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.