Tag Archives: graphing

Hint Cards

Hint CardI added two components to a lesson task on rational equations.The first was an idea called, Hint Cards, shared by Michael Pershan at Shadow Con at NCTM 2015.  The second, which I will discuss in another post, was having students use Desmos to confirm answers that they found algebraically. There were so many surprising positives from this lesson that I have to share.

As always, lesson awesomeness starts with a good task. The class had studied the simplifying, solving and graphing of rational functions. It was time to write and apply them. My school’s Algebra 2 team decided on a common task titled Optimum Bait Company. I’m not sure where the task came from, but it offered the following context followed by six prompts.

My brother Matt owns Optimum Bait Company. Optimum Bait Company manufactures fishing lures. The monthly cost to run the factory is $4200 and the cost of producing each lure is an additional $0.25 per lure.

  1. If he produces 1000 lures in one month, what is the average production cost per lure?
  2. Create a function, C(x), that models the average production cost per lure.
  3. Calculate the average production cost per lure if he produces 4000 lures in one month? 8000 lures? 12000 lures? 420000 lures?
  4. As he produces more lures what price does the average cost of production approach? Why?
  5. If he wants the average cost of production to be $1, how many lures would he have to produce in one month?
  6. If he wants to make a profit of at least $4000 per month, what is the minimum number of lures he would have to produce if he sells every lure he produces for $4?

I was thinking that the students would need a lot of help on this, so I created a set of six Hint Cards. Each card gave some assistance for one of the prompts.

Front of Card

Back of Card

#1: Average Cost of 100 lures

Average = Total Cost/Total Number

#2: Create C(x)

Let x = number of lures

#3: Average Cost per Lure

C(4000) = (4200 + 0.25(4000))/4000

#4: Limit of Average Cost

The Ratio of the Leading Coefficients

#5: Average Cost of $1

C(x) = 1, instead of   x = 1

#6: Profit of $4000

Profit = Income – Expenses

As an incentive, I announced the following scoring system.

  • Like all other tasks, this will be worth 5 points.
  • There are 6 prompts. Every wrong answer to a prompt costs a point.
  • There are also 6 hints. Every hint used costs a point.
  • Yes, that means you either have one free pass on a wrong answer, or a free hint.
  • The only thing that you may ask of the teacher is for a hint card to a specific prompt.
  • 30 minutes will be allotted to complete the task.

I was pleasantly surprised how little they needed or wanted help. Many groups didn’t even take advantage of their free hint. In fact, for all of eight groups, I gave out only seven Hint Cards… total. The most common hints asked for were for #1 and #5. The most commonly missed questions were the last two. I suspect that if I had given more time on the task, students may have ask for more hints and given more effort on what I consider to be the hardest questions for the task. That’s a lesson for next time; there will definitely be a next time because of the benefits that resulted from the Hint Cards technique:

  • The time crunch spurred a hyper-focus in the students.
  • The level and intensity of the student discourse was heightened tremendously.
  • A common dynamic was having one student raise a hand for a hint, while another group member protested, “we don’t need it, yet.”
  • The average score on the task was 4.2 out of 5.

Hint Cards delivered a terrific learning experience for my students. One that I will be sure to give them again.

Graph of the Week (New Site)

Kelly teensI am getting the word out on this awesome site: Turner’s Graph of the Week. My friend Kelly Turner did a presentation at the Great San Diego Math Conference last spring and I loved her idea of having students analyze graphs from magazines and newspapers. These are mostly one-quadrant graphs with a natural context. This ties in directly to the Common Core’s call for applications and for reading non-fictional text. I was so impressed that I encouraged her to go public with the idea. I am serving as her megaphone.

Kelly does this activity once a week with her students, thus the name. The site offers several features:

    • Graphs. You don’t have to find your own. Kelly has already posted 12, and will post more on the GOWS page of the site as the school year progresses.
    • Submissions. If you like the activity and have graphs of your own that would serve others, email them to turner_k@auhsd.us. Kelly will screen the submissions and build the online collection.
    • Templates.  There is a generic worksheet template with writing prompts to guide students in reading, interpreting and analyzing the graphs.
    • Samples. On the home page Kelly will offer the graph that she is currently using for the week. Directly below that will be a student sample of the previous week’s graph.

Try it. If you like it, share with your colleagues. The same graph can be used at multiple course levels, with the level of questioning being adjusted to the level of students.

Kelly thanks for all the work on this. You have made teachers’ work easier and students’ education better.

Mr Cornelius’ Desmos Lesson

This lesson on graphing conic sections rocked on multiple levels. For the students, it involved concrete mastery of standards, conceptual understanding of several topics, higher order thinking skills, student autonomy and intellectual need. For the teacher, Mr. Cornelius of Great Oak High School, it was a week’s worth of experimenting with new software and pedagogy. The genesis of the lesson was a combination of an email and a diagram. I had sent to my Math Department a link to the free online graphing calculator Desmos.com; a mutual colleague, Michael White, shared the idea of having students use their knowledge of equations to graph a smiley face. Mr. Cornelius merged these ideas into a new 5-day lesson in the computer lab. That week produced a multitude of pleasant surprises.

Desmos smile 2

Michael started with a whole-class demonstration of Demos at the end of the period on a Friday. He posed the Smiley Face graph (shown above) as the minimal requirement for passing the assignment. The strength of this lesson is two-fold: 1) There are a variety of equations involved (circle, ellipse, parabola, absolute value, as well as linear), and 2) repeated restriction of the domain and range.

Michael invited students to create their own designs for a higher grade. He expected only a few takers, but in the end only a few decided to produce the Smiley Face, and this is where the richness of the lesson was truly found. During the week-long lab session, I observed one of the days and took a few pictures of some works-in-progress.

Desmos spiderDesmos MinnieDesmos CatDesmos House Alien
As you can see, the students independently chose to include inequalities in order to produce the shading. Here was my favorite use of shading.

Desmos Arnold

What really impressed me about the lesson was the examples of students who asked to learn something new in order to produce something they chose to create. In the example below, a student wanted a curly (wavy) tail for her pig. Mr. Cornelius taught her how to graph sine and cosine waves. Granted, this was a superficial lesson, but to see someone wanting to learn a skill from next year’s course was a treat.

Desmos Pig 1

The rigor that the students imposed upon themselves, again as demanded by their creative idea, was remarkable. Look at the detail of the door handle on this house.

Desmos House Desmos Hinge

Desmos Lesson

My favorite moment was this one with Michael and a handful of students. It is not as sexy as the pictures that the students were producing, but it was far more significant. Three students all had a similar question, so Mr. Cornelius conducted a mini-lesson on the board while the rest of the class worked away on their graphs. The topic on the board was not part of Michael’s lesson plan. It was sheer improvisation. For me, this interaction was the treasured gem of the lesson experience: A teachable moment generated by an intellectual need.

This was the first run of Michael’s lesson and in a conversation that we had while he was grading the assignments he conceded that he needed a scoring rubric. We also discussed how this idea could be woven throughout both Algebra 1 and 2 courses. The idea of Graphing Designs could span linear, exponential, quadratic and conic equations. I smell a lesson plan brewing!

(P.S. For those of you that get hooked on Desmos, I suggest you also check out the Daily Desmos Challenge)

Interpreting the Graph of a Helicopter Flight

A colleague of mine at Great Oak HS, Reuben Villar, found this wicked cool app at Absorb Learning.Helicopter
Click below to access the free online version of the app, by Adrian Watt.

Helicopter App

We incorporated this app in our latest lesson, Tubicopter (sample page here). It intensely challenges student understanding of graphing by directly contrasting the physical flight path of the helicopter and abstract shape of the graph of the relationship between time and the helicopters altitude. Toy with it and leave your comments here.

3 Cool Sites That I Discovered

I have used three web sites for the first time at school over the last couple of weeks.

Estimation 180, Andrew Stadel

Elevator EstimationThe premise here is very interesting: Students acquire number sense better by making mental estimations, than from direct instruction. Since I teach an Algebra class to a large group of high-needs students, who have proven to lack number sense, I thought I would give this one a go. While the name of the site implies estimations for 180 days of the school year, we entered at day 75. The students were hooked right away.

The process that Mr Stadel offers is even more useful than the pictures that drive the site. I have my classes participate in the following manner. My students each record their own estimates, then pair up and record on a lapboard, and then as they hold up their boards, I announce the minimum and maximum values that I see. On the Estimation 180 site, I record either the median of these values or the mode if there is preponderance of one value. Depending on the spread, I decide the level of confidence (1-5), and then submit our collective response under “Great Oak” (our high school). This committment raises the level of engagement of the students, who really want to see how close we get to the actual answer.

The site offers a handout for students to record their estimates, and their margins of error for 20 days on each side of the sheet. The students are to average this margin of error at the end each page. This serves two great purposes: 1) Students must add and divide positive and negative numbers as well as practice calculating a mean, and 2) as students progress through the year, they can see if their estimations are getting anymore accurate (average margin of error getting smaller?).  In only three weeks, I have already seen my students posing more accurate numbers.

I have other processes that I also use as warm-ups, so I won’t be using all 180 days, but the mathematical gains and enthusiasm that I am seeing in my students will encourage me to use this site as often as possible. (Chris Shore’s 180Blog)

Graphing Stories, Dan Meyer & Buzz Math

The premise of this site is that students will develop understanding of graphing through visual contexts, in this case, through 15 second video vignettes. The genius of the site is the consistency of its structure.

Time GraphEvery coordinate plane is a one-quadrant grid with time as the domain, from 0-15 seconds. The range and its scale is left to be defined for each video. Each video is shown with a clock tracking the 15 seconds, then the video and clock are replayed at half speed. The answer is revealed by superimposing the grid over the video. The graph is drawn in real-time as the video plays out. There is a variety of the types of functions offered, as well as various degrees of difficulty.

In my class I used this as remediation for the most commonly missed question on the semester final, graphing from a verbal context. So I used only about 7 of the 24 videos offered, over the course of a few days. On the next quiz, students showed a drastic improvement in their ability to, graph both from verbal context as well as from given equations. (Chris Shore’s 180Blog)

Math Mistakes, Michael Persan

This site is intended for teacher use, rather than student use. Its purpose reflects the hyper-focus of its author: self-improvement. I used this site in my most recent math department meeting. I posed two entries from the site. One sample dealt with fractions, the other with graphing. The discussion ensued around two questions: 1) Why might the students be making these mistakes, and 2) How should we as teachers respond if this were occurring in our classes?

MM Number LineMM Graph

The conversation was brief, but very rich. I used it to encourage our PLC meetings to focus more on instructional decisions. It was very well received by my teachers.