# Tag Info

14

Here's a silly example: Give students collections of the same type of thing, where each collection contains "good" objects and "bad" objects -- for example, a stack of Pokemon cards with both rare and common cards. We might assume common cards are worth $1$ and rare cards are worth $5$. Have ready another stack of cards that are all common -- call this the "...

12

I wouldn't do that. The parenthesis in use are also used for legal expressions within equations. So you can end with one line containing the same parenthesis meaning different things, what looks like a seed of confusion to me. If you prefer, you can replace the GeoGebra parenthesis with some others i.e. <>, or [] etc. to keep the notation similar to ...

11

I found out about the Prisoners' Dilemma as a kid from a book about the Harry Potter phenomenon, which had a chapter about the problem, but presented as a story about Harry and Draco being accused of breaking school rules. Each was offered the same deal as in the original problem, formulated with House Points being taken away instead of a prison sentence. ...

10

The problem with $(4x+7=6x+2)-6x$ is that there is no subtraction operation that involves subtracting a term from an equation. Subtraction involves subtracting a term from a term. So the correct notation is $(4x+7)-6x=(6x+2)-6x$

10

Speaking as someone who has taught college precalculus several times, I have an intense dislike for the way that Geogebra writes this step. In my opinion, it is very important to emphasize to students that we are subtracting 6x from both sides of the equation, which means of course that we are subtracting 6x from two different expressions. If we write "-6x"...

8

Your question is kind of two parts: one about a convention Is the constant term a "coefficient" and one about a philosophy, which I perhaps find to be a more important question to answer. Isn't mathematics supposed to be non-arbitrary and consistent? Different fields of mathematics have different conventions; this can lead to some mathematicians ...

8

I like Nick C's idea more than modifying the typical formulation. The notion of snitching on a friend, regardless of the severity of the "crime", has real-world ramifications beyond the punishment put out by the authorities. Depending on your student population, that is possibly going to spur a conversation that will overshadow the objectives of your ...

7

I'm going to use GeoGebra to teach equations. Is it OK to let the students write the steps just like GeoGebra (I mean, with parenthesis)? I would not allow this in my class, but it would depend what you think is good/clear/consistent notation for your students. The developers at GeoGebra made a choice for showing algebra steps, but that doesn't mean we ...

7

You might explain that BEDMAS is not the whole story when it comes to the order of operations. There is an operation called negation. It reverses the sign on numerical quantifies. It gives the additive inverse of any number, i.e. for all $x \in R$, we have $x + (-x) = 0$. Unfortunately, most textbooks use the same symbol for both subtraction and negation. ...

7

I would say that the word of is pretty important here, and as long as you get that correct, the rest is less important. Using the word of consistently when a function is being applied to an input, and talking about the fact that you are doing this, will help ward off a lot of the following kinds of mistakes that I see in my (much later) classes: Question:...

5

Since I recently had a math teacher from another institution observe me during an Algebra 2 lesson and comment positively on my notation ("I'm going to steal that!") here is a worked example: Find all values of $x$ that satisfy $3(x-4)^2 + 8 = 23$.   $-8: \hspace{20 mm}3(x-4)^2 = 15$ $\div3: \hspace{20 mm} (x-4)^2 = 5$ $\sqrt {}:... 5 Thinkeye's answer is good in that it easily extends to dividing two related equations and similar, more advanced operations. On the other hand, for the sake of brevity, I would suggest the way I have been taught to explicitly specify operations: $$6x + 14y = 4x + 12y \quad | -4x$$ A vertical line to separate the equation from the intended operation. Such ... 4 It is not incorrect to think through all of the cases, and it is not unreasonable to make even a middle schooler think through the possibilities, but you do have to be careful about it, and keep track of the assumptions that have been made. If you are working over the real numbers, then the set of all possible solutions is$\mathscr{U} = \mathbb{R}\setminus ...

4

I do recall some elementary texts that do this. Subtraction $$6 - 5$$ written in a different way than a negative number $${}^-8$$ so we can do calculations $$5 - {}^-2 = 5 + 2 = 7$$ Presumably at some point, the students are switched to the conventional notation.

4

Here is one I enjoyed from middle school. This was a project: I think we had a whole week to experiment, and discuss, and come up with a solution. Consider a rectangle a 231 by 84 rectangle which is tiled with 1 by 1 squares. How many squares does a diagonal of this rectangle pass through? I think this was phrased in terms of a mouse running from one ...

4

It sometimes happens that slightly different definitions of the same word each have advantages and disadvantages. In such cases, I wouldn't be surprised to see some people supporting one definition and other people supporting a different definition. In the case at hand, though, I can't think of any advantages for defining "coefficient" to exclude the ...

4

I have to admit I was skeptical of the OP's claim that contemporary textbooks do not identify the constant term as a coefficient, so I checked the first book that I had handy -- and indeed it does seem to be the case, in at least my sample of 1. Here is some evidence: (Source: McDougal Littell Algebra 2, 2004.) Note however that 2004 precedes the Common ...

4

The earliest mathematical insight I remember from childhood is that the word "of" almost always means "times." Half of a dozen $= \frac12 \cdot 12 = 6$ Three-fourths of a mile $= \frac34 \cdot 5280$ feet $= 3960$ feet. I'll take 6 of those thousand-count boxes $= 6 \cdot 1000 = 6000$. I remember feeling like I had secret knowledge that no one ...

3

If I had to communicate the solution to this problem to someone else in writing, I'd probably write ...gives $$4x+7=6x+2\text{.}$$ Subtracting $6x$ from both sides of this equation gives $$-2x+7 = 2\text{.}$$ If the equality were one in a long chain, I'd write $$\begin{split}&\ldots \\ 4x+7&=6x+2 \\ -2x+7&\stackrel{\text{(a)}}{=}2 \\ ... 3 A nice third problem arises by continuing from the second problem to show the power of parity arithmetic for solving problems in integer arithmetic. For example, we can use it to show that large classes of polynomials have no integer roots. Let's consider a simple example. f(x) = x(x\!+\!1)+2n\!+\!1\, has no integer roots since \,x(x\!+\!1)\, is even, ... 3 I'm not aware of any "official" version, the text that you quoted not withstanding, so any response is going to be mostly just personal preference. With that said, I wouldn't read either expression the way your textbook wrote them. "the positive square root" is redundant. The square root of a number, in this context, is always positive so I would just ... 3 If your students already understand that exponents precede multiplication, and that multiplying by -1 is the "negation" operator, then you should be able to convince them that$$-5^2 = -1*5^2 = -1*25 = -25 is a reasonable way to interpret this expression.

3

I tutored a student who had a hard time understanding this, and the way that helped him to understand it was this: Any time there is a negative sign on a number, we can read it as $(-1)$ . So $-5 \equiv (-1)5$ and $-3^2 \equiv (-1)3^2 = (-1)9 \equiv -9$. In the case you mention then, $-1^2 \equiv (-1)1^2 = (-1)1 \equiv -1$ The key to helping him ...

3

Let's keep in mind that "mathematics" and "mathematics education" are different subjects. This question brings forth this distinction. At one end of the "Piaget" spectrum of mathematical stages of development, explaining that the constant term $c$ is the same as $cx^0$ might be too much cognitive overload. However, towards the other end of the spectrum, say ...

3

The word "of" means $\times$. For example "half of $a$" means $\frac12\times a$. Note that the product of $a$ by a number isn't necessarily less than $a$ ; it'll be less than $a$ iff the number is less than $1$ ; in that case we can write the number as $\frac{m}{n}$ ($m<n$) so that $\frac{m}{n}\times a$ means to divide $a$ into $n$ equal parts and take $m$...

3

Perhaps: You and another classmate are together in an obstacle course. If you both make it to the end within a minute, you each get a free day off from school. If just one of you makes it to the end in a minute, they get a whole week off. You know that if you had the other student's help, you could easily finish in a minute, but without it, you ...

2

I like the presentation on the NCTM Math Forum/Dr. Math website: We don't usually list unary operators in PEMDAS because they're thought of as being implied by the rules for binary operations. You can think of the minus sign as either subtraction -3^2 = 0 - 3^2 = 0 - 9 = -9 or multiplication -3^2 = -1 * 3^2 = -1 * 9 = -9 and in ...

2

If you're main concern is students' writing, you could take the route of avoiding the issue altogether and tell students that $-1^2$, regardless of what it should be equal to according to BEDMAS, is just bad notation and that it can be avoided. The purpose of writing mathematics, of writing anything really, is to communicate effectively to a reader. The ...

2

It is perhaps worth noting that on most graphing calculators there are already different symbols for subtraction and negation, instantiated on different keys. For example on the TI-84 Plus (see image below) there is a key labeled (-) at the bottom right corner, for negation, and a separate key along the right-hand side for subtraction. Is this "better" ...

2

The premise of your question seems to indicate an undesirable precedent. When the students are able to solve a problem correctly it's immaterial what approach they followed. In fact, giving positive feedback to those who conjure creative methods (instead of following conventional approach taught in the class) is imperative. But, to be fair, your question ...

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