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 Ethan Gomez. His topic, from Pre-Algebra: absolute value.

How does this topic extend what your students should have learned in previous courses?

This topic extends students understanding of distance relative to positive and negative integers. First, students learn the positive integers — the counting numbers. Then, students are introduced to negative numbers. Visually, positive integers are to the right of zero, and negative integers are to the left of zero; students understand that these numbers exist and where they lie relative to each other. Essentially, students start by having a directional sense of numbers. Also, students also have a good understand of distance. With the concept of absolute value, students are able to associate distance with positive/negative numbers. Negative numbers aren’t just randomly placed but are rather a certain unit away from the number zero. For example, the absolute value of -5 is 5. So, -5 is not just a number that happens to be to the left of zero, but it is also 5 units away from zero. We now have a spatial sense of integers along with the directional intuition, making the numbers feel a bit more tangible and less abstract.

How can technology be used to effectively engage students with this topic?

Technology can be used to effectively engage students with the concept of absolute value since it allows students to explore its meaning. Students can discover the connection between distance and integers on their own, which reinforces the meaning-making process that teachers strive to provide students. For example, Gizmos has a wonderful tool that displays integers on a number line. On this gizmo, students are provided a visual that portrays the spatial and directional aspect of integers. This gizmo also makes students take note of the similarities between the absolute value of positive and negative numbers, forcing them to think about why they happen to be the same number sometimes.

https://gizmos.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=210

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

I could create a documentation sheet for students to keep track of what time they get home from school. They will keep track of this information for a week. The first time that they record will be the “reference time.” Every day after that, the students will document the time they get home, and how many minutes off it was from the first time, as well as if it was earlier or later than the first time. Having students think about “how many minutes off” they were from the first recorded time get them used to the idea of a magnitude, and how the number they are using tends to always be positive; the only difference is in the description of that number, which can be associated with the positive and negative characteristic of integers.

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 Haley Higginbotham. Her topic, from Pre-Algebra: absolute value.

A2. How could you as a teacher create an activity or project that involves your topic?

A fun activity to do would be to have a graph on the floor taped out and divide people into pairs and give them sets of points to graph. Then, they would have to measure how far away they were from origin. I would ask if it mattered that the x and y values were sometimes negative, and why or why not. Hopefully they’ll respond that since they were measuring distance, and distance isn’t negative, then it didn’t matter if the x and y values were negative. And that would lead into the idea that absolute value refers to the distance from origin, and it doesn’t just “make the negative a positive number.” If I were to teach absolute value, I would very much want to emphasize this point because even though it seems like the absolute value just magically gets rid of negative signs, it is important to know what it actually is.

 

D2. How was this topic adopted by the mathematical community?

Originally, the term absolute value came from Jean-Robert Argand’s term ‘module’ (unit of measure in French). The term wasn’t commonly used in English until about 1857. The standard notation of vertical bars came from Karl Weierstrass in the time intermediate time. Now, the notation of vertical bars is used for different purposes in other areas of mathematics, like determinants and cardinality, which don’t relate to distance. However, the idea of absolute value (or magnitude) extends to the realm of physics, and science in general. Generally, when you want to know how far an object has traveled, but it has returned to its original position, you take the magnitude of the distance. In physics, you often want to find the magnitude of a vector, in order to know the distance. It’s also helpful because you can extend this idea into multiple dimensions, even though the calculations can become longer than just removing the negative sign.

 

E1. How can technology be used?

GeoGebra’s graphing calculator is fantastic for math in general because it has a wide range of functionality besides just graphing. In terms of absolute value, you can graph the absolute value function easily and it will actually pop up with the vertical bars next to it and not just abs(x) which is good since then student can get more familiar with the notation. GeoGebra allows you to measure distance between points, which is really the important tool in this case. You can easily plot different points and measure the distances to verify more accurately that the distances are the same regardless of sign. GeoGebra is also fairly intuitive to use, which is good if you have students who aren’t very familiar with using technology. Plus, it’s just plain fun to play with and students will love the fact they don’t have to graph a bunch of points and functions by hand.

References:
en.wikipedia.org/wiki/Absolute_value
geogebra.org/graphing

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 Deanna Cravens. Her topic, from Pre-Algebra: absolute value.

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

A great way to teach absolute value is to do a discovery activity. A blogger and teacher, Rachel, posted on her blog, called Idea Galaxy, a great step by step on how to do a discovery activity for absolute value of integers. First the students will start out by showing the distance between two numbers on a number line, such as the distance between one and three.

They will do a few of these examples to build upon the prior knowledge of the students. Then the class will transition to another page. This one will also have number lines and will ask them problems like ‘what does negative four and four have in common?’ Some scaffolding can also be used like asking them to mark both numbers on the number line and look for similarities related to distance. After completion, students will discuss with one another about the observations they noticed. Lastly, the teacher will give them the term of absolute value and then ask students to rewrite it and put it into their own words.

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

This short video YouTube video discusses absolute value and then explains one standard way that absolute value is used in real world applications. First it explains absolute value in terms of distance away from zero. It gives a few concrete examples to display, for instance -4 and 4 both have a distance from zero that is 4. So the absolute value bars will always make the number positive. Next, the video uses an example that shows a real world example. It shows a student, Lucy, who is traveling to go to a tuba lesson. She accidentally drops her sheet music and has to go back to get it. This video does a great job of showing what it would the distance would be in terms of number of blocks walked, and how far she is from where she started or her displacement. This can easily be shown at the beginning of class either as an introduction or a review. It can spark more discussion by asking for other real world examples to help show that math really is relevant and needed for every day use.

 

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

Absolute value can show up in many areas of future math classes. It comes up when learning about the absolute value function, working with inequalities, proofs and so much more. One specific way that absolute value is used, is in calculus. After students have learned how to take derivatives, they will learn how to take antiderivatives. If a student is given ∫1/x dx, they need to find the antiderivative. Students will know that the derivative of ln x is 1/x, however this is not the case when you take the antiderivative of 1/x. The domain of 1/x is everything except zero, so negative numbers must be taken into consideration. However, if one was to say the antiderivative is lnx, it only accounts for positive numbers. Thus, in order to make the domain match 1/x, the absolute value must be brought in. Therefore, the ∫1/x dx = ln|x|+c. Thus a very basic concept becomes for important within calculations at higher level mathematics.

References:

Teaching Absolute Value Through Discovery

https://www.youtube.com/watch?v=wrof6Dw63Es
https://www.khanacademy.org/math/ap-calculus-ab/ab-antiderivatives-ftc/ab-common-indefinite-int/v/antiderivative-of-x-1

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 Dalia Rodriguez. Her topic, from Pre-Algebra: absolute value.

A2. How could you as a teacher create an activity or project that involves your topic?

Get a deck of cards and take the Ace cards out, as students walk through the door give them a card. The red cards will represent a negative number, the black will represent a positive number, Jacks will represent the number 11, Queen will represent the number 12, and King will represent the number 13. From the student roster call out two students at a time and ask them their number, the two students will then decide which card has the highest value, do this until all students are called. Then ask the students, “What if I told you that the red cards represented a negative number?” This will engage the student because they will feel confident about their answer until they hear that the red cards represented a negative integer. Then the student will start thinking and coming up with conceptions on how a negative number affects which integer is higher. The teacher can then ask another pair of students what their numbers are, and follow up by asking which integer is higher. The students will most likely answer incorrectly so this would be a time to ask other students what their thoughts are. All the students will be participating and thinking. The last question the teacher would ask before beginning the lesson would be, “What if I told you that the color of your card does not matter, or affect the number on the car?” Allowing all the students to participate by calling on them, at the beginning, will break at least a small barrier and open the doors for them to share their opinion. Also, asking scaffolding question to let the students start thinking about properties of absolute value will let the students remember the activity and acknowledge that even thought the number is negative or positive absolute values is the distance away from zero and it will always be positive.

 

In Finding Dory, her parents laid out sea shells on the ocean floor that lead to her parents. The sea shells were spread out in lines going around the house, the distance from the beginning shell to the house is always positive, even though they are in the left side (negative side). The teacher can tell the student that each sea shell represents 1 unit, as they see the length of the sea shells lines the students will think of these lines as positive numbers, no matter what direction the sea shells are coming from.

 

Students should have already learned about positive, negative integers, and distances. You can engage your student by asking them question and having a class discussion. Questions like:

 

“What is a positive integer?”

 

“What is a negative integer?”

 

“How do you measure distance?”

 

“Can distance be negative?”

 

These types of questions will scaffold student to get a base line idea of what absolute value is, but also allow them to remember what they already have learned. Allowing students to realize that their connections from past knowledge to new knowledge will let them better understand what they are learning. Having a class discussion on their previous knowledge will allow a teacher to see where there might be misconceptions and also see a base line where the students are at, or what they might need help at. A small review lesson from the teacher, after a discussion, will then clear up any final misconceptions and allow the class to move forward from the same starting position.

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:

if ;

if .

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