Cling to the main vine, not the loose one.
Kei hopu tōu ringa ki te aka tāepa, engari kia mau ki te aka matua

Thoughts on Teaching and Learning of Mathematics
Lesson #5 • Revised 1/1/20

Geometry 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 the universe.

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 dimensions.

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. (Since WW2).

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 Powers of Two.

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 studied.

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 Lesson #4.

Axiom #0
One can be anything I choose one to be.

Axiom #1
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.

Task
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 future chapters.

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 cares?

2. Create a line equal in length to π = 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.


The Deep 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 good ones!]

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!

Sample Pattern

There are hundreds of patterns like this and all use light construction lines and careful selection of the required lines to colour.

This book is one of a series and this link should take you to Amazon and you can explore.

Experience is 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.

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 "straight".

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 language.

 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 so.

3D 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 started.

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 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.

Teacher TASK
Describe your approach to using geometry and measurement to explore mathematics.
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 Earth.
Write your version of the Deep Understandings applying across the early years of learning Measurement and Geometry.

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