the main vine, not the loose one.
Kei hopu tōu ringa kei te aka tāepa, engari
kia mau te aka matua
Thoughts on Teaching and Learning of Mathematics
Lesson #5 • Revised 4/1/19
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 soem of the powerful connections in
mathematics. Hidden are the axioms of knowing what "one" is and
"comparing the same size bits" described elsewhere, the patterning
and generalisation of algebra. Geometry is a great place to
develop algebraic skills, reasoning and proof. It is essential
geometry is part of all learning programmes in mathematics.
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, and all the visuals of number we
create, 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 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 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 the same
law from the microscopic or quantum direction. 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. 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?
Negatives in measures usually involves the concept of
direction. Negative temperature is cooler, positive warmer.
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. Vectors are an amazing
construct and at one time I studied Tensor Calculus which involves
vectors in 3d.
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.
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 little
more. For evidence of truth to what I say just ask a carpenter,
engineer or cakemaker 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 efficent,
but not as efficient as the Powers of Two.
We need to reason with measures as well. Lengths become 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!
In all the above the dimension called time is connected 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.
The Deep Understanding of Measurement
-Being enabled to quantify size of shape, reason 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".
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 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 and
always encourage and support creative thought. 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. What of the multichoice below shows 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 3.14159265359 cm long, approximately!
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) and connect ideas to solve
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 but 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.
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 "To lean by doing"
[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
The picture (up and right) shows the use of dividers, another
tool. The set squares 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.
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 a message across.
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 and 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
meanings. In mathematics the metaphor is almost non-existent.
We need exact communicatiuon of ideas and concepts, 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
squirrils 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 "nearly",
"typical" and "it appears" in a later chapter.
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!
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.
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.