Place Value
Place Value
Rita Thompson
Place Value An understanding of place value is central to the study of mathematics in the later stages, and one of the most important concepts to be developed in the Primary School. Place value is an aspect of mathematics that is difficult both to teach and to learn, and is therefore worthy of detailed consideration. Young children should not be expected to move too quickly into formal written calculations : this is more likely to hinder than to enhance progress and could cause confusion, it is essential to give children extensive practical experience with a wide variety of concrete materials, beginning with experience in grouping.
Begin with 1st groupings only (no remainders) Use things from the environment e.g. leaves or shells before using representational structural apparatus.
Then introduce examples that require the understanding that remainders are objects which cannot be grouped. The remainder can be referred to as a unit’ or a single object.
Initial groupings should be done with bases less than ten, so that the concept of grouping can be developed with small numbers of objects. Vary the number of objects in the universal set, but only introduce one number base at any one time.
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e.g.
Base 3
1st grouping
2nd grouping
A variety of boxes and containers could be used to develop the understanding of successive groupings.
Put 3 apples in each bag
Put 3 bags in each box
Recording can initially be done verbally, discussing the various stages. Position is unimportant at first. As soon as the concept of grouping is understood, we can begin to introduce recording and the idea of columns and movement from right to left.
Base 4
The small base activities are not meant to give practice in formal operations but to provide children with a range of experiences needed to fully understand the important ideas which are common to any base, including ten. Children learn a pattern of behavior to follow regardless of the number base. Grouping in tens can be included as soon as the children can group with ease. Sticks can be bundled together in tens using elastic bands, or matchsticks put into boxes of ten, and then further grouped into ten bundles of ten, or ten matchboxes into a larger box.
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It is useful to label groupings This example of a base 3 grouping illustrates the importance of zero as a place keeper. Exchange The idea of exchange should be introduced gradually, preferably when children fully understand the concept of grouping in a variety of bases. Again, this should be done practically, using concrete materials. e.g. three small pebbles can be exchanged for a shell and three shells for a cone.
Cubes can be matched to a variety of objects and grouped using a variety of bases, or base ten only.
Children should be encouraged to decompose their groupings to establish idea that they are composed of individual (or unit) cubes. The cubes can then be regrouped. These activities also aid the understanding of addition and subtraction. When the concept of grouping is fully understood, and the idea of exchange has been introduced, M.A.B. blocks can be introduced to’ speed up’ the process.
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3rd grouping
2nd grouping
4 flats exchanged for one block
1st grouping
4 longs exchanged for one flat
unit cubes
4 cubes exchanged for one long
When introducing Multibase materials, involving exchange, care should be taken to remind children that there are ‘hidden cubes’. If children have previously constructed their own groupings, they should have a better understanding of place value. A Spike Abacus is very useful. Each spike will only hold nine beads. Ten beads can be exchanged for one bead of a different colour and moved into a new place on the left.
Games are invaluable at this stage. 1.
A game for two or more players Materials: red, yellow and blue multilink/unifix cubes; A die Work in base 10 (or adapt to any other base) Players should take turns to roll the i.e. and take the number of red cubes shown on the uppermost face of the die. The cubes should be linked together to form a rod. Rods must be no more than 9 cubes long. (A spike abacus is useful for this game) Ten red cubes can be exchanged for a yellow cube and ten yellow cubes can be exchanged for a blue cube. The winner is the first player to get a blue cube(or is the player with the longest blue rod when the game ends.)
2. Players are given 3 yellow cubes each. The aim is to dispose of all the cubes. The first player to dispose of all the cubes is the winner. Players take turns to roll the die and are allowed to return the number of cubes shown on the face of the die This of course, will inevitably involve a certain amount of exchange and decomposition.
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Boxes This game can help children to read, write and order numbers, to use the knowledge that the tens digit indicates the number of tens and to compare greater and smaller numbers. Equipment A set of ten cards, numbered zero to nine and a board for each child as shown. Tens
Units
How to Play Playing in pairs, the children shuffle the cards carefully and place them face down on the table. They then take turns to select one card at a time and place it on the board. The aim is to make the largest possible number, and once a card is placed it cannot be removed. In this example, then second child is the winner. Tens Units Tens Units
3 Tens Units 3 Tens Units 3 6 Tens Units
3
6
First Child
Tens Units 1 Tens Units 1 Tens Units
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1
Second Child
Variations *Try making the smallest number possible *Includes hundreds in the game *Use more boards for each player *Include addition in the game *Instead of boards, use two chairs for children to sit on (one tens and one units) Cards can be selected in the same way and held up by the children.
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More Place Value Games Subtraction You need Dice, Dienes M.A.B. or Cuisenaire Each player starts with 29 ( 2 longs and 9 units) They take turns to throw the die and subtract the number from their own total The first player to reach 0 is the winner (Longs must be decomposed) Variations *Use a die with numbers - 4,5,6,7,8,9. and use a higher starting number *Start with 100 (a flat) and the winner is the first to reach 50 *play the game with money, starting with 25p, 50p etc.. Ladder Game You need a 10 rung ladder for each player 1 set of 100 number cards to fit on rungs Score sheet and pencils
1. Shuffle the number cards and place them face down in the centre. 2 First player takes top card and places it on his/her ladder in a space. Once card is placed, it cannot be removed 3. Other player takes next number card and places it on his/her ladder. 4. Players continue to take cards and attempt to place them in order, lowest at the bottom, highest at the top. If there is no suitable space for a number card, it is kept as a dead card. 5.The game ends when each player has taken 10 cards. Each player scores a point for each card placed. The winner can be the one with the highest score or the game can be repeated until one player scores 25.
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FIVE CARD CHOICE The children should be in groups of 4 5 or 6 to a table but the game may be played as a class activity under the direction of the teacher or as a group game for one group only WHAT YOU NEED 1. An ordinary pack of playing cards for each group. The picture cards and the tens should be removed. Aces count one. 2. A three Section baseboard (in the Resource Pack) for each child. This is similar to the one used in the Practical Place Value booklet and the subsequent Resource pack for this booklet published by NORMAC. The baseboard must he large enough to fit a play ing card on each section. 3. A collection of counters or tokens for each table. These will help to make scoring easy. 4. A list of challenges for the teacher or for the Group Leader of each group. (List in the Resource Pack). THE GAME IN ACTION. VERSION A . Teacher controlled class version. 1. Each group has its own pack of cards and 5 cards arc dealt to each player. Each player looks at the cards in his hand. 2.
The teacher calls out the first challenge on the list “the largest 3 digit number”
3. Each player selects 3 of the 5 cards in his hand and places them on the baseboard, one in each section, to form a 3 digit number. For example,
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1. 2.
When the group has .agreed who, within the group has the largest 3 digit number, that player collects two counters or tokens from the central collection. The player with the next highest number can collect one counter.
3.
The 3 cards are removed from the baseboard and the teacher calls out the next challenge.
NOTE THAT EACH PLAYER RETAINS THE SAME FIVE CARDS THROUGHOUT ALL THE CHALLENGES THE WINNER is the Player who collects the most counters when all the challenges have been attempted The list of possible challenges is as follows:—
Tokens Won
(a) largest 3 digit number
largest takes 2 tokens next takes 1 token
(b) smallest 3 digit number
smallest takes 2 tokens next takes 1 token
(c) nearest number to 150
nearest takes 2 tokens next takes 1 token
(d)furthest from 835
as above
(e)largest odd number
as above
(f)largest even number
as above
(g)nearest to any century
as above
(h) any multiple of three
any valid number gets I token -for the player (all players could score here)
(i) any even multiple of 3
as above
j) any multiple of six
as above
(k) any multiple of
as above
(I) any multiple oh 9
as above
(m) any even multiple of 9
as above
(n) any multiple of 11
as above
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FIVE CARD CHOICE Photocopy and make 2 Baseboards Hundreds Tens Units Hundreds
Hundreds
Tens
Tens
10
Units
Units
FIVE CARD CHOICE 14
Photocopy and Cut
2 Largest 3 digit number
4
3 Smallest 3 digit number
5 Farthest number rom 835
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6 Largest odd number
8 Nearest to any century
10
Any multiple of 3
Any multiple of 6
Any even multiple of 3 12
Any multiple of 5 14
Any even multiple of 9
Largest even number 9
11
13
Nearest number to 450
Any multiple of 11
11
Any multiple of 9
Working with a calculator Examples of activities connected with place value An understanding of place value is vital if children are to use the number system with confidence. It is important that you give children plenty of opportunities for working with the abacus or with base ten blocks, for playing grouping games with counters or exchange games with money, and for activities involving tens and hundreds, tenths and hundredths on the number line. In addition, the calculator offers a powerful means of extending the range of activities available. • Give a group of children a calculator each and say, Put one thousand and twenty four in your display. Show each other. Have you all got the same? Now try ten thousand and twenty. • Put 1040 in the display yourself and ask the children, What number is that? • Ask the children to use their calculators to work out 28 + 37. Then ask, What would 280 + 370 be? And 2.8 + 3.7? • Say to the children, Enter 7 + 1 0 = =
=.What is happening?
Go on pressing the button, but stop at 97. What will the next number be? Now try =
8 7
-
1
0
= =
Stop at 7. What will the next number be? • Ask them to enter any one-digit number - Multiply by 10, then again, then again. What is happening? Then try a harder one. Enter a one-digit number. Multiply by 100, then again, then again. What happens? • Ask the children to enter any single digit number — Guess what will happen if one of you multiples by 1000. What if someone else multiplies it by 10 000? Now try it and see. What will happen if you multiply by 0.1 or 0.01? A similar activity can be developed for division. • Try these comparisons. What happens and why? Enter 28 x 0.1 . Compare with 28 ÷ 10 Enter 28 x 0.01 Compare with 28 ÷ 100 Enter 28 x 0.001 Compare with 28 ÷ 1000
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CALCULATOR NUMBERS
Using matchsticks or lolly sticks, can you make all the digits as you see them on a calculator? Remember to make them all the same height
ACTIVITIES Which digit uses the most sticks? How many sticks does it use?
Which digits require the same amount of sticks?
Which digit uses the least sticks? Using only 7 sticks, what is the biggest number you can make? How do you know that is the biggest number? Using 7 sticks, can you make odd and even numbers?
Using 7 sticks, can you make a multiple of 5 or 3?
Using 7 sticks how many numbers between 20 and 60, can you make?
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Making Numbers Can you make your calculator show all these numbers?
How many of the numbers can you make with matchsticks?
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LOLLY STICKS a game for two players —
You will need: 14 lolly sticks
In this game your numbers must look like the numbers on a calculator.
Decide who goes first.
Use the lolly sticks to make the number 88.
The first player moves two lolly sticks to make a higher number, e.g. 138.
The second player also moves two lolly sticks to make an even higher number, e.g. 381.
From now on take turns to tell your opponent to make a higher or lower number. Your opponent may choose to move either one or two sticks to do SO. Numbers made previously cannot be repeated.
The game continues until one of you cannot go. The other player is the winner.
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What is Place Value? How can it be assessed and where do calculators fit in? Pat Cannell describes how the responses of year 3 children to an activity got her thinking.
IT AIN’T WHAT YOU DO - IT’S THE WAY THAT YOU DO IT! The original idea was to develop an activity that could provide calculator use in the classroom: an activity where the calculator could help develop strategies for the user by aiding number manipulation. I also wanted it to show me something of the children’s understanding of place value. We had been doing a lot of work on place value and I found the level of understanding achieved difficult to assess. Most of my evidence came out in snippets of conversation. Maybe I could look at how the children used numbers in this activity and get an idea of what they had taken on board. A calculator is very quick at throwing up numbers. Do the children know the order of magnitude of those numbers? The game was to find four numbers totalling a given target. Each target was arranged as shown. The outside squares were to be filled in to meet the target. I provided a work sheet with twelve such targets on, the first six being between 10 and 100, the latter six being all 100.
The class in question were Year 3, divided into three groups. Group 1 did the activity without a calculator or equipment. They could use scrap paper if they wanted to. Group 2 were given access to calculators but no direction. Group 3 were given a supply of squared paper. The three groups did not carry out the activity simultaneously. Within a few seconds of giving the first group their sheets I had to introduce some rules! These were: No two numbers are to be the same. No noughts are to be used. Group 1 set to and manipulated numbers with the aid of a rubber. They displayed a good grasp of place value, particularly with the repeated targets of 100, where they formed a strategy of building on the previous answer to make new totals. Some of their solutions followed a pattern.
In some cases the strategy was handled better than the accuracy of the answers. Group 2 were the lucky ones with the calculators, who thought they were well away, and so they pushed buttons and thought life was great. Then they got back to the task in hand and found that an initial strategy had to be formed manually. Mostly they picked a number then built up on it to meet the target. None of this group started at the total and worked backwards, which surprised me as that is how I would have gone about it! After hitting the first couple of targets they found the calculator helpful but it added neither to their speed or their accuracy.
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Chris’s reasoning for this answer was 50 add 30 is 80, add 10 gets to 90 and 1 more ten gets me to 100.
Group three were given the squared paper. This was my mistake. How easy, I thought, to count out the squares (I even gave them 100 squares), divide into four and then count each section. The children thought I was round the bend and proceeded in a way very similar to group 1. Some did use the ‘apparatus’ in the latter section, as can be seen from this example, where the number is broken down in a less convenient way.
I felt that the majority of the children showed a good grasp of place value, at least to 100, and they were developing strategies which involved breaking down the numbers into the largest working unit, in this case ‘tens’ and then shuffling those numbers about. I also learnt that children’s calculator use was not as effective as it might be - more activities required! As a bonus there were some revealing results. Stephen completed the sheet in ten minutes without a calculator. As he was thirsting far more I quickly gave him a variety of, as I hoped, awkward numbers, a calculator and the instruction to get each number to zero in four moves. He did this very quickly by subtracting 100s, then 10s, splitting if necessary, and finally units. 1278 - 300 – 900 – 78 - 0 = 0 Neatly takes care of 1200 At the other extreme Sam showed a lack of understanding of the conservation of number. As he was obviously stuck on the first square I suggested that he used Multilink. On breaking a stick of 17 cubes into two parts he was unable to say how many altogether. He also shows what a quandary he is in by the fact that one of his numbers around the edge is higher than the target.
Hayley, desperate to please and not be any trouble, dared not move far away from the target number, in case she lost her way; so the constituent numbers bear no relation to the exercise intended, but stay safely close to the target.
When this activity was tried with different older children a variety of more stringent rules were introduced, such as that no number may be divisible by 10. When we were planning extensions the immediate thought was that large numbers would be difficult. We
have now come to the opposite conclusion. If place value (upwards) is securely in place a large number presents no problem. However, consider 3 as the target number. This could take you in several directions according to the rules imposed: • only integers can be used (negative numbers) • parts of numbers can be used (fractions) • all the numbers must be the same (interpreting a calculator display) This has proved to be a versatile and revealing activity. My next task is to refine the targets and rules to make better use of a calculator. Pat Cannell teaches at Southfield CP School, Brackley Extract from MT140 1992
The National Curriculum (a prose poem) We did tables with our last teacher We've done computers, it’s ticked off miss We did calculators with the supply I don’t have to do estimates, I got them all right last time I asked ‘What if.. .‘ in Year 4 My mum says I’m not allowed to test my hypotheses We’ve all done rulers before... Anne Watson
