Investigating The Answer When The Products Of Opposite Corners on Number Grids Are Subtracted
Introduction
The purpose of this investigation is to explore the answer when the products of opposite corners on number grids are subtracted and to discover a formula, which will give the answer in all cases. I hope to learn some aspects of mathematics that I previously did not know.
The product is when two numbers are multiplied together.
There is one main rule: the product of the top left number and the bottom right number must be subtracted from the product of the top right and bottom left numbers. It cannot be done the other way around.
I plan to achieve my aim by attempting a series of different
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(15 x 19) – (13 x 21) 285 – 273 = 12 Answer = 12 (The same) 13
14
15
16
17
18
19
20
21
The formula will not be the same for these. There will be a different formula for each kind. So the easiest way is to substitute the numbers in the grid for 1,2,3,4 and so on. Then you can use the formula for that type of grid to work out the answer. However this will only be easy for smaller number grids.
Sequences
Now I have tried grids with number sequences. First where the numbers increase by 2.
Text Box: (4 x 6) – (2 x 8) 24 – 16 = 8 Answer = 8 The answer is four times that of the previous 2 x 2 grids I have done. 2 x 2 Grid
2
4
6
8
I have checked this by doing another 2 x 2 grid but with different numbers.
Text Box: (5 x 7) – (3 x 9) 35 – 27 = 8 Answer = 8 (The same) 2 x 2 Grid
3
5
7
9
I have now done a 3 x 3 grid just to check this rule.
Text Box: (6 x 14) – (2 x 18) 84 – 36 = 48 Answer = 48 The answer to the previous 3 x 3 grid was 12 and 4 x 12 =
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In a 2 x 2 grid with consecutive numbers the answer is 2.
The formula however will not be exactly the same. Instead of having D it will be replaced by W, because it is a square grid.
It will be:
W (W²) + W - W² - W²
= W³ + W - 2W²
= 2³ + 2 - 2 x 2²
= 8 + 2 - 8
= 2 : Which is correct.
I have checked this with a 4 x 4 grid. The answer should be 36.
W³ + W - 2W²
= 4³ + 4 - 2 x 4²
= 64 + 4 - 32
= 68 - 32
= 36: Which is correct
Therefore this formula does work.
Even though the formulas I got before I simplified them look different, when simplified they are the same:
N (1 + N² - N) - N²(1 + N - N)
= N³ - 2N² + N
W (W²) + W - W² - W²
= W³ - 2W² + W
Decimals
Now I have decided to investigate decimals in number grids.
Text Box: (0.2 x 0.3) – (0.1 x 0.4) 0.06 – 0.04 = 0.02 Answer = 0.02 0.1
0.2
0.3
The lesson is about knowing the concept of place value, and to familiarize first grade students with double digits. The students have a daily routine where they place a straw for each day of school in the one’s bin. After collecting ten straws, they bundle them up and move them to the tens bin. The teacher gives a lecture on place value modeling the daily routine. First, she asks a student her age (6), and adds it to another student’s age (7). Next, she asks a different student how they are going to add them. The students respond that they have to put them on the ten’s side. After, they move a bundle and place them on the ten’s side. When the teacher is done with the lesson, she has the students engage in four different centers, where they get to work in pairs. When the students done at least three of the independent centers, she has a class review. During the review she calls on different students and ask them about their findings, thus determining if the students were able to learn about place value.
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This is the product difference of 2 x 2 in a 5 x 5 grid.
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