Transcript Number Lines

Mathematics
Kings Worthy Primary School
1st May 2012
Brenda Robertson
Teaching and Learning Advisor
Primary Mathematics
When you were at
school…..
• How did you feel about maths?
• What were your experiences of
maths like?
• How do you feel about maths
now?
Maths in school today….
• Maths is fun!
• Children are encouraged to “have
a go”!
• They are not afraid to be wrong –
they learn from mistakes.
• Maths is made purposeful and
interesting
Why has maths changed?
• Children used to learn “standard methods” –
we were shown what to do.
• If you couldn’t remember, or hadn’t
understood the methods…you went wrong.
• Today, we teach methods that help children
to understand the underlying maths and the
basic concepts involved
• Children need to develop “number sense” –
more insight into mathematics
Today’s techniques…
• …are not just about getting the right answer
– but about knowing and understanding how
you got there
• …are not new – many pre-date the techniques
you learned
• …eventually join up with the ones you did in
school – but the children understand them
thoroughly
• …reduce the chance of mistakes being made
• …build a firm foundation for understanding
more complicated mathematics later on
A balanced mathematical diet…
• Using and applying mathematics
• Counting and understanding
number
• Knowing and using number facts
• Calculating
• Understanding shape
• Measuring
• Handling data
Place Value
• Our number system consists of ten digits
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• The place of each digit determines its value.
• For example, the “6” digit can represent for
6, sixty, six hundred, six thousand…it
depends where we place it…
6….65….653…6782…
Number Lines
• They help to develop an ability to order numbers, and
give children a sense of where numbers sit in our
number system
•
They allow children to draw a picture – or model – in
their heads when calculating
• They allow us to see the strategy being used – and the
refinements that occur.
• Can be used in a range of aspects of mathematics –
numbers and the number system, fractions,
decimals, percentages, addition, subtraction,
multiplication, division, measures, handling data
The four types of number
tracks or number lines
Number tracks
1
2
3
4
5
6
7
10
11
12
8
9
10
Numbered lines
5
6
7
8
9
13
14
15
16
The four types of number tracks or
number lines…
• Partly numbered line
5
10
• The empty number line
15
Jump 10’s first
26 + 23
+10
+10
+1
26
36
46
+1
47
+1
48
49
-10
-10
44 - 25
19
20
21
22
23
24
34
44
Jump and adjust
26 + 9
“Compensating”
+10
26
44 - 9
34
+1
35
35
-10
44
36
-1
number lines give a clear image of
repeated addition (multiplication)
2
0
+
2
2
+
4
2
+
6
2
+
8
2
10
This image can be expressed as 2 multiplied by 5, two
five times, 5 groups of 2, 5 lots of 2 and 5 hops of 2 on
a number line.
Multiplication on a number line
There are 5 cakes in one box. How many cakes in 4 boxes?
Division on a number line
I have 20 cakes, I can fit 5 cakes in a box.
How many boxes will I need?
Mental skills and strategies
We constantly draw upon mental strategies that we
know and have internalized when we carry out
calculations
•
•
•
•
•
doubling
adding multiples of ten
Partitioning
compensating
applying known facts
Children need to learn and understand these strategies
– and know how and when to apply them.
Partitioning
• This means breaking up numbers
into smaller numbers
• All numbers can be partitioned in
many different ways
• If children understand about
partitioning numbers in different
ways, they will calculate more
efficiently – and with
understanding.
Partitioning to add…
452
+289
400 50 2
+200 80 9
600+130+ 11
Partitioning to subtract
60
70 + 4
- 20 + 7
14
70 + 4
- 20 + 7
40 + 7
60 14
74
- 27
47
Arrays
Arrays are important because they provide a good
visual image of the multiplication that links closely
to the concept of repeated addition.
2+2+2+2
4x2=8
2x4=8
They are the visual image which leads directly into grid multiplication
a key strategy in KS2
Arrays
• They show the link between
multiplication and division:
2x4=8
4x2=8
8÷2=4
8÷4=2
Using arrays
35 x 2
x
30
5
2
60
10
The grid method
Division… “grouping” or “sharing”?
• I have 48 toffees and I want to
share them equally into 8 bags.
How many toffees will be in each
bag?
• I have 48 toffees and I want to
put them into bags of 8. How
many bags do I need?
Division - Sharing
I share 9 sweets between 3 children how
many sweets each do they each have?
Division - Grouping
9 fish are put into bowls. If 3 fish
go in each bowl how many bowls
are there?
Division – formal written methods
Let’s look at an example…
Eleanor wanted to cut a piece of
ribbon that measured 749cm long into
7 equal pieces. How long would each
piece be?
What went wrong for these pupils?
17
7 749
7
101 r 2
749
Division by “chunking”
7 749
700
49
X 100
X7
107
How many sevens can I get
out of 749?
Well, 100 sevens is 700 – so
I know I can get 100.
That leaves me with 49.
I know 7 sevens are 49, so
that’s another seven lots of
seven…
So that’s 107 lots of seven
Altogether!
Division by “chunking” or “grouping”
• This is an extended method that
allows children to use what they know
to solve the problem – and to develop
understanding about the process
• Large “chunks” or “groups” are being
subtracted from the number being
divided into.
• Most children will eventually
understand how long division works if
they refine and use this extended
method first.
• Resources, models and images help
children to visualise and understand
mathematical concepts
• They build up – and remember - the
mental picture in their minds
• They should be available throughout
the primary years. Children will rely on
them less and less
When tackling a mathematical problem,
children will ask themselves…
• What do I know about these numbers?
•
Can I do this in my head?
• Do I know the approximate size of the
answer?
• If I can’t do it all in my head, what do I need
to write down to help me?
“Above all – have fun! Children
often say that maths is their
favourite subject and they get
quicker and better at it when
they understand what they are
doing. There’s no need for you to
“teach” your child: you help by
helping them to explain their
thinking and their understanding”
From “Count on me – 200 ways to help with mathematics”
• “It’s the school’s job to provide
the structured learning….Your
role is to nurture and support
your child’s mathematical
knowledge away from school, to
bring it into their real lives and,
most important of all, to turn it
into an exciting adventure”
From “Maths for Mums and Dads” – Rob Eastaway and
Mike Askew