# How do I teach algebra?

I find that soon I'll be working with high school students that are struggling with math. In particular, we'll be talking a lot about algebra and some basic trigonometry. The latter I have experience with (via working with students in calculus and "pre-calculus"), but I have legitimately no idea how one would teach algebra. If I see $3x+5=14$, it's obvious to me what to do, and unlike, say, calculus, I can't really even see how someone would get confused on that (even though I know they do!)

This is a bit broad, but how do you teach introductory algebra? Do you have any references for new teachers?

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Part of your problem is what is called "expert blindness" or similar: the subject is so familiar to you that the trouble your students have becomes incomprehensible. First step is obviously to see the phenomenon, next step is to find out what specific problems are common and how to handle them. – vonbrand Apr 21 '14 at 4:25
@vonbrand I'm unfortunately aware. My struggle is that I don't know how to solve it. I have to admit no experience teaching algebra in the past, and I'm a bit worried I'll show up and do poorly without some practice/background. – Mike Miller Apr 21 '14 at 4:56
For the symbol-manipulation side, I would recommend having them play with DragonBox ( dragonboxapp.com ). It doesn't explain any of the theory behind why the rules are what they are, so it's not sufficient by itself, but it's fantastically good in teaching the rules and making it seem fun. I once saw a 5-year old solving (with assistance, but still) about a hundred first-degree equations within a couple of hours when playing with it, and also later on some older kids arguing over who gets to play and solve algebraic equations next. – Kaj_Sotala Apr 21 '14 at 8:34
All of the technical advice offered here is golden. I wouldn't change a syllable! On some level, I envy you. My own life was changed 40 years ago by a man who had the patience to do the job you now face. His name was Mr. Shetler. He taught me that I wasn't an idiot and that this stuff isn't magic. There are simple rules that we apply to do algebra. Learn the rules and the problems solve themselves. Above all I council patience. – user1168 Apr 22 '14 at 9:34
Remarkably, when I was in middle school, I went half a year effectively solving these problems via the bisection method (en.wikipedia.org/wiki/Bisection_method). The teacher at one point pulled me aside when I was getting answers like 9.97 when the actual answer was 10, and asked me if I was being a wise guy. The idea of subtracting a number from both sides is extremely foreign, so much so, that I effectively developed an algorithm from numerical analysis to avoid it! – MHH Jul 27 '14 at 16:14

As a personal tutor, I’ve been teaching algebra to kids from ages 8 to 16 for many years. Mostly I find myself in the position of picking up the pieces when the kids are failing and fearing more failure.

The root of the problem, in my experience, is the way algebra is taught as something alien, and in particular, different from arithmetic, which it really isn’t (at least in the early years).

So first off, constant emphasis on the fact that “$x$ is just a number you don’t know yet”. So it behaves like a number, and you can do all the stuff to it, that you can do to numbers.

Next, the nature of equality $2 + 3 = 4 + 1$.

And from there, the fact that when you do the same thing to both sides, you still end up with two things that are equal.

Always “do the same thing to both sides” (since this is clearly based on the nature of equality), never “move this from one side to the other and change the sign” (which is a magic rule that makes no sense until you have a deeper understanding).

Once you get them happy with the idea that doing the same thing to both sides is the way to go, you can give them suggestions for which things to do in which order, but stress that provided they rigorously write down the consequence of the thing they decide to do to both sides, they won't go wrong (although some ways are harder – look out for these as a pointer that choosing another way will be easier).

The manipulation of each line is easy, once you’ve got them to decide what they’re going to do at each stage.

For instance, in the example, $3x + 5 = 14$:

• First, decide what to do to both sides (subtract 5)
• Write down first what you have ($3x + 5$), then do what you’ve decided. So you get $3x + 5 - 5$, and on the RHS, $14 - 5$.
• Then collect terms and simplify to get $3x = 9$.
• Then repeat for division by 3.

Emphasise that once you’ve decided what to do at each stage, there’s very little thinking, since you’re just writing – starting with what you had on the previous line, and adding on the chosen operation.

Figuring out what to do (add or multiply, subtract or divide) needs to come after they are truly grounded in the principle that doing the same thing to both sides is the key.

They will also need help with things like why $3x/3 = x$. Again, use numbers to illustrate, and stress that $x$ is just a number, so it behaves the same way as a number.

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In school, we had to put a long vertical line to the right of the equation and had to write next to it what we did to go to the next line ($-5$ and $/3$ in this case). – user11235 Apr 21 '14 at 14:47
Exactly. I think you may have misunderstood me. The habit of writing the vertical line, and what you've done to get to the next line, is what they need to get out of. My point is that in going from 3x + 5 = 14, to 3x + 5 - 5 = 14 - 5, to 3x = 9, it becomes a natural progression to miss out the middle line, more so than no longer writing the vertical line stuff. You may find something else easier - that's fine. I'm only commenting on my experience. – ChrisA Apr 21 '14 at 22:32
constant emphasis on the fact that “x is just a number you don’t know yet <-- be careful with this -- this emphasis on "solving for x" and that x has only one value can really confuse kids when faced with learning about linear equations, where x can take an infinite number of different values. – PurpleVermont Jul 27 '14 at 21:24
@21820 - First, you've conflated my example with the general principle I stated. The objective is to remove the fear of manipulating equations, not to give a complete and exception-free language syntax that will be complete forever. Second, in high school algebra, x generally is a number. The implicit 'if x is a number' can of course be restated in rare special cases eg 1/x=0. The whole point is to give them confidence. Once they have this, the finer points can be added. – ChrisA Jan 25 '15 at 8:36
@PurpleVermont: Again, you don't get the kids to eat an elephant by stuffing the whole thing down their throat - it's one bite at a time. Simply plotting a graph will introduce them to the fact that x can take many different values. But when they can't solve 3x+5=14 yet, all these technically true but educationally pointless facts will not help. One thing at a time, and in the right order. – ChrisA Jan 25 '15 at 8:39

For some students, the difficulty with solving $3x+5=14$ is even more basic than figuring out what operations to do in what order in order to reach the goal. Before getting to that, they need to know what the goal is. "Everybody knows" that, when solving an equation with one variable $x$, the goal is to end up with a statement of the form $x=$ some specific number. Unfortunately, this "everybody" doesn't really include everybody; some students have never had the goal made clear. Moreover, in some cases, once they understand the goal, they're remarkably good at finding strategies for working toward it.

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Indeed. I remember the big sigh of relief ("Ohhhh!") when I told one of my students that "Solve" the equation means "Find the value(s) of x for which the equation is true". – ChrisA Apr 21 '14 at 18:54

I saw this question and laughed, "That is way too broad!", but I've been in your position. I was a classroom teacher for 10yrs in the public school system and was often tasked with teaching something that I hadn't had training in.

What you are looking for initially is a "Scope and Sequence" - a guide showing the steps in teaching a subject. Your 'expert blindness' makes it hard to make one on your own, but it is also redundant - experts have already done this. You can put together a S&S by look at roughly 2 sources:

1. State or privately developed curriculums - some states offer there curriculum online in the form of "Standards". You can look at what is required at each grade level and get an idea of what you need to teach. You'll need to assess your student against current grade level requirements and then work backwards until you get to the point they understand. That reveals their 'deficiency'. Then you remediate. So, the curriculum will tell you at 8th grade they need to know 'this' and at ninth, 'this'. You teach what they sequentially through the curriculum.

2. Books and guides - academic textbook are often set up in a proper sequence that will show you a framework of what needs to be learned first. You can obtain these often at libraries, but you may need to dig. Ideally you can find the books the students have used in their classes. Homeschooling resources are also readily available and can be found to meet a lot of different special needs.

This is a tough nut to crack. It really highlights the fact that 'Teaching' is far more than knowledge of a subject! Teaching is its own skill. Good luck!

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I think this is the most useful answer to such a broad question. We all have our favorite tips and tricks and points of view of what is important, but the first thing a new teacher needs to know and understand is scope and sequence. Understanding scope and sequence will give the framework around which to develop a point of view about what is important to emphasize. I would only add that there is a third resource that the questioner should seek out: excellent veteran teachers. – jbaldus Apr 25 '14 at 2:00

First, I want to comment on something that ChrisA seemed to have glossed over in his detailed description.

For instance, in the example, $3x+5=14$:

• First, decide what to do to both sides (subtract 5)

In my experience as a teacher and tutor, I have noticed that this is not easy for novice Algebraists. However, I have found that there is a way that you can help to make this "decision."

We are all familiar with the order of operations and many of us use the mnemonic PEMDAS to determine which operations to perform in which order when evaluating an expression. This is applicable for evaluating expressions and complicated/fabricated arithmetic problems.

This can be used in Algebra as well, though it is not something that I have seen often. The decision on what to do is the reverse of the order of operations. The two operations which are acting upon $x$ in the given example are multiplication and addition.

(As an aside, multiplication is represented with the $3$ immediately in front of it; multiplication has many forms and these forms may be something that you may want to discuss with your students as well if they have trouble recognizing each of the forms. Addition is represented with the $+5$, though you may wish to have a deeper discussion when dealing with a problem with subtraction, as it can be thought of as either subtraction or addition of a negative number.)

According to the order of operations and PEMDAS, multiplication comes before addition, and if this were a problem with only numerals, then that is the order that you would have to do things. However in Algebra when we solve for a variable, we are attempting to unravel the operations being performed on the variable so we can read the variable alone and use the property of equality to determine its equivalent value. This unraveling is done by observing what is happening to the variable, and performing the inverse operation to be left with only an identity (in the case of multiplication and division, that identity is $1$ while in addition and subtraction it is $0$; this is another topic you may wish to go into more detail about). Because identities provide equivalent values, we often do not write these. It is this reason that in Algebra we reversing the steps of the order of operations and PEMDAS, and because these operations are reversed, so too is the order of the operations. This is why we need to do the opposite of addition (subtraction) first, and the opposite of multiplication (division) second. I have found that making this thought process explicit has helped some of my students more easily determine this "decision."

Do not be afraid to use the technical terminology either (e.g., inverse, identity), as I have found that this actually helps to clarify things.

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Second, there is a lot of value in rewriting the equations in two different ways. I have seen students who prefer each style, so you may want to try both:

Method 1:

$$3x+5=14$$

$$\qquad \color{red}-\color{red}5 = \color{red}-\color{red}5$$

$$\quad 3x=9$$

Method 2:

$$3x+5=14$$

$$3x+5\color{red}-\color{red}5 =14\color{red}-\color{red}5$$

$$\quad 3x =9$$

Using color here with this second method is particularly helpful, and is something that I use whether in a room with a white board or a blackboard.

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You're right, I did gloss over that in the interests of brevity, and I agree that it's often not obvious to novices. Reversing the order of operations is certainly helpful. I try to build in the understanding of what to do from much simpler examples, eg x + 1 - 1 = x (with several numerical examples of x), and (x/3).3 = x, again with numbers as examples. If they can be persuaded to grasp that, remembering a rule (which I'm usually dead against!!) becomes unnecessary. – ChrisA Apr 22 '14 at 15:54
YES! You have no idea how many people I teach who are so very grateful for an organising principle. Suddenly they can track the decisions their lecturer makes when solving equations and do it themselves. @ChrisA, I find it's not enough that they know how to get rid of a "/3" or a "+1$. They still need guidance of which to focus on first when there are both. – DavidButlerUofA Aug 27 '14 at 16:44 Just a note: Not all of us know what PEMDAS is. Wikipedia does, luckily. – Tommi Brander Oct 22 '14 at 8:57 I think you will need to be very cognizant of student conceptions of how to solve algebraic problems. It may be useful to not try and immediately show them how to solve problems, but rather to ask them how they would go about solving the problems. This will enable you to learn about their mathematical thinking and possible misconceptions they may have. In the example you gave, a student my try to divide both sides by 3 but then simplify it to x + 5 = 14/3. Students solving linear equations often forget to apply the operation to both entire sides and are very focused on eliminating a particular coefficient or term. - Emphasize word problems. If I have \$14 to spend on 3 toys and a hat, and the hat will cost me \$5, how much can I spend on each toy? - Nice. I'll use that. – ChrisA Apr 21 '14 at 18:43 Problem is that makes things worse for dyslexic people! – kjetil b halvorsen Apr 29 '14 at 15:56 A practical introduction is always a good idea. The box method. Here we have three sealed boxes. Each box contains the same number of counters. I will label each box with an x. I give Anna the three boxes and five counters. I give Bob fourteen counters. Now I will tell you that if Anna was allowed to open the boxes she would have the same number of counters as Bob. Without opening the boxes how can we work out how many counters there are in the box? See if the class can come up with a way of solving this. EDIT Further to this idea I have a blog that is developing it further - I use a similar box method when teaching the concept of variables in programming. I heard about it from a Professor who believes that you should try to involve all senses when teaching. If the students see an actual box with the label x the idea becomes more concrete because neural connections involving touch are also reinforced. To my surprise it worked. Generally if I can reduce something abstract into something concrete that they can touch, hear, smell etc. - I get better receptiveness from my class. – tls Dec 18 '14 at 8:32 Teach the students that they can (and should) check their own work. A student who knows that their check-by-substitution worked will be a lot more confident that they learned that day's lesson than a student who is waiting until the next day to find out they got some answers wrong. Also, it is great practice for professions (like accounting and programming) that need to "tie out" or "unit test" their work. Here is how I was taught to check my work. In the examples, most of the "·" signs are optional: 1) Write out my answer, such as x = 3 If it is the answer to a story problem, include a note about what the answer means, such as x = 3 \$/toy. Each toy can cost an average of 3 dollars.

2) Circle the answer in a fluffy cloud.

3) Write "CBS:" below the answer. (CBS stands for check-by-substitution.)

4) Substitute in the answer into the original problem. Put a question mark over the equals sign. For example,

3 toys · 3 $/toy + 1 hat · 5$/hat   ≟ 14 $ 5) Do the math on both sides of the equals sign. Keep the question mark over the equals sign until it is obvious that the equation is true. Put each version of the equation on a following line, and try to line up the equals signs. For example, 3 · 3$ · toy/toy + 1 · 5 $· hat/hat ≟ 14$
3 · 3 $+ 1 · 5$           ≟ 14 $9$           +     5 $≟ 14$
14 $= 14$


6) When/if it becomes obvious that the equation is true, put a check over the equals sign, and congratulate yourself.

7) If it becomes obvious that the equation is not true, either try to find the mistake, or start over, or try a different problem.

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What does CBS stand for? Check by substitution? – Tommi Brander Oct 22 '14 at 8:59

If your curriculum allows you the flexibility to do this, I prefer to start with what are variables (a letter representing a number that varies), then what are expressions (a plan what you'll do once you know the variable's value), then how can we evaluate the expression for a particular variable value.

Stick with various expressions for at least a few days before turning the page to equations. With all this practice evaluating expressions, the guessing-game nature of equations will be clear: you guess the value of $x$, evaluate the LH expression, evaluate the RH expression, and see if they're equal, meaning the $x$ you guessed is a valid solution to the original equation.

Once the problems get too hard to solve by guessing, finally follow ChrisA's answer to teach a methodical way to solve equations, always preserving equality by doing the same thing to both sides.

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In algebra, a variable (confusingly) doesn't vary, it is fixed but unknown (or we just don't want to commit on a particular value). – vonbrand Dec 25 '15 at 23:06

I would recommend looking at Dan Chazan's excellent book, Beyond Formulas in Mathematics and Teaching: Dynamics of the High School Algebra Classroom, which grapples with many of the issues you raise. In particular Chazan narrates the challenges of working with struggling students like the one you anticipate working with, and he spends a lot of time unpacking fundamental issues like "What does an equation mean?".

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here is another way of looking at solving the problem $$3x + 5 = 14$$ break it up into two simpler problems solve $u + 5 = 14$ for $u$ and solve $3x = 9$ for x; see if they can solve that. in fact, this is how the student ought to have arrived at the problem.

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@JoeTaxpayer, you want me to introduce more variables? i have introduced an extra variable to make it clearer. – abel Dec 18 '14 at 3:57
@JoeTaxpayer, have trouble subtracting and this is not the first time either. – abel Dec 18 '14 at 12:02
@JoeTaxpayer, dont erase your comment. i should have made a separate edit. it is ok for the students to see we make silly arithmetical errors too. i should have error checked as i tell my students. you make mistakes and learn. – abel Dec 18 '14 at 22:17
Too late, I can't undelete. On a general site, it's worth seeing we all make mistakes. Here, I'm more concerned for the clean Q&A with no need to see that history. If it were more characters, I'da just have edited. But only author can edit 2 characters. I +1 this, I like the idea and will use it when the opportunity arises. – JoeTaxpayer Dec 18 '14 at 22:33