developing mathematical thinking in number : focus on multiplication
TRANSCRIPT
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?
What would be efficient ways of finding out
how many cubes there
are?
Group in 2s and Count in
2s?
Group in 2s and Count in
2s?
Group in 5s and Count in
5s?
Group in 5s and Count in
5s?
9
When children have mastered the facts ofeg x2, x3, x4, x5, x10,
children have only
10 more x facts to learn!
Multiplication Facts
Discuss!
Multiplication Facts
Using commutative property.
The 10 more facts to learn are
ie 6x6, 6x7, 6x8, 6x9 Why?7x7, 7x8, 7x9
8x8, 8x9
9x9
=
How well do children calculate?
6x6
Square numbers
5x54x43x32x2
Any other
patterns?
Any other
patterns?
Why are they called
square numbers?
Why are they called
square numbers?
How do we
encourage pupils to investigat
e?
How do we
encourage pupils to investigat
e?
What is the most sensible order for
teaching times tables?
What is the most sensible order for
teaching times tables? How can we
help children see the links between the 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 !
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)
Making the links between the tables
What about x10? What tables does this help with?
What about x10? What tables does this help with?
From repeated addition to multiplication as array and as area
3+3+3+3 4+4+4
4 rows of 3 = 4 x 3
3 rows of 4 = 3 x 4
How do these images help children’s
understanding?
How do these images help children’s
understanding?
20 4 20 4 20 4
20 24 44 48 68 72
3 x 24 = 24 + 24 + 24
Multiplication as repeated addition
20 20 20 4 4 4
20 40 60 64 68 72
3 x 24 = (3 x 20) + (3 x 4)
Using the distributive property of multiplication
Progression 2nd level – ‘ using their knowledge of commutative, associative and distributive properties to simplify calculations’
24p
Illustrating the distributive law using money 3 x 24p = (3x20p) + (3x4p)
How do these images help children’s
understanding?
How do these images help children’s
understanding?
What might be an added challenge in
this example?
What might be an added challenge in
this example?
24p
24p
14
30
Area = 30 x 14
Multiplication as area
1410
4
30
30 x 10 = 300
30 x 4 = 120
30 x 14 = (30 x 10) + (30 x 4) = 300 + 120 = 420
Area models for multiplication
14
10
4
30
30 x 10 = 300
30 x 4 = 120
38 x 14
8 x 10 = 80
8 x 4 = 32
8
30 x 10 = 300 8 x 10 = 8030 x 4 = 120 8 x 4 = 32
38 x 14 = 532
Area models for multiplication
What is the explanation for the algorithm
values ?Why
include the zero?
A challenge ...
Draw a similar diagram
to explain what is happening
in the calculation
48 x 34 ?
Solution
34
30
4
40
40 x 30 = 1200
30 x 4 = 120
8 x 30 = 240
8 x 4 = 32
8
2
x
2 x x
2 (x + 3) = 2x + 6
3 x 2
3
Area models for multiplication
x
2
x
X2
2x
(x + 3) (x + 2) = x2 + 3x + 2x + 3x2 = x2 + 5x + 6
3x
3 x 2
3
Area models for multiplication
y
b
x
xy
bx
(x + a) (y + b) = xy + ay + bx + ab
ay
ab
a
Area models for multiplication
Further support for progression in mathematics
http://www.ltscotland.org.uk/curriculumforexcellence/mathematics/outcomes/moreinformation/developmentandprogression.asp
Make the links
3x4=12
12÷3=4
12÷4=3
¼ of 12 = 330 x 4= 120
30 x 40 = 1200
0.3x 4= 1.2
0.4x 3= 1.2
25% of 120 = 30
Next stepsWhat
information will you
share with
colleagues?
What might you or your
staff do differently in
the classroom?
What else can you do as to improve learning and
teaching about number
What impact will this have on your
practice?
What impact will this have on your
practice?
Developing Mathematical Thinking In Number : Focus on Multiplication