# Research ideas for the classroom: high school mathematics by Sigrid Wagner; Robert J Jensen; Douglas T Owens; Patricia S

By Sigrid Wagner; Robert J Jensen; Douglas T Owens; Patricia S Wilson; National Council of Teachers of Mathematics. Research Interpretation Project (ed.)

The Golden part has performed an element because antiquity in lots of components of geometry, structure, track, paintings and philosophy. although, it additionally looks within the more recent domain names of know-how and fractals. during this approach, the Golden part is not any remoted phenomenon yet relatively, in lots of circumstances. the 1st and likewise the easiest non-trivial instance within the context of generalisations resulting in extra advancements. it's the objective of this booklet, at the one hand, to explain examples of the Golden part, and at the different, to teach a few paths to extra extensions. The remedy is casual and the textual content is enriched via the presence of very illuminating diagrams. Questions are posed at really common durations and the solutions to those questions, maybe simply within the kind of very huge tricks for his or her answer, are accumulated jointly on the finish of the textual content Dépouillement: [v.1]. Early adolescence arithmetic. -- [v.2]. center grades arithmetic. -- [v.3]. highschool arithmetic

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This is, of course, a ridiculously simple example, but it makes the point. Equations that have this property are called identities. Some examples of identities would be 2x = x + x 3=3 (x – 2)(x + 2) = x2 – 4 All of these equations are true for any value of x. The second example, 3 = 3, is interesting because it does not even contain an x, so obviously its truthfulness cannot depend on the value of x! When you are attempting to solve an equation algebraically and you end up with an obvious identity (like 3 = 3), then you know that the original equation must also be an identity, and therefore it has an infinite number of solutions.

Infinite Number of Solutions Consider the equation x=x 45 Chapter 2: Introduction to Algebra This equation is obviously true for every possible value of x. This is, of course, a ridiculously simple example, but it makes the point. Equations that have this property are called identities. Some examples of identities would be 2x = x + x 3=3 (x – 2)(x + 2) = x2 – 4 All of these equations are true for any value of x. The second example, 3 = 3, is interesting because it does not even contain an x, so obviously its truthfulness cannot depend on the value of x!

46 Chapter 2: Introduction to Algebra What we want to do when we solve an equation is to produce an equivalent equation that tells us the solution directly. Going back to our previous example, 2x + 3 = 7, we can say that the equation x=2 is an equivalent equation, because they both have the same solution, namely x = 2. We need to have some way to convert an equation like 2x + 3 = 7 into an equivalent equation like x = 2 that tells us the solution. We solve equations by using methods that rearrange the equation in a manner that does not change the solution set, with a goal of getting the variable by itself on one side of the equal sign.