Tag Archives: Geometry

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.

Graphing Geometry

On Tuesday a question bubbled up in me for no apparent reason, saying, “What does happen when you vary the side lengths of a rectangle but keep the perimeter constant?” To others in the math teaching blogosphere, I’m sure this question is old hat, and even I recognize it from the days of my own high school homework. I can picture myself reaching a question about varying side lengths at the end of an assignment (you know, in the word problem section), weighing the point value of the problem against the inconvenience of figuring it out, and opting to not worry too much about leaving it half done.) But on Tuesday I flipped over the answer key in my hand, made a table of values, and graphed area vs. height.

After noting and confirming that a perfect square yields the maximum area, another question struck me and I thought, “I wonder if I can write the equation for that graph.” It looked similar to a y=–x2, so I wrote in the adjustments that would put the max of that graph in the same location as my max, (6.5, 42.25). I expected to need to make additional adjustments to fit my curve, but what do you know, it was exactly right!

So, said I, I bet I can do that for right triangles with the same give and take between height and base (for every increase in height, with a  max height of 12, the base decreases by the same amount). Looking at it now, of course it makes sense that you’d divide the rectangle equation by two, but I went through a process of fitting the curve to see how it would turn out.

With success number two in the bag, I went on to holding the area of those right triangles constant at 50 and relating the base length to the height.

Then, finally, using the same data about right triangles with area set at 50, I related perimeter to height and truly offered myself a challenge. I quickly pegged it as developing from y=1/x, but had no clue how to make it fit. After a fair bit of graphical exploration, it suddenly dawned on me: all I have to do is write the process of finding the perimeter as a function of height! Connection made!

What a triumph. This is what I want for my students, and how to “give” it to them is clearer to me now. My students know how to use area and height to solve for a right triangle’s base (b=2A/h); they know how to use base and height to find the hypotenuse (a2+b2=c2); they know how to use all three sides to find the perimeter. But they may not know that they can use these geometric pieces to explore totally cool graphical relationships.

Stalling

The goal of teaching LD students is to make information, etc., accessible, but there is a risk of becoming too eager to give “help” when these students struggle. The result is a cluster of students who plead for help even when they can succeed without it, who throw their hands up at the slightest sign of struggle, who demand assurance after every thought, who insist that the teacher stand beside them while they work, just in case.

Realizing that I could do more to discourage such dependency, I’ve adopted a new habit of stalling when certain students ask for help with their work. I protest that it will take me some time to finish what I’m doing and that they should solve the problem as best they can without me—even if they’re unsure—and that we’ll discuss how they solved it when I make my way over to them. Or, at the very least, that they should skip that problem and move on rather than sitting immobile, waiting for teacher.

Whiteboard Overdose

In a class of nine geometry students, I am assigned to work specifically with three tenth graders. They attend lectures and class activities with the other students, then typically leave to work independently or go over homework with me. This allows students in both groups to get more attention from a teacher, and provides me with a sort of teaching apprenticeship. The students, like their classmates, have learning disabilities that cause me to continually reassess what I think I know about how they think.

My latest realization is that I’ve been overusing the whiteboard. At the beginning of the year we introduced angles (acute, obtuse, right and straight) and I found colored markers a great asset as we practiced identifying angles on the board. The students enjoyed working problems on the board with such a small group and even grew comfortable enough to work on the board in front of the whole class when the groups were together. Based on that experience I continued to use the whiteboard each day, expecting that a good method in one case would be good in all cases.

Only this week I accidentally let the students sit down and get to work on their own without rolling out the whiteboard. They worked at individual paces, and I was soon circulating to answer questions on paper instead of addressing each question with the whole group. I was able to see how each student progressed from question to question and was pleased to find them frequently succeeding. Clearly the whiteboard is an invaluable tool but should be given a rest from time to time.