Tag Archives: quadratic

Project 8: Function Art

Instead of a journal, I assigned this Desmos art project to get kids working and experimenting with graphs. Next time I think I’ll make this first graphing project less free-form, asking them to make a specific image (a face?) or recreate a given pattern.

I improved the assignment sheet for a similar project given later in the year, but this is what I had managed to put together in time for Project 8.

Editable version here: Ch. 8 Desmos Graphing Project

Ch. 8 Desmos Graphing Project

Advertisements

Journal 6: Rank the Solving Methods

Journal 6 asked students to rank the three quadratic solving methods they’d been working on: factoring, graphing, and the quadratic formula. This was the most heavily curriculum-related writing they’d had to do, which is probably why they struggled to describe the math with any specificity.

I liked having written evidence that preferences for solving methods differed, so I posted opposing excerpts in the classroom (for example, one student talked about preferring the factoring method, while another described how he hated that method the most).

Editable version: Ch. 7 Pre-test Rank Solving Methods Journal

Screen shot 2014-05-29 at 3.37.02 PM

Philosophy of the Final 2013-14

I organized this year’s trigonometry final according to theme. The sections of the final were:

  1. Vocabulary [synonyms, examples, descriptions, comparisons]
  2. Equivalence [simplifying expressions, factoring expressions]
  3. Functions [linear, quadratic, exponential, trigonometric] and Graphing [linear, quadratic, sinusoidal; domain and range]
  4. Interpreting Graphs [distance vs. time]
  5. Solving Quadratic Equations
  6. Exponents and Logarithms
  7. Angles and Trigonometric Values [sine and cosine; deriving the values for angles in quadrant one, providing the values for other standard angles]

I find I’m a big supporter of the cumulative final, even and especially for students who struggle with long-term retention. How else will they train their minds to hold on to things? I’m an equal proponent of intelligently designed cumulative finals. My final this year was not a test designed to congratulate those with natural retention and punish those without it. We spent time throughout the year, plus a good chunk there at the end, building student retention of important skills and information, making my final an opportunity for students to take pride in having actually learned things.

Not that my final was a cake walk. The expectations in the test were high to match the value I intended it to have.

Two Highlights and a Lowlight of Quadratics, 2012-13

  • Highlight: Having students use arms to form the shape of a parabola. Engaging large muscle groups in class gets a little extra blood flowing and promotes retention–I don’t do it as much as I’d like. One student turned the “arms parabola” into a dance move to go along with her hit new song, “I’m a Parabola!” When we got into graphing quadratics in standard form, a student stood in front of a projected Cartesian plane and others instructed him how to position himself and his arms to represent the function.
  • Highlight: The quadratic formula song.
  • Lowlight: Major struggles with simplifying the quadratic formula once they had all the values in the right places.

Goals Jotted Down

I remember drafting this post in May, but I guess it never made it out of my drafts folder. Until now, that is!

05/23/13 Here are some goals I jotted down for improving Trigonometry next year.

  1. Set better sections for student notebooks. The notebook sections I required this year were basically useless. Here’s what they should be:
    1. Notes and Classwork
    2. Homework
    3. Quizzes and Tests
    4. Journals and Projects
    5. Miscellaneous
  2. Trigonometry is not bathroom and drinking fountain time (yeah, it was a problem this year)
  3. Support the “function box” concept better*
  4. Make better intro learning activity for quadratic functions**
  5. Emphasize slope as a rate of change
  6. Contrast rate of change vs. accumulation
  7. More “discuss/work with a partner”
  8. Follow through on promise of notebook checks

7/10/13 Of course there are others, it seems hundreds more. Tweak that activity. Improve those notes. Allow them to critique their own presentations. Make sure homework assignments are worthwhile. But at the end of the school year, the bulleted eight were the goals that seemed big, essential.

*7/10/13 After coming across the illustration of inverse functions that I featured here, I’m not sure if I’ll continue to use the function box. I might.

**Number 4 is my primary curricular goal. At the end of last year I set a primary goal to improve the way I taught logarithms, with the result that this year the logarithm unit was one of the best of the whole year. Quadratics are the first non-linear functions my students study so their introduction deserves to be genuinely meaningful. Thus, it receives “primary goal” status.

Introducing Quadratics

Quadratics were the first non-linear functions my students ever encountered. From reading different blogs I had the idea that it would be cool for the non-linearity of the relationship to be a surprise that they would discover and therefore take to heart by exploring the relationship between two quadratically related variables. This plan was also influenced by my own experience in 9th grade of being utterly confounded by seeing the teacher graph a parabola for the first time. I was convinced it was wrong. When you graph a function, it has to be straight! This can’t be right.

As for the two quadratically related variables, I figured it would be good to work with a physical shape, so I went for the base-length of a rectangle with constant perimeter and the area of that shape.

The collective student response I anticipated:

“Gee, I wonder how the base length of a rectangle with constant perimeter is related to the rectangle’s area. This is a neat-o activity. Okay, now I’ll plot the ordered pairs I found and…wait a minute! Something’s weird here. My data points don’t lie in a straight line! How can this be? This is like nothing I’ve seen before! Well, since I’m working with a physical shape with which I’m quite familiar, and have proven for myself that these two variables are related in this curved way, I must accept the reality and strength of this kind of relationship. I wonder what other variables are related this way. Wowzers, this is fascinating!”

The actual collective student response:

“Why are we doing geometry? What’s the point of this? What do I do with the values in the table? How do I graph them? And then I connect the dots? Okay.”

Not even the slightest hint of interest that they had graphed a curve instead of a line.

Problems:

1) The apparent aimlessness and disconnectedness of the rectangle activity. Blame this on my thirst for surprising them with the quadratic relationship. I should have introduced the activity with something like, “These two values are related quadratically. This activity will help you explore what that means.” I hadn’t wanted to taint their discovery process with my teacher-y information, but I ended up robbing them of the sense of purpose that they needed.

2) Many students lacked automaticity for graphing a table of values. A problem indeed.

3) Many students showed no meaningful comprehension of the concepts of mathematical relationships and functions (related to problem 2). While certainly not everyone will be intrigued by the same mathematical discoveries that I was, I’m sure some of the lack of interest in the curve was caused by the meaninglessness of even line graphs to these students. If a graphed line communicates no information to you, a graphed curve is hardly different.

Improvements for next time:

1) Give an intro to the rectangle activity. Okey dokes.

2) Provide physical rectangles that allow them to adjust and measure the side lengths; allow for non-whole number side lengths. (This time through, the rectangle was only hypothetical, and they were dealing with whole numbers exclusively.)

3) Consider spending time practicing reading graphs before teaching quadratics, and moving the review of domain and range up from where it currently is (right before teaching inverse trig functions). Although a more realistic expectation would be to review domain and range before quadratics and before inverse trig functions.

4) Prep self for the potential dismay of the pre-calc teacher that my students have covered so little material because I was forever adding preparatory and enrichment activities to the material we did cover. Dismay or no, I’m convinced that it’s better for these students to understand some things than to have seen many things and understand none.