Place Value Workshop
Friday, 27th September
University of Greenwich
Place Value Workshop
Objectives
• Understand issues & progressions in
recording larger numbers
• Use effectively a range of
manipulatives, reflecting place value
• Know common misconceptions linked
with place value
• Recognise the cultural and historical
aspects of place value
Pre-requisites for learning Place Value
Identify a set given the number
• For example select a set of say four objects from a collection
of different sized sets when asked to pick out the set of four.
Create a set given the number
• For example when asked to put out six objects can do so.
Correctly name the number of objects in a set
• For example shown a selection of eight objects can say that
there are eight.
Can do all the above but presented with numbers in a written
form rather than spoken, and can record as number symbols
sets of 0-9 objects.
Can count from 1 through 10 both with and without objects.
Askew, M. (1998) Teaching Primary Mathematics. London: Hodder & Stoughton
Why is it so important that children
understand place value?
• They can develop mental calculation methods
that are effective and efficient
• Paper and pencil methods of calculation can be
carried out with understanding
• Multiplying and dividing by 10 or multiples of 10
become simple
• Decimal fractions and percentages can be
understood as extension of the place value
system.
Askew, M. (1998) Teaching Primary Mathematics.
London: Hodder & Stoughton
How does our number system
work?
Work with a partner to fill in the gaps in the
Chinese and Bengali number square.
How did you work out the missing numbers?
How does this link to our number system?
Principles of our number system
• All numbers are made up of digits ( 1 – 9)
• Zero is used as a place holder to represent an
empty column
• The column the digit is placed in determines
its value.
• Each column is 10x bigger / 10x smaller to the
one next it depending on the direction of travel
18.732
10x smaller
10x bigger
Teaching Maths with Diversity
http://webarchive.nationalarchives.gov.uk/201010
21152907/http://www.Multiverse.ac.uk/ViewArticle
2.aspx?anchorId=131&selectedId=149&menu=178
77&expanded=False&ContentId=523
This link above looks at other PV written systems.
You may want to look at it with children – especially
in a cross curricular context.
What about Roman Numerals? …
Roman Numerals have now been
introduced into the new NC
Units
Tens
Hundreds
Thousands
I
One
1
X
Ten
10
C
One
hundred
100
M
One
thousand
1000
II
Two
2
XX
Twenty
20
CC
Two
hundred
200
MM
Two
thousand
2000
III
Three
3
XXX
Thirty
30
CCC
Three
hundred
300
MMM
Three
thousand
3000
IV
Four
4
XL
Forty
40
CD
Four
hundred
400
MMMM
Four
thousand
4000
V
Five
5
L
Fifty
50
D
Five
hundred
500
MMMMM
Five
thousand
5000
VI
Six
6
LX
Sixty
60
DC
Six hundred
600
VII
Seven
7
LXX
Seventy
70
DCC
Seven
hundred
700
VIII
Eight
8
LXXX
Eighty
80
DCCC
Eight
hundred
800
IX
Nine
9
XC
Ninety
90
CM
Nine
hundred
900
National Curriulum (2014) KS1
Year 1: Number and Place Value
Count to and across 100, forwards and backwards
beginning with any number
Count, read and write numbers to 100 in numerals
Count in different multiples – 1s, 2s, 5s and 10s
Given a number, give one more and one less
Identify and represent numbers using concrete
objects and representations including numberlines
Read and write numbers from 1 to 20
National Curriculum (2014) KS1
Year 2: Number and Place Value
Count in steps of 2, 3 and 5 from 0, count in 10s from
any number, forward and backward
Recognise place value of each digit in a 2 digit
number
Identify, represent and estimate numbers using
representations including number line
Compare and order numbers from 0 to 100
Read and write numbers to at least 100 and in words
Use place value to solve problems
example
National Curriculum: Lower KS2
Year 3: Number, place value and rounding
Count from 0 in multiples of 4, 8, 50 and 100, give
10 or 100 more or less of a given number
Recognise place value of each digit in a 3 digit
number
Compare and order numbers up to 1000
Identify, represent and estimate numbers in different
representations
Read and write numbers to 1000 in numerals and
words (ie. 768 = seven hundred and sixty eight).
Solve number and practical problems
National Curriculum(2014) : Lower
KS2
Year 4:
Count in multiples of 6, 7, 9, 25 and 1000
Find 1000 more an less of a given number
Count backwards through zero to negative numbers
Recognise place value of digits in 4 digit number
Order and compare numbers beyond 1000
Round numbers to nearest 10, 100, 1000
Read and write numbers to 2 decimal places
Round decimal numbers to nearest whole number
Compare two decimal numbers with the same decimal places
Solve problems
Read Roman numerals to 100 and understand how number
systems have changed over time and include the concept of zero
and place value
National curriulum (2014) : Upper
KS2
Year 5:
Read, write, order and compare numbers to 1,000,000 and
determine value of each digit
Count forwards and backwards in powers of 10 up to 1,000,000
Interpret negative numbers in context and count forward and
backwards through zero
Round any number up to 1,000,000 to nearest 10, 100, 1000,
100,000
Round decimals to nearest whole number and one decimal place
Read, write, order and compare numbers with 3 decimal places
Read Roman numerals up to 1000, recognise year written in Roman
numerals
Solve problems
National Curriculum (2014) : Upper
KS2
Year 6:
Read, write, order and compare numbers up to
1,000,000 and determine value of each digit
Round whole numbers
Use negative numbers in context
Identify value of each digit to 3 decimal places and
multiple numbers by 10, 100, 1000 answering up to 3
decimal places
Solve problems
Misconceptions linked to teaching
Place Value
•
•
•
•
Naming and writing numerals
Calculating with large numbers
Multiplying or dividing by 10
Not understanding zero as a place holder
Naming and writing numbers 1
• Why isn’t seventeen written as 71 as the 7 is said
first?
• The naming system we use becomes clearer with larger
numbers. Should we confine children to low numbers when
investigating our number system?
They will be able to interpret larger numbers, even though they
cannot yet calculate with them
Research suggests that children in Japan develop an
understanding of PV younger, this appears to be because
number names are explicit (Stigler et al, 1990)
Number Spellings
0 – zero
1 – one
2 – two
3 – three
4 – four
5 – five
6 – six
7 – seven
8 – eight
9 – nine
10 – ten
11 – eleven
12 – twelve
13 – thirteen
14 – fourteen
15 – fifteen
16 – sixteen
17 – seventeen
18 – eighteen
19 - nineteen
20 – twenty
30 – thirty
40 – forty
50 – fifty
60 – sixty
70 – seventy
80 – eighty
90 – ninety
100 – hundred
1000 – thousand
Naming and writing numbers 2
Why isn’t 32 written as 302 … 361 as 300601?
http://www.bbc.co.uk/learningzone/clips/understandinghundreds-tens-and-units-dave-and-the-penguinsanimation/2918.html
Interactive Teaching
Programs
• Place value (arrow cards)
• Beads
Place Value in larger numbers
Children who cannot understand groups as units
are confined to counting in ones
• eg a group of 7 and a group of 3 makes 10 –
this is more efficient than counting 7 in ones
and then counting on 3 more.
Children who have learnt traditional calculations
by rote can be hindered if they cannot think
about the value of digits when calculating
Multiplying or Dividing by 10
What happens when
you multiply / divide by
10?
Children are often
taught that when
multiplying or dividing
by 10, they add or take
away the 0…..is this
true?
Does the decimal point
move?
Can the above cause
misconceptions?
I think it should be
0.740
7.4 ÷ 10 =
To divide by
10, move
the digits
one place
to the right
to make
0.74
To divide by 10, you just
take a zero off, so it is 7.4
You move the digits
one place to the left
so it is 74.0
What do YOU think?
Zero as a place holder
• Children may not understand that zero is
needed to indicate the position of say the tens
when no tens are actually present.
• In the number three hundred, the two zeros
do not indicate hundreds – they indicate an
absence of any tens or units (ones).
• Can the above cause misconceptions?
Zero needed….Zero not needed…
• Two hundred and fifty
• Two point five zero
Confusion – consider interpretation – i.e
money on a calculator – when calculator gives
monetary answer of 2.5 – children need to
know that this is £2.50 (SATs)
Other Uses of zero where zero
has a meaning
No score
As a label
A numerical
value in a
measure
Introducing Negative Numbers
• Needs to have a meaning
Can you think of any real life
situations where negative numbers
are used?
• Is seen as an extension of the numberline
Negative numbers
Look at some resources to support the
understanding of place value
•
•
•
•
•
Numicom
Arrow Cards
Money
Straws
Unifix /
multilink
• 100 beads on
string
• PV hats
• 100 grid
• Base 10 blocks
(Dienes)
• Gattegno chart
And finally: Imagery
Visualisation helps to bridge the gap
between concrete and abstract.
Now try this exercise.