Reflection What have you tried since our last workshop? Number knowledge teaching Addition/subtraction teaching Group rotations Nzmaths What’s gone well for you / your children? What has not gone so well for you / your children?

Effective Pedagogy in Mathematics Guiding principles for effective mathematics teaching. What does it mean for you as a classroom teacher?

Scenarios – Addition and Subtraction Work with a partner – read the scenarios and assign a stage to each one. Cut them out and order them according to stages. Be prepared to justify your decisions

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: 24!

Purpose of this session… Understand Number Framework Progressions for multiplication and division Develop personal content knowledge Reinforce the teaching model to teach strategy Continue to develop an effective classroom maths programme

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

Understanding the Number Framework for Multiplication and Division Stage Name / Number Description Strategy Example

Show 6 x 5 on your Happy Hundreds Board GLOSS Question How many cups in each row? How many rows of cups? How many cups are there altogether? The convention in New Zealand is to regard 6 x 5 as 6 groups of 5

Don’t assume children understand multiplication just because they know their basic facts. Develop conceptual understanding When asked to show 3 x 4 on the happy hundreds board, a Y6 child in the top group could only show this – Why?

Materials Slavonic Abacus Happy Hundreds Board Unifix cubes Multilink

Counting All From One (Stage2-3) 6 x 5

Skip Counting AC (Stage 4): 6 x

Repeated Addition EA (Stage 5): 6 x = = 30 So how do you think they would solve 4 x 9?

Derived Multiplication AA (Stage 6) 8 x 6 8 x 5 = 40 8 x 1 = 8 So =48 What does deriving mean? How could you derive 8 x 6?

Derived Multiplication AA (Stage 6) 8 x 6 10 x 6 = (2x6) =48 2 x 6 = 12 8 x 6 So how could you solve 18 x 6?

Multiplication & Division Basic Facts Unknown basic facts, e.g. 8x7 can be effectively taught with understanding through strategy. StrategiesKnowledge Mult/div basic facts knowledge is then useful for application to harder problems.

Where are most of your class? StageStrategy used to solve a multiplication/division problem Basic facts being learnt for recall 2/3CA Counting all the objects, making groupsDoubles to 10 Skip counting in 2’s,5’s,10’s 4AC Skip counting for x2, x5, x10 (equal sharing for division) doubles and halves to 20 5EA Repeated addition and known facts (skip counting & repeated addition for division) x2, x5, x10 multiplication and division facts 6AA Deriving (by splitting/doubling/ rounding) (reversing/inverse operations for division) x3, x4, x6, x7, x8, x9 7AM Choosing efficiently from a range of strategies (place, value, tidy numbers, proportional adjustment) and written form with larger whole numbers ÷3, ÷4, ÷6, ÷7, ÷8, ÷9 Square numbers & roots 8AP Choosing efficiently from a range of strategies and written form with decimals and fractions Framework Revision

Developing Stage 7 Content Knowledge (AM = NZC Level 4) book 6 Page 41 Take a moment to read….. Required Knowledge Knowledge being developed Key Ideas

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

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

Compensation with tidy numbers (rounding) (3 x 20) - (3 x 2) Place Value Partitioning (splitting) (3 x 10) + (3 x 8) Proportional Adjustment 6 x 9 Splitting Factors 3 x (3 x 3 x2) 3 x 18 How would you use the teaching model to teach these strategies? Book 6 p.52 onwards Standard Written Forms

Place Value Partitioning: 3 x 18 (Multiplication Smorgasboard, p x 10 = 30 3 x 8 = = 54

Tidy Numbers 3 x 18 Multiplication Smorgasboard p x 20 = (3 x 2) = 54

6 x 4 = 3 x 8 Proportional Adjustment Cut and Paste: Book 6 page 49

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

13 x 16 – Using dotty array

13 x 16 = =

13 x x x The algorithm is essentially the same as this place value method

Practice

4 x 23 6 x 292 x 26 9 x x 8

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)

Multiplication & Division Basic Facts Unknown basic facts, e.g. 8x7 can be effectively taught with understanding through strategy. StrategiesKnowledge Mult/div basic facts knowledge is then useful for application to harder problems.

Sums and Products sum product sum Product

Multiplying by 10 (Stage 6) ThousandsHundredsTensOnes 23 Not just “add a zero” The numbers move one place value along How do you describe what happens?

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

Division

Write a division story problem 8 ÷ 2 = 4 “shared between” “put into groups of”

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?

Now write two division story problems for the following equation using both a sharing and grouping context 18 ÷ 3 = 6

Division Stage 5 EA: Use skip counting/repeated addition, 24 ÷ = 4 times 6, 12, 18, 24 Stage 6 AA: Use reversibility from known multiplication fact 24 ÷ 6 6 x ? = 24

Using Skip Counting or Repeated Addition 24 shared between 6 24 put into groups of ÷ = 24 or 6, 12, 18, Stage 5

Using a number line Stage 5 24 ÷ 6 Using Skip Counting or Repeated Addition

Using reversibility: 24 ÷ 6 = ? ? groups of 6 = 24 Goesintas p. 6 How many groups of 6 make 24? ? X 6 = 24 So….. 24 ÷ 6 4 x 6 = 24 Stage 6

Compensation with tidy numbers (rounding) Place Value Partitioning Proportional Adjustment (adjust the divisor, the dividend or both) Splitting Factors 72 ÷ 4 Standard Written Form Stage 7 Division

NZ Curriculum and Number Framework -Knowing what to teach based on assessment data Effective Pedagogy -Knowing how to teach it -How you respond to students and their misconceptions -Notice, Understand, Respond

Activities Junior – Marilyn Holmes activities based on work of Richard Skemp Senior – FIO books – find activities that you could use in your class, work together on a plan to use in your classroom. Basic Facts?