How to Solve Math Problems

Update

I’ve expanded this blog post into a 16-page booklet. You can get your free copy here, along with some writing tips for young authors-in-training:

That’s a Tough One!

What can you do when you are stumped? Too many students sit and stare at the page, waiting for inspiration to strike — and when the solution doesn’t crack their heads open and step out, fully formed, they complain: “Math is too hard!”

So this year I have given my Math Club students a couple of mini-posters to put up on the wall above their desk, or wherever they do their math homework. The first gives four questions to ask yourself as you think through a math problem, and the second is a list of problem-solving strategies.

How to Solve a Tough Problem

1. What do I know?

• List the facts or information given in the problem.
• Underline or circle any key words, such as factor, multiple, area, or perimeter.
• Watch out for mixed units!
• Express the facts in math symbols, if you can.

2. What do I want?

• Describe the goal, what the problem is asking you to find.
• Underline or circle any key words, such as sum, product, next, or not. (Small words are easy to miss!)
• Express the goal in math symbols, if you can.

3. What can I do?

• Combine the given facts. Can you get closer to the goal?
• Try a tool from your Problem Solving Tool Box.
• Do one little step at a time.

4. Does it make sense?

• When you get an answer, always look back at the original problem one more time.
• Do you have the correct units (inches, cm2, kg, etc.)?
• Can you think of a way to confirm that your answer is right?

Problem Solving Tool Box

• Draw a diagram or picture.
• Act the problem out, step by step.
• Make a systematic list, chart, or table.
• Look for a pattern.
• Simplify the problem.
(Try it with smaller numbers.)
• Restate the problem in another way, or look for a related problem.
• Think about “Before” and “After” situations.
• Work backwards.
• Guess and check.
(Try something and see if it works.)

Sharing the Fun

In case you were wondering, my 4-step method is based rather loosely on the recommendations of George Polya (see the third quote here) in his classic book, How to Solve It.

32 thoughts on “How to Solve Math Problems”

1. THANK YOU! My son and I just started homeschooling and he hates math right now. I have to retrain him from the “NEW” math they were teaching him in the school system. This will be a great help!

2. These directions are for “word problems” in general?

I like to distinguish between “problems” (necessary approach is not obvious) and exercises (the opposite). Most “word problems” fall into the latter category.

I do quite a bit of work training kids on what to do when they encounter a problem that seems new, and where a direct route to solution is not obvious.

The first step, of course, is letting them encounter that sort of problem.

3. JD, to me these look like things to think about when solving “problems” rather than “exercises”. With the latter, you don’t normally need all this scaffolding.

4. Embarrassing (maybe) question: what does ‘scaffolding’ mean?

5. “Scaffolding” is a metaphor from building — it’s just a support structure. It’s the “Problem Solving Toolbox” that I’m mainly seeing as ‘scaffolding’ for solving challenging problems, that would not be used for simple exercises.

6. Part of my trouble is that what I think should be a mere exercise turns out to be a real problem for the students. The 4 steps in “How to solve…” should help the kids who just stare at their homework and don’t know where to begin. The “Toolbox” is for when they get thoroughly stumped. It is oriented toward elementary Math Olympiads or other challenge problems.

7. Anthony says:

If Sally can paint a house in 4 hours,and john can paint the same house in 6 hour,how long will it take for both of them to paint the house together

8. Anthony, the way to approach this is to think about what each of them could do in 1 hour, and then figure out what they could do together in 1 hour.

9. Cora says:

How to solve -8(t – 4)(2t + 1) = h

10. Hi, Cora!
This post was really about how to solve story problems, but yours is a good question. The difficulty with your equation is that you have two unknown quantities: t and h. You have a relationship between them, which would be enough to graph a curve of possible values, but you can’t narrow it down to any specific numbers without more information.

11. Do you have a simple, step-by-step system you can share with me to use with my students?

12. If you want a method that will work on almost any math problem, I don’t know if I can simplify it more than the 4-step method above. For certain types of problems, there are specific steps that work most of the time. For instance, arithmetic word problems can often be solved with the 8-Step Model Drawing method.

13. Mugabi Ronald says:

I would like to be getting some mathematical problems from you for me to solve and then you grade me and help me correct the wrong ones. otherwise your programme is good.

14. Being a Math teacher and a perpetual student of mathematics I am convinced that to be able to solve any Math problem the most important thing is to be able to visualize. Clearer the images faster you would be able to solve it. Thus students should be encouraged to sketch a rough diagram and put all the given data on it. This way at a glance they will know what is given, what needs to be found out and what are the possible ways of finding it.

Here is another wonderful math website on Faster Math that will help us instantly power-up the math muscle.

Fast mental Math tricks n techniques ( secret ) to end daily mental math problems. Fun Mental Math for kids. Vedic Mathematics tricks website with math videos

15. Normally, I would delete comments linking to hard-sell websites, but you are clearly persistent. And you have a good point — that many math problems are easier to visualize by sketching a rough diagram.

For the future, however, please notice that your name becomes a link to your website. You don’t need to include the spammy-sounding advertising paragraphs. People who like your comment will click through on your name to see what else you have to say.

16. Great stuff. Thanks

17. NATHANIEL says:

A COMPANY THAT PRODUCES CALCULATORS HAS OBSERVED THAT THE COST OF PRODUCTION OF X CALCULATORS(in hundreds) IS GIVEN BY THE FUNCTION Y=-2x+20.ALSO THE SELLING PRICE HAS BEEN MODELED INTO THE Y=2x+4.THE MAXIMUM NUMBER OF CALCULATORS THAT CAN BE PRODUCED IS 400.
SOLVE AND PLOT BOTH ON THE GRAPH

If this problem is copied from your homework, I can see why you are confused. The problem is poorly worded. Not very realistic, either — that second equation certainly flunks Econ 101!

I think this is what your teacher wants you to do:

Plot both equations on your graph, with y representing money (dollars or whatever you use) and x showing the number of calculators produced, in hundreds.

Also, plot the line x = 4, because 400 is the maximum number of calculators your company can make.

Find where the lines meet.

The two slanty lines on your graph will meet at the point where the selling price exactly equals the cost of production. You can’t work there without going out of business, and you certainly can’t sell calculators for less than they cost to make. It would make more sense to look for where that second line (selling price) has a greater y value than the first (cost), and x is still less than or equal to your maximum production capacity — but the data given in your problem has no such solution.

19. Min says:

here is a question i need to know easy way to solve it…please

Find the sum of all the two-integers, XY , between 10 and 99, which have the property that

XY
x XY
_____
….XY
_____

please tell me an easy way to find it..

20. Min says:

its XY times by XY which equal to something XY at the back or end

21. Min says:

25 n 76
101

22. The easiest way I know to solve something like this is with educated “Guess and Check” — also known as trial and error. The ones digits give you a big clue: Y $\times$ Y = _Y, which is only true for 0, 1, 5, or 6. So you need to try the 2-digit numbers that end in these digits.

If necessary, you could further analyze the problem. The tens digit of your answer must equal the ONES digit of XY plus the TENS digit of Y squared. But I wouldn’t bother with that — I would just grab a calculator and try numbers.

23. tomas says:

i like math but its tricky sometimes

24. You’re right, tomas, math problems can be tricky. But that is part of the fun — trying to solve the puzzles.

25. i need to know a question there a 7 lillypads and 3 frogs on each side how do you get the three frogs on 1 side to the other and the other side frogs to the other side. the rules are you can only jump 1 space and over 1 frog but cant go back what is the least moves you can do it in????
anyone now please let me know thank you

26. Look at the Problem Solving Toolbox above, and try the tool called “Act it out.” Use pennies or something for the frogs and circles on paper for the lily pads.

I’m sorry, people, but I don’t want to do your homework for you. Try one of the forums listed on the Free Resources page. I will not respond to any future homework problem comments.

27. Yurni says:

When I ask my student to solve problem, they thought it is hard problem even though it’s easy. How can I help them?

28. Tigran says:

I want to sell new math problems for olympiads.
Who can help me??

29. john says:

forestry:a tree casts a shadow 75 feet long at the same time that an 8-foot song casts a shadow 10 feet long.How tall is the tree

30. samuel says:

pls help me i want lean

31. I can’t help you with your homework, but if you really want to learn math, try this:

(1) Print out the pdf pages linked in the article above. Hang them on the wall by your desk, or wherever you work on math. When you get stumped on a problem, ask yourself those questions, and think carefully about your answers.

(2) Check out the homework help and online resources listed on my resource page:
* Worksheets and math practice
* Study on your own with online math lessons
* Forums where you can ask for help

(3) Don’t give up! Some things may come easily, while others may be very difficult for you. But if you work at it, you can learn just about anything.

I wish you the best of luck!