# Mayfield High School

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Introduction

Jamie B

Maths Coursework - Hidden Faces

Results:

Number of cubes in a row | Number of hidden faces |

1 | 1 |

2 | 4 |

3 | 7 |

4 | 10 |

5 | 13 |

6 | 16 |

7 | 19 |

8 | 22 |

Rule: h=3n–2

Key: h=hidden faces, n=number of cubes

Explanation of the rule:

The way in which I worked out my rule was by looking at the number of hidden faces on the row of cubes. For each cube in the centre of the row, there are three faces seen and three faces hidden so that’s why the first part of my rule was 3n. At the end of each row there are only 2 hidden faces so that’s the reason you have to subtract -2 from the total number of hidden faces.

Test and predict:

Predict:

Number of cubes=7

Using the rule, I predict that for 7 cubes there will be 19 hidden faces.

7x3=21-2=19

Test:

I used 3-D cubes (7 cubes) and counted the hidden faces, which gave the same answer as my prediction, which proves that my rule is correct

Predict:

Number of cubes=8

Using the rule, I predict that for 8 cubes there will be 22 hidden faces.

8x3=24-2=22

Test: I used 3-D cubes (8 cubes this time) and counted the hidden faces, So yet again the prediction I made was correct and shows this was a consistent rule.

Diagrams:

Middle

Test and predict:

Predict:

Number of cubes in cuboid =12 Using the rule, I predict that for a 12 cubed cuboid there will be 46 hidden faces.

6x12=72, 72-26-46 (Counted up seen faces)

Test:

I sketched out a 12-cubed cuboids that shows all the hidden faces, which gave the same number of hidden faces as my predicted as my predicted number of hidden faces. Therefore my rule for working out the amount of hidden faces of a cuboid is correct, however I had to count the shown faces, which took me so long. This means that the rule I have is still not to see prefect, so therefore I’m going to use the expression h=6n-s as well as dimensions to work out another equation, which makes calculating the hidden and showing faces so much quicker and easier.

(B)

Results:

Number of cubes | Hidden faces | Seen faces | Length | Width | Height |

30 | 133 | (3x5)+2(2x3)+2(5x2)= 47 (2 identical sides) | 5 | 3 | 2 |

30 | 128 | (5x2)+2(3x5)+2(2x3)=52 | 2 | 5 | 3 |

30 | 124 | (2x3)+2(5x2)+2(3x5)=56 | 3 | 2 | 5 |

36 | 162 | (3x6)+2(2x3)+2(6x2)=54 | 6 | 3 | 2 |

36 | 156 | (6x2)+2(3x6)+2(2x3)=60 | 2 | 6 | 3 |

36 | 150 | (2x3)+2(6x2)+2(3x6)=66 | 3 | 2 | 6 |

The calculations above show how you would go around working out the seen faces on a cuboid, by using dimensions.

Key: S=showing faces, W=width, L=length, H= height

Explanation of rule:

Conclusion

6(2x5x3) – ((5x2) + 2(3x5) = 2(2x3)) = 128

6(L x W x H) – ((W x L) + 2(H x W) + 2(L x H))

(24 x 6) – (8 + 24 + 66) = hidden faces

144 – 98 = 46 hidden faces

Test and predict:

Predict:

Number of cubes=24, length=2, width=4, height=3 Using the final rule and the dimensions of the cuboid, I predicted that there will be 46 hidden faces:

H=6(L x W x H) – ((W x L) + 2(H x W) + 2(L x H)) = 128 H=6(2 x 4 x 3) – ((4 x 2) + 2(3 x 4) + 22x3)) (24 x 6) – (8+24+66) = hidden faces 144-98 = 46 hidden faces

Test:

I sketched out the 24-cubed cuboid with the length=2, width =4, height=3. I counted the total number of faces, (144) within the cuboid, I then counted the number of seen faces on the cuboid, (98); the numbers were the same as my prediction. I then worked out the calculation of the hidden faces (144-98) according to the faces I had on the cuboid. 46 hidden faces was the answer; this was the same according to the prediction I made. This proves that my final rule works and that you can work out the total number of faces and seen faces the easy way by using dimensions and not using the long method ‘counting’.

This student written piece of work is one of many that can be found in our GCSE Height and Weight of Pupils and other Mayfield High School investigations section.

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