# Teaching Solving Linear Equations before teaching evaluating expressions

Traditionally, I have always taught evaluating expressions before teaching linear equations. But, I was recently given a remedial class of students that have to cover the bare minimums (and we have until mid-December to finish). Luckily, I have great flexibility with what I can do to the syllabus, so for the first time ever, I have completely cut out evaluating expressions since they won´t even be tested on this on the final exam.

My question is more if anyone else has done this, or thinks this is not a good way to go. Most of my students in that particular class have ZERO to little formal math background, a lot of them did not even finish high school, and they barely get by with mean, median, mode, rounding, etc. I started equations with them today, and they seemed ¨fine" for the most part. Of course, I also have spent the past week emphasizing positive and negative integer operations, so they are pretty OK with that so far. The textbook itself does not cover linear equations until after the section on evaluating expressions.

EDIT: Upon request for "non-native" English speakers:

Evaluating expressions simply means in the US to plug in numbers for the given variable values of the algebraic expression. Thus, for example, an exercise would be:

"Evaluate a + b + c" if a = 1, b = 2, c = 3."

The expressions can be as simple as that, or very much more complicated/interesting/beautiful. But, you get the picture.

Linear equations simply mean basic equations where you solve for an unknown variable.

Example: x + 5 = 10. What is x? Or 2x + 20 = 40, what is x? Etc.

• Done. As for the content, it is a remedial class (middle school level, but they are college students, older adults mostly).
– Wasp
Nov 18 '21 at 11:35

I hope this does not come across as overly harsh: I do not think that thinking of teaching as "covering material" in a particular order is a useful framework.

If your students are solving equations like $$3x+4 = 19$$, but are unable to evaluate $$3x+4$$ at $$x = 5$$ to check to see if they are correct (or, if they get an incorrect answer, to see for themselves that their answer is incorrect), then they are not doing anything of intellectual value. It is impossible to "solve" the equation without understanding how to evaluate the expression on the lefthand side.

One can manipulate the equation according to some rules, obtain $$x=5$$, and circle it, but this would just be mimicry of mathematics, not the genuine article.

So my answer is: you do not need to specifically devote a day to "evaluating expressions", but you had better be sure that the students do achieve this outcome in tandem with solving equations. If you do not, then all of the work you have done will be meaningless.

• Thank you for this. I do not think it is harsh. I have taught them how to "check" the answer by plugging it back, I just don't spend the entire chapter on evaluating multiple different types of algebraic expressions, particularly when the only equations they will encounter in this specific course do not even include fractional equations or anything beyond mx + b = c type equations.
– Wasp
Nov 18 '21 at 17:49
• @Wasp: I think requiring students to show work (that you can look at when grading) in checking the solution is a good idea, and also plenty enough for them. This is pedagogically useful as well as practically useful (the latter in case they have extra time on the final exam to check some/all of their answers). As for evaluating expressions, my understanding is that this will involve order of operations conventions and usage (e.g. exponents) beyond what they strictly need, and thus in a pinch (as you appear to be in) can be dropped. Nov 18 '21 at 19:15
• Regarding my last comment, for an example of following my own advice, see "Important Note" and "Safety Check for Values of Constants" near the end of this answer. Nov 18 '21 at 19:22
• @Wasp It would seem from your comment on this question - you are expecting them to evaluate simple expressions before teaching linear equations. Perhaps you should clarify this in your question. Nov 18 '21 at 19:35

Your context is students who have not learned much from the mathematics education they have been exposed to thus far. I guess, but can not be sure, that they have already been exposed to someone showing them algorithms and asking them to repeat. Because they are performing badly now, likely the teachers have tried to take this into account by teaching these things to them very slowly and thoroughly.

How has it been working thus far?

Given this, it is a tempting idea that you should do something, anything, else. What exactly this something else is seems to vary a bit, but the literature I have been exposed to recently has been all about open questions and questions with low threshold to doing something but with lots of room to expand.

Some that might be relevant for your students are (and this is speculation on my part, so a truckful of salt etc.)

• https://en.wikipedia.org/wiki/Figurate_number : have a pattern, ask them to continue it, figure out a rule, evaluate to check. This should practice evaluation of expressions, though mostly with positive integers. But then you could ask if we can make sense of the two-and-halft figure or zeroth figure or minus tenth figure; what might such mean?
• how complex an equation can you make that has a given answer?
• how much do different telephone contracts cost in terms of fixed costs and use of data, calls etc.? Which ones are cheapest for particular amounts of use? At which point do the costs of different contracts break even? Maybe use the contracts the students have. Or take something more relevant for their everyday life. Obviously, make sure to explicitly draw the connection to the more abstract mathematics, or the skills won't translate outside the context. Note that this problem trains both of the skills you mention.

This perspective would suggest approaching the issue more in terms of interesting and open questions and less from the perspective of a staircase of simple skills.

• I love all of these suggestions! I am part of a facebook group called "Low floor, high ceiling mathematics" where I have seen many such tasks. Nov 18 '21 at 19:58
• The Norwegian term is LIST, which stands for lav/låg inngansterskel, stor takhøgde/høyde. I did not bother finding out the English term, so translated this in a freeform manner instead. Nov 19 '21 at 8:40