Month: July 2017

New look, new title

/ Bill McCallum / 3 Comments

I have updated to a new WordPress theme, partly because I thought it was time for a makeover, and partly to see if it would cure some of the problems people have had commenting. In the process I decided to change the title of the blog. My recent writings have been about school mathematics generally, and I hope they are of use to teachers everywhere, whatever their standards. I will still write occasionally about the Common Core, and I will still answer questions over in the forums. I have changed the settings in the forums to allow anonymous posting for people who have trouble logging in. I may have to change that back again if it causes security problems. And, speaking of security, the site has been protected by SiteLock since last summer’s hacking, which means that you will occasionally encounter a captcha screen.

And, by the way, the url mathematicalmusings.org also points to this blog.

Truth and consequences: talking about solving equations

/ Bill McCallum / 1 Comment

The language we use when we talk about solving equations can be a bit of a minefield. It seems obvious to talk about an equation such as $3x + 2 = x + 5$ as saying that $3x+2$ is equal to $x + 5$, and that’s probably a good place to start. But there is a hidden assumption in there that the equation is true. In the Illustrative Mathematics middle school curriculum coming out this month we start students out with hanger diagrams to represent such equations:

The fact that the hanger is balanced embodies the hidden assumption that the equation is true. It is helpful for explaining why you have to perform the same operation on each side when solving equations; if you take two triangles from the left side you have to take two triangles from the right side as well in order to preserve the balance. This leads to a discussion of how performing the same operation on each side of an equation preserves the truth of the equal sign.

But what happens with an equation like $3x + 2 = 3x + 5$? In this case, the hanger diagram is a physical impossibility: the right hand side will always be heavier than the left hand side. I can imagine that students who have an idea of an equation as “the left hand side is equal to the right hand side” might be confused by this situation, and think this is not a proper equation. Especially when they reduce it to $2 = 5$. Students learn to say that this means there are no solutions, but it’s hard to make sense of that response rule without understanding what’s really going on with equations.

The fact is, an equation with a variable in it is neither true nor false, because it is merely a phrase in a longer sentence, such as “If $3x + 2 = x + 5$ then $x = \frac32$.” This sentence is true, but the phrases within it are not sentences and have no inherent truth or falsity. When we perform the same operation on each side of an equation, we are not only preserving the truth of the equal sign but also preserving the consequences of the equal sign. If we use if-then language when talking about equations, then we can make sense of equations with no solutions. A sentence like “If $x$ is a number satisfying $3x + 2 = 3x + 5$ then $2 = 5$” makes perfect sense. It’s the mathematical equivalent of “If the moon is green cheese, then I’m a monkey’s uncle.” It’s a way of saying the moon is not green cheese . . . or that there is no solution to the equation.

The middle schooler’s version of if-then language might not always use the words “if” and “then.” You might say “Imagine there is a number $x$ such that $3x + 2 = x + 5$. What can you say about $x$?” Just as you say “Imagine this hanger is balanced and the green triangles weigh one gram. How much do the blue squares weigh?” I think it’s a useful approach with students to remember that equations are a matter not just of truth, but of truth and consequences.

Why is a negative times a negative a positive?

/ Bill McCallum / 15 Comments

OK, I can hear the groans already. There are many contexts for answering this question and they are dubious in varying degrees because the real answer is “because I said so.” That is to say, the rule for multiplying negatives is a convention; adopted for good reasons, but a convention nonetheless. Those good reasons are mathematical: we want to make sure that when we extend multiplication and addition to negative numbers the properties of operations still apply. In particular, we want the distributive property to apply. Meditate on this:
$$
3\cdot(5 + (-5)) = 3\cdot5 + 3 \cdot (-5).
$$
The left side is really $3 \cdot 0$, so it had better be zero. So the right side had better be zero as well. The first term on the right side is 15, so the other term had better be $-15$. So $3 \cdot (-5) = -15$. We want the commutative law to hold, so we had better say $(-5)\cdot 3 = -15$ as well. Now meditate on
$$
(-5)\cdot(3 + (-3)) = (-5)\cdot 3 + (-5)\cdot(-3).
$$
The same reasoning tells us that $(-5)\cdot(-3) = 15$.

Trouble is, all this is really hard to explain to middle schoolers, so people invent contexts. One context I’ve seen has something to do with sending out bills. If you receive 5 bills for 3 dollars then you have $5 \cdot (-3) = -15$ dollars. Sending out is the opposite of receiving, so if you send out 5 bills for 3 dollars, you have $(-5)(-3)$ dollars. But once you receive payment, you have \$15. So $(-5)(-3) = 15$.

One problem with this is that you have to buy more conventions to believe it: the convention about negative amounts of money representing debt, the convention about negative receiving being kinda sorta like sending out. That’s a lot of conventions to prove something that is, as I said, a convention itself. Another problem is that all this context really shows is that $-(-3) = 3$, five times. The multiplication in this context is really just repeated addition; it doesn’t work for numbers that are not integers. You can’t send out 5.6 bills.

There is one context that I think does a better job here, and that is $\mbox{distance} = \mbox{speed} \times \mbox{time}$. This does work with non-integers, and you can make sense of all of the quantities involved as negative numbers. Let’s assume that an object is moving along the number line, and that you measure its position at different times, setting your stop watch to 0 when it passes through the origin. Negative distance is distance to the left; negative speed is speed from right to left; and negative time is time before you started measuring. (Later we use the terms displacement and velocity, but there’s no need to introduce them right away.)

So if the object is moving at $-5$ m/sec, where is it at time $-3$ seconds? Well, it’s moving from right to left and it has 3 seconds before it hits the origin, so it is 15 m to the right of the origin. So $(-5)(-3) = 15$.

Was I cheating there? Is this context subject to the same objections I made about the money context? Didn’t I just make up a whole bunch of conventions about negative distance, time, and speed? I think these conventions pass the cognitive sniff test better. They don’t seem as artificial to me. You can really make quantitative sense of negative distance, speed, and time. It feels more like the real world and less like an accountant’s convention. (No offense to accountants intended.) In a way, we have replaced the mathematician’s desire to have the properties of operations continue to hold with the physicist’s desire to have the laws of physics continue to hold.

So where is the distributive property in all of this? I think it is built into our physical intuition about this context. If I travel for 3 hours, and then for another 2 hours, I can figure out how far I have gone by just adding the times and multiplying by my speed, or I can add the distances traveled in each time period. That’s the distributive property. If you dig into the reasoning I gave for the object moving at $-5$ m/sec in the light of this common sense, questioning each claim, you end up with something not too far from the mathematical reasoning I gave earlier.

By the way, this is the approach we take in the Illustrative Mathematics middle school curriculum. Finding contexts for mathematical ideas that are faithful to the mathematics is difficult and requires real sensitivity to both the mathematics and the way students think. Our brilliant curriculum writing team is up to that challenge.