a lesson…

• Stage 6/7 Multiplication Smorgasbord; p52

• Stage 5/6 A Little Bit more, A little bit less; p32

• Stage 4/5 Animal Arrays; p15

• Stage 3/4 Number Strips; p8

• Stage 2/3 Pre Book

Let’s look at Division:

• In your thinking groups make up a division problem for the following:

6 x 3 = 18

The Different Types of Division:

• Division by Sharing (partitive): 18 lollies to share equally into 3 bags. How many lollies in there in each bag?

• Division by Measuring/Grouping (quotitive): John has 18 lollies, he puts 6 lollies into each bag.

How many bags of lollies will he have?

Pirate Crews:

• Sharing – “There are two pirates in the crew. If they have 20 pieces of gold to share out, how many coins will each pirate get?”

Twenty coins shared between two pirates is ???

• Grouping -

Modelling Books:

• Share your modeling books with your thinking group.– What are the common features?– What have you seen that you would like to

develop?

Using a Modelling Book: • Helps capture students thinking and is a powerful

tool for modeling written recording.

• Is a useful record of work that both teacher and students can refer back to during the lesson or in future lessons.

• Helps teachers keep track of students progress by writing names next to strategies or by attaching the student profiles at the front of the book.

Next Time:

• We will be looking at Place Value, Fractions, Decimals, Ratio and Proportion.

• Please bring Books 1 and 7 and SHARED LUNCH.

• Try a knowledge check and diagnostic snapshot or recording in your modelling book or some equal grouping/sharing.

• See you on Wednesday, 2nd November.