(A)
Results:
Rule: h=6n-s Key: h=hidden faces, n=number of cubes, s=seen faces
Explanation of the rule:
In the rule above it shows that you multiply the number of cubes by six because it gives us a total number of faces on the cuboid. The reason being for why x 6 gives us this total is because each cube has 6 faces, e.g. if a 24-cubed cuboid is x 6, you get the answer of 144; this is the total amount of faces on a cuboid made up of 24 cubes. Then you take the toal amount of seen faces on a cuboid away from the total amount of faces, so therefore you get the amount of hidden faces, e.g.24-cubed cuboid 144-40=104. (6 x number of cubes – number of faces showing=number of hidden faces)
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:
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:
To work out the number of seen faces you need to work out how many faces there are on each side of a cuboid. In the equation, to give the total amount of seen faces you multiply the total of 2 of the brackets by 2. This, in turn, gives the total amount of seen faces on two of the identical sides of the cuboid. The reason for not multiplying the first bracket (W x L) by 2 is 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 seen faces it has. The shorter the length, the less hidden faces, and more seen faces it has. The reason why there is more hidden faces when the length and width of a cuboid are longer is because there are more faces touching the table-the base of a cuboid. Even when the numbers of cubes remain the same the hidden faces change; the reason being is because the dimensions of a cuboid change depending on how you place the cuboid on the table. For example, by using dimensions of a cuboid, I was able to work out the seen faces of a 30-cubed cuboid, (length=5, width=3, height=2):
(3x5) + 2(2x3) +2(5x2) = 47
(W x L) + 2(H x W) +2(L x H) = Seen faces
Test and predict:
Predict:
Number of cubes=24, length=4, width=3, height=2 Using the rule and the dimensions of the cuboid, I predicted that there will be 40 seen faces.
S= (W x L) + 2(H x W) + 2(L x H) S= (3 x 4) + 2(2 x 3) + 2(L x H)
12+12=16=40 seen faces
Test: I sketched out a 24-cubed cuboid, by using the dimensions above, and by counting the amount of seen faces on the cuboid-this then gave me the same answer as my prediction, this shows that my rule was therefore correct. The alternative is by working out the amount of hidden faces the 24-cubed cuboid had once I knew the amount of showing faces the 24-cubed cuboid had once I knew the amount of seen faces of the cuboid; 24x6=144-40= 104 hidden faces.
(C)
Results:
Key: W=width, L=length, H=height
Explanation of the final rule:
The final rule came to h=6(L x W x H) – ((W x L) + 2(H x W) + 2(L x H)), this is a more detailed version of the rule h=6n-s. The purpose of this rule is to make it easier for you to work out the number of cubes and seen faces as you don’t have to spend all that time counting all the faces but use the dimensions of a cuboid instead to work it out. This rule works by taking away the total amount of showing faces away from the total amount of faces on every cube in a cuboid. This final rule consists of the rule h=6n-s, and the previous rule S= (W x L) + 2(H x W) + 2(H x W) + 2(L x H). When you use the two rules in one expression, you can find out the number of hidden faces in a cuboid without counting, but instead using the dimensions of a cuboid. E.g. by using the final I was able to find out number of hidden faces of a 30-cubed cuboid, (length=2, width=5, height=3) without counting, there fore 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))
(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’.
