Categories What aspect of solving linear equations is this student struggling with?  From the student’s point of view, why does the moves made seem like the right ones?

Are there any interesting things that you do to help students with linear equations? How do you make the right moves seem intuitive? John Achesonsays:

I too see this all the time. The student obviously does not understand that each and every element needs to be addressed. I tried to focus on combining like terms before they start moving terms to get the variables on one side and the constants on the other. Hawkesays:

I wouldn’t touch the mistakes on #5 yet; the underlying misunderstandings indicated by #4 need to be addressed first.

This student’s simplification in #4 seems to indicate a desire to “get rid of” terms. First, the student gets rid of the 8x on the right side, then decides to get rid of the 4x. This is done through a trick that the student has learned, namely, to subtract the same term from here and from there (which are supposed to be on opposite sides of the equation, but which he/she forgets in the second step, because the student isn’t particularly attentive to the fact that this trick exists to maintain equivalence of both sides of the equation.)

To address this, I would ask the student to first forget any tricks he/she may have learned to solve linear equations, and first try to build a conceptual understanding of what an equation like this represents. I’d take it down a notch on the ladder of abstraction by having the student represent these quantities on a balance, showing two rods, three units, and four more rods on the left. On the right, I’d put nine units, eight blocks, then remove five units. Instead of skipping right to algebraic methods, I’d ask the student to use guess-and-check to find the value of each rod. Is each rod (x) five? Let’s see, that would give us 33 on the left and 44 on the right. So 33=44…true or false? Let’s try something else. And so on.

After spending some time here, I’d then show the student how, sometimes, a faster method of guessing and checking is to first simplify the picture/balance by combining rods, removing the same number of rods from both sides, etc.

None of these are my ideas; they’re all taken directly from the Hands-On Equations series. I’ve tried ignoring conceptual gaps like this and just plowing through with better tricks, and it hasn’t ever ended well. The only real long-term success I’ve ever had with this type of issue has been through building strong conceptual understanding and introducing procedural tricks only when kids understand how the tricks themselves are coherent with the math they were already performing. secretseasonssays:

Before even getting to the algebra that takes up the bulk of the work in #4, I’d want to know what happened at the very end, when the student concludes that 1/6 = -6. Was this a careless mistake or does it indicate a deep problem?

So I think I’d ask the student to solve things like 4x=1 and 7x=1 and maybe throw in something like 8x=3 and see what they do with that before we step back up to the linear equations issues. Dave Radcliffesays:

I’ve seen the suggestion that students should be taught to move all of the terms to the left side, i.e. write the equation in the form ax+b=0. This might be less confusing than moving the variable terms to one side and the constant terms to the other side. Has anyone tried this approach?