Engaging students: Absolute value

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 Andrew Wignall. His topic, from Pre-Algebra: absolute value.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

In some sense, absolute value has been with us for a long time, but it’s also relatively recent. Distances have always been measured as a positive value – Denton and Dallas are 39 miles apart, for instance. It’s not that one is 39 miles away, and the other is -39 miles away – they’re both the same distance apart. We take negative numbers for granted in our lives now, and have learned to accept them relatively early in our advancing math education in schools. Absolute value developed as a way to “remove” the negative from negative numbers for calculation and discussion.

In fact, mathematicians didn’t discuss absolute value much until the 1800s. Karl Weierstrass is credited with formalizing our notation for absolute value in 1841! However, this is because negative numbers were not given serious consideration by mathematicians until the 19^th century, when the concept of negative numbers was more formally defined. With negative numbers, mathematicians needed a way to talk about the magnitude of the negative numbers – and so entered absolute value!

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

The concept of absolute value is used in many places in many math and science classes. In geometry, volume and area are almost always positive – if you are dealing with figures of variable size, you’ll need to use an absolute value to ensure the volume/area is positive. When dealing with square roots of squared figures, we often have to deal with two possible answers, positive and negative – but absolute value simplifies this complication in many calculations. In physics, time and distance are always positive, so we again need absolute value. In chemistry and statistics, percentage error is often expressed as a positive value. Calculus uses absolute value when dealing with derivatives and logarithms.

How could you as a teacher create an activity or project that involves your topic?

It’s important to address absolute value as not just removing the negative sign from negative numbers, but also that it functions as a measurement of magnitude, or distance from zero. Springboard Mathematics with Meaning suggests an activity where a number line is placed on the floor and students are lined up along the number line. Students record their position, and then measure their distance. Their position is positive or negative, but their distance from 0 is always positive – the absolute value of their position!

Students can also work backward, and place two students so they are each a distance of 4 from 0. Students can also express inequalities, with any students more than 5 away from 0, or any students less than 3 units from 0.

By having students on the positive and negative side of the number line, they can see how absolute value is calculated:

$|x| = x$ if $x \ge 0$ ;

$|x| = -x$ if $x < 0$ .

There are several benefits to this activity. First, it is a physical activity, which gets students out of their chairs and physically active and awake. Second, it can be used to demonstrate how absolute value is distance from zero (by measuring distance), the magnitude (length of distance), and students can derive a formal definition for how absolute value is determined analytically. It allows students to think about absolute value abstractly, concretely, or theoretically. The activity can be referenced any time in the future curriculum when absolute value is required for a quick refresher.

References

Barnett, B. (2010). Springboard algebra I: Mathematics with meaning. New York: CollegeBoard. http://moodlehigh.bcsc.k12.in.us/pluginfile.php/8095/mod_resource/content/1/1.7%20Absolute%20Value.pdf

Rogers, L. (n.d.). The History of Negative Numbers. : NRICH. Retrieved January 22, 2014, from http://nrich.maths.org/5961

Tanton, J. (2009). A brief guide to ‘absolute value’ for high-school students. Thinking Mathematics. Retrieved January 22, 2014, from http://www.jamestanton.com/wp-content/uploads/2009/09/absolute-value-guide_docfile.pdf

Importance of labeling axes

Source: http://www.xkcd.com/833/

How signed integers are represented using 16 bits

Source: http://www.xkcd.com/571/

This probably requires a little explanation with the nuances of how integers are stored in computers. A 16-bit signed integer simulates binary digits with 16 digits, where the first digit represents either positive ( $0$ ) or negative ( $1$ ). Because this first digit represents positive or negative, the counting system is a little different than regular counting in binary.

For starters,

$0000000000000000$ represents $0$

$0000000000000001$ represents $1$

$0000000000000010$ represents $2$

$\vdots$

$0111111111111111111$ represents $2^{16}-1 = 32767$

For the next number, $1000000000000000$ , there’s a catch. The first $1$ means that this should represent a negative number. However, there’s no need for this to stand for $-0$ , since we already have a representation for $0$ . So, to prevent representing the same number twice, we’ll say that this number represents $0 - 32768 = -32768$ , and we’ll follow this rule for all representations starting with $1$ . So

$0000000000000000$ represents $0-32768 = -32768$

$0000000000000001$ represents $1 - 32768 = -32767$

$0000000000000010$ represents $2 - 32768 = -32766$

$\vdots$

$0111111111111111111$ represents $32767-32768 = -1$

Because of this nuance, the following C computer program will result in the unexpected answer of $-37268$ (symbolized by the sheep going backwards in the comic strip).

main()

{

short x = 32767;

printf(“%d \n”, x + 1);

}

For more details, see http://en.wikipedia.org/wiki/Integer_%28computer_science%29

The Loneliest Number

Courtesy of Math with Bad Drawings: http://mathwithbaddrawings.com/2014/02/12/the-loneliest-number/

How did the NSA hack our emails?

The following is a reasonably accessible description of the mathematics of encryption, courtesy of Numberphile.

How to Study as a Mathematics Major

This is an interesting book review that I just read:

This small paperback book is directed at U.S. undergraduate students who may be wondering how to get along in, or even survive, a mathematics major…

The full review can be found at http://digitaleditions.walsworthprintgroup.com/publication/?i=187509&p=38.

Getting the right answer the wrong way

I just read “But My Physics Teacher Said… A Mathematical Approach to a Physical Problem,” which was a very interesting pedagogical article concerning the teaching of calculus. Here’s the central problem:

I included on their exam a question involving average velocity. I gave the students a quadratic function and asked them to calculate the average velocity over a given interval… One of my students… got the final numerical answer correct, but he hadn’t used the average velocity formula he had learned in our course. Instead… he had calculated the average of the velocities at the end points of the given interval. When I explained this to him, he stated that he didn’t understand the difference because he had learned the latter formula to calculate average velocity in his physics class.

It turns out that this alternative approach always work under the condition of constant acceleration (i.e., a quadratic function), and since constant acceleration is such an important special case in freshman physics, the formula was presented and the student remembered the formula. Of course, the student probably was not aware of the formula was only generally true under this specific circumstance.

After some pedagogical reflection, the author concluded

My student and I both learned from this experience. He gave me the opportunity to look at a familiar topic with the eye of a physicist, and I taught him the importance of context when using a formula. Specific adventures such as the one my student and I encountered will undoubtedly strengthen my approach to teaching this course and my students’ ability to think like mathematicians.

The full article can be found at http://digitaleditions.walsworthprintgroup.com/publication/?i=187509&p=19.