It can be a challenge to teach change of base in middle school or high school. After explaining the concept most students will look at me in confusion and have difficulty converting between different numeral systems. Introducing the binary numbers is a great first step and can provide tie ins with many other math concepts. In this blog I want to introduce a wonderful “math-e-magic” trick that you can perform to get your students excited about binary numbers.
Instructions:
The trick consists of 5 special cards each containing a subset of the numbers from 1 to 31 written on them (see image). Your goal will be to guess a student’s secret number. This can be performed on one student but I find that kids are more impressed when you demonstrate it in a group setting.
Begin by asking one volunteer to pick a secret number between 1 and 31 (or perhaps the day of the month in their birthday). Then ask them to accurately point out all of the cards that contain their secret number. To guess their secret number total the upper left hand numbers from each of the selected cards (see image). I recommend using some showmanship at this point, for example you might want to tell the students you need to clear your mind first in order to pick up the psychic signals.
The Math Behind the Magic:
Notice the numbers in the left hand corners are all distinct powers of two. Coincidence? Of course not! What is really going on here is that the students are converting their secret number into base 2 by selecting the cards; by adding up the upper left hand powers of two we can recover the number! Let’s try a concrete example:
19=16+2+1=100112.
You can check and see that 19 belongs to exactly three of the cards, the cards chosen correspond to the 1’s in the binary representation and those not chosen correspond to the 0’s.
It is worthwhile to ask the students to look for patterns in the cards before giving away the trick. For example the first card consists of only oddnumbers, and the second consists of numbers with remainder 2 and 3 when divided by four. This corresponds to having a one in the “1’s place” and “2’s place” in the binary representation respectively.
Conclusion and Beyond:
Binary numbers and change of base are some of the more abstract concepts that are introduced in the middle/high school curriculum. The simple idea that any quantity can be represented by system with two symbols has very deep consequences and is essentially the foundation of computer science. It can be intimidating for a student to imagine manipulating numbers outside of base 10, I hope this lesson will remove some of that anxiety and replacement with wonderment and curiosity.
The binary representation is also a jumping point into many other interrelated topics. Some examples are: binomial theorem, information theory, geometric series, as well as number theory and logic.
This trick is available for iPhone and iPad on the App Store: Birthday Genie App.
My roommate and I built this app and we really appreciate your support. We recommend hooking it up to an external display and performing the trick on a group of students. If you have any questions or feedback please leave us some comments below!
