Multiplication and Division Workshop Developing Multiplicative Thinking (through the Multiplication and Division Domain) Lisa Heap and Anuja Singh Mathematics Facilitators

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

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

Objectives: Understand the progressive strategy stages of multiplication & division Explore the properties of multiplication. Know how to use numeracy book six and other resources to help teach multiplication and division

Discuss these expectations in your groups: Teaching model used for strategy group teaching. Number knowledge being taught whole class, in groups as well as through independent activities. Modelling book used. Group boxes established. Planning from assessment to meet identified needs. Maths routines well established. Wait time for students to process their thinking. Listening to and beginning to respond to student’s thinking. Where are you at now? What’s your next step? Reflection on Numeracy Teaching:

The Development of Multiplicative Thinking: There are 6 minivans outside the 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? Hint…..use your Framework as a reference.

Strategy Framework Revision 2/3CACounts all the objects 4ACUses skip counting 5EARepeated addition or using known facts 6AADerived multiplication 7AMChoosing efficiently from a range of strategies and written form with whole numbers 8AP Choosing efficiently from a range of strategies with decimals and fractions

The convention in New Zealand is to regard 8 x 6 as 8 groups of 6 Make 8 x 6 using animal strips or happy faces

8 x Stage 4 - Skip Counting AC

8 x = =48 Stage 5 - Repeated Addition EA

8 x 6 8 x 5 = 40 8 x 1 = 8 Stage 6 - Derived Multiplication AA

8 x 6 10 x 6 = (2x6) =48 2 x 6 = 12 Stage 7 - Derived Multiplication AM

Multiplicative Thinking: Multiplicative thinking is not about the type of problems you solve but how you solve it. E.g. Although 3 x 23 is a multiplication problem, if it is solved by adding then you are not thinking multiplicatively but are using an additive strategy. Similarly an addition problem e.g can be solved multiplicatively by doing (3 x 9) + (6 x 9) = 9 x 9 What is multiplicative thinking?

Using Number Properties Using Imaging Using Materials New Knowledge & Strategies Existing Knowledge & Strategies Using Materials The Strategy Teaching Model

3 x x 10 = 30 Place Value Partitioning 3 x 8 = = 54

3 x x 20 = (3 x 2) = 54 Tidy Numbers using Compensation

Proportional Adjustment:

3 x 18 3 x 9

6 x 9 = 54 3 x 18 6 x 9 x 2÷ 2 Proportional Adjustment:

Proportional Adjustment Place value Tidy Numbers A B C Using Number Lines:

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?

Lets 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) = = 180 The 36 was split (distributed) into

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 the factors: So in 36 x 5, the 36 was split multiplicatively: 36 x 5 = (18 x 2) x 5 = 18 x 10 = 180

Using the Associative Property: There were 12 children. Each had 33 marbles. How many marbles are 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 99EASY!!

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 throughout the lesson? What mathematical language was being developed? When were the mathematic symbols introduced? How did written recording support the student’s 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” etc..

Exploring lessons from Book 6: Stage 2- 3 Birthday Cakes Feed the Elephants Stage 3-4 Number Strip (pg.8) Stage 4-5 Animal Arrays (pg.15) Pirate Crews (pg. 17) Biscuit Boxes (pg.19)

Exploring Lessons in Book 6: Stage 4-5 Animal Arrays (pg. 15) Biscuit Boxes Stage 5-6 Fun with Fives (pg.28) A little Bit More, A little Bit Less (pg. 32) Stage 6-7 Cut and Paste (pg. 49) Proportional Packets (pg.54)

Solve 3 x 18 Stage 7 Advanced Multiplicative: I have 3 children. I give them 18 lollies each, how many lollies do I need to buy altogether?

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 6 bags. How many lollies in each bag? Division by Measuring/Grouping (quotitive): John has 18 lollies, he puts them 3 lollies to a bag. How many bags of lollies will he have?

Why is this important? Try solving this problem….. Is it division by sharing or by measuring and grouping? 2 ½ ÷ ½ =

Division Delights FIO N3-4; 18 The Goodwill gang get paid $54 for picking blueberries. There are 3 people in the gang. What is each person’s share of the money?

54 ÷ 3 Using Place Value: 30 ÷ 3 = 1024 ÷ 3 = = 18 Using Tidy Numbers: 60 ÷ 3 = 206 ÷ 3 = = 18 Using Proportional Adjustment: 54 ÷ 6 = 954 ÷ 3 = 2 x 9= 18

Solving a Division Problem: A sheep station has eight paddocks and 296 sheep. How many sheep are there in each paddock?

296 ÷ 8 Reversibility 8 x 30 = x 7 = 56 Place Value: 240 ÷ 8 = ÷ 8 = = 37 Tidy Numbers 4000 ÷ 8 = (320 ÷ 8)= = 460 Rounding and Compensating 296 ÷ 8 = 148 ÷ 4 = 74 ÷ 2 = 37 Proportional Adjustment: Algorithm 320 ÷ 8 = (24 ÷ 8)= = 37

Review Objectives: Understand the progressive strategy stages of multiplication & division Explore the properties of multiplication and division Know how to use numeracy book six and other resources to help teach multiplication and division

Thought for the day: “Success is…. getting up one more time than you fell down!”