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Look for a Pattern, Find a rule, Test.

Mathematics Lessons for Remote Learning

Strand: Geometry and Measurement

Target: Y7, 8, 9, 10 –NZC Level 3/4

Topic: GM Two Dimensional Hexagon Construction Challenges

Starter – Odd One Out

hexagon

pentagon

octagon

square

The idea here is to select the odd one.

I choose __________ because _____________________

Learn by Doing

THE DEEP UNDERSTANDING OF GEOMETRY

· Being enabled to describe shapes and their properties in the world around us, to reason, to generalise (prove), to make sense of and connect ideas to solve problems.

Task for Experts...

Using a compass, ruler and pencils, draw and colour the patterns above. These are based on a hexagon and all the diameters. These construction lines can be seen in both illustrations above. Do not read on!

Learning to draw a hexagon and construct some colourful patterns.

Reminder. All these illustrations use two key constructions. Bisecting an angle and bisecting a line segment . These two skills should become second nature. It is also a good idea to think about why they actually work and prove this to yourself. A proof is to find two congruent triangles in each diagram. Congruency means identical in every way. In the right diagram I can see four right angled and conguent triangles.

and

Bisecting and angle Bisecting a line segment

Constructing a Hexagon

Here are three similar ways to construct a regular hexagon. A regular hexagon has 6 sides of the same length and 6 axies of symmetry (mirror lines) and a rotation symmetry of 6 as well making a total symmetry of 12. This is quite high and makes a hexagon perfect for many drawings and 3d illusions.

The left hexagon is made by drawing the diameter 1 to 4 through the center. Arcs are made from 1 and the center to find 2, from 4 and the center to find 3 and likewise 5 and 6. This has created a nest of 6 equilateral triangles. All sides are the same because they are the same radii. The only difference with the middle construction is that the complete circle was drawn first. The construction on the right starts with a center and the circle, then using the same radius on the compass stepping around the circle 6 times. I find this method is not quite as accurate as the other two probably because there is a cummulative error that sneeks in on each new step. Find a method that works for you.

The Diameters of the Hexagon and selecting patterns.

Connect each point to every other point and you will end up with (6x5)/2 or 5+4+3+2+1 diameters.

The left diagram shows two kites interlocked by selecting two similar (not same size but same angles).

The middle diagram shows a hexagon inside the big hexagon by finding the crucifiction points.

The right diagram uses the crucification points as well and hints at two interlocking equilateral triangles.

Here are the finished illustrations.

The center illustration creates an illusion showing and your brain sees either a cube with a cube cut out or a cube sitting in a corner by turning the image upside down.

TASK

Draw and colour all these designs and make up 6 more of your own.

Constructing a Square

You can probably do this already and I challenge you to do so.

The angle of my camera made the square look a bit wonky but it is square. To make a 5cm square set the compass to a 5cm radius. Draw a line and mark the bottom left corner O on the line. Use the compass to find B. Draw an arc around O and from each intersection with the line make a longer arc to find X above A. Reset the compass to 5cm and from O arc to find A, and from A and B makes arcs to find X. This creates the square OAXB.

The secret to the square is it has four right angles, 4 equal sides and a lot of symmetry like the hexagon and octagon.

The centre pattern is the TANGRAM PUZZLE pieces and here is a great resource for puzzles and finding out more. https://www.tangram-channel.com/. You could make a 1m Tangram square from plywood and use as a garden puzzle.

The Puzzle on the right is a curious contruction and can be formed into a triangle. The three cuts across the square are all the same length. Copy the image and print it, cut and play. The kite is not symmetric!

Challenge – What fraction of this square is shaded?

I found this little gem on https://brilliant.org/daily-problems/. The side at the top is bisected.

This website has become a favourite for me and has small courses in maths on every topic and every level of difficulty you might imagine. A recent course I did was about logic and how to make an algorithm to recognise human writing. It is about artificial intelligence and is already in action around the world. An example is facial recognition.

There is lots to learn in geometry. Geometry gives a real and visual context to learn how to prove things. Geometry develops the power of logic.

Starter – Odd One Out

hexagon

pentagon

octagon

square

The idea here is to select the odd one.

I choose _____pentagon_____ because ______it has an odd number of sides _____

The honey bee is an expert with hexagons.

The hexagon shape tesselates. This means it can be translated to fill the complete 2D plane with no gaps. This makes a sheet like the following illustration.

And can be used to make many fascinating patterns.

Here is a link to printable isometric papers. “Iso” means “constant” and “metric” means “measure”. Isometric paper measures teh same distances. Here is an example pattern made with isometric paper.

Search the internet with “isometric patterns” for more.

Journalling

Today I learned ________________________________________________________________

And I would like to know about ___________________________________________________

Comments

Make any comment you feel like making here.

Math Language: List all the math words you can find in this document and write what you think it means beside the word. Eg subtraction means to take away or to find the difference. Keeping a list of these words is a very good idea.

Answers

Starter – Odd One Out

ray

line

segment

point

The idea here is to select the odd one .

I choose _____point_____ because ______the others contain many points. _____

How many diameters?

The general formula is n(n-1)/2 so for a hexagon it will be 15. This problem is teh same as teh handshake problem for 6 people and teh numbre of intersection problem for 6 lines. Geometry mirrors many number problems and allows mathematics to visualise solutions. Who does not draw a problem to try and explain it?

Feedback

Students and teachers are welcome to email [email protected] with comments. This was a lesson that could be given to a NZC Level 2, 3, 4, 5 student for some placevalue learning and revision. Students should select a set time each day and perhaps using the timer on a cell phone set 45 minutes or so to learn and practice mathematics. Keep trying on problems and expect to struggle. Persevering and struggling are great competencies to develop. You can learn more about these from https://www.youcubed.org/resource/growth-mindset/. We have a great math website in Nzwith a special resource called e-AKO https://nzmaths.co.nz/information-about-e-ako-pld-360 .

Learn by doing and using your hands, persevere.