Getting Started

1. Brainstorm with the class ”The properties of triangles”.  
   Possible responses might include:  
   Three sides  
   Three angles/corners  
   Angles add to 180^o  
   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 emphasised.
3. Make connections between students’ knowledge of triangles and contexts that are relevant, real-life contexts. You could ask students to find images of triangles around the school, or online, or in their home environments. 
4. Ask students how they could construct an equilateral triangle. Discuss their ideas – their advantages and disadvantages. There are three relatively easy methods. Model these and give students the opportunity to try each (or demonstrate each to the class on the board).
   1. Method 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.  
          
   2. Method 2: 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.  
          
   3. Method 3: 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.  
          
5. 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.
6. Give students the opportunity to practise constructing a variety of triangles using all three methods. Whilst some students may feel confident enough to do this independently, others might benefit from working in pairs, or in a more-structured, teacher-led group. Consider which approach will best ensure that the learning content is accessible to all of your learners. For extension, students could create a paper or digital presentation (e.g. video, poster, set of slides, infographic) that demonstrates and explains each of these construction methods.

Exploring

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. Ask students, working in groups, to use the toothpicks/popsicle sticks and Blu-tack to create a 3D model of a tetrahedron. As they build these shapes, roam and support them to identify the properties of the shape:  
   How many sides does the shape have?  
   What 2D shape forms the base of the shape?  
   If you unfolded this shape, what would it look like?   
   How many different arrangements of the unfolded tetrahedron could you make?
4. Pose the above questions to the whole class, and give students time to discuss their ideas, before sharing back to the rest of the class.
5. Distribute paper. Challenge students, in the same groups, to construct the net, without tabs, for a tetrahedron. You might need to show students nets for other 3D shapes (e.g. cube, cylinder, cuboid) to ensure they understand what a successful net will look like. Remind students to think about the base shape of their tetrahedron models, how many sides there were in the completed shape, and what the “folded out” tetrahedron looked like.
6. 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).  
   These nets will make tetrahedron:  
    
   This net won’t make tetrahedron:  
   
7. Show students the examples of the nets with tabs. Allow them to make a model of a tetrahedron from card. Students should either join edges with tape, or if they wish could include tabs on their net and glue the edges. 

8. Challenge students to make a different three-dimensional object from only equilateral triangles. 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. Students ready for extension could be asked to see how many different objects they can make, and name, from only equilateral triangles. 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)

    

9. Discuss the three platonic solids that can be made from equilateral triangles. How many of these has your class made?  
   Some nets you can print from the attached files and use are: tetrahedron, octahedron, icosahedron. 

   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).

    

10. Look at some of the other objects students have made. Try to name them.

11. 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. Support students, as necessary, to create at least one of the following shapes. 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. Students ready for extension could be challenged to see how many different objects they can make and name, and could find examples of these shapes from real-life contexts (e.g. diamonds, dice, terrariums, footballs, sculptures).  
    Some nets you can print from the attached files and use are: triangular prism, pyramid, pentagonal pyramid, cuboctahedron, truncated tetrahedron, truncated cube, icosidodecahedron. 

    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)

     

12. 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.
13. Look at some of the other objects students have made. Try to name them. Ensure that all students have time and opportunity to share their thinking and work.

Reflection

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 in the construction of the shape, 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.

This is a good activity for the students to work in pairs to find and discuss patterns. The real fascination with the activity will probably emerge in question 3, where the students find patterns in the lists of numbers and factors. They may notice, for example, that the numbers with only two factors are all the prime numbers (which makes sense when they think about it). They may also notice that the numbers with three factors are all square numbers and that the five-factor number (16) is also a square number. The students could investigate how many factors the next square number (36) has. (It has 9.) They may notice that in the case of numbers with four factors, the first three always multiply together to give the fourth factor. For example,
with 15, the factors are 1, 3, 5, and 15 and 1 x 3 x 5 = 15.

Investigation

The students will hopefully be intrigued by the relationship between squares of triangular numbers and cubed numbers. This is an investigation into pure number because there is no ready way that the relationship can be modelled with materials. The students will need to list some of the other triangular numbers to tackle
this. If they sketch the numbers like this:

they should see that the fifth triangular number is 15. They could write the relationship between the square of this number and cubed numbers like this:
(1 + 2 + 3 + 4 + 5)² = 15²
= 225
= 1³ + 2³ + 3³ + 4³ + 5³.
Again, to find the cube of the higher numbers (such as the 4 and 5), the students could simply use the constant function on their calculator (that is, 4 x = = or 4 x x = = ) or the x^y button on a scientific calculator, as was suggested in an earlier activity.

Answers to Activity

1. Practical activity. Your factor tower city should be similar to this one:

2 . a. 2, 3, 5, 7, 11, 13, 17, 19, 23
b. Three factors: 4, 9, 25
Four factors: 6, 8, 10, 14, 15, 21, 22
Five factors: 16
Six factors: 12, 18, 20
Eight factors: 24
3. a. Responses will vary. They could include:
The numbers with two factors are prime numbers. The numbers with three factors are square numbers. For the numbers with four factors, the first three factors (in order) multiply together to give the fourth factor. For example, factors of 10 are 1, 2, 5, and 10. 1 x 2 x 5 = 10
b. Responses will vary. They could include:
All numbers have at least 1 and themselves as factors.
Numbers with an odd number of factors are square numbers.
24 has the most factors (8), and 1 has the least (1).
investigation
Yes, it does. For example, 10² = 100, which is the same as 1³ + 2³ + 3³ + 4³ = 100.

Square numbers are one type of polygonal number. They are called square numbers because when a pattern of evenly spaced dots is drawn to represent that number, the dots form a perfect square. This activity will enable students to investigate the square pattern and become familiar with how squares can be represented using exponents. For example, the “²” is always used to mean squared.

By building patterns with counters or square tiles or by shading squares up to 52 on grid paper, the students should be able to see a pattern developing. They can extend the sequence by building onto a smaller square to create the next largest square. They will thereby see that each larger square is formed by adding two side lengths and one more to the previous square total.

To look at the bigger picture in question 2, a table would be useful alongside the models.

In question 3, encourage your students to use additive and multiplicative strategies as they think out their answers.
Question 3a i can be reasoned using the formula “add two sides and one more to the previous square total”, for example, 10² + (10 x 2 + 1) = 112, or from the progression of odd-numbered differences (3, 5, 7, … 21).
For question 3a ii, the students could look for two consecutive numbers that total 19. However, you could encourage the students to look at it in terms of a similar patterning response to question 3a i or to use the additive inverse to subtraction. You could ask: “What square number added to 19 will give the next highest square number in this sequence?” and so on.

The inverse of this is 10² – 9² = 19.
Encourage students who wish to extend this investigation further to look for other patterns in consecutive square numbers. For example, they could observe that the square numbers alternate between even and odd.
You could ask them questions such as:
“Is there a pattern in the ones digits?”
“Are there more square numbers in the second hundred (101–200) than in the first hundred (1–100)?”
“What patterns can you find between consecutive triangular numbers?” (1, 3, 6, 10, …)
“What triangular numbers can you combine to make square numbers?” (For  example, 3 + 6 = 9)

Answers to Activity

1. a. 2² is known as “two squared” because it gives the area of a square with sides of two units:

b. 7. (To make a 4 x 4 square, 16 tiles will be needed altogether.)
2. a. Practical activity
b. The differences are consecutive odd numbers (+ 3, + 5, + 7, + 9, …) or double the side length of each square plus 1.
3. a..

(When you subtract a perfect square from the next largest perfect square, the answer is the sum of the two numbers.)
b. i. Answers will vary. You can show 3² + 7 = 4² as You can also show this as

You can show this as a diagram. To build the next biggest square from a x a (a²):

you need to add two lengths of the current side length plus 1. That explains why the
differences are odd.
ii. Answers will vary. The simplest explanation, following the pattern shown, is to look for two consecutive numbers that add up to the solution.

The students stand in a circle. Decide on a multiple of two or five that will be the “beep”numbers. Select a student to start counting from one. It is important that all the students countaloud. For example, for counting in fives: “1, 2, 3, 4, beep, 6, 7, 8, 9, beep, 11 ...” When a studentsays “beep”, they sit down. The game continues until only one student is left standing. Thisactivity can be used to reinforce the forwards and backwards counting sequences. Use ahundreds board to assist the students to visualise the patterns. Flip over the spoken numbers butleave the “beep” numbers unflipped.

Extension Activity

Have two multiples going at the same time. For example, threes (say “beep”) and fives (say“buzz”). If the number is a multiple of both three and five, then the person says “buzz-beep”. Sothe sequence goes “1, 2, beep, 4, buzz, beep, 7, 8, ... 11, beep, 13, 14, buzz-beep ... “Begin the counting sequences at different starting numbers. For example, “3, 7, 11 ...” or “100, 97,94, 91 ...” These patterns will help the students to recognise algebraic relationships.

Extension Activity

Repeat “Beep” for multiples of three, four, and six. Develop increasingly complex sequences byusing counts like hundreds, twenties and fifties, in fractions (on multiples of four quarters), like,one-quarter, two-quarters, three-quarters, beep, five-quarters ..., in decimals (on multiples of 0.25), like 0.05, 0.1, 0.15, 0.2, beep ...
 

Set a problem involving a multiplication fact that the students know by instant recall: “Five children have 15 sweets altogether. How many does each have?” Let the students model 15 objects in pairs or small groups. Ask how many groups have to be created (5). Without touching the 15 objects, ask the students to imagine how many each child gets and to discuss why they think this is so. Check by sharing out the objects. Record 15 ÷ 5 = 3 on the board or modelling book. Link this problem to the multiplication fact 5 x 3 = 15 (not 3 x 5 = 15).

Repeat for other multiplication facts that the students know by instant recall.

Activity – Advanced Additive Onwards

Set a problem like: “Seventeen lollies are shared among three children. How many will each child get?” Suppose an answer like two comes up, record 5 x 2 = 10 on the board or modelling book and ask, “How many will be left over?” “Is there enough for everyone to have another lolly?” (Yes.) “How many does everyone now have?” (Three.) “Now is there enough for everyone to have another lolly?” (No.) “Why not?”

Now record 17 ÷ 5 = 3 with 2 left over on the board or modelling book.

Repeat with similar examples.

In this activity, students identify multiples and factors and solve problems that involve finding highest common factors and lowest common multiples. Students will need a good recall of multiplication basic facts in order to be able to do these activities.

Activity One and Game

Discuss the definitions of multiples and factors (on the student book page) before your students begin this activity. Make sure that they understand that every number is a factor of itself, because if they divide a number by itself, there is no remainder. For example, 12 ÷ 12 = 1 without a remainder, so 12 is a factor of 12.
A prime number is a number that has only two factors, itself and 1, for example: 5, 7, 13, and 29. (Note that 1 itself is not considered to be a prime number.)
Before the students play the game, ask the following questions:
• Imagine you threw a 4 and a 6. Which squares could you choose to cover with your counter?(a number with more than two factors, a factor of 24, a multiple of 2, a multiple of 3, a multiple of 4, a multiple of 8, an even number, or a multiple of 6)
• Imagine you need a multiple of 5 to get four counters in a row. Which throws of the dice would give you a multiple of 5? (1 and 5, 2 and 5, 3 and 5, 4 and 5, 5 and 5, 6 and 5, 7 and 5, 8 and 5, or 9 and 5)
 

This game could be extended by asking:

• What are all the different products you could throw with the two game dice, one labelled 1–6 and the other 4–9? (4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 54)

• There are two different ways of getting a product of 12: throwing a 3 and a 4 or a 2 and a 6. Which other products can you throw more than one way using the game dice? (8: 1 x 8 or 2 x 4; 16: 2 x 8 or 4 x 4; 18: 2 x 9 or 3 x 6; 20: 4 x 5 or 5 x 4; 24: 3 x 8 or 4 x 6 or 6 x 4; 30: 5 x 6 or 6 x 5; 36: 4 x 9 or 6 x 6)

• What’s the probability of throwing a double? (There are 36 possible combinations that can be thrown with these dice, and only 3 of these are doubles: double 4, 5, or 6. So the probability of throwing a double is or .)

• Which squares in the game are easier/harder to cover? Can you use the information you have about the possible products that can be thrown to explain why? (Easier to cover: a number with more than two factors [34 out of 36 possible combinations have more than 2 factors; only 5 and 7 don’t], an even number, a multiple of 2 [27 out of 36 possible combinations are even and are therefore also multiples of 2], and a multiple of 3 [20 out of 36 possible combinations].

Harder to cover: a prime number [only 2 out of 36 combinations] and a multiple of 7 [only 6 chances out of 36].)

Activity Two

These problems ask students to find highest common factors and lowest common multiples. An understanding of these ideas is important for working with problems involving fractions and in algebra.
It may help the students if they make a list of all the possible products that can be thrown with the two game dice so that they can then compare this list with the factors and multiples needed in the questions.

Answers to Activities

Activity One
1. 7, 14, 21, 28, 35, 42, 49, and so on 2. 1, 2, 3, 4, 6, 8, 12, 24
Game
A game involving factors and multiples
Activity Two
1. a. 8, because 8 is the highest factor that 16 and 24 have in common (highest common factor). The dice throws would be 1 and 8 or 2 and 4.
b. 4 is the highest common factor of 12, 16, and 60. (The other common factors are 1 and 2.)
2. a. 21. (3 x 7 = 21)
b. 30. (2 x 3 x 5 = 30. 15 is a multiple of 3 and 5 but not of 2, and the only even numbers less than 30 that are multiples of 5 are 10 and 20, neither of which is a multiple of 3.)

This game gives students practice in quickly identifying common multiples of a single-digit number.
The students will need to have good recall of basic multiplication facts. They can find the common multiples, and therefore find the best position for the thrown factor, by mentally dividing each number in a column or row to see if there is a remainder.

Activity One

You could introduce the students to this page by asking what numbers on the dice are factors of 48 and 54.
This activity will help the students to understand how to play the game that follows. Review the scoring system carefully.
Discuss other possible outcomes. For example, “What might have been the outcome if Gina had put the 7 in the box at the top of the right-hand column?”

Game

Whether or not the students beat Gina’s score depends on what they throw on the dice. It is possible to beat Gina’s score. The highest possible score is 15.
For example:
Top boxes 4 6 9
Side boxes 9 9 4
Points: 2 + 3 + 3 + 2 + 2 + 3 = 15

Activity Two

The students can play the game with these two grids several times. Ask them: “Where is the best place to put the first factor?”
When they have placed the first factor, they can work systematically to work out the highest possible score. For example, for the box above the left column in a, they need to find a number on the dice that is a factor of as many of the numbers in that column (18, 25, and 40) as possible:
6 x 3 = 18
9 x 2 = 18
5 x 5 = 25
4 x 10 = 40
5 x 8 = 40
None of the numbers on the dice is a factor of all three numbers in the column. Five is a factor of two of the numbers, 25 and 40. So the highest score that they can get for that box is two points if they put 5 in it.
As an extension, the students could try to design a matrix that will produce a result of more than 15 points or extend the size of the grid to a 4 x 4 matrix for further games.
To create a matrix that produces a score of more than 15 points means that the  numbers used in the grid must have more common factors. In grid b, the bottom left-hand corner has 41. But 41 is a prime number, so no score can be generated from this entry in the grid. An easy way to improve the possible top score for this grid would be to replace 41 with a number that has a factor shown on the dice. A profitable replacement would be 48 because this has 4, 6, and 8 as factors. On the
other hand, 84 is a bonus since it has 4, 6, 7, and 8 as factors!
The highest possible score that can be gained from a well-planned 3 x 3 grid is 18. You can achieve this by being able to score three points on every row and every column of the matrix.

Answers to Activities

Activity One
1. With the 6, 9, and 4 in their best boxes, the final grid should look like this:

2. Her total score is: 2 + 2 + 1 + 3 + 3 + 1 = 12.
(Gina gets 3 points from the 6, 3 from the 9, and 1 from the 4.)
Game
A game of factors

Activity Two

a. The highest possible score for a is 13. There are only four ways to get this:

b. There are 72 ways to get the highest possible score of 11 points. Two of these are: