S L E O A V S L I M P E R P R O B E L M When We read a math problem and recognize that it will be very hard to solve it directly, then it is time to solve a simpler problem. Most of the time, when we solve a simpler problem or a series of simpler problem, a pattern will emerge that we can extend to solve the larger problem. Another thing that can happen when we solve a simpler problem is that we will see the path we need to take to get the solution to the larger problem. Here is an example: A polygon has 6 times the number of diagonals as it has sides. How many sides does this polygon have?

We solved one like this before, when we were asked to find how many diagonals a decagon had. We can use our method to solve this one. We will begin by reviewing the number of diagonals some of the smaller polygons have, finding the pattern, and extending it out to answer the question. If you remember, we were able to draw out the 3, 4, 5, and 6 sided polygons and count the diagonals they had. From there, we found the pattern and extended it out. Here is our chart. We can see from the chart that the polygon with six times the number of diagonals as sides has 15 sides. sides 3 4 5 6 7 8 9 10 11 12 13 14 15 diagonals 2 20 27 35 44 54 65 77 90 multiple? 1

Solve a simpler problem to find the solution. Find the sum of the first 50 odd numbers. Find the sum of the whole numbers from 1 to 120. There are 250 pennies laid out in a row and all of them are heads up. Suppose the first person comes along and flips all of the pennies over. Then the second person flips over every second penny. The third person flips every third penny. The fourth person flips every fourth penny, the fifth person flips every fifth penny, and so on, until the 25oth person flips the last penny. At the end of this, which pennies will be tails up? A pyramid of wooden blocks is stacked up against the wall so that there are 2 blocks in the top row, 4 in the second row, 6 in the third row, and so on. How many rows are there if the pyramid contains 756 blocks? How many squares of any size are on an 8x8 chessboard?

1) Find the sum of the first 50 odd numbers. Well, let’s begin small and see if we can find a pattern: 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 Wait a minute – we’ve seen the numbers 1, 4, 9, 16, and 25 before – they are the perfect squares. So, the sum of the first 1 odd number = 1 squared; the sum of the first 2 odd numbers is 2 squared; the sum of the first three odd numbers is 3 squared; and so on. This means the sum of the first 50 odd numbers is 5o squared or 50 x 50. This number is 2500.

2) Find the sum of the whole numbers from 1 to 120. If we try to use a pattern like in the last problem, we won’t get very far. (In the interest of time and space, I will leave that to you if you want to try it.) So, let’s look a little deeper and think outside the box. 120 is an even number, so we can pair up everything. If we pair up (1+2), (3+4), (5+6), that won’t get us anywhere. How about it we pair up (1 + 120), (2 + 119), (3+ 118), and so on? If we do that, each of the pairs has a sum of 121. Now we’re getting somewhere! How many pair of numbers will we have? 120 ÷ 2 = 60, so we have 60 pairs with a sum of 121. our answer will be 60 x 121, which is 7260.

3) There are 250 pennies laid out in a row and all of them are heads up. Suppose the first person comes along and flips all of the pennies over. Then the second person flips over every second penny. The third person flips every third penny. The fourth person flips every fourth penny, the fifth person flips every fifth penny, and so on, until the 25oth person flips the last penny. At the end of this, which pennies will be tails up? Hmmm . . . 250 is too many to try and simulate a solution to. Let’s try it with 20 pennies and see what happens. Begin: H H H H H H H H H H H H H H H H H H H H PERSON 1: T T T T T T T T T T T T T T T T T T T T PERSON 2: T H T H T H T H T H T H T H T H T H T H Person 3: t h h h t t t h h h t t t h h h t t t h Person 4: T h h t t t t t h h t h t h h t t t t t Person 5: t h h t h t t t h t t h t h t t t t t h Person 6: t h h t h h t t h t t t t h t t t h t h Person 7: t h h t h h h t h t t t t t t t t h t h Person 8: t h h t h h h h h t t t t t t h t h t h Person 9: t h h t h h h h t t t t t t t h t t t h Person 10: T h h t h h h h t h t t t t t h t t t t Person 11: t h h t h h h h t h h t t t t h t t t t Person 12: t h h t h h h h t h h h t t t h t t t t Person 13: t h h t h h h h t h h h h t t h t t t t Person 14: t h h t h h h h t h h h h h t h t t t t Person 15: t h h t h h h h t h h h h h h h t t t t Person 16: t h h t h h h h t h h h h h h t t t t t Person 17: t h h t h h h h t h h h h h h t h t t t Person 18: t h h t h h h h t h h h h h h t h h t t Person 19: t h h t h h h h t h h h h h h t h h h t Person 20: t h h t h h h h t h h h h h h t h h h t At the end of our simulation, we see that pennies 1, 4, 9, and 16 have tails up. Once again, we have seen these Numbers – they are the squares. So, all the square numbers up to 250 will be tails up. They are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

4) A pyramid of wooden blocks is stacked up against the wall so that there are 2 blocks in the top row, 4 in the second row, 6 in the third row, and so on. How many rows are there if the pyramid contains 756 blocks? Once again, we will work this row by row and look for a pattern. Row 1 = 2 Row 2 = 2 + 4 = 6 Row 3 = 2 + 4 + 6 = 12 Row 4 = 2 + 4 + 6 + 8 = 20 Row 5 = 2 + 4 + 6 + 8 + 10 = 30 ‘ Now, if we keep going, we will have to do this until we get a total of 756, which could take a long time. So, we will look back at the row numbers compared to the sum and see if we can make a connection. If we think about it, every row’s sum equals the row number times (the row number + 1). So, row 1 = 1x2 = 2, row 2 = 2x3 = 6, row 3 = 3x4 = 12, row 4 = 4x5 = 20, row 5 = 5x6 = 30. We can use this to find our answer. For row x, we would have x(x+1) = 756. This means two numbers right next to each other will multiply to give us 756 and we want the smallest one. So, let’s look at 756. the Square root of 756 (its middle for multiplication) is 27.49545417 so we will check for 27 and go down until we get the pair. This pair is 27 and 28. That means the number we want is the smaller one – 27. The pyramid contains 27 rows.

5) How many squares of any size are on an 8x8 chessboard? Again, if we tried to do this outright, we might lose our sanity! Let’s look at a 1x1 board, then a 2x2 board, etc. and see if any patterns emerge. A 1 x 1 board has only 1 large square. (1 = 12) A 2 x 2 board had 1 large square and 4 small squares. (4 = 22) A 3 x 3 board had 1 large square, 4 medium squares, and 9 small squares. (9 = 32) We can already see a pattern emerging here, and it involves the squares again (funny how they keep popping up!) Here is a summary and then on to our solution: 1x1 = 12, 2x2 = 12 + 22, 3x3 = 12 + 22 + 32, therefore, 8x8 = 12 + 22 + 32 +42 + 52 + 62 +72 + 82 = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204 total squares.

More solve a simpler problem problems and resources online http://www.studyzone.org/testprep/math4/d/simplerproblem4l.cfm http://fcit.usf.edu/math/resource/fcat/strat.htm https://www.youtube.com/watch?v=Ji0giVL0Xr0 https://www.youtube.com/watch?v=9qkdR85Pd30 https://www.youtube.com/watch?v=BbL3zoaKOBw https://www.youtube.com/watch?v=MOPzQNpklJo https://www.teachervision.com/math/problem-solving/48932.html