Day 42: Questions to Consider When Graphing

Quote of the Day: “The word compete comes from the Latin competere, which means ‘to seek together; strive in common; coincide.’ True competition means two (or more) rivals are playing the game they love together.” - Jim Murphy

Question of the Day: Wouldn't a squared plus b squared equal c to the fourth power?

Regular Math Objective: Use the Pythagorean Theorem to calculate the leg of a triangle when the hypotenuse and one leg is given.

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. Pepper. I loved kicking the class off with pepper today. My heavy review topics were the square roots of numbers that are perfect squares up to 400 and defining why rational numbers were rational numbers. The kids were dead - especially in the morning with the sun not rising yet and the weekend still fresh in their minds. 
2. QSSQ - I included a recap of the previous lesson today in QSSQ because we were coming off of a weekend. Well worth it.
3. Marker boards to try two problems of a rational and irrational hypotenuse. Students had all sorts of unique answers such as the hypotenuse is 100 for a triangle that had legs of 6 and 8. I only did this in two out of four classes, but when it came time for the exit ticket this proved to be very helpful. We only did two review problems with the boards, but it makes all the difference. The student in the picture below tried to argue that the side was exactly 7 and 1/7 because she added it 7 times. Her group actually corrected her by saying she was not multiplying the same number by itself but was doing 7 times 7 and 1/7. 
   
4. Exploration of finding the legs of a triangle. I really enjoyed this. Some students were able to do it on their own without my instruction. Other students needed some prompting. It was also interesting to see that no students dared to leave the answer as an irrational in a square root sign because we were taught to find approximations earlier this year. Leaving it in a square root is virtually unheard of for them so I had to introduce that. In some classes I referenced the fact that we write pi as the symbol not 3.14. 
5. Exit ticket. One triangle needed the leg solved and one needed the hypotenuse solved. One was irrational and one was rational.

The student above froze once they saw that it was irrational. The student below just sort of ditched the c squared part of the equation. 

Honors Math Objective: Derive an equation of a graph

Honors Math Standards: A1.F-BF 1 Write linear, quadratic, and exponential functions that describe a relationship between two quantities.

Honors Math Lesson Sequence:

1. I had the students work on a graph for a problem that had the number of workers on the x-axis and the amount of time it takes to paint a bridge as the y-axis.
2. We discussed the problem in small groups and then as a large group.
3. I went over four questions worth considering as we were comparing and contrasting different graphs. Is it linear or nonlinear? Is it discrete or continuous? Is it increasing or decreasing? Where are the intercepts?

 This picture just frustrated me. Then again, it does help the students line up the numbers on certain points. I suppose marker board graph paper (such a product does exist) would be useful.

 It was cool to see the student's explanation in the corner. That said I was wondering why the y-axis was horizontal and the x-axis was vertical.

Here this student is about to erase the continuous function and make it discrete. I did not say anything. I think she knew by virtue of the camera coming out it needed to be tended to. 

Eventually in going over the four questions we needed all of these students seemed to have an uh-huh moment in terms of the fact that it was decreasing, discrete, non-linear and had to avoid the intercepts. This lesson is taking a much longer time than I would have anticipated, but I'm finding it engaging for the students and fun as a teacher. We have been lacking a good deal of real-world connection to this point in the year and so this is a little refreshing.