place value workshop friday, 27 th september university of greenwich
TRANSCRIPT
Place Value WorkshopFriday, 27th SeptemberUniversity of Greenwich
Place Value WorkshopObjectives
• 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
1 8 . 7 3 210x smaller
10x bigger
Teaching Maths with Diversity
http://webarchive.nationalarchives.gov.uk/20101021152907/http://www.Multiverse.ac.uk/ViewArticle2.aspx?anchorId=131&selectedId=149&menu=17877&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 COne hundred
100 MOne thousand
1000
II Two 2 XX Twenty 20 CCTwo hundred
200 MMTwo thousand
2000
III Three 3 XXX Thirty 30 CCCThree hundred
300 M M MThree thousand
3000
IV Four 4 XL Forty 40 CDFour hundred
400 M M M MFour thousand
4000
V Five 5 L Fifty 50 DFive hundred
500 M M M M MFive thousand
5000
VI Six 6 LX Sixty 60 DC Six hundred 600
VII Seven 7 LXX Seventy 70 DCCSeven hundred
700
VIII Eight 8 LXXX Eighty 80 DCCCEight hundred
800
IX Nine 9 XC Ninety 90 CMNine 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 KS2Year 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 KS2Year 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/understanding-hundreds-tens-and-units-dave-and-the-penguins-animation/2918.html
• Place value (arrow cards)
• Beads
Interactive Teaching Programs
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?
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
I think it should be 0.740
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
A numerical value in a measure
As a label
Introducing Negative Numbers
• Needs to have a meaning
• Is seen as an extension of the numberline
Can you think of any real life situations where negative numbers are used?
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.