2. Vision is important. For the past few weeks, we have been working on our online common core geometry course with varying degrees of structure. Some days, we would know exactly what to do, other days it was a struggle to stay together. However, after a bit of a discussion today, I think we are on the right track for the remainder of PCMI.

3. Wendy gave a presentation on her end of year assignment for her geometry class. She created a grid with all of the units they had covered for the year on the top of the grid and various levels of Bloom's Taxonomy on the side. The center was filled with various activities, and students had to pick one from each column and each row. In this way, students were exposed to a wide variety of topics and modalities.

4. I learned strategies to make students into teachers. Ben gave a presentation on his initiative to turn his class into cooperative group of student-teachers. He created videos to model what effective tutoring looks like and discussed common pitfalls such as steamrolling. He explained how he sorted the class into student-teachers and not-yet-student-teachers via exit slips from the day before. Ben also shared some of his tactics for when students struggle with being the teacher, such as only allowing student-teachers to ask questions or collaborating as a class to come up with question stems.

5. I’m going to try using giant 2 ft by 3 ft whiteboards in class. Earlier in the week, I had a discussion about how writing either horizontally (as part of a group) or vertically (teaching a group) can affect learning. In a separate talk, I thought about how important it is for students to reflect and revise their work - especially when working in groups. Lastly, I realized how expensive sticky-note poster paper is. Taken together, it makes sense to have students write and present with over sized whiteboards rather than disposable poster paper. They’re reusable, revisable, can be placed on the chalk sill or window sill, and make great student work displays.

6. (Extra Credit) In projective space, parallel lines intersect at infinity. This is because projective space can be represented by all complex numbers

*(x,y)*such that

*(hx, hy)*is equivalent to

*(x,y)*for any non-zero value of h. In other words, all pairs of numbers that are proportional to each other are equivalent. We could also say that the pair

*(x,y)*is equivalent to another pair when the ratio

*y/x*is equivalent to another ratio. But, what happens when

*x=0*? Normally we can’t divide by zero, but in this case it simply stands for the pair

*(y, 0)*which we label as the point at infinity. Projective space can also be represented as the set of all lines through the origin (where the vertical line represents infinity) or as a circle (where the north pole represents infinity). Thanks to Dagan Karp from Harvey Mudd for his lecture, both in the pool and at the chalk board.

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