Faceted Curriculum Project/B1/Teachers Facet

Looks at::Faceted Curriculum Project/B1

Introduction to the Problem
Before learning the material of this block, it is assumed that students can "solve linear equations and inequalities". The meaning of this expression is often taught as follows: one is given such an equation or inequality, and one is asked to perform a sequence of symbol manipulations (following strict rules), in order to arrive at an equation with $$x$$ on one side, and a number on the other side. For example, if one is asked to "solve the equation $$ 2x + 3 = 9$$", one is expected to undertake the following steps:
 * $$2x + 3 = 9$$.
 * $$2x = 6$$ (by subtracting 3 from both sides).
 * $$x = 3$$ (by dividing both sides by 2).

In this block, one "twist" is introduced into such a problem: the equations or inequalities may also include "absolute values". It is assumed that students know what absolute values are -- for example, they know that $$ \vert -3 \vert = 3 $$ -- but this does not necessarily help them to solve an equation such as $$ \vert 2x + 3 \vert = 9$$.

Multiple solutions
Perhaps the greatest hurdle to be overcome, and one which is very important for later studies (such as quadratic equations), is the following: students probably think of "solving" a math problem as finding the unique correct numerical answer. However, when solving an equation such as $$ \vert 2x + 3 \vert = 9 $$, there are "two solutions", as many people would say. This runs harshly against the following two preconceptions: As a teacher, how should one explain that the equation $$ \vert 2x + 3 \vert = 9 $$ has the two "solutions": $$ x = 3 $$ or $$ x = -6 $$?
 * 1) That a solution is an answer to a math problem.
 * 2) That a math problem has exactly one right answer.

Overcoming the hurdle
One suggestion for overcoming this hurdle is to take great care in the language used. Consider the following ways of asking about the equation $$ \vert 2x + 3 \vert = 9 $$:
 * 1) Solve for $$ x $$: $$ \vert 2x + 3 \vert = 9 $$
 * 2) Solve the equation $$ \vert 2x + 3 \vert = 9 $$.
 * 3) If $$ \vert 2x + 3 \vert = 9 $$, then what is $$ x $$?
 * 4) If $$ \vert 2x + 3 \vert = 9 $$, then what could $$ x $$ be?
 * 5) For which values of $$ x $$, is it true that $$ \vert 2x + 3 \vert = 9 $$?
 * 6) Suppose that $$ \vert 2x + 3 \vert = 9 $$. How many values of $$ x $$ are possible?  What are they?
 * 7) Describe the set of solutions to the equation $$ \vert 2x + 3 \vert = 9 $$.

The goal should be for students to understand and express the following mathematical ideas, perhaps using different language:
 * 1) If $$ \vert 2x + 3 \vert = 9 $$, then $$ x = 3 $$ or $$ x = -6 $$.
 * 2) The set of numbers $$ x $$, for which $$ \vert 2x + 3 \vert = 9 $$, is $$ \{ 3, -6 \} $$.

Given that the above ideas are desired from students, which ways of phrasing the question are the clearest? Some ways of asking a question encourage students to describe their answer as a "solution set". Other ways of asking a question encourage students to perform deduction. As a teacher, you should ask yourself the following: which way (among those listed above) of asking a question would lead the student to given an answer such as the two desired answers given above?

If you use a question such as "Solve for $$ x $$: $$ \vert 2x + 3 \vert = 9 $$", you must make sure that students know that their answer should be a set of numbers, not a single number. There can be only one correct answer to the question, though there are multiple solutions to the equation.

Boolean combinations
Whenever one encounters an equation with absolute values, the solution process "branches". The first step in solving the equation $$ \vert 2x + 3 \vert = 9 $$ should be:
 * $$ 2x + 3 = 9 $$ or $$ 2x + 3 = -9 $$.

This step should not be taught as a simple symbol manipulation. Rather, students should understand that "the absolute value of 'something' equals 9, if and only if the 'something' equals 9 or equals -9". Indeed, if students were taught a simple symbol manipulation in this branching step, then they could make frequent mistakes; for example, a student, starting from $$ \vert x \vert = -2 $$, could incorrectly write $$ x = -2 $$ or $$ x = 2 $$. Although it is possible (as one may see from the Math Facet) to make the branching step "algorithmic", it is recommended that this step be approached by thinking about each problem independently.

In this way, the linear deductive thread, which students should be accustomed to, branches into two separate threads joined by the "boolean operator" "or". Unlike its common usage in spoken English, the word "or" in mathematics typically means "and/or". This will become more important when studying inequalities involving absolute values.

After a branching, the individual branches may be followed using the familiar deduction and symbol manipulation. From the previous step, we deduce: In this way, we deduce from the initial statement $$ \vert 2x + 3 \vert = 9 $$, that $$ x = 3 $$ or $$ x = -6 $$. In other words, the solution set is $$ \{ 3, -6 \} $$.
 * $$ 2x = 6 $$ or $$ 2x = -12 $$ (by subtracting 3 from both sides, in both equations).
 * $$ x = 3 $$ or $$ x = -6 $$ (by dividing 2 from both sides, in both equations).

Multiple branching
Every time an absolute value occurs within an equation or inequality, another branching will occur in the deductive process. Such multiple branchings can yield multiple (more than two) solutions. When inequalities are involved, multiple branchings can yield a solution set composed of zero, one, or many intervals of real numbers.