Teaching CCSS.Math.Content.HSA-APR.D.6

Instructions for You

Objective: Who's up for a little cryptography? The objective of this activity is to get students rewriting fractions in different forms. A series of rational functions will hide a code word, and by simplifying these functions, the students will figure out the code. The main mechanic of the activity requires students to rewrite rational functions as the sum of a polynomial and a rational function, but they'll also have to do this process backwards, as they make their own secret passwords.

Activity Length: 1 class period
Activity Type: Individual
Materials Needed: Paper, pencils/pens, board or document camera

Step 1: You'll start by presenting the core idea of using rational functions as a code. By this point, students should know that any rational function can be rewritten in the form . That means, for any rational function , there's hidden function q(x) tucked within it. And if you have a legend, where different polynomials q(x) represent different letters, you can hide a code word in a series of rational functions.

This idea becomes a lot clearer with an example. Using your board or document camera, introduce the following two rational functions:

Also put up this truncated legend:

A: 2x + 1
B: 2x + 2
C: 2x + 3
D: 2x + 4
E: 2x + 5
F: 3x + 1
G: 3x + 2
H: 3x + 3
I: 3x + 4
J: 3x + 5

To figure out the secret password, you'll have to simplify these two rational functions, and check the resulting q(x) polynomial against the legend. Since this is meant to get the students into the swing of things, perform the divisions as a class, so that everyone can see and understand this process.

By the way, in case you were wondering, the simplified version of the first rational function is . Since 3x + 3 is H, that means the first letter is H. Similarly, the simplified version of the second rational function is , and since 3x + 4 is I, the second letter is I. So the secret password is: HI. (Hi!)

Step 2: Now that everyone knows how these passwords work, it's now your students' turn to think one up. It's their job to come up with a secret password that's at least six letters long. That means there are at least six rational functions that will encode the password, and they need a legend, too, so that someone else can solve it.

Encourage your students to work their way up from their predetermined password to the bigger rational functions that hide it. For instance, if they want to use the password "ROSEBUD," they should first decide on seven polynomials (one to represent each letter). Then, for each polynomial, they should add a rational expression (y'know, a fraction), and then group this sum together as one big rational expression.

Of course, a legend with only 7 letters in it gives the game away rather easily, so they'll need to fill it out with some red herrings. They can fill out their legend with whatever polynomials they want; the closer they are to the actual answers, the better.

Once they've figured out their codes and their legends, have them take a clean piece of paper, and write out the rational expressions that make up their code, as well as their legend.

Step 3: Once all the students have come up with their secret passwords, have them trade sheets with each other. Now it's up to each student to solve a password that one of their peers devised. At this point, they should know the process inside out, so it should be a nice, relaxing puzzle to solve. Like a Sudoku, but with even more numbers somehow.

Instructions for Your Students

Student intro: Given how confusing math can sometimes be, it might not be too surprising to hear that math can be a great way to hide information. World powers have been using mathematical methods to code and decode messages for a long time. There are tons of ways to do it, but in this activity, we'll show you a (relatively) simple method. You'll have to do more than just learn this method, though: you'll be using it, to hide your very own secret password. The fate of the free world won't depend on it or anything, but your grade might.

Step 1: Your teacher will present the core idea of using rational functions as a code. By this point, you should know that any rational function can be rewritten in the form . That means, for any rational function , there's a hidden function q(x) tucked within it. And if you have a legend, where different polynomials q(x) represent different letters, you can hide a code word in a series of rational functions.

This might seem a bit fuzzy and abstract, but your teacher will also run through an example of a secret code that's hidden in a pair of rational functions, and once your see the method in action, these ideas should be a lot clearer.

Step 2: Now that you know how these passwords work, it's your turn to think one up. You've got to come up with a secret password that's at least six letters long. That means you'll need at least six rational functions to encode the letters, and a legend, so that someone else can solve it.

It's a good idea to work your way up from a predetermined password to the bigger rational functions. For instance, if you want to use the password "ROSEBUD," you should first decide on seven simple polynomials (one to represent each letter). Then, for each polynomial, you should add a rational expression, and then group this sum together to make one big rational expression.

Of course, a legend with only 7 letters in it gives the game away rather easily, so you'll need to fill it out with some false leads. You can then fill out your legend with whatever polynomials you want, but the closer they are to the actual answers, the better.

Once you've got your code and your legend, take out a clean piece of paper, and write out the rational expressions that make up your code, as well as the legend. Almost as if it were a puzzle, for someone else to solve. Which brings us to…

Step 3: Trade your paper with just the code and legend with a classmate. Now it's time to solve your classmate's password. At this point, you should know the process inside out, so it should be a nice, relaxing puzzle to solve. Like a Sudoku, but with even more numbers somehow.