Problem Solving Strategies Algebra/Use a Variable Look for a Pattern Make a List Solve a Simpler Problem

Using Algebra/Variables How to use this strategy: Look for a pattern that can be described in words and/or a general formula Finding Patterns: Look at the sequences of numbers Look at input/output table for rule Start with simpler case first Use finite differences

Using Algebra/Variables When to use this strategy: The phrase ‘any number’ is used There is a list of numbers that can be generated from the problem There is an input/output that can be described as a rule A large number of cases are involved

Example 1 Determine the sum of the first 50 odd counting numbers. Step 1: odd counting numbers are 1, 3, 5, 7, 9 … and so on. I am adding together the first 50 of these odd numbers starting with 1+3+5… Step 2: The plan will be to start with simpler problems and look for a pattern in the smaller/simpler sums

Example 1 Step 3: Use an input/output table to keep track of the simpler sums and look for a pattern. Input-how many odd #’s am I adding? Output-what is the sum? Do you see the pattern? In What odd #’s I am adding Out 1 2 1+3 4 3 1+3+5 9 1+3+5+7 16 5 1+3+5+7+9 25

Example 1 The pattern that related the input to the output is: Sum of n number of odd numbers is n x n (the sum of the first 3 odd numbers was 3 x 3) So the sum of the first 50 0dd numbers is 50 x 50, or 2500 Step 4: Look back and verify answer

Example 1 Imagine each odd number represented by tiles in the following shapes (note the number of tiles used represents the odd number, show below are 1, 3 and 5)

Example 1 Now add the shapes: 1 + 3 1+3+5 forms a square of how many tiles?

Pattern Strategy When to use: A list of data is given A list or sequence of numbers is involved Trying to predict or generalize

Make a List Strategy When to use: When looking for a pattern/rule Data can be easily generated and organized Listing results from ‘Guess and Test’

Solve a Simpler Problem When to use: Problem involves complex or complicated computation Involves very large or small numbers Looking for a pattern/rule

Example 2 Carl Friedrich Gauss (1777-1885) was a mathematician that made many contributions to math, physics, astronomy and related sciences. A story of his knowledge of math at a young age is attributed to his exasperated third grade teacher. The teacher was constantly trying to find things for Gauss to do since he always finished his math lessons way before his classmates. The teacher gave him the assignment of adding the counting numbers 1 through 100 figuring that would keep Gauss busy for awhile. The teacher was astounded when Gauss was able to complete the task in only a few minutes. How did he do this so quickly? (this is before calculators remember)

Example 2 Step 1: we are adding the counting numbers 1,2,3,4… and so on, up to 100 Step 2: combine strategies of making a list (table), solving simpler problems, looking for a pattern, and using a variable Input will be how many numbers I am adding Output is the sum of the numbers

Example 2 Input Output 1 2 3 6 4 10 5 15 What #’s are added 1+2 1+2+3 1+2+3+4 10 5 1+2+3+4+5 15

Example 2 There seems to be no operation that clearly relates the input to the output (input times 2 for example) The finite difference does not give much insight Gauss tried a new approach as follows: Write the numbers you are adding in reverse order Compare each pair top # paired to bottom# and notice they make the same sum in every pair

Input Output 1 2 3 6 4 10 5 15 What #’s are added 1+2 have 2 pairs of 3’s 2+1 3 1+2+3 have 3 pairs of 4’s 3+2+1 6 4 1+2+3+4 have 4 pairs of 5’s 4+3+2+1 10 5 1+2+3+4+5 have 5 pairs of 6’s 5+4+3+2+1 15

Input Output 1 2 3 6 4 10 5 15 What #’s are added 1+2 have 2 pairs of 3’s 2+1 2 x 3 = 6 3 1+2+3 have 3 pairs of 4’s 3+2+1 3 x 4 = 12 6 4 1+2+3+4 have 4 pairs of 5’s 4+3+2+1 4 x 5 = 20 10 5 1+2+3+4+5 have 5 pairs of 6’s 5+4+3+2+1 5 x 6=30 15

Example 2 What is the relationship now? If you are adding 2 pairs of 3’s you have exactly twice the output for the sum of the first two numbers If adding 3 pairs of 4’s, you get 12 which is double the output for the sum of the first 3 numbers (1+2+3) If adding 4 pairs of 5’s, you get 20, which is double the sum of the first 4 numbers (1+2+3+4)

Example 2 So if I want to add the numbers 1,2,…,10 using this pattern I would take the list: 1+2+3+4+5+6+7+8+9+10 10+9+8+7+6+5+4+3+2+1 Reversing the list gives you 10 pairs of 11’s which is =10 x 11 or 110. But this is double what I need, so divide 110 by 2 to get 55. 55 is the sum of 1+2+…+10 How do you write this using variables? What is the sum of 1+2+…+100?

Example 2 Adding n, number of the counting numbers can be found by (when n was 10, n+1 was 11) Sum of first 100 numbers is: