Transcript Mult Div Pickup

Developing Multiplicative
Multiplication and
Thinking
Division
(through the Multiplication and
Division Domain)
Workshop
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
Reflection on Numeracy Teaching:
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?
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
• 2/3 CA
• 4 AC
• 5 EA
• 6
• 7
• 8
Counts all the objects
Uses skip counting
Repeated addition or using known
facts
AA
Derived multiplication
AM Choosing efficiently from a range of
strategies and written form with whole
numbers
AP
Choosing efficiently from a range of
strategies with decimals and fractions
Make 8 x 6 using animal strips
or happy faces
The convention in
New Zealand is to
regard 8 x 6 as 8
groups of 6
8Stage
x 6 4 - Skip Counting AC
6
12
18
24
30
36
42
48
5 - Repeated Addition EA
8 xStage
6
12 + 12 = 24
24 + 24 =48
Stage
8 x 6 6 - Derived Multiplication AA
8 x 5 = 40
8x1=8
Stage
8 x 6 7 - Derived Multiplication AM
10 x 6 = 60
60- (2x6) =48
2 x 6 = 12
Multiplicative Thinking:
What is 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 23 + 23 + 23 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 9
The Strategy Teaching Model
Existing Knowledge &
Strategies
UsingUsing
Materials
Materials
Using Imaging
Using Number Properties
New Knowledge &
Strategies
Place Value Partitioning
3 x 18
3 x 10 = 30
30 + 24 = 54
3 x 8 = 24
10
10
10
Tidy Numbers using
Compensation
3 x 18
3 x 20 = 60
60 - (3 x 2) = 54
10 10
10 10
10 10
Proportional Adjustment:
3 x 18
3x9
3x9
Proportional Adjustment:
6 x 9 = 54
3 x 18
x2
6x9
÷
2
Using Number Lines:
18
0
9
A
9
18
9
9
18
9
54
9
54
3 x 10
Place value
3x8
30
0
54
B
Tidy
Numbers
3 x 20
- (3 x 2)
0
C
Proportional
Adjustment
54
60
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)
•
= 150 + 30
•
= 180
• The 36 was split (distributed) into 30 + 6
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 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 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)
Stage 7 Advanced
Multiplicative:
I have 3 children. I give them 18 lollies
each, how many lollies do I need to buy
altogether?
Solve 3 x 18
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…..
2½÷½ =
• Is it division by sharing or by measuring
and grouping?
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 = 10
24 ÷ 3 = 8
10 + 8 = 18
Using Tidy Numbers:
60 ÷ 3 = 20
6÷3=2
20 - 2 = 18
Using Proportional Adjustment:
54 ÷ 6 = 9
54 ÷ 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?
Reversibility
296 ÷ 8
8 x 30 = 240
8 x 7 = 56
Proportional
Adjustment:
296 ÷ 8 =
Place Value:
240 ÷ 8 = 30
56 ÷ 8 = 7
30 + 7 = 37
148 ÷ 4 =
74 ÷ 2 = 37
Algorithm
Tidy Numbers
Rounding
and
Compensating
4000
÷ 8 = 500
320 ÷ 8 =40
500 - (320 ÷ 8)=
40 - (24 ÷ 8)=
500 - 40 = 460
40 - 3= 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!”