California 7th Grade standard, Algebra & Functions:
4.1 Solve two-step linear equations and inequalities in one variable1 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.
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.