The Problem:
How many triangles are in the figure above?
Problem Solving Step 1: Write what you know.
Hint: No solving is needed for this step, just simply state the facts. In this step, it helps to start your sentences with the words "I know..." For this problem, you would write:
I know the problem is a figure with triangles.
As you can see, no extra information is needed. Just find the facts given to you and write them in a sentence. Always use words from the problem.
Problem Solving Step 2: Write what you are looking for.
Hint: In this step, you write a sentence telling us specifically what the question is asking you. Again, don't do any solving here. In this step, it helps to start your sentences with the words, "I am looking to find..." For this problem, you would write:
I am looking to find how many triangles are in the figure.
This helps you to focus on the question being asked. Like step 1, always use words from the problem.
Problem Solving Step 3: All your strategies. (The fun step!)
Hint: In this step, this is where you'll do the work to solve the problem. There are many, many, many, many strategies that are available to solve one problem. Different people may use different strategies to get the same answer! Sometimes, if you're creative, two people may use different strategies, get two different answers, and both be correct. So, be creative and don't be afraid to think in a different way! Show lots of details in this step. Explain every single, little thing you do. Remember, we can't see you or your work so you'll have to write sentences to explain what you've done! For example, if you solve a problem by using a calculator, tell us you have a calculator and what you did with it. Several strategies could have been used to solve this problem. Here's a few:
1) I took the figure and numbered the triangles. I did this by putting the numbers inside each triangle. This strategy is called "labeling." There were three sizes of triangles in this problem: small, medium and large. The first time I did this problem, I didn't see the medium triangles. So, I used colored pencils to help me see the different triangles and then I labeled them so I could count them all. When I was done, I found all the triangles.
2) The first thing I did was count the triangles I saw by pointing at them and using marks to count them up. I found six. But something didn't seem right. So, I tried a different strategy. I traced the triangles with my finger and realized there was one big triangle holding all the other triangles. When I realized there was triangles inside of triangles, I looked harder and was able to trace triangles inside of other triangles. Finally, I figured out how many triangles there were.
Did you notice that they didn't state the answer? That's because the answer goes in Step 4. It's okay to put it here, but you need to repeat it in the next step. Also, these two students made a mistake the first time they tried to solve the problem. Mistakes are beautiful! Don't erase them, just write about them! Explain your mistake and how you caught it. Then explain your new work.
If you draw something, be sure to explain it in words. If you create a model, be sure to explain it in words. Give us a lot of details so that we can imagine it.
Problem Solving Step 4: Solution(s)
Hint: Sometimes problems have more than one solution. Or, you may be so creative that you were able to come up with more than one solution for one problem using different strategies. The cool part is that your answer here is usually only one sentence long and it starts with the words "My solution is.." or "My solutions are..." For this step, only write the answer that you came up with. Don't tell us how you solved it, you already did that in step 3. Just tell us the solution. Here's an example:
The solution is thirteen (13) triangles.
Simple, huh?
Problem Solving Step 5: Check your work.
Hint: This is where you check to make sure your answer is correct. Sometimes, all you have to do is count again or use a calculator to check your addition, mutiplication, subtraction or division. If you came up with steps, this is where you make sure the steps make sense and work them through one more time. For this step, explain to us how you checked the work. It's really smart to try to use a different method to check your work than you did to solve the work in step 3. That way you can be sure you have the right answer. Your sentences should begin with, "I checked my work by..." For this problem, your answer could look like:
I checked my work by writing "small," "medium," "large," across the top of the paper. Then, I printed out a copy of the figure and put it underneath my paper. Under the "small" heading, I traced the small triangles. Under the "medium" heading, I traced the medium trangles. Under the "large" heading, I traced the large triangle. After I did this, I counted how many there were in all and I came up with thirteen. So, my answer is correct.
This is another place to give lots of details. If you don't get the same answer, or find that you made a mistake, simply go back to step 3 and try again. Be sure not to erase what you've already done because that's part of the problem-solving process. Tell us everything, mistakes and all! Then continue through the next steps like usual.
Problem Solving Step 6: Extend (The Final Step!)
Hint: This is such a cool step! Here's where you make up a brand new problem based on the problem you solved before. Be creative and make it challenging! But, make it similar to the problem you've already solved. The start of your sentence should look like this, "I extended the problem by..."Here's what one might look for the problem we just solved:
I extended the problem by asking how many squares are in the figure?
As you can see, it's very much like the other problem but with a little difference.
There you have it! The problem-solving steps should help you to handle any problem that comes your way. You can even use these steps to solve problems in other areas besides math.
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Copyright 1998-2002 Carolynn S. Mortensen