Engaging students: Defining the terms perpendicular and parallel

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Allison Metlzer. Her topic, from Geometry: defining the terms perpendicular and parallel.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

The concepts of perpendicular and parallel will be implemented in many of my students’ future mathematics courses not only in high school, but also in college. In algebra, the students are asked to find the slope or the rate of change. In looking at the slope, students are asked to find if it’s parallel or perpendicular to another function’s slope.

In geometry, many shapes have properties that define them as having parallel or perpendicular sides (i.e. squares, rectangles, parallelograms, etc.). Also, in order to decide if triangles are similar, their corresponding sides must be parallel. In order to use the Pythagorean Theorem, the triangle must be right angled or have the two legs perpendicular to one another.

In calculus, students are asked to find orthogonal vectors which are also defined as perpendicular vectors. Also, calculus incorporates concepts from algebra and geometry which in turn, include parallel and perpendicular lines.

Therefore, many, if not all of my students’ future math courses will use the topics parallel and perpendicular. Thus, it would be important for me to teach them the two concepts correctly now so that there wouldn’t be any misconceptions in the future.


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C3. How has this topic appeared in the news?

One big thing the news talks about every two years is the Olympics. Using the concept of parallel and perpendicular, the constructions are made for all of the different events. Apparent examples of events incorporating parallel lines are track, speed skating, and swimming. The one I will focus on is swimming, namely because it is a very popular Olympic event and one of my favorites. Pictured below is an Olympic swimming pool of 8 lanes. Do the lanes appear to be parallel? Two things that are parallel are defined as never intersecting while also being continuously equidistant apart. One can clearly see the lanes of the pool never intersect. If they did, then the contestants could interfere with one another. Also, because the Olympics is a fair competition, the lanes are equidistant in order to give each contestant a fair and equal amount of room.

Because the Olympics is a well-known event featured in newspapers, articles, and on TV, the students will be able to understand this real world application of parallel and perpendicular.


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E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Before I would play the video, https://www.youtube.com/watch?v=vnnwfcDcNlY, I would first ask the students to think of as many examples they can of parallel and perpendicular in the real world. After about a couple of minutes, I would tell them to keep those in mind and see if the video included any they didn’t think of. I would play the video from 1:25 to 3:05 which is the portion that displays all of the examples. It has clear pictures of recognizable objects which incorporate parallel or perpendicular lines. Also, the video has labels on the pictures to even more clearly describe where the components of parallel and perpendicular lines are. I believe that the initial brainstorm along with this video would get the students thinking about the importance of parallel and perpendicular lines. Also, I would make the connection that those examples would not be considered parallel or perpendicular unless they met the following definitions. Then I could explicitly define both parallel and perpendicular.

Thinking of real world examples, and seeing pictures of them will help the students understand what parallel and perpendicular lines should look like. After they have this initial understanding, they then could get a better grasp of the definitions. Also, they would recognize the importance of following the definitions to correctly construct objects involving parallel and perpendicular lines.


Detwiler, dir. Intro to Parallel and Perpendicular Line. YouTube, 2010. Web. <https://www.youtube.com/watch?v=vnnwfcDcNlY >.

Medicine’s Uncomfortable Relationship With Math: Calculating Positive Predictive Value

Taken from: http://archinte.jamanetwork.com/article.aspx?articleid=1861033&utm_source=silverchair+information+systems&utm_medium=email&utm_campaign=archivesofinternalmedicine%3aonlinefirst04%2f21%2f2014

In 1978, Casscells et al1 published a small but important study showing that the majority of physicians, house officers, and students overestimated the positive predictive value (PPV) of a laboratory test result using prevalence and false positive rate. Today, interpretation of diagnostic tests is even more critical with the increasing use of medical technology in health care. Accordingly, we replicated the study by Casscells et al1 by asking a convenience sample of physicians, house officers, and students the same question: “If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs?”

Approximately three-quarters of respondents answered the question incorrectly (95% CI, 65% to 87%). In our study, 14 of 61 respondents (23%) gave a correct response, not significantly different from the 11 of 60 correct responses (18%) in the Casscells study (difference, 5%; 95% CI, −11% to 21%). In both studies the most common answer was “95%,” given by 27 of 61 respondents (44%) in our study and 27 of 60 (45%) in the study by Casscells et al1 (Figure).

Encouraging Students to Tinker

A recent blog post from Math Ed Matters had the following pedagogical insight:

How do we encourage students to tinker with mathematics? As a culture, it seems we are afraid of making mistakes. This seems especially bad when it comes to how most students approach mathematics. But making and then reflecting on mistakes is a huge part of learning. Just think about learning to walk or riding a bike. Babies are brave enough to take a first step even though they have no idea what will happen. My kids fell down a lot while learning to walk. But they kept trying.

I want my students to approach mathematics in the same way. Try stuff, see what happens, and if necessary, try again. But many of them resist tinkering. Too many students have been programmed to think that all problems are solvable, that there is exactly one way to approach each problem, and that if they can’t solve a problem in five minutes or less, they must be doing something wrong. But these are myths, and we must find ways to remove the misconceptions. The first step is to encourage risk taking.

A few months ago, Stan Yoshinobu addressed this topic over on The IBL Blog in a post titled “Destigmatizing Mistakes.” I encourage you to read his whole post, but here is a highlight:

Productive mistakes and experimentation are necessary ingredients of curiosity and creativity. A person cannot develop dispositions to seek new ideas and create new ways of thinking without being willing to make mistakes and experiment. Instructors can provide frequent, engaging in-class activities that dispel negative connotations of mistakes, and simultaneously elevate them to their rightful place as a necessary component in the process of learning.

Here are a few related questions I have:

  • How do we encourage students to tinker with mathematics?
  • How do we destigmatize mistakes in the mathematics classroom?
  • How do we encourage and/or reward risk taking?
  • What are the obstacles to addressing the items above and how do we remove these obstacles?

Source: http://maamathedmatters.blogspot.com/2014/04/encouraging-students-to-tinker.html

An unorthodox way of solving quadratic equations

This post concerns an unorthodox but logically correct technique for solving a quadratic equation via factoring. I showed this to some senior math majors as well as graduate students in mathematics; none of them had ever seen this before. Suppose that we want to solve

6x^2 - 13x - 5 = 0

without using the quadratic formula. Trying to solve this by factoring looks like a pain in the neck, as there are several possibilities:

(x + \underline{\quad})(6x - \underline{\quad}) = 0,

(x - \underline{\quad})(6x + \underline{\quad}) = 0,

(2x + \underline{\quad})(3x - \underline{\quad}) = 0,


(2x - \underline{\quad})(3x + \underline{\quad}) = 0.

So instead, let’s replace the original equation with a new equation. I’ll get rid of the leading coefficient and multiply the constant term by the leading coefficient:

t^2 - 13t - (5)(6) = 0, or

t^2 - 13t - 30 =0.

This is a lot easier to factor:

(t - 15)(t+ 2) = 0

t = 15 \quad \hbox{or} \quad t = -2

So, to solve for x, divide by the original leading coefficient, which was 6:

x = 15/6 = 5/2 \quad \hbox{or} \quad x = -2/6 = -1/3.

As you can check, those are indeed the roots of the original equation.

This technique always works if the quadratic polynomial has rational roots. But why does it work? I’ll give the answer after the thought bubble.

green_speech_bubbleThe original quadratic equation was

6x^2 - 13x - 5 = 0

Let’s make the substitution x = t/6:

6 \displaystyle \left( \frac{t}{6} \right)^2 - 13 \left( \frac{t}{6} \right) - 5 = 0

\displaystyle \frac{t^2}{6} - \frac{13t}{6} - 5 = 0

Multiply both sides by 6, and we get the transformed equation:

t^2 - 13t - 30 = 0

Although I personally love this technique, I have mixed feelings about the pedagogical usefulness of this trick… mostly because, to students, it probably feels like exactly that: a trick to follow without any conceptual understanding. Perhaps this trick is best reserved for talented students who could use an enrichment activity in Algebra II.

Opting Out of High-Stakes Assessments

In response to the growing movement of parents who have opted out of high-stakes testing, Michelle Rhee wrote a defense of the (commercial) enterprise in the Washington Post. This op-ed piece was brilliantly deconstructed, point by point, at http://curmudgucation.blogspot.com/2014/04/wapo-wastes-space-on-that-woman.html. I encourage you to read the whole thing. A few excerpts:

[Michelle Rhee]: No, tests are not fun — but they’re necessary. Stepping on the bathroom scale can be nerve-racking, but it tells us if that exercise routine is working. Going to the dentist for a checkup every six months might be unpleasant, but it lets us know if there are cavities to address. In education, tests provide an objective measurement of how students are progressing — information that’s critical to improving public schools.

Except that the current crop of Standardized Tests are not like stepping on a scale or going to the dentist. They are like trying to find out a child’s weight by waterboarding him. They are like having your teeth checked by a blind blacksmith. Because, in education, tests NEVER provide an objective measure of anything, because tests are made by people. Yes, tests are useful– but only good tests. And do you know what good tests are useful for? They are useful for providing information critical to helping further the education of students.

I am not a Systems True Devotee. STDs believe that we just have to create a well-oiled precision machine and it will spit out Smarterer Student Products like toasters off an assembly line. I would stop to further develop the point, but we’re only one paragraph in. These woods are dark and deep, but we have miles to go.

From this diving board, That Woman proceeds to register her stunned amazement that in various places, there’s a movement that is convincing parents to pull kids out of these tests! Really!!! These marvelous tests that will tell us how schools are doing!! What in the name of God are they thinking!?!?!!?


[Michelle Rhee:] This makes no sense. All parents want to know how their children are progressing and how good the teachers are in the classroom. Good educators also want an assessment of how well they are serving students, because they want kids to have the skills and knowledge to succeed.

Allow to help you comprehend this, O She. You are correct that parents and educators do want to know these things. Your mistake is in believing that they can only know this by looking at standardized test results.

Yes, the Great and Powerful Woman Who No Longer Has a Curtain To Hide Behind imagines a world where parents sit at home after eight months of school, wringing their hands and saying, “Oh, jehosephat, I wish we knew how Janey was doing in school. But we have no idea.” Meanwhile, at school, teachers sit and the lounge and say, “Yeah, I’ve been with this kid for eight months but I just don’t know how he’s doing. Thank God we’re going to be giving a high stakes high pressure badly written unproven standardized test soon so that I’ll know how it’s going.”

In That Woman’s universe, parents and teachers (sorry– public school parents and teachers) are dumber than dirt. In fact, the list of People Standing in the Way of Educational Excellence gets longer and longer. Parents, teachers, democratically elected school boards– reformy fans have an enemies list that keeps lengthening.



[Michelle Rhee:] We don’t need to opt out of standardized tests; we need better and more rigorous standardized tests in public schools. 

Yes!! When you’re doing something stupid and bad and non-productive, do it More Harder!!


[Michelle Rhee:] We also shouldn’t accept the false argument that testing restricts educators too much, stifles innovation in the classroom or takes the joy out of teaching. That line of thought assumes that the test is the be-all and end-all — and if that’s the perspective, the joy is already long gone. 

Here’s a multiple choice test for you, dear, exhausted reader. Select which statement best reflects the meaning of the above excerpt:

1) Do not assume that the test is the be-all and end-all. It will just be-all the way we decide to end-all teaching careers, school existence, and student futures.

2) You cannot claim that this year’s testing is sucking up all the joy of teaching, because we actually drained that lake long ago and killed the fish flopping in the mud with fire and big pointy sticks.

Jonathan Katz on Some Problems of Common Core Mathematics

Courtesy of Diane Ravitch:

Jonathan Katz taught mathematics in grades 6-12 for 24 years and has coached math teachers for the past nine years.

He prepared this essay for the New York Performance Standards Consortium, a group of high schools that evaluates students by exhibitions, portfolios, and other examples of student work. The Consortium takes a full array of students and has demonstrated superior results as compared to schools judged solely by test scores.

What is of special concern is his description of the mismatch between the Common Core’s expectations for ninth-grade Algebra and students’ readiness for those expectations.

Here is a key excerpt…

[The Common Core standards seem] to honor the idea of problem solving and the many ways a student might engage with a problem. It seems to value the process of problem solving, the ins and outs one goes through as one tries to solve a problem and that different students will engage in different processes.

To implement such a standard, a teacher would need to present students with problems that allow for and encourage different approaches and different ways to think about a solution—what we call “open-ended problems.” Yet, when you look at the sample questions from the Fall 2013 NY State document you would be hard pressed to find an example of a real open-ended problem. Here is one example in which a situation is presented and three questions are then posed.

Max purchased a box of green tea mints. The nutrition label on the box stated that a serving of three mints contains a total of 10 Calories.

a) On the axes below, graph the function, C, where C (x) represents the number of Calories in x mints.

b) Write an equation that represents C (x).

c) A full box of mints contains 180 Calories. Use the equation to determine the total number of mints in the box.

A situation is presented to the students but then they are told how to solve it and via a method that in reality few people would even employ (who would create a graph then a function to find out the number of full mints in the box?). If you are told what to do, how can we call this solving a problem? (This would have been a very easy problem for most students if they were able to solve it any way they chose which is what we do in real life.) In fact, all eight problems in the same of Regents questions follow the same pattern. Students are told they have to create the equation (or inequality or system of inequalities or graph) to answer the question. Thus there is no real problem solving going on—merely the following of a particular procedure or the answering of a bunch of questions. Why don’t we use problems where there is a real need for an algebraic approach? Why would we ask students to look at a simple situation then force them to use an algebraic approach, which complicates the situation? We should be helping students to see that the power of algebra is that is gives us the means of solving problems that we would have great difficulty solving arithmetically.

If we were truly trying to find out if our students are developing the ability to problem solve, we would never create questions of this nature. They would be more open-ended so students had the chance to show how they think and approach a problematic situation. But that can’t happen on a test where everyone is instructed to do the same thing so we can “measure” each student’s understanding of a particular standard. This is not real mathematics and a contradiction of the Common Core Standards of Mathematical Practice!

Why does this matter? The consequences are huge, and not just for students. Consider the message we are sending to teachers. Since students will be assessed on following given procedures rather than how they strategize and reason through a problem, then teachers’ lessons will become all about following procedures to prepare their students for an exam they must pass in order to graduate. This will simply perpetuate the same failing math teaching practices we had in the past, will compound the dislike that students already have for math class, and will not in any way help our students to develop mathematical thinking.

Source: http://dianeravitch.net/2014/04/02/jonathan-katz-on-some-problems-of-common-core-mathematics/

How the Texas Testing Bubble Popped

The Dallas Morning News recently ran a three-part long-form article on the passing of HB 5, which significantly rolled back the number of high-stakes exams that are administered in Texas. From the concluding paragraphs:

So in a relatively short time, a Legislature that had been the most all-in in the nation about high-stakes testing as the key tool for accountability became almost as all-out as federal law would allow.

As inevitable as it may look in retrospect, however, the shift was anything but at the time. Politics, policy and more than 30 years of history pushed hard against the change in course. As House Speaker Straus put it recently:

“We got as close as we could to something not happening, but it happened.”

HB 5 did not have my unequivocal support, as it removed the requirement that all high school students take Algebra 2 before graduating from high school. But, on balance, I think HB 5 definitely helps more than it harms.

Part 1: http://res.dallasnews.com/interactives/2014_March/standardized_tests/part1/

Part 2: http://res.dallasnews.com/interactives/2014_March/standardized_tests/part2/

Part 3: http://res.dallasnews.com/interactives/2014_March/standardized_tests/part3/