MEASUREMENT and GEOMETRY
Concept Development
The world is full of two-dimensional shapes and three-dimensional objects. People began to
measure the length - one dimension - of these shapes and objects. A line is one-dimensional because
it takes up space along one direction
Length
Names for specific lengths grew out of necessity and began using body parts, for example foot, hand
and el (elbow). Multiple units were used to measure longer lengths and yards, furlongs, miles etc
were created. Units were divided to measure shorter lengths and inches etc were created
Around 300 years ago, the French decided to create a new system for measuring length, which they
called the metric system. They based the metric system on place value - multiplying and dividing by
10 to make longer or shorter units
They began with the metre. A metre was designed to be one-ten-millionth of a quarter of a meridian
- A meridian is a circular line around the Earth passing through both North and South Poles - how
amazing is that? The word metre means to measure
To measure shorter lengths, the metre was divided by 10 to make a decimetre (used in Europe). To
measure even shorter lengths, the decimetre was divided by 10 to make a centimetre. To measure
even shorter lengths, the centimetre was divided by 10 to make a millimetre. The prefixes 'deci',
'centi' and 'milli' are used for all units of measurement in the metric system. Deci means tenth - a
decimetre is a tenth of a metre (10 of them are as long as a metre so one is a tenth as long as a
metre). Centi means hundredth - a centimetre is a hundredth of a metre (100 of them are as long
as a metre so one is a hundredth as long as a metre). Milli means thousandth - a millimetre is a
thousandth of a metre (1000 of them are as long as a metre so one is a thousandth as long as a
metre)
To measure longer lengths, the metre was multiplied by 10 to make a decametre (used extensively
in Europe). To measure even longer lengths, the decametre was multiplied by 10 to make a
hectometre. To measure even longer lengths, the hectometre was multiplied by 10 to make a
kilometre. The prefixes 'deca', 'hecto' and 'kilo' are used for all units of measurement in the metric
system. Deca means ten - decametre literally means ten metres. Hecto means 100 - hectometre
literally means 100 metres. Kilo means 1000 - kilometre literally means 1000 metres
This means that we can name really long lengths or really short lengths using words rather than
fractions to name the lengths!
Because the metric system is based on a decimal
system, learning to convert between units of
measurement makes sense when relating it to
multiplicative place value
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Area
Area is amount of space a two-dimensional shape takes up. A shape is two-dimensional because it
takes up space along 2 directions - that means it has length in two directions. Name we give the
lengths include length, width, depth, height, breadth ...
When children first begin to investigate area, they investigate tessellation and shapes. By the end
of Year 2, they find that the best shape to measure area is the square because it takes up the same
space no matter which way it is oriented (a word which has the same meaning as orientated!)
In Year 3, they begin to investigate measuring the dimensions of their square and find that a square
with dimensions of 1cm is a square centimetre. There is a huge difference between a square
centimetre and a centimetre square(d) as this example demonstrates:
This shape is 2 square centimetres and has an area of 2 square centimetres
This 2 centimetre square(d) describes a square with sides 2cm in length
and has an area of 4 square centimetres!
A square centimetre is a unit of measurement for measuring area
The units of measurement for measuring area grew out of the units of measurement used to
measure length. Squares are created from metres (square metres), centimetres (square
centimetres), hectometres (hectares), kilometres (square kilometres)
As children develop their understanding of the features and properties of shapes, they use these
understandings to measure and calculate area. Children investigate the relationship between the
area of rectangles and triangles. This then allows them to investigate dividing straight-sided shapes
into rectangles and triangles to find the area - no need to rote learn formulas!
Children investigate the relationship between the circumference (perimeter) of a circle and the
radius of the circle (pi) allows children to find the area of circles
Volume and Capacity
Volume is amount of space something three-dimensional takes up. An object is three-dimensional
because it takes up space along 3 directions - that means it has length in three directions. Name we
give the lengths include length, width, depth, height, breadth ...
Volume includes capacity. Capacity is the total amount of space inside a container - think of a
container of water 'filled to capacity'. If the container of water is not filled to capacity, the volume
of water (amount of space the water takes up) will be less than the capacity of the container
Volume and capacity can be measured in cubic units or in liquid units
When children first begin to investigate volume and capacity, they investigate tessellation and
objects. By the end of Year 2, they find that the best shaped object to measure volume and capacity
is the cube because it takes up the same space no matter which way it is oriented (a word which has
the same meaning as orientated!)
In Year 3, they begin to investigate measuring the dimensions of their cube and find that a cube with
dimensions of 1cm is a cubic centimetre. There is a huge difference between a cubic centimetre and
a centimetre cube(d) as this example demonstrates:
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This object is 2 cubic centimetres and has a volume of 2 cubic centimetres
This 2 centimetre cube(d) describes a cube with sides 2cm in length and
has a volume of 8 cubic centimetres!
A cubic centimetre is a unit of measurement for measuring volume
and capacity
The units of measurement for measuring volume and capacity grew out of the units of measurement
used to measure length and area. Cubes are created from metres (cubic metres), centimetres (cubic
centimetres), hectometres (cubic hectometres), kilometres (cubic kilometres)
As children develop their understanding of the features and properties of objects, they use these
understandings to measure and calculate volume and capacity.
Objects and containers with straight edges and 90⁰ vertices can be measured using cubic units.
Objects and containers without straight edges and 90⁰ vertices need to be measured using a liquid
measure
The creators of the metric system created a liquid unit of measure that had the same volume as a
cubic centimetre - and called it a millilitre. Liquid units of measurement for measuring volume and
capacity are the millilitre (cubic centimetre), litre (1000 cubic centimetres - visualise a MAB block =
1/1000 cubic metre), kilolitre (1 000 000 cubic centimetres = 1 cubic metre)
Using this fabulous relationship, we can work out the volume and capacity of any three-dimensional
object in both cubic and liquid units!
When the creators of the metric system multiplied a millilitre by 1000 to get a litre - they used the
mass of a litre of water to create a unit of measurement of mass
Mass
Mass is a measure of the amount of matter. We often use the terms mass and weight
interchangeably. They are different: The amount of matter does not change if gravity changes. For
example the amount of matter in a person is the same on Earth or on the Moon. Weight changes if
gravity changes. For example a person's weight is greater on Earth than on the Moon, because the
Earth's gravity is greater
When children first begin to investigate mass, they investigate through hefting and equal-arm
balances
The units of measurement for measuring mass grew out of the units of measurement used to
measure length, area and volume and capacity.
The creators of the metric system called the mass of a litre of water, a kilogram. This means that the
mass of a millilitre of water is 1 gram. (Technically water at 4⁰C - the temperature water is at its
densest) Kilograms were then divided by 10 to create decigrams, the decigram divided by 10 to
create centigrams. Kilograms were multiplied by 1000 to create tonnes
So 1 litre of water has a mass of 1 kilogram - no wonder 'heavy' rain can cause so much damage!'
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PLACE VALUE
Before Place Value number systems were created, people used Symbol Value number systems. This
means a symbol retained its value regardless of its place in the number. In Roman Numerals the
value of the symbol X is always 10, for example XI, IX, CCCDXVII - the value of the X is always 10.
Symbol Value number systems are additive - they use addition and subtraction to calculate the value
of the number. For example, CCCDXVII is 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1 = 367
SYMBOL NUMBER SYSTEMS ARE ADDITIVE!
In Place Value number systems, the value of a symbol changes dependent on its place in the number.
For example in the current day place value number system (Hindu Arabic), the symbol (numeral) 3
has a value of 3 ones in 23 and 3 tens in 32. In Place Value number systems, we also add the values
of the symbols to calculate the value of the number. For example, 23 is 20 + 3 and 32 is 30 + 2
PLACE VALUE NUMBER SYSTEMS ARE BOTH ADDITIVE AND MULTIPLICATIVE!
PLACE VALUE NUMBER SYSTEMS - A BRIEF HISTORY
The Place Value number system the world uses today is one of many created and used by different
civilisations. Some of the different Place Value number systems in use include:
Origin
Base
Hindu Arabic
Chinese (Abacus)
10
Sumerian
Babylonian
Mesopotamian
60
The base refers to the number the civilisation decided to
multiply by to calculate the value of subsequent places, for
example in the Mayan Place Value number system, the
first place had a value of 1, and the next place had a value
of 20 (1 x 20). Their symbols included
Numbers were written from bottom to top. Below you can
see how the number 32 was written:
Mayan
Mesoamerican
North American
Danish
Irish
Gaulish
20
North America
Central America
4
Yuki Tribe(California)
8
Papua New Guinea
27
20s
1s

(1 x 20 = 20)
(2 x 1 + 2 x 5 = 20)
Place Value number systems were created by civilisations
who were involved in scientific study, particularly
Astronomy. Famously, the Mayans created calendars
using their number systems which traced the path of
planets through the skies. Interestingly, some have
interpreted the end of the Mayan calendar to be an
ominous prediction! Our own calendar ends every year on December 31 - and then a new calendar begins the next
day!
The Hindu Arabic Place Value number system is the one that has survived and is used internationally
today. This is not because it was the best one (Base 12 would make multiplication easier as 12 has
more factors than 10!), but because of the Hindu's proximity to other civilisations who were also
involved in scientific study. The Hindu's Place Value number system, created between 300 and
500CE, soon spread west through Muslim civilisations, where the numerals were changed - thus the
number system became known as the Hindu Arabic number system.
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The number system allowed the Hindu and Muslim civilisations to make huge advances in
Mathematics and Scientific study during the European 'dark ages' - so called partly due to the lack
of progress of European civilisations (some would say regression) during the same period.
Following time spent in Muslim Algiers, Italian Mathematician Leonardo Fibonacci recognised the
superiority of the Hindu Arabic Place Value number system and introduced it to Europe through the
publication in 1202CE of his Liber Abaci (Book of Calculation) - and the dark ages ended and the
renaissance began!
From then it was widely used in European mathematics, and with the invention of the printing press
in Europe around 1440CE, it replaced Roman Numerals in general use. As the Chinese were also
using a Base 10 number system, transition to the Hindu Arabic Place Value number system was
natural
Initially, the Hindu Arabic Place Value number system was used solely for whole numbers. The value
of the lowest place was ones. In this way, all other values could be calculated from the ones place
(see Place Value is Multiplicative below). Fractions were used using different notations - including
numerator and denominator
Extending the Place Value system to include fractions, began with Jewish and Persian
Mathematicians who recognised that using different systems for recoding whole numbers and
fractions made calculation difficult. When the Place Value number system was extended to include
fractions, it soon became less obvious which place was the ones place and so some kind of notation
was needed to identify the ones place (and to separate the whole numbers from the fractions). For
example 13 as 1 ten and 3 ones, and 13 as 1 one and 3 tenths, look the same
Initially different people used different notations to identify the ones place. In his booklet De
Thiende ('the art of tenths'), first published in Dutch in 1585, Flemish Mathematician Simon Stevin
used numbered circles, for example 65∙728, he recorded as 65728. Others used a bar over
the ones place to identify it. European countries finally settled on a comma(,) however English
speaking countries used the comma to separate groups of numbers (for example 1765293 was
recorded 1,765,293) so Britain used a mid dot (∙) and the United States used period (dot) on the
base line. Australia adopted the British mid dot. All notations are 'decimal' points because of the
decimal nature of our numbers system (see Place Value is Multiplicative below). Modern
computers generally use a dot on the base line
TEACHING PLACE VALUE
Place Value is Additive, which means that the values of the digits are added together to
determine the value of the number. Simply naming the place of a digit is not demonstration of an
understanding of additive place value. For example, identifying that the in 23 the 2 is in the tens
place and the 3 is in the ones place does not demonstrate understanding that we have 2 tens and 3
ones - and that we are adding the 2 tens to the 3 ones to calculate the value of the number
Additive Place Value begins in Kindergarten as students develop their understanding that teen
means 10 and… It continues into Year 1 and Year 2 as students develop their understanding that 10
can be seen and described in 2 ways - as 1 ten and 10 ones. From Year 1, students develop their
capacity to understand and describe place value flexibly as they partition two- and three-digit whole
numbers into standard and non-standard place value parts. For example, 54 can be seen and
described as 5 tens and 4 ones, 4 tens and 14 ones, 3 tens and 24 ones, 2 tens and 34 ones, 1 ten
and 44 ones, and 54 ones! 327 can be seen and described as 3 hundreds and 2 tens and 7 ones, 32
tens and 7 ones, 327 ones, 2 hundreds and 12 tens and 7 ones, 2 hundreds and 2 tens and 107
ones...!
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Additive Place Value continues into Year 3 and beyond, as students extend their capacity to
understand and describe place value flexibly as they partition two- and three-digit numbers
including decimal fractions into standard and non-standard place value parts. For example, 5∙4 can
4
14
be seen and described as 5 ones and 4 tenths (5 ones and 10), 4 ones and 14 tenths (4 ones and 10),
24
34
3 ones and 24 tenths (3 ones and 10), 2 ones and 34 tenths (2 ones and 10), 1 one and 44 tenths (1
one and
44
54
), and 54 tenths (10)! 3∙27 can be seen and described as 3 ones and 2 tenths and 7
10
2
7
7
hundredths (3 ones and 10 and 100), 32 tenths and 7 hundredths (32 tenths and 100), 327 hundredths
327
(100), 2 ones and 2 tenths and 107 hundredths (2 ones and
Multiplicative below for more on decimal fractions)
2
10
and
107
)...! (see Place Value is
100
Understanding of additive place value is essential for students to add, subtract, multiply, divide and
create fractions intelligently using partitioning (in both standard and non-standard place value
parts), number relationships and properties
Place Value is also Multiplicative which means that the values of places are 10 times
greater as we move to the left and 10 times lower as we move to the right. The decision to increase
10 times as we move places to the left and to decrease 10 times as we move places to the right is
not a mathematical one - the other way would have worked just as well! As long as we all agree that
this is how we will assign values to numbers, we will all read numbers in the same way
This understanding that we are multiply and dividing by 10 as we move between places is vital to
understanding decimal fractions. The value of the place to the right of the ones place is calculated
1
1
by dividing 1 by 10 to get (tenths). Further dividing by 10 gives the value of the next place to
1
10
10
the right - 100 (hundredth). Symbols were needed and created to identify the ones place, (see Place
Value Number Systems - A Brief History above) resulting in a decimal point
The decimal point has no greater importance than to identify the ones place, thus separating the
whole numbers from the fractions. It is a decimal point simply because our Place Value number
system is decimal. The prefix (first letters) in decimal are 'dec' - meaning 10 (think decade - 10 years,
and decagon - a shape with 10 angles and sides). Values of places to the right of the decimal point
are calculated in exactly the same way as values of places to the left - we simply divide by 10 as we
move places to the right and multiply by 10 as we move places to the left
The decimal point cannot move - as then it would no longer identify the ones place nor separate
whole numbers from fractions! Understanding that the values of the places increase 10 times as we
move to the left, and decrease 10 times as we move places to the right develops understanding that
when multiplying and dividing by 10, the numbers move between places - regardless of whether we
are multiplying or dividing a whole number or a decimal fraction!
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