Did you miss the information that the student is 8 years old?

No, I wrote it for Dave -- who asked the question -- to read and understand, step by step, then translate to whatever form is appropriate and best for the 8 year old.

The only language I know of how to do that for math, is math; it is a precise language for exactly that purpose.

The form I used above is the form I still use when grasping new mathematical techniques, and it works. And I do computational material physics, and often play with classical potential models and stuff; and am definitely not a mathematician.

I would have never guessed that being precise would be frowned upon here, of all places.

It would be interesting if you forgot you ever learned algebra and then think about how you would solve them with only a basic knowledge of arithmetic? Would you draw sketches? Would you work in terms of "smaller angle" and "bigger angle"? Use language like "if one part is four times the other part, then there are five small parts in total", then "let's divide 180 into five small parts", and so on?

Whatever you do, never underestimate kids capability of understanding things. Just because they are immature, does not mean they cannot grasp logic and rules. Heck, they are masters at working out working rule sets from first principles, when inventing new games they play with friends.

Elementary algebra, for equations, is really a small extension to basic arithmetic:

**=** is the equals sign. It means the value or expression on its left side is equal to the one on its right side.- You can add or substract any value from both sides of the equals sign.
- You can multiply or divide both sides of the equals sign by any value except zero.
- When a value is unknown, you can use a
**variable** to represent it. Typically, we use single letters like *x* for variables. - When you substitute a variable with a known value or expression, you need to put the value or expression in parenthesis to not accidentally break operator precedence rules.
- To find out the unknown value, you use the above operations to isolate the variable to one side; that gives you its value on the other.

Elementary linear algebra, or pairs or systems of equations, add a couple more points:

- You need exactly as many independent equations as there are unknowns.

(Independent means you cannot "cheat" by just modify an existing one to get a new one. Technically, the proper definition is linear independence, but you don't generally need to care until you do calculus or linear algebra.) - Simple systems can be solved by isolating each unknown in one equation, then substituting it into the next one, in any order you see fit, until you find the actual value for the (last) unknown. Then, you work backwards, substituting that value to the unknown in the previous equation, one by one, until you have defined the value for each of them.
- More complex systems require linear algebra to solve. There, you find new tools, vectors and matrices, that make solving these much easier.

Would I explain the above to an 8 year old? I probably would, but not to expect them to learn it then and there; more to give their subconscious a gentle nod in the correct direction.

(In case anyone "adult" wonders, "simple system" above means one that can be described using a diagonal, bidiagonal, or tridiagonal matrix, in matrix form.)

Some people are visually oriented, some are more quantitatively oriented. So, it would definitely depend on the person whether I'd use visual sketches or countable tangible items to describe how to solve these problems.

As an example, round or cylindrical-shaped objects, like Lego figurine heads or axle nuts, can be used to form a half-circle. If you work out how many of them you need, you can put them on a half-circle arc (the radius depending on how many of them you have), and grasp the solution-finding process that way: then, each object represents a part of the arc -- if you have

*N* objects, each one of them represents one

*N*th of 180 degrees.

Some people are more visually oriented (in the sense that they gain more insight from drawing descriptions of the problem than handling countable items describing the problem), but the problem here is that dividing a half-circle into a number of sectors is nontrivial. That is, this particular kind of problem is a lot of work to visualize when solving it; the visualization itself becomes a problem, and can easily frustrate a learner.

Using a computer, for example a simple application that shows you a half circle, and lets you divide it into any number of equal segments, showing the angle in degrees as well, can actually

*hinder* the learning process. The learner can skip the entire solution method, and just find the solution via brute-force trial-and-error. It becomes easy repetitive mechanical work, and does nothing to learn the solution method at all.

In my reply (above), I assumed that I was "talking" to Mr. Jones but I was trying to model the way I think you should approach teaching the method. Basically, step one is to translate the English into mathematics. Then step two is to translate the resulting mathematics into the answer.

I agree, except that step zero is to make sure Mr. Jones understands the exact methods these types of problems can be solved with.

(You could argue that isn't that step 1.5 instead, but thing is, each problem can be described in many equally valid ways. Just compare

Snell's law and

Fermat's principle, which describe the exact same thing using completely different mathematical tools. So, if you want to solve a problem, you should make sure you translate the problem into mathematical expressions you theoretically know how to solve.)