Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.
Pólya’s How to Solve It
George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1
1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY
In 1945, Pólya published the short book How to Solve It, which gave a four-step method for solving mathematical problems:
First, you have to understand the problem.
After understanding, then make a plan.
Carry out the plan.
Look back on your work. How could it be better?
This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.
Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!
Problem Solving Strategy 1 (Guess and Test)
Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.
Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?
Step 1: Understanding the problem
We are given in the problem that there are 25 chickens and cows.
All together there are 76 feet.
Chickens have 2 feet and cows have 4 feet.
We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.
Step 2: Devise a plan
Going to use Guess and test along with making a tab
Many times the strategy below is used with guess and test.
Make a table and look for a pattern:
Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.
Step 3: Carry out the plan:
|Chickens||Cows||Number of chicken feet||Number of cow feet||Total number of feet|
Notice we are going in the wrong direction! The total number of feet is decreasing!
Better! The total number of feet are increasing!
Step 4: Looking back:
Check: 12 + 13 = 25 heads
24 + 52 = 76 feet.
We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.
Videos to watch:
1. Click on this link to see an example of “Guess and Test”
Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!
Videos to watch demonstrating how to use "Draw a Picture".
1. Click on this link to see an example of “Draw a Picture”
Problem Solving Strategy 3 (Using a variable to find the sum of a sequence.)
Problem Solving Strategy 4 (Working Backwards)
Problem Solving Strategy 5 (Looking for a Pattern)
Definition: A sequence is a pattern involving an ordered arrangement of numbers.
We first need to find a pattern.
Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?
Example 1: 1, 4, 7, 10, 13…
Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.
Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.
4 + 5 = 9
9 + 7 = 16
So the next number would be
16 +9 = 25
25 + 11 = 36
Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.
In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5
-5 – 3 = -8
Example 4: 1, 2, 4, 8 … find the next two numbers.
This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?
1 x 2 = 2
2 x 2 = 4
4 x 2 =8
So each number is being multiplied by 2.
8 x 2 = 16
16 x 2 = 32
1. Click on this link to see an example of “Looking for a Pattern”
Example 1: Can perfect squares end in a 2 or a 3?
List all the squares of the numbers 1 to 20.
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.
Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.
How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?
0 3 30 cents
1 2 45 cents
2 1 60 cents
3 0 75 cents
Videos demonstrating "Make a List"
How would we find the nth term?
Solve a simpler problem:
1, 3, 9, 27.
1. To get from 1 to 3 what did we do?
2. To get from 3 to 9 what did we do?
Let’s set up a table:
Term Number what did we do
1 1 1 * 3
2 3 1* 3
3 9 1* 3
Looking back: How would you find the nth term?
Nth term = 1 times
Find the 10th term of the above sequence.
Let L = the tenth term
L = 1 times = 19,683
Problem Solving Strategy 8 (Process of Elimination)
I’m thinking of a number.
The number is odd.
It is more than 1 but less than 100.
It is greater than 20.
It is less than 5 times 7.
The sum of the digits is 7.
It is evenly divisible by 5.