Problem of the Month - RADAR

Submitted By John Bookston

I used to assign a Problem of the Month to my secondary students, at the start of the month, four or five times each year. Here is a nice “starter problem.”

The Puzzle

In this puzzle, you can move up, down, left, right, and diagonally up or down. You can start at any R and you can use a letter more than once.  















Explain why there are 144 ways to spell RADAR  

My System for Assigning a Problem of the Month (PoM)

In the process of solving this problem, your students will experience growth in organization, working smart, and explaining their work -- i.e. problem solving. There is no one right way to solve the puzzle. Students try various approaches and learn from false starts not to guess wildly.

Working smart means looking for invariants (notice that all spellings must go through the only D) and symmetries (once you have calculated the number of ways to spell RADAR starting at the top R, you need not repeat those steps for the R at the bottom, the R at the far left or the R at the far right). Similar logic allows you only to count the number of ways to spell RADAR starting from any one of the remaining four R’s.

Further examination, allows you to work even smarter. If you count all the ways to get “to the center D,” there must be the same number of ways to get “back to an R.”

It is best to give the students the answer, so they do not race to be the first to finish. Additionally, they will know they must find their mistake if they come up with a different number.

I always suggest that students do at least some of the work in groups but insist that each student submits his or her own write-up as if explaining a solution to a younger sibling. I seat my students in groups of four and change the seating every month, so everyone has worked on many different teams by the end of the year.

I give students the three documents below at the beginning of the year:

  1. The process that must be followed to get credit

  2. How their submittals will be graded

  3. Time management sheet


Document 1

Problem of the Month Required Procedure

I do not expect anyone to get a complete solution. The purpose of a problem is to strengthen your problem-solving skills and habits. Difficult problems in the real world take long periods of time and many contributors to solve. Do not get discouraged or give up if you are not making good progress, just take a break. You will be GRADED on the process. You can receive full credit by showing six hours of work at home (alone or with others) following the guidelines. You can get a poor grade if you do not follow the guidelines even if you submit a “correct” answer.

  1. Calendar six hours (over the month) to work on the problem before you start.

Time “thinking” without recording what you think is wasted and does NOT count.

2. Date your work as you go. Head each entry with the date (Record time you start

and stop.) Keep your work in chronological order. Use plenty of paper.

3. Work only when rested and not pressured for time. Do not throw away false starts.

Do explain how you found the start was “false.”

The work during the month will be scheduled as follows.

Week 1

Day 1 – In Class: read the problem aloud with others. Write about general thoughts and various ways you might generate and record data.

At Home: spend 15 minutes recording ideas about ways you could begin your work.

Day 2 or 3 – In Class: Re-read the problem at your table. Make sure you understand the problem. Generate and record only a few pieces of data.

At Home: Start small Generate and record a few pieces of data. Explain your thinking process. Always record places you made mistakes.

Day 4 or 5 – In Class: Talk to others about their ways to generate and record data and double check (with them) the data you each collected and the methods used.

At Home: Re-visit how you will allocate the rest of your six hours working on the problem. Begin to look for patterns in the data that you are SURE are correct.

Week 2

Day 1 or 2 – At Home: Record more data. Explain your thinking process.    

Day 3 – Check your data with others. Give credit in your write-up for specific ideas and data.  

Day 4 or 5 – At Home: Carefully record a substantial amount of data using your patterns.

Week 3

Day 1 or 2 – At Home: Finish the parts of the problem you have worked on or come to a contradiction and find your mistake(s).           

Day 4 or 5 – At Home: 1st Draft– A reflection on your journey: your growth as an efficient problem solver and things you will try on the next PoM.

Week 4

Day 1, 2 or 3 – At Home: Do a careful re-write of your paper.

Day 4 – At Home: Re-read your paper. Correct spelling, grammar, and flow.

Turn in (on time) ALL your work, not just your final write-up.  

Document 2

Scoring Rubric for each Problem of the Month

Date the 6 hours of work as you go. Work neatly. Keep in chronological order. (25 pts)

Step 1. Read the problem aloud with others. Jot down a few general thoughts. (5 pts)

Step 2. Re-read and discuss the problem. Make sure you understand what it is asking for. Decide how you might approach the problem; carefully record a little data. (10 pts)

Step 3. Talk to others about ways to record data. Check your data with others. Carefully generate more data. (10 pts)     

Step 4. Check and combine your data with others' data that has been verified. Search for patterns in your data. Record your thinking as you work. (20 pts)    

Step 5. Write a draft report explaining your thinking process. Add your thoughts on how the problem helped you grow as a problem solver. (15 pts)

Step 7. Carefully edit your draft into a final report and turn it in on time. (15 pts)

Difficult problems in the real world take long periods of time, careful recording of data and many contributors to progress toward a solution. When you feel discouraged and want to give up, just take a break. See me anytime for getting back on track.

Document 3

Time Management Sheet


3 to 4:30








engl / hist

math / sci


R and R


read / misc

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2 Tues

PoM / read

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3 Wed

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7 Sun

PoM / math

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PoM / music

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10 Wed

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11 Thur

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PoM / math

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PoM in back-


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© ATMIM (Assoc. of Teachers of Mathematics in Mass.)

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