Day 41: Pythagorean Theorem Exploration

Quote of the Day: “Mark Twain said, ‘I can live for two months on one good compliment.’ That’s the way most people feel. They are willing to work hard if they receive recognition for their efforts.” - John Maxwell

Regular Math Objective: Use area of squares and triangles to find an unknown side length

Regular Math Standards: 8.G. 6a. Understand the relationship among the sides of a right triangle. b. Analyze and justify the Pythagorean Theorem and its converse using pictures, diagrams, narratives, or models.

Regular Math Lesson Sequence:

1. WODB #28
2. Review the first four questions from the Pythagorean theorem exploration. I got the idea for this exploration from Math is Fun. I modified it a little. Instead of using a and b for the legs of the right triangle, I used 3 and 4 (color pic in word) to make the problem less abstract. I feel like students at this age are still having a hard time with variables. I kept one side a variable so that we were discovering something in this problem. As we were uncovering what c was, I would give students three minutes to discuss in groups and go over problems in groups of two every two to three minutes. As students were exploring I circumvented to classroom manage, but also to direct the conversation where necessary. It was really cool when they got to the final questions.
3. In classes where time allowed, I had students cut out the four triangles so we could see what it looked like to subtract visually. 
4. After the exploration, I used the term hypotenuse. I had students write it at the bottom of the exploration as a picture. I then let the cat out of the bag and put the a-squared plus b-squared equals c-squared jargon on the board. I felt bad, but I also remember taking the test to become a math teacher and finding the origins of this theory to be baffling as a college graduate, so I wanted students to see it. After seeing it we could break it down again. 
5. For an exit ticket (again where time allowed) I had groups find the hypotenuse of a right triangle with 6 and 8 as the legs. 

Overall I loved this lesson. Students were asking me questions like does it matter if it's a right angle and why a squared plus b squared did not equal c to the fourth power. I also had a student try to make sense of all it algebraically (which we will revisit on Monday). 

I also saw many rich mistakes with units and square root calculation. 

Honors Math Objective: Graph everyday situations

Honors Math Standards: A1.A-SSE Interpret expressions that represent a quantity in terms of its context.

A1.F-LLE Distinguish between situations that can be modeled with linear functions and with exponential functions.

Honors Math Lesson Sequence:

1. WODB (same as above). 
2. Revisiting the exit ticket from yesterday. Today I let students talk by discussing what they wrote on the exit ticket. I think the conversation was vastly improved because students had the opportunity to try it on their own and then could compare. As we discussed a candle burning over time, one student mentioned that she had thought she was graphing how many inches had been burnt over time and not the height of the candle. The conversations were really powerful. 

Ultimately as I was taking part in the conversations with students, I realized it boiled down to four questions. Is the graph increasing or decreasing? Is it discrete or continuous? Is it linear or nonlinear? What should the intercepts look like? We only really broke down two graphs, but that is how rich this content is for where these students are in their learning. Next class, we can continue the conversation with how long it will take to paint a bridge.