I begin class today by going over the unit review that students completed for homework yesterday. I will put the answers on the document camera and have students go over answers with their table group while I go around and check it in while asking for questions.
After a few minutes of this we will go over some questions as a whole class. I find that my students will still have some questions about partial fraction decomposition, so I will be sure to talk about that. The main issue is setting up the partial fractions when there are repeated factors or non-factorable quadratic factors. Usually if a student can see the correct partial fractions from another student, they can find the numerators on their own.
Question #5d is another one that my students will ask about. Because I asked for an exact answer, they will know that they have to solve algebraically, but they may get bogged down by the algebraic steps and working with quadratic equations. Some students may also try to solve this using matrices. One they get their answer they may notice that it does not work. Then we can have a discussion about why we cannot use the same matrix method to solve systems with quadratic terms.
After we go through the exam review, this cryptography workhseet is a good way to end the unit and to give students a new context to used matrices and inverses. I will give students the worksheet and offer a few points of extra credit if they can figure out the uncoded message and bring it in to class tomorrow. They must also show some work of how they got the answer.
On the worksheet I direct students to read a portion from their textbook (Precalculus with Limits: A Graphing Approach, 3rd Ed. Larson, R., Hostetler, R.P., & Edwards, B.H. (2001)) that gives some information about cryptography. If your textbook does not have similar information you could look online or watch the video below to get a quick rundown of how it works.
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As mentioned in the video above, the number represents what number each letter of the alphabet is (A is 1, B is 2, C is 3, etc.). An entry of zero represents a blank space.
When students set up this problem, they will likely set up an equation with (coded matrix)(encoding matrix) = (uncoded matrix). They know the coded matrix and uncoded matrix, and will try to find the encoding matrix. The tricky part is that students will likely only use 1 by 3 matrices for the coded and uncoded matrices, and it will not be enough information to find the 9 entries of the 3 by 3 encoding matrix. My students will often try to set up a system of 9 equations to figure out all of the entries.
Students will have to be a little bit more savvy and realize that they can do 9 letters at once by making the coded and uncoded matrices both 3 by 3. Since they know the first 9 letters of the message ("Greetings"), it will be possible to do this. Then they can easily find out the encoding matrix.
Since this is an extra credit opportunity, I will not give too much guidance. However, I will encourage them to work together and keep focused on what we know about matrices in order to get the correct message.