2 diggers
Students spent the last week solving the problem - "Will the diggers meet? If yes, when do they meet? If not, when do they pass?" They viewed a video of two diggers approaching each other from a distance and made some calculations based on their positions and rate of speed. Students have used many different concepts in investigating this problem - some have developed distance and midpoint formulas to find out the distance between the two points and then done rate calculations and proportional relationships based on the amount each digger moves per day, some have derived linear equations to model the position of each digger as a function of their position in the x-y plane, some have translated each digger's path so that they are on the same line, and some have used sequences and series in order to express the minutes and seconds just before the two diggers passed each other. Students are still completing their final answers and preparing their presentations, so check back in for more to come!
Student Work
Students interpreted and approached this problem in a variety of interesting ways. We all started by superimposing a grid over the problem:
By extending the pattern, students quickly came to the conclusion that the diggers do not meet each other. The second part of the question, when the 2 diggers will pass each other, became more complicated. Students had to decide what that meant. Here are some of their interpretations:
This group translated one of the lines down so that the two trajectories formed a single line, and they wrote an algebraic expression for the position of each digger for any given day, and made an equation that modeled the distance between the two diggers. When that distance is "zero," the group determined that the two diggers would meet.
- They pass each other when they have the same x-coordinate
- They pass each other when they have the same y-coordinate
- They pass each other when they become the CLOSEST they will ever get to each other
- Since the two trajectories have the same slope, translate the two lines down so that they make up the same line. They pass each other when they meet on this line.
This group translated one of the lines down so that the two trajectories formed a single line, and they wrote an algebraic expression for the position of each digger for any given day, and made an equation that modeled the distance between the two diggers. When that distance is "zero," the group determined that the two diggers would meet.
A picture is coming!!!
This group determined that the closest the two diggers would ever get would be "square root of 2" units apart. They found out which day the two diggers would have the same x-coordinate, which day the two diggers would have the same y-coordinate, and then they made a square out of those 4 data points. They used the midpoint formula to find the (x,y) coordinates each digger would have when they were "square root of 2" units apart from each other, and then they worked backwards to find the exact day and time that they first become closest.
This group decided that they would pass when they had the same x-coordinate. They found how far in the x-direction each digger went in a day, and then in half a day, and then in a quarter of a day, and then in and 8th of a day, and so on. They determined that the two would met each other in 23 days and 1/2 + 1/8 + 1/32 + 1/128 +... and so on. This is an infinite summation, and the group investigated sequences, series, and summation notation to determine their answer.
The next question asked how each digger would have to change their slope in order to meet up along the same line. This group developed an expression modeling the position of each digger at any given time, and then they used that expression to develop a formula for the slope each digger would have to change to in order to meet up on any given day.