Who Takes 5 Hours to Mow a Lawn?

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

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.


Factoring – Sending Out the Bat Signal!

One of the joys of my job is having mathematically interesting chats with my colleagues about how they approach  specific problems with their classes.  These conversations often begin as one-on-one discussions, but usually evolve into calling multiple people into the fray to give their two cents.  This semester, a teacher in my department is tackling an Accelerated Algebra II class for the first time.  Having taught academically talented kids for many years, my advice to him was to constantly challenge his students, perhaps using problems like those from the American Mathematics Competitions as openers.  But while offering up academic challenges can keep a teacher’s mind sharp, there is the risk of having that “hmmmm…” moment…..that uncomfortable feeling where you’re not quite sure what the correct response to a student question is.

The discussion today came from a review of factoring, and a problem which seems innocent enough:

Factor x6– 64

Take a moment and think about how you would factor this….show all work for full credit…

Enjoy a few lines of free space as you consider your work….

And…time….pencils down….

The interesting aspect of x6– 64 is that it is both a difference of cubes and a difference of squares.  I used the neat algebraic interface on to do some screen captures and make the algebra look pretty here.  In this case, the calculator factors this expression as a difference of squares, (x3– 8 ) (x3 + 8),  which then become both a sum and difference of cubes and can both factor further:


But, the initial expression is also a difference of cubes, and can be factored as such.  It is verified below:


The plot thickens as the discussion then centers about the “remnants” we get when we factor a difference of cubes.  We can verify that the two “remnants” (underlined in red) from the first factorization are factors of the remnant of the second method (underlined in green):


So, what’s happening here?

The extra, messy, factor we get when we factor a sum or difference of cubes is up for discussion here.
According to Purplemath:

The quadratic part of each cube formula does not factor, so don’t attempt it.

But we don’t have a quadratic here (though we could perform a quick substitution and consider it is one), we have a 4th degree polynomial.  Even the algebra calculator on the site doesn’t care for this quirky 4th power expression:


So, I am looking to my math peeps for some thoughts:

  1. Is there an order to consider when a polynomial meets 2 special cases?  Should we look at sum of cubes or squares first?
  2. Does anyone have any insight on x4+4x2 + 16?

Good night, and good factoring…

Algebra Middle School

Tapping Into the Addiction of Bubble Wrap

Engaging students in discussion of mathematics in applied situations is a rewarding experience. Seeing students immerse themselves in a task and offering to share their results makes a math class hum with excitement. But finding the right scenario, the right “hook” which will drive discussion can be an effort. While we hope to link math to real-life science and engineering, sometimes the silliest data collection experiments create a buzz in class.

I give you the Bubble Wrap Challenge.

The past week, I worked with a 7th grade teacher on a slope activity. By the end of the unit, students would be expected to compute the slope of a line via a formula. In my experience, students tend to understand indivdual aspects of slope,as they are often taught in pieces, but have difficulty shifting between meanings. What do we want students to understand about slope?

Slope between data points can be computed using a formula
Slope can indicate steepness
Slope can indicate a rate of change

As students entered the class, and the teacher took attendance, I was playing on a SMART Board with a Virtual Bubble Wrap Applet. You’ll need Java for it.   I challenge you to play with it for less than 10 minutes, and without calling 3 friends over to play.  Can’t be done.  Try Manic Mode for the extra-special dose of stress relief.


After some initial playing, we sought to find the Inter-Galactic Bubble Wrap Champion of period 2.  Each group was given an Ipad loaded with a similar app and 60 seconds to play the game.  While a student played, a partner wrote down the player’s score every 10 seconds.  Results were then plotted and a connected line-graph made.


Discussion then centered around finding not only the overall bubble-popping rate, but debating the 10-seconds intervals when Aiden was the most, and least successful, at bubble-popping. A second contestant was then added to the mix…


Which player was the fastest popper? Who was the best in a short period? Students were soon able to compute rates for segments, without prior knowledge of the slope formula. The teacher later introduced the formal formula. The payoff comes when students volunteer that we can identify the “best” popping rates by looking for steep segments, and lower popping rates in shallow segments.

Now back to popping some bubbles…..