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# Elementary School Math Problem 4

How Many Rectangles

by

Tweet## Andrea Ustick

on 27 March 2013#### Transcript of Elementary School Math Problem 4

How Many Rectangles? By: Andrea, Christina & David What about a 7x7 square? What if we had a 6x6 square? Going back to the 5x5 square... How many squares do you see in a 6x6 panel?

Can anyone find the General Rule for 6x6? How many rectangles can you find? How many rectangles can you find in this square? How can we solve this problem using the information we achieved in our 5x5 square? How many different rectangles can you find in this square? How can you use our work from the previous problem to answer this question? Can you Predict? How many rectangles are in an 8x8 square? Not by looking at a grid but looking at our patterns in our tables? The Square family saw a solar grid consisting of 25 small squares in a 5x5 grid. How many squares do we see? 4 2x2 squares 3x3 squares: 9 4x4 squares: 16 5x5 square: 25 Total squares: 55 1 1 2x2: 4 3x3: 9 4x4: 16 5x5: 25 6x6 square:36 Total Squares: 91 These are all the ares of rectangles that can be in a 5x5 square.___CHROME_BUG___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 Total: 225 rectangles How to get the number of rectangles for each area:

Try finding a method for counting the amount of rectangles in a 5x5 Square To find how many rectangles we can find in a 6x6 square, all we have to to do is follow what we learned from the 5x5 square and add a row of 6.

1x6 is 6 6x1 is 6

2x6 is 5 6x2 is 5

3x6 is 4 6x3 is 4

4x6 is 3 6x4 is 3

5x6 is 2 6x5 is 2

It's still 2x6 is 5 up x 1 across is 5 Total is 103 You follow the same pattern:

1x7 is 7 7x1 is 7

2x7 is 6 7x2 is 6 Total:130

3x7 is 5 7x3 is 5

4x7 is 4 7x4 is 4

5x7 is 3 7x5 is 3

6x7 is 2 7x6 is 2 1x8 is 8 8x1 is 8

2x8 is 7 8x2 is 7

3x8 is 6 8x3 is 6

4x8 is 5 8x4is 5

5x8 is 4 8x5 is 4

6x8 is 3 8x6 is 3

7x8 is 2 7x8 is 2 Total:165 "Different" Any individual rectangle is formed by 2 distinct points in a row matched with

2 distinct points in a column. It is choosing the number of ways of n points

in a row taken two at a time, multiplied by the number of ways of n points

in a column taken two at a time.

So, for the question of the total number of rectangles in a 4 X 4 square grid,

you are looking at a 5 x 5 grid of points (vertices of each

vertical and horizontal rectangle).

And (5 choose 2) multiplied by (5 choose 2) = 1x2 1x6

Full transcriptCan anyone find the General Rule for 6x6? How many rectangles can you find? How many rectangles can you find in this square? How can we solve this problem using the information we achieved in our 5x5 square? How many different rectangles can you find in this square? How can you use our work from the previous problem to answer this question? Can you Predict? How many rectangles are in an 8x8 square? Not by looking at a grid but looking at our patterns in our tables? The Square family saw a solar grid consisting of 25 small squares in a 5x5 grid. How many squares do we see? 4 2x2 squares 3x3 squares: 9 4x4 squares: 16 5x5 square: 25 Total squares: 55 1 1 2x2: 4 3x3: 9 4x4: 16 5x5: 25 6x6 square:36 Total Squares: 91 These are all the ares of rectangles that can be in a 5x5 square.___CHROME_BUG___d2cd99cc64cb05c3bf5e5080919ea7852af441deca7d47b0210c26e04d14eb45140e81de68e0043fc5becbcf22d842272a04759cac191115c99be632568ef9415d5e0da041c595cc113202c7c650bad5ac3e1048505964b3a3ce9e4f56a8e9b049abffa4af98382e343435373837323138333537323937652d31362220733d2231302e333933343238313734393437343938223e3c736f7572636520773d223130392220683d22313234223e3436353936363530323c75726c3e687474703a2f2f303130332e7374617469632e7072657a692e636f6d2e73332e616d617a6f6e6177732e636f6d2f636f6e7665727465642f622f392f322f313064663866373566636138663832323634623130616233316532356366643436613538612e6769663c2f75726c3e3c2f736f757263653e3c2f6f626a6563743e3c2f7a75692d7461626c653e Total: 225 rectangles How to get the number of rectangles for each area:

Try finding a method for counting the amount of rectangles in a 5x5 Square To find how many rectangles we can find in a 6x6 square, all we have to to do is follow what we learned from the 5x5 square and add a row of 6.

1x6 is 6 6x1 is 6

2x6 is 5 6x2 is 5

3x6 is 4 6x3 is 4

4x6 is 3 6x4 is 3

5x6 is 2 6x5 is 2

It's still 2x6 is 5 up x 1 across is 5 Total is 103 You follow the same pattern:

1x7 is 7 7x1 is 7

2x7 is 6 7x2 is 6 Total:130

3x7 is 5 7x3 is 5

4x7 is 4 7x4 is 4

5x7 is 3 7x5 is 3

6x7 is 2 7x6 is 2 1x8 is 8 8x1 is 8

2x8 is 7 8x2 is 7

3x8 is 6 8x3 is 6

4x8 is 5 8x4is 5

5x8 is 4 8x5 is 4

6x8 is 3 8x6 is 3

7x8 is 2 7x8 is 2 Total:165 "Different" Any individual rectangle is formed by 2 distinct points in a row matched with

2 distinct points in a column. It is choosing the number of ways of n points

in a row taken two at a time, multiplied by the number of ways of n points

in a column taken two at a time.

So, for the question of the total number of rectangles in a 4 X 4 square grid,

you are looking at a 5 x 5 grid of points (vertices of each

vertical and horizontal rectangle).

And (5 choose 2) multiplied by (5 choose 2) = 1x2 1x6