Transcript Fractional Framework

Workshop 4:
Multiplication, Division
and Fractions
Presented by CW
Building the knowledge foundations
Factor of 10 between columns
Groups to 20
Decade Pairs to 100
10’s in any number
Name columns
10+
Counting
Using decades
5+
Numerals
X 2, x5, x10
Larger numbers
Importance of 10
Doubles to 20
Groups to 5
Using 10 x
Using facts to10
Count in 2’s 5’s 10’s
Groups to 10
Sequence
Doubles to 10
Language
Sets
Multiplication type problems
I have 3 bags of lollies and there are 5 lollies
in each bag.
How many lollies have I got altogether?
Can I solve the problem if I am stage 0 or stage 1?
How would I solve the problem if I am stage 2 or
stage 3?
How could I solve the problem if I am at stage 4 and
what knowledge MUST I have in my tool box?
The key Idea to use Skip Counting
 Skip Counting like counting can be just rote
learnt.
 Realisation that the skip counting sequence
relates to putting the same sets of numbers
together and the end count measures the
set and that the rote count forwards gives
the result of adding the set number of
objects and backwards gives the result of
subtracting the set number of objects.
Stage 4 - 5
 Key Idea: Children are learning to use
addition strategies to solve problems that
could be solved by multiplication.
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Multiplication as repeated addition
Commutativity of multiplication
Sharing into equal sets
Grouping, how many sets can be made.
Stage 5: Using known facts
 2 x doubles
 10 x place value
 5 x place value and doubles (halves)
X
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
4
6
8
10
12
14
16
18
20
6
8
12
14
16
18
15
20
30
35
40
45
30
40
10
15
20
25
30
35
40
45
50
60
70
80
90
20
30
40
50
60
70
80
90
100
Stage 5 - 6
 Children are learning to derive further
multiplication facts using addition and subtraction
strategies from multiplication facts they already
know.
 4 x = 2 x + 2 x (doubling)
 3x=2x+1x
 6x=5x+1x
( or double 3 x when 3 x are known facts)
X
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
4
6
8
10
12
14
16
18
20
6
9
12
15
18
21
24
27
30
8
12
16
20
24
28
32
36
40
10
15
20
25
30
35
40
45
50
12
18
24
30
36
42
48
54
60
14
21
28
35
42
16
24
32
40
48
18
27
36
45
54
70
80
90
20
30
40
50
60
70
80
90
100
Not many left to learn
 7x =5x+2x
 8 x = 5 x + 3 x or 10 x – 2 x
 9 x = 10 x – 1 x
Children need to understand and be able to use the
relationships between the multiplication tables.
When working with bigger numbers the “tables”
become the knowledge tools for multiplicative
thinking in the same way as basic addition facts
are the knowledge tools for additive thinking.
Tools in the tool box
Counting
Plastic tools, good to start with but
not very efficient
Addition and Subtraction
Increasing range of more sophisticated
tools
Multiplication and Division
The power tools.
Increasingly efficient and sophisticated.
Fractions at ENP
Why? When ? How?
Fractional Misconceptions
 No acceptance of the existence of numbers
between numbers
Share 5 biscuits between 4 children …. You
can’t do it!
Fractional Misconceptions
 The numerator relates to the whole
½ is 1 shared between 2
¼ is 1 shared between 4
Fractional Misconceptions
 Half is a variable hence all fractions are
variable
½ of 12 is 6
½ of 8 is 4
½ of 4 is 2……so the half keeps changing
Fractional Misconceptions
 The symbol can be either way up
 You can’t have fractions greater than 1
4 = 6
6
4
never been challenged
if fractions greater than
haven’t been explored
Fractional Framework
Regions
Knowing how
many parts
make a whole
Number
Sets
Recognising the Using half as an
symbol
operator on
Knowing where other numbers
half lives on a
number line
Stage 2/3
Regions
Know half is 2
equal parts.
Know quarter is
4 equal parts.
Number
Recognise
symbol ½
Recognise
symbol ¼
Sets
Doubles and
halves
Stage 4
Regions
Number
Sets
1 half, 2 halves,
3 halves
1 quarter,
2 quarters,
3 quarters
4 quarters
5 quarters
Place of ½ on a
number line.
Place of 2
halves 3 halves
etc.
Find half and
quarter of a set
by equal
sharing.
Stage 5
Regions
Number
Sets
Comparison and
ordering of unit
fractions by region.
½>¼>⅛
Extend knowledge to
other fractions: Thirds,
fifths, tenths etc.
Meaning of – ths
Identify symbols for
fractions including
those greater than 1
Place of ½ on a
number line
2x½=1
4x½=2
Link to repeated
addition for
multiplication.
Comparison and
ordering of unit
fractions by sets of
½ of 12 = 6
¼ of 12 = 3
Link to repeated
addition and
commutativity of
multiplication.
½ x 12 = 12 x ½ = 6