# Using Open Questions to Teach Simplest and Equivalent Forms

In September, while taking that Jo Boaler class I’ve mentioned, I finished a section about asking open questions that allow students to use different approaches and learn from each other. That’s a valuable idea, I thought. Unfortunately it’s just not compatible with my upcoming unit. Simplifying expressions is all about getting a single answer and there’s really just one way to do it.

The idea of asking open questions kept turning in my mind, though, until an application was suddenly obvious. I would keep the goal of having students write expressions in simplest form intact, but would add a dimension by having students also write expressions in non-simplest equivalent forms. This would allow students to follow their own thought processes and provide individualized responses, while also developing the concept of equivalence.

I wrote an expression in simplest form on the board, then asked students to write it in equivalent ways using

• Subtraction (the requirement of like terms for addition and subtraction was a worthwhile challenge)
• Multiplication
• Division
• Negative Exponents*

We used this activity during the simplifying unit, and returned to it during reviews of previous material. Some students understood the meaning of the activity quickly, while at least one didn’t perceive  it until just before the spring final: “Oh! We’re just writing things that will give us that first thing.” Bingo. Sometimes comprehension takes time. That’s why I’m in favor of continual review and intelligently designed cumulative finals rather than the take-a-test-and-forget-about-it model.

*I didn’t think to include negative exponents in the list until late in the year.

# 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.