Method of Showing Algebraic Work

Question

I have seen two different methods of showing algebraic work when solving equations. I show both of them below for the same simple math problem:

\begin{alignat}{8} x+3 &\;=&\; 5 \qquad&&\qquad x+3 &= 5 \\ -3 &\;&\;-3 \qquad&\text{or}&\qquad x+3-3 &= 5-3 \\ x &\;=&\; 2 \qquad&&\qquad x &= 2 \end{alignat}

Which method do you show when teaching and require of your students when they show work? Why do you feel one method has merit over the other?

Answer 1

At my institution, your 2$^{nd}$ option is the standard in the algebra sequence. This was chosen for three reasons:

1. [Most convincing for me] Immediately before solving equations, students practice simplifying an expression by combining like-terms. They are already accustomed to simplifying things like $$3x+5+(-3x),$$ so adding a term to both sides of an equation, in-line with everything else, reduces to just simplifying two expressions -- something they should be comfortable doing.
2. Writing $+(-3x)$ under each side of an equation may not appear to some students as being the same as what they've been practicing, because we emphasize horizontal expression-writing $$13x + 25x$$ over stacked notation: $$\begin{array}{c} \phantom{\99}13x\\ \underline{+\phantom{9}25x}\\ \phantom{\times9}\\ \end{array}$$ Better to keep consistent with the prevailing notation.
3. It preserves notation. While many people add or subtract terms underneath expressions, students often write these changes too low, not low enough or sometimes up above at an angle. These variations often leave students (and instructor) wondering if a value is really a term being added, a subscript on a term or an exponent.

All of that said, many of my students have writing underneath ingrained in their math habits. I will only make them change their habit if they are consistently making mistakes and need a new approach. For students who don't have any algebra habits developed, I suggest it as the standard in the class, and recommend that they learn this approach. In later courses, I am less pedantic about how one shows their algebra, provided they show something complete and mathematically correct.

Finally, I demonstrate that method with colored chalk/pens, and many students follow suit by bringing their own colored pens.

Answer 2

I like the left-hand example, because it highlights the step, which is something being done to both sides. This is how I add or subtract. Multiplying and dividing look different, of course.

If you were doing the right-hand version on a whiteboard, you could do it in a different color. But that isn't something the students could easily do in the same way.

Answer 3

I'm not sure that I actually like either presentation all that much. If I were explaining the work on the board, I would probably start by writing $$ x + 3 \phantom{{}-3} = 5 \phantom{{}-3} $$ leaving a little bit of space where I know that I am going to have to fill something in a later. Then, in a different color, fill in the blank to get $$ x + 3 \color{red}{-3} = 5 \color{red}{-3}. $$ Finally, the last statement needs to be connected to what is already written. In this case, an implication suffices \begin{align} x + 3 \color{red}{-3} &= 5 \color{red}{-3} \\ \implies x &= 2. \end{align} This may be pedantic and nit-picky, but I generally insist that statements not be written one atop another without some kind of logical connection. This becomes particularly important when, for example, working with equations that can have extraneous solutions. For example \begin{align} &\frac{1}{x-2} = \frac{3}{x+2} - \frac{6x}{x^2 - 4} \\ &\qquad\implies x\ne 2,\quad x\ne -2,\quad \text{and}\quad x + 2 \color{red}{{}+3x-2} = -3x-6 \color{red}{{}+3x-2} \\ &\qquad\implies x\ne 2,\quad x\ne -2,\quad \text{and}\quad 4x = -8 \\ &\qquad\implies x\ne 2,\quad x\ne -2,\quad \text{and}\quad x=-2. \\ \end{align} Since we arrive at a contradiction, we know that the original statement must have been false, i.e. the equation has no solutions. By carefully keeping track of the argument and writing down the details (including the way in which statements are connected to each other, as well as the full set of implications of a particular identity), there is no need to come back and check for extraneous solutions.

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The situation is slightly different for the work that student turn in to me (since they don't know, ahead of time, how much space they might need to add or multiply). For the original problem, I might expect to see something like \begin{align} x+3 &= \ 5 \\ {\small -3} &\ \ \ \ {\small{-}3} \\ \implies x \phantom{{}+3} &= \ 2. \end{align} My preference would be that they write out the extra line (primarily for the third point that Nick C makes in his answer), but I recognize that students often lack the time to write more clearly (say, in an exam setting), and I think that it is more important for them to be comfortable with the big picture idea of keeping equations "balanced" than it is to nit-pick their notation too much.

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Multiplication and division, as well as exponentiation and taking logarithms, follow a similar pattern. On the board, I use color to emphasize what is changing from step-to-step, whereas in student work, I rather expect to see everything done in-line. For example, on the board I might write $$ \color{red}{\tfrac{1}{3}(} 3x \color{red}{)} = \color{red}{(} 15 \color{red}{)\tfrac{1}{3}} = 5 \implies x = 5 $$ or $$ \color{red}{\frac{\color{black}{3x}}{3}} \color{white}{\frac{\color{black}{=}}{-}} \color{red}{\frac{\color{black}{15}}{3}} \color{white}{\frac{\color{black}{=}}{-}} \color{white}{\frac{\color{black}{5.}}{-}} $$ I would expect students to write basically the same thing in their work, except without the use of color.

Answer 4

I do not mandate either method over the other, only that the students are able to demonstrate they understand how/why each step in their solution is done. I usually make my students go to the whiteboard, show all steps, and be prepared to present/explain each step. I like the geometry way of using a "t-gram" with each step on one side and justification (usually in the form of mathematical property) on the the other. Another thing, in order to make sure mere memorization is not at hand, the students know they may be challenged or asked for evidential answers to challenging questions either by classmates or myself. A bit slow at first, but soon enough, speed is comparable to using only verbal explanations; and a writing element (as well as critical thinking) is now included.