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