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Grade 8 - patterns investigation - Rene Rix

Module by: Pinelands High School. E-mail the author

Summary: An investigation of patterns of squares and triangles

Grade 8: Mathematics: Investigation one: patterns

General instructions

This paper consists of SIX tasks. Attempt all tasks.

You may use a calculator.

Work with a partner. Write your answers in pencil. Submit ONE completed set of answers per pair.

Task 1: Matchstick triangles

Consider the following pattern created with matches:

Figure 1
Figure 1 (graphics1.png)

Activity 1.1:

Use the diagram above to help you complete the table below. You may draw additional sketches if you would like to.

Table 1
t (number of triangles) 1 2 3 4 6 n 20
m(number of matches) 3 6          

Information section

Flow diagrams

One way to represent the relationship between the number of triangles and the number of matches is with a flow diagram, like the one given below.

Figure 2
Figure 2 (graphics2.png)

The flow diagram shows us that if we multiply the number of triangles by three we find the number of matches in the construction.

Equations

This flow diagram may be represented as an equation (or formula): m=t×3m=t×3m = t times 3.

Instead of t×3t×3t times 3 or 3×t3×t3 times t we can write 3t3t3t and then the equation becomes: m=3tm=3tm = 3t.

We can use this equation to determine the number of matches required to build a row of 50 triangles. We know that m=3tm=3tm = 3t and t=50t=50t = 50. So we substitute 50 in place of t and then we can calculate the value of m:

m=3(50)=150m=3(50)=150m = 3(50) = 150, therefore 150 matches would be required.

Working in reverse

Working in reverse we have the flow diagram below which shows how to work out the number of triangles that can be formed given the number of matches.

Figure 3
Figure 3 (graphics3.png)

The flow diagram shows us that if we divide the number of matches by three we get the number of triangles we could create in the construction.

This flow diagram may be represented as an equation: t=m÷3t=m÷3t = m div 3 or t=m3t=m3t = m over 3.

We can use the equation to determine the maximum number of triangles we could build with 210 matches.

We know that t=m3t=m3t = m over 3 and m = 210. So we substitute the value of 210 in the place of m and then we can calculate the value of t.

t=2103=70t=2103=70t = 210 over 3 = 70 . So we could create 70 triangles.

Activity 1.2:

Use the equations given in the information section to assist you in completing the following table:

Table 2
t(number of triangles) 45  
m(number of matches)   600

Task 2: Matchstick squares

Consider the following pattern created with matches.

Figure 4
Figure 4 (graphics4.png)

Activity 2.1

Use the diagram to help you complete the following table. You may draw additional diagrams to help you.

Table 3
s(number of squares) 1 2 3 4 6 n 20
m(number of matches) 4 8          

Activity 2.2

Complete the following flow diagrams, representing the relationship between the number of squares and the number of matches.

Figure 5
Figure 5 (graphics5.png)
Figure 6
Figure 6 (graphics6.png)

Activity 2.3

  1. 1) Write down the equation/ formula for calculating the number of matches required if we want to create s squares. Write it in the form m = …
  2. 2) Write down the equation/ formula for calculating the maximum number of squares we could create if we have m matches. Write it in the form s = …

Activity 2.4

Use your equations/ formulae from the previous activity to assist you in completing the following table:

Table 4
s(number of square) 36  
m(number of matches)   480

Task 3: Linking matches

Consider the following pattern created with matches.

Matches have been joined together with blobs of prestik.

Figure 7
Figure 7 (graphics7.png)

Activity 3.1

Use the diagram above to help you fill in the table below. You may draw additional sketches if you wish.

Table 5
m(number of matches) 1 2 3 4 6 n 20
b(number of blobs) 0 1          

Activity 3.2

Complete the following flow diagrams, representing the relationship between the number of matches and the number of blobs.

Figure 8
Figure 8 (graphics8.png)

Activity 3.3

Write down the equation/ formula for calculating the number of blobs required if you want to link m matches. Write it in the form b = …

Write down the equation/ formula for calculating the maximum number of matches you can join if you have b blobs. Write it in the form m = …

Activity 3.4

Use your equations/ formulae form the previous activity to help you fill in the following table.

Table 6
m(number of matches) 55  
b(number of blocks)   127

Task four: More matchstick triangles

Consider the following pattern created with matches:

Figure 9
Figure 9 (graphics9.png)

Activity 4.1

Use the diagram above to help you fill in the table below. You may draw additional sketches if you wish.

Table 7
t(number of triangles) 1 2 3 4 6 n 20
m(number of matches) 3 5          

Information section

More complex flow diagrams

One way to represent the relationship between the number of triangles and the number of matches is with a flow diagram, like the one given below. Two steps are required to determine the number of matches required to create a certain number of triangles.

Figure 10
Figure 10 (graphics10.png)

The flow diagram shows us that if we multiply the number of triangles by two and add one to our answer we find the number of matches in the construction.

Equations

This flow diagram may be represented as an equation: m=t×2+1m=t×2+1m = t times 2 + 1.

Instead of t×2t×2t times 2 or 2×t2×t2 times t we can write 2t and the equation becomes: m=2t+1m=2t+1m = 2t + 1.

We can use the equation to determine the number of matches required to build a row of 50 triangles. We know that m=2t+1m=2t+1m = 2t + 1 and t = 50. So:

m=2(50)+1=101m=2(50)+1=101m = 2(50) + 1 = 101, so 101 matches would be required.

Working in reverse

Working in reverse we have the flow diagram below, which shows how to work out the number of triangles that can be formed with a certain number of matches.

Figure 11
Figure 11 (graphics11.png)

The flow diagram shows us that if we subtract 1 from the number of matches and then divide the answer by two to get the number of triangles we could create in the construction.

The flow diagram may be represented as an equation: t=(m−1)÷2t=(m−1)÷2t = (m - 1) div 2 or t=m−12t=m−12t = {m - 1} over 2.

We can use the equation to determine the maximum number of triangles we could build with 211 matches. We substitute 211 in place of m and then calculate t:

t=211−12=2102=105t=211−12=2102=105t = {211 - 1} over 2 = 210 over 2 = 105.

So we could create 105 triangles.

Activity 4.2

Use the equations given in the information section to assist you in completing the following table:

Table 8
t(number of triangles) 70  
m(number of matches)   253

Task five: More matchstick squares

Consider the following pattern created with matches:

Figure 12
Figure 12 (graphics12.png)

Activiy 5.1

Use the diagram to help you complete the following table. You may draw additional diagrams to help you.

Table 9
s(number of squares) 1 2 3 4 6 n 20
m(number of matches) 4 7          

Activity 5.2

Complete the following flow diagrams, representing the relationship between the number of squares and the number of matches.

Figure 13
Figure 13 (graphics13.png)

Activity 5.3

  1. 1) Write down the equation/ formula for calculating the number of matches required if we want to create s squares. Write it in the form m = …
  2. 2) Write down the equation/ formula for calculating the maximum number of squares we could create if we have m matches. Write it in the form s = …

Activity 5.4

Use your equations/ formulae from the previous activity to assist you in completing the following table:

Table 10
s(number of square) 234  
m(number of matches)   463

Task 6: Matchstick houses

Consider the following pattern created with matches.

Figure 14
Figure 14 (graphics14.png)

Activity 6.1

Use the diagram above to help you fill in the table below. You may draw additional sketches if you wish.

Table 11
m(number of matches) 1 2 3 4 6 n 20
b(number of blobs) 6 11          

Activity 6.2

Complete the following flow diagrams, representing the relationship between the number of matches and the number of blobs.

Figure 15
Figure 15 (graphics15.png)

Activity 6.3

Write down the equation/ formula for calculating the number of matches required if you want to create h houses. Write it in the form m = …

Write down the equation/ formula for calculating the maximum number of houses you can create if you have m matches. Write it in the form h = …

Activity 6.4

Use your equations/ formulae form the previous activity to help you fill in the following table.

Table 12
h(number of houses) 120  
m(number of matches)   456

Extension Activities

Extension activity one

Investigate the following relationships in the pattern shown below.

Figure 16
Figure 16 (graphics16.png)

  1. 1) Investigate the relationship between the number of matches along one side of the big square and the number of small squares (with sides one matchstick in length) formed inside the biggest square.
Table 13
m(length of side of big square) 1 2 3 4 6 n 20
s (number of small squares inside) 1 4          
  1. 1) Investigate the relationship between the number of matches along one side of the big square and the number of matches required altogether.
Table 14
m(length of side of big square) 1 2 3 4 6 n 20
t(total number of matches) 4 12          

Extension activity two

Investigate the following relationships in the pattern below.

Figure 17
Figure 17 (graphics17.png)

1) Investigate the relationship between the number of matches along one side of the big triangle and the number of small triangles (with sides one matchstick in length) formed inside the biggest triangle.

Table 15
m(length of side of big triangle) 1 2 3 4 6 n 20
s (number of small triangles inside) 1 4          

2) Investigate the relationship between the number of matches along one side of the big triangle and the number of matches required altogether.

Table 16
m(length of side of big triangle) 1 2 3 4 6 n 20
t(total number of matches) 3 9          

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