Inverse Functions and Cryptography

Here are some teaching notes on a recent lesson I did on inverse functions with my group of grade 10 students. 

Learning Objective: Understand the definition of inverse functions and their relevance to everyday life. 

1. Setting the Stage with Office Supplies
Beginning the day’s lesson with a “picture talk” can be a good way to get students to practice using subject-specific terminology. The pictures can be controversial, or you may choose to pick analogies relating to the topic of discussion, like the one below for instance. 


Inspired by a photo I found on Pintrest. (Click to follow link)

I find that leaving the questions open-ended gives students more freedom to get creative with their answers.  

A series of open-ended questions you might ask: 
What do you notice about this photo? 
What do you think  ‘x’ represents?
Predict: How do you think this photo relates to our new topic? 

Teaching Notes
A lot of my students like to shout their answers out loud when they get excited about a topic, which tends to drown out the quieter students and does not give them enough time to think. But sometimes, I get excited too and I basically just end up shouting “YES!” while enthusiastically pointing and staring  wide-eyed at the students who yelled out the answer I was looking for. “The answer I was looking for” – which meant ignoring all other answers that may have added important insight to our discussion. Such habits are dangerous because they tend towards a classroom environment in which it is not safe to take chances or make mistakes. When I  dismiss wait time and only acknowledge the quick answers, I am effectively giving everyone else the permission to shut down and stop thinking.

To combat the issue of the shouting-the-answers-out-loud thing, I introduced this image by inviting students to take ten seconds (any more and they would have shouted the answers anyway)  to silently look at the picture and gather their thoughts about it. When ten seconds had passed, I asked them to share their ideas with the person sitting next to them when they were ready.   

2. Lesson Objective and the Enigma Machine
​​By now, it is likely that students have guessed that the lesson has something to do with “functions” and “opposites.” At this stage, I presented the day’s lesson objectives and key terms (with translations), along with a dashing photo of Benedict Cumberbatch in “The Imitation Game.”  I gave a brief synopsis of the events in history in which the movie is based, and explained how it related to our topic of inverse functions. In retrospect, I probably should have posed this as a question instead: “How do you think this relates to our topic?” 

3. Intro to Cryptography
Next came the fun part, and what formed the bulk of the lesson. The students worked  pairs and were asked to  decode a hidden message within a twenty minute timeline. The less hints you give, the more challenging the activity becomes. (You can download the activity page along with my teaching notes below). 
Once the twenty minutes are up, you’ll see some students scrambling to finish decoding the message. Debrief the activity with them. Discuss strategy- for instance, how were they able to determine the cipher? Follow up- how do you think Alan Turing and the British Intelligence were able to crack the unbreakable code? (May choose to show video clip of the relevant scene in the movie for dramatic effect). Extension- What is the probability of randomly guessing the code correctly? What assumptions do you have to make in order to do so? (Can also relate topic to permutations and combinations).

Teaching Notes:
I downloaded a 20 minute digital countdown timer and added it to my ppt so students can keep track of how much time they had left for the task. If you have any English Language Learners in the class, it may be helpful to post the English alphabet on the board as an added hint. 

4. Consolidation
I ended the lesson by asking students to write a short journal entry relating to the picture shown at the beginning of class. I think a better journal prompt would have incorporated a debriefing of the cryptography activity. The students’ journal entries  gave me individualized feedback on how well they understood inverse functions and the composition of functions. A common mistake I noticed was mistaking the concept of reciprocals with inverse. Some students wrote that 2 and ½ were “opposites”, and therefore the inverse of each other. Something to address in my next lesson. 

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