Place Value Workshop Friday, 27 th 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

1 8 . 7 3 2

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? …

I Units

One

II III IV V VI VII VIII IX Roman Numerals have now been introduced into the new NC

Two Three Four 1 2 3 4 Five Six Seven 5 6 7 Eight Nine 8 9

Tens X

Ten

XX XXX XL L LX LXX LXXX XC

Thirty Forty 10 Twenty 20 30 40 Fifty 50 Sixty 60 Seventy 70 Eighty Ninety 80 90

Hundreds C

One hundred

CC

Two hundred 100 200

CCC

Three hundred 300

CD D DC DCC DCCC CM

Four hundred 400 Five hundred 500 Six hundred 600 Seven hundred Eight hundred Nine hundred 700 800 900

Thousands M

One thousand 1000

MM

Two thousand 2000

M M M

Three thousand 3000

M M M M

Four thousand 4000

M M M M M

Five thousand 5000

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/understanding hundreds-tens-and-units-dave-and-the-penguins animation/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

To divide by 10, move the digits one place to the right to make 0.74

7.4

÷

10 =

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.