52

"Because we said so" is a bit of a conversation closer, I agree. But "Because some people agreed a long time ago to define it that way so we could have conversations where we all understood each other. Does that seem like it would be a good idea?" is both more inclusive and more correct. I don't even think it's that hairy to talk through the FTA with ...


34

There was a multiplication table posted on the wall. Like this \begin{alignat}4 1 &\quad 2 &\quad 3 &\quad 4 &\quad\cdots\\ 2 &\quad 4 &\quad 6 &\quad 8 &\quad\cdots\\ 3&\quad 6 &\quad 9 &\quad 12 &\quad\cdots\\ 4&\quad 8 &\quad 12 &\quad 16 &\quad\cdots\\ \vdots&\quad \vdots &\quad \...


15

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 ...


12

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"...


10

FYI: here's some pro and con: http://primefan.tripod.com/Prime1ProCon.html One was originally considered prime. It is prime with the most convenient ("natural") definition. It got excluded from prime-ness because many other higher theorems would be complicated by leaving it as prime. Essentially "prime" -> "prime*". The definition of primeness was ...


9

It might work better, with this age level, not to be concerned about how large n is when the apparent pattern falls apart. My favorite example is the problem of making all possible straight-line segments between n points on a circle, and then counting the regions. 2 points makes 2 regions, 3 makes 4, 4 makes 8, and 5 makes 16. It sure looks like doubling...


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 ...


8

How should one talk about the question of 1 or 0 being prime ... with primary or middle school children? Depending on what you did before you will have an easy or a hard task: If the children were told: A prime number is a natural number which cannot be divided by other numbers than by 1 and by itself. ... you will have problems explaining why 1 is not ...


8

I second Sue Van Hattum's suggestion that you should not be so concerned with how large the $n$ is where the pattern eventually fails. I'll go one step further and recommend an example where that $n$ is not only fairly "small" but also such a situation where the students can see why that $n$ makes the pattern fail. Consider the function $f(n) = n^2+n+41$. ...


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'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

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 ...


5

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 ...


5

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 ...


5

A good way to lead to the uniqueness of prime factorization and the convention that $1$ is not a prime is to build factor trees (that's common in elementary school these days in fourth grade, sometimes third grade). 24 24 24 8 3 6 4 2 12 2 4 3 2 2 2 3 ...


5

This one is really very simple. First, tell them what a prime number is: A prime number has exactly two different factors. (If they don't know what factors are, and they ask about primes, the correct answer is "well, first you have to know about factors...") With that definition, it is very easy to figure out 0 and 1. Is 1 a prime? No, because it only ...


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

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 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 $m$...


4

If you build each number n using n square blocks in rectangular configurations, there are multiple configurations for each composite number. (4 is 4 by 1 or 2 by 2.) The primes are the ones that can only be built as a 1 by n rectangle. It seems clear that 0 would be neither prime nor composite, when looked at this way. The easiest way to understand why we ...


4

I'm kind of restating what other answers have said, but I wanted to practice expressing it in the clearest, most concise way I could think of. (Coincidentally, this came up with my partner tonight, so I got a test-run with it, and got an entirely satisfying result). Consider only natural numbers (i.e, positive integers). It seems like the number of ...


4

This happened to me once on a topic that should be accessible to middle-school students. As an undergraduate, I formed the following conjecture: An antiprime (also called a highly composite number) is a positive integer that has more divisors than any number less than it. The first few antiprimes are $$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360,.....


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