**Goodie:** “A technique/question that can be applied in many places and teaches thinking.”

**Baddie: **“A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

**January 2009 Goodie of the Month** – A Good Question for Algebra 1

Two ships are sailing in the fog and are being monitored by tracing equipment. As they come into the observer’s rectangular radar screen, one ship, the Rusty Tube, is at a point 900 mm to the right of the bottom left corner of the radar screen along the lower edge. The other ship, the Bucket of Bolts, is located at a point 100 mm above the lower left corner of that screen. One minute later, both ships’ positions have changed. The Rusty Tube has moved to a position on the screen 3 mm left and 2 mm above its previous position on the radar screen. Meanwhile, the Bucket of Bolts has moved to a position 4 mm right and 1 mm above its previous location on that screen.

Assume that both ships continue to move at a constant speed on their respective linear courses. Using graphs and equations, find out if the two ship will collide.

Why do I like this question?

- Students can understand it and it is fun.
- They can graph it on paper or using a graphing program.
- It involves finding the equation of a line through 2 points (twice) – good reinforcement.

**Why do I think it is a “good question”?**

- The graph
*looks like*every 2×2 system of linear equations they have solved in Algebra 1.- It looks like the boats collide at the intersection point (see below).
- It seems like all they need to do is solve the system and be done.
- … until you say “Where is time on the graph?”.

- The student can build a animated simulator that “shows time” –
**easily**!- Then they can see that the boats do not collide.
- Below is a simulator I built using the freeware GeoGebra.
- Here are step-by-step directions.

- The kids can make the boats collide – what fun!.
- They can move the starting points until they get the boats to collide.
- They can also adapt the simulator so that they can change the slopes and get the boats to collide. Directions here.
- My thanks to David Cox for seeing this!

- You can get all kinds of mathematics out of them.
- You can get them to calculate when each of the boats reaches the intersection point in the original question.
- You can get them to check the math on their “colliding simulator” to see if the boats really do collide, where and when.
- You can ask them about a 3D graph and what this would look like when the boats don’t collide and when they do.

To animate, click on the play button at bottom left of **graph**.

To animate manually, right-click on slider and deselect “Animation on”. Then, click and drag the point on the slider.

Source: I found this question asked on answers.yahoo.com. My webpage for this question is: mathcasts.org/mtwiki/Gq/BoatCollide