This video presents an idea for introducing quadratic functions — instead of starting with the definition or the standard form, give them a variety of examples and non-examples of quadratics and let students discern the difference! It also addresses a cultural connection with regard to why we call quadratics quadratics.
CCSS.Math.Content.HSF.IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
CCSS.Math.Content.HSA.SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24-34. http://www.sciencedirect.com/science/article/pii/S0959475212000904
Guo, J. P., & Pang, M. F. (2011). Learning a mathematical concept from comparing examples: The importance of variation and prior knowledge. European Journal of Psychology of Education, 26(4), 495-525. https://link.springer.com/article/10.1007/s10212-011-0060-y
As mathematics teachers, all of us have experience with quadratic functions but lots of our students don’t. So our key question is how do students learn what is and what isn’t a quadratic function.
Lots of textbooks begin with clearly defining quadratic functions in terms of the standard form.
This is kind of like teaching a child what a dog is by just showing an amazing pure bred labrador. What happens when they see a terrier or a schnauzer or a great dane? Do they still recognize it as a dog? What about this critter? Is she a dog?
Just like seeing a variety of dogs helps you learn the essence of what makes a dog a dog, students need to get to the essence of what makes a quadratic function, well, quadratic.
One way to start is by giving students lots of examples of quadratic functions. You can start with the classic quadratic functions (the labradors, if you will). You’ve got your parent function of course, and perhaps some other nice functions with integer coefficients, but then you want to add in some with non-integer coefficients, in non-standard order, and perhaps even functions that are in factored-form (too far?). You can also get creative with placing terms on either side of the equation. You could even put the dependent variable, y, on the right hand side (gasp) or use variables other than x and y (double gasp).
Back to dogs, to further solidify their essence you might throw in some close but not quite examples like these. For quadratic equations, give them some non-examples. Students have recently learned about linear functions, so why not start there? Then you can move to other polynomials or absolute value functions. What about a constant function? Sure, why not? Feel free to be sly as a fox with your non-examples.
Students can look over these examples and non-examples to try to pick out what symbolically makes a quadratic function quadratic. They should notice things like all of the examples have a quadratic term or that the degree of the non-examples is not 2. As a class, you can come up with a definition. This allows students to be flexible in their views of quadratics, beyond just standard form. Hooray!
There’s a student stumble that can happen when students use their textbooks and see lots of examples of quadratics in standard form. Students often think that it’s not quadratic if the squared term doesn’t come first [left equation], or they may think that it’s not quadratic if there’s no linear term [right equation; show students’ confused faces here], so these are good issues to follow up on so that you can trigger and resolve these misconceptions.
Also, for extra fun you might ask students why the leading coefficient can’t be 0 instead of just telling them. (“Because I said so” is not a great reason).
Finally, we want to mention a cultural connection. Why are quadratics called quadratics? That’s a weird name, right? Because quad means 4, not 2. The names made more sense back with linear functions because they produce lines.
But if you have any Spanish speakers, ask them if they know any words that are similar to quadratic and if so, what they mean. They may say cuadrado or cuadro which means square. The connection is that all quadratic functions have a squared term as the highest degree.
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