Tag Archives: applications

Confirming Answers with Graphing Software

I added two components to a lesson task on rational equations. The first was an idea called Hint Cards which I discussed in a previous post. The second one was having students use Desmos to confirm numerical answers that they found algebraically. This lesson is worth sharing, because of what I discovered that the students didn’t know how to do and what they learned from the activity.

The students had just completed the  Optimum Bait Company task. I gave them 5 minutes to check their answers with Desmos. Mathematically speaking, they needed to confirm that for the equation, , the following were true:

  • C(1000) = 4.45
  • C(4000) = 1.3, C(8000) = 0.775, C(12000) = 0.6 and C(42000) = 0.26
  • The horizontal asymptote is y = 0.25
  • C(5600) = 1

I was struck by how utterly stumped they were on using a graph to check these answers, even though they possessed all the prerequisite knowledge necessary. They understood the context of their answers (The average production cost of 1000 lures is $4.45). They knew how to find a y-value from a graph given the x-value (C(1000) = 4.45). They also were experienced at graphing equations with Desmos (y = (4200 + 0.25x)/x), and they knew how to establish a domain and range for axis in context. While my students had each of these four connected skills, they were missing the connection between them all, so I embraced the teachable moment.

I started with the basic idea that the equation we need to graph was the one for the average cost:  C(x) = (4200 + 0.25x)/x. A major problem faced them, though, when they first graphed this equation… they couldn’t see the rational function in the default window.

Bait Desmos Blank

So my next move with the class, was to use the numeric results to determine the domain and range. The range was simple since we all of the monetary answers were between 0 and $5.00. The domain needed a bit more discussion because the one value of 42000 lures compressed the graph so significantly that the class thought it better to leave it out of the visible domain, so we agreed upon 0 < x < 13000.

Bait Desmos Window

From here I could have just traced with finger along the screen to show where a point with a value of x would be located on the curve, but I wanted to tie in the writing of horizontal and vertical equations, and solutions of systems. Therefore, I had the students enter the additional equation of x = 1000, and click on the point of intersection.

Bait Desmos 1K

The students were getting happier and more confident so we kept rolling by entering a table with the additional increasing values of x, representing the number of lures.

Bait Desmos Table

The table confirmed the students answers and supported their common response that the values were approaching a limit of 25 cents. This appealed to their sense of context that the cost per lure could not drop below the original 25 cents per lure. That made for an easy connection to the finding the horizontal asymptote of the rational function for which the degree of the numerator and denominator are equal, which in this case was also 0.25. So we graphed this as an asymptote. It turns out that it was easier to see how closely the curve approached the asymptote if we temporarily increased our visible domain to the 42000 lures.

Bait Desmos Asymptote

Finally, we entered the equation y = 1 in order to see the number of lures required to drive the average cost down to $1 a lure.

Bait Desmos 1 Dollar

This serendipitous exercise was amazingly productive for reinforcing understanding of graphing in general, as well as the connection between numerical values, algebraic formulas and context.

Millionaires and Their Cars?

Car FerrariMy teenage son is preoccupied with three things these days: water polo, his girlfriend and expensive cars. He has fantastic talent in water polo, and has a wonderful girlfriend. He does not have an expensive car.

Currently, he is saving for his first car and knows that he will have to start with a used, low-end model, but he dreams big. He is always talking about Ferrari’s and Rolls-Royces. We like to talk about them together and point them out on the road whenever we are driving. He is convinced that he will own one someday. When I respond to his talk of grandeur, I want to sound like the Encouraging Dad (“Terrific, what kind of successful job do see you see yourself having, so you can afford that kind of car?”), but I worry that I sound like the Practical Dad (“That’s nice, it might be more realistic to set your sights on a cheaper car.”). The truth is that my words usually come out somewhere in the middle, which led to our very interesting math conversation the other day.

On a long drive back from a water polo game, we were talking about reasonable incomes (Practical Dad ruling the moment). He is a Junior in high school, so his interest is peaking about how much money is to be made as an adult. The conversation went like this:

Me: Guess what the average annual income is in America.

Preston: I know, because we talked about this in History class. $36,000 a year, but if I make $100,000 a year, a can save half of that and buy a rich car in five years.

Me: Do you know what percentage of Americans make over $100,000 a year?

Preston: 25%?

Me: It is actually about 4%. I know several people who make that kind of money. None of them drive a Ferrari, so you are going to have to make more than that. (Encouraging Dad trying to break through.)

Preston: If I made a million dollars a year, I could buy it in one year and still have enough to live on.

Me: With enough left over to care of me and your Mom. That would be awesome, but you are going to have to do something special, because less than one-half of one percent of Americans make a million dollars a year.

Preston: It has to be more than that. Look at how many rappers there are making bank.

Me: And think about how many are making just a normal living or how many are standing on a street corner singing while they hold their hat out for tips. Very few earn “Checks that look like phone numbers.”

Preston: Look at how many millionaires we know.

Me: I would say less than 5, off the top of my head.

Preston: Yeah, see?

Car LamborghiniInspired Math Question #1: If you know 5 millionaires, what percentage is that of all the people you know?

Inspired Math Question #2: If one-half of one percent of the people you know are millionaires, how many people would that be?

Preston: I bet there are at least a million millionaires in the country.

Inspired Math Question #3: Given that there are 300 million people in the U.S., and that 75% are adults, would one-half of one percent of American adults be more than a million people? (to be estimated while driving without a calculator)

Me: I am guessing that we are both correct on this one.

Preston: I still say it has to be more than that then. (Whether I am encouraging or practical, I am still Dad, so he must win!) Look at how many expensive cars we saw just today. There was a Ferrari, a Lamborghini and a Bentley.

Me: Yes, and think of how many other cars we saw today.

BentleyInspired Math Question #4: Approximately how many total cars might you see driving on a freeway for an hour on a Sunday afternoon? (must explain your reasoning on this one)

Inspired Math Question #5: If you see three expensive sport cars on that same trip, what percentage of all the cars would that be?

As we arrived home, Preston was still seeking victory. He is very good with mental math, so he knew where I was going with all the number crunching. In order to get the upper hand, he needed to bring in an expert, and what better expert in the world of teenagerdom to call upon than the internet? He Googled on his smart phone, “How many millionaires are in America?” and got an answer of over 3,000,000. He loudly reveled in glory. I countered with the age-old math argument of the importance of definitions. In this case, there was a difference between annual income and net worth. He was having no part of it. He was to busy flexing and bragging to Mom about how he just “owned” Dad in a math debate.