Multplication and Division - Bruce Moody

There are two views regarding multplication and division. One view is that multiplication comes from repeated addition. The other view comes from a different way of thinking about number; it isn't subsequential. It is repetitive. 
It takes years to develop this. 

When talking about doubles, get the students to see them as copies rather than 4+4 =
e.g. I have 4 and Sarah has the same. 

In multiplication we have two counts going on. It is important that we drop skip couningt and turn it into multiplication in Year 3. 
5 x 6 = 30 not 5, 10 etc.
If we asked the problem "I have 3 kete and there are 2 kumara in each kete. How many kumara are there?"
Our eyes see 3 objects (kete), but the second count is inside - there are two kumara (second count is 2,4,6). 
The inside (kumara) and the outside (kete) count. 
The first check point is "Does the students see both counts?"
If they cannot think multiplicatively the student will not be able to do division.

Arrays 
The curriculum says wait until Level 3 to use arrays.
For early stories, use the terms 'groups of', 'bags of', 'teams of' … this is the natural (this is where we start). 

Students need to have 2x and 5x  This is a must have. (until the students have this without skip counting you cannot move on.)
They can then use isomorphism (same structure). 

Use a T chart - 
Hands     Fingers
    4               20
    5               25

7                        35
    8               40

6 hands must be 30 because it is in the middle of the two - the students do not need to go back and recount in 5s. Using a T Chart will break the cycle of skip counting especially if you challenge them to a race. It's like a holding a mirror up. Tell the students you will solve the probelm using the old way of skip counting. They can solve it using the new way. Which way is faster?

Suggested scenarios:
Buying dominoes pizzas at $5 each.
Players on the court in a basketball game (5 on at a time). 
Bags of potatoes 5kgs in a bag. How many kg altogether? 

Challenge the students to races - teacher skip counts and students use facts. This show the students that skip counting is slow. Then turn the modelling book over without the answers. Students will continue to beat the teacher. 
This is the aim for the end of Year 3.

For Year 4s (deriving multplication facts):
For 3 times use two groups with an extra group.
Use the commutative law.
Use the 2s to build the 3s and 4s. 

Bring out material again for these extensions. 
If I gave 4 people $7 each (physically hand then a$5 note and $2 each) in pairs how much have you got? $14 
Link to doubles.
How much for all 4 of you? 28. How many groups of $7 have I got? 4 x 7 = 28 
Maybe only 1 or two scenarios in one lesson. 

Think - in my story as a teacher what does it look like? We don’t want students to misunderstand. Have the model before the deriving. If the model was incorrect, the deriving will be wrong too. 
Show me what 6 groups of 2 looks like. 

For Year 5 & 6s (deriving and extending):
4 groups of 9 lollies story - Provide 4 film cannisters with 10 "lollies" in each cannister. How many lollies do I take out? - 4  how many will be left? 36 
4 x 9 = 36 
6 tens - 6 = 54 
Get students to see the pattern. 
Can extend up to 17 x 9 for older students 170-17 = 153. 
Using the model to extend. 

Year 6s need to be able to handle 
6x 17 

Division with reminders - needed for Year 6s 
"Violet was a terrible tagger and she wrote her name everywhere - what was the 53rd letter she wrote?" (6 letters in her name 8x6 =48 5th letter would be e) 
Stop students who start to write it - "Stop, how many letters are there?" Scaffold with questions.

Year 6s 7x8 = 56 
70 x 8 = 560 
(powers of 10 problems) 

Non unit fraction of a set - ⅗ of 30 
7/10 of 70 

They could get to 23x 34 but this is more for Year 7s.

Triangle Facts Handout.
Students see that they know their facts or they don’t.
Students highlight the facts they already know. 
There isn’t that many to learn is there.


Make the connection between 6x2 and 2x6 - investigate the communicative law of multiplication. 

9 squares handout

The students can make all times tables in order 

3

6

9

12

Division:

Addition and subtraction are opposites of each other so are division and multiplication

12 blocks all green

What are they - broccoli

Each student can have different colours 

Put them in bags by joining them together - students can decide how many go in each bag. 

Think about 12 multiplicatively. Hold up your bag, how many broccoli in a bag, how many bags have you got? How many brocolic did you start with? 

Division does not make the number smaller, it repackages them. 

I walked into a class the other day and I counted 30 fingers in the air, how many hands were there?

Use the T chart they have made in the past - get them to make the link.