Generalising whole number place value properties
Value depends on place: column names and sizes
Ten-
thou-
sands
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Thou-
sands
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Hun-
dreds
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Tens
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Ones .
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tenths
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hun-
dredths
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thou-
sandths
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ten-
thou-
sandths
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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
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plus |
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5 tens
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plus |
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6 ones
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plus |
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7 tenths
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plus |
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8 hundredths
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plus |
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9 thousandths
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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
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thousandths
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Hundreds
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hundredths
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Tens
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tenths
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Ones
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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.
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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.....
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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.
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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.
Top
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
follow rules learned without understanding.
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
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hundreds |
0.04 |
tens |
0.4 |
ones |
4 |
tenths |
40 |
hundredths |
400 |
thousandths |
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0.00376 |
hundreds |
0.0376 |
tens |
0.376 |
ones |
3.76 |
tenths |
37.6 |
hundredths |
376 |
thousandths |
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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.
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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
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639 thousandths |
Fully expanded form |
0.6 +0.03 + 0.009
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6 tenths +3 hundredths + 9 thousandths |
Partially expanded form |
0.63 +0.009
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63 hundredths + 9 thousandths
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Partially expanded form |
0.6 + 0.039
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6 tenths +39 thousandths |
Unusual partially expanded form |
0.609 + 0.03
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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?
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The ones column must always be shown. (Marked by decimal
point, except in whole numbers)
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The zeros that matter are those between other digits and the
ones column.
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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.
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Answers to Exercises
(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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