When I taught middle years Mathematics, I found myself intrigued by the different ways students solved puzzles. Classic puzzles, such as the following problem, use letters to represent numbers which can be displayed as simple calculation problems in disguise. One must replace each letter by a single 0 to 9 digit. The same digit must be used to represent the same letter. So, if you believe that 4 is being used to represent the letter A, then 4 must be used for all A’s in the same problem. Of course, one must use a different digit for different letters. After one substitutes all the letters for the numbers, a perfectly valid calculation problem will result.
Although some students might approach the problem using a “brute-force” method in which guesswork was employed, I was fascinated by those who approached the problem logically or employed a variety of strategies. I found that it was wise to provide my students with an opportunity to share with the class what assumptions were made and what strategies were used. For example, most would try to solve the above puzzle as shown in its subtraction format. However, some students found this puzzle easier to solve when the problem was rewritten in its corresponding addition format. Such “working backwards” is an effective strategy for solving problems and one that works well in this case. In addition, unlike most mathematical problems where there is only one correct answer, I encouraged students to see if they were able to find more than one series of letter substitutions that would solve each particular puzzle problem.
Excel Spreadsheet Format:
In the January, 2001 issue of “Bits and Bytes”, I wrote an article entitled “Spreadsheets: A Problem Solving Puzzle Creator“. This extensive article, provided the reader with insights into how I designed the spreadsheet component using Microsoft “Works” to create problems such as the above. Recently, I decided to improve and upgrade these same six problems so that they could be used by those readers using the 2007 (or later versions) of Microsoft Office. Educators may download a single Excel 2007 workbook which contains each of the six puzzles as individual worksheets at the end of this blog post.
Word Document Format:
As an educator, I wanted to provide students with an opportunity to identify the hunches and strategies that they used when attempting to solve these problem/puzzles. In order to facilitate this important documentation step, I created a Word document containing each puzzle as an embedded spreadsheet followed by a “What I/we learned” portion at the bottom of the page where students could list the various steps and strategies they took in solving a particular problem. The individual six Word problem/puzzles can be downloaded as “freebies” at the end of this article.
Suggested Lesson Steps:
I would use the following Word puzzle activities in the following manner:
1. Divide up the classroom into teams consisting of two or three students of similar abilities. I believe that the collaborative interaction in the team approach, as well as a reflective “What I/we learned?” nature of this project gives the best results for learning. Since the puzzles are arranged in order by relative difficulty, I would assign particular puzzles to challenge teams while providing a “best-fit” scenario.
2. Make a backup of all Word document puzzles (and keep them in a safe place) in case student puzzles get saved over the originals.
3. Transfer a copy of each Word document puzzle to a computer server drive location to which the students have access.
4. Open the first Word Puzzle #1 (ONE + ONE = TWO) on a workstation and project the image onto a screen so the entire class can see how the activity will work.
5. Demonstrate where team members will enter their names at the top of the document and stress that student teams must frequently save their progress using a file name format and drive location with which they are familiar.
6. Stress the need to document the team strategies and hunches and have students in class suggest why certain letters can take on certain values. For the initial demonstration, I would only choose values for E within the range 1-4 so that there is no need to carry over a 1 to the adjacent column. You might ask students if E could take on the value of 5 and to explain their rational to support their decision. Likewise, why can E not equal zero?
7. Before attempting to activate the spreadsheet component, enter a few of the class suggestions in the “What I/we learned” area at the bottom of the page and save the Word document (using the agreed upon location and file name convention). Possible sample comments can be viewed by clicking on the image near the end of this article.
8. Demonstrate how the embedded spreadsheet is activated by double-clicking within the yellow frame. Inform students that one will know that the spreadsheet is activated when the Excel column letters and row numbers appear along the top and left hand edge.
9. When the embedded spreadsheet is activated, one can position the mouse over the red-coloured “DIRECTIONS” link to review the puzzle-solving process.
10. Using the suggestions and the strategies listed by the students, enter a value for E in the “Guess & Test Area” and observe the changes in the “Results Area”. Remind students that the <Enter> key must be pressed after inserting any value into the “Guess & Test Area”.
11. Students must regularly update and enter strategies at the bottom of the Word document by deactivating the spreadsheet. To do so, one must click outside the yellow framed embedded spreadsheet. When the spreadsheet column letters and row numbers disappear, one is now in the Word document format.
If the information in the yellow frame becomes lost or distorted, immediately click the "Undo Object" (reverse arrow button) or enter the "Ctrl-Z" keystroke combination to recover.
WARNING: Sometimes when one returns to the Word document, the yellow frame disappears or is not completely displayed. If this happens, students must immediately click on the “Undo Object” (the “reverse arrow”) button or enter the <Ctrl-Z> keystroke combination. This “undo” step will always recover the original Word document display with the complete yellow framed puzzle/problem and team comments at the bottom.
12. Remind students to take turns so that different team members alternate tasks between entering strategies and hunches at the bottom of the Word document and activating the spreadsheet component to enter another letter value to observe the results and progress.
13. Continue with the demonstration on Puzzle #1 until the spreadsheet feedback area finally displays “CORRECT!”
14. Return to the Word document (using the Undo process) to re-draw the Word document and finish entering any additional strategies or hunches that the class agrees upon. Remind the class that it is important that students not only document strategies that worked but also hunches that need more refinement. Stress that we often learn more from our failures than our successes.
15. Invite students to suggest whether Puzzle #1 can have other solutions. Discuss such possibilities, without demonstration, particularly if you wish to use this puzzle with a team as an easier entry into this activity.
16. Save the Word document, one last time, and demonstrate how you wish this completed Word puzzle file to be submitted to the teacher.
17. Now ask the students to arrange themselves into the teams that you have chosen and advise each team which puzzle is their responsibility. Select a particular member of each team to open their assigned Word document puzzle.
18. Instruct another team member to enter their team or individual names at the top of the Word document puzzle and save this named document in an appropriate location using their student names as a file identifier.
19. If one of your teams is working on Puzzle #1 (which was used during the demonstration) tell these team members that they have to find a different solution than the one that was demonstrated.
20. Direct team members to examine their puzzle (in the word processing format) and discuss any beginning steps or strategies that they think might be used to help solve the puzzle. Have a new team member type their initial strategies/thoughts in the “What I/we learned” area at the bottom and save their team Word puzzle document before proceeding.
21. Ask a new team member, to start the puzzle by double-clicking within the yellow frame to activate the embedded spreadsheet. Remind the teams that the spreadsheet is “activated” when the Excel column letters and row numbers appear along the top and left side boundaries.
22. If necessary, team members can move the mouse over the red-coloured “DIRECTIONS” link, when the spreadsheet is activated.
23. Have each student, in turn, enter a value in the “Guess & Test Area” and observe the results in the “Results Area”.
24. Frequently remind teams, to return to update their strategies at the bottom of the Word document. To do so, one must click outside the yellow frame to deactivate the spreadsheet. Once the spreadsheet column letters and row numbers disappear, one may update the strategies and hunches at the bottom of the Word document. Remind students that if they return to the Word format (where no Excel column or row indicators are displayed) and the yellow frame spreadsheet does not display properly, one must immediately click the “Undo Object” (reverse arrow) or enter the <Ctrl-Z> keystroke combination to recover the Word document without any distortion.
25. Have students alternate between activating the spreadsheet, entering the next number, observing the results, returning to the Word document (using the “Undo” process to re-draw the Word document) and refining or typing in the new hunch or strategy to be tested.
26. Once the problem/puzzle feedback indicates that the team is “CORRECT!”, have the team complete the “What I/we learned … Area”, save the final version of the Word document, and transfer the resulting Word document to the teacher.
27. After finding their first solution, suggest that student teams, re-open the same puzzle, repeat the same steps to see if they can find more than one solution to their particular problem or to determine if their solution unique.
28. This problem solving activity could be used again by ensuring that different puzzles were assigned to different teams.
Sample Team Word Document
< Click the above image to view the display in a larger format. >
Take care & keep smiling