Test: I drew a diagram showing that a row of 10 cubes has 28 hidden faces, so yet again this confirms the fact that the rule h=3n-2 is correct.
2nd Rule: h=2(n-1)+n
Key: h=hidden faces, n=number of cubes
Predict:
Number of cubes=11
Using this rule, I predict that for 11 cubes there will be 31 hidden faces.
2(11-1) =22-2=20+11=31
Test: I drew another diagram showing that a row of 11 cubes has 31 hidden faces, so this confirms the fact that the rule 2(n-1) +n also works.
Explanation of rule:
The same rules and explanations apply to this rule as the first rule- it is just written in a more simplified way with brackets. You still have to x the number of cubes in a row by 3 and –2 because the end of a row has 2 hidden faces instead of 3.
Part 2
Introduction:
This part of the investigation is about how many hidden faces there are in cuboids made up of individual cubes. The investigation will show how the amounts of hidden faces in cuboids change just by the way you place a cuboid on the table (change in dimensions). At first, I will be using simple expressions in algebra to calculate how many hidden faces there are in a cuboid, but I will have to develop a more advanced expression if I do not want to count the amount of seen and unseen faces. I will also try to investigate how dimensions, (width, length and height) and algebra can make an expression that will easily calculate the total amount of hidden faces without you having to count the seen and unseen faces on a cuboid. To prove that the various expressions are correct I will test and predict the amount of hidden faces in a cuboid by using the rules in algebra. If the rules work then I will be able to explain them and draw diagrams to test the expressions.
(a) Part 2- Investigating the number of hidden faces in cuboids
Results:
Rule: h=6n-s
Key: h=hidden faces, n=number of cubes, s=shown faces
Explanation of rule:
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 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 of a cuboid faster and easier.
(b) Part 2- Investigating the number of hidden faces in cuboids
Results:
Rule: S= (W x L) + 2(H x W) + 2(L x H)
Key: S=showing faces, W=width, L=length, H=height
Explanation of rule:
To work out the number of showing faces you need to work out how many faces each side of the cuboid show. In the equation, to give the total amount of showing faces, you must multiply the total of 2 of the brackets by 2. This, in turn, gives the total amount of shown faces on two of the identical sides of the cuboid. You don’t multiply the first bracket (W x L) by 2 because the faces on the base of the cuboid are hidden-the side of the cuboid that touches the table.
The longer the length of a cuboid, the more hidden faces, and less showing faces it has. The shorter the length, the less hidden faces, and more showing faces it has. When the length and width of a cuboid are longer there are more hidden faces because there are more faces touching the table-the base of the cuboid. Even when the number of cubes stays the same the hidden faces change; this is because the dimensions of a cuboid change depending on how you place the cuboid on the table. For example, by using the dimensions of a cuboid, I was able to work out the shown faces of a 30-cubed cuboid, (length=5, width=3, height=2):
(3 x 5)+ 2(2x3) + 2(5x2) = 47
(W x L)+ 2(H x W) + 2(L x H) = showing faces
3.) Test and predict:
Predict: Number of cubes=24, length=4, width=3, height=2
Using the rule and the dimensions of the cuboid, I predict that there will be 40 showing faces.
S= (W x L) + 2(H x W) + 2(L x H)
S= (3x4) + 2(2x3) + 2(4x2)
12+12+16=40 showing faces
Test: I drew out the 24-cubed cuboid, whilst using the dimensions above, and counted the amount of shown faces on the cuboid-this gave me the same answer as my prediction, which proves that the rule is correct. I could also work out the amount of hidden faces the 24-cubed cuboid had once I knew the amount of showing faces of the cuboid: 24x6=144 – 40=104 hidden faces.
(c) Part 2- Investigating the number of hidden faces in cuboids
Results:
Key: w=width, l=length, h=height
Explanation of final rule:
The final rule, h=6(l x w x h) – ((w x l) + 2(h x w) + 2(l x h)), is a more detailed version of the rule h=6n-s. It uses dimensions so that you don’t have to count out the number of cubes and showing faces, but instead use the dimensions of a cuboid to work out the total number of hidden faces. This rule works because all you are doing is taking away the total amount of showing faces away from the total amount of faces on every cube in a cuboid. The final rule consists of the rule h=6n-s, and the previous rule 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.