# 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.

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.

As you can see, the students independently chose to include inequalities in order to produce the shading. Here was my favorite use of shading.

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.

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.

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.
Click below to access the free online version of the app, by Adrian Watt.

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.

# Lesson: Get Your Kicks on Route 66 (Preview)

This is an example of the many lessons available only to MPJ Monthly subscribers.

### Rapid Roy drives the highway to demonstrate that absolute value is the distance from zero.

SUBJECT: Pre-Algebra
TOPICS: Absolute value equations, inequalities
PAGES: 3
Preview Page 1

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# Activity: Tic-Tacs and Kisses

### Minty breath will score both kisses and an understanding of slope and y-intercept.

SUBJECT: Activity
TOPICS: Rates, slope, y-intercept
PAGES: 1

# Lesson: The Tortoise and The Hare

### Who wins this race depends on careful manipulation of data, algebraic equations and graphs.

SUBJECT: Algebra
TOPICS: Writing, graphing and solving systems of equations. Rate and unit conversion
PAGES: 3

# Lesson: Stixture Problems

### Multi-link cubes allow the students to SEE the solutions to mixtures problems.

SUBJECT: Algebra
TOPICS: Mixture problems; fractions and percentages; geometric and algebraic modeling
PAGES: 2

# Lesson: The Coin Fountain

### Study parabolic curves through the design or water arcs

TOPICS: Writing, graphing and solving quadratic equations; finding the roots, vertex, axis of symmetry, directrix, and focus; solving by factoring, completing the square, and the quadratic formula; systems of quadratics
PAGES: 3

# Lesson: Rising Water

### Show that words, data, equations and graphs ares different ways to represent the same thing.

SUBJECT: Algebra
TOPICS: Linear relationships, dependent and independent variables, linear equations and graphs, slope, x- and y-intercepts, slope-intercept form of a linear equation
PAGES: 4