Transcript Pickups 2010 - mathsleadteachers

Developing Multiplicative
Multiplication and
Thinking
Division
(through the Multiplication and
Division Domain)
Workshop
Fiona Fox & Lisa Heap
Mathematics Facilitator
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:
• Reflect on addition & subtraction strategy
teaching.
• Understand the progressive strategy stages of
multiplication & division
• Explore the properties of multiplication.
• Know how to use planning sheets, numeracy book
six and other supportive resources to help teach
multiplication and division
Reflection on Numeracy Teaching:
Discuss these expectations in your groups:
•
Teams sharing planning and teaching ideas.
•
Teaching model used for strategy group teaching.
•
Number knowledge being taught whole class as well
as through independent activities.
•
Modelling book used.
•
Tried some Figure It Out or website activities.
•
Using the long term planning units.
•
Maths routines well established
•
Listening 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
Counts all the objects
Uses skip counting
Repeated addition or using known
facts
6 AA
Derived multiplication
7 AM Choosing efficiently from a range of
strategies and written form with whole numbers
• 8 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
8 Skip
x 6 Counting AC
6
12
18
24
30
36
42
48
Addition EA
8 xRepeated
6
12 + 12 = 24
24 + 24 =48
Derived
8 x 6 Multiplication AA
8 x 5 = 40
8x1=8
Derived
8 x 6 Multiplication AA
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
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..
A Strategy Teaching Lesson:
Start of Lesson Consider strategy stage of group
-
Required knowledge check
-
Diagnostic Snapshot (Can they already do it?)
Teaching
- Teaching Model (materials, imaging, number
properties)
-
Consider thinking groups & modelling book
End of Lesson
-
Feedback: Who’s got it? Who hasn’t? Set
practice.
Consider next teaching and learning steps.
Exploring a lesson from Book 6:
Break into 3 groups to explore a mult/div
lesson:
Stage 3-4
Number Strips Pg.8
Stage 4-5
Animal Arrays Pg.15
Stage 6-7
Cut & Paste Pg.49
Stage 7 Advanced
Multiplicative:
I give 3 lollies to every child in my class. I
have 18 children in my class, how many
lollies did I need to buy altogether?
Solve 3 x 18
Tidy Numbers using
Compensation
3 x 18
3 x 20 = 60
60 - (3 x 2) = 54
10 10
10 10
10 10
Place Value Partitioning
3 x 18
3 x 10 = 30
30 + 24 = 54
3 x 8 = 24
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
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 3 bags. How many lollies in
each bag?
• Division by Measuring/Grouping (quotitive):
John has 18 lollies, he puts them 6 lollies to a bag.
How many bags of lollies will he have?
Why is this important?
• Try solving this problem…..
2 1/2 divided by 1/2
=
• 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
The Strategy Teaching Model
Existing Knowledge &
Strategies
UsingUsing
Materials
Materials
Using Imaging
Using Number Properties
New Knowledge &
Strategies
Review Objectives:
• Revise addition & subtraction strategy teaching
• Understand the progressive strategy stages of
multiplication & division
• Explore the properties of multiplication and
division
• Know how to use planning sheets, numeracy book
six and other supportive resources to help teach
multiplication and division
Thought for the day:
“Success is….
getting up one more time than you
fell down!”
A Strategy Teaching Lesson:
Start of Lesson Consider strategy stage of group
-
Required knowledge check
-
Diagnostic Snapshot (Can they already do it?)
Teaching
- Teaching Model (materials, imaging, number
properties)
-
Consider thinking groups & modelling book
End of Lesson
-
Feedback: Who’s got it? Who hasn’t? Set
practice.
Consider next teaching and learning steps.