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:
Leave time for class discussion, even if students have not yet managed to decipher the code at that time.
Here is an encoded message, a message that has been converted into a code.
WRGDB LV D JRRG GDB.
Can you figure out what it says in English? How was the original message encoded?
The message says: TODAY IS A GOOD DAY. It was encoded by adding 3 to the position number of each letter of the alphabet: A becomes D, T becomes W, and so on.
If one or more students were able to decode the message, ask them to share their result 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:
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 after. 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.
| plain text | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| cipher text | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z | A | B | C |
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.
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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:
Leave time for class discussion, even if students have not yet managed to decipher the code at that time.
Here is an encoded message, a message that has been converted into a code.
WRGDB LV D JRRG GDB.
Can you figure out what it says in English? How was the original message encoded?
The message says: TODAY IS A GOOD DAY. It was encoded by adding 3 to the position number of each letter of the alphabet: A becomes D, T becomes W, and so on.
If one or more students were able to decode the message, ask them to share their result 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:
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 after. 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.
| plain text | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| cipher text | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z | A | B | C |
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.