Elementary POW, January 8-12, 1996
Elementary POW Problems || January-March, 1996 Problems || Elementary POW Main Page
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Elementary Problem of the Week, January 8-12, 1996
How many squares of different sizes are there in a 7x7 square?
Hint: Using graph paper, mark off a 7x7 square region (49 small squares).
Use the table below to record the number of squares of different sizes in
squares of size 1x1, 2x2, 3x3, etc. to help find the solution.
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Number of Squares of Different Sizes
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1x1 2x2 3x3 4x4 5x5 6x6 7x7 Total
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Size of Square
1x1 1 - - - - - - 1
2x2
3x3
4x4
5x5
6x6
7x7
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Look for patterns in your chart. Explain any patterns that you've found.Can
you use any of these patterns to find the number of squares in an 8x8 and a
9x9 square without actually having to count them?
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Hi all, I just wanted to comment on some of the solutions sent. Most of you are getting much better at explaining how you arrived at your answer. Remember, pretend that you are writing your solution for another student who could not figure out the problem. Let them know what you were thinking each step of the way. I would like to give special recognition to Mrs. Pensa's 3rd Grade Class - Center School, Stow, MA. Your students are a real credit to you. Their explanations were great! I was especially impressed by the solution and diagram sent by Kristin Squires. Keep up the great work! -Ruth ********************* Center School, Stow, MA Mrs. Pensa's 3rd Grade Class Kristin Squires, Chaz Rosenberg, Mrs Palmbach's 3rd grade class Center School - Stow, MA Joanna Brench Mr Ellsworth's 4th grade class Center School - Stow, MA Sam DeLuca 5th Grade Mrs. Bach Joyce Kilmer School Mahwah, NJ Spencer Singer, Mike Tybursky, and Cheryl Verblaauw Jeremy Wolland, captain Will Gaybrick, co captain Linda Prueter's 5th grade Class Georgetown Day School Washington D.C. Underhill School Maidstone, Kent, England Mr. Grant Whitaker's class Holly Stamp,Annabel Bates, Elizabeth Ireland, Richard Gibson, Rozanah Brown, Edward Glass, Duncan Crooks, Anwen Cornell, Victoria Higginson, Caroline Court, Carly Crockford, Ashley Phillips, Caroline Leaver, Katie Horne, Laura Gower, Chloe Kerrigan, Angus Rouse Jefferson Road School, Pittsford, New York Ms. Pat Gaborski William Munsey Park School, Manhasse, NY Mrs. Moran's third grade class Eileen Ward and John McGill Munsey Park School, Manhasset, NY Mrs.Hirn's 4 th grade class Anita Minakyan, Matt Burke & Ryan Drago Lincoln Elementary School Mrs. Kaye's 3rd Grade class Christopher William Howard Taft Elementary Boise, Idaho Mrs.Crisp/Coles grade 4 Adam and Chris
Kristin Squires - Mrs. Pensa's 3rd grade class - Center School Stow MA The way that I solved this problem was by getting some graph paper and then I drew a seven by seven square. I would put my pencil on the point where the 7x7 square would end and say that there is only one way to do a seven by seven square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7x7 x shows the bottom right corner that the . . . . . . . 7 x 7 square can fit in . . . . . . . . . . . . . x Then I would find the point in the 7x7 square where a 6 by 6 square would fit in and put a dot everywhere where i could put the 6x6 square in it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6x6 x shows the bottom right corner that the . . . . . . . 6 x 6 square can fit in . . . . . x x . . . . . x x Then I repeated this way a few more times for the 5x5 square and the 4x4 square and I found a pattern that helped me solve the rest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 x 5 x shows the bottom right corner that the . . . . x x x 5 x 5 square can fit in . . . . x x x . . . . x x x If I had five dots on the out side of my "x-ed" surface then I would add two to it and then I would add those two numbers together and get n answer. And my answer adding up 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140 squares. ************************************** Chaz Rosenberg - Mrs. Pensa's 3rd Grade Class - Center School there are 140 squares in a 7 by 7 square. the pattern is: you go opposite the number and square it to get the number. For instance, 7 x 7 = 49 and that goes in the 1x1 column; 6 x 6 = 36 and that's what goes in the 2x2 column; etc. For an 8x8 square you would have 204 squares that would fit inside. For a 9x9 square you would have 285 squares that would fit inside. ************************************* Joanna Brench - Mrs Palmbach's 3rd grade class - Center School - Stow MA There are 140 squares in a 7x7 square. I made a graph and I drew the squares of each kind (1x1, 2x2, 3x3....) and counted how many would fit in and then I added them all up. 1x1 there were 49 2x2 there were 36 3x3 there were 25 4x4 there were 16 5x5 there were 9 6x6 there were 4 7x7 there was 1 If there was an 8x8 square there would be 204 smaller squares to fit inside. Adding 8x8= 64 to the 140 gives you 204. If there was a 9x9 square there would be 285 smaller squares to fit inside. Adding 9x9= 81 to the 204 gives you 285. ********************************* Answer: 140 The way we got the answer to this problem was by multiplying 7x7+6x6+5x5+4x4+3x3+2x2+1x1=140 For an 8x8 you should add 8x8+7x7+6x6+5x5+4x4+3x3+2x2+1x1=204 For a 9x9 square you can just add 81 because the square of nine equals 81. Jeremy Wolland, captain Will Gaybrick, co captain Linda Prueter's 5th grade Class Georgetown Day School Washington D.C. **************************** Dear Ruth, The following pupils came up with the correct solution: Holly Stamp,Annabel Bates, Elizabeth Ireland, Richard Gibson, Rozanah Brown, Edward Glass, Duncan Crooks, Anwen Cornell, Victoria Higginson, Caroline Court, Carly Crockford, Ashley Phillips, Caroline Leaver, Katie Horne, Laura Gower, Chloe Kerrigan, Angus Rouse. The sequence of numbers for squares in a square: 1x1 = 1, 2x2 = 5, 3x3 = 14, 4x4 = 30, 5x5 = 55, 6x6 = 91 and 7x7 = 140. Ashley offered this explanation: When I had worked out the first three in the sequence it was easy to work out the rest! I worked out that the number of squares went up in square numbers. Anabel said she found this out because of the square numbers in the middle of each squares, add them up to make the next number. Grant Whitaker ************************ Munsey Park School in Manhasset Mrs. Moran's third grade class Eileen Ward and John McGill We found out that if you go diagonal it is the same number on the chart you sent to help us. We got 140 by adding all the numbers in the row of 7 by 7 squares. We really figured out how many of each size there was in a 7 by 7 and added them together. Your chart helped a lot. From Eileen Ward and John McGill ******************** This is my chart: 1x1 sq. 2x2 sq. 3x3 sq. 4x4 sq. 5x5 sq. 6x6 sq. 7x7 sq. 8x8 sq. 9x9 sq. l unit l unit l unit l unit l unit l unit l unit l unit l unit l l l l l l l l l 7x7 sq.l 49 l 36 l 25 l 16 l 9 l 4 l 1 l - l - l l l l l l l l l 8x8 sq.l 64 l 49 l 36 l 25 l 16 l 9 l 4 l 1 l - l l l l l l l l l 9x9 sq.l 81 l 64 l 49 l 36 l 25 l 16 l 9 l 4 l 1 I figured out a pattern that the 1's slant down diagonally and the 4's slant down diagonally below the 1's. Then the 9's slant down diagonally below the 4's. The 16's come down diagonally below the 9's. after that come the 25's, 36's, 49's, 68's and the 81's. Also if I did a 10x10 sq. after the 81's there would be the 100's. Christopher 3rd Grade Lincoln Elementary School Mrs. Kaye's class **************** Try this. Forget about solving the 7x7 square problem for a minute. Pretend that you only want to find out how many squares are in a 1x1 square (One box on graph paper). The answer to this is 1. I put that answer in your chart below. Next pretend that you only want to solve for the number of squares in a 2x2 square (4 blocks, 2 on top of each other on graph paper). Well, there will be tow different types of squares in this problem. Squares that are 1x1 in size and squares that are 2x2 in size. There are 4 (1x1) squares inside the 2x2 square (I put this number in your chart). There is also 1 (2x2) square - the whole square. That makes a total of 5 different squares altogether. Now for the 3x3 square (9 blocks on computer paper - 3 rows & 3 columns). There are 9 little 1x1 blocks. Now for the 2x2's (this gets a bit trickier because the squares can overlap. I'll number each little box on the graph paper to try to explain 1 2 3 4 5 6 7 8 9 One 2x2 square is boxes (1,2,4,5); another is boxes (4,5,7,8); another is boxes (2,3,5,6) and the fourth is boxes (5,6,8,9). I hope this makes sense to you. There is also only 1 (3x3) square, the whole thing. Before you go on to looking at the 4x4 square, you should examine your chart. Do you see any possible patterns? Try to guess what the next row in the chart would be and then count to see if you're correct. (Don't forget about overlapping squares). Once you've verified the pattern, use it to fill in the rest of the chart. I hope this helps. I'm looking forward to hearing your answer. -Your math pal, Shannon F., Grade 9 Mount St. Joseph Academy _________________________________________________________________________ Number of Squares of Different Sizes _____________________________________ 1x1 2x2 3x3 4x4 5x5 6x6 7x7 Total _________________________________________________________________________ Size of Square 1x1 1 - - - - - - 1 2x2 4 1 - - - - - 5 3x3 9 4 1 - - - - 14 4x4 5x5 6x6 7x7 __________________________________________________________________________
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