**In class** today we worked on a tricky motion problem and started to learn the algebraic methods for solving a system of equations. By the end of the module students should understand how to solve a system of equations by a variety of methods including graphing, substitution, and elimination. I think it is also an opportunity to study situations involving motion and rates.

**Today's Motion Problem**

Aryahi and Sophia are about to start a race around a 400m track. They plan to run in opposite directions starting from the same spot. Aryahi can run a lap in 1.25 minutes and Sophia can run a lap in .8 minutes. How far from the start will Sophia be when she passes Aryahi for the third time?

I solved this problem with the class and had students discuss the steps at their tables. It would be a great chance for you to ask them to explain it back to you. **Some questions you can ask:**

- How do we find that Aryahi can run 320 meters per minute and Sophia can run 500 meters per minute?
**Use the formula d = rt, substitute d = 400m and t = 1.25 min for Aryahi, and d = 400m and t = 0.8 min for Sophia. Solving the equation 400m = (r) * (1.25 min) we get r = (320 m)/(1 min) so 320 meters per minute. Similarly for Sophia, 500 meters/min.** - What is the significance of 820m per minute?
**Since they are running in opposite directions, 820 m per minute is the speed at which they are covering the distance between them around the 400 meter track.** - $400 m div 820 m/min approx 0.4878$ what are the units for the answer here?
**They are minutes. So 0.4878 min times 60 seconds per minute is 29.268 seconds. It takes the two runners about 29 seconds to cross paths the first time.** - How would you explain the significance of the moments at which they cross paths?
**When they cross paths for the first time, the total distance run by the two runners must be***exactly*400 m in total. - Can you show me how to finish solving the problem?
**The correct answer in the end is about 68.3 meters**

**Today's assignment**: complete all problems from 5.7 Task, Set, and Go.