Archive for July, 2011

Recently, a high school teacher asked me about the idea related to slopes of perpendicular lines. As the conversation went on, it became clear to us that we don’t just want students to “know” that the slopes of perpendicular lines are negative reciprocals. Rather, we want students to be able to make sense of relationships between lines. Certainly, we could tell students about this relationship…we could ask them to memorize it…we could check to see if they memorized this bit of trivia by putting some item on the next test. We decided that it would be better to allow students the opportunity to try to make sense of this idea and that this could be done in class in small groups or even as a HW assignment – assuming that students have previously worked to make sense of the idea of slope. Here is a first draft of an activity/assignment. Notice the quest to guide students towards making sense and then demonstrating the sense that they have made by preparing a convincing argument. Imagine creating a classroom where students are engaged in making sense and communicating their thinking to one another. Any ideas for improving this?

1. Find the slope of the line in the graph below.

2. We now rotating the line 90o about point B. Note that the marks on the line segments are to help you to make the connections between the original line and the rotated line. Find the slope of the rotated line.

3. Describe the relationship between the slope of the original (black) line and the rotated (red) line. Do not focus on formulas but rather on the geometry seen in the graphs.

4. Confirm or adjust your ideas from number 3 by sketching the graphs of linear functions and finding the slopes of these lines. Then, rotate the line about a point and find the slope of the new line.

5. Prepare a convincing argument that describes, in general, the relationship between the slopes of perpendicular lines. Use graphs, formulas, and words to help make your argument. Be prepared to present your argument in class tomorrow.



At the community college, it has been an on-going discussion and dilemma to determine how to diagnose a student’s mathematical skills so that they might be placed into the proper course. Diagnosing student mathematical ability is not an easy thing to do!

At the heart of the issue, in my opinion, is this…do we want to assess student ability to reason, make sense, problem solve, etc. or do we want to know what skills and procedures that they can remember how to do?

It seems to me that diagnostic tests only serve to reveal how little students can remember! Why is this so? I contend that because students have not developed a well connected understanding of the kinds of skills and procedures that are on the diagnostic or placement tests, they cannot remember well and/or cannot re-construct or problem solve their way through the diagnostic situation.

My strategy has been to not worry about diagnosing students and rather to work to develop thinking and reasoning skills and to work on students’ beliefs concerning sense making in mathematics. Then, when algebraic issues arise, I try a “just in time” mini-lesson to address that issue.

For example, recently in my Calculus II course, students needed to evaluate sin(0). Several students had no idea and their only recourse was to push buttons on the calculator. I used this as a teachable moment to do a mini-lesson on the meaning of sin(A). Most students do not have a mental image of sine being the vertical position of the endpoint of the corresponding arc on the unit circle. If they did, sin(0) would not really have to be something to memorize. Rather, they could draw on their understanding of the meaning of sin(0) and figure it out!

This is an important topic (diagnostic testing) that is worthy of discussion and debate. I encourage everyone to share their experiences, strategies, and successes!


Inverse Functions: What Our Teachers Didn’t Tell Us was the engaging cover article of the March 2011 Mathematics Teacher journal written by the author team. The article was identified as an Editor and Panel Pick for Summer 2011.  Editor Albert Goetz wrote, “This article took a topic that is very difficult to teach and hard for students to understand and made it just the opposite: easy to teach and crystal clear for students.” The article may be previewed here:


Constant Rate of Change

Posted: July 14, 2011 in Math Concepts

When we say a function has a constant rate of change m, we mean that whenever input value changes the output value changes by m times as much. For example, Brunswick Kyrene Lanes in Chandler, Arizona, offered the following pricing for Monday through Friday daytime bowling in 2011.

Shoe rental: $4.25

Price per game: $3.49

In this context, the independent variable g represents the number of games played. The dependent variable C represents the total cost of playing g games (in dollars) including the cost of shoe rental. Since the price per game is $3.49 and does not change, we know the cost function has a constant rate of change of m = 3.49. We consider a table of values for the cost function.

Games Played


Cost (dollars)



3.49(1) + 4.25 = 7.74


3.49(2) + 4.25 = 11.23


3.49(3) + 4.25 = 14.72


3.49(4) + 4.25 = 18.21

Let’s verify that whenever the number of games increases the cost increases by 3.49 times as much.

In each of these cases, 3.49 times the change in the number of games equaled the change in cost.