Curricular Microteaching Lesson Plan

 Chris, Snow, Danielle 

Lesson Plan:

• Subject: Pre-Calculus 11, Powers with Rational Exponents

• Total time 15 mins

• Lesson breakdown: short lecture followed by a hands on activity and gallery walk to see other students solutions

Lesson Objective:

• By the end of this lesson you will be able to evaluate an expression that has a rational number in the exponent

Intro: (2 mins)

• Let’s recall what a rational number is (give some examples and some non-examples)

• Can give the class the opportunity to show what they know and define a rational number if they remember

• Definition: a number that can be written as a fraction of two integers where the denominator (bottom number) is not zero

• Example: 5/2, -3/1, non-example: sqrt(2), pi

Short Lecture: (8 mins)

• Let’s remember our exponent rules

• Can give the class the opportunity to show what they know and share exponent rules they remember (ask them which ones they remember)

Product Rule	am • an = am+n
Quotient Rule	am / an = am-n
Power of a Power Rule	(am)n = amn
Power of a Product Rule	(ab)m = am • bm
Power of a Quotient Rule	(a / b)m = am / bm
Zero Exponent Rule	a0 = 1, a≠0
Negative Exponent Rule	a-m = 1 / am

• Next, introduce rational numbers in the exponents. Lead an investigation with the students and help them discover a reasonable definition for these values

• Start by drawing a graph of y = 2x and plotting what we know (integer exponents)

• Ask students to think about 21/2. What could that mean? Is it nonsense? If it exists, where should it be plotted? Can we make it consistent with the above properties? Give them time to discuss, then collect them for an answer (It is √2. They can recognize this with either the product rule or the Power of a Power rule)

• Extend to any unitary fraction exponent (1/n). How can they use the Power of a Power rule to determine this value? (nth root of 2)

• Finally, consider any fractional exponent (m/n). How can they again use the Power of a Power rule to define any such value? (Split the m and the 1/n up to get the rational exponent law. There are two ways to do this.)

• Write out the law explicitly; add it to our list of other laws

• x takes the place of 2 from our investigation. It must be positive though (mention that we will discuss why “at a later date”)

• Mention that it is consistent with the rules that we already know!

• Do some examples on the board, maybe one easy one more difficult: 62/5, 84/3

• Questions for extending learners: What issues arise if x is negative? How could we come up with a reasonable definition of an irrational exponent?

Hands on Activity: (5 mins)

• Break students into three groups

• Each group will have the same set of expressions with rational numbers in their exponent

• The goal of the activity is to get students to place the numbers in ascending order from lowest to highest

• After about 4-5 minutes stop and get students to walk and look at other groups solutions

• At the very end share the correct solution on the board with all three groups

• During this time encourage questions, all three instructors are walking around talking with groups and answering questions

Backup Exercise:

The largest square in this diagram has side length 8 cm. Calculate the side length and area of each of the two smaller squares. Write the radicals in simplest form.

Battleground Schools Reflection

 I learned a lot from reading this article. The first thing that stopped me was in the first few line - the three movements of math education in the twentieth century. I had never heard of that before - I'd always thought of math being taught the same way for the past hundred years. It was quite interesting to learn that there was this progressivist period of math education that was inquiry based. I also found it ironic that that's the model of education that we're aiming for now. Trends are so cyclical, I suppose that education is not immune to these cyclical trends either. 

The second thing that stopped me, simply because I found it so funny, was this line "those who like mathematics are (generally male) eggheads, nerds, absent-minded professors, and mad scientists, unable to cope with the world of human interactions, and not fully mentally competent (ranging from the mildly autistic to the completely mad)".  I do agree that the public has this perception of mathematicians. It reminds me of an interaction my sister had at work that she told me about. She was talking to one of her coworkers and she mentioned that her sister and her sisters husband were mathematicians. Her coworker looked at her wide eyed and said "... I wonder what that house is like.". I had a good laugh, what must he have pictured our house was like? Equations written on every wall? Non-stop talk of mathematics? Mad scientists with crazy hair? 

The third thing that stopped me in the article was "Conservative attitudes towards mathematics education were adopted by a number of right-wing evangelistic Christian religious lobby groups. Traditionalist, antiprogressive views on mathematics teaching and learning quickly became affiliated with a patriarchal, authoritarian, fundamentalist religious agenda, including antigay rights, antiabortion, and pro school prayer lobbies." I knew that in the past math was influenced by religion, but I had no idea that in the 1990's it was still being influenced by religion. I think of math as such a secular subject, it's hard for me to grasp why religious groups had opinions about it. It's very interesting to me that they chose math as something to latch onto. 

Lockhart’s Lament Reflection

 This article was an …. interesting read. I liked the beginning of the article, but by the halfway point Lockhart had lost me. I resonated with his beliefs on how math is being taught in school, and how different that math is from say university or research mathematics. I found his analogies about music and art class reasonable. I have always felt that the general public has the wrong impression of math as a whole, and it’s my goal as a teacher to bring creativity, problem solving, and beauty back into the math we teach in secondary school. 

With that being said, I didn’t appreciate the tone of the article, particularly the way he spoke about math teachers. At one point he went as far as to say that math teachers didn’t want to teach the subject properly because it would be too much work. To me, as a future teacher, I found this offensive. I think teachers as a group are very hardworking individuals, and it put me off his writing that he thought of us that way. As a whole, I felt that if the goal of the article was to bring about change, then he missed the mark. The tone was “calling out” instead of “calling in”, which I think puts people off of his message. It read more as an angry rant than a constructive message to teachers.

I think this article connects to Skemp in that Lockhart is frustrated with the instrumental teaching of mathematics, and that he wants the understanding to be deeper. 

Micro - Teaching Reflections

 Overall I thought my micro teaching went well! I was happy with how the lesson went, and I thought that it was successful because everyone knew how to tie the figure eight knot by the end. 

For my own review I thought that I could have improved upon the timing and pacing of the lesson a little bit. At the end I had one minute left, which is an awkward amount of time. Luckily I had a video ready that they could watch which expanded on the figure eight knot, but it wasn't enough time for them to try out the new knot. 

I appreciated the feedback that I got. I do agree with Sissie that I could have done a better job explaining exactly how we used this knot in rock climbing. I think that if I was going to do it again, I would either show a video or diagram of where the knot attaches to the rock climbing harness. It would also be cool to show a video of a rock climber falling with this knot, just to show how stable and secure it is. I think these two things would have made the understanding of the knot much stronger. I also like the feedback of adding more applications. I could have done some research to see what other uses of figure eight knots were. In terms of if someone in the group had already known how to tie one, I don't know what I would have done. This is something that I could have thought about more. Otherwise I feel like I received pretty positive feedback.