How do I teach algebra?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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!

Answer 4

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:

\begin{equation} 3x+5=14 \end{equation}

\begin{equation} \qquad \color{red}-\color{red}5 = \color{red}-\color{red}5 \end{equation}

\begin{equation} \quad 3x=9 \end{equation}

Method 2:

\begin{equation} 3x+5=14 \end{equation}

\begin{equation} 3x+5\color{red}-\color{red}5 =14\color{red}-\color{red}5 \end{equation}

\begin{equation} \quad 3x =9 \end{equation}

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.

Answer 5

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

Answer 6

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.

Answer 7

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?

Answer 8

Some students that I've encountered, especially those who have struggled with solving linear equations in the past, have taken well to what is sometimes called "backtracking" [in CME Project: Algebra 1] or "the arrow method" [in LINCT Transition Course curriculum (note: this is the program I work for)]. The illustration below may be enough for a master algebraist, but if you seek more description, I recommend searching for either of the two curricula mentioned above.

First the black text and arrows is written in, from left to right. Generally students are able to follow the order of operations that occur to the variable and can generate this step with relative ease. Next the green text is handled, where students just need to focus on the inverse operation from the black arrows. Last, the red text allows students to follow along with the operations from the green arrows. I like this method particularly because it also opens the door for fruitful conversations around inverse operations and their relationship to solving algebra equations of this type.

Alternatively, I have also seen instances where students and/or teachers prefer to leave out the black text below the arrows, and just leave empty circles (or "bubbles") to be filled in where the red text is. If the only goal is to solve a particular equation, then I think this is fine; however, I think there are many more benefits to seeing the expression grow and its equivalent values with the complete picture.

There are limitations with this method, notably when there is more than one variable in the equation, but your initial question doesn't seem to ask about this. I think this is a viable option for students to see who struggle with solving basic linear equations that provides a visual representation for inverses that is otherwise obscured in many other methods. It does not work universally, but is a good place to start and can develop some very strong conversations as well.

Answer 9

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.

Answer 10

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.

Answer 11

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?".

Answer 12

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.

Answer 13

If the students are weak, lots and lots of tangibles: get them into the habit of drawing pictures and moving stuff around ("paper is cheap").

For example, integer arithmetic can either be a bunch of rules, or you can use the "chip model" to really get across certain ideas of how it works:

https://youtu.be/EA8j7V677Z4?list=PLKXdxQAT3tCuSrZNwMUIXbfmpmYyS9bfy

Multiplication of polynomials can be a bunch of rules, or you can use the "area model" to perform products:

https://youtu.be/7tK2DjxZQeY?list=PLKXdxQAT3tCvNbJUuFSqhXPfQ_53yskfg

Absolute value can be a bunch of rules, or you can use its geometric meaning as "distance between":

https://youtu.be/jl9BtnBFRJA?list=PLKXdxQAT3tCvNbJUuFSqhXPfQ_53yskfg

Graphing in the coordinate plane can either be a bunch of rules, or you can use the idea of it as a set of directions.

https://youtu.be/OYCfmHpYXlA?list=PLKXdxQAT3tCvNbJUuFSqhXPfQ_53yskfg

Answer 14

For students a problem such as $3x + 4 = 16$ presents several conceptual difficulties even before the question of how to solve it is addressed.

1. The notion of a variable/indeterminate is somehow quite abstrct. The idea that $x$ is a placeholder for a putative solution (which in general may or may not solve the equation once $x$ is replaced by the putative solution) requires a substantial mental leap when compared with operating only with concrete numbers, e.g. $3(5) + 4 = 19$.

2. The notion of equality of formal expressions is also quite abstract. In the equation $3x + 4 = 16$, the left-hand side is a formula involving a formal variable, while the right-hand side is a particular number. These are objects that do not occupy the same mental space. The difficulty in some sense is that the particular number, in this case $16$, has to be promoted to be viewed on an equal footing as the formal expressioN $3x + 4$. This is not a straightforward step.

3. The identification of equations whose solutions are the same presents the fundamental difficulty, encountered at many stages of learning mathematics, of the difference between equality an identification. The equations $3x + 4 = 16$, $3x = 12$, and $x = 4$ are all identified in the sense that their sets of solutions are the same, but they are (obviously!) not equal as formal expressions (the symbols appearing in them are different!). Whatever algorithmic procedure is used for solving the equation requires identifying the different equations obtained at each step of the procedure. Further compounding the difficulty is that the final equation, $x = 4$, is usually conflated with its set of solutions (consisting of the number $4$), and this is done without any comment.

The issue of understanding an abstract variable is the most easily addressed. Repetition of problems of a similar form, e.g. $3x + 4 = 16$, $3x + 4 = 19$, $3x + 4 = 22$, $2x + 4 = 16$, $2x + 4 = 18$, etc. (minor variations) helps develop a sense of the formal meaning of $x$. Later these same problems can be formulated in words - I have $4$ widgets and I buy some boxes each containing $3$ widgets, how many boxes need I buy to have $16$ widgets in total? - and the exercise of identifying the relation of the formal equation $3x + 4 = 16$ with the word problem will help understand what $x$ is (the student can be asked to formulate a word problem that corresponds to the equation).

The issue of the identification of equations whose solution sets are the same seems to me much harder to teach and explain. Even at the university level, with students who are quite practiced at solving single variable equations, the issue reoccurs when one tries to teach them methods for solving systems of equations (e.g. Gaussian elimination). Part of the problem is that instructors rarely give enough credit to the notion that $3x + 4 = 16$ and $3x = 12$ are entirely different equations. On the one hand this is so obvious that it seems to require no comment at all, on the other hand, we treat them (operationally) as the same when they occur in the series of steps leading to $x = 4$. Probably one needs to be quite explicit that, although different, these equations have the same solution (their interpretations might be quite different, if, for example, the coefficients have physical or economic meaning - if I have no widgets and I buy $3$ boxes looking to have $12$ the situation is not the same as that described in the word problem above).

That a smallish child can readily assimilate the notion of a variable as a placeholder is an indicator of certain talent for the sort of abstraction that makes for a good mathematics student. Many, maybe most, children don't have this talent, or at least require that it be developed and directed, usually via a lot of drill with minor variations of a simple theme. Even with university engineering students (generally excellent students in high school!) one frequently encounters a lack of understanding relating to formal substitution of variables and its role vis-a-vis the function concept; this is part of why so many have trouble with things like the chain rule and changes of variables in integrals.

Answer 15

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

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

Funny that no one mentioned balance scale, because equation is exactly this: balance scale being in balance. "Do the same thing to both sides" meaning add the same number (like the same weight) to both sides, remove the same number (weight), divide or multiplie by the same number (multiplying an apple by three means replacing the apple with three times as many) to keep the scale in balance. For anyone who have seen balance scale daily by going into a corner store to buy groceries or by going to a farmer's market, this concept is extremely easy to grasp.

From there you proceed with what you want to do: to isolate x on one side of the scale. You do this by collecting like terms first, then moving all constants to another side (that is, adding or subtracting to both sides to isolate a monomial), then getting rid of the coefficient by dividing both sides by it. The process is highly algorithmical, at least for simple equations studied in middle school.

I would suggest introducing the notions of monomial, polynomial, and like terms before introducing solving equations. On another hand, kids as young as four-grades can easily solve systems of linear equations without knowing they are doing it if you phrase it somewhat like "a bar of soap and two tubes of toothpaste cost 2.50, while one tube of toothpaste costs 0.80, how much the bar of soap costs?" and you can go from here. The idea or substitution seems to be not the most obvious one for many kids, and one has to get used to the idea that things can be substituted with equal things. After it, solving equations becomes much easier.