I am going to multiply the numbers from the opposite corners of the square, then find the difference.

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The difference is 90, I am now going to investigate other 4 x 4 squares. I have tried 3 examples of 4 x 4 squares in a 10 x 10 grid they and the differences of the products of the opposite corners always seems to equal I am going to try a different sized square in a 10 x 10 grid.

First I will try a 2 x2 square.

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I have tried 3 examples of 2 x 2 squares in a 10 x 10 grid the difference is always Now I will try a 3 x 3 square in a 10 x The difference of the products of the opposite corners is always 40 for a 3 x 3 square. Now I will try a 5 x 5 square. The difference of the products of the opposite corners is always in a 5 x 5 square. Now I will try a 6 x 6 square. The difference of the products of the opposite corners of a 6 x 6 is always I am going to put all the results that I have collected in a table. I have noticed from the results that the differences are all square numbers multiplied by 10, eg.

Also I have noticed that the square number which is multiplied by 10 is 1 less than the length of the square, eg. From this I can work out a formula for the difference of the products of the opposite corners of any size square in a 10 x 10 grid. This formula shows how to find the difference of any products of opposite corners of any size square in a 10 x 10 grid.

### Pond Borders - Mark Greenaway

I will use n to represent the original number in top left of the square inside the grid. It is always the smallest number in the square. To work out the difference of the products of the opposite corners you have to multiply the opposite numbers in the opposite corners by each other then subtract.

I will write these numbers in terms of n. Now I am going to investigate what will happen if I change the grid sixe. First I will try a 5 x 5 grid. First I will try a 2 x 2 square. The difference for a 2 x 2 square in a 5 x 5 grid always seems to be 5. Now I will try a 3 x 3 square. The difference always seems to be 20 for a 3 x 3 square in a 5 x 5 grid.

Now I will try a 4 x 4 square.

## Free S1 & S2 Maths

The difference is always I am going to compare the results of the 10 x 10 grid to the results form the 5 x 5 grid. I can see from this table that the differences of the products of the opposite corners from the 5 x 5 grid are half of the differences from the 10 x 10 grid. This is because the 5 x 5 grid is half the size of the 10 x Now I am going to try a different size grid, first I will try a 6 x 6 grid, then a 7 x 7 grid. The difference is always 6 in a 2 x 2 square in a 6 x 6 grid. The difference of the product of the opposite corners of a 3 x 3 square in a 6x 6 grid is always Now I am going to change the grid size to 7 x 7.

The difference is always 28 in a 3 x 3 square in a 7 x 7 grid. The difference is always 63 in a 4 x 4 square in a 7x 7 grid. Now I have tried some different sized grids I am going to put the results into a table. I think the formula to work out the difference of the products of the opposite corners for any square in any sized grid, is connect I found for any sized square in a 10 x 10 grid; 10 n — 1.

I have worked out that the formula is n — 1 x Grid size. I will check the formula by putting in some numbers.

To prove the formula works these numbers of the example with algebra. I have highlighted it red in the grid above. This is the same answer that I got in the example above, and this is also the same answer that is in the grid. To further the investigation I am investigate the difference of the products of the opposite corners of rectangles inside a square grid. First I will investigate different sized rectangles inside a 10 x 10 grid. The difference of the products of the opposite corners is always 20 in a 2 x 3 rectangle in a 10 x 10 grid.

The difference of the products of the opposite corners is always 30 for a 2 x 4 rectangle in a 10 x 10 grid. The difference of the products of the opposite corners is always 40 in a 2 x 5 rectangle in a 10 x 10 grid. The difference of the products of the opposite corners is always 60 in a 3 x 4 rectangle in a 10 x 10 grid.

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The difference of the products of the opposite corners is always in a 5 x 4 rectangle in a 10 x 10 grid. The difference of the products of the opposite corners is always in a 5 x 6 rectangle in a 10 x 10 grid. I will put my results into a table. I have noticed that if you subtract 1 from each length of the rectangle then multiply them both by 10 then you get the difference. Counting: How many combinations are possible given certain conditions.

For example, if there are eight chairs and eight guests, how many orders could the guests sit in? Elementary Number Theory: Properties of integers, factorization, prime factors, etc. Properties of Functions: You'll need to be able to identify the following kinds of functions and understand how they work, how they look when graphed, and how to factor them. Polynomial: Functions in which variables are elevated to exponential powers.

Rational: Functions in which polynomial expressions appear in the numerator and the denominator of a fraction. You could also skip standardized testing and go live alone in the desert. Note that plane geometry concepts are addressed on Math 2 via coordinate and 3-D geometry. Coordinate: Equations and properties of ellipses and hyperbolas in the coordinate plane and polar coordinates.

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Three-Dimensional: Plotting lines and determining distances between points in three dimensions. You must know how to convert to and from degrees. Law of Cosines and Law of Sines: Trigonometric formulas that allow you to determine the length of a triangle side when one of the angles and two of the sides are known. You'll need to know the formulas and how to use them. Double Angle Formulas: Formulas that allow you to find information on an angle twice as large as the given angle measure. Logarithmic: Functions that involve taking the log of a variable.

Trigonometric Functions: Graphs of sine, cosine, tangent, etc. Inverse Trigonometric Functions: Graphs of the inverse of sine, cosine, tangent, and other trig identities. Periodic: Any function that repeats its values over an interval; trigonometric functions are periodic.

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However, Math 2 also tests more advanced versions of the topics tested on Math 1. It leaves off directly testing plane Euclidean geometry, though the concepts are indirectly tested through coordinate and 3-D geometry topics. Math 2 also covers a much broader swath of topics than Math 1 does. This means that question styles for Math 2 and Math 1 can be pretty different, even though many of the same topics are addressed see the next section for elaboration on this. Given that Math 2 covers more advanced topics than Math 1 does, you might think that Math 1 is going to be the easier exam.

But this is not necessarily true. Since Math 1 tests fewer concepts, you can expect more abstract and multi-step problems to test the same core math concepts in a variety of ways.

The College Board needs to fill up 50 questions, after all! Below is an example of a tricky question you might see on the Math 1 test.