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I am a student teacher trying to brainstorm some effective lesson plans for combinations and permutations for a high school statistics course. My master teacher has decided that he will introduce combinations, so I'm left with presenting counting problems involving permutations with nondistinct items and probabilities involving combinations. For example, by the end of the lesson I expect the students to find the total number of distinct arrangements of the word TATTER. I also expect them to solve problems like these:

A manufacturing site produces 120 computers, 4 of which are defective. The quality control manager selects 5 computers. What is the probability that exactly one of them is defective?

In this problem, I expect them to reason that they need 1 defective computer from 4 defective computers: $ \binom{4}{1}$ and that the number of choosing 4 nondefective computers from 116 is $\binom{116}{4}$. They are then supposed to use the multiplication rule and then divide the result by $\binom{120}{5}$.

What activities/worksheets/teaching strategies would be helpful for this lesson? I've looked up on-line and there aren't that many resources available. The target audience are 11th/12th graders who at least passed Algebra II with a C and Precalculus with a D.

My main concern is the presentation of the materials. Many of them are expected to have problems with manipulating with factorial expressions and a significant % are developmentally not ready yet to handle questions that seem open-ended or ask for something that requires additional thinking beyond the lesson. The majority of them are familiar with teachers showing them how to do a math problem and then copying a procedure to similar questions.

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Let me try to answer this.

I think that one of the most important things you have to teach your students is the intuition behind problem-solving. The art of problem-solving, in my opinion, consists of writing down what you do/don't know, breaking down the problem in smaller, digestible pieces, and then trying to solve a simpler version of the problem first before solving the original problem. Throughout my experiences, some of the best problem solvers I've met are people who can articulate well what they do and don't know, and seek help on things that they don't know.

Writing down what you don't know is, in my opinion, a key trait commonly found among good problem solvers. Going back to the example, if I am to teacher, I would first ask them to do a think-aloud exercise. First, I would tell them to re-read the question and ask themselves what terminologies they do or don't understand. This can yield several insights for you, such as are most of the students struggling trying to understand what a permutation is? Are students having trouble understanding the term "exactly one..." in terms of probability? Or are students having trouble translating words in the problem into mathematical terms? Also, this can help you teach better by preventing the "theater effect", where I define it as "I've attended a great lecture, I felt like I understand what the teacher is saying, but when I solve a conceptual problem, I can't do it." Writing down what you don't know can be beneficial to both the students and the teachers because it helps identify gaps in students' conceptual understandings and how teachers can improve their teaching methods.

Once you've identified the problems that students encounter, ask the students to write down what they do know. Writing down what they do know, in my opinion, is useful because then you have a way to measure how much they understand about the background knowledge needed to solve the problem. If the students are completely oblivious about probabilities and combinatorics, then you need to review what they are. On the other hand, if the students can recite the definitions of permutations and combinatorics but don't understand how to translate them into operationalizable knowledge, then it's a mapping problem that you are dealing with.

Finally, instead of telling them that they have to solve the original problem, try to teach them how to reduce this problem to a similar problem that they've solved before. Did they solve a problem where instead of having 4 defective computers, all of the computers were defective? Teaching them problem reduction can help students understand the connections between problems they've solved before to new problems that they've never seen before.

To summarize, I think that you should conduct a think-aloud exercise where they write down what they do/don't know, and teach them how to reduce an unfamiliar problem to something that they are familiar with through recalling past exercises.

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