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Introduction

Mathematics GCSE coursework: hidden faces              Tina Harris

Aim:

To find a rule in algebra to reveal how many hidden faces there are in a row of cubes, and to use dimensions and the number of shown faces to work out the number of hidden faces in cuboids.

Part 1-Investigating the number of hidden faces in rows of cubes

Introduction:

This part of the investigation is about how many hidden faces there are in a row of cubes. I will be investigating how to work out the number of hidden faces in a row of cubes in algebra, whilst using the number of cubes in a row in the expression, e.g. Number of hidden faces = number of cubesa specific number. I will also be showing how to work out the number of hidden faces in a row without counting, but instead by using an expression. To prove the expression works I will test and predict the amount of hidden face in a row of cubes by using the rule in algebra. If the rule works then I will be able to explain it and draw diagrams to test the rule in algebra.

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 rule:

Middle

In the rule above, you multiply the number of cubes by six because it gives the total number of faces on the cuboid. The reason for why x6 reveals this total is because each cube has 6 faces, e.g. if a 30-cubed cuboid is x6, you get the answer of 180; this is the total amount of faces on a cuboid made up of 30 cubes. You then take the total amount of shown faces on a cuboid away from the total amount of faces, so you then get the amount of hidden faces, e.g.30-cubed cuboid 180-47=133.

6 x number of cubes - number of faces showing=Number of hidden faces.

2.) 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 showing faces)

Test: I drew out a 12-cubed cuboid that shows all the hidden faces, which gave the same number of hidden faces as my predicted number of hidden faces. This means that my rule for working out the hidden faces of a cuboid is correct, but I had to count the shown faces, which took considerably long. This means that the rule still isn’t perfected, so

Conclusion

S= (W x L) + 2(H x W) + 2(L x H). When you use both these rules in one expression, you can work out the number hidden faces in a cuboid with out counting, but instead by using the dimensions of the cuboid.

For example, by using the final rule I was able to work out the number of hidden faces of a 30-cubed cuboid, (length=2, width=5, height=3) without counting, but by using dimensions:

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)) =hidden faces

4.) 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 predict 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))

H=6(2x4x3) – ((4x2) + 2(3x4) + 22x3))

(24x6) – (8+24+66) =hidden faces

144 – 98 = 46 hidden faces

Test: I drew out the 24-cubed cuboid with the length=2, width=4, height=3. I counted the total number of faces, (144) within the cuboid, and then I counted the number of shown faces on the cuboid, (98); the numbers were the same as my prediction. I then did the calculation to work out the hidden faces (144-98) according to the faces I had counted on the cuboid. The answer was 46 hidden faces; this was the same as my prediction. This proves that my final rule works and that you can work out the total number of faces and showing faces by using dimensions and not counting.

This student written piece of work is one of many that can be found in our GCSE Hidden Faces and Cubes section.

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