California 7th Grade standard, Algebra & Functions:

4.1 Solve two-step linear equations and inequalities in one variable

^{1}over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.

This is the introduction to algebra that most kids stumble on. It is fraught with peril, and heavily dependent on good prior understanding of fractions, negative numbers, and order of operations. Even with a good base, there is enough new knowledge here that kids get lost, some of them forever. Wrestling with the mechanics without understanding the abstraction, or vica versa, dooms them. The following lesson is how I’ve best found to establish enough of the mechanics that they can be successful enough to then worry at the abstraction aspect.

**WARNING: Half assing this lesson, or not bringing it to completion, will earn you the ire of any math teacher who needs to teach your kids after you are done with them.[2]**

That warning is only half in jest. Kids love this method, and hate to leave it behind. I have seen teachers glom onto it because the kids love it so much, only to screw it up so that the kids can solve these problems, but never use it to transition to anything else. I usually preface it with a training wheels warning, and then follow it immediately with 3 step problems (for which this method doesn’t work) to prove to them that they will have to use the more traditional methods. This method makes for a heck of a bridge – you just have to make sure it’s anchored on both ends.

So, here it is:

Start off with the one step variation of what I’ll do below. The method is the same, except that it uses one less box, and will help cement the idea of “doing something to a variable. For instance, in the expression 3x, x is being multiplied by 3. In the expression x – 4, 4 is being subtracted from x.

Always start of with the question “How many numbers are there?” Immediately follow it with “How many boxes do I draw?” This is a verbal cue to get the kids started on the problem, and is the kick that starts off the avalanche for the rest of the problem. After they’ve drawn the boxes, label the first one as x, and draw some arrows between them as follows:

(The one step version of this, having only two numbers in the equation, would have only two boxes).

The arrows are then labeled with the “what you do to x”. This is a great time to reinforce order of operations, and mix things up with parentheses. The lone number on the right side of the equals sign goes in the box at the end.

This is where it is easy to screw up. The boxes, as drawn right now, represent the problem. The following steps represent the solution. It is easy to perform some of the following steps while setting up what’s been done so far, which would paint you into a corner when it comes time to transition to the more traditional method of solving equations. In fact, after about a half or a dozen of these, I’ll throw up a set of boxes with steps, and ask kids to create the equation from them, just to reinforce the difference between setting up the problem and solving it.

So, we know how to get from x to the answer. Problem is, the box at the front is what we want, the box at the end is what we have, and all the arrows go the wrong way. We need arrows to go the opposite way. That’s fine, as long as we use the inverse operations. So, from the box that has the number in it, we drw an arrow going back, with the inverse operation:

Then we do it again, for the final box:

Kids love this. The lower performing they are, the more they love this. I have a couple in my class who would happily do this day after day for weeks. This is hard math, and they’re doing it. They don’t need candy to solve these – they’re high on their own success. I don’t even need to tell them what the right answers are: follow the arrows along the top, and not only can they tell if they got the answer right, they can tell where the mistake, if any, is.

So why not just teach this and nothing else?

Two problems: This doesn’t transition to 3 step problems. It won’t work if you’re subtracting x. It’s a dead end.

So, why teach this?

The answer is in those little red circles up there. Those operations, going backward? Those are the operations that you apply to each side of the equation to solve in the traditional method. The numbers in the boxes? They’re what you get on the right side of the equation after each step.

After a day of this, (and that’s more than enough, actually), I have them do this in two columns. On the right side of the paper, they get to do the boxes. Then they have to solve the equations traditionally on the left, using the red circled operations from the right side. There is never a question of which operation they should be doing. There are only complaints about too much writing – so I let them leave out whatever they don’t want to draw on the right side.

Coming up: two more lessons to further wean them from the boxes.

^{1} This always kills me. How can you have a linear equation with one variable? Linear implies a relationship between two variables.

^{2} Further warning – I tried teaching this to gifted kids. Once. Backfired horribly – those kids already get what this is intended to teach, and will hate you for the extra work.

## { 30 } Comments

Interesting method. I’ve had some success with the pan balance method (which I guess I should write up). I try to stress “undoing” each operation too.

Your first footnote brought a smile – I agree.

That’s one of the nice things about this method – not only does it show why the order of undoing is the opposite of the normal order of operations, it shows that the opposite operation is what you use to undo.

I am definitely going to try doing this next year. I don’t think I provided enough training wheels this year. Students were so immediate in knowing the answer to one-step equations (without knowing what they did), but by the time we got to three-step equations I couldn’t motivate them to try because “that problem looks hard.”

this is AMAZING!!! im from seattle but this made things so much easier than what my teacher does not explain!

hope it works,

LAURA

Hi, I just wanted to let you know that I tried this method with my summer school class this week. The first lesson, we used just the box method. I gave lots of examples like

3x + 2 = 8

-5x – 7 = 13

12 = 8 – 4x

8 = 12 – x

I really liked using different examples. It didn’t take long before EVERY single student could set up even the trickiest of equations.

In the second lesson we solved two-step equations using the box method and algebraically. I forced them to use both methods. By the third lesson (solving multi-step equations), I had broken them away from the box, but it seemed they had a really good grip on problems like 4 = -7 + 2x. In the past, I have found many students just don’t know how to get started when the problem is arranged “backwards”. After using the box method, they seemed to be very comfortable knowing that the first step was to add 7.

I worked up a template with sample problems and an assignment if you are interested. I am going to include it in my curriculum next year. I actually worked up two templates. In the second one I used columns to separate the algebraic column and the box column.

I will be glad to email them to you if you are interested in seeing how your post inspired me!

It didn’t take long before EVERY single studentThat’s part of what i find great about this – it really does get every single student to do it. The tricky part is transitioning to leave it behind, and it sounds like that worked well for you too.

I’d love to see your templates – you can contact me here

Trying this next week. About half the class already knows how to solve equations, though, and I’ll have to think of something sensible to do for those students who are repeating the course due to not knowing second-semester material while being fine with simple equations.

Let me know how it goes – I just had that same situation.

I ended up asking the kids who already knew how to do it properly to follow along and see if they could figure out why this worked. I also sped up the process to fit into two days. It’s going to take a while to wean the kids have just recently gotten it, but I didn’t want to bore the heck out of the other kids.

Hey, i was trying to help my gf in her pre calculus class and i was just wondering how to add and multiply linear functions.

I think this might just work. My students are struggling a bit, so I am going to show this to them tomorrow. Thanks.

i whant to lor how to do two step equation…

I really like this method. I am going to pass it on to the other teachers in my department. The method will work if you subtract x as long as you consider something like “3 – x = 5” to actually mean

“3 + -1x = 5.”

mrs terri is my math teacher and she is so nice this days we are see equations and it so fun i like mrs terri because she is so nice my name is ivon trejo 11/4/09

This stuff is so hard i cant even do it. show some examples

I love it! I was trying to find an easy way to explain this to my nephew. This is it! He is really struggling with Algebra I and this is why. You explained the problem to a tee. I’m going to try this out tonight. I would love to see the templates Ms. Hughey came up with as well.

Also I need to know how the transition works. What do I do?

I teach Kindergarten so please explain in detail.

Thanks for your help!!

this is very easy and you should try my technique that i got my from my tutor this is very cool and easy after i learned.

The 2 step equations are very easy and I can do these equations without any problems when i have to take tests. This is and example i like to give you:

2x+6 =20 i first do this

-6 -6then you have to cross out the 6 and you subtract

__________ 20 by 6 which equal -4.

2x 14 then you put the 14 over 2 and put 2x over 2

___ ___ and then you cross out the 2x over 2.

2 2 Then you divide 14 by 2 which equals 7 so x=7. so the equation would look like this 2(7)+6=20

This seems way too complicated. I don’t care for it.

I absolutely hate this “method.” It is so confusing.

How do you solve a linear equation like this?

h -3 =2 – 9

I started my journey through equations this year conceptually, using wonderful pan balance equations and problems from Teachscape. The kids did well until there were no pictures. I kept bringing them back to the pictures but couldn’t once we had begun dividing. Subtracting didn’t even work since having a (-n) on one side was abstract. I’m ready to move on to linear inequalities and am frustrated because many of my kids still can’t do equations, despite my best efforts and teaching strategies. I have an unusual method of my own and understand it when you say that certain kids will really get frustrated when you give them a “gimmick” that they really don’t need. However, my less than gifted kids are still being left behind. I like this method better than my method, which wasn’t effective on the group I have this year anyway.

I also think that I’m going to use this not with linear equations since I’ve technically already passed those, but with linear inequalities which is my next objective. I think doing this will strengthen the equation piece for those kids. I also think it will give me an opportunity to really emphasize when we’ve divided/multiplied by a negative number so those signs can be flipped in the proper direction.

Thank you for freely sharing your good ideas with others! I’m excited about tomorrow now. I needed this!

This, I do not understand. I’m in SEVENTH grade, and I learned this recently, I understood it JUST FINE. This makes it worse! I got all A’s on these papers, and this is just HORRIBLE!

I have never seen a strategy like this. I think it might be worth trying with my intervention group to see if they can handle it better than the method we used.

This year we used dixie cups, counters, and pipe cleaners to model solving equations and it worked great. We were even able to this model on mutli-step equations.

This is so cool! I’m excited to try this method on Monday!!

I just ran through a two-step INEQUALITY using the boxes. No problem at all!! Only addition for inequalities: if you multiply/divide by a negative number, you must switch that inequality around! (

i.e., change > to <). As for what to do when__x_+_5_ < 3

-2

Simply place the appropriate Order of Operations step between the boxes (+5, then /2) then run it backwards on the bottom to solve (Read this RIGHT to LEFT: -5, *2)

Dear heavens when I said balance i meant like 6(k-12)=7(9×12k) nothing like this. Heaven’s this is how my daughter says, bull shit?

i have istep tomorrow &my head is spinningg!! im trying to go over this as easily as possible but its very confusing me ***i suck at math so dont judge me!! :( pleasee help ☺

You can do it when you are subtracting x if you stress that -x is the same as x*-1, it makes a good learning block for when they get more advanced and they need to divide -x by -1 to get just x. I reviewed BODMAS before and stressed how the top need to follow BODMAS where the bottom need to do it backwards. It works just fine!

I am teaching 2-step equations to group of low-functioning 5th graders and want to thank you! This is brilliant!

Thank you for this. I can’t wait to present it in the Fall. I, too, can see my low kids loving the mechanical nature of the method.