Transcript 24 June Mult Div Workshop

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
The convention in New
Zealand is to regard
6 x 5 as 6 groups of 5
GLOSS Question
How many cups in each row?
How many rows of cups?
How many cups are there
altogether?
When asked to show 3 x 4 on the happy hundreds board, a
Y6 child in the top group could only show this – Why?
Don’t assume
children
understand
multiplication just
because they
know their basic
facts.
Develop
conceptual
understanding
Materials
•
•
•
•
Slavonic Abacus
Happy Hundreds Board
Unifix cubes
Multilink
Counting All From One (Stage2-3)
6x5
Skip Counting AC (Stage 4):
6x5
5
10
15
20
25
30
Repeated Addition EA (Stage 5):
6x5
10 5
++
105+=10
10 = 30
So how do you
think they
10 solve
would
4 x 9?
10
Derived Multiplication AA (Stage 6)
8x6
What does
deriving
mean?
8
x
5
=
40
How could you
derive 8 x 6?
8x1=8
So 40 + 8 =48
Derived Multiplication AA (Stage 6)
8x6
8x6
10 x 6 = 60
60- (2x6) =48
2 x 6 = 12
So how could you
solve 18 x 6?
Multiplication & Division Basic Facts
Knowledge
Mult/div basic facts
knowledge is then useful for
application to harder
problems.
Strategies
Unknown basic facts, e.g.
8x7 can be effectively taught
with understanding through
strategy.
Framework Revision
Stage
Strategy used to solve a
multiplication/division problem
Basic facts being
learnt for recall
2/3 CA
Counting all the objects, making groups
Doubles to 10
Skip counting in 2’s,5’s,10’s
4
AC
Skip counting for x2, x5, x10
(equal sharing for division)
doubles and halves to 20
5
EA
Repeated addition and known facts
(skip counting & repeated addition for
division)
x2, x5, x10 multiplication
and division facts
6
AA
Deriving (by splitting/doubling/ rounding)
(reversing/inverse operations for division)
x3, x4, x6, x7, x8, x9
7
AM
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
8
AP
Choosing efficiently from a range of
strategies and written form with decimals
and fractions
Where are most of your class?
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 18 + 18 + 18 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
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)
Place Value Partitioning
(splitting)
(3 x 20) - (3 x 2)
(3 x 10) + (3 x 8)
3 x 18
Proportional
Adjustment
6x9
Splitting Factors
3 x (3 x 3 x2)
Standard Written Forms
How would you use the teaching model to teach these
strategies? Book 6 p.52 onwards
Place Value Partitioning: 3 x 18
(Multiplication Smorgasboard, p.
3 x 10 = 30
30 + 24 = 54
3 x 8 = 24
10
10
10
Tidy Numbers
Multiplication Smorgasboard p.
3 x 18
3 x 20 = 60
60 - (3 x 2) = 54
10 10
10 10
10 10
Proportional Adjustment
Cut and Paste: Book 6 page 49
6x4=3x8
Using Imaging for 3 x 18
3 x 18
3x9
3x9
3 x 18 = 6 x 9
3 x 18
x2
÷
6x9 2
Generalise using number properties:
Proportional Adjustment
6482 x 5
is about re-arranging the
factors to create a simpler
problem
(Associative Property)
12 x 33
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 18
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)
C
0
Proportional
Adjustment
54
60
13 x 16 – Using dotty array
13 x 16 = 100+60+30+18
= 208
10
10
6
100
60
30
18
3
13 x 16
10
6
10
100
60
3
30
18
The algorithm is essentially the
same as this place value method
16
16
x 13 x 13
18
48
30 160
60 208
100
208
Practice
99 x 8
4 x 23
2 x 26
9 x 31
6 x 29
Multiplication Roundabout
(MM6-6)
Start
42
28
13
59
51
34
48
17
Multiplication Roundabout
(MM6-6)
E.g. Roll a 3. Move 3 places then multiply the number by 3
42
13
28
59
51
34
48
17
Multiplication Roundabout
(MM6-6)
E.g. Roll a 3. Move 3 places then multiply the number by 3
42
13
28
59
51
34
48
17
Multiplication Roundabout
(MM6-6)
59 x 3 =
180 - 3 = 177
(place counter between 150 & 200)
42
13
28
59
51
34
48
17
Multiplication & Division Basic Facts
Knowledge
Mult/div basic facts
knowledge is then useful for
application to harder
problems.
Strategies
Unknown basic facts, e.g.
8x7 can be effectively taught
with understanding through
strategy.
Sums and Products
Product
42
product
6
7
13
sum
15
3
5
8
sum
Multiplying by 10 (Stage 6)
How do you describe what happens?
Not just “add a zero”
The numbers move one place value along
Thousands
Hundreds
2
Tens
Ones
2
3
3
0
Arithmefacts
6
7
+
x
÷
+
x
÷
3
4
6+3= 9
6-3= 3
6 x 3 = 18
6÷3= 2
“6 - 3 = 3, and
7 - 4 = 3”
Division
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 ÷ 6
6 + 6 + 6 + 6 = 4 times
6, 12, 18, 24
Stage 6 AA:
Use reversibility from known multiplication fact
24 ÷ 6
6 x ? = 24
Stage 5
Using Skip Counting or Repeated Addition
24 ÷ 6
6 + 6 + 6 + 6 = 24
6, 12, 18, 24
24 shared between 6
or
24 put into groups of 6
6
6
6
6
6
6
6
6
Stage 5
Using Skip Counting or Repeated Addition
Using a number line
6
12
24 ÷ 6
18
24
Stage 6
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 7 Division
Compensation with tidy
numbers (rounding)
Place Value Partitioning
72 ÷
4
Proportional Adjustment
(adjust the divisor, the
dividend or both)
Splitting Factors
Standard Written
Form
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?