Number, Quantities, Equations and Expressions

In this lesson you will: Read an expression's structure — terms, factors, coefficients, chunks — before rewriting it; Let the shape of an equation choose the solving method; Carry units and a sensible level of precision all the way to the written answer.

Rewriting: every form answers a different question

An expression has a structure before it has a value. A.SSE asks you to see it: in 1200+45w the 45 is a rate and the 1200 is a starting amount; in 5000(0.88)^t the 0.88 is a factor, so the quantity falls by 12% each period.

Factoring is the same skill pointed the other way. Take out the greatest common factor first, always. Then look for a special form — a difference of squares or a perfect square trinomial — and only then hunt for a factor pair.

The single most common error in this whole category is a sign. (x-7)(x+2) and (x+7)(x-2) differ by one character and by nine in the middle term.

Solving: four routes, and the discriminant tells you what to expect

A linear equation in one variable has one solution unless the variable cancels, in which case a true statement means every real number and a false statement means none.

A quadratic has four routes. If there is no linear term, take a square root and keep both signs. If the constant is missing, factor out the variable. If it factors cleanly, factor. Otherwise use the formula.

Before you start, the discriminant b^2-4ac tells you how many real solutions are coming: positive means two, zero means one, negative means none.

Quantities: the units and the last digit are part of the answer

A modeling question is scored on the number you write down, not on the algebra that produced it. Two things sink otherwise-correct work: a unit that was never converted, and a level of precision the situation cannot support.

An inequality that models a budget or a capacity usually has a second constraint hiding in the situation — a count cannot be negative, and it is often a whole number. Solve, then ask what the situation allows.

Common mistake — Taking a square root and writing only the positive value

Asked to solve 18x^2=50, a student divides to get x^2=(25)/(9) and writes x=5/3.

The habit comes from the square-root key, which returns one value, and from years of measuring things that could not be negative. Both are reasonable and both are the wrong instinct here.

Squaring destroys sign information. (-5/3)^2 and (5/3)^2 are the same number, so undoing the square has to put both signs back.

How to catch it: Count your solutions against the degree. A quadratic has two, one, or none — never exactly one unless the discriminant is zero. If you produced one solution and the discriminant is positive, you dropped a sign.