Lesson 15

This warm-up challenges students to decode a short message, prompting them to think about what was done to produce the coded message so that it could be undone. The reasoning they do here paves the way for thinking about reversing the process that defines a function and about using outputs as inputs. This will get students ready to make sense of the inverse of a function.

It is not essential that students decode the message. What is important is the awareness that cracking the code involves a reversal process.

Some students may be unfamiliar with the idea of ciphers or coded messages. Offer a brief introduction, if needed.

Give students a moment of quiet think time. If students struggle to get started after some time, consider giving a clue or two:

• The letters A and I are the only two letters that can stand on their own (each letter can be a word).
• The word “is” is the most common two-letter word in the English language.

Leave time for class discussion, even if students have not yet managed to decipher the code at that time.

If one or more students were able to decode the message, ask them to share their finding and how they went about decoding it. Otherwise, solicit some comments on the strategies they tried and any hypotheses on how the message was encoded. (Students are likely to hypothesize that the code is related to the position number of each letter in the alphabet.)

Then, reveal the original message. Give students a brief moment to think about how it was coded. Discuss questions such as:

• "Suppose you found out that each letter in the message was encoded by using the letter 3 places down the alphabet, or 3 places after the original letter. How would you decode the secret message?" (By using the letter 3 places up the alphabet, or 3 places before that coded letter.)
• "The code JRRG IRRG is produced with the same method. What does it say?" (GOOD FOOD) "What about EDEB?" (BABY)

Introduce Caesar shift cipher (or shift cipher) as a way to encrypt a message by shifting its alphabet position a certain number of places. The message in the warm-up is called "a shift of 3" because it substitutes each letter in the original message with the letter 3 places down. A table could be used as a key. It enables us to easily see the plain-text alphabet and the cipher-text alphabet. Here is an example for a shift of 3.

A similar table could be used as a key for a shift of 5, 2, -3, or any other number.

Tell students that they will use the idea of writing and decoding a cypher to think about functions.

In this activity, students are introduced to inverse functions. They begin by creating their own shift cipher and using it to encode a message. After exchanging messages with a partner and decoding each other's messages, students describe the encoding and decoding process in terms of mathematical functions.

The encoding and decoding portion of the activity can be simplified by assigning a shift number to students.

The activity statement includes a table students can use to create a key to help with encoding and decoding.

Another tool that can be used as a key for multiple shift cipher is a cipher wheel, as shown here. If desired and if time permits, consider constructing one or asking students to do so. A template and instructions are included in the blackline master.

Arrange students in groups of 2. Tell partners that they are each to use a shift cipher to write a short secret message, exchange it with their partner's, and try to decode each other's secret message.

Focus the discussion on the mathematical representations of ciphering and deciphering a message and their connections to functions.

Select students to share the equations they wrote for the last two questions. Display them for all to see and ask students to make some observations. Discuss with students:

• "How are the two equations alike?" (Each pair of equations have the same letters and number.)
• "How are they different?" (One equation seems to "undo" the other. In each pair of equations, a different variable is isolated.)
• "Can we think of the process of encoding a message (going from \(m\) to \(c\)) as a function? Why or why not?" (Yes. For every input letter, there is only one possible output letter.) "What would be the input and output?" (In encoding a message, the plain-text letters are the inputs. The ciphered letters are the outputs.)
• "Can we think of the process of decoding a secret code (going from \(c\) to \(m\)) as a function? Why or why not?" (Yes. Every coded letter used as an input has only one output.) "What would be the input and output?" (The ciphered letters are the inputs. The plain-text letters are the outputs.)

Explain to students that, if the rule for encoding is a function, then the rule for decoding is its inverse function. Two functions are inverses to each other if their input-output pairs are reversed, so that if one functions takes \(a\) as its input and gives \(b\) as an output, then the other function takes \(b\) as its input and gives \(a\) as an output.

In this activity, students continue to develop an understanding of inverse functions, using currency exchange as a context. Students convert values in one currency into another and then the other way around. They make sense of the former as a function and the latter as the inverse of the first function.

The reasoning here builds on students' prior work on solving equations for a variable. Earlier in the course, students had isolated a variable to highlight a quantity or make it easier to compute the values of the variable. What is new here is the idea of inverse function: the recognition that a variable that is an output in one function (the one that is isolated) is an input in the inverse function, and vice versa.

Some students may choose to graph the first function (\(p=19.32d\)) and use the graph to find all the unknown values in either currency. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Give students a few minutes of quiet work time and follow with a whole-class discussion. Provide access to scientific or four-function calculators.

Discuss with students how they found the dollar value of the amounts given in pesos. Ask questions such as:

• "How did you find the inverse function?" (By reversing the steps used to find the amount in pesos when we know the dollar amount. By solving for \(d\).)
• "In the first function, which variable is the input and which is the output?" (Amount in dollars is the input and amount in pesos is the output.)
• "In the inverse function, which variable is the input and which is the output?" (Amount in pesos is the input and amount in dollars is the output.)

To emphasize the two quantities switching roles, display the graphs of both functions.

Give students a moment to observe the graphs and invite them to share something they notice and something they wonder.

If not mentioned by students, highlight that the labels of the axes have switched places, as have the first and second values in the coordinate pair. Explain that if we trace the graphs using graphing technology, we will see that all the values of all the coordinate pairs are reversed.