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
Ask yourself these 4 questions:
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.
- Does your answer make sense?
- 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
If you would like to download these handouts for your students, here are the files:
How to solve a tough problem (pdf 63KB)
Problem solving toolbox (pdf 98KB)
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.
For this and many other excellent books about teaching math, check out the Let’s play math! bookstore.
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Have more fun on Let’s Play Math! blog:
16 responses so far ↓
Greta // October 10, 2007 at 8:04 am
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!
jd2718 // October 13, 2007 at 8:23 am
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.
mathmom // October 13, 2007 at 2:08 pm
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.
jd2718 // October 13, 2007 at 7:21 pm
Embarrassing (maybe) question: what does ’scaffolding’ mean?
mathmom // October 13, 2007 at 11:23 pm
“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.
Denise // October 15, 2007 at 8:01 am
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.
Anthony // February 16, 2008 at 4:34 pm
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
mathmom // February 16, 2008 at 6:44 pm
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.
Cora // February 21, 2008 at 10:15 pm
How to solve -8(t - 4)(2t + 1) = h
Denise // February 22, 2008 at 9:03 am
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.
Stacey // February 22, 2008 at 6:31 pm
Do you have a simple, step-by-step system you can share with me to use with my students?
Denise // February 23, 2008 at 9:48 am
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.
Mugabi Ronald // March 6, 2008 at 12:49 am
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.
Denise // March 6, 2008 at 7:48 am
I am not set up to do online tutoring, but I am sure you can find help on my Free Resources page. Try the Study on your own with online math lessons section.
glad2teach // May 2, 2008 at 12:08 am
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.
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Denise // May 2, 2008 at 10:09 am
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.
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