Day 27: Password Hacking

Quote of the Day: “Try to encourage a kid to learn math by paying her for each workbook page she completes - and she’ll almost certainly become more diligent in the short term and lose interest in math in the long term.” - Daniel Pink

Question of the Day: "Why is it that 0.7 is 7/10 but 0.7 repeating is not 7 repeating over 10?"

Regular Math Objective: Give an example of where exponents are used in everyday life.

Regular Math Standards: 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. 

Regular Math Lesson Sequence:

1. Students completed a self-assessment. I asked them two things. "Do you believe at some date in the future you can answer all ten questions?" I wanted to see who among them was brave enough to say no to this question. These are the students that are truly crushed when it comes to confidence in math. From there I know that I need to make it my own personal mission to prove them wrong and get these kids to see that it is possible to change their mindset in math. The other question was "Can you think of some real-world examples of exponents?" This has kind of been lost on me in teaching this entire unit. Many students struggled with this question. The thing I heard all day was space exploration or some version of that, but we missed other obvious answers. 
2. Review quiz errors in pairs. I really liked this idea. Students sat in partners and had to put the correct answer to a problem that they had answered incorrectly on a marker board. They exchanged ideas to one another. I heard great discussion as it was taking place. It was a great way to ditch stand and deliver in going over the quiz and a much more measurable way of knowing that students are recognizing their misconceptions than a fist of five or asking if they understand now. They always say yes when you ask that question.
3. Password hacking. 

I knew that students were lacking a basis for why we are teaching exponents and saw that on their self-assessments. Robert Kaplinsky's lesson really hits home especially on a level that students can relate to. What I did to introduce this was have students leave their seats and tell each other the five vowels (we did not include y). It might seem silly to have students rehearse what the vowels are, but it got them out of their seat and for students that are WIDA level 1 or 2, knowing the vowels is not a given.

Second, I had students write a password on a marker board so that their partner could not see it. The password had to be only one letter that was a vowel. Students then took turns guessing one another's password. It's an amazingly simple password, but I wanted them to guess it quickly so that we could move to a more advanced password and see the value in that. I asked what the maximum guesses are for a password are under these constraints. Students recognized that five guesses would be the maximum assuming of course that the same letter was not guessed twice.

Third, I told students to pick a new password that was two vowels long and then exchange guesses with their partner to see if they could hack the password. Some students could do it while others could not. As the students were guessing, I was putting all possible combinations on the board. After about a minute or two, I had students raise their hands if they got hacked. Only a handful had been hacked. We discussed why. Students were quick to point out by looking at my chart that there were now 25 password combinations instead of only 5 with two letters.

Fourth, I had a chart which compared characters to password combinations. I asked students based on the pattern, how many password combinations would there be if we added a third character. Students discussed with their partners and then were able to tell me that 125 password combinations were possible. For the skeptics in the room, I showed all 25 passwords that would have started with the letter A.

After all of this, we were able to get a deeper and more invested conversation of the idea that there are actually 70 potential characters when we consider uppercase, lowercase, numeric, and special characters. The lesson got somewhat frustrating from here as I had to rehearse for students that a computer could check for 350 billion passwords in one second. To students (and teacher) that have no background knowledge of this it's hard to say if this is really all that impressive. We live in an era where technology has seemingly no limit so some students even guessed that a computer could guess an infinite number of passwords in one second. We eventually closed the lesson with the Desmos link and formula. I manipulate the numbers of characters that were allowed so that it would be an exceedingly long password and explained to students that it would take centuries to crack this password. Of course, computers are probably capable of testing even more passwords now than they were two years ago.

Honors Math Objective: Graph absolute value inequalities of the form |x| is greater than or less than an integer c

Honors Math Standards: A1-A-REI B3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. a. Solve linear equations and inequalities in one variable involving absolute value.

Honors Math Lesson Sequence:

1. QSSQ
2. We looked at yesterday's exit ticket which was a compound inequality. I then had students do a similar problem and when it seemed that the room felt comfortable (I did a fist of five and circumvented to certain students that I knew had struggled yesterday) we moved to absolute value. 
3. Students were put in groups of three to four and given a two foot by two foot marker board. 
4. I gave them the instructions to pass the marker when I told them to and that whoever the writer was could not talk and could only write what the others in the group asked. I had done something similar two days prior with great success. Today I wish I had not done this because students really struggled with the concept. As a result, I had to go around and do cueing questions and failed to pass the marker as much as I would have liked. I think students would have benefitted from having everyone in the group be able to communicate at all times as a result of the level of difficulty absolute value inequalities gave us. 
5. The first two problems were extremely basic in terms of absolute value, but they were deep enough to last the entire class. They were in the form that the objective states. Students had problems with graphing and setting the inequality to the negative with a flipped sign. The mistakes graphing was perplexing. The inequality |x| < 3 was graphed correctly on the positive side, but then students were putting an open dot at -2.