Some units and chapters in algebra lend themselves naturally to interesting openers. Interesting scenarios to discuss slopes, systems of equations or quadratic functions are abundant. Finding examples for topics like radicals and complex numbers or rational expressions can be a bit more of a challenge. Addition and subtraction of rational expressions mean that shared work problems can’t be far behind, like this nugget from algebra.com:

One good use for rational equations is the shared work problem. This solution would be of great help in scheduling employees. For example, If Bob can mow a lawn in 3 hours and Joe can do it in 5 hours, how long would it take them together?

A few thoughts come to mind:

- I’m doubting that the personnel schedulers at WalMart or Jiffy Lube are using rational expressions to schedule their employees.
- How many of our kids would guess 8 hours, or even 4 hours, as their initial guess?
- Joe needs to stop lollygagging on the job.

I set out to make a video to encourage discussion of these problems. In a first attempt, my sister and nephew were recruited to each build a Lego tower separately, then together.

Working together had little effect on the overall time, as the partnership tripped over each other digging into the bucket for Legos, and had trouble coordinating the overall tower construction. This leads to a nice discussion of the assumed independence of the two volunteers in these problems, but made for a pretty bad video.

In the video below, teachers Christine and John were recruited to staple index cards to a stack of 50 “top secret” papers. A shared work ending was also produced. But in a version that was later eliminated, Christine passed papers to John, who then stapled. In order to maintain independence, a new ending was shot where they worked separately, yet simultaneously.

Christine’s final time was 4:50, while John’s final time was 4:29

To find the ideal shared time, we let x = the number of seconds required to complete the job together.

- Christine’s rate is 1 / 290 of the job completed per second

- John’s rate is 1 / 269 of the job completed per second

Since we want one job to be completed, this leads to the equation:

Solving for x yields an ideal solution of 2:19, so the partnership’s time of 2:10 is not too surprising. The subjects admitted that they were a bit more competitive to do well working together than when they were separated. Also, my quick appearance during the shared portion on the video is due to the team needing more index cards, and not any funny business! What would happen if Christine showed up a minute late? How long would it take them to complete 2000 cards?

Hopefully, we can encourage some discussion and debate, and move away from Joe and his 5-hour lawns.