Te Whānau Taparau - The Polygon Family

Purpose

This unit examines the properties of polygons and how these are related. It also gives the names in both Māori and English.

Achievement Objectives
GM3-3: Classify plane shapes and prisms by their spatial features.
Specific Learning Outcomes
  • Investigate properties of symmetry in shapes.
  • Investigate spatial features of shapes.
  • Use both English and Te Reo Māori to describe different polygonal shapes.
Description of Mathematics

This unit allows students to develop an understanding of the geometrical features of regular polygons. It also aims to develop aspects of symmetry (reflective and rotational) through a problem solving approach.

Mathematical language is also explored particularly in terms of Māori. It is envisaged that such an exploration will give rise to descriptions that can incorporate both languages in order to make sense of the various aspects of geometry examined in this unit.

The unit begins with string geometry to set the scene for investigating shapes and their properties using folding and possible turning techniques. It then progresses to an examination of regular polygons where Māori terms are introduced. The concept of whānau or family is introduced to reinforce the fact that polygons are linked in a range of ways.

For this unit you will need to know, and the students will need to find out, the following:

porohita = circle
whānau = family
taparau = polygons
tapatoru = triangle
tapawhā = square
whakarara = parallelogram
rangiwhitu = diameter
putoro = radius
pae = circumference
tapatoru rite nui = a large equilateral triangle
puku = tummy
e toru, nga tapatoru rite = made from 3 smaller equilateral triangles
whakarara rite = a rhombus
taparara = trapezium
tapawhā rite = a square tapawhā
hāngai = an oblong
koeko tapatoru = a triangular pyramid
ahu 3 = 3 dimensions
tapa ono rite = regular hexagon
kai = food
e ono, tapatoru rite = 6 equilateral triangles

Required Resource Materials
  • String
  • Compasses
Activity

Getting Started

In this session we use loops of string to form shapes. We then prove to others that our shape is what we said it was.

  1. Put students into groups of 4 and hand each group a length of string - approximately 2 to 3 meters long. Have them tie the ends of the string in order to make a loop.
  2. Pose the following problem:
    Use the members of your group and the loop of string to make a shape that is a square. When you are satisfied that you have finished, be ready to demonstrate to another group that you can prove that the shape that you have made satisfies all the requirements of a square.
  3. Allow the students to experiment, but provide questions that focus on the symmetry of the shape and the ‘squareness’ of the corners. For example:
    How did you find each of the equal length sides?
    Using the string, how might you show that the shape has got 4 lines of reflective (line) symmetry?
    How might you show that the corners are right angles (without using a protractor)?
  4. Now challenge the groups to use the loop of string to make an oblong. When they are satisfied that they have made an oblong, they have to be ready to demonstrate to the others that their shape satisfies all the requirements of an oblong.
  5. Ask the students to provide different arguments for the symmetry of the oblong than they used for the symmetry of the square.
  6. Use your loop of string to make the following triangles:
    - a triangle that has no sides that are the same (scalene);
    - a triangle that has only 2 sides the same (isosceles);
    - an equilateral triangle.
  7. After making each type of triangle, the group has to be prepared to prove that the necessary conditions for each triangle are met. It may be a good idea to have the students plan on a piece of paper how they might approach each of the shapes and demonstrate how they might go about justifying their shape.
  8. Besides focusing the students on the sides of the different triangles, also ask them to consider the angle measures of the corners. In particular, provide opportunities for them to make links between the number of sides and the angular measures of the corners.
  9. Give each group another length of string and challenge them to make a cube. However, before they do so, each group must plan their ‘construction strategy’.

Exploring

Over the next couple of days we create Te Whanau Taparau (The Polygon Family) by transforming a circle.

The Circle

  1. This is a story about a shape from the porohita (circle) whanau (family) who wanted to be part of the taparau (polygon) whanau. He was a special porohita because he was able to transform himself into other shapes by folding and unfolding. However, he needed the help of a mathematician who knew some things about geometry ... that person is you!
  2. Either give the students a set of porohita, or give the students a piece of A4 paper and ask them to make the largest possible circle using compasses. Discuss how the students worked out the biggest circle.
  3. Discuss the attributes of a circle.
    What makes a circle different from other polygonal shapes, for example the triangle (tapatoru), square (tapawhā) or the parallelogram (whakarara)?
    What are some names that are used to describe parts of a circle?
  4. Focus the students on:
    - diameter (rangiwhitu), radius (putoro) and circumference (pae);
    - the relationship between te rangiwhitu and te putoro.
    The relationship between pae and rangiwhitu may be an extension or extra investigation.
  5. This activity can be extended to the following exploration:
    Given a piece of A4 paper, what is the largest circle you can make?

The circle becomes an equilateral triangle

  1. We now continue the activity to transform the circle:

    The porohita looked in the mirror one day and decided he was getting bored of being a porohita. Porohita decided that he wanted to look just like tapatoru rite nui. But, he needs your help.
    Using the circles (provided or made), fold Porohita to make the biggest possible tapatoru rite (equilateral triangle). You may need to experiment with different ways of folding to get the biggest.

  2. Develop a method that you will use in order to check that the shape you have made satisfies the requirements needed for a tapatoru rite - and that it is the biggest one possible.

  3. If this is too challenging for the students, the teacher may need to give a hint (refer to the diagram) or show one that has already been made - just to convince them that it can be done.
     circles.
  4. The teacher will need to ensure that the discussion and follow-up activity includes elements of rotational or reflective symmetry, ‘size’ of the corners or ‘size’ of the angles of each of the corners, lengths of each of the 3 sides. It would be worthwhile getting the students to develop methods to check for other tapatoru rite, and in so doing, develop the relationship between length of the sides and size of the angles.

Triangle to trapezium

  1. Porohita not only changes his shape, but also changes his name to Tapatoru rite nui. At the end of the week Tapatoru rite nui felt cramped - he couldn’t roll around like he use to when he was a porohita. His aching sides and corners needed to be massaged in order to get rid of some of his pain. While lying on his puku reading the Geometers’ Weekly magazine, he spotted his favourite sports hero, Jonah Trapz who is ‘built like a taparara’ (trapezium). Immediately, Tapatoru rite nui wished he could transform himself into a taparara. But he needs your help.
  2. As a possible way of introducing students to a trapezium, have them explore attributes of various trapezoidal shapes. Have them compare these with non-trapezoidal shapes including quadrilaterals that are oblongs, squares and that are ‘nearly’ trapezoidal in shape. One approach is illustrated as follows.  
    Here is the work of one student who we will call Hannah.  She has sorted some quadrilateral shapes into two different categories: trapezia and not trapezia.

    Trapezia

    Not Trapezia

    quadrilateral shapes.

     quadrilateral shapes.

     

    Hannah states that her sorting procedure is based on the relationship between one pair of sides. If one pair of sides is parallel and not of the same length, then it’s a trapezium. It doesn’t matter about the other pair of sides.

  3. Ask the students to explore Hannah’s table of shapes and explain her sorting procedure.
    Ask questions that encourage the students to focus on quadrilaterals that have one pair of parallel sides. Some students might observe that a trapezium often looks like a triangle with the top cut off leaving an edge that is parallel to the side opposite.
  4. Back to Tapatoru rite a nui who wants to look like his sports hero Jonah Trapz.
    Explore how you might fold Tapatoru rite nui to make a taparara that is made up of e toru, ngā tapatoru rite (i.e. make a trapezium from the big equilateral triangle in such a way that the trapezium could be said to be made from 3 smaller equilateral triangles).
    Using the taparara that you have made:
    - How would you check that a pair of sides is parallel?
    - How would you use your understanding about the equilateral triangle to help you make a convincing argument for a pair of parallel sides?
    - How would you check that each of the smaller tapatoru rite are of equal size and shape, that is they are congruent?
  5. The following diagram illustrates the change to a trapezium.
    trapezium.trapezium.

Trapezium to rhombus

  1. To go with his new shape, Tapatoru rite nui changes his name to Taparara. One day, he attempts to enter the Diamond Exhibition at the Geometrical Museum - diamonds are one of Taparara’s favourite shapes. He is stopped at the door and told that only whakarara shapes (parallelograms) are allowed to go in. Taparara really wants to get in, so he decides to change shape, but he doesn’t want to be just any whakarara, he wants to be a whakarara rite (rhombus). He needs your help.
  2. Ask the students to fold Taparara to make a whakarara rite.
  3. Using the whakarara rite made ask:
    - How would you check that the sides are parallel?
    - How would you check that all the sides are of equal length?
    - How would you use your understanding of equilateral triangles to help you make a convincing argument for parallel and equal length sides?
    - What might the relationship be between the angle measure of the smaller corner and the angle measure of the larger corner?
  4. Of course, Taparara changes his identity and name and becomes known as Whakarara rite. Some of his friends call him Rhombus.
  5. The following diagram illustrates the change to a rhombus.
    rhombus.

Rhombus to a triangular pyramid

  1. Once in the Geometrical Museum, Whakarara rite meets up with some of his friends, Tapawhā rite (square) and her sister Tapawhā hāngai (oblong) .
  2. At this point the students could do some personal research and write a report, using diagrams if needed, that illustrate the difference between the rhombus alias whakarara rite, the square alias tapawhā rite, and the oblong alias tapawhā hāngai.
  3. Whakarara rite gets so excited by all the different diamond shapes that he folds in to a particular diamond shape in Ahu 3 (that is in 3 dimensions) called a koeko tapatoru (a triangular pyramid).
  4. By folding the whakarara rite out and in, make the koeko tapatoru.
  5. The following diagram illustrates the change to a triangular pyramid.

 triangular pyramid.

Triangular prism to hexagon

  1. He stays like this for only a short time before changing back into a whakarara rite. A little while later, Whakarara rite stops in at his favourite food place called Geo Flatworld Takeaways to get a kai of tapa ono rite (rectangular hexagonal shape) chips.

    "Awesome", he says. "I can look like a tapa ono rite too. All I have to do is fold down my 3 sides and.... ".

  2. Whakarara rite forgot what else he had to do. He has told you the first step, but needs your help to complete his transformation into a tapa ono rite. Explore a way of folding to make a tapa ono rite that might be said to be made out of e ono, tapa toru rite (i.e. 6 equilateral triangles).
  3. Ask the students to check whether all the sides are equal. Encourage them to use both rotational and reflection symmetry.
  4. The following diagram illustrates the change to a tapa ono rite.

 diagram.

Reflecting

In the final session we reflect on the shapes that we have explored during the week.

  1. At home Tapa ono rite feeling tired after a long day, sits on the floor, closes his weary eyes, curls up into the frustum of a triangular pyramid and dreams about all the different shapes he was able to make ... with your expert help of course!
  2. Tell the students that a frustum of any solid shape is made by making a cut parallel to the base and removing the top of the cone.solid. Show them the example of the cone. 
  3. Next get the chidlren to explore how they might ‘curl’ their tapa ono rite up so that they can make such a frustum.
  4. Pose the following problems:
    Name the different shapes of the faces of your frustum.
    Make some statements about some of the attributes of the shapes that go to make up the frustum.
    If Tapa ono rite had sides of length two, how long were the sides of the open top and the base of the frustum?

Teaching notes:

The following diagram illustrates how to change a hexagon into the frustum of a triangular pyramid.

 hexagon.

Fold B to O, then A to O, then C to O to make a triangle shape. Lift the flaps so D and E touch, F and G touch and H and I touch.

The open top has sides of length two because these sides are the same as Tapa ono rite’s sides. The base sides are of length three. You can find this by measuring. (If you are clever you can use right angled triangles to work it out without measuring.)


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