If I was forced to put my money where my mouth is about alternative paths to Algebra, I would start with a very simple, yet significant puzzle. I call it The Famous Jinx puzzle. It was inspired by Harold Jacob’s introductory lesson “A Number Trick.” (See pages 1-4 in his Algebra book (1).) Here is a description of how I do it with kids.
I start off by saying, “Today, I’m going to jinx everyone. Take out a piece of paper and do the following steps.” (I read the instructions one at a time and wait an appropriate amount of time for them to complete each step.)
1. Choose a number from 1 to 10
2. Add 11
3. Multiply by 6
4. Subtract 3
5. Divide by 3
6. Add 5
7. Divide by 2
8. Subtract the original number that you chose.
9. Your answer is. Write it down. Don’t yell it out.
When I see that most of them are done, I ask one of those students what they got. If the student says 13, I ask another student the same question. I continue until at least 3 students said 13.
Note: For the time being I acknowledge other answers but don’t make a fuss about it. I just move on and ask other students the same question. The students who get a different answer have made a mistake somewhere in their calculations. So for the time being I ignore the problem though I return to it with those students on an individual basis because, at this point, I don’t want to change the focus of the lesson to correcting computational errors. This is also a good assessment whether the kids can do this kind of computation. What I do with that information is different lesson.
Then I say playing "dumb" (2): "Thirteen? So that must mean you all started with the same number?" Negative head shakes. So what numbers did you start with?" I list 4 or 5 of their numbers on the board. “Do you think all numbers from 1 to 10 will end up jinxed?” I ask. Not being sure, the students now try it with the missing numbers on the board.
Note: While the students are working on the remaining numbers, I check in with those students that didn't get 13 and help them spot their mistakes. Obviously, if most of the kids were having trouble with the computation, then this lesson could have easily bogged down since we can't go on without them knowing how to compute. (But this was not the case here, so I continued and left the few computation concerns for another time.)
“How about other numbers? Bigger ones? What do you think? Or what about fractions? Say ½?” I continue. The students respond with mixed results. Some kids plow ahead and try it using paper and pencil. Others grab a calculator. If it's a calculator with fractional entry capability then they proceed without much difficulty. But others with scientific calculators (who are in the know) will enter .5 and continue from there.
Note: It's very interesting to watch from a teacher’s standpoint their facility with numbers. Each time I teach it I'm surprised by what happens. One student once told me you can't enter fractions into a scientific calculator. You can only use decimal numbers. This started a wonderful conversation about whether fractions are decimals or not. We came up with the fact that fractions are masters of disguise and can take on an infinite number of forms including decimals!
Since it's time-consuming to do this puzzle by paper and pencil or with a calculator tedium starts to set in. Sooner or later (timing is important here) I stop and say: "Would it help if we had a calculating machine that would do all the calculations for us? I usually get a resounding “yes”. That's when I turn on the projection device that’s connected to my laptop and show this file (figure 1) on the screen and introduce them to the Jinx calculator. (Actually, it’s really a spreadsheet file in disguise that does the calculations for them so they can try more ambitious numbers. An Excel version is available here.)
I start by showing them only the left side (column A) in Figure 1. Then I drag the lower right hand corner of the window to reveal the numbers in column B. I then replace the number B1 with another whole number. They see the results of the calculations change, but not the 13 in cell B8.
Figure 2 shows what happens when I enter .25 in B1. Decimals appear everywhere except in B8 where 13 holds steady.
Note: This Excel file has all the instructions encoded in cells B2 through B8. Figure 3 shows what they look like. After a few tries I ask Do you think it matters what number you started with? (The “correct” answer of course is no, but we will need the Algebra to prove it.)
After a few more tries I ask the class if they think that this puzzle works for all numbers. Most of my students answered "Yes". The others weren’t sure. One student said “Don’t we have to test all the numbers?” “Yes,” I responded emphatically. “But can do we do that?” It was pretty obvious that that was not possible. "Instead of testing all the numbers," I said, "let’s see if we can find just one number that will "break" the Jinx puzzle, then we can say it doesn’t always work.” With the convenience of the spreadsheet Students try larger numbers, fractions, negative numbers, etc. but come away frustrated giving up declaring that “it always works” . But strange things did start to happen when the numbers got very large. One student tried 30 billion (3 followed by 10 zeros.) Notice a couple of strange results (Figure 4.). The “Multiply by 6” command results in an approximation of the answer in Excel’s version of scientific notation. Same thing for Subtract 3. But the computing recovered in the next step and finished with the expected result.
Another student noted that 3 followed by 15 zeros produced the results in figure 5.
What happened here? Was it something mathematical that broke down here? No, it was the spreadsheet that misfired. It's not that the puzzle doesn't work for a number three followed by 16 zeros, it's just that this version of the spreadsheet doesn't handle such big numbers. So it appears that the spreadsheet is not good enough to test all numbers. But even if we could does that mean we can guarantee that it would work for all numbers? Unfortunately, math is very fussy sometimes. To prove that it works for all numbers Algebra needs to come to the rescue. We have to test all the numbers but how do you do that?
Another number that might come up and challenging the jinx calculator is Pi. Here the best you can do is enter an approximation like 3.14 since Pi has an infinite string of digits after the decimal point. We need a better way to show that it works for all numbers. This is where using a bag which symbolically can contain any number we choose including Pi and any other pesky irrational numbers. I use a bag metaphor (inspired by W. W. Sawyer (3)) for younger students because it's more intuitive, since a bag usually contains something while using X is more abstract. To add to the confusion the variable X is the name of the “container” that holds an unknown quantity usually a number. You can see why algebra can become messy, very quickly for young minds. Here is a suggested approach to the doing the puzzle.
Step 1: Pick a number. Instead of picking a specific number, let’s choose a bag to represent any number.
Step 2: Add 11 . To show 11 we will use 11 small circles or “marbles”. Now we have a bag and 11 “marbles.”
Step 3: Multiply by 6. We now have 6 bags and 66 marbles.
Step 4: Subtract 3. Now we have 6 bags and 63 marbles.
Step 5. Divide by 3. What's left? 2 bags and 21 marbles.
Step 6. Add 5. There are 2 bags and 26 marbles.
Step 7. Divide by 2 and we have 1 bag and 13 marbles.
Step 8: Subtract the number you picked in step 1. But that’s the bag! You subtract the bag. So you are left with only 13 marbles or just plain 13.
In my teaching I used bags, boxes, and triangles to represent variables. I wanted to reinforce the notion that there is a distinction between the picture or name of the variable and the contents that it holds so I used boxes instead of X even after Harold Jacobs stopped using them in his book. I eventually switched to using Xs after my students told me to stop with the "baby boxes" because they "get it." But even then on occasion I had to remind them that X means the "value of X" quite often.
1. Harold Jacobs. Elementary Algebra. Freeman and Co. (1979)
2. Robert Berkman defines Kimp in his February, 1994 article "Teacher as Kimp" in the Arithmetic Teacher as someone who is so stupid that they are unaware of how stupid they are. Here I'm pretending to not know the answers to the questions I'm asking.
3. W. W. Sawyer. Vision in Elementary Mathematics. London: Penguin Books. (1964). Chap. 4. P. 65.
4. Online lesson the Famous Jinx Puzzle