Note: If you would like to leave a comment, please don't give away the solution.
<SPOILER ALERT> My own handwritten work for the problem appears at the bottom of this post.
At our high school, we have a 35 minute period on Tuesday, Wednesday, Thursday, and Friday. This period allows us to stagger lunches, one for the freshmen, the other for the upperclassmen. I have a "CATS" class devoted to Math Club. This class is distinct from my problem-based learning course called Mathematical Theory and Problem Solving (MTPS). In Math Club CATS today, I posed the following problem to the students:
All my students are struggling with this problem. Many make assumptions about congruences that may or may not be true because the triangle 'looks' a certain way. Below are some photos of students actively working to find a solution.
Exhibit A: Solving a quadratic system of two equations
Exhibit B: Making mistakes in a good direction... Pythagorean theorem, auxiliary lines inside and outside the diagram, attempts at establishing congruences between triangles in the diagram
My students are still working to find a solution. As a whole class, they collectively agree on a numeric solution for s^2 but are having trouble reconciling this value with the answer choices.
Question: From a pedagogical perspective, how does a math teacher support students when performing operations on nested radicals?
The problem below is taken from the PROBE I Exam from University of Nebraska - Lincoln Math Day 2000. The acronym PROBE refers to Problems Requiring Outstanding or Brilliant Effort. If you want more information on the UNL Math Day contest, click here.
I use this problem with students and in-service teachers alike to challenge procedural thinking. This problem frustrated me when I first encountered it in 2004. I was a first year teacher teaching at a school in Omaha at that time. I remember feeling the same way many teachers feel when they get the wrong answer: Am I in the right profession? Am I not a good math teacher? Shouldn't I know this? However, good mathematics is knowing how to behave when one isn't sure what to do next.
Feel free to post questions or comments about work on this problem. How exactly should a math teacher pose the question to optimize the classroom conditions for learning?
On the first day of school this year, I gave students different numbers of Starbursts based on the number of characters in each student's given first name. Students did not know how I assigned the differing numbers of Starbursts to each student. I asked students to "fairly" allocate the Starbursts, disallowing the students to cut the Starbursts up. I wanted students to have a conceptual hook on which to hang the notions of mean and median. In this case, the arithmetic mean isn't worth much; the median tends to be what students choose to allocate.
After a discussion about this unfair allocation, I gave each student a post-it note to place on the number line I drew on the back dry erase board. The photo features four classes worth of data. We then had discussions for the following week in each class, both regular stats and AP Stats, about the inferences we could make regarding the typical number of characters in a student's first name at the high school. The photographed sample size, pooling the four classes together, is 79 students out of our approximately 800 students at the high school.
I took this photo while shopping with my wife. Students in my MTPS class worked on mathematical investigations related to information from the photo. I will post more about the students' work as it develops. The activity ran like this:
4 minutes: Present the image to students. Formulate as many questions as you can as an individual. What does this photo make you wonder?
5 minutes: Partner up. Write down questions and observations you each had in common.
7 minutes: Groups of 3 to 4. Decide on 3 questions the group will use as a basis for mathematical investigation.
20 minutes: Limited resources (peers, textbooks, pencil, graph paper, ruler) Come up with a plan for how you will address these questions mathematically. How will you maximize on your time in the computer lab?
THEN we go to the computer lab.
My primary point of emphasis with this image is to investigate the temperature conversion. There's something going on there. I will post student work examples before Monday 9/30. Feel free to leave comments or questions.