Measurement
The Language of Measurement
Of all the mathematical language that young students first encounter measurement is the most
demanding; one reason is that a wide variety of words are used that mean the same thing. For example, all
these nouns refer to the measurement of length:
• width
•
breadth
•
amplitude
• height
•
base
•
gap
• distance
•
thickness
•
dimension
• girth
•
radius
•
circumference
• diameter
Even more extensive than the nouns related to measurement are adjectives and adverbs that they often
have both comparative and superlative versions.
Comparatives and Superlatives
Comparative words or phrases always involve comparing two things. These words are often used
wrongly in everyday speech. For example sports commentators will often compare two halves of a game by
saying “In the test match the All Blacks had the best of the first half”. This cannot be correct as best is a
superlative that is reserved for three or more things. So the corrected sentence needs better not best:
• In the test match the All Blacks had the better of the first half
English has a number of rules for creating comparison words:
Rule 1
Add –er to a word. Example:
• When I compared the height of the two children I saw Ronnie was taller
Rule 2
Place more before a word. Example:
• The swimming pool was more crowded in the afternoon - crowdeder sounds crazy!
Rule 3
For words where the last three letters are consonant, vowel, consonant, the last consonant is doubled.
Example:
Big becomes bigger not biger
Rule 4
For adjectives and adverbs ending in ly sometimes change the y to i and add er. For example:
• Early becomes earlier
Rule 5
For adjectives and adverbs ending in ly sometimes delete the ly and add er. For example:
• Quickly becomes quicker
Rule 6
For adjectives and adverbs ending in ly sometimes add more. For example:
•
Quickly becomes more quickly
Rule 7
Place more before words that have three or more syllables. Example:
• Enjoyable becomes more enjoyable not enjoyabler
Rule 8
Watch out for the exceptions to the above rules. Examples:
• Good, better, best
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• Bad, worse, worst
• Less, lesser, least
Rule 9
Even the best set of rules may not cover every possibility. When in doubt, and experience should help
suggest that something may be wrong, check with a good dictionary.
The superlative words follow the same rules as the comparative words with these changes:
• –er is replaced by –est
• more is replaced by most
Clearly teachers should not teach these rules; this is never the way grammar is best acquired. Students
need to be exposed constantly to comparative and superlative words.
Countable and Continuous Quantities
Students who use counting methods as their best cognitive strategy will find all measuring difficult
because they need to understand what continuous measurement means, which requires understanding of
fractional numbers. When ready students will find movement helpful in understanding continuity. For
example, using a ruler with an object whose length is to be measured, and imagine an ant crawling along it
starting at zero. The ant passes continuously over an infinite number of points and finally ends up at the
point corresponding to the length being measured. This is a powerful image of continuity for students, and
it is continuity that characterises all measurement whether dealing with lengths, weights, times and so on.
Here are some examples of measurements that make sense:
• The bag of potatoes weighed 3· 655 kilograms
• Manu drank a bottle of orange juice that contained 600 millilitres
• Sarah took 13· 8 seconds to run the 100 metres
• The Silver Ferns’ goal-shoot is 1· 93 metres tall
Yet there are many problems where the decimal numbers that represent measurements above would
no sense. Consider these impossible situations:
• 56· 7 people attended the football match
• Mary had 12· 87623 blouses
In both these case measures of continuous situation is absurd.
In summary we can say that measurement is always about continuous quantities, and this must exclude
countable quantities.
Misuse and Use of Less, Lesser, Least, Fewer, and Fewest
In order to use the special words less, lesser, least, fewer and fewest, students need to understand the
difference between measuring and counting. Less and least than apply to measurable quantities but not
countable ones. Examples:
• ‘Jerry drank less water than Thomas’ (measuring) is correct but
• ‘Jerry drank fewer water than Thomas’ (measuring) is wrong
• ‘Maria had the fewest apples from the orchard to take home’ (counting) is correct, but
• ‘Maria had the least apples from the orchard to take home’ (counting) is wrong
Confusingly there is no similar rule for more and most; here there is no difference between measuring and
counting. Examples:
• ‘Harry ate the most apples’ (counting) is correct and
• ‘Miranda ate the most soup’ (measuring) is also correct
• ’Jane had three more children than Kate’ (counting) is correct and
• ‘Kumar had mixed more concrete than he could fit on to his truck’ (measuring) is also correct
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Learning to Measure
Assuming that students have had a rich diet of stories and situations using the language of length and its
comparison words teachers will move on to measuring. Students initial measuring activities should not use
standard measuring devices like rulers or scales; rather they should use non-standard measures that are
often based on the students themselves. For example the width of a room can be measured by the number
of ‘student-feet’ needed before any attempt at using a ruler is attempted.
When teachers think that their students are ready to move on to using standard measuring devices like
rulers and other devices that have scales, they should check that the highest cognitive level that students
have available is beyond involve counting i.e. they must be part-whole thinkers. Figure 1 illustrate a typical
problem that students who always use counting methods run into. They say the length of the rod is 7 units
0
1
2
3
4
5
6
7
8
9
10
Figure 1
long. The fact that the rod needed to be position against the zero on the scale is too much for them to
cope with.
There is deeper problem in reading scales. Students need to understand fractions to read scales
correctly. For example, Figure 2 shows is a more realistic measurement task than the rod in Figure 1
because it the answer is near to 6· 6 units; to read this students have to understand the place-value system
0
1
2
3
4
5
6
7
8
9
10
Figure 2
that has been extended from whole numbers, which counters can understand to some extent, to decimal
fractions. And, because most year sixes have very little working understanding of decimal numbers, they
will struggle be to read scales properly.
Given this general background to teaching all measurement concepts the specifics of teaching length,
area, volume and angle are now addressed.
Learning About Length
In selecting material for activities involving length it is important that initially the material is essentially
one-dimensional. Of course it is impossible for any material not to have the second and third dimensions,
but if these dimensions are small enough student will effectively discount them and concentrate on length
only. For example a thin rod is good for measuring length because two of its dimensions are ignored;
however a thin flat rectangle has two dimensions rather that one dimension that distracts the user, so it is
a poor choice to measure length. And, of course, a three dimensional block is even a poorer tool for
measuring length.
Activities initially need to concentrate on comparisons. For example, suppose there is a large box of
various length sticks in the classroom. The activity could be to find the longest stick in the box . Though no
longer popular Piaget’s concept of conservation of length is useful here. Not to conserve length means the
students will not realise that the length does not change as the position off the object changes as the
students sort through the box. However, it is a very serious error for the teacher to try to teach the
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students that length does not change as the position of objects in space change. This is because the
students have to make this connection themselves and cannot be trained into making it. What this suggests,
and it is true for all measurement not just length, is that the teacher’s role is to provide a rich variety of
tasks and observe when students understand that length is invariant when an object is moved in space.
They can do no more than this.
Once comparisons of length are well understood students may move on to measuring by counting using,
for example, the foot of a student, the width of a student hand, a ‘ten rod’ from place-value material, and
any kind of rods that are all the same length.
When teachers are satisfied that part-whole thinking has emerged in their
students, and they can measure lengths with non-standard units rulers should be
built by the students themselves using a repeating unit. For example, a landscape
Figure 3
gardener's uses tiles shown in Figure 3. The plan in Figure 4 shows about four and a
quarter tiles are needed. The task is now to create a ruler that will measure the length of the tiles.
Figure 4
Students can create their own rulers by transferring from a line of tiles to the ruler and adding a scale as
(Figure 5). Initially ‘Halves’ can be added as this is enough for a first construction of a ruler. Also notice the
zero on a ruler is highly problematic so is best omitted on the scale; this is a common feature of many, but
not all, commercial rulers.
1
2
3
4
5
6
Figure 5
This is a vital task to stimulant students’ understanding of how to measure lengths with rulers.
Students need to understand a very important idea about length, namely that curves are longer when
pulled straight. Figure 6 illustrates the point. Here students would often measure a piece of string with a
ruler, and say the length is 4·5 units. However, they fail to realise that finding the length requires them to
pull the string straight.
0
1
2
3
4
5
6
7
8
9
10
Figure 6
In a similar vein students often misunderstand that sloping straight lines are longer than apparently same
length lines that are horizontal or vertical. Figure 7 shows an example where many students will argue that
both red lines are equal in length because they both start and end at the same horizontal points.
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Figure 7
It is particularly important to realise that the relatively small vertical component in the second red line only
lengthens the line slightly. Here measuring the length of the lines is unlikely to detect that they are of
different length because the sloping line is only half a millimetre longer; the important idea here is that
students understand in principle the sloping line must be longer.
Perimeter
The word perimeter is derived from the Greek ‘peri’ meaning ‘around, about, or beyond’, and ‘meter’
meaning a device that measures time, distance, etcetera. Students can be usefully taught this. ‘Peri’ can be
linked to ‘periscope’ in a submarine, which literally means ‘view around’. A good way to teach the meaning
of perimeter is for students to engage in walking around a variety of buildings, fields, etcetera that can be
found in school.
Anomalously the perimeter of a circle has its own special word, namely circumference, whereas no
other shape has this privilege.
Standard units like millimetres, centimetres, metres, and kilometres can now be introduced.
Teaching Area
Experiences involving coverage of a surface are needed for students to understand what area means,
and they are the essential starting point for learning this concept. In these experiences students need to use
the word ‘area’ appropriately; this is a vital aid in learning what area means.
Unfortunately unlike ‘length’, which can be introduced using comparison
language like ‘longer’ and ‘shorter’, beginning area with comparative area language
is problematic. Figure 8 illustrates why; if asked which rectangle has the larger
area students attention will initially slip to looking for the longest side, and choose
the rectangle on the left even though the rectangle on the right has the bigger
area. This suggests we ignore the comparative language for area and immediately
proceed to using non-standard tessellating shapes to measure area.
In junior classes a common activity is to cover a shape with non-tessellating
Figure 8
objects like beans. This is a bad idea since beans leave gaps between them, and
therefore always underestimate the area; it is not desirable to imply that gaps are acceptable when
measuring area.
Before introducing tessellating materials it is important to remember that, just as materials used in
length activities have one clear dimension with the other minimised to avoid confusion, area materials need
two obvious dimensions with the third reduced so as not to distract students. This argues against using
materials like Cuisenaire rods, but instead uses thin flat ‘tiles’ of various shapes. Importantly students
should explore which tiles shown in Figure 9 are suitable to measure area i.e. they find which shapes
tessellate.
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Figure 9
Older students familiar with suitable software can copy rotate, flip and translate a basic shape to see
whether it tessellates and if it does create colourful and interesting ‘tiled’ surfaces on screen. Details of this
are covered in the previous chapter.
Once the students have discovered suitable tiles that tessellate they can use these to find the area of a
surface by tiling and counting the number of tiles needed - this gives the area of the surface. For example
Figure 10 shows a kitchen table that has been
tiled with regular hexagons. Initially, for first
time learners about area, a good answer would
be 140 tiles plus enough part of tiles to fill the
spaces. A more sophisticated answer that can
come later would be 140 tiles plus an estimate
of the sum of the fractional tiles that have had
to be cut to create the edges plus an allowance
for breakages.
It may be reasonably asked why not start
Figure 10
teaching area with square tiles and stick with them. The answer is that students
need to understand that any tessellating shape is suitable for coverage of a
surface and not just squares, that is to say, area is a concept independent of the
unit or tile used to measure it.
Of course eventually area activities involving squares will be introduced since
the square is the most convenient unit – try finding area formulas using
hexagons rather than squares and you will see how hard this is. Figure 11 shows
a good activity in which the area of an island is found by counting squares where
each square has an area of 1 hectare. Importantly such counting activities need
to precede any formulas for area that are based on the square as the countable
unit.
Figure 11
The Area of a Rectangle Formula
An important part of learning about area for older students is the development of area formulas. The
obvious first one to discover is the area of a rectangle (and therefore also the area of a square). Crucial in
all students’ learning is that they construct the formula for themselves rather than being provided the
formula by the teacher after which endless practice by substitution occurs. And this construction, to be
understood properly, requires teachers to take great care.
Starting with squares tiles Figure 12a shows a rectangular shape with area 12 tiles. Figure 12b shows the
same the same tiles spread out in a manner that allows students to see why the area is the number of rows
times the number of tiles in each row. Figure 12c shows the same tiles spread to show why the area is the
number of columns times the number of tiles in each column. These both gives the conceptually correct
formula for the area of a rectangle rather than rote ‘length times width’ formua.
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Figure 12a
Figure 12b
Figure 12c
Students should be encouraged to solve similar problems using all three ways. For example for a 4 by 5
tiled area students should understand all these methods:
• Count from 1 to 20
• Show four rows with five tiles in each row and connect this to 4 x 5
• Show five columns with four tiles in each column and connect this to 5 x 4
Students then should be given larger numbers to work out the area of a rectangle where it is obvious that
the formula is the best way to solve it. For example, the number of tiles needed for a 45 by 10 area is 450
because:
• 45 rows with 10 tiles in each row = 45 x 10 = 450 or alternatively
• 10 columns rows with 45 tiles in each column = 45 x 10 = 450
A very useful kind of diagnostic question to see whether students understand the area of a rectangle
formula is illustrated in Figure 13. The problem can be presented this way: A bench has been partially
covered in square tiles. There are 20 tiles along the longer side and 5 tiles down the shorter side. When
the tiling is finished how many tiles will there be on the bench?
20
Figure 13
A common error is to add – either 20 + 4 or 20 + 5. In either case the students probably do not
understand that area is ‘coverage’ even though they may sometimes be able to apply by rote the formula
‘length times breadth’, and get the correct answer
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Confusing Area and Perimeter
Large numbers of students, including many who are at secondary school, confuse the concepts of area
and perimeter. This is obvious when they claim that if an area is fixed then the perimeter is also. So a very
significant set of tasks should involve creating sets of different shapes for a fixed area, and also creating
shapes with a fixed perimeter but different area. Figure 14 shows some possible shapes with fixed area (A)
of 12 tiles with perimeter (P) varying.
A = 12
P = 14
A = 12
P = 16
A = 12
P = 24
Figure 14
Another useful set of tasks is to reverse the problems above by fixing the perimeter and finding out that
the area now changes with various shapes.
100 metres
Standard Units for Area
So far it has been important to use tiles in establishing the meaning of area in
100 metres
non-standard units. When we accept that the only tiling shape that is going to be
used is a square the time comes to use standard lengths for the dimensions of
the squares. In the metric system these will typically lead to the basic tessellating
units being square centimetres, square metres or squares kilometres.
10 000 square
metres
There is one very strange area unit, the hectare, that is the area of a 100
metre by 100 metre square. Students need this image of what a hectare looks
like, and further know that a hectare is the same 10 000 square metres
(Figure 15). This is about the area of one and half football pitches.
Figure 15
A curiosity is that the prefix hecto means a hundred so a hectare is literally
one hundred ares. And an are is the area of a ten metre by ten metre square.
International recognition for the are as a unit of area was withdrawn in the 1960s.
A somewhat pedantic point here is to distinguish between, say, ten square metres and ten metres squared;
ten square metres is an area covered by ten tiles each of which is 1 metre by 1 metre, whereas ten metres
squared means the area of a square with each side ten metres, so its area is a hundred square metres. To
avoid confusion it is better for students to write initially, say, 10 square metres rather than 10 m2.
Volume
While length deals with one dimension where the second and third dimension are essentially ignored,
and area deals with two dimensions where the third dimension is discounted, finally we arrive at having to
use all three dimensions in volume problems. We have seen with area, when students compare a pair of
shapes, they may well be distracted by one dimension being the longest into thinking that that must have
the larger area when this is not true. It is even more problematic for volume. The famous Piaget
experiment about pouring water from a fat glass in to a long thin one indicates the difficulty students may
have is realising that the tall thin glass does not contain more water than the original. Appearances around
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volume are very deceptive; Piaget suggests understanding volume will come later than understanding length
and area.
Volume should be taught by building three-dimensional shapes out of cuboid (box shaped) blocks. Unlike
area, where there are many suitable tiling shapes, students engaging in filling space with materials will be
essentially restricted to cuboids. It is highly desirable that the cuboids can be joined together to assist in the
building. Here are some examples using commercial names – there are many more:
• Unifix
•
Multilink
•
Lego
Teachers should be aware that the shapes that students build initially to avoid the temptation to teach
the formula ‘length times breadth times height’; it is better to build shapes with uneven heights because this
will force students to find the volume by counting rather than using rote formulas. Significantly students
should engage in building and imaging 3D shapes from pictures then find the volume.
A Unifying Volume Formula
There is just one formula to find the volume of a prism:
• Volume equals the area of the either end times the height
This can be expressed concisely with algebra:
€
• V = Ah
This formula is powerful because it applies to an infinite numbers of shapes; this is the
motivation to give a general definition of prism that is initially outside students’
experiences. Students typically will think a prism is a glass device that bends light through
it leading to rainbow colours at the edges. However, a more general mathematical
definition is needed so we can use the unifying volume formula:
• Any 3D shape is called a prism if slices parallel to the ends produce cross sections that
Figure 16
all look the same
So, by this definition, a cylinder is a prism (Figure 16) because all parallel slices look like the ends. Of course
this definition of a prism is of no use unless it offers the learner some advantages, which it does.
Consider a student who does not understand the prism-volume formula but must learn a formula for
the volume of a cylinder for the exams. At best the student will end trying to remember this formula:
• V = πr 2h
Yet this formula is easy to remember and reconstruct if forgotten provided the student uses V = Ah .
Then:
•
A = πr 2
€
So for cylinders, which are a kind of prism, this is true:
!
•
V = Ah = πr 2h
The teaching point here for teachers is that understanding underlying general principles will makes formulae
€
easy to bring to mind without having to do many sets problems of by rote.
Students need to understand clearly what a prism is because there a powerful simple formula for the
€
volume of any prism. Initially this formula can be developed using non-regular faces that are built into three
dimensions. For example, Figure 17a shows the first layer of a prism that student can build that contains
eight volume counting units, while Figure 17b shows two layers needing two time eight counting units. This
generalises to the formula that the number of counting units is the number in one layer times the number
of layers.
Figure 17a
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This formula can be applied volume of a shoebox
(Figure 18). This is best found as a special case of the
volume of a prism formula rather that using the often
this meaningless formula:
• V = lbw
Instead using the formula V = Ah on the largest face we
have:
€
• 12 centimetres times 4 centimetres = 48 square centimetres,
which is the€area of the face.
So the volume is:
Length = 12 centimetres
Breadth = 2 centimetres
Height = 4 centimetres
Figure 18
• 48 square centimetres times 2 centimetres = 96 cubic centimetres
It is important that students realise that any pair of sides could be regarded as the face with the third
dimension regarded as the height. This way of working out the volume of a cuboid is far superior to the
rote formula volume equals length times breadth times height.
Volume and Capacity
Students can easily confuse the ideas of volume and capacity. It is worthwhile to teach that volume
means actual space occupied; this is usually applied to solids and liquids. However capacity refers to the
maximum amount that something can contain. For example the capacity of a carton of milk might be one
litre but, because it is half empty, the volume of milk is 500 millilitres.
Standard Measures
Students need to carry with them a set of standard measures that they can use for comparison in real
situations. For example, when buying about three kilograms of potatoes from the supermarket everyone
needs to have sense of what a kilogram in weight feels like. We shall return to standard measures in the
metric system section that follows soon.
Mass and Weight
In their profession physicists and engineers need to distinguish mass and weight. For them mass is,
roughly speaking, the amount of matter; so a person with mass 100 kilograms would have this mass on
Earth and the Moon even though the gravitational forces are very different. But the person’s weight, due to
lower gravity on the Moon, would be roughly one tenth of his or her weight on Earth. However this
distinction is not important in mathematics: no one is going to say the potatoes ‘massed’ three kilograms,
nor would they say the weight of a one kilogram mass on Earth is nearly 30 Newtons though both things
are technically correct.
Learning About Time
With all notions of space and time students learn what a concept means through experiences not by the
teacher ‘teaching’ them; for example we have seen already that area is understood through tiling surfaces
not by learning area formulas. However time is somewhat different from length, area, and volume that we
have discussed to this point because students are already immersed in the world where time passes.
It is important for young children to be actively involved in the geometry of movement, which has time
imbedded in it, before attempting to measure time.
Later teachers should be aware that students may learn to measure time before they are cognitively
ready. For example measuring time by reading digital and analogue clocks for such students inevitably
becomes an exercise in rule-following with little idea why they should attempt such tasks.
There are many interesting time problems associated with the Euclidean geometry of the Earth but
stage seven multiplicative thinking is a prerequisite for this. For example, able students find it curious that a
traveller can leave Rarotonga in the Cook Islands on Tuesday at 11:00 pm local time, fly four hours to
Auckland and, after crossing the International Date Line, arrive in Auckland on Thursday early in the
morning thereby missing out on a Wednesday entirely!
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A small technical point about writing times needs to be made. It is very common for a decimal point to
separate hours and minutes instead of a colon. So 1.30 pm is often wrongly regarded as thirty minutes past
1 pm, whereas the correct notation here is 1: 30 pm. This adds one more place where the ‘point’ is used
where it does not signal the presence of a decimal number. There seems little hope of correcting the
mixed messages that students are receiving from everywhere about the use of the point and colon for time
but at least teachers can require them to use the correct notation in class.
The Metric System and Système International
Originally the metric system of measures was designed to replace a hotch-potch of varying ways of
measuring weights and lengths that varied from country to country. Beginning in France in 1799 over time
the system was adopted virtually everywhere in the world, with the notable exception of the United States.
Originally a systematic series of prefixes was introduced so that ten units was equal to one new unit.
For example, here are the original units based on the metre – it could just as well be litres, or kilograms:
• 1 kilometre = 10 hectometres;
• 1 hectometre = 10 decametres.
• 1 decametre = 10 metres
• 1 metre = 10 decimetres
• 1 decimetre = 10 centimetres
• 1 centimetre = 10 millimetres
Outside these common prefixes the system a ‘times a thousand’ was used:
• 1 megametre = 1 000 kilometres
• 1 millimetres = 1 000 micrometres
The meanings of the most common prefixes is summarised in Figure 24.
mega:
kilo:
hecto:
deca or deka:
deci:
centi:
milli:
micro:
million
thousand
hundred
ten
tenth
hundredth
thousandth
millionth
Figure 24
Over time somewhat inconsistent metric systems of measurement used in science caused confusion, so
in 1948 the Ninth General Conference on Weights and Measures (CGPM) asked the International
Committee for Weights and Measures to conduct an international study of measurement system to cope
with the needs of scientific, technical, and educational communities around the world. Based on the findings
of this study, in 1954 the Tenth CGPM decided that an international system should be derived from six
base units for all scientific measurements. In 1960, the Eleventh CGPM named the system the International
System of Units; this was abbreviated to SI from the French name: Le Système International d'Unités. A
seventh basic unit, the mole, was added in 1971 by the 14th CGPM. Figure 25 shows the current seven
units used worldwide in science. (Curiously only one of the seven basic units, the kilogram, has a prefix
because a gram is not a suitable basic unit.)
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Length:
Mass:
Time:
Electric current:
Temperature:
Intensity of light
Amount of a substance:
metre
kilogram
second
ampere
kelvin
candela
mole
Figure 25
The common unit used for everyday rather than scientific temperature is degrees Celsius; typically for
the SI it has a capital letter as it is named after a real person. Absolute zero, which is 0 degrees Kelvin is
the same as −273° Celsius. So 273 degrees Kelvin and 0 degrees Celsius are the same. (Google ‘Celsius’ for
somewhat more exact temperatures details - these are not needed for most students.) Originally
centigrade, which later became Celsius was based on the freezing and boiling points of water:
• Water freezes at 0° Celsius,
• It boils at 100° Celsius.
Because boiling points are affected by air pressure so the boiling and freezing points of water have to be
measured at one standard atmosphere. (Lower air pressure explains why making a good cup of tea at the
top of a high mountain is hard – the water boils at too low a temperature.)
Density
It would be normal to expect the subject of density to be raised in a science class rather than in
mathematics, however all teachers need to understand that students need to be at least stage 7
multiplicative thinkers to cope with the concept. And it is difficult.
Crucially teachers need to realise that they cannot talk students into understanding what density means.
Only experiences can lead to students constructing a useful meaning that is beyond merely repeating a rule
for its calculation. As an example, experiments involving floatation may be very helpful. Consider a variety
solid blocks of the materials, not hollowed out to prevent situations like steel boats floating. Students can
investigate why these things are true:
• A light iron bar sinks in water but a much heavier wooden block floats
This idea should be investigated with a variety of objects. The inference drawn here is that weight does not
predict whether an object floats or sinks in water. Further experiments will show this important fact:
• If any block made from a material sinks then all blocks made from that material sink
• If any block made from a material floats then all blocks made from that material float
So neither volume nor weight alone can be used to predict whether an object floats or sinks. We need to
construct a measure that is constant for all objects made from the same material that involves weight and
volume that can helps us decide whether they float or not. Consider a block of wood that weighs 80 grams
and has a volume 100 cubic centimetres:
• By the division 80 ÷ 100, one cubic centimetre would weigh 0· 8 grams
Suppose the volume of a similar wooden block weighed 240 grams:
• Its volume would be 3 x 100 = 300 cubic centimetres
• By the division 240 ÷ 300, one cubic centimetre again would weigh 0· 8 grams
In fact weight divided by volume is the same for all wooden blocks made from the same kind of wood. This
is a useful measure of density. So this is the definition we use:
•
Density = weight ÷ volume
Teachers should expect most year elevens to struggle to understand density because they usually lack
the multiplicative reasoning necessary. Such students often wrongly reason additively. For example, suppose
3 eggs need 2 kilogram of flour in a recipe and we want to find the number of eggs needed when using 6
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kilograms of flour, additive thinker will say 3 + 4 = 7 eggs are needed because 2 kilograms of flour plus 4
kilograms of flour is 6 kilograms of flour. But the correct answer needs multiplication: 2 x 2 eggs = 4 eggs.
Systeme Intèrnationale
From a student point of view the important different feature of SI compared from the original metric
system is all units are a factor of 1000 apart. So, for example, 1000 millilitres equals 1 litre or, equivalently,
1 thousandth of a litre is 1 millilitre. This means that SI commonly uses the prefixes micro, milli, kilo, and
mega, so the original metric system units centi, deci, and deca have been dropped. Outside science the only
major exception to the use of these standard prefixes is centi which is only used in centimetres; the
suggested reason for this is that the clothing trade did not find millimetres very suitable for making clothes
preferring centimetres. Another very minor exception is French wine bottles that contain 75 centilitres of
wine but this is not SI; most of the world writes this as 750 mL.
Since something like half the world use a decimal comma rather than a decimal point it was agreed that
SI would avoid this confusion by omitting the comma from numbers completely, and instead we should use
half spaces every three places in front of the decimal point. (Google ‘decimal comma wiki’ for details.) So,
for example, 456 897 098· 56 is SI compliant, whereas the same number with commas 456,897,098· 56 is
not.
Notice that SI allows both version of four digit whole numbers one with a half space and the other not.
So 7 903 and 7903 are both acceptable.
The Mathematics of Conversions
Because the general SI pattern linking units together is that one unit is a thousand times bigger, or
smaller than, the next unit the flowchart in Figure 26 shows how conversions between pairs of units may
readily be made. For example, to convert metres into millimetres students need to fill in the flowchart
using the following steps:
÷ 1000
metres
millimetres
x 1000
Figure 26
•
Fill in metres and millimetres in the ovals
•
Realise the number of millimetres in a measurement in metres is found by multiplying by 1000
•
Look at the suitable arrow directions and place ‘x 1000’ in the bottom box
•
Add ‘÷ 1000’ in the top box
The flowchart now shows us how to solve these conversions:
• 45· 78 metres to millimetres: 1000 x 45· 78 = 45 780 millimetres
• 7903 millimetres to metres: 7903 ÷ 1000 = 7· 903 metres
This flowchart method of calculation is always very useful provided the following conversions are
known:
• 1000 micrometres = 1 millimetre
• 1000 micrograms = 1 milligram
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• 1000 milligrams = 1 gram
• 1000 grams = 1 kilogram
• 1000 kilograms = 1 tonne
• 1000 millilitres = 1 litre
The one exception to this ‘times or divide by 1000’ pattern occurs with centimetres. Conversions
involving centimetres need these facts:
• 10 millimetres = 1 centimetre
• 100 centimetres = 1 metre
Conversions between millimetres and centimetres will involve multiplying or dividing by 10, and converting
between metres and centimetres will involve multiplying or dividing by 100.
Important Facts about Metric System
Probably the most important facts for students to know about the metric system is the relationship
between volumes, lengths, and weights of water. The metric system was deliberately designed to have the
following features related to water:
• The volume of one gram of water is one cubic centimetre
• One millilitre of fluid is the same as one cubic centimetre
• The volume of one gram of water is one millilitre as well as one cubic centimetre
These facts have an important consequence for creating a standard reference set for a kilogram:
• 1000 millilitres, or one litre of water weighs one kilogram
Another reference set for one kilogram could be the weight of a litre of milk. Notice also a standard
pack of butter weighs 500 grams so a kilogram could also be imagined as this:
• One kilogram is the weight of two packs of butter
A tonne - the metric ton - is a thousand kilograms. This is the weight of one cubic metre of water,
which is also 2205 pounds. Up to 1985 in the United Kingdom, the ton was 2240 pounds, so a British ton
and a metric tonne are very nearly the same. However in the United States and Canada the ton is still used,
and a ton is 2,000 pounds or 907 kg. This is all somewhat confusing in the few countries that have not
swapped over to using SI.
A standard school ruler is 30 centimetres long. This can be used by students as a reference to estimate
short lengths. A metre can be used for longer lengths, and a reference for a metre can be imagined in a
number of ways:
•
A large adult pace is about a metre
•
A metre is a bit more than the length of three standard rulers
•
Two paces of a seven year old is about a metre
Imagining a kilometre is very challenging because it is usually cannot be seen. However this may help:
•
A kilometre is about the length of ten football fields
A common feature of the metric system in New Zealand is to hear tradespeople and others say ‘mils’
for ‘millimetres’ where sstyrictly speaking ‘mils’ means millilitres. In passing it is significant to note the
abbreviation of millilitre as ml can be confusing as the ‘l’ looks like a ‘one’ in many fonts. One way around
this was to use a capital L for litres. So a millilitre can be abbreviated to mL rather than ml. SI allows both.
Another way is use a ‘curly’ ℓ, so millilitres becomes mℓ. While this is easy for students to write finding
ℓ in a word processor students need to find the ‘Unicode script el’, a task that is tedious. Note, however,
SI does not permit the use of the curly ℓ, so it is now seldom used.
SI and the Trades
One important feature of the Systeme Intèrnational of measurements normally allows builders,
plumbers, and in fact most tradespeople to avoid the use of the decimal point. This means length etcetera,
are measured in whole numbers in two ways. For example, a builder would not use 1· 655 metres but
rather chose to see it as one of the following two measurements:
• 1 metre plus 655 millimetres or
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• 1655 millimetres
Centimetres are not used by carpenters, builders, and cabinetmakers as measuring to the nearest
centimetre rather than nearest millimetre can lead to serious errors in the construction process. The
clothing trade is the main user of centimetres. Here millimetres are more accurate than is warranted.
References
Lehrer, R. & Chazan, D. (1998). Designing learning environments for developing understanding of geometry
and space. New Jersey: Lawrence Erlbaum Associates.
Craine, T. & Rubenstein, R. (2009). Understanding geometry for a changing world: seventy-first yearbook.
United States: National Council of Teachers of Mathematics.
National Research Council (1989). Everyone Counts. Washington D.C.: National Academy Press.
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Appendix A
More Area Formulas
A very powerful way to develop a set of area formulas related to quadrilaterals and triangles is to relate
them to just one result – the area of a rectangle. The key is to understand how a rectangle can be
transformed to a number of shapes of related area that lead to their area formulae. The first example
shown is a parallelogram. This is best done on geoboards where the transformations become obvious.
Figure 28 shows the steps:
h
h
h
b
b
Figure 28a
b
Figure 28c
Figure 28b
• The rectangle becomes a parallelogram – Figure 28a becomes Figure 28b
• This parallelogram in turn can be transformed to Figure 28c
It is obvious that the area of the rectangle in Figure 28a is the same as the area of the parallelogram in
Figure 28c. Therefore the area of the parallelogram is b times h.
Similarly Figure 29 shows why the two triangles equal a parallelogram, so the area of the triangle is half
the area of the parallelogram i.e. the formula for the area of a triangle is half b times h.
h
h
b
b
Figure 29
Finally the area of a trapezium is equal to half the area of the parallelogram created by making a copy
and rotating it around the mid-point of the indicated side. This is shown in Figure 30.
b
a+b
+ copy
h
then rotate
180º
a
h
a+b
Figure 30
So the area of a trapezium is the sum of the sides times the height divided by 2.
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Appendix B
Radians - For Secondary Teachers Only
A perpetually puzzling idea for all year 12 students is why mathematicians delight in using 2π radians for
a revolution instead of 360º. The main point to understand is that the choice of 360 units for a revolution is
arbitrary – any number will do. But once we have agreed to measure angles in degrees it is essential in
communicating with others that we stick to this arbitrary choice. It is a complete surprise to senior
secondary students that from literally nowhere a very strange new way of measuring angle, radians,
emerges. And very peculiarly one revolution is 2π radians!
To overcome this real problem a central learning principle needs stating:
• Definitions before experiences are meaningless
So what experiences lead to radians?
A helpful step in getting students to understand why we sometimes use radians it is to useful to use
another different choice for measuring angle before introducing calculus, namely that 400 gradians equals
one revolution. The reason that this was invented around the time of the French Revolution was to create
the ‘metric’ right angle of 100 gradians. This is surely more sensible than a 90 degree right angle but history
has not been kind to gradians. Nevertheless the idea survives; all scientific calculators the equivalence of
degrees and gradians can be seen when students find sin 90 in degrees equals sin 100 in gradians. Likewise
tan 45 in degrees and tan 50 in gradians are the same.
Having established there is another sensible, though seldom used, arbitrary measure for angles there is a
clearer path to understand radians. So how could we introduce radians? The first place radians are actually
required in school mathematics as a measure of angle is year 13 where the calculus of trigonometrical
functions is first introduced. Up to that point degrees are entirely adequate; quite simply there is no good
case for teaching radians in year 12. Arguments about the usefulness the area of a sector and length of an
arc of circles formulae in radians namely A = 12 r 2θ and s = rθ are flawed. In fact the length of an arc is best
seen as the fraction of the circumference of the whole circle no matter what unit of angle measurement is
used. Similarly the area of a sector is best seen as the fraction of the area of the whole circle. In both these
cases what we decide to measure one revolution does not alter the formulae: suppose there are k units in
€
one revolution of a circle, where k is completely arbitrary, then the formulae for segment area and arc
length become:
A=
•
θ
θ
× πr 2 and s = × 2πr .
k
k
C
Before exploits these formulae to understand radians we
need a little calculus. Suppose f(x) = sin x then after some
€
sinθ
.
θ →0 θ
fiddling we find that f ʹ′(x ) = cos x × lim
D
In order to differentiate trig functions, all of which can be
derived from this derivative we need to find the value of
sinθ
€
for any arbitrary choice of k.
θ →0 θ
lim
r
In Figure 31 AD < Arc DB < BC
€
O
€
θ
A
Figure 31
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B
And AD = r sinθ , Arc DB =
θ
× 2πr, and BC = r tanθ
k
θ
× 2πr < r tanθ
k
Divide through by r sinθ :
2π
θ
1
1<
×
<
k sinθ cos θ
So r sinθ <
Doing a little tidying and taking limits we have:
sinθ 2π
=
θ →0 θ
k
lim
€
€
So this vital limit depends on the arbitrary choice of k for one revolution.
Now suppose we choose to let k = 360, which is natural as it the only sensible measure that students
have so far seen for measuring angles, then the derivative of sin x is:
2π
cos x = 0 ⋅ 017453292 cos x .
360
And even worse the second derivative of sin x is 0· 003046174198 sin x.
This is just not sensible. So what would be a better choice of k? Letting k = 2π then:
•
sinθ
=1
θ →0 θ
lim
€
Nice and simple. In fact simply the best! Now everything is harmonious; the derivative of sin x is cos x, and
the second derivative is sin x. This is why 2π radians for a revolution is the best arbitrary choice to
€ measure a one revolution angle when there is calculus involved.
As a postscript the invention of radians is credited to the English mathematician Roger Coates (1682 1716). And he needed it for calculus. Prior to this there was no need.
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Appendix C
1· 84 metres
A Secondary and Tertiary Education Issue: Sensible Rounding
A difficult area in secondary and tertiary education is how to round numbers that have been calculated
from measurements. The normal practice in secondary school is to not worry about it much as students do
not have to build anything as a result of their calculation.
The source of potentially serious errors in using calculated values is that the initial measurements are
inevitably rounded to the nearest value on the measuring scale. For
2· 56 metres
example, suppose the dimensions of a plywood sheet are 2· 56 metres
and 1· 84 metres (Figure 32). This implies that the measurements have
been made to the nearest centimetre in each case. Now, if the area
were required the calculation is 2· 56 x 1· 54, which gives 3· 9424
square metres that has apparently five significant figures. Anyone
working in a job where such calculations are made will know this answer
is much too accurate – the original lengths were both measured to three
Figure 32
significant figures so the answer at best is accurate to three significant
figures. We can prove this by considering the minimum lengths. Because
the measurements were made to the nearest centimetres the sides might be as short as these:
• 2· 555 metres and 1· 535 metres
And 2· 555 x 1· 535 = 3· 921925
Rounding this to two decimal places gives 3· 92. Similarly the measurements might be as large as:
• 2· 565 metres and 1· 545 metres
And 2· 565 x 1· 545 = 3· 962925
Rounding this to three significant figures gives 3· 96. So a sensible answer would be 3· 94 ± 0· 02 square
metres not the 3· 9424 square metres calculated earlier. The decision workers must now make is whether
in the context of the real problem they face the degree of uncertainty - and uncertainty is inevitable if
measured quantities are used in calculations - is tolerable.
€ to suppress unnecessary
Calculators are programmed to never accept unnecessary zeroes as input, and
zeroes in answers. So, for example, 00045 will be accepted – a calculator will always show 45. It will,
however, permit 45· 0000 to be entered because non-zero digits may be following. Notice, however, that if
45·0000 + 23 is entered immediately the plus sign is entered the zeroes in 45·0000 on screen are
suppressed to show 45.
It is therefore reasonable for students to think that 1· 60 always means the same thing as 1· 6, which it
does unless the numbers refer to measurements; 1· 60 and 1· 6 do mean different things. For example, how
accurate are measurements 1· 60 metre and 1· 6 metre?
The measurement 1· 60 metres implies that it has been measured to the nearest hundredth of a metre,
so the exact length is anywhere between 1· 595 metres and 1· 605 metres. But the measurement 1· 6
metres implies that it has been measured to the nearest tenth of a metre so the exact length is anywhere
between 1· 55 metres and 1· 65 metres. So the zeroes at the right hand end of a number representing a
measurement are significant. Students find the notion that, say, 1· 6 kg and 1· 600 kg do not represent the
same weight very challenging.
So any calculated measurements should be treated with the greatest suspicion; in fact the practical
situations in which trades-people find themselves such as cutting timber, will determine how accurate
calculated measurements need to be.
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