Date: 03/06/97 at 16:28:11 From: Michael McDonel Subject: Quadrilaterals...how to find a pattern when increasing the size of a grid by one unit (each side) We are studying quadrilaterals in 4th grade and I cannot seem to find a pattern when you increase a square grid by one unit on each side. For instance, a 1x1 grid has one quadrilateral, a 2x2 has 9, and so forth. And when you start asking about 5x5's and 6x6's, it really gets complicated. Please help in any way you can. Thank you.
Date: 04/30/97 at 03:31:01 From: Doctor Mike Subject: Re: Quadrilaterals...how to find a pattern when increasing the size of a grid by one unit (each side) Hello Michael, Your problem has made me think, and the answer turns out to be pretty simple. So here goes. I will show most of the details of the proof that my answer is right. This is not often done in 4th grade, but I believe you can take it. By a quadrilateral, I think you mean one with horizontal and vertical sides that go along the grid lines. These are like the rectangles you can make using some of the squares on a checker or chess board. I have a formula for the number of rectangles in an NxN grid, but I want to start you out thinking with me about something similar, but which is simpler. Take the same question as yours, but instead of a square grid use a grid that is 1xN, that is, just a straight string of N grid squares like this: _____________________________________ | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | |_____|_____|_____|_____|_____|_____| This is a 1x6 grid, and it will help to look at this in detail. How many rectangles are there in this grid with 6 squares? Yes, just one. How many rectangles are there in this grid with 5 squares? I count that there are 2 of those, the one consisting of the left five squares and the one consisting of the right five squares. Now it gets a bit more complicated. How many rectangles are there with 4 squares? I will list them all: the left 4, the middle 4, and the right 4, for a total of 3. Do you see a trend yet? Let's make a table of values. How many squares in a rectangle How many different rectangles --------------------------------|------------------------------ 6 | 1 5 | 2 4 | 3 3 | 4 2 | 5 1 | 6 ------- | ---- Total | 21 The total number of possible rectangles you can make is 1+2+3+4+5+6, which is 21. If it were a 1xN grid the total is 1+2+3+4+....+N or the sum of all numbers from 1 to N. There is a formula, which I will not prove here because this e-mail is already long, that this sum is equal to N*(N+1)/2. In our case N = 6 so the formula is 6*7/2 = 42/2 = 21. Now, finally, let's get to YOUR problem. We want to count the number of rectangles we can make on an NxN grid. Let's imagine one. Let's imagine the grid is an 8x8 chess board, and the rectangle is 5 squares wide and 3 squares high. This 5x3 rectangle can be located anywhere on the chess board, so there are many possible locations. How can we possibly count them all? How? We can use what we already know from the simpler case! See if you can answer these two questions about rectangles on the chess board: Q1 -- For an AxB rectangle on the chess board, how many ways are there to choose both the number "A", and also the location (left to right) on the chess board? Q2 -- For an AxB rectangle on the chess board, how many ways are there to choose both the number "B", and also the location (up and down) on the chess board? The answers to these two questions are the same, since you can count all the possibilities in exactly the same way as we did the simple 1xN example. Both answers are 1+2+3+4+5+6+7+8 = (8*9)/2 = 72/2 = 36. Now for the final step. Because there are 36 ways to choose A and place the AxB rectangle left/right, and because there are 36 ways to choose B and place the AxB rectangle up/down, that is a total of 36*36 or 36 squared ways to choose the size and exact location of a rectangle. Now for the general version of the formula: If you have an NxN grid of squares, then the number of possible rectangles you can draw in that grid is the square of the number N*(N+1)/2. You have waited a really long time for this answer, so I will give you a bonus, namely, the more general formula for when the whole grid itself is a rectangle. If you have an NxM grid of squares, then the number of rectangles that can be drawn on that grid is given by the following expression: N*(N+1) M*(M+1) --------- * --------- = N*(N+1)*M*(M+1)/4 2 2 For example, if N = 5 and M = 3, then there are 90 possibilities! I will leave you with two thoughts. First, the formula for adding up the first N counting numbers can be proven by a method known as mathematical induction. You might want to look that up in a few years. Second, this e-mail would probably be considered complicated by most high school students. You may have to read it several times before it starts to come together for you. Please do that. I think you will enjoy thinking about this problem and solution. Also, I suggest you get out a real physical chess (or checker) board, and some paper and scissors to cut out rectangles of different shapes and move them around the grid of the board. Math may seem mental, but some of the best mathematicians have a good understanding of how numbers connect with the physical real world. This is especially true for geometry and for applying math to solve real world problems. I hope this helps. -Doctor Mike, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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