We began our morning by drying off and (in some cases) taking some drastic measures to dry our shoes.

Then we reviewed the homework revisiting the Kayak, Film, and Currency problems. We considered how the concepts of x-intercept, y-intercept, slope, and inverse function relates to each of those problem contexts. Josh got us started by showing and explaining his work on the Kayak problem. (Thanks Josh!)

We then moved into the discussion of the “What isn’t a linear relation?” sheet. We came up with some key features that linear relations have (e.g., the constant rate of increase or decrease). We also discussed proportions, and how to tell whether a linear relation is a proportion (in particular, the two quantities maintain the same ratio throughout the table). Andrew admitted to providing a “rule that expired” in checking only whether the two quantities were zero at the same time (this broke down in the octagon problem). We went down several side roads, including continuous vs discrete quantities, additive vs multiplicative comparisons, and how proportions are fundamentally multiplicative relationships.

Before lunch, Andrea and Fran recommended more books and resources:

- Seeley, C. (2014).
*Smarter than we think: More messages about math, teaching, and learning in the 21st century.*Sausalito, CA: Math Solutions. - Shumway, J. (2011).
*Number sense routines: Building numerical literacy every day in Grades K-3.*Portland, ME: Stenhouse. *Teaching Children Mathematics-*a monthly magazine for early childhood and elementary mathematics teachers (Go here for more info.)- Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006).
*Teaching number in the classroom with 4-8 year-olds.*London: Paul Chapman. - Van de Walle, J., Karp, K., & Bay-Williams, J. (2010).
*Elementary and middle school mathematics: Teaching developmentally*(7th ed.). Boston: Allyn & Bacon. - NCTM’s Navigating through…. .. (Part of the
*Principles and Standards for School Mathematics*Navigations Series.) Go here for more info.

After lunch, we used Andrew’s adaptation of the Candle Burning problem as an opportunity to discuss ideas related to tasks and cognitive demands, funneling vs. focusing, scaffolding, productive struggle, and curriculum materials. We moved into a discussion of equity and access by discussing last night’s reading from *Principles to Actions *and the graphic of kids standing on boxes. We concluded the pedagogy discussion by posting our grade-level tasks in a timeline on the wall.

Then we worked on the “Fox’s Furniture Problem.” Andrew started by presenting the scenario without a given question, eliciting participants to ask questions that struck them. Participants worked on solving the problem and explore other possibilities. We found commutativity and associativity provide a way to justify why a “30%, then 10%” discount yields the same final price as a “10%, then 30%” discount. Terri showed how using a variable can demonstrate that the 40% discount will always be better than “10% then 30%” (or the other way), regardless of the original price.

We ended by analyzing some students’ work to solve the problem, and judged them based on accuracy and efficiency. We found that a correct answer does not always imply a correct process, but also that a “weird” process is not necessarily inaccurate. We discussed how re-trying the purported method for another set of numbers can often reveal that the process is wrong, but also that trying just one more example might lead to a weird coincidence (as happened in class).

**Pedagogy Homework**

Read the “Professionalism” section from *Principles to Actions* (pages 99-108).

**Math Homework**

Review your materials from the past two weeks in preparation for the course evaluation (also the post-test, I guess).

Also, at lunchtime, Andrew insisted that Fran and Andrea pose for this picture with him and the Lion:

Enjoy your last evening in State College!