"There is no one way"

Thursday, October 9, 2014

Make These Designs

Among the activities I have developed, "Make These Designs" is among the most popular. Students enjoy it, and teachers appreciate how it can be used as an engaging introduction to, or interesting review of, an important topic: graphs of linear functions, and the parameters m and b in y=mx+b.

When electronic graphing first came onto the math education scene, one reaction was that it should not be allowed, because it would undermine student understanding of graphs. This is not unlike the earlier idea that basic calculators would prevent students from developing their arithmetic prowess, and the more recent one that computer algebra systems will undermine students' ability to learn algebra. However, there is one difference: unlike basic calculators in elementary school, and CAS in secondary school, electronic graphing has been accepted by a substantial majority of high school teachers. That's probably because it is allowed on the SAT. (At some schools, students are not allowed to use electronic graphing until Algebra 2.)

Among teachers and curriculum developers who were early adopters of electronic graphing, the standard lesson was "graph this, graph that, graph this, graph that -- what do you notice?" That approach works for a few students, but for many, it is just a game of "guess what the teacher wants me to say". The reason is that graphing this and then graphing that does not engage them intellectually. No thinking, no learning. The teacher or curriculum developer has done all the thinking, choosing functions that will supposedly guide the students to a given conclusion. Alas, students may not notice anything, or they may not notice what we want them to notice.

For example, after graphing y=x, y=2x, y=3x, and so on, on an old graphing calculator, a student may notice that the line is getting "more jagged", as the pixelated nature of the screen is not as obvious in the first graph. Or after graphing y=4x, y=4x+1, y=4x+2, and so on, a student may notice that the lines move to the left. Not what we were hoping!

Alas, "what do you notice?" is still going strong, and it is a terrible misuse of this technology. "Make These Designs" provides an alternative.Here is how it works: students are given figures such as these:

      

and they are asked to find the parameters m and b in y=mx+b that are needed to create the designs. They, not the author of the worksheet, are doing the thinking. They are making something, not following directions. They correct their own mistakes because the software gives immediate and non-judgmental feedback. They are proud of what they achieve. Perhaps with the teacher's help, they set their own level of challenge: should they make an exact replica of of the design, or just get the general idea? And finally, the teacher can ask for an illustrated write-up of what they learned, to help make it explicit, and to make it stick.

What prompted me to write this post is that I just updated that page on my Web site, and I created a GeoGebra version of the worksheet, from which the above figures are excerpted. Find the worksheets for different platforms, an in-depth article about the activity, and links to possible extensions in Algebra 2 and beyond, by clicking here.

--Henri

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