Math-ic Prediction

Tricks like these are often presented as a “one-off” in middle school mathematics classrooms. The truth is, they can be used to develop deeper understandings about mathematical concepts and skills such as variables, constants, combining like terms, solving equations, and problem-solving. This Math-ic Prediction trick was one of the first I used for these purposes.

Click on the video below to see a presentation of this trick:

If you use coding with your students, I also created a version of this using Scratch: A Math-ic Prediction. Using this, students can actually “look inside” the code to see how it works, then use their coding skills to make a version of their own. If they do, please share!

The cool thing about this trick is not that everyone gets the same answer (although that’s pretty cool). The cool thing is that this can, and should be a visual exploration. So, grab some bags or envelopes and some counters and let kids explore this visually and concretely to figure out what’s going on. Then run through it again, so they can work out what the math might look like.

The Lesson:

Provide each group of students with some bags or small envelopes and have them put the same number of counters in each. This represents their number. As we go through each of the steps of the trick, students should write down the step they are on, then build what it looks like using the envelopes and counters. Work through the whole Math-ic Prediction trick this way.

Next, students will do two things: 1) draw a representation of the envelopes and counters for each step, and 2) work to determine what the algebraic expression would look like to describe each step.

Close the lesson by connecting students’s expressions at different steps in the trick. For example, in step 3 students double what they had in step 2. Some students may write 2x + 6. Others may write 2(x + 3). The concrete and visual representations will help students make the connection that these are equivalent expressions because of the distributive property.

Follow up with more tricks like this, where students follow steps and created physical, visual, and algebraic representations for each. As students become more adept at writing the expressions, shift their thinking a little with something like this:

Luisa should be the most familiar since this is what students have been doing in previous explorations. Julia is only one step away from the first number, but she chose a number that’s a bit different from what was probably used in previous explorations like this. Jake and Andre require the most steps to solve.

This leads to the idea of using inverses to undo what is being done in solving equations. Students should explain their reasoning for how they are determining the numbers in the table. Facilitate discussions about what works and what doesn’t work when trying to fill in the table. Encourage students to use visuals and concrete representations as needed. Close the lesson by helping students connect concrete/visual representations to the algebraic expressions used. Additionally, highlight equivalent expressions during the discussion.

More Resources for Algebraic Magic Tricks as problems to solve: