The Five Big Ideas of First Year Algebra:

- Identifying the slope and y-intercept of a line from a linear equation, and graphing a linear equation provided in slope-intercept form.
- Solving multistep, single-variable equations that involve distribution and combination of like terms.
- Using substitution or elimination to solve a system of equations.
- Binomial multiplication
- Factoring a quadratic equation (a=1)

(You middle-school algebra teachers are saying “Wait, what about graphing a parabola? What about point-slope and standard form for linear equations? What about….” Stop right there. I teach the kids who didn’t make it through your classes. Some winnowing is necessary. Furthermore, I said the five BIG ideas, not the ONLY ideas.)

I’ve taught some form of algebra every year. From my first year on, I’ve nailed factoring quadratics. I do it with the generic rectangle, which has the added feature of helping out when a > 1. The integrated method I use to teach both binomial multiplication and factoring really seems to help the students put it all together.

Towards the end of my first year teaching algebra, I noticed a weird thing. My kids were bombing multi-step equations, and I couldn’t see why. I’d taught them distribution, combination, and isolating—and they’d all done well. Then they were utterly discombobulated when faced with an equation like 3x + 5(2x-7) = 4. They’d add the 3x and 5 to get 8x, then multiply it by 2x, get 16x….it was insane. Yet when I walked them through the problem breaking down distribution and combination, they got each step individually.

Then, early in my second year, I saw the same problem. Kids who had shown solid mastery of distribution, combination of like terms, and solving for x were crashing and burning when I gave them a multi-step equation that mixed and matched everything. I suddenly got it. Multi-steps up the cognitive load considerably. The kids had to take each step in the context of a larger task, and they were losing track. They couldn’t look at the problem and break it down into parts.

So I created the Distribute-Combine-Isolate worksheet, one of the best worksheets I’ve ever done. First distribute, then combine, then isolate. It gave them a sequence to follow. The improvement was tremendous. My first year students, who had much lower incoming test scores than other classes, topped all the other classes in a course-alike assessment on the multi-step equation. This year, I used the same worksheet with any Algebra 2 students who struggled with multi-steps. Again, working multi-step equations has been a major success area; I don’t have to review it, and I can put a tricky question on a test and know that all the students will either get it right or make a few minor mistakes. I am pleased.

Systems: This is the lowest priority of the Big Five when I’m working with struggling students, but it’s a high priority item for my stronger students. I find the challenge comes in when I want them to *recognize* a system problem. They get the technique, but the overall solution approach is still iffy. But then, this is tough. I don’t feel any real frustration or energy about it.

Leaving linear equations and binomial multiplication, arguably the easiest of the Big Five, as the most challenging and mindbogglingly crazy-making. They get it and forget it. Get it and forget it. Over and over and over and over………….[bam bam bam bam bam]

Slope: You teach them how to plot points. You teach them to see the line. You use manipulatives, transparencies with lines on them, that they can use to match up two points and see how the slope and y-intercept change. You show them how different types of situations map to different slopes. And of course, you give them endless practice.

And then you sketch a line, clearly mark the slope and the y-intercept, and ask any kid who isn’t acing the class, “So, is the slope of this line positive or negative?” and wait, and wait, and wait, and wait and sure as a villain in a Bond film, the kid will say “Um, negative?” when the slope is positive and “Positive” affirmatively when the slope is negative.

So you teach them how to model equations quickly, which works a charm and gives them all sorts of new skills. You see them become much more proficient at word problems, at seeing an equation like 8x +3y = 24 and thinking “Burgers for $8, hotdogs for $3, total of $24” and by god, it’s awesome. All this keeps, beautifully; months later, they are still showing increased competency at word problems and linear equations. You also give them endless practice worksheets where all they have to do is identify + or – on a slope image—nothing more, and they do it cheerfully and successfully. You give them the “N” rule (negative slopes form an N).

And then, you give them a test, in which they have to identify a simple system of inequalities, and a student, a mid-level student calls you over and says, “I have no idea how to do these problems.”

“Well,” you say, “look at one of the lines in the system.” The student points to a line. “Positive or negative slope?” and wait and wait and wait and wait and sure enough, the student says “Positive” when it’s negative and “Um, negative?” when it’s positive and you gnash your teeth and try to figure out how to help them without making them feel hopeless.

And later, when the same thing happens again during the test review, and you start beating yourself over the head with a whiteboard (they make them student-sized, did you know? Like slates in Laura Ingalls’ day) and then you get up and say, carefully,

“Look. When you see me beating myself over the head with a whiteboard, it’s because I am wondering what other way I could teach you this HUGE, SINGLE MOST IMPORTANT idea in first year algebra, something that I’ve told you fifty times, and believe me when I say that I’m not angry or disgusted when people don’t get it. I just can’t figure out how to make it clearer. And I think the real problem is NOT that I can’t make it clearer, but that I can’t get you to stop and think about the many, many many ways to determine the direction of a slope. All you need to do is stop and think about it and remember what you’ve done. And for some reason, many of you don’t. Let me say it again: I am not blaming you. I don’t think you’re dumb. I JUST WANT YOU TO STOP DOING IT SO I WON’T HAVE TO BEAT MYSELF OVER THE HEAD ANYMORE.”

And the class laughs, and you remind them again to stop the minute they see a line. What is the direction? What methods do they have for making that determination? Do NOT simply look at it and say “Heads, positive. Tails, negative” and guess. Please?

Lather, rinse, repeat.

As bad as slope is–and it’s terrible, horrible, the single most frustrating thing about teaching algebra to kids who struggle with math–it doesn’t have the short sharp shock value of the Binomial Multiplication Middle Term Miss.

Last week I gave my kids a geometry test and one of the questions was:

BOTH CLASSES. Every single kid (except the top 6 students, who took a different test) took x^{2} + x + 9, meaning that they squared (x + 3) and got x^{2} + 9. WHY? WHY? WHY?

I tell them that this makes baby Jesus cry. It’s the math equivalent of clubbing cute little seals. THEY MUST STOP. It hurts. And we review it, with the rectangle, which they use for factoring and SHOULD MAKE IT CLEARER, DAMMIT! and they learn it again. But I know, very soon, they will forget. If only to make me crazy.

May 22nd, 2012 at 2:31 am

Reblogged this on nebusresearch and commented:

educationrealist here writes up “the five big ideas of first year algebra”, which I likely could have used as guide to the algebra class I plunged into this past term. I’m reblogging as part of my hope to remember this when I teach algebra again.

May 23rd, 2012 at 5:48 am

I like the Talladega Nights reference. I use “if you ain’t first, your last” quite frequently in my algebra classes. And my head hurts quite a bit when I slam my palm into my forehead when I hear similar comments from my students…I’m surprised my forehead is not sticking out the back of my head by now…

June 17th, 2012 at 6:25 pm

[…] earlier posts that expand on these issues: Teaching Algebra I and Teaching Algebra, or Banging Your Head with a Whiteboard Share this:TwitterFacebookLike this:LikeBe the first to like […]

July 5th, 2012 at 5:06 am

Last week I gave my kids a geometry test and one of the questions was: A right triangle has sides of [square root of x] and (x-3). You are going to use these terms in the Pythagorean theorem to solve for x (along with the hypotenuse, which I’ll give you in the next question). After you square both sides and combine like terms, what will you end up with?

BOTH CLASSES. Every single kid (except the top 6 students, who took a different test) took x2 + 7x + 9, meaning that they squared (x + 3) and got x2 + 9. WHY? WHY? WHY?

********************************************************************************************

I’m super confused by the above.

The sides are (x-3) and sqrt(x)? how did (x+3) appear in the next paragraph? what about the x^2 + 7x + 9? Where is that from?

July 5th, 2012 at 5:43 am

Just a typo on my part. I’ll fix. When I’m in rant mode, things happen.

October 10th, 2012 at 3:47 pm

I’m coming late to the conversation, but I enjoyed your rant, and plan to crib some of your ideas for my students (I teach math methods in a teacher ed program.)

I have a speculation about why the kids do x^2 + 9 for (x + 3)^2. Assuming they don’t leave out the middle term when they see something like (x + 4)(x + 7), which you didn’t discuss. If that’s the case–I bet they are doing the above because they only see one binomial, with an exponent. I’d try making them write out both binomials when they see a binomial squared, before doing anything else, and then multiply just like they do with any binomial times a binomial, no shortcuts, no doing it in your head. With some kids, out of sight is out of mind, if I don’t see it it isn’t there, so the idea is to make it visible first.

Of course, if they ignore middle terms for all binomials, this won’t work.

October 10th, 2012 at 5:03 pm

They ignore it for all. I’m convinced they do it to make me crazy. It’s not just the squares, it’s everything. And I teach the generic rectangle method; I have been known to just draw the rectangle on the board and look ferociously at the students at the beginning of a test.

Glad you liked the methods! You may want to read up on the modeling work I did; it’s been very effective at helping kids retain linear equation concepts. Even if they get the slope direction wrong. (sob)

December 22nd, 2015 at 3:42 pm

If they are forgetting it regularly, you could try teaching it with “double distribution” in addition to the rectangle. I don’t teach the rectangle, but I do teach them from the beginning that to multiply (x+2)(x-5) you have to distribute x onto (x-5) and +2 onto (x-5). The first thing we write is x(x-5)+2(x-5). From there they can get it. In the end of the year most of them are FOILing, but some stick with double distribution all year. (Admittedly I’m teaching it to 8th graders…)

December 22nd, 2015 at 9:23 pm

I don’t usually teach FOIL, but the rectangle method, which is nothing more than a visual representation of double distribution (I show them that as well). Doesn’t matter.

May 22nd, 2016 at 7:36 pm

Maybe the problem is the rectangle method. Maybe it is easier to remember (and apply) an algorithm than a picture. I never learned the rectangle. FOIL or distribution make much more sens to me.

Don’t even know what rectangle is. Maybe that confuses the kids. Maybe its too intuitive and something that is more crunch along step by step would in the end be better.

October 27th, 2012 at 9:30 pm

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January 27th, 2013 at 12:21 am

[…] are clubbing orphan seals and making baby Jesus cry when you forget the middle term” while banging myself on the head with a whiteboard. And some of them just forgot […]

April 28th, 2013 at 1:20 am

Show slope dude from you tube over and over until they beg you not to show it again. Rise up run out is another I show over and over until the students start saying ok now I understand! I introduce, insert in the middle, and finish with good youtube songs for most topics and it works!!!! Repetition in disguise.

May 22nd, 2016 at 7:41 pm

Yep. Show them that video after every experience with bad slopes. They will be so annoyed by the repeated cheesiness, they will learn it just as a mantra.

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June 23rd, 2013 at 1:26 am

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September 15th, 2013 at 10:02 pm

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November 19th, 2013 at 4:31 pm

I just got out of testing my first year Algebra class. The whole time I could hear, “I don’t remember this.” “Do we do this for this problem?” And these were all students who the exact day before led the class and TAUGHT the study guide to everyone else. A couple of the problems were from the ACTUAL study guide that THEY worked. So believe me, when I say, thank God it’s not just me having this problem.

January 1st, 2014 at 3:12 pm

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November 1st, 2015 at 9:04 am

[…] change my approach to curriculum, I also flatly punted on coverage from that point on, focusing on the big five for every subject. As I improve at introducing and explaining concepts, my students become capable […]

December 29th, 2016 at 7:22 am

[…] They’ve held on through binomial multiplication, to hang on, at least temporarily, to the linear term so that (x+3)2 doesn’t become x2 + 9. They’ve mastered factoring quadratics, to their […]

October 29th, 2019 at 4:42 am

I’m not a teacher, but I have a suggestion on how to help kids ‘get’ slope. The problem is is that ‘slope’ is a difficult, arbitrary word. ‘Positive’ and ‘negative’ are also difficult, arbitrary words. And, the words aren’t really necessary, what is needed is the concept: they don’t need to know if the slope is positive or negative, they need to know if y goes up when x goes up, or if y goes down when x goes up, and they need to know how much y goes up when x goes up.

Instead of asking them ‘is the slope positive or negative’ ask them: “does y goes up when x goes up?” — y/n? if they are staring at the graph, they can’t get this wrong! They are going to say, yes! y goes up! or no! y goes down!

Then ask ‘how MUCH does y go up (down) when x goes up one? Again, they can read it off the graph. up 3! up 5! down 2! or whatever.

Don’t use the WORD ‘slope’, use the WORDS ‘how much does y go up when x goes up one’. Don’t use the WORDS ‘positive’ or ‘negative’ use the WORDS ‘up’ and ‘down’ (or ‘bigger’ and ‘smaller’).

Then ask ‘so do we add 3 to y when x goes up one or do we subtract 3 from y when x goes up one?’ they will know this too, if they are staring at the graph and come back with yes yes yes! or no no no! up 3! down 3! It’s a concrete question.

And it feeds right into the equation y1 = y0 + m(x1-x0). Or rather, y1 = m(x1-x0) + y0.

Don’t skip over that format of the equation and try to jam the kids right into y = mx + b straight from the graphs. y1 = m(x1-x0) + y0 is a much more comprehensible equation than y = mx + b, one that is much easier to generate from the graph, and to understand. Filling in the blanks for equations like y1 = y0 + m(x1-x0) or y2 = y1 + m(x2-x1), etc. is very easy to do, with nothing but concrete questions. (eg ‘does y go up or down when x goes up?’ ‘how much does y go up (down) when x goes up 1?’ ‘what is y when x=0 (y0)?’)

And, when I did introduce y=mx+b, as the ‘general case’ of y1 = m(x1-x0) + y0, I wouldn’t use ‘b’ in the y= mx+b equation at all, I would use ‘y0’. In other words, write the equation as y=mx + y0, with ‘y0’ as the constant, not with ‘b’. Why introduce a new and arbitrary symbol ‘b’? and then make the kids memorize (or rather, fail to memorize) that b is ‘the y intercept’? That’s just another arbitrary and confusing word, with an arbitrary and confusing symbol, that they are going to forget the definition of. b is just y0, y when x=0, just like y1, is y when x=1, etc. Why make them learn and remember a new, arbitrary symbol for it?

So, I’m saying the kids are confused because the terminology is confusing–the words are arbitrary and abstract and their definitions are hard to remember. They don’t make it easier to convey your meaning, they make it harder because it is hard to remember what they mean, so, just avoid them altogether, they were obviously poorly chosen to begin with, and you’d be much better off if you just used more concrete, comprehensible terms like ‘rise over run’ or ‘how much does y increase’ rather than ‘slope’, and ‘y when x=0’ rather than ‘y-intercept’. They’re not really that many more syllables to say.

Then, at the very end, when actually doing the math has become turn the crank, THEN define the terms of ‘slope’ and ‘positive’ and ‘negative’ and ‘y-intercept’, and add in the additional step of translating the terminology.