Category Archives: 180 Blog Algebra 2

Bumping Airlines

Out of 615 million airline passengers last year, half a million were bumped from flights. 9 out of 10 of those were voluntary. What percentage of all booked passengers were involuntarily bumped from a flight?

Yes, I had fun at the expense of United Airlines’ most recent viral embarrassment, but I had two serious questions that I needed answered:

  1. Should I change my air travel habits?
  2. How many of my Algebra 2 students could correctly answer this question?

I had my class answer both questions for me. I started class by handing them the prompt and this now famous video clip:

United Air Clip

I then shared what I learned about the law in regards to this incident. United Airlines and law enforcement officials were legally in the right to remove the passenger from the plane. When overbooked, an airline has the right to randomly bump passengers, but they must first offer an adequate incentive for volunteers, which United did. These regulations are in the contract rules that we all agree to, but never read, when buying an airline ticket. The law also states that any passenger must comply with directions given by airline personnel or law enforcement officers. Since the unfortunate gentleman on the plane resisted the directions of the authorities, the airline and the police had the legal right to forcibly remove him.

It was the third part of the law, however, that was the most disconcerting for me. If an airline involuntarily bumps you, they must guarantee your arrival at your intended destination within 24 hours. But that is not good enough for me. I often travel to places where I am expected to be working with teachers early the next morning. A 24-hour delay would be far too late. So my new burning question is: Should I leave greater leeway in time when I am traveling? That is what I needed to answer. The students helped me think through it.

Nine out of 10 voluntarily bumped means only 10% of the 500,000 bumped passengers, or 50,000 passengers were removed involuntarily. That 50,000 out of 615 million is a whopping 0.008% of all booked passengers last year. So what does that mean in terms of my flight habits? How many times would I have to fly in order to expect being bumped at least once? 0.008% of what number equals one (1 = 0.00008x)? It turns out that I would need to fly 12,500 times. In over 40 years of an active adult travel life, I would have to board a plane nearly every day of my life to expect this to happen. Of course, probably and possibility are utterly different, so I could be bumped on my next flight, but I am not ready to start adding an extra day to every travel trip for such a small chance.

So how did my students do with this calculation? My prediction of one student was an underestimation. Five actually calculated correctly, with 3 others getting close, showing appropriate work. Why was such a simple math topic (calculating percentage) such a challenge for a group of 15 & 16 year olds? In talking with the students, I came to realize that this was the classic case of “making sense of problems.” There were multiple layers in unpacking the prompt,  as well as the added layer of interpreting such a small fraction of a single percentage point and the need to make a decision based on that numerical interpretation.

It is noteworthy to reveal that I gave this problem to four adults. All four answered correctly (0.008%), and all four struggled to make sense of what was being asked.

So how do we get students more proficient at making sense of problems that require basic math? Easy. We pose those problems more often. Which I intend to do.

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.

Neuron Problems & Classroom Norms in Algebra 2

Day 3, Fri Aug 12, 2016

A vs Don-stepmom-shoulderTarget: Recognize that  Voice = Choice when it comes to having a growth mindset as we solve problems about our amazing brains.

Entrance Ticket
I greeted the students at the door, but today I was checking homework. They only had to do one problem of their choosing from the Neuron Facts last night. If they did not have it, they had to quickly do one outside. Message sent: You are doing your homework in this class.

Growth Mindset
On the growth mindset web site they make a point of the “voice = choice,” meaning that we have a choice whether or not to listen to the fixed mindset thoughts that we all have, They give a 4-step breakdown of how to shift from a fixed to a growth mindset. I had fun soliciting the help of a very ancient visual of a devil and an angel on your shoulder.

Voices choices

Neuron Fact Problems
So then came time to practice recognizing the fixed voice and talking to ourselves in the growth voice, while doing challenging math problems. They already sit in groups of four, so I had them spend the rest of the period working through the Neuron Fact Problems, which I created from the Facts on the front side of the paper.  They were to call out any fixed mindset words or actions demonstrated by their partners. They actually did. I worked the room with Neuron stickers and Nicki. I honored about half the groups. I was pleasantly surprised at how well my crew worked.

During the lesson, as I worked the groups, I asked  one student how she got her answer, and she told me that she had copied from her partner. I praised her for her honesty, then paused the class and brought their attention to our classroom norms.


These were originally shared with us by Dr. Juli Dixon (@thestrokeofluck) in a math training at our district. They became very popular among our teachers. Our new principal has implemented them schoolwide, providing posters for every classroom. I drew the students attention, that we “Share, Don’t Copy.” When we share, one person explains, the other listens, then question follow if we don’t understand or if we disagree. If these three norms are occurring then writing down someone else’s solution is not copying.

After a half hour of solid work, we debriefed where we saw evidence of a fixed mindset and where we saw evidence of a growth mindset. This whole activity was very well received by the students. I gave them advance notice that Monday we will be debriefing their actual solutions to the problems.

Wrinkle Sprinkle

  • Share, Don’t Copy.
  • The equals about 3 lbs.

Introductions & Neuron Facts in Algebra 2

neuron vertical
Day 2, Thurs Aug 11, 2016

The Brain Surgeon
Today, we began my regular routine of designating a daily Brain Surgeon. Since this was our first day of the Brain Surgeon, I introduced the routines of the Drum Roll, Reading of the Dual Target, Music Cues, and the Wrinkle Sprinkle. The students seem to embrace the spirit of of it all.

Student Introductions
As with every new school year, I had each student briefly state their name and something interesting about themselves. When they were all done,  I recited all their names. That always impresses a class. Then I told them things about myself. I state that yesterday we started with math, because that is what we are all about here. But since I teach math to them, they are also important and I need to know who they are.

Growth Mindset
Most of our Course Teams across the district agreed to do some kind of growth mindset activity. Here was mine.

I started by summarizing the plethora of lists of fixed vs growth mind set statements with two pictures. I told the students that research in student learning is showing that self-perception of talent as a limit or as a starting point has a tremendous influence on their learning.

Talent Wall

Then I shared that scans of the brain of someone with a fixed mindset versus a growth mindset, shows something very interesting. When faced with a challenge, the fixed mindset brain “goes cold.” It literally shuts down. However, when faced with the same challenge, the growth mindset brain “fires up.” It knows that more is being asked of it, so it kicks into high gear to meet the challenge, rather than duck it.
Brain MindsetsNow it was time to test out where we see ourselves demonstrating  a fixed or growth mindset.

Neuron Facts
I gave the students the worksheet with the Neuron Facts on the front side. I found these on the internet and thought they would make for a good lesson since they highlight the amazing function of our brains. I added the subheadings of Fast, Crowded ,etc. I started with a common practice of mine Notice & Wonder popularized by Annie Fetter (@MFAnnie) of Math Forum.  My Gradual Reel-In process looked something like this:

  1. You Do: Independent response.
  2. Ya’ll Do: Each member of the group shares both their notice and wonder.
  3. We Do: Each group decides on one Notice and one Wonder from those shared. These get shared out by each group as I write them on the board.
  4. I Do: I summarize the major point(s) that I want all students walking out with. Here it was the process of Noticing and Wondering and how we facilitate group discussion in class… And of course how amazing our brains are.

The groups were then tasked with doing one problem together. Homework was to do one more.

Wrinkle Sprinkle
Tying into the concept of the plasticity of the brain, I joke that when we learn we get a new wrinkle on the brain. Each class then concludes with what we learned that day. The brain surgeon leads and records the discussion. The students today stated that they learned…

  • Negative thoughts shut down your brain
  • Speed of the brain cell
  • The amount of oxygen the brain uses

First Day – Algebra 2

Day 1, Wed  Aug 10, 2016

{I am new to Chaparral High School, having transferred within my district as a Math Specialist.}

Opening Quiz Alg 2 on the 6Cs: After greeting each student at the door with a high-5, I started the year by answering the transformation question: “How will you (the students) be different in June than you are now, because of my class?” I am still answering that question with the same 6Cs that I launched 2 years ago. My Claims-Based grading system and the students portfolios are structured as such also.

6 Cs PicAs I do with all classes each year, I gave the students the blank copy of the quiz below, and told them this was not to be graded nor was it a test of their previous knowledge. It was like a movie trailer of things to come, but I still wanted them to give me their best shot. I then gave them my standard 3-response speech.

As a mathematician I cannot always give an accurate response; I cannot always give a complete response; but I can always, always, always give an intelligent response. Blank is not intelligent.

I pressed them to give me something… numbers, equations, drawings … anything intelligent.

Opening Quiz Alg 2 Pic

They worked on these independently, then in groups, then as a class, followed by my summary. I wanted to model this process of “gradual reel-in” (as opposed to gradual release) right away, because I use it often.

During the class discussion, one senior claimed out loud, “This is the 5th time that I have taken this class!” (She had failed two semesters as a junior, then 2 semesters in summer school.) I told her that this year she will pass because, “You are that smart, and I am that good.” I had the students repeat this:

Me: “You are that …”
Class: …smart!”
Me: “I am that …”
Class: “…good!”

This was a set-up for the Growth Mindset discussion that was coming over the next few days. In the meantime, I hope I sent the message that I believe in them, and that I believe in my ability to teach them (The 3 Growth Mindsets).

The students brought some terrific energy. I’m so looking forward to my first year as a Puma.