Here are my predictions for other rectangles with different lengths.
Rectangles No.2
I will now do the same as the first lot of rectangles but instead of (t) being 2 it will be 3.
Here is a table showing my results:
For the second column ( ) each rectangle has 4 because the symbol represents the corners and all rectangles have 4 corners. The nth term is n= 4
For the third column ( ) the numbers go up in 2’s because the length increases by one and an extra T-shape symbol has to be added to both sides. The nth term is (n+1) x2.
For the fourth Column (+) the numbers go up in 2’s because when the length is increased by one an extra 2 +’s are added to the middle. The nth term is (n-1) x2.
Here are my predictions for other rectangles with different lengths.
Rectangles No.3
I will now do the same as the first and second lot of rectangles but instead of (t) being 2 or 3 it will be 4.
Here is a table showing my results:
For the second column ( ) each rectangle has 4 because the symbol represents the corners and all rectangles have 4 corners. The nth term is n= 4
For the third column ( ) the numbers go up by 2’s because the length increases by one and an extra T-shape symbol has to be added to both sides. The nth term is (n+2) x2.
For the fourth column (+)the numbers go up by 3’s because when the length is increased by one an extra 3 +’s are added to the middle. The nth term is (n-1) x3.
Here are my predictions for other rectangles with different lengths.
I noticed with the first rectangles the nth term for + was n-1, for the second lot of rectangles it was (n -1) x2 and for the third lot (n -1) x3.
Also I noticed with the first rectangles the nth term for was n x 2, for the second lot of rectangles it was (n +1) x2 and for the third lot (n +2) x2
Plus with any rectangle at all will always be 4.
So with this bit of information I predict:
When t = 5, is n = 4, is (n + 3) x2 and + is (n – 1) x4
When t = 10, is n = 4, is (n + 3) x2 and + is (n – 1) x4.
Triangles
I will now move on to an even harder challenge doing triangles.
Here is a table showing my results:
For the second column ( ) each triangle has 3 because the symbol represents the corners and all triangle have 3 corners. The nth term is n= 3.
For the third column ( ) the numbers go up by 3’s because the length increases by one and an extra T-shape symbol has to be added to all 3 sides. The nth term is (n-1) x3.
For the fourth Column (*) the numbers go up in triangle numbers because when the length is increased by one an extra 3 *’s are added to the middle. The nth term is n-13.
Here are my predictions for other triangles with different lengths.
To find the nth term for which are triangles numbers I had to use this equation ax3+bx2+cx+4d .
Hexagons
I will now move on to something more complicated than triangles, hexagons.
Here is a table showing my results:
For the third column ( ) the numbers go up in 6’s because when each side increases it’s length by on an extra symbol is needed on all of the 6 sides. The nth term is n x 6.
For the fourth Column ( ) the numbers go up 12 then 24 then 36, every time the nth number goes up the number increases by an extra 12.
Here are my predictions for other hexagons with different lengths.
Cubes
I will now move from 2-d shapes to 3-d shapes.
Here is a table showing my results:
For the second column ( ) each cube has 8 because the symbol represents the corners and all cubes have 8 corners. The nth term is n= 8.
For the third column ( ) the amount goes up in 8’s because an extra symbol is needed 1 more time on each of the 8 sides. The nth term for this is (n-1) x12.
For the fourth Column ( ) the numbers go up 6 then 18 then 30, every time the nth number goes up the number increases by an extra 12.
For the fifth column the numbers cubed numbers because when you increase the lengths of the cube the amount of go up by n – 2. The nth term is (n-1)3 .
Here are my predictions for other cubes with different lengths.
Cuboids
Another example of 3-d shapes are the cuboids.
Here is a table showing my results:
For the second column ( ) each cuboid has 8 because the symbol represents the corners and all cubes have 8 corners. The nth term is n= 4.
For the third column ( ) the numbers go up in 4’s because an extra symbol is needed 1 more time on each of the 4 sides. The nth term for this is (n +1) x4.
For the fourth column the numbers ( ) the numbers go up in 4’s because an extra symbol is needed 1 more time on each of the 4 sides. The nth term for this is (n x 4) -2.
For the fifth column the numbers go up in ones because an extra symbol is added to the middle when the cuboid size increases. The nth term is n-1
Here are my predictions for other cubes with different lengths.
Cuboids 2
Another set of cuboids.
Here is a table showing my results:
For the second column ( ) each cuboid has 8 because the symbol represents the corners and all cubes have 8 corners. The nth term is n= 8.
For the third column ( ) the numbers go up in 4’s because an extra symbol is needed 1 more time on each of the 4 sides. The nth term for this is (n +3) x4.
For the fourth column the numbers ( ) the numbers go up in 4’s because an extra symbol is needed 1 more time on each of the 4 sides. The nth term for this is n x 8.
For the fifth column the numbers go up in ones because an extra symbol is added to the middle when the cuboid size increases. The nth term is (n+1) x4.
Here are my predictions for other cubes with different lengths.
Cuboid 3
Another set of cuboids.
Here is a table showing my results:
For the second column ( ) each cuboid has 8 because the symbol represents the corners and all cubes have 8 corners. The nth term is n= 8.
For the third column ( ) the numbers go up in 4’s because an extra symbol is needed 1 more time on each of the 4 sides. The nth term for this is (n +5) x4.
For the fourth column the numbers ( ) the numbers go up in 4’s because an extra symbol is needed 1 more time on each of the 4 sides. The nth term for this is (nx12) +6.
For the fifth column the numbers go up in ones because an extra symbol is added to the middle when the cuboid size increases. The nth term is (n-1) x9.
Here are my predictions for other cubes with different lengths.