# Wed. Feb 22

Hi All —

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.