Double-Digit Card Trick

The best thing about this problem/trick is that it requires no figuring on the part of the performer. It’s self working and students do all of the computation. And it’s one of the mathematical magic tricks from which gets positive reactions from students and teachers. This is a great algebraic algebra trick to use with 7th and 8th grade mathematics or algebra students. The task is to find out how and why this trick works, which students get the opportunity to do when they ask, “How did you do that?”

In order for students to solve this problem, they need to see this performed 2-3 times. The reason for this is that there are two variables and they will most likely be able to write the algebraic expression up until the last step.

How and Why this Works (For Teachers):

The Lesson:

Introduce this problem, by performing the trick. Students write down and discuss with a partner what they notice. Discuss as a class, then perform again. When you perform this a second time, students will likely notice that the last step (the subtraction changed). They may ask for another performance. Record the student and teacher cards for each performance.

The task is to find out how this trick works and to also try to write the steps in mathematical language (algebraic expressions). Students should work together write these expressions.

As students work, facilitate discussions with groups and ask questions like those below to keep students in the flow of work:

Possible “least helpful” questions:

  • What does it mean when you are asked to double?
  • How does the teacher know what to subtract?

Students’ mathematics (algebraic expressions) may look a bit different than what is seen on the explanation video above (they may use different variables or they may only use one form of the expression 5(2x + 2) vs (10x + 10). If/when this happens, it might be a good time to discuss equivalence and the distributive property. Students using 5(2x + 2) may not see the connection to the end result, so having them share first during the closing would be a great idea. Follow up with a group who used 10x + 10 to explicitly show how the end result answers the question about how this is done.