Building with triangles

Purpose

In this unit, students will identify the properties of triangles, construct both equilateral and irregular triangles using either ruler and protractor or ruler and compass, and make nets.

Achievement Objectives
GM4-1: Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time.
GM4-6: Relate three-dimensional models to two-dimensional representations, and vice versa.
Specific Learning Outcomes
  • construct triangles with specified dimensions using two different techniques
  • design and construct nets for three-dimensional objects
  • name basic three-dimensional objects, especially those made with equilateral triangles
Description of Mathematics

This unit explores some aspects of three-dimensional geometry as well as techniques for the construction of triangles.

Required Resource Materials

card or heavy paper for three dimensional constructions

rulers/compasses/pencils/protractors/scissors/selotape

paper for two dimensional constructions

copymasters of shapes

Activity

Getting Started

A review of knowledge about properties of triangles, followed by a discussion of ways to construct triangles with specified dimensions.
  1. Brainstorm with the class ”The properties of triangles”.
    Facts you should get include;
    Three sides
    Three angles/corners
    Angles add to 180o
    Different types (scalene/isosceles/equilateral/right-angled)
  2. Beware of students saying that triangles have all sides the same length or all angles the same. This is not a property of triangles, but a property of regular polygons (including equilateral triangles). Ensure that this is emphasized.
  3. Ask students how they could construct an equilateral triangle. Discuss their ideas – their advantages and disadvantages. There are three relatively easy methods, give students the opportunity to try each (or demonstrate each to the class on the board).
  1. Using ruler and protractor
  1. Draw one side to the required length, using a ruler.
  2. Measure an angle of 60 degrees from one end of the first side.
    Draw a second side to the same length as the first, again using a ruler.
  3. Connect the ends, measuring to ensure the third side is the same length as the first two.
     protractor and ruler.
  1. Using ruler and protractor
  1. Draw one side to the required length, using a ruler.
  2. Measure an angle of 60 degrees.
  3. Draw a second side through your mark at 60 degrees.
  4. Measure another angle of 60 degrees from the other end of your first side.
  5. Draw the third side and rub out any extra lines.
     angles.
  1. Using ruler and compass
  1. Draw one side to the required length, using a ruler.
  2. Set your compass to a radius equal to the length of the side.  Put the point of your compass on one end of the side you have drawn and lightly draw a small arc about where you think the third corner should be.
  3. Move the point of your compass to the other end of your first side and draw another arc, which should cross the first.
  4. The point where the arcs cross is the third corner – draw in the remaining two sides, checking they are the correct length as you do so.
    creating arcs 
  1. Discuss whether you could use each method to construct triangles if the sides are not all the same. Get students to list advantages and disadvantages of each method, and decide which method is the best.
    Method a) can be used to construct triangles as long as you know two side lengths and one angle.
    Method b) can be used to construct triangles as long as you know two angles and one side length.
    Method c) can be used to construct triangles as long as you know the lengths of all the sides.
    Method c) is probably the easiest and most accurate, since measuring with a protractor is likely to produce errors.
  2. Give students the opportunity to practice constructing a variety of triangles using all three methods.
Exploring
Using the knowledge gained of how to construct triangles to create three dimensional objects made up of triangles.
  1. Show students a model of a tetrahedron.
  2. Ask them to identify what kind of shape and how many it is made up of (4 equilateral triangles).
  3. Challenge them to construct the net for a tetrahedron (on paper first to avoid wasting card).
  4. Check the nets that are drawn. There are two layouts of four equilateral triangles that will fold to make a tetrahedron and one that will not. Get students to identify why (two of the triangles fold to be on top of each other, so there is an open side).
triangles.
Will make tetrahedron
tetrahedron.
Won’t make tetrahedron
  1. Allow students to make a model of a tetrahedron from card. Students should either join edges with selotape, or if they wish could include flaps on their net and glue the edges. If you have access to plastic shapes that create nets they could construct their models with these. 
  2. Challenge students to make a different three-dimensional object from only equilateral triangles.
More able students should be challenged to see how many different objects they can make from only equilateral triangles. Can they name them? There are at least 9!

 

4 sides: tetrahedron
6 sides: triangular dipyramid (glue two tetrahedra together)
8 sides: octahedron
10 sides: pentagonal dipyramid (glue two pentagonal pyramids together)
12 sides: snub disphenoid (split a tetrahedron into two wedges and join them with a band of eight triangles)
14 sides: triaugmented triangular prism (attach three square pyramids to a triangular prism)
16 sides: gyroelongated square dipyramid (attach two square pyramids to a square antiprism)
20 sides: icosahedron
24 sides: stellated octahedron (attach a tetrahedron to each face of an octahedron)
Students requiring more teacher  support can either design the net to make an octahedron (8 equilateral triangles – fold to make a shape like two square based pyramids joined together), or they could be provided with the net to cut out, fold and stick together.
Some nets you can print from the attached files and use are: tetrahedron, octahedron, icosahedron.
  1. Discuss the three platonic solids that can be made from equilateral triangles. How many of these has your class made?

 

A platonic solid is a polyhedron all of whose faces are congruent regular polygons, and where the same number of faces meet at every vertex.
In other words, a platonic solid is a three dimensional shape where each face is an identical flat shape with all sides and angles the same, and the same number of these faces meet at each corner.
There are 5 platonic solids, the cube (6 squares, 3 meeting at each vertex), the tetrahedron (4 triangles, 3 meeting at each vertex), the octahedron (8 triangles, 4 meeting at each vertex), the dodecahedron (12 pentagons, 3 meeting at each vertex), and the icosahedron (20 triangles, 5 meeting at each vertex).
  1. Look at some of the other objects students have made. Try to name them.
  2. Challenge students to make an object that uses equilateral triangles in combination with a different shape. Point out that all the side lengths will need to be the same, even if the shapes are different.
More able students could again be challenged to see how many different objects they can make. Can they name them?

 

pyramid (4 triangles and 1 square)
pentagonal pyramid (5 triangles and 1 pentagon)
triangular prism (2 triangles and 3 squares (or rectangles))
cuboctahedron (8 triangles and 6 squares)
icosidodecahedron (12 pentagons and 20 triangles)
truncated tetrahedron (4 hexagons and 4 triangles)
truncated cube (6 octagons and 8 triangles)
square antiprism (2 squares and 8 triangles)
Students requiring more teacher  support can either be helped to design the net to make a triangular prism (2 equilateral triangles joined by three squares), or they could be provided with the net to cut out, fold and stick together.
Some nets you can print from the attached files and use are: triangular prism, pyramid, pentagonal pyramid, cuboctahedron, truncated tetrahedron, truncated cube, icosidodecahedron.
  1. Talk about the three types of pyramid (bases: triangle, square, pentagon). Has your class made all three? What do you call a pyramid with a triangle base? (tetrahedron) Could you make a hexagonal pyramid? (No) Why not? (the six equilateral triangles would lie flat on the hexagon and the object would be flat. Note: You can make a hexagonal pyramid, but not with equilateral triangles, you need to use isosceles triangles.
  2. Look at some of the other objects students have made. Try to name them.
Reflection
Reflecting on the objects that the class has made and what they know about them.
  1. Ask students to name as many three-dimensional objects as they can which can be made using equilateral triangles.
  2. Can anyone think of ways we could group these objects? (Number of sides, other shapes used, symmetry)
    The most obvious way is to group the objects into those that use exclusively triangles and those which include other shapes in combination with equilateral triangles.
  3. Try grouping the models the class has made using some of these criteria.
    There is an abundance of information on polyhedra on the internet. If you require more background or extension material, try these sites:

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