Middle Maths Focus Group

Number Sense and Place Value

9 March 2011

Number Sense

• Having a good intuition about numbers and their relationships.

• Develops gradually as a result of exploring numbers, visualising numbers in a range of contexts, and relating them in ways that are not limited by traditional algorithms.

• Grows more complex as children learn more, eventually being able to think flexibly about fractions, decimals, percent and integers.

Big ideas

• Numbers are related to each other through a variety of number relationships - more than, less than, composed of

• “Really big” numbers possess the same place-value structure as smaller numbers. Best understood in terms of real- world contexts

• Whole numbers can be described by different characteristics, even and odd, prime and composite, square, understanding characteristics increases flexibility when working with numbers

Key Mathematical Ideas

Developing Meanings for the operations• Addition and subtraction are related. Addition names the

whole in terms of the parts, subtraction names a missing part• Multiplication is related to addition• Multiplication involves counting groups of like size and

determining how many there are in all. Multiplicative thinking• Multiplication and Division are related. Division names a

missing factor in terms of the known factor and the product. • Models can be used to solve contextual problems for all

operations, regardless of the size of the numbers. They can be used to give meaning to number sentences.

Van de Walle & LouvinTeaching Student Centred Mathematics,

Fifty and some more

• Say a number between 50 and 100. Children respond with “50 and ____.

• For 63, the response is “50 and 13”

• Use other numbers that end in fifty such as 350, 650

Read, say, do - Peter Hughes

Say the numeral oneway, e.g. 63 is sixty three

Read 63 as “sixty three” and Sixty three as “sixty three”

Model the numeral as ones

e.g. as 63 ones

Model the PV form of thenumeral e.g. 63

is 6 tens and 3 ones

Say the numeral in the other way, e.g. 63 is 6 tens

and three ones

Extend - read, say, do

• If we have 243 lollies, how many packets of ten would we have?

• How many loose ones?• If we pack ten packets into a box, how many boxes

will we have, how many packets and how many loose ones?

• Continue the model – crates, pallets, container and so on

Peter Hughes

Canon of Place Valueone for ten or ten for one

the canon of place value means…

Ten ones must be exchanged for tenTen tens must be exchanged for one hundredTen hundreds must be exchanged for one thousand...

Modeling with Place value money

• Children need to be able to verbalise the ten for one exchange in problems such as

• I have $1003 and I owe my friend $7 – must follow ten for one rule.

• Try the same with problems like $998 + 6

• Aim for children to be able to verbalise the ten for one exchange fluently.

Peter Hughes

Equipment for developing place value concepts

Stage Equipment

Stage 2- 4Concrete representation of ones

Bundling sticks, beans and containersCounters and plastic bags, Slavonic Abacus

Stage 5 Non representational

Place value money, place value blocks, arrow cards, place value houses

Stage 6 Number Lines

Stage 7 Decimal Fraction Mats

Important activities for developing place valueEarlyReading numbers as wordsMove flexibly from sixty to six tens to 60Making sense of 20 + seventy = ___ tensExplaining where the tens are in 67, 17, 127.ThenReading, writing and saying numbers up to a million and beyondDevelop the concept of really big numbers – what does a thousand look like?Familiar real world contexts for big numbers

Listening/Watching to learn

• Ask children to write numbers such as twenty three thousand and four –

What might children write if they don’t understand?

• Place Value Play Money and Place value Houses

• How many tens in 124? How do you know?• How many ways can you show 243?

Bundling To Ten

Bundles of ten board, Ice block sticks, Dice Pipe cleaners

Roll the dice - put the number of ice block sticks in ones column - in tens frame pattern.

Roll again add ice block sticks - what happens when we get to ten? Bundle the 10 put into tens column - Part whole thinking

Record the storyIntroduce to groupPlay in pairs - first to 100.Remember to unbundle too!

Hundred or Bust

You need - a dice

Aim is to make 100 in seven throws or be the nearest to 100.

Children draw a table with 8 rows, 3 columns write tens, ones and total.

Roll the dice, children choose to put the number in the tens or ones column. Keep a running total.

Tens Ones Total

Activities that reinforce place value knowledge• 100 or bust• Rocket - change numbers to suit the level.• Traffic Light - book 4 page 25

• Thousands book• Digit Cards - playing cards deal out three card , make the

largest, smallest number you can - order them in your group. Who has the highest? Lowest? How do you know?

• Staircase/Grid and a dice - strategy game. Make the highest lowest number