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I'm a private tutor working with a 7th grader who is struggling with solving equations. Given a simple equation, he is able to solve it using a formulaic procedure, but it is very obvious that he has no idea what the solution really means. Hence, if he gets a problem that's slightly different from ones he's solved before, he's completely lost.

Working with him yesterday, I realized that he doesn't understand what a variable is, or what he's doing when he's solving an equation -- he's just following the steps his teacher told him for that specific problem.

I'm trying to think of a way to visually demonstrate what's happening when he's solving an equation -- something visual that he can see. Kind of an algebraic equivalent to putting four coins on the table and adding one more to show 4 + 1 = 5.

Any ideas? Thanks! -Ian

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  • $\begingroup$ Word problems will clarify what the solutions mean, and make it easier to find other ways to solve the problem. E.g.: matheducators.stackexchange.com/a/1852/173 $\endgroup$ – user173 Jan 23 '15 at 10:14
  • $\begingroup$ The idea of "what a variable is" can be found here. $\endgroup$ – skullpatrol Jan 23 '15 at 10:54
  • $\begingroup$ Why does it have to be visual? Have you tried just explaining it to him? Using examples? $\endgroup$ – Jack M Feb 6 '15 at 20:27
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I don't know how feasible this is or what kind of resources you have, but try finding a balance scale, some standard weights, and a collection of identical objects. You can set up the scale to represent a problem and ask the student to determine the unknown weight. The only rule is that the scale can never become unbalanced in the process (i.e. what they do to one side they must do to the other).

For example, to demonstrate solving $3x+6=5x+2$ start with 3 identical objects and 6 unit weights on one side and 5 of the same identical objects and 2 unit weights on the other.

Then the solutions is: Remove two units from each side. Remove 3 objects from each side. Split each side into two equal groups and remove a group from each side. This should leave one object on the right and 2 unit weights on the left.

Of course, to get the scale to balance out from the beginning, you, the teacher, will have to know the weight of the objects. And you'll have to watch out for fractions and negatives.

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  • $\begingroup$ scales were the first things that came to mind. there are also some web browser scale animations that show this concept in action. i'll see if i can work something out. thanks! $\endgroup$ – Ian Taylor Jan 23 '15 at 5:22
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I would backtrack and revisit substition, solving equations should fall out as a bi-product or the student hasn't grasped that topic fully. With regards to solving equations I believe choosing the numbers to provide a rich example is extremely important. Consider for example $$\frac{x}{4}=12$$

This is far more likely to be answered incorrectly than$$\frac{x}{3}=7$$

Why? Multiplication is the only 'sensible' option for the student in the second question while the opposite is true in the first example. Force them to confront such addictive number bonds head on.

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  • $\begingroup$ Excellent example. Do you have any more information or references for "addictive number bonds" $\endgroup$ – Richard Jan 22 '15 at 22:25
  • $\begingroup$ Thanks for the comment. Addictive number bonds are just one of things I've noticed during my teaching. I'd be glad to give more info if you have any specific questions. $\endgroup$ – Karl Jan 23 '15 at 19:07
  • $\begingroup$ I have previously read about problems that students have reasoning about shapes when they are presented at unusual orientations, ie. a square rotated 45degrees ceases to be a square. This appears to be a numerical equivalent. Is "addictive number bond" your own term, or have you seen it mentioned elsewhere? $\endgroup$ – Richard Jan 24 '15 at 0:48
  • $\begingroup$ As far as I know it's mine. Geometry is a strange thing. Many students can correctly identify an alternate angle in a picture where the parallel lines are horizontal and there are two transversals. Rotate the same picture so that the parallel lines are vertical and the success rate drops dramatically. That's the example I always use teaching. $\endgroup$ – Karl Jan 24 '15 at 7:45
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You may find this Mathematics Models I webpage by Paul Griffith useful, although it targets a lower educational level:


      Factors
Another source is this Oregon web page with many links, including:
          AlgVis


Caveat: I have not used any of these materials in the classroom myself.

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I would use Algebra tiles (like these). I've found that as students see the difference between an x tile and a unit tile, for example, they tend to make fewer mistakes with combining like terms. When they have to physically remove the same amount from both sides of the "equal sign", it's harder for them to make careless mistakes, especially with negatives. It takes a little practice (and more so for students who have a traditional method sort of stuck in their heads), but moving to abstraction seems to take less time.

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