Answering to Sir What is xNovember 11, 2010 11:16 AM   Subscribe

But Sir, what is x? Teaching students algebra for the first time. They keep wanting to put values in for x, and write that down. E.g. x + x = . Becomes 1 + 1 = 2, in their books. Whereas I want x + x = 2x. Does anyone have any strategies, or techniques to overcome this?

My understanding of algebra came years later around 16-17, due to properly understanding Sets. Unfortunately, the students don't have any sort of Set knowledge, neither do I have the time to teach it them.

Thank you.
posted by 92_elements to Education (23 answers total) 3 users marked this as a favorite

My first algebra teacher had us think of the letters as animals, so 'x' could be an elephant and 'y' a tiger. 1 elephant + 1 elephant = 2 elephants. It helped me get the basics.
posted by IanMorr at 11:23 AM on November 11, 2010 [5 favorites]

IANAMT, but off the top of my head, can you have them thing of x's as--I don't know, ducks, or apples, or something along those lines. so if you have a duck plus a duck, you get not 2, but 2 ducks. And you can add a duck plus a duck to get 2 ducks, but when you add a duck plus a goose, you have kids chasing each other around a circle 1 duck and 1 goose, not 2 of anything.
posted by drlith at 11:23 AM on November 11, 2010

Algebra tiles might help.

How old are the kids?
posted by sleepingcbw at 11:26 AM on November 11, 2010

Which age? I remember that when I was 10 I was extremely interested in algebra but I couldn't really wrap my head around it until a couple of years later when I started messing around on computers. Maybe the material is too abstract... perhaps it might helpful to try to convey the subject through simple word problems and explain the equations that way.
posted by crapmatic at 11:27 AM on November 11, 2010

I remember reading somewhere that people learning algebra often have trouble with the symbols. They did much better when equations were presented initially as having "blanks":

eg

_ + _ = .

x + x = .

with the rider that the two values were equal. Then symbols for unknowns were introduced after the "idea" of unknowns in equations had sunk in.
posted by pharm at 11:32 AM on November 11, 2010 [1 favorite]

Remember that you can illustrate x as 1x for the sake of clarity. It looks weird but it might help them take the concept in. 1x + 1x = 2x
posted by Gainesvillain at 11:33 AM on November 11, 2010 [3 favorites]

My algebra teacher used non-letter symbols, like Box and Triangle (drawn as the shapes, not written out as words), and then later on replaced Box and Triangle with letters x, y, etc.
posted by number9dream at 11:39 AM on November 11, 2010

First, map to something they already know and understand. Encourage kids to think of variables as blanks, as in ___, fill-in-the-blank. Only here, you have different blanks that happen to have names like x or z attached to them.

`x + 5 = 7` now looks like `__ + 5 = 7`, and kids are very, very used to filling in the blanks.

Second, explain some reasoning: by naming these blanks, we can have more than one blank and still tell them apart.

Finally (and here is where you will want a practical example), describe the why of this as a way to produce formulas and ways to "plug-in" so they can solve a whole bunch of problems at once. A particularly nerdy example is your GPA for that semester. You know that you are getting three As and a B, and two unknown grades. How can you figure out what range of GPAs you could get?

You could definitely run the numbers for an F, a D, a C, a B, or an A grade (and you should make them do so that they might see how tedious it is), or you can show them that this can be done with algebra and suddenly they have answered a bunch of permutations of concrete numbers with a single equation.
posted by adipocere at 11:39 AM on November 11, 2010 [4 favorites]

If they are big on assigning x a value, maybe focus on assigning it a value (or multiple values), but after you do any calculations. So start with 5x + 2(x+y), simplify to 7x + 2y, then start plugging in values for x and y and showing that the original and simplified expressions are the same. Plugging in the values would hopefully help them understand that x and y are placeholders that can take various values and still be consistent with your calculations.
posted by burnmp3s at 11:39 AM on November 11, 2010

E.g. x + x = . Becomes 1 + 1 = 2, in their books.

Depending on their age and their previous texts, they may be doing exactly what they've been taught to do. Like, lots of younger grades show a picture of an apple, then a picture of a plus sign, then a picture of an apple, then a picture of an equals sign, and they want the kids to write down 1 + 1 = 2. They kids seem to be doing the same thing, but with x's.

If they're old enough and it doesn't seem confusing, perhaps jump right in to equations that have more than one variable. That might force them to label the things they are counting.

2x + 2x + 4y + 3y =
posted by 23skidoo at 11:45 AM on November 11, 2010

I like Box or Squiggle or something---x is a little abstract, sometimes.

As in, ok, you've got one BLAH and one BLAH, so how many total BLAHs do you have?

(Also, sometimes I ask for names of variables. You know, like Joe. So then we'll do a problem for a while using Joe. Say:

If 3 Joe + 2 Lisa = 5, and Lisa = 17, what's Joe?

It's good for a laugh, at least. But probably it should wait until they're comfortable with the notion of variables, first.)
posted by leahwrenn at 11:48 AM on November 11, 2010

Good idea for the 1x Gainesvillain.
The students are 11 years old. Slightly below average for their cohort.

They can solve questions like 2 + ? = 5, fairly easily. Which I tested using a prymid puzzle, similar to this. Though I had filled in some of the numbers.

Looking good so far keep the ideas coming.
posted by 92_elements at 11:48 AM on November 11, 2010

I do the same thing as the lions and tigers comment above. I say 'one xylophone plus one xylophone gives you two xylophones'. (or one yak plus three yaks equals four yaks etc). Not sure if my students have any idea what a xylophone is (or even if that is the correct spelling!) but it seems to work. Just the concept of adding two (or more) similar items, regardless of what they are. Then I use a similar concept for adding radicals later... then I tell my students to stop throwing paper!:) It's an ongoing process...
posted by bquarters at 11:53 AM on November 11, 2010

Kind of an aside: The whole concept of "doing the same thing on both sides" was really tough for me to get my head around. Like, my dad would say x + 2 = 7 and I'd know that x was 5 because it's obvious that 2 + 5 = 7, and not because x + 2 - 2 = 7 - 2, which is the algebraic way.

Anyway, I nth the symbols as variables idea. Simply drawn on the chalkboard symbols helps a lot. Snakes, stars, stick dogs and so forth.
posted by papayaninja at 12:11 PM on November 11, 2010

People who are familiar with the conventions of mathematics often don't realize all the hidden knowledge underlying the most simple things. for example: 1/2 Is that a number or a request to divide something? X is used in various ways. In X + 5 = 7, it turns out to be just the number 2 disguised as a letter. X + X = 2X is a statement about all numbers.

The reason you "do the same thing" to both sides of an equation is because of what "=" means. And some things you do may turn a false statement into a true one, such as if you square both sides of -1 = 1,

In X + X = 2X, you can teach it as an abstraction of apple + apple = 2 apples, pear + pear = 2 pairs. Those individual statements say nothing about bananas, but how can you say a general principal that incorporates the principal that you can then apply to other fruits, not to mention vegetables.
posted by Obscure Reference at 1:32 PM on November 11, 2010

That should have been "principle."
posted by Obscure Reference at 1:33 PM on November 11, 2010

For me the thing that helped was having a father who absolutely categorically would not let me leave my homework until I had mastered the "same thing on both sides" method. Dedicate an entire lesson to 'write and repeat' and have your students do the grunt work of memorizing what X is. X is something they are solving for, and if they can't show their work of how they got there THE ALGEBRAIC way, they're not going to get full credit. I have tutored many students in algebra and I myself struggled with math for years before someone (my dad) just insisted on breaking my bad habits of writing in the blanks and taking shortcuts. Now I would feel weird deviating from the standard solve-for-x method because it's a foundational thing that's necessary to get you through all the other higher maths. If X can become a pictorial symbol, though, that's definitely a great idea for the younger kids.
posted by patronuscharms at 1:36 PM on November 11, 2010 [1 favorite]

I am speaking as someone who never really excelled at math.

I always found the substituting things or symbols for x distracting. I favorited the 1x + 1x = 2x example because to my mediocre math mind, that is the most commonsensical approach to the point you want to get across to your students.
posted by vincele at 4:14 PM on November 11, 2010

You might want to look through the free! workbooks/ lesson plans from the Mathematics Enhancement Programme. The ideas of algebra are gradually introduced throughout the primary grades. The workbooks also have many pyramid- like and general algebraic puzzles.
posted by oceano at 4:59 PM on November 11, 2010 [1 favorite]

When I teach this to underprepared adults, I have already taught them how to add integers in terms of money. So, -3 + 5 = 2 because if you owe someone three dollars and you have five dollars, you would have two dollars left. When we get to x's, I tell them to imagine an island where they use x's for money instead of dollars. This often works, in my experience.
posted by wittgenstein at 5:03 PM on November 11, 2010

Perhaps you can say: X is a present. It's wrapped up in X paper. We can't know what's in the present until we unwrap it. In math, unwrapping X is called solving for X.
posted by chairface at 9:48 PM on November 11, 2010

For me, the key thing I needed to understand with algebra was that when I asked a question like "But sir, what is x?", the answer was "We don't know yet! That's why we're calling it 'x' instead of '3' or '7'. 'x' is what we're trying to find out!" And then the teacher went on to look at ways we could figure out what 'x' actually represented.

He also got through to a few people by using money. He'd do something like the following:

"I have a coin in my hand. You don't know which denomination of coin it is, so let's call it 'X' instead. 'X' means 'a mystery coin'. Now I'm going to show a penny coin. Now I'm going to tell you that X - the mystery coin - is the same value as 5 pennies. So what coin am I holding? That's right, a 5p piece. By telling you what the mystery coin was worth in terms of other, non-mystery coins you could figure out what the mystery coin was, right? This is why we call things we don't know 'X' or 'Y' or 'Z' or some other label, and then see if we know about other things that work out to the same value. Here, we knew that X = 5 pennies'. And because there's only one coin that's worth 5 pennies, we figured out what it must be. We figured out what 'X' was."

From there he went on to call pennies "Y' and then write equations like Z = 10Y and get the kids to tell him what coin Z must be. And so on. Yeah, there's the whole issue of equality/equivalence being conflated but this was early days. One step at a time. :-)
posted by Decani at 2:49 AM on November 12, 2010

Some great ideas in this thread. Thank you everybody.
posted by 92_elements at 6:26 AM on November 21, 2010

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