This video addresses the problem of calling parabolas “U-shaped curves” and talks about accommodating learners by letting them develop meaningful vocabulary. It also discusses the importance of creating equations from graphs, not just graphing equations.
CCSS.MATH.CONTENT.HSF.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
CCSS.MATH.CONTENT.HSA.REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Roberts, N. S., & Truxaw, M. P. (2013). For ELLs: Vocabulary beyond the definitions. Mathematics Teacher, 107, 28-34.
The graph of a quadratic function is a special curve called a parabola. A common student stumble is that after students learn parabolas are U-shaped, they mistakenly believe any U-shaped curve is a parabola. This is kind of tricky, because yes, parabolas vary quite a bit — they can be narrow or wide, concave up or concave down, they can even be askew if you define them with a focus and directrix — but nevertheless, parabolas are a special U-shape.
And there are lots of other U-shapes that are not parabolas, such as higher-degree polynomials, catenary curves, or the letter U itself. So let’s try to keep parabolas special, shall we, by letting students know about these other U-shaped curves so they don’t overgeneralize.
Don’t Just Draw Curves from Equations, Find Equations for Curves
Now, most lessons about graphing quadratic functions will include, as you might expect, lots of graphing of quadratic functions. Students usually take a quadratic equation, plot the points, and draw the curve. Equation, points, curve. Equation, points, curve.
But the goal of this lesson isn’t to learn how to plot points. They already did that, plus technology can do it for them. The goal is to learn about features of parabolas and the connections between quadratic equations and their graphs. So how can we get students thinking about features and connections?
One idea is to give them a curve and have them produce equations. See what we did there? You could do this with one graph and one equation at a time, or you could have students explore and generate several equations and graphs at the same time. For example, give students one parabola to start with — maybe the parent function, y equals x-squared. And have them figure out some equations for new quadratics that are related to the original in some way — maybe find some that are wider than the original, narrower, higher, opens down. Students should pretty easily sketch curves that have these characteristics, but the real work will come in finding equations that produce the graphical results they’re looking for.
As they try to come up with the equations, they’ll have to tinker around with the components of quadratic equations — the leading coefficient, the constant term, and anything else they want to mess around with. And as they’re tinkering, they are thinking about how the equation relates to its parabola, and they can uncover the relationships between the equation and the graph.
As the students are talking quadratics, you can look for opportunities to accommodate learners through rich vocabulary development. Rather than front-loading vocabulary at the beginning of the lesson, a better way is to build on student talk that comes out as they are working. They will need a way to refer to that turning point in the curve that keeps moving around, and they will have to talk about how y equals x-squared is different than y equals negative x-squared. But they might start with their own terms like bottom, opens up, and crossing points, or with the equation they might talk about the front number or the number by itself.
This is a great opportunity for you to connect their vocabulary with more formal vocabulary like vertex, minimum and maximum, concave up and concave down, roots, leading coefficient, and constant term. Other words to keep an ear out for would be parent function, line of symmetry, and y-intercept.