the main vine, not the loose one.
hopu tōu ringa ki te aka tāepa, engari kia mau ki te
Thoughts on Teaching and Learning of Mathematics
Lesson #5 • Revised 1/1/20
and Measurement (and Algebra)
- to know how geometry
connects to all mathematics
- to know how measure quantifies our world
- to learn to reason with size and shape
Measurement quantifies Geometry by using Number. This
statement illustrates a powerful connection of mathematics. Hidden
are the axioms of knowing what "one" is and "comparing the same
size bits" described elsewhere, the patterning and generalisation
of algebra and the vats context of geometry. Geometry is a
great place to develop algebraic skills, reasoning and proof. It
is essential geometry is part of all learning programmes in
mathematics at all levels, practically and theoretically.
Geometry is the world of Size and Shape. Size is Measurement
and Shape is Geometry. I am fascinated by shape and how
important it is in many engineering problems. The way a bullet is
shaped, a boat is curved, a rocket is sleeked, a chimney is
moulded, a wheel is crafted, the visuals of number we create all expose
shape applications. In electronics an experimental "V" cut
made on the base npn semi-conductor material
created a whole V-MOS technology that allowed easy high power
control in a transistor. The shape of lenses, antennas, wedges,
screws, fish hooks, coil springs all show how shape is very
important. Did you know a screw is a wedge wrapped in a circular
helix? Rotating the screw makes a wedge which climbs powerfully
into the material.
Length has been measured in cubits, feet, inches and metres among
many other units. Mass has been pounds and kilograms. Charge is
measures in Coulomb. Each unit has its history and is a worthy
study alone. The history of measurement is a history of the
world. The world was an interesting place when the unit
called a "peck" was used. What is a firkin? The history of
measurement is a great context for building literacy.
The capital letters in some units are in honour of a
significant researcher or person involved in that area of
knowledge. Sir Issac Newton is acknowledged with force (N).
Coulomb developed the macroscopic law that represents how charges,
separation and force inter-relate. In the middle of last century
(1950's) Richard Feymann lead his team to develop QED or the
Quantum Electo-Dynamic Theory of Electricity and came to Coulomb's
Law from the microscopic or quantum/photon path. Feymann explained
what was really happening using the fundamental particles of
matter which we still do not fully understand! Hence my comment in
Number and Algebra about a binary mathematical system underpinning
Measures use units and we have the choice of selecting how big our
"one unit" is. Axiom #0 = "One can be anything we choose one
to be". We make rulers (scales) that repeat the size of the
unit and use the number line to quantify so we can compare
sensibly. Most measures have infinitely small and infinitely big
limits. Some measures like temperature have an Absolute Zero but
no upper limit. The slowest speed you can have is zero and the
fastest is the speed of light, the limiting speed of the universe
(Einstein). On 16 OCT 2017 an astronomical event of two neutron
stars meeting up 130 Million years ago and collided transmitting
100 seconds of powerful gravitational waves that were measured on
Earth. The gravitational waves were measured to travel at the
speed of light. Google "Neutron star collision Oct 2017".
Why should gravitational waves travel at this speed? Did you feel
the Earth move that day? Did anyone?
A December 2019 observation of Betelgeuse, a large red giant star
about 10xthe mass of our sun dimmed and appears to have run out of
the elemental hydrogen fuel that keeps it huge bulk [Jupiter to
Jupiter orbit sized] pumped up. The star will now collapse and
will do so surprisingly quickly to go super-nova. Luckily it is 60
Light Years (another distance measure and a good maths question)
from Earth, but still well within the Milky Way. When Betelgeuse
goes supernova it will be visible in the daylight sky and shine
like the sun in the night sky for a few months. It would be very
cool if it happened while I was alive! New elemental molecules
would be produced and we would witness creation. WEBSITE = https://earthsky.org/space/betelgeuse-dimming-late-2019-early-2020-supernova
Negatives in measures usually involves the concept of
direction. Negative temperature is cooler, positive warmer
and increasinly negative temperature means "getting colder".
Negative velocity is in the opposite direction to positive
velocity. Positive and negative angles simply mean anticlockwise
or clockwise. [Note - Anticlockwise is positive angle, chosen by
early mathematicians]. We join direction and size to make vectors
to represent velocity such as 3km/hr North. A scalar, like mass
has no directional sense, such as 4kg and negative mass has no
sense. We do have a concept called "antimatter" that parallels
negative mass in quantum physics. [See Angels
and Demons movie]. Vectors are an amazing construct and at
one time I studied Tensor Calculus which involves vectors in many
Time seems to travel in one direction which is explained by
our universe tending to a state of more disorder (Thermodynamic
Entropy Law]. It is where we get the energy from to live. Light
and heat from the sun powers our plants and animals which are our
fuel. All life energy can be traced back to sunlight. The universe
is continually increasing its state of disorder and using up its
energy. This Law of Thermodynamics predicts a very long slowing
down and dimming of our universe long past the extinction of our
Sun and destruction of our Solar System. Do not panic! The
predicted life span of our Sun is at least another 10 billion
years and human's have had an ordered civilization for only about
50,000 years, and a technological civilisation for about 70 years.
To be a master of measurement is vital. Measurement and
it's importance is all too often overlooked in mathematics
programmes and almost trivialised. Do this at your peril!
It is important and is a place we need to make sure all students
are competent. Include a comprehensive programme of measure in all
mathematics programmes and every year develop this world a little
more. For evidence of truth to what I say just ask a carpenter,
engineer or cake maker if measurement is important. Measure
twice, cut once. Normalise this adage! It is astonishing
how much waste is generated by not measuring correctly. Ask any
one in the construction industry. It would be a very confident
person to say they have never made a measuring error.
Physics lives inside measurement. Strangely only 43 powers of
10 are required to contain the smallest length we can see to the
largest length we know about and imagine. That is the power
of number and in this case, the powers of 10. Just ask the
internet for "Powers of 10".
The base 10 system is very efficient, but not as efficient as the
We need to reason with measures as well. Lengths combine become
perimeter, area and volume and derived units like speed emerge
when blended with time. Density compares mass and volume. Ideas
like rotation becomes circular measures and new units called arcs
and solid angles appear. Comparing two different units form a rate
and when we compare two units that are the same we make a ratio.
Scale then emerges and we see "scale models" and a"scale drawings"
and in the different ways a unit is divided to make different
"scales" to measure the same thing! Language! All this has to be
explored practically and theoretically.
In all the above the dimension called time is connected,
obvious and hidden as well. Time is history and all the wonderful
ways we have connected history and time to make a year, months,
the month names, calendars and how we measure time is a complete
study as well. Make a clock! Explore some relativity with Einstein
and see that even time is a relative concept. Time must be
The Deep Understanding of Measurement
-Being enabled to quantify size of shape, reason make
sense of and connect ideas to solve problems.
Measurement works because of Axiom #0 and #1, repeated here from
One can be anything I choose one to be.
When we combine or compare anything in mathematics we do
so using the same size bits.
I make a unit using Axiom #0 and then a ruler is marked out with
units the same size using Axiom #1.
On a 30cm ruler are 30 identical line segments or intervals that
can be compared to another object. The count begins at 0 and end
at 30. This all might seem obvious but a very interesting task
for a new group of Year 7 to 10 students is to have them
"Make a ruler". That is the complete instruction. They are
allowed clarification of course but "NO TELLING!" Most select
length but a few select other dimensions and even bizzare measures
like viscosity, bounciness, springiness, colour, sound and light.
Make a ruler.
Clarifications may include - of what?, how
long, how big, to do what?, what with? Dear Liza, Dear Liza and
so on...many questions.
Answers include... "Of anything you like" and "To measure
anything you choose".
Teacher Hint - Do not TELL!
As a teacher , I am wanting to find out what students
know about measurement. Do they start from zero?; are the
gaps the same size?; does it measure length or area, or volume, or
capacity, or angle or weight, or time or whatever they wish to
measure? Is it fit for purpose?
They could measure current, bounciness, colour, agility, IQ,
roundness, stickiness, catch-ability, bend or density. No
restrictions. Creative thinking opportunities.
What do I usually find?... everyone chooses length or the usual
type of what we might call a ruler. Relevance and personal
experience inform these decisions. Someone who has never been
flyfishing would never choose a way to measure the sinking rates
of different nymphs used as bait. Experience is everything.
Some start from 0, some from 1, most use card or paper and often
the intervals are not all the same size, the divisions (if anyone
has been game enough) are often confused. Did their previous
teachers understand and teach the fundamentals of measurement?
Measurement and decimals, fractions are multiplicative so probably
not. But that is the "wicked problem" presented, so "grin and bear
it!". Work with what students know and build!
The freedom to create their own measuring device to measure any
quantity is as invisible "as a thought" [G Eliot]. Later I ask
them to make a measure to grade the 'sinkiness' of weighted fly
fishing nymphs, to measure viscosity, magnetism, voltage, angle,
bounciness, colour, pitch, loudness, bitterness, sweetness,
smoothness and sharpness. You can teach creative thinking. Always
encourage and support creative thought, it is our greatest gift.
Doing these sorts of activities answers many questions without
teaching anything. Brilliant! This is problem based learning.
I just love it when I create situations when students do not
realise they are learning. Problem based learning causes
this to happen. In solving a problem students also explore many
other things and learning. Do we ever know what students are
actually learning? These and other ideas are explored in
Sample measurement questions.
1. A voltmeter has 4 divisions (or graduations) between the 2V and
3V marks. Which multichoice below show this scale?
(a) 1V, 2V, 3V 4V (b) 2.1V, 2.2V, 2.3V,
2.4V (c) nothing (d) 2.25V,
2.5V. 2.75V (e) 2.2V, 2.4V. 2.6V. 2.8V (f) Who
2. Create a line equal in length to π = 3.14159265359... cm long,
3. Here is a broken ruler starting near 3cm and ending near 13cm.
There are usually plenty of broken ones around.
Use the broken ruler to measure a short length and a long
length, measure your desk, measure the width your index finger,
measure the length of your shoe. Create a question.
Understanding of Geometry
Being enabled to describe shape in the world around us, to
reason, to generalise (prove), make sense of and connect ideas
to solve problems.
Geometry is all about points, lines and angles. It
was a natural human curiosity from early beginnings and remains
that way today thousands of years on. The ancient Babylonians and
the Chinese recorded what became known as Pythagorus's Theorem for
right angled triangles.
For me, geometry starts with a compass and a ruler.
The Greeks used a straight-edge. I like numbers and the way
measurement works. That is the physicist in me...measure, measure,
measure. Then there is the hyperbolic geometry of Nikolai
Lobachevsky and spherical geometry of many people including
Euler. Euler must have had a lot of spare time and we know
he did not have TV to waste his waking hours.
Geometry is a hands-on experience. There are some wonderful
geometry software programs like Geogebra with amazing
features and accuracy. However here is my warning... Use this software too early and be the maker
of your own peril. Like calculators and statistics
software, use them too early and the "hands on" learning is lost.
You will destroy the need to find a better way. I like to think
that learning travels from your hands to your arms and to your
brain and every class I have taught had that view explained. There
are reasons you have hands and one of them is "Learn by Doing"
[My primary Lytton St School Motto!"]
To learn geometry start with a ruler and compass. Practice
drawing a circle. Learning the skill of a slight lean in the
direction of rotation soon makes everyone confident compass
users. Learn also how to rule a line. Now to make the
hexagon patterns and start developing 90 degree constructions,
perpendicular, parallel, bisectors and centroids. Octagons...
overlapped squares...pentagons! Develop the language, know the
words. [Hint for keeping a nice sharp set of compasses is to also
have a smallscrewdriver for keeping the angle screw tight and buy
The picture (up and right) shows the use of dividers, another
tool. The set square triangles (30,60,90) and (45,45,90)
are vital as well because of the connections to Pythagorus and
irrational numbers, standard triangles and the common angles of
30, 45 and 60 degrees. There are now apps for cellphones that
measure angles, position, distance and time.
As with all subjects the language is the geometry so use
at every opportunity. Here is a sample! http://www.varsitytutors.com/aplusmath/homework-helper/geoterms.
If you do not speak the language students will not learn
the language. Use the language of students and develop the
use of the language of mathematics to make it their language.
Normalise mathematical language. I like the word normalise.
Normalise language, writing, speaking, listening and reading.
Normalise doing. It is what we do!
There are hundreds of patterns like this and all use light
construction lines and careful selection of the required lines to
everything in geometry and having constructed an angle bisector we
have oiled the link to proof and why it works. Similar and
congruent triangles likewise.
This book is one of a series and this link
should take you to Amazon and you can explore.
All students ask why algebra is important or grandstand with "When
am I ever going to use THIS!" Adding algebra to geometry,
however, creates a relevant context. It is an easy step then,
after learning how to label vertices and sides of triangles, to
prove that "In any triangle the exterior angle equals the sum
of the two interior opposite angles."
Add a circle to the triangle and the basis of all circle theorems
"The angle at the center of circle is equal to twice the
subtended at the circumference" is established. "L at
Centre = 2x L at Circum" or some similar abbreviation because
"Mathematicians write the least possible and always in the most
efficient manner". They do not use metaphor being careful to
convey the meaning as precisely as possible. This summarises the
literacy of mathematics.
No apologies here to literacy pundits who I notice have a
propensity of using as many words as possible to say the same
thing as many times as possible and in multiple ways and fill as
much space, time and silence as possible to get any message
across. I have more important things to do with my lifespan than
to listen to or read repetitive dribble. In mathematics we
go to extraordinary lengths to ensure concepts are identical in
everyone's mind and do so using the smallest and most precise
number of words. EG, A circle is the locus of all points a
distance r from a single point, called the center. Nothing
more is needed.
When I say "a point" the same dimensionless spot appears in
your imagination as it does in mine. The shortest line
joining two such points is a straight line. There are many
lines lines joining two such points but only one of them is
The "metaphor" and acceptance that you "make your own meaning from
spoken and written language" I accept. We all make our own
meaning. In mathematics the metaphor is almost non-existent. We
need exact communication of ideas and concepts and not a quagmire
of confusion and misunderstanding and many meanings. The Māori
language is full of metaphor and it is also acceptable in this
language to make your own meaning. When teaching mathematics know
that metaphor does not exist and say why. This does not mean we
have to think the same way, but it does mean we use the same
Try explaining "a location" to someone and you will
understand the issues of communication and multiple meaning.
Different meaning is easy to expose. In the situation of two
squirrels running around a tree always keeping opposite one
another we can ask "Do the squirrels go around the tree?" and
"Does the tree go around the squirrels?" The answer of course to
both questions is YES! You can ponder why.
We do our best in mathematics to avoid these misunderstandings
by having a carefully defined language. Not so in statistics
which is a literacy lovers delight and filled with what I can
"floppy language". More later on this wonderful world of "suggest,
"could", "nearly", "typical" and "it appears" in a later chapter.
Statistics aims to make sense of data and uses mathematics to do
models are dramatic and fun, require team work and
accuracy. They can be the Stellated Platonic Solids and can
be made from 120gm card. I print sheets of 6 circles with either
equilateral triangles, squares or pentagons shaped on the circle.
Ruling along the fold lines and spot gluing with PVA creates a
strong mess free join.
The stellations are made from arcing the compass around a corner
of card and creating 3, 4 or 5 sided stellations, all with a glue
tab. Varying the length of the stellations and using a couple of
colours to highlight the body and the points make startling eye
catching models that can be hung around the classroom. A visiting
parent one day when I was running a 3D modelling workshop in a
class stayed and made a large stellated icosahedron which was to
be hung in the lounge of her house. She enjoyed every minute
of the whole day session and was a persevering inspiration to my
young students. She did not want to leave my class! See
elsewhere on my website for all the templates to help you get
This blue "football" of
pentagons and hexagons was created as a Y9 class group project and
is about 1m across.
This pink 12 sided dodecahedron model
shows different sized stellations.
The proof that only 5 Platonic solids exist is quite easy. These
are the tetrahedron (4 sides of ∆), the octohedron (8 sides of ∆),
the icosahedron (20 sides of equilateral triangles), the
hexahedron (cube, 6 sides of squares) and the dodecahedron (12
sides of pentagons). The proof involves the sum of angles at
one vertex and what is possible. Again, visually proving a
truth about the world we live in and all accessible to all
students. It is curious that many pollen, fungi and bacteria
shapes are based upon these Platonic solids, and only these
Art can be brought into the mathematics classroom with flowers and
the leaf structures of plants. Flowers are often based upon 3 and
5 which are both Fibonacci numbers. I enjoyed the project of "Use
a camera, phone or otherwise and find as many different geometric
shapes as you can in the flowers and plants of the botanical
gardens." WOW - a .ppt of all the shapes was a stunning
presentation when accompanied with some words and music.
Hence Lesson #5
"Hands-on" means making circles and using tools to measure and
make geometry. Learn by doing. Movement in the classroom,
opportunities to talk about maths, work together, create
products, collaborate and participate are all worthy key
competencies that can be developed by model making in the
mathematics classroom. It is always surprising to me how doing
mathematics often causes the learning to be embedded. The main
vine is intact and perseverance, curiosity, participation and
self management are all present.
Describe your approach to using geometry and measurement to
How do you link number and algebra?
Can you link probability?
What does the LOGO for the NZAMT
bring to mind?
Describe 5 or more different ways to measure the size of the
Write your version of the Deep Understandings applying across
the early years of learning Measurement and Geometry.