Generalising whole number place value properties

 Value depends on place: column names and sizes Common problems in learning the decimal place value column names and values. The Endless Base Ten Chain Base ten links between the place value columns. Multiplying and dividing by ten and its powers. Overflow from a column. Expanded form: reunitising tenths, hundredths etc Understanding when and how zeros change place value. Which zeros matter? One of the most important things to teach.

### Value depends on place: column names and sizes

 Ten- thou- sands Thou- sands Hun- dreds Tens Ones . tenths hun- dredths thou- sandths ten- thou- sandths

One of the first tasks of understanding whole number notation is to learn the names of the columns and to appreciate their sizes: ones, tens, hundreds, thousands etc. A number such as 456 is 4 hundreds plus 5 tens plus 6 ones - each digit contributes according to its place value and the contributions are added together.

The structure of decimal numbers is the same. A number such 456.789 is

 4 hundreds plus 5 tens plus 6 ones plus 7 tenths plus 8 hundredths plus 9 thousandths

Difficulties arising from symmetry around the decimal point or ones column
It is very helpful for children to note the near symmetry in the place values of the columns - tens to hundreds to thousands going up to the left and tenths to hundredths to thousandths going down to the right. However, this can be misinterpreted in several ways. The whole number knowledge helps, but can also interfere with learning the names for the columns in a decimal. The diagram below stresses that the symmetry is about the ones column, not the decimal point, but it does not indicate the relative sizes well.

 Thousands thousandths Hundreds hundredths Tens tenths Ones

Error: Adding a "oneths" column to increase the symmetry.
Some children implicitly think that there will be a "oneths" column immediately after the decimal point. They imagine that because tens are the second whole number column from the decimal point, tenths will be second on the left etc. They may write 3 tenths as 0.03. Sue, in year 7, wrote the number 237 hundredths as a decimal as 0.00732. Sue had started in the third column from the decimal point for hundredths, and also reversed the number (see below). Irwin (1996) in New Zealand also records children discussing a "oneths" column.

This error can be minimized if teachers stress that the decimal point is really a marker to indicate where the ones column is. To emphasise this, the decimal point is in the ones column in the chart above, not in a column by itself.

 OUR MOTTO: The decimal point marks the ones column.

Error: Thinking there are whole numbers on both sides of the decimal point.
Some children do not know that the place value of the columns decreases when we move to the right. They know the column names are similar, but assume they are the same. This can be because the child has not got even a basic understanding of the meaning of fractions, or often because of hearing and language difficulties. Children who think that the names for the place value columns are the same on both sides of the decimal point:

 .....thousands, hundreds, tens, ones, tens, hundreds, thousands.....

might then be a reverse thinker and select 0.35 as larger than 0.41 because 53 is larger than 14.

Other common problems with column names and values
Error: Analogy with money confuses tens and tenths.
Some children are confused about whether the first column after the decimal point is tens or tenths. This can come from analogies with money. Veronica in Year 5, for example, seemed to routinely think about this. She read the number 0.3 as "oh point three, that's thirty" and 3.5 as "three point five, that's fifty".

 Point of confusion: In \$4.65, the 6 indicates sixty cents (i.e. 6 tens) whereas in 4.65, the 6 indicates 6 tenths.

This is sometimes a language problem, sometimes arises by false analogy and sometimes is a reunitising problem.

Error: Knowledge of limited number of columns.
Some children are only familiar with tenths while others are only familiar with tenths and hundredths. They know nothing about the names or values of other place value columns. In younger children, this is simply a stage of teaching. However, older students must be taken beyond two decimal places, so they understand that the place value relationships continue indefinitely; we call it the endless base ten chain and it is described below. Students cannot generalise properties from just one or two instances. More details of money thinking.

Error: Analogy with the number line confuses decimals and negatives.
As noted above, the place value names are (nearly)symmetric around the ones column, although with a twist! This seems to remind some older students of the way in which the positive and negative parts of the number line are symmetric about zero. This may dispose some of them to interpret decimals as negative numbers.

### Endless base ten chain

In addition to knowing the names and size of the place value columns, students need to know the relative value of the columns. They need to know that the value of each column is ten times the value of the column to the right (including across the decimal point) and that the value of each column is one tenth of the value of the column to the left. The illustration below shows this endless base ten chain.

Multiplying and dividing by ten and its powers
One of the great advantages of the base ten system is that multiplying or
dividing by ten or the powers of ten (glossary) is achieved by shifting
digits into adjacent place value columns. Although this is very easy to
carry out, it is not well understood and consequently many children cannot
do it reliably. They often get confused about which way to "move the
decimal point" and when to "add or take away zeros" - the result of trying to

This problem is widespread. For example Bell (1983) reports that only 47% of a very large sample of British 11 year olds correctly answered "How many times is 0.1 greater than 0.01?" and only 34% correctly answered "What number is 10 times 0.5?" Performance on the item "ten times 100" was 71%, much better but not as good as might be expected. Equivalencies like those shown below for 4 tenths and 376 thousandths are crucial to understanding. For example, 4 tenths is equivalent to 40 hundredths, which is equivalent to 400 thousandths etc.

Tables of Equivalencies:
 0.004 hundreds 0.04 tens 0.4 ones 4 tenths 40 hundredths 400 thousandths

 0.00376 hundreds 0.0376 tens 0.376 ones 3.76 tenths 37.6 hundredths 376 thousandths

Overflow from a column
A critical feature of the relationship between the place value of columns is the very simple way in which "overflow" from a column is dealt with.

• Only one digit (from 0 to 9) ever goes in one column
• Because all the columns have place value ten times as great as the column to the right, ten in any column gives one in the next column to the left.

Students who have not mastered this cluster of ideas will sometimes exhibit column overflow thinking. They interpret decimals as if more than one digit can go in each column. For example, Brad in Year 6, would interpret

• 0.35 as 35 tenths,
• 0.678912 as 678912 tenths,
• 0.035 as 35 hundredths,
• 0.0149 as 149 hundredths and
• 0.0043 as 43 thousandths.

Brad's interpretation of 0.35 as 35 tenths instead of 35 hundredths may also have arisen simply because he has "forgotten" which column name to take when describing the decimal as a fraction. Instead of getting the name from the rightmost column (in this case the hundredths, as 0.35 is 35 hundredths) he may just take the name from the leftmost column (the tenths). This is an important idea that needs definite consolidation, so that students are very secure with it. It is related to understanding equivalent fractions.

### Expanded form: reunitising tenths, hundredths etc

An idea central to dealing with the relative size of decimal numbers is to be able to interpret them in expanded form as decimals and as fractions. This section demonstrates the challenge of the cognitive processing involved. A decimal such as 0.639 can be interpreted in all the ways shown below. All these forms, except the last, can be obtained with a number expander.

 Condensed form 0.639 639 thousandths Fully expanded form 0.6 +0.03 + 0.009 6 tenths +3 hundredths + 9 thousandths Partially expanded form 0.63 +0.009 63 hundredths + 9 thousandths Partially expanded form 0.6 + 0.039 6 tenths +39 thousandths Unusual partially expanded form 0.609 + 0.03 609 thousandths + 3 hundredths

Exercise:
Write down all the expanded and partially expanded forms that you can for: (a) 2.3 (b) 5.82 (c) 0.7411 (Answers)

To think of a number 0.639 is all the ways shown above requires a student to be able to deal with units made out of other units (unitising and reunitising).

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### Which zeros matter?

Many children have difficulty deciding how or whether zeros change the value of a number. With decimals, children need to know that the zeros which affect the place value of the figures are on the left and the insignificant ones are on the right: 3.250 and 3.2500000 are the same as 3.25 and 003.25 but 3.025 or 3.205 are different. The essential understanding of which zeros affect the place value of the digits and which ones do not is best demonstrated using concrete materials such as Multi-Base Arithmetic Blocks (MAB) or Linear Arithmetic Blocks (LAB). Making 3.25 and then 3.025, 3.250, 3.205 and 03.25 from blocks clearly demonstrates which zeros affect the size of 3.25, rather than just providing a list of rules for students to learn. Using the ideas of expanded notation are crucial.

 Basic Principle - Which Zeros Matter? The ones column must always be shown. (Marked by decimal point, except in whole numbers) The zeros that matter are those between other digits and the ones column.

It is very hard for children who do not understand the place value basis of decimal numbers to memorize the rules. Children generally decide which zeros change the value of a number according to their own interpretations of decimal notation. The erroneous misconceptions show many examples of this. However, the one principle above applies to both whole numbers and decimals.

(a) 2.3 = 2 ones + 3 tenths
= 23 tenths

(b) 5.82 = 5 ones + 8 tenths + 2 hundredths
= 5 ones + 82 hundredths
= 58 tenths + 2 hundredths
= 582 hundredths

(c) 0.7411 = 7 tenths + 4 hundredths + 1 thousandth + 1 ten-thousandth
= 7 tenths + 4 hundredths + 11 ten-thousandths
= 7 tenths + 411 ten-thousandths
= 7 tenths + 41 thousandths + 1 ten-thousandth
= 74 hundredths + 1 thousandth + 1 ten-thousandth
= 74 hundredths + 11 ten-thousandths
= 741 thousandths + 1 ten-thousandth
= 7411 ten-thousandths

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