#2586. The Magic of Algebra

Mathematics, level: Senior
Posted Wed May 15 00:24:09 PDT 2002 by Eric Hutchinson (hutchinson_math@hotmail.com).
UNLV, Las Vegas, NV
Materials Required: Script (given below)
Activity Time: 20-25 minutes depending on group time
Concepts Taught: Forming algebraic equations from computational instructions, simplifying algebraic expressions

The teacher will read the following script to the students: "I am going to read your mind! This is a large class, so I'm not sure if I can get it exactly right, but I'll try. I want you all to think of any number. Since you will be doing some mental math, you may want to choose a smaller number -- one or two digit. Write this number down on a scratch piece of paper. (Teacher will write down these steps on the board). We will call this the original number. Now take this number and double it. Add the number 8 to that result. Now divide this result by 2. The number you have now we will call the new number. Now subtract this new number from your original number. Find the letter in the alphabet that corresponds to that number. That is, if you have a 1, the letter you should be thinking of is 'A'. If the number you have now is 2, use the letter 'B', 3 is 'C', 4 is 'D', 5 is 'E', and so forth. Continue through the alphabet until you reach your number. Think of a country whose name begins with the letter you are at. Take the second letter in that country name, and find an animal that lives in that country. Now think of the color of your animal. You should all be thinking of the same animal. Now let's see. Let me think a second. Could it be possible that you are thinking of a gray elephant in Denmark? Now it is your job to figure out how the trick works by forming algebraic expressions for each step of the trick I went through."
After the teacher performs that trick, students should be put in groups and are asked to use algebra to figure out how the trick works by writing algebraic expressions to match the steps written on the board. Use x as your number, and perform the adding and dividing steps as used in the trick. The algebra should work out so that you end up on the same number, 4, no matter which one you start with.
The teacher will be moving around the room offering help for students if they are not getting this concept. At the end of the class, the teacher will explain the above steps to make sure everyone gets it. The following are the mathematical algebra steps. First start with x and build the algebra expression: Each line of this proof represents one of the instructional steps given in the script:
x
2x
(2x + 8)
(2x + 8) / 2
(2x + 8) / 2 - x

Simplifying:
2x/2 + 8/2 - x
x + 4 - x
= 4
Notice no matter what x is, you will always get 4!
The letter 'D' is the letter that is used, and Denmark is one of the only countries students can think of that begins with 'D'. The second letter in Denmark is 'E', and an elephant is usually the only animal that students can think of.
For homework, students will be asked to come up with a math magic trick like the one performed in class where the number 7 is the number you will end up on. This will allow students to review what was gone over in class, and also they can perform it on their friends. The teacher can even tell them to think of something that begins with the 7th letter in the alphabet and see if it is something common that everyone would think of, like Denmark. A giraffe comes to mind for an animal. Be creative!