Dec 21 2013

Math topics that need to be put out of their misery. Part 2: Absolute value

The old math standards, say common core defenders, were “a mile wide and an inch deep.”  I’m inclined to agree with this.  Too many topics and too little time led to teachers having no choice but to teach many topics at a superficial level.  As time in a math class is a somewhat fixed quantity (unless you want to do ‘double blocks’ in math which, of course, is taking away from something else which is equally, if not more, valuable), the only feasible solution is to cut out some, if not many, topics.

The common core writers missed a good opportunity to do this, however.  The standards do specifically describe certain ‘topics’ that are required, but not all topics are explicitly described.  This leaves the teacher, school, or district, in a strange predicament:  If they skip a ‘topic’ which was not specifically described in the standards and then that topic appears on the nationally ‘aligned’ common core tests, that teacher, school, or district may find themselves punished for not being ‘accountable’ enough.

One such topic which is ambiguously hinted at in the standards is part of this second part of my (probably ten part) series about math topics that somehow were put into the curriculum years ago and continue despite the fact that they serve no purpose whatsoever.  That topic, which plagues kids starting in about 6th grade, is ‘absolute value.’

For those of you who have forgotten (or maybe have just repressed memories of it) the official definition of ‘absolute value’ is “the distance a number is from zero on the number line.”  So the number five has, we say, an absolute value of five since five is five away from zero.  But five is not the only number that has an absolute value of five.  Negative five also has an absolute value of five since negative five is also five away from zero (on the other side of zero) on the number line. In symbols, we say:

For sixth graders, kids generally ignore the whole “distance from zero” thing and remember absolute value as the easiest thing in the world:  If there is one, get rid of the negative sign.

Later on, maybe in 7th grade, when teachers are teaching how to add positive and negative integers together, they might even use the concept of absolute value by teaching that the rule for combining signed numbers, like (-8)+5 is to “Subtract the number that has the smaller absolute value from the number that has the larger absolute value.  The answer takes the sign of the number with the larger absolute value.”  So some people might say that we ‘need’ to teach absolute value in sixth grade so that the students will be ‘ready’ for combining signed numbers as the procedure requires an understanding of absolute value.  To this I’d say that I’ve taught combining signed numbers to hundreds of kids and have never once used the term ‘absolute value.’  Yes, the idea of the ‘bigger’ number was used and even though positive numbers are ‘greater than’ negative ones, kids were easily able to understand what I meant and master the topic without getting confused by technical jargon.

Then, in 9th and 10th grade, student learn absolute value equations and absolute value inequalities, and then things really start going awry.  Maybe they start with an equation like,

which is novel because it has two solutions, five and negative five and then move on to more complicated equations like,

and even,

Students memorize “copy the equation, but omit the absolute value bars, then write the word ‘or,’ then copy the equation again, but change the thirteen to a negative thirteen, and solve the two equations.”

Sometimes they ‘plot’ the two answers on a number line to create a ‘graph’ of the solution set, which generally looks like,

It is hard for kids to get excited about these two dots.  I try to inject some thought provoking ideas into every lesson, so for this one I show them how on the number line there are only two points that are exactly five away from zero, while on a plane there are an infinite number of points which are all on a two dimensional circle with radius five.  So while most people look at the number line with two dots on it and see just two dots, I see a one dimensional circle.  This doesn’t exactly ‘blow’ their minds, but it does add something interesting to an otherwise awful topic.

In tenth grade, maybe they move on to two new aspects of absolute value.  There are absolute value inequalities, like,

The most obedient students ‘master’ this by remembering that you first copy the equation, but without the absolute value bars, then if it is a less than sign you put ‘and’ while if it is a greater sign you put ‘or’ (famous math teacher mnemonic:  great’or’ than vs. less th’and’) and then for the second equation ‘flip the direction of the inequality sign’ and also ‘flip the sign of the number after the inequality sign.’

Ironically, there is something meaningful that can be taught with absolute value inequalities, and which I’ve always taught along with this, in an attempt to make the topic more interesting.  A situation like, “to be in the army, you must be between 60 and 74 inches tall” can be expressed as the absolute value inequality above.  On the SAT they actually have a modelling equation like this from time to time.  I do like that aspect of absolute value, and that seems to be the most ‘common core’ type of application of it, so maybe this can be part of the high school standards working up to a question like this, while the sixth, seventh, and eighth graders can be spared from ‘exposure’ to the topic as some kind of ‘set up’ for an interesting use of it in the future.

Finally, students learn to graph the ‘function’,

which they learn is a function, but not a one-to-one function, and also learn to transform the infamous ‘V’ shape in different ways.

All in all, the months spent on absolute value between 6th and 10th grade are a waste of time.

Yes, I am aware that eventually they will ‘need’ the ‘distance from zero’ definition in order to understand how to determine the absolute value of a complex number in eleventh grade.  I have no problem with the concept being introduced five minutes before it is time for that lesson.  And, yes, I know that absolute value is important in BC Calculus epsilon-delta proofs.  I am not convinced, though, that by setting them up for it five years earlier, it will help them to understand it more later.  Not everything in school needs to follow the Chekhov’s formula where the gun shown in act I needs to go off in act III.  No kid ever, when learning about the absolute value of complex numbers, jumped up and said “Oh, that’s why back in sixth grade we were told not to think of absolute value as just taking away the negative sign, but to think of it as distance from zero!”

Now, as I mentioned in the beginning of this post, the common core doesn’t specifically mention absolute value equations and inequalities.  There are references to absolute value in sixth and seventh grade and also to graphing absolute value functions in high school (thanks to mathed.net for researching this):

CCSS.Math.Content.6.NS.C.7 Understand ordering and absolute value of rational numbers.
CCSS.Math.Content.6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
CCSS.Math.Content.6.NS.C.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 degrees Celsius > –7 degrees Celsius to express the fact that –3 degrees Celsius is warmer than –7 degrees Celsius.
CCSS.Math.Content.6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
CCSS.Math.Content.6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
CCSS.Math.Content.6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

CCSS.Math.Content.7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

CCSS.Math.Content.7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

CCSS.Math.Content.HSF-IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

But even without an explicit mention of absolute value equations and inequalities, textbook publishers, like this one published by Glencoe McGraw-Hill, called Algebra 2 Common Core edition, is definitely playing it safe by including it in the book.  Since we don’t know what topics are fair game for the national common core tests, it would be wise to expose kids to this topic.

So will teachers and students continue to suffer through this infamous topic?  Absolutely.

3 Responses

  1. educator

    Diane Ravitch posted an insider essay on her blog showing how bad the Klein/Black/Walcott education policies were
    http://dianeravitch.net/2013/12/20/tweed-insider-where-the-bloomberg-administration-went-wrong-on-education/#comments

  2. KTeacher

    One more thought (a bit of a tangent) on the whole discussion of “mile wide – inch deep” issues in prior state math standards, and CCCS allegedly fixing this without really cutting much from the math standards. I am hearing from many elementary teachers that one thing that is definitely cut (at least in the new CCSS friendly curriculum in our district) is substantial review / revisiting / re-teaching of subjects that have been previously taught. So the assumption is that students have mastered all skills from prior years (this differs from the “spiral learning” that was the basis of earlier curriculums). This leaves teachers in a big quandary when they get students (particularly a significant portion of their class) who do not have the foundational skills needed for the new skills for the current year. Given the terrible job that the CCSS writers did of getting input from those who teach at the early years (as well as the non-relationship between child development and some of the standards), this could put many students in the position of falling behind at one grade level, and never catching up.

  3. J Choi

    Gary,

    I agree that the key to the absolute value is that it is a distance. Great example with the one-dimensional circle – keep that up, but also please plot the center point as well so the distance concept becomes strengthened. Absolute value is the distance from point, and since distances cannot be negative, it is assigned a postive value regardless of the order of the points.

    (Be sure to emphasize that the order of the points is irrelevant in a distance calculation, and this is precisely what the a.v. calculation does! – also check with your physics teachers, and tie in the idea to their physics curriculum: the difference between distance and displacement often appears on tenth grade standardized exams. Solidify all the ideas and tie them together as often as possible.)

    It is when one strays from the distance meaning that both teaching the concept and learning the concept become bogged down in a quagmire. The seventh grade student then gets stuck memorizing rules. Indeed, if it is taught as a series of rules, it becomes very difficult for the student to reverse this and realize that its original intention is simply as a way to represent distance. Therefore using it as a tool to teach adding and subtracting negative numbers may be ill-advised. (sorry, CCSS!) The beauty of the distance idea shows up during the ‘more difficult’ problems /x-5/=3, which the students should be taught to read as “the distance from the point to 5”. My concern is that it is not taught this way and not emphasized this way. In most cases, a review of the textbook reveals that the textbook does not emphasize the thought process that a mathematician would use – instead, a bunch of rules and non sequiturs fall out. This is what we teach the students, and they quit. If we show them the beauty and the logic, and they may learn to love it.

    Similarly, the inequalities must be taught graphically first, in which case the student should reason out the ‘and’ or ‘or’. Without the graphical understanding of distance, the student is forced to memorize, and all is lost, for there is no reasoning process.

    Setting up the form of the equation /x-x1/=d is so very important, and you will do your students and yourself a favor, because they will see this again in Algebra II. Why? All the equations of conic sections can be written as an offset from the origin. We start by teaching them the equation of, say, a circle, centered at the origin:

    x^2 + y^2 = r^2

    When they have mastered that, we show them that the whole thing can be shifted up, down, or sideways by simply subtracting the offset from each coordinate:

    (x-x1)^2 + (y-y1)^2 = r^2

    The same is true for any conic section.

    Further, the idea of offsetting from the origin should first be introduced when teaching the equation of a line: to shift the line vertically subtract y1, to shift it horizontally subtract x1:

    y = mx (+b) , but start with b=0; the line goes through the origin

    Let the student pick the slope, m. The line goes through the origin, but can have any slope that the student wants. The student is god. Make sure the student feels like god.
    (write THAT in your lesson plan! CCSS.7.1 Make sure the student feels like …)

    Now let the student pick any point on the plane, and give its coordinates (x1,y1).
    Show them by offsetting the x-coordinate (x-x1) that THEY have shifted the line left or right.
    Then by offsetting the y-coordinate (y-y1) THEY have shifted the line up or down.

    y-y1 = m(x-x1)

    have the students expand it to determine b. Then have them plot it and show that it indeed goes exactly through the point that THEY have chosen, with the slope that THEY have chosen. This is very powerful to a struggling seventh grade student, and they may actually come away with a sense of mastery. (!) Tell the student that THEY are the masters, and have the ability to move the line anywhere on the plane that THEY determine. It empowers them. (I love doing this…)

    Now back to the absolute value. As you mention, one starts with the definition of distance from zero:
    Obviously this is the special case where the offset, x1 = 0.

    /x-x1/=d, (distance from x1)
    /x/=d (distance from 0)

    Be sure that the student understands this, and make sure they practice this idea so that it becomes solid, and not a collection of rules. Let them pick the point x1, emphasize that THEY may pick any point, and that THEY may pick any distance, then let them solve for the actual two possible answers of x. When they see that the mathematics does in fact automatically get BOTH solutions, they should begin to understand the power of what they are doing.

    It is rare to find a math text, or test that appreciates this. It is very frustrating when both the textbook writers, and the test writers have lost the point, and reduce math to a bizarre collection of rules. I’d hate math too.

    There is a reason our seventh graders cannot do algebra, and the fault is largely ours.

    J Choi,
    Worcester, MA

    I’d rather see it taught right, than not at all.

    Teaching tends to follow the textbooks. The textbooks follow the standards. There is a huge opportunity to fix this right now, that has not happened and will not happen for another 50 years.

    Should we get NCTM involved here?

About this Blog

By a somewhat frustrated 1991 alum

Region
Houston
Grade
High School
Subject
Math

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