Haylock, Derek and Manning, Derek
Numbers and Place Value
Haylock, Derek and Manning, Derek, (2014) "Numbers and Place Value" from Haylock, Derek and
Manning, Ralph, Mathematics Explained for Primary Teachers 5th Ed. pp.65-86, Los Angeles: Sage ©
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6
Numbers and Place Value
In this chapter there are explanations of
•
the difference between numerals and numbers;
•
the cardinal and ordinal aspects of number;
•
natural numbers and integers;
•
rational, irrational and real numbers;
•
the Hindu-Arabic system of numeration and the principles of place value;
•
some contrasts with numeration systems from other cultures, including Roman
numerals;
•
digits and powers of ten;
•
two ways of demonstrating
•
how the number line supports understanding of place value;
•
the role of zero as a place holder;
•
comparing and ordering positive integers; and
•
rounding a positive integer to the nearest 10 or 100.
LINK TO NATIONAL
CURRICULUM
place value with materials;
What is the difference between a
'numeral' and a 'number',?
A numeral is the symbol, or collection of symbols, that we use to represent a number. The number is the concept represented by the numeral, and therefore consists
of a whole network of connections between symbols, pictures, language and real-life
situations. The same number (for example, the one we call 'three hundred and sixtysix') can be represented by different numerals - such as 366 in our Hindu-Arabic ,
place-value system, and CCCLXVIusing Roman numerals (see Figure 6.5 later in this
MATHEMATICS
EXPLAINED for PRIMARY TEACHERS
hapt r and the accompanying commentary). Because the Hindu-Arabic system of
numeration is now more or less universal, the distinction between the numeral and
th numb r is easily lost.
What are the cardinal and ordinal aspects of number?
A numeral, such as 3, together with the associated
word 'three', has a wide range of situations and
contexts to which it can be connected. The two
The fact that two sets of three objects most significant for young children are the cardisuch asthree cupsand three spoons - share
nal and ordinal aspect of number.
the property of 'threeness' can be experiThe learner's first experience of number is
enced by the processof one-to-one matching; for example, the three spoons can be
likely to be as an adjective describing a small set
allocated one to each of the three cups.
of objects: two brothers, three sweets, five fingers,
This is a key practical processfor the young
three blocks, and so on. This idea of a number
child that makes what is the same about
being a description of a set of things is called the
three cups and three spoons explicit.
cardinal aspect of number. By the process of
one-to-one matching between sets containing the
sam number, as shown in Figure 6.1, the learner is able to recognize that there is
111 thing the arne about the sets; in other words, they identify an equivalence. The
pr P rty that is shared by all sets of three things, for example, is then abstracted to
f rrn th
ncept of 'three' as a cardinal number, existing in its own right, independent
f any sp if
ontext.
LEARNING.
,TEACHING POINT
this is not by any means the only aspect of number that the young learner
unt rs. Numbers are much more than just a way of describing sets of things.
Yi Ling hildren also encounter numbers used as labels to put things in order. For example,
BLIt
n
Figure 6.1
One-to-one matching
NUMBERS and PLACE VALUE
they turn to page 3 in a book. They play games
LEARNING· • Tl!ACHlNG POINT
on the number strip in the playground and find
themselves standing on the space labelled 3. They
It is in counting the number of objects in
learn that they are 3 years old and that next birtha set that the cardinal and ordinal aspects
day they are going to be 4. One of the tricycles in
come together. In pointing to each item
the playground is labelled 3 and this has to be
in turn and numbering them, one, two,
parked in the space labelled 3, which is once
three, and so on, the child is using the
again between 2 and 4. The numerals and words
ordinal aspect. The child has to learn that
the ordinal number of the last number
being used here do not represent cardinal numcounted is the cardinal number of the set.
bers, because they are not referring to sets of
This is a significant step in the developthree things. In these examples 'three' is one
ment of the young child's understanding
thing, which is labelled three because of the posiof number and counting.
tion in which it lies in some ordering process.
This is called the ordinal aspect of number.
Numbers in this sense tell you what order things come in: which thing is first, which
is second, which is third, and so on. The most important experience of the ordinal
aspect of number is when we represent numbers as locations on a number strip (see
Figure 3.5 in Chapter 3) or as points on a number
line, as shown in Figure 6.2. We shall make conLEARNING· • 1IACHING POINT
siderable use of this image of number as we
explore understanding of number operations in
To help young children to develop undersubsequent chapters.
standing of number, provide opportunities
There is a further way in which numerals are
for them to make connections between
used, sometimes called the nominal aspect. This
the symbols for numbers (numerals), the
language of number, such as 'four' and
is where the numeral is used as a label or a name,
'fourth', real-life situations where numbers
without any ordering implied. The usual example
are used in both the cardinal sense (recogto give here would be a number 7 bus. Calling it
nizing sets of two, three, and so on) and
number 7 is not much different from calling it the
the ordinal sense (numbering items in
East Acton bus. It just identifies the bus and disorder), the process of counting, pictures
tinguishes it from buses on other routes. When
such as set diagrams and, especially, number
we see a number 7 bus, we do not expect it to be
strips and number lines.
followed by a number 8 and then a number 9 - in
fact, we may well expect it to be followed by two
more number 7s, as is the habit of buses. Having said that, I should make clear that
when various bus services are listed in numerical order in a timetable their numbers
are then being used in an ordinal way.
O
Figure 6.2
2
3
4
5
Numbers as points on a line
6
7
8
9
10
11
MATHEMATICS EXPLAINED for PRIMARY TEACHERS
_=-t are natural numbers and integers'?
How many numbers are there between 10 and 20? This is a question I like to ask
primary trainee teachers when we start to think about understanding number. The
111 st common response is nine: namely, the numbers Ll, 12, 13, 14, 15, 16, 17, 18 and
19. orne trainees answer a different question and give the answer ten, which is the
clifferen b tween 10 and 20. Others give the answer eleven, choosing to include the
10 and the 20, in an unorthodox use of the word 'between'. All of these answers
a sume that when I say 'number' I mean the numbers we use for counting: {I, 2, 3, 4,
5, 6, ... J, going on forever. These are what mathematicians choose to call the set of
natural numbers. As we have seen above, natural numbers can have both cardinal
and ordinal interpretations. But there are other kinds of numbers that children will
neounter in primary schools and which will feature in later chapters in this book.
TIere I introduce them briefly.
How many numbers are there that are less than
lO? That's another interesting question I like to
discuss with trainee teachers! Some say nine, just
counting the natural numbers from 1 to 9. Most
Negative integers cannot be understood
include O (zero) and give the answer ten. But othif we think of numbers only in the cardinal
ers have the insight to include negative numbers
sense,as sets of things. You cannot have a
in their understanding of 'numbers', and give
set of 'negative three' things. We have to
responses such as 'there is an infinite number' or
make the connection with numbers used
in the ordinal sense, as labels for putting
'they go on for ever'. In this way we can extend
things in order. This is done most effecour understanding of what constitutes a number
tively through the image of the number
to what mathematicians call the set of integers:
line. This shows the importance of teach{... , -5, -4, -3, -2, -I, O, I, 2, 3, 4, 5, ...} now
ers using number strips and number lines
going
on for ever in both directions. Integers
with young children at every opportunity,
build
on
the ordinal aspect of number, by extendso they begin to visualize numbers in this
way and not just as sets of things.
ing the number line in the other direction, as
shown in Figure 6.3, labelling the points to the
left of zero as negative numbers.
The mathematical word 'integer' is related to words such as 'integral' (forming a
whol ) and 'integrity' (wholeness). So the set of integers is simply the set of all whole
numbers. BUl thi includes both positive integers (whole numbers greater than zero)
and negative integers (whole numbers less than zero), and zero itself. The integer -4
is PI' perly named 'negative four', rather than 'minus four' as is the habit of weather
2
Figure 6.3
Extending the number line
3
4
5
6 ...
NUMBERS
and PLACE VALUE
forecasters; minus is an alternative word for subtraction. Likewise, the integer +4 is
named 'positive four', not 'plus four'; plus is an alternative word for addition. Of course,
the integer +4 is another way of referring to the natural number 4, so we would not
normally write +4, or say 'positive four', but would simply write 4 and say 'four' - unless
in the context it were particularly helpful to signal the distinction between the negative
and the positive integers. So we note that the set of integers includes the set of natural
numbers: to be precise, natural numbers are positive integers. Integers are explained
in greater detail in Chapter 14.
What are rational and real numbers?
When you read earlier in this chapter the question that asked how many numbers
are there between 10 and 20, you may have been bursting to say, 'It's an infinite number!' Yes, of course, there is no limit to how many numbers there are between 10 and
20. There's 141/2 for a start; and 16.07 and 19.9999999; and endless other numbers
using fractions and decimals. So 'number' can also include numbers like these, as well
as all the integers. When we extend our concept of what is a number to include fractions and decimal numbers (which, as we shall see, are a particular kind of fraction)
we get the set of rational numbers.
The term 'rational' derives from the idea that a fraction represents a ratio. The
technical definition of a rational number is any number that is the ratio of two integers. We shall see in Chapter 10 that the ratio of two numbers is a way of comparing
them by dividing one by the other. So, for example, we could say that the ratio of 6
to 2 is 3, because 6 is 3 times larger than 2, and 6 -ê- 2 = 3. Fractions and ratios are
explained in Chapter 15 and decimal numbers are explained in Chapters 16 and 17.
I include a few examples here that may help to illustrate the concept of a rational
number:
is a rational number, because it is the ratio of 3 to 8 C3 divided by 8).
0.8 is a rational number, because it is the ratio of 8 to 10 (8 divided by 10).
141/2 is a rational number, because it is the ratio of 29 to 2 (29 divided by 2).
16.07 is a rational number, because it is the ratio of 1607 to 100 (1607 divided by 100).
23 is a rational number, because it is the ratio of 23 to 1 (23 divided by 1).
-7 is a rational number, because it is the ratio of -7 to 1 (-7 divided by 1).
3/8
All the mathematics involved in these examples is explained later in the book. At
this stage the reader should just get hold of the basic idea that the set of rational
numbers includes all fractions, including decimal fractions (which are just tenths,
hundredths, thousandths, and so on), as well as all the integers themselves.
Rational numbers enable us to subdivide the sections of the number line between
the integers and to label the points in between, as shown in Figure 6.4; here the
MATHEMATICS
6
6
Figure 6.4
EXPLAINED for PRIMARY TEACHERS
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6Y,¡
7
7
Some rational numbers between 6 and 7
interval between 6 and 7 is divided first into ten equal parts (tenths) and second
into four equals parts (quarters).
Now, the reader may be thinking that the set of rational numbers must include all
th numbers there are. But, in fact, there are other real numbers that cannot be written down as exact fractions or decimals - and are therefore not rational. Believe it or
not, there is a limitless number of points on the number line that cannot be repreented by rational numbers. These are numbers like square roots or cube roots (see
hapt r 19) that do not work out exactly.
ITer is an example to illustrate this point, although again some readers should be
feas ured that the mathematics involved here is al! explained later in the book.
Ther is no fraction or decimal that is exactly equal to the square root of 50 (written
as ~50). This means there is no rational number that when multiplied by itself gives
exacrlv the answer 50. We can get close. In fact, we can get as close as we want.
But we annot get exactly 50. Using a calculator, I could discover that ~50 is somewhere b tween 7.07 and 7.08. I find that 7.07 x 7.07 (= 49.9849) is just less than 50
and 7.08 x 7.08 (= 50.1264) is just greater than 50. If we went to further decimal
pia's, we could d cide that it lies somewhere between 7.0710678 and 7.0710679.
But neith r of these rational numbers is the square root of 50. Neither of them when
multipli d by itself would give 50 exactly. And however many decimal places We
wcm i - Y u will just have to believe me about this - we could never get a number
that gay us 50 exactly when we squared it. But ~50 is a real number _ in the sense
thal it r 'pre ents a real point on a continuous number line, somewhere between
7 and 8. lt represents a real length. For example, using Pythagoras's theorem, We
' ul I W rk LItthat the length of the diagonal of a square of side 5 units is '-150
unit: .
this i a real length, a real number, but it is not a rational number. It is
all \ I an irrational number.
Th ire is n nd of irrational numbers, all of them representing real lengths and real
pints on th number line. Some examples would be: V8, ~17.3, 3 50 (the cube root
r 50), and that fav urite number of mathematicians, 1t (pi: see Chapter 23). So, what
math emau ians all the set of real numbers includes all rational numbers _ which
in .lude integers, which in turn include natural numbers - and all irrational numbers.
Wi' think r il as the set of all numbers that can be represented by real lengths or b
.
y
P .mts n a onunuous
num b er l'me.
v
NUMBERS
and PLACE VALUE
I can imagine that some readers are now wondering if there are
than real numbers. If your appetite for number theory is really that
will have to look elsewhere to find out how mathematicians use
imaginary number (like the square root of -1) to construct things
numbers other
insatiable, you
the idea of an
called complex
numbers.
What is meant by ·place value·?
--------------------
The system of numeration we use today is derived from an ancient Hindu system. It
was picked up and developed by Arab traders in the ninth and tenth centuries and
quickly spread through Europe. Of course, there have been many other systems developed by various cultures through the centuries, each with their particular features.
Comparing some of these with the way we write numbers today enables us to appreciate the power and elegance of the Hindu-Arabic legacy. There is not space here to
go into much detail, but the history of different numeration systems is a fascinating
topic, with considerable potential for cross-curriculum work in schools, which will
repay further study by the reader.
The Egyptian hieroglyphic system, used as
LEARNING. • TeAéMING POINT
long ago as 3000 BC, for example, had separate
symbols for ten, a hundred, a thousand, ten
When teaching about place value, give
thousand, a hundred thousand and a million.
appropriate credit to the non-European
The Romans, some 3000 years later, in spite of
cultures that have contributed so much to
all their other achievements, were using a
the development of numeration.
numeration system which was still based on the
same principle as the Egyptians, but simply had
symbols for a few extra numbers, including 5, 50 and 500. Figure 6.5 illustrates how
various numerals are written in these systems and, in particular, how the numeral
366 would be constructed. Looking at these three different ways of writing 366 demonstrates clearly that the Hindu-Arabic system we use today is far more economic
Egyptian hieroglyphics
I
IIIII
V
X
rYVVV\
L
9
C
5
10
50
100
D
500
999 rYVVVY\ IIIIII
Figure 6.5
Hindu-Arabic
r.
99999
,
Roman numerals
CCCLXVI
Same numbers written in different numeration
366
systems
MATHEMATICS EXPLAINED for PRIMARY TEACHERS
The common use today of IV, IX and XC in
Roman numerals instead of 1111,Villi and
LXXXX to represent 4, 9 and 90, respectively, was a later variation introduced to
avoid having to write a string of four
identical symbols. When introducing
Roman numerals in the classroom follow
the historical development; so start with
the earlier expanded form. For example,
29 will be XXVIIlI and 194 will be
CLXXXXIlII. When children are confident
with this form, show how these can be
written in their shorter forms as XXIX and
CXCIV. (See self-assessment question 6.4
at the end of the chapter.)
in its use of symbols. The reason for this is that
it is based on the highly sophisticated concept of
place value.
In Roman numerals, for example, to represent three hundreds, three Cs are needed, and
each of these symbols represents the same
quantity, namely, a hundred. Likewise, in the
Egyptian system, three 'scrolls' are needed,
each representing a hundred. But, in the HinduArabic system we do not use a symbol representing a hundred to construct three hundreds:
we use a symbol representing three! Just this
one symbol is needed to represent three hundreds, and we know that it represents three
hundreds, rather than three tens or three ones,
because of the place in which it is written. The
two sixes in 366, for example, do not stand for
the same number: reading from left to right, the
first stands for six tens and the second for six ones, because of the places in which
th "yare written.
o,
our Hindu-Arable place-value system, all numbers can be represented using
a finit set of digits, namely, O, l, 2, 3, 4, 5, 6, 7, 8, 9. Like most numeration systems,
no d ubi be ause of the availability of our ten fingers for counting purposes, the
syst 'm uses ten as a base. Larger whole numbers than 9 are constructed using powers r th ' base: ten, a hundred, a thousand, and so on. Of course, these powers of
t 'o ar' not limited and can continue indefinitely with higher powers. This is how
s me f these powers are named, written as numerals, constructed from tens, and
' pr 'ss ed as powers of ten in symbols and in words:
A
in
million
A hun Ir .d thousand
.~'n thousand
A th usand
A hundre I
Ten
1 000 000 = 10 x 10 x 10 x 10 x 10 x 10 = 106 (ten to the
100 000 == 10 x 10 x 10 x 10 x 10
10 000 = 10 x 10 x 10 x 10
1000 = 10 x 10 x 10
100 = 10 x 10
10 == 10
=
=
=
=
=
power six)
10 (ten to the
power five)
104 (ten to the
power four)
103 (ten to the
power three)
102 (ten to the
power two)
101 (ten to the
pOwer one)
5
NUMBERS and PLACE VALUE
The place in which a digit is written, then, represents that number of one of these
powers of ten. So, for example, working from right to left, in the numeral 2345 the
5 represents 5 ones (or units), the 4 represents 4 tens, the 3 represents 3 hundreds
and the 2 represents 2 thousands. Perversely, we work from right to left in determining the place values, with increasing powers of ten as we move in this direction. But,
since we read from left to right, the numeral is read with the largest place value first:
'2 thousands, 3 hundreds, 4 tens, and 5 ones'. Certain conventions of language then
transform this into the customary form, 'two thousand, three hundred and forty-five'.
So, the numeral 2345 is essentially a clever piece of shorthand, condensing a complicated mathematical expression into four symbols, as follows:
Similarly, the numeral 2 345 678 is made up of 2 millions, 3 hundred-thousands,
4 ten-thousands, 5 thousands, 6 hundreds, 7 tens and 8 ones. Note that the '3 hundred-thousands, 4 ten-thousands, 5 thousands' part of this number is equivalent to
345 thousands. This enables us to read the number more concisely as 'two million,
three hundred and forty-five thousand, six hundred and seventy-eight'. Again, we
note that the numeral 2 345 678 is a clever piece of shorthand, condensing a very
complicated mathematical expression into just seven symbols:
Notice that each of the powers of ten is equal to ten
times the one below: a hundred equals 10 tens, a
thousand equals 10 hundreds, and so on. This introChildren have to be taught how to write
duces the principle of exchange. This means that
down a number as a numeral when it is
whenever you have accumulated ten in one place
read out to them in the conventional way;
and to say the number in the conventhis can be exchanged for one in the next place to
tional way when written down as a
the left. This principle of being able to 'exchange
numeral. These skills have to be extended
one of these for ten of those' as you move left to
gradually over numbers with increasing
right along the powers of ten, or to 'exchange ten of
numbers of digits.
these for one of those' as you move right to left, is
a very significant feature of the place-value system.
It is essential for understanding the way in which we count. For example, when counting
in ones the next number after 56, 57, 58, 59 ... is 60, because we fill up the units position
with ten ones and these are exchanged for an extra ten in the next column.
This principle of exchanging is also fundamental to the ways we do calculations with
numbers. It is the principle of 'carrying one' in addition (see Chapter 9). It also means
that when necessary we can exchange one in any place for ten in the next place on the
right, for example, when doing subtraction by decomposition (see Chapter 9). The same
MATHEMATICS
EXPLAINED for PRIMARY TEACHERS
principle of exchanging 'one of these for ten of
those' extends to decimal numbers, where positions after the decimal point represent tenths, hundredths, thousandths, and so on (see Chapter 16).
I should mention here that the proper convention for writing numerals with more than four
digits is to group the digits in threes from the
right, using spaces to separate the groups. So, for
example, a million would be written 1 000 000.
In explaining place value to children use
the language of 'exchanging one of these
for ten of those' as you move left to right
along the powers of ten, and 'exchanging
ten of these for one of those' asyou move
right to left.
What are the best ways of explaining
place value in concrete terms?
---------_
There are two sets of materials that provide particularly effective concrete embodiments of the
place-value principle and therefore help us to
explain the way our number system works.
They are (1) base-ten blocks and (2) Ip, lOp
and £,1 coins.
Use coins (tp, lOp and £1) and base-ten
blocks to develop children's understandIng of the place-value system, particularly
to reinforce the principle of exchange.
Figure 6.6 shows how the basic place-value
principle of exchanging one for ten is built into
th se materials' for ones , tens and hundreds. Note that the ones in the base-ten blocks
arc metimes referred to as units, the tens as longs and the hundreds as flats. With
the bl k, of ourse, ten of one kind of block can actually be put together to make
ne of th n xt kind. With the coins it is simply that ten ones are worth the same as
on l en, and o on.
a hundred (a flat)
a ten (a long)
i+~++++~è---This
block
is made up of
This block
is made up of
ten of these
ten of these
a hundred (a pound)
---"'0
a ten (a ten-pence)
This coin is
worth the same
as ten of these
Figure 6.6
a one (a unit)
-.8..
Materials for explaining place value
a one (a penny)
This coin is
worth the same
as ten of these
-
NUMBERS and PLACE VALUE
hundreds
tens
ones
O
O O
DO
O
hundreds
tens
ones
G)p~plp
G) lPG)
Figure 6.7
The number 366 in base-ten blocks and in coins
Figure 6.7 shows the number 366 represented with these materials. Notice that with
both the blocks and the coins we have 3 hundreds, 6 tens and 6 ones; this collection
of blocks is equivalent to 366 units; and the collection of coins is worth the same as
366 of the lp coins. Representing numbers with these materials enables us to build
up images which can help to make sense of the way we do calculations such as addition and subtraction by written methods, as will be seen in Chapter 9.
How does the number line support
understanding of place value'?
As we have seen already, the number line is an
LEARNING· • TEACHING POINT
important image that is particularly helpful for
appreciating where a number is positioned in
Making the connection between numbers
relation to other numbers. This ordinal aspect of
and points on the number line provides
a number is much less overt in the representation
children with a powerful image to support
of numbers using base-ten materials. Figure 6.8
their understanding of number, emphasizshows how the number 366 is located on the
ing particularly the position of a number
number line. The number-line image shows
in relation to other numbers.
clearly: that it comes between 300 and 400; that it
comes between 360 and 370; and that it comes
between 365 and 367. The significant mental processes involved in locating the position of the number on the number line are: counting in lOOs; counting in lOs; and
counting in Is. First you count from zero in lOOsuntil you get to 300: 100, 200, 300;
MATHEMATICS EXPLAINED for PRIMARY TEACHERS
300 310 320 330 340 350 360
Figure 6.8
370 380
390
400
The number 366 located on the number line
then from here in lOs until you get to 360: 310, 320, 330, 340, 350, 360; and then in
1 from here until you get to 366: 361, 362, 363, 364, 365, 366. The number-line image
i also particularly significant in supporting mental strategies for addition and subtracti n calculations, as will be seen in Chapter 8.
What is meant by saying that zero is a place holder?
The Hindu-Arabic system was not the only one to use a place-value concept.
R markably, about the same time as the Egyptians, the Babylonians had developed
a system that incorporated this principle, although it used sixty as a base as well as
ten. But a problem with their system was that
you could not easily distinguish between, say,
three and three sixties. They did not have a symbol for zero. It is generally thought that the
Incorporate some study of numeration
Mayan civilization of South America was the first
systems into history-focused topics such as
to develop a numeration system that included
Egyptian and Mayan civilizations, and use
this to highlight the advantages and sigboth the concept of place value and the consistnificance of the place-value system we use
ent use of a symbol for zero.
today.
Figure 6.9 shows 'three hundred and seven'
represented in base-ten blocks. Translated into
hundreds
tens
ones
O
O
O
O
DD
O
Figure 6.9
Three hundred and seven
In
base-ten blocks
NUMBERS and PLACE VALUE
symbols, without the use of a zero, this would
easily be confused with thirty-seven: 37. The zero
is used therefore as a place holder; that is, to
indicate the position of the tens' place, even
though there are no tens there: 307. It is worth
noting, therefore, that when we see a numeral
such as 300, we should not think to ourselves that
the 00 means 'hundred'. It is the position of the 3
that indicates that it stands for 'three hundred'; the
function of the zeros is to make this position clear
whilst indicating that there are no tens and no
ones. This may seem a little pedantic, but it is the
basis of the confusion that leads some children to
write, for example, 30045 for 'three hundred and
forty-five'.
Arrow cards, as shown in Figure 6.10, provide a
strong visual image to help children to understand
how a number like 452 is made up of 400 (4 hundreds), 50 (5 tens) and 2 (2 ones). By placing the
three cards for 2, 50 and 400 one on top of the
other, so that the arrows at the ends of the cards
line up, the numeral 452 is constructed. Clearly,
this idea can be extended to four-digit numerals.
~4~~0~
LEARNING·
• TlACHM POINT
Give particular attention to the function
and meaning of zero when writing and
explaining numbers to children. The zero
in 307 does not say 'hundred'. The 3 says
'three hundred' because of the position it
is in. The zero indicates an absence of
tens; it says'no tens'.
LEARNING·
• TEACHING POINT
Use arrow cards to construct numerals
with three digits from hundreds, tens and
ones; and numerals with four digits from
thousands, hundreds, tens and ones.
Discussion of how these cards make 3and 4-digit numerals and what gets hidden when the cards are stacked enables
children to make helpful connections
between the visual image of the cards,
language and symbols.
__0~ __ ~)
ARROWCARDS
,--5--1._0---,_--,)
c__2
Figure 6.10
,--4__.__5-,-_2__.__~>
..____.)
Arrow cards showing the values represented by the digits in the
numeral452
How is understanding of place value used
in ordering numbers'?
Being able to put numbers in order requires a good understanding of the basic principles of place value. For example, given the set of numbers {906, 2345, 97, 967}
someone understanding place value should be able to rearrange them in order from
MATHEMATICS EXPLAINED for PRIMARY TEACHERS
smallest to largest: {97, 906, 967, 2345}. They will
also be able to visualize these numbers as being
positioned from left to right on a conventional
One of the best indicators of children's
horizontal number line.
developing understanding of place value is
To do this you need to be able to decide
that they can arrange a set of numbers in
instantly which is the greater or which is the
order, from smallest to largest or from
largest to smallest. The starting point for
smaller of two numbers. For example, think about
this is to be able to state whether one
how it is that you immediately know that 2345 is
number is greater or less than another.
greater than 967? I would guess that you just use
These are important skills on which to
the fact that any 4-digit positive integer is greater
focus, whether working with younger chilthan any 3-digit positive integer. This probably
dren and numbers up to 20 or with older
seems
very obvious but it is actually a very
children and numbers into the millions.
sophisticated idea, showing just how powerful is
the place-value system for numeration. It is
always the first digit in a numeral that is most
significant in determining the size of the number.
So the first digits in these two numerals tell us that
In due course children can record their
the 4-digit number is into the thousands whereas
comparisons of numbers using the maththe 3-digit number is only into the hundreds.
ematical symbols > (greater than) and
Similarly,how do you know that 487 is less than
< (less than). For example, 25> 16 (25 is
609?
Again the first digits tell us all we need to
greater than 16) and 16 < 25 (16 is less
know: the first number has 4 complete hundreds _
than 25). These two symbols are called
inequality signs.
which must be smaller than the second number,
which has 6 complete hundreds. But what about
comparing, say, 3456 and 3701, where they are both
4-digit numbers and the first digits are the same? In this case they both have 3 complete
thousands. You will now find yourself looking at the next most signillcant digit, which tells
y u how many complete hundreds there are to go with the 3 complete thousands. The
number 3456 has 3 complete thousands and 4 complete hundreds; the 3701 is greater than
this, having 3 complete thousands and 7 complete hundreds.
The xtensive language used to compare quantities in a wide range of measuring
on texts is discussed further in Chapter 7. Significant digits are discussed further in
hapter 16. Putting three or more numbers in order also makes use of the principle
f tran itivity, which is explained in the context of measurement in Chapter 22.
How are numbers rounded to the nearest
10 or the nearest 1007
Rounding is an important skill in handling numbers, particularly, as we shall see in
hapter 16, in handling the results of calculations involving decimals and in many
practical ontexts. One skill to be learnt is to round a number or quantity to the
NUMBERS and PLACE VALUE
nearest something. For example, someone might state their annual salary to the
nearest thousand pounds, or a parent might measure a child's height to the nearest
centimetre (see Chapter 22).
The first step in developing this skill is to be
able to round a 2-digit number to the nearest ten.
To find what 67 is when rounded to the nearest
When rounding to the nearest ten, help
ten is to find which multiple of 10 is nearest to 67.
children to seethat if the units digit is less
The word 'nearest' implies that this is a spatial
than 5 then the number rounds down to
idea, so the image of a number as a point on a
the preceding multiple of 10; but if the
number line is important here. Figure 6.11 - with
units digit is greater than 5 then the number rounds up to the next multiple of 10.
multiples of 10 marked on a number line - shows
Key to this is seeing the number visually
how 67 lies between 60 and 70, but is nearer to
sitting on a number line between two
70. This is exactly what we mean when we say
multiples of 10 and identifying which one
that 67 rounded to the nearest ten is 70. This can
is nearest to it.
extend to numbers with more digits of course: so,
for example, 167 rounded to the nearest ten is 170
and 4163 rounded to the nearest ten is 4160. I should just mention that if the number
ends in a 5 and therefore lies halfway between two multiples of 10 then there is not
a nearest multiple of lO! Note also that a positive integer less than 5 (that is, 1, 2, 3 or
4) rounded to the nearest ten is O.
40
Figure 6.11
50
60
70
80
The nearest multiple of 10 to 67 is 70
We can then round numbers to the nearest hundred, using the same two steps. First,
between which two multiples of 100 does the number lie? Then, which of these is it
nearest to? So, for example, 765 lies between 700 and 800, it is nearer to 800 than 700,
so we could say that it is 800 rounded to the nearest hundred. The principle is that if the
Lasttwo digits are less than 50 then you round down to the previous multiple of 100, and
if they are greater than 50 then you round up to the next multiple of 100. For example
4126 (ending in 26) is rounded down to 4100; and 4156 (ending in 56) is rounded up to
4200. If the final two digits are actually 50 then there is no nearest hundred.
A tricky example would be to round, say, 8967 to the nearest hundred. The two
multiples of 100 that this number lies between are 8900 and 9000. Since it ends in 67
it must be rounded up. So 8967 is 9000 to the nearest hundred.
Clearly, these principles can extend to rounding to the nearest thousand, the
nearest ten thousand, the nearest hundred thousand, or the nearest million. For example,
MATHEMATICS
EXPLAINED for PRIMARY TEACHERS
at th time of writing, rounded to the nearest thousand there are about 21 000
maintained primary schools in the UK; and rounded to the nearest ten thousand,
the population of the beautiful county of Norfolk, where I am writing this book, is
about 730000.
Research focus: place value
in e the essence of our number system is the principle of place value, it seems
natural to as ume that a thorough grasp of place value is essential for young children
I ef r they can successfully move on to calculations with two- or three-digit numbers.
Th mps n ha undertaken a critical appraisal of this traditional view (Thompson,
2000. onsid ring the place-value principle from a variety of perspectives, Thompson
n ludes that the principle is too sophisticated for many young children to grasp. He
argues that many of the mental calculation strategies used by children for two-digit
addition and subtraction are based not on a proper understanding of place value but
n what h ails quantity value. This is being able to think of, say, 47 as a combinati fi of 40 and 7, rather than 4 tens and 7 units. Similar findings are reported by Price
20 1), whose research focused on the development of place-value understanding in
Year
hildren. I-I proposed an independent-place construct: that children tend to
make singl -dirnensíonal associations between a place, a set of number words and a
ligit, rather than taking account of groups of 10. A later research study by Thompson
and Bramald (2002) with 144 children aged 7 to 9 years demonstrated that only 19 of
th '91 hildr n who had successful strategies for adding two-digit numbers had a
g d understanding of the place-value principle. Approaches to teaching calculations
witl: y uriger hildren that are consistent with these findings would include: delaying
the Intro lu lion of column-based written calculation methods; emphasis on the posíLion r numbers in relation to other numbers through spatial images such as hundred
squar JS and numl er lines; practice of counting backwards and forwards in Is, lOs,
J 005; an I m rual alculation strategies based on the ideas of quantity value.
Sugg est I a '[ivilies f r your lesson plans related to this chapter can be accessed
via your interactive eBook. We have provided an example in this chapter of
th ' r .sourc 's that arc available for later chapters.
J
INClUD£
UIION
IN
PIAN
Activities for Lesson Plans
Years 1-2: Place race
Learning intention
Recognize the place value of each digit in a two-digit number (tens, ones).
organization
Children play against one another in pairs.
Equipment
per pair
See 'Guidance and Resources for Lesson Activities'
•
•
•
•
•
One set of place-value Carrow') cards representing tens and units for each pair of children;
Base-ten or Dienes' apparatus (9 tens and 9 units only);
Place value mat (TU is sufficient, or fold back the 'H' on an HTU mat);
lOO-square (0-99 or 1-100);
Counters in two colours.
Activity
Luke and Emily agree which colour of counters each will use. They shuffle the p.v. cards and set them
out randomly, face down on the table. The base-ten apparatus is in a pile the other side of the p.v.
cards. Luke calls any uncovered two-digit number from the hundred square. The pair then race each
other:
•
•
Luke searches for the correct tens and unit p.v. cards needed to comprize the number and assembles these face up on the table; while '"
Emily grabs the correct numbers of tens and unit base-ten pieces and places these in the correct
columns on the p.v. mat.
Thc first to complete their task calls out the number and claims the number by placing their coloured
counter over it, then waits for the other to finish (and helps them if necessary). They both check each
has the correct cards/numbers of base-ten pieces. Then they return the base-ten pieces to the pile and
put the p.v. cards face down to the table, to begin again. The next time Emily calls the number and
searches for the p.v. cards, while Luke grabs the base-ten pieces for the p.v. mat.
Teaching points and notes
•
•
•
•
Children get to match the correct number of base-ten pieces to each of the p.v. cards in the number.
They have to recognize the equivalence of 'two tens' and'twenty'.
Emphasize the correct use of the irregular English vocabulary for the numbers eleven up to
twenty.
The activity can be simplified by limiting the range to 11-20, or it can be extended to three-digit
numbers to make it more challenging.
Years 3-4: King of the castle
Learning intention
I~ , .ogn¡z ' th ' plac e-value of each digit in a three-digit number (hundreds,
tens, ones).
Add and subtract numbers mentally, including a three-digit number and tens.
Organization
hil Ir 'n play against one an ther in pairs.
Equipment per pair
SeC''Ciuidance Gild Resourcesfor Lesson Activities'
•
•
•
Two sets of plac -value Carrow') cards representing tens and units for each pair of children;
Bas' I in or Di 'nes' apparatus (up to 3 hundreds, 19 tens and 19 units);
Pla" va lue mat (I IT ).
Activity
Shell 'Y and H han separate the p.v, cards into two separate, shuffled piles, face down: a tens pile and a
untts pll '. Start with 200 (two hundred base-ten blocks) in the 'H' column of the p.v. mat: Shelley intends
to subtra '1 basc-i .n blocks to make the pile less than 100; Rohan intends to add base-ten blocks to make
th ' pil ' bigger than 300. Th' winner of the game is the first to reach their target. Here is an example:
•
Sh'II'y turns over the top p.v, card [rom each pile. She combines these to assemble a two-digit number,
in this xample 37 and writes down the calculation she is going to carry out:
200
•
•
•
37 =
Sh 'II 'Y 'arri 's out th i physi al subtraction, by exchanging one base-ten hundred block for 10 fens,
and on ' of the tens for lO units as needed. At the end of her turn, she writes the result of the
calculation whi h is repr s ented on the p.v. mat:
200 37 = 1 )3.
Rohan war .h 's al' efully and checks that SheJJey's answer is correct.
Hollan turns over th ' n ext two p.v, cards, and combines
56.
II ' writ 's down the calculation
163 + 5)
•
=
these to assemble a two-digit number, 'ay
he is to carry out given the present value represented on the mat:
I le 'arri·s out th, physical addition,
exchanging
10 units for 1 ten, and 10 tens for
I hlllldr.·cI as need xl, whenever h has more than 9 of the pieces for a particular place. At the end
of Hohan's turn, he writ es the result of the calculation, which is represented on the p.V. mat:
1';+5).::2J7.
•
Sh 'II 'y ward) 's carefully and checks that Rohan's
•
Th.'y 'ontinlI' to alt 'mat' turns, Shelley subtracting and then Rohan adding until either Shelley
nnw's al li numb 'I' below LOa or Rohan gets lo a number greater than 300. When this happens the
winn 'I' gains 5 p iints, and they begin again, swapping the subtract/add roles.
Wh -n th 'Y nav . LIS 'd all the p.v cards, they shuffle them and place them again to continue the game
An extra p lint is award id to Rohan if he shows Shelley that she made an incorrect calculation
(and vic' versa).
•
•
•
answer
is correct.
An .xrra point is awarded to Shelley if she spots that Rohan has left more than 9 pieces
th' units r I .ns olurnn of the p.v, mat (and vice versa).
in one or
Teaching points and notes
•
Emphasising
lhat 'JO of these
. . '(.I
nLlIll IX~r .IS mall1tall1
is one of those'
and ensuring
that the integrity
of each
digit in ~
\
•
•
•
Emphasising the term exchange.
Using the blocks to physically place value in the calculation physically, rather than reverting to an
empty number line.
To simplify the game, work with addition and subtraction of single-digit numbers starting with a
value of, say 50, and aiming for targets of below 10 and above 90.
Years 5-6: Place invaders
Learning intention
Read, write order and compare numbers up to 10000 000 and determine the value of each digit.
Identify the value of each digit in numbers given to three decimal places.
Organization
Children play against one another in pairs.
Equipment per pair
'>impIe (non-scientific) calculator.
Activity
space Invaders was an arcade video game from the 1970s. The aim was to shoot down as many alien
spacecraft as possible. Versions of this game are available on the internet today. 'Place invaders' is
based on this idea, to 'shoot' specified digits in a number displayed on a calculator, by making them
zero (or blank for the most and least significant digits in the number) whilst leaving the other digits
unchanged. Here is an example:
•
•
•
•
•
•
•
•
Meena clears the calculator and enters a number, say, with up to 6 integer digits and 2 decimal
digits, for example 76398l.72.
Charlie selects a digit for Meena to shoot: 'Take out the 3, Meena'.
Meena has to enter the correct subtraction into the calculator to do this:
<: 3000 ='
They both check that the calculator display has been correctly changed to 76098l.72, and Meena
gets 1 point.
Meena selects a target digit: 'Shoot the 7 on the right, Charlie'.
Charlie has to enter the correct subtraction:
,_ 0.7 ='
The calculator now shows 76098l.02 and Charlie gets 1 point.
They continue alternating turns until the display is O.
If the display is incorrect after a subtractíon, the player does not get a point, but they can continue
LO work with the changed number.
Teaching points and notes
•
•
•
Simplify the game by reducing the number of digits, and/or removing decimal places.
Extend the challenge, by giving an additional point to the target-setter if they describe the target in
terms of its actual place value, for example: 'Take out the three thousand, Meena', or 'Shoot the seven
tenths, Charlie.'
Extend it further with more digits in the integer and/or decimal.
MATHEMATICS EXPLAINED for PRIMARY TEACHERS
Self-assessment questions
B 'for' trying the self-assessment questions below, you should complete the interactive
s 'If-ass essmeni questions for this chapter that are available on the companion website:
study.sagepub.com/haylock5e
Ifli
ASI(IIM[NT
OUfITlON\
6.l:
6.2:
6.:3:
6.1t:
/
ç
6.'):
6.6:
6.7:
What is the next number after 199?
(a) How many numbers are there between Oand 20? Cb) How many integers are there between O and 20?
Arrange these numbers in order from the smallest to the largest, without
converting them lO Hindu-Arabic numbers: DCXIII, CCLXVII,CLXXXVIII,
()
. Then onvert them to Hindu-Arabic, repeat the exercise and
not' <lnysignif ant differences in the process.
In th • lat er form of Roman numerals, writing a lower value symbol in front of
another symbol meant that this value had to be subtracted. For example, N
represents 4. (= 5 - 1), XC represents 90 C=100 - 10) and CD represents 400
(= 500 - 100). What years are represented by (a) MCMLXXXIV?
Cb) MCDXCll?
Add on' to four thousand and ninety-nine.
Write th 'se numbers in Hindu-Arabic numerals, and then write them out in
full using powers of len: (a) five hundred and sixteen; (b) three thousand
an I sixty; and (c) two million, three hundred and five thousand and foul'.
I have :34one-penny coins, 29 ten-penny coins and 3 one-pound coins.
Apply the principle of 'exchanging ten of these for one of those' to reduce
this '011 'Clion of coios to the smallest number of lp, lOp and £1 coins.
Further practice
From the Student Workbook
LI
LI
LI
Checking understanding (numbers and place value)
'sti ns Ol5-O22: Reasoning and problem solving (numbers and place value)
istions 023-033: Learning and teaching (numbers and place value)
estions 001-014:
Glossary of key terms introduced in Chapter 6
Numeral th' symbol used to represent a number; for example, the number of chilIren il a class might be represented by the numeral 30.
Cardinal aspect of number the idea of a number as representing a set of thing
s.
This idea of number has meaning only in terms of non-negative integers.
NUMBERS
and PLACE VALUE
Ordinal aspect of number the idea of a number as representing a point on a
number line. This idea of number as a label for putting things in order has meaning for negative as well as positive numbers.
Natural numbers the set of numbers that we use for counting, 1, 2, 3, 4, 5, and so
on, going on for ever.
Integer
a whole number, positive, negative or zero.
Positive integer an integer greater than zero. The integer +4 is correctly referred to
as 'positive four'. Usually the + sign is understood and the integer is just written as
4 and referred to as 'four'.
Negative integer
'negative four'.
a number less than zero. The integer -4 is correctly referred to as
Minus and Plus synonyms for 'subtract' and 'add' respectively. Strictly speaking, it
is incorrect to refer to negative integers and positive integers as 'minus numbers'
and 'plus numbers', as is often done by weather forecasters.
Rational number a number that can be expressed as the ratio of two integers
(whole numbers). All whole numbers and fractions are rational numbers, as are all
numbers that can be written as exact decimals.
Real number any number that can be represented by a length or by a point on a
continuous number line. The set of real numbers consists of all rational and all
irrational numbers.
Irrational number a number that is not rational; for example ..)2 is irrational
because it cannot be written exactly as one whole number divided by another.
Place value the principle underpinning the Hindu-Arabic system of numeration, in
which the position of a digit in a numeral determines its value; for example, '6' can
represent six, sixty, six hundred, six tenths, six hundredths, and so on, depending
on where it is written in the numeral.
Roman numerals a system of numeration that does not use the principle of place
value, so the value represented by one of the symbols used (I, V, X, L, e, and so
on) is not dependent on the position in which it is written; e represents a hundred,
for example, wherever it is written.
Digits the individual symbols used to build up numerals in a numeration system; in
our Hindu-Arabic system the digits are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Base the number whose powers are used for the values of the various places in the
place-value system of numeration; in our system the base is ten, so the places
represent powers of ten, namely, units, tens, hundreds, thousands, and so on.
MATHEMATICS
EXPLAINED
for PRIMARY
TEACHERS
Power a way of referring to a number repeatedly multiplied by itself; for example,
10 x 10 x 10 x 10 is referred to as '10 to the power 4', abbreviated to 104
Exchange
the principle at the heart of our place-value system of numeration, in
whi h t n in one place can be exchanged for one in the next place to the left, and
vi versa; for example, 10 hundreds can be exchanged for 1 thousand, and 1
th u and can be exchanged for 10 hundreds.
Number line a traight line in which points on the line are used to represent numb ers, emphasizing particularly the order of numbers and their positions in relation
t
a
h
ther,
the role of zero in the place-value system of numeration; for example,
in th ~ numeral 507 the O holds the tens place to indicate that there are no tens
h r '. Without the us of zero as a place holder there would just be a gap between
the 5 and th 7.
Place holder
Rounding (to the nearest 10 or 100) done to find which multiple of 10 (or 100)
is nearest t th given number and to use this as an approximate value for the
numl r.