Multiplication & Division Workshop Teacher: Who can tell me what 7 x 6 is? Pupil: 42! Teacher: Very good. Now who can tell me what 6 x 7 is? Pupil: It’s 24!

Add Sub Teaching - Where are you at? - What are your next steps? Talk about one child in your class. - what are they at - What are their next steps?

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

Multiplication Madness (Book 4 Page 36, MM4-16) Four in a Row Multiplication (Book 6 Page 36, MM6-6a)

Developing Multiplicative Thinking Early thinking to solve unknown basic facts: There are 8 minivans outside the school, they are going on a school trip. There are six children in each minivan. How many children are going on the trip? How would a student at different stages of the framework solve this problem? Hint…..use your Framework as a reference.

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

Skip Counting AC (Stage 4)

Repeated Addition EA (Stage 5) x 6 = 24

Derived Multiplication AA (Stage 6) 8 x 5 = 40 8 x 1 = 8 So =48

Derived Multiplication AA (Stage 6) 10 x 6 = (2x6) =48 2 x 6 = 12 8 x 6 = ?

Strategy Framework Revision Stage 2/3CACounts all the objects Stage 4ACUsing skip counting Stage 5EARepeated addition or known facts Stage 6AADerived multiplication Stage 7AMChoosing efficiently from a range of strategies and written form with whole numbers Stage 8APChoosing efficiently from a range of strategies with decimals and fractions Where were most of your class?

Multiplication snapshots 1. Here is a forest of trees. There are five trees in each row and eight rows. How many trees are in the forest altogether? If I planted 15 more trees, how many rows of five would there be then? STAGE 2 –3 Count All Solved by counting all the trees STAGE 4 Advanced Counting Solved by using skip counting, e.g. 5, 10, 15, 20, 25, …40 Stage 5 Early Additive Part-Whole Solved by using repeated addition e.g. (5 + 5 = 10, so = 40) or by forming the factors and knowing the fact, e.g. (8x5=40)

Multiplication 2. What is 3 x 20? If 3 x 20 is 60, what is 3 x 18? 3. What is 8 x 5 If 8 x 5 is 40, what is 16 x 5? STAGE 6 Advanced Additive Part-Whole Solved by deriving the answer from a known fact. (adding some on/taking some off, or doubling) and reversibility for division x 6 = ÷ 4 = STAGE 7 Advanced Multiplicative Part- Whole Using a range of strategies with larger numbers, e.g. place value partitioning or rounding and compensating, using reversibility

Stage 6 AA (NC Level 3) Using book 6 Page 24 Required Knowledge: Twos, fives and tens facts Knowledge being developed: -recalling all multiplication facts for 10 x 10 and some division facts Key Ideas: -Derive unknown multiplication facts efficiently -Solve division problems by reversing to multiplication. -Diagnostic Snapshot ( for each question)

X Study this grid

Deriving Multiplication Facts with Understanding 0 x 7 = 1 x 7 = 2 x 7 = (doubling / 7 + 7) 3 x 7 = 4 x 7 = (double then double again) 5 x 7 = (half of 10 x 7) 6 x 7 = 7 x 7 = 8 x 7 =(double, then double, then double again) 9 x 7 =(10 x 7) - (1 x 7) using tidy numbers 10 x 7 = (7 tens = seventy, “ty means tens”)

A point to note about division… Write a division story problem for the following: 8 ÷ 2 = 4

Different Types of Division 8 ÷ 2 = 4 Division by Sharing: 8 lollies shared between 2 people. How many lollies does each person have? Division by Grouping: John has 8 lollies, he puts 2 lollies into each bag. How many bags of lollies will he have?

Why is this important? Try solving this problem….. Is it division by sharing or by grouping? 2 1 / 2 ÷ 3 / 4 =

Dividing Fractions: Make 2½ or 5 / 2 with your fraction strips. 11 ½ How many groups of ¾ fit into 2½? 11 ½

Contexts for Multiplication and Division Equal groups Rate Comparison Part-Whole (ratio) Cartesian Product Rectangular Area 8 ÷ 2 = 4 Page 5 Book 6 Where does your story problem fit?

Arithmefacts x ÷ = = 3 6 x 3 = 18 6 ÷ 3 = x ÷ “6 - 3 = 3, and = 3”

What is Multiplicative Thinking? Multiplicative thinking is not about the type of problems you solve but how you solve it. Although 3 x 18 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

Stage 7 AM (NC Level 4) book 6 Page 41 Required Knowledge? Knowledge being developed? Key Ideas?

3 x 18 There were 3 minivans each with 18 children on them going on a school trip. How many children were there altogether?

Place Value (distributive property) 3 x 10 = 30. And 3 x 8 = 24. So = 54 Proportional Adjustment Double 3=6, half 18=9. So 6 x 9 = 54 Written Form 18 X 3 54 Tidy Numbers I know 3 x 20 = (3 x 2) = x 18 =

Using the Teaching Model How would you use the place value equipment to solve 3 x 18 by; -using tidy numbers? -using place value? (Multiplication Smorgasboard Book 6 p.52)

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

Tidy Numbers using Compensation 10 3 x 20 = (3 x 2) = 54

Proportional Adjustment Cut and Paste Book 6 p.49

6 x 4 = 3 x 8

Using Imaging for 3 x 18

3 x 18 3 x 9

3 x 18 = 6 x 9 3 x 18 6 x 9 x 2÷ 2

Generalise using number properties: 6482 x 5 12 x 33 Proportional Adjustment is about re-arranging the factors to create a simpler problem (Associative Property) 12 x 33 (2 x 6) x 33 2 x 2 x 3 x 33 4 x 99

Using Number Lines to show 3 x Proportional Adjustment Place value Tidy Numbers A B C

Multiplication Roundabout (MM6-6) Start

Multiplication Roundabout (MM6-6) E.g. Roll a 3. Move 3 places then multiply the number by 3

Multiplication Roundabout (MM6-6) E.g. Roll a 3. Move 3 places then multiply the number by 3

Multiplication Roundabout (MM6-6) x 3 = = 177(place counter between 150 & 200)

Let’s look at Division… 72 ÷ 4 Tidy Numbers Place Value Proportional Adjustment Reversing

Which strategy will you choose? 3680 ÷ 8 = A sheep station has eight paddocks and 3,680 sheep. How many sheep are there in each paddock?

3, 680 ÷ 8 Place Value: 3200 ÷ 8 = ÷ 8 = 60 Tidy Numbers 4000 ÷ 8 = (320 ÷ 8)= = 460 Proportional Adjustment: 3680 ÷ 8 = 1840 ÷ 4 = 920 ÷ 2 = 460 Algorithm Reversibility 8 x ? =

Getting to know Book 6 Choose an activity from Book 6. Find this activity highlighted on the long term unit plan Key Mathematical Ideas? Key Mathematical Knowledge? Diagnostic Snapshot? Teaching Model (materials, imaging, number properties) Any Follow up independent activity?

What now? Use snapshots if you are unsure of grouping for teaching mult div Source appropriate long term planning units for mult/div. Ask for support for planning and locating resources if needed Ask Lead Teachers for you to observe strategy teaching in your school. In Class Visits

Thought for the day Human beings share 99.4% of their DNA with the chimpanzee and 50% of their DNA with the cabbage.