Transcript Developing Mathematical Thinking In Number : Focus on Multiplication

Developing Mathematical Thinking In
Number : Focus on Multiplication
Aim of presentation
To encourage staff reflection on approaches to teaching
number.
To stimulate professional dialogue.
To use as a CPD activity for staff individually or collegiately.
Experiences and Outcomes
I can use addition, subtraction, multiplication and division when solving problems, making
best use of the mental strategies and written skills I have developed. MNU 1-03a
Having determined which calculations are needed, I can solve problems involving whole
numbers using a range of methods, sharing my approaches and solutions with
others. MNU 2-03a
I can use a variety of methods to solve number problems in familiar contexts, clearly
communicating my processes and solutions. MNU 3-03a
Having recognised similarities between new problems and problems I have solved before, I
can carry out the necessary calculations to solve problems set in unfamiliar contexts. MNU
4-03a
Progression
Building up
times tables
How many cubes?
What would be
efficient ways of
finding out how
many cubes there
are?
Group in 2s and
Count in 2s?
Group in 5s and
Count in 5s?
Multiplication Facts
When children have mastered the facts of
eg x2, x3, x4, x5, x10,
children have only
10 more x facts to learn!
Discuss!
9
How well do children calculate?
Multiplication Facts
Using commutative property.
=
The 10 more facts to learn are
ie
6x6,
6x7,
6x8,
7x7,
7x8,
7x9
8x8,
8x9
9x9
6x9
Why?
Square numbers
2x2
3x3
Why are they
called square
numbers?
4x4
5x5
Any other
patterns?
6x6
How do we
encourage
pupils to
investigate?
What is the most
sensible order for
teaching times tables?
How can we
help children
see the links
between the
times tables?
“I know the 2x and 3x table. My
teacher tells me I know the rest.”
Discuss !
Making the links between the tables
From x2
x4 and x8 (doubling)
From x3
x6 (x2x3) and x9 (x3x3)
From x2 and x3
x5 (x2+x3)
From x3 and x4
x7 (x3+x4)
What about x10?
What tables does this help with?
From repeated addition to multiplication as array and as
area
3+3+3+3
4 rows of 3
= 4x3
4+4+4
How do these
images help
children’s
understanding?
3 rows of 4
= 3x4
3 x 24 = 24 + 24 + 24
Multiplication as repeated addition
20
4
20
20
24
3 x 24 = (3 x 20) + (3 x 4)
20
44
20
4
48
68
72
Using the distributive property of
multiplication
20
20
4
20
40
4
60
4
64
4
68
72
Progression 2nd level – ‘ using their knowledge of commutative,
associative and distributive properties to simplify calculations’
Illustrating the distributive law using money
3 x 24p = (3x20p) + (3x4p)
What might be
an added
challenge in
this example?
How do these
images help
children’s
understanding?
24p
24p
24p
Multiplication as area
30
14
Area = 30 x 14
Area models for multiplication
30 x 14 = (30 x 10) + (30 x 4)
= 300 + 120
= 420
10
14
4
30
30 x 10 = 300
30 x 4 = 120
Area models for multiplication
38 x 14
30 x 10 = 300
8 x 10 = 80
30 x 4 = 120
8 x 4 = 32
38 x 14 = 532
What is the
explanation for the
algorithm values ?
30
10
30 x 10 = 300
4
30 x 4 = 120
38
X14
152
380
532
Why include
the zero?
8
8 x 10 = 80
14
8 x 4 = 32
A challenge ...
Draw a similar diagram
to explain what is happening
in the calculation
48 x 34 ?
Solution
30
40
40 x 30 = 1200
8
8 x 30 = 240
34
4
30 x 4 = 120
8 x 4 = 32
Area models for multiplication
2 (x + 3) = 2x + 6
x
2
2 x x
3
3 x 2
Area models for multiplication
(x + 3) (x + 2) = x2 + 3x + 2x + 3x2
= x2 + 5x + 6
x
x
2
X2
2x
3
3x
3x2
Area models for multiplication
(x + a) (y + b) = xy + ay + bx + ab
x
y
b
xy
bx
a
ay
ab
Further support for progression in mathematics
http://www.ltscotland.org.uk/curriculumforexcellence/mathematics/outcomes/
moreinformation/developmentandprogression.asp
Make the links
12÷4=3
0.4x 3= 1.2
0.3x 4= 1.2
¼ of 12 = 3
3x4=12
12÷3=4
30 x 4= 120
30 x 40 = 1200
25% of 120 = 30
Next steps
What might you
or your staff do
differently in the
classroom?
What impact will this
have on your practice?
What else can you do as to
improve learning and teaching
about number
What
information
will you
share with
colleagues?
Developing Mathematical Thinking In
Number : Focus on Multiplication