Pre-calculus readiness, measured in 43 prerequisite concepts

Why one alternating sequence is the target

The diagnostic is scoped to alternating sequence patterns of the form (-1)^n, the opening concept of the Sequences section that starts Consiliae's pre-calculus chapter. It is a deliberately small target, chosen because three separate strands have to work at once before that term means anything: exponent notation, integer sign rules, and the definitions of a function and a sequence. Miss any one of the three and the term is a symbol you copy rather than a value you can evaluate.

Its prerequisite closure is 43 concepts across 10 sections in four chapters, and the check adapts through them in six to ten questions. Named, they are: the definition of a sequence and the definition of a function; the Cartesian plane and plotting an ordered pair; exponent notation and positive integer exponents; what a variable is; order of operations; and the whole integers block, which means negative numbers on the number line, the opposite of a number, adding a positive to a negative, subtracting integers, the sign rules for multiplying and dividing, and order of operations once negatives are present.

The gap that hides behind a decent score

The most common result by a distance is that everything holds except the integers block: seven concepts, all taught around age eleven, all of them things an adult will tell you they already know. And they usually do know them, in the sense that they get them right most of the time. That is the problem. A rule you apply correctly most of the time feels like competence and behaves like a coin flip across a six-line derivation.

This is why an alternating sign was picked as the target rather than something that looks harder. A sequence carrying (-1)^n forces the sign rule to fire on every single term, in a context where the answer is not a number you can sanity-check by eye. A sign rule that is right nine times in ten leaves a ten-term expansion correct only about a third of the time, and the error is invisible to re-reading.

The second profile is a missing coordinate plane or a missing definition of a function. It is less common and more consequential than it looks, because every graph in the rest of pre-calculus assumes both, and because neither gets re-taught. If your result names either one, fix it first regardless of what else is on the list.

Reading your result, gap by gap

Integers block. Work it as one unit rather than as six separate lessons, and finish on order of operations with negatives, which is the concept that ties the other five together. The measure of done is not accuracy on a worksheet, it is that you stopped narrating the rule to yourself while applying it.

Exponent notation. This is rarely ignorance of powers. It is unfamiliarity with a base that is negative or parenthesised. Being able to say why (-2)^4 is 16 while -2^4 is -16, without hedging, is most of the test.

Function or sequence definitions. Short lessons, and load-bearing. A sequence is a function whose inputs are the positive integers; once that sentence carries meaning, index notation stops looking like a new language and subscripts start reading as inputs.

Nothing missing. Then the Sequences section is a safe entry point, and you should start there rather than at the front of the chapter, which will bore you into quitting.

Where this check deliberately stops

It does not probe quadratics, radicals, rational expressions, logarithms, complex numbers, combinatorics or trigonometry. All of those appear somewhere in a pre-calculus syllabus, and none of them sits upstream of the target concept, so none is tested. This is a check on the entry point of the chapter, not a survey of the course.

It is also silent on pace. Two people with identical gap maps can be a term apart in how quickly they close them, and nothing in six to ten questions predicts which of them you are. What you get instead is an ordered list and a starting position, which is the part a syllabus cannot give you.

And it is scoped to Consiliae's curriculum graph. If your school's pre-calculus course opens with conic sections or with a trigonometry review, week one will test things this check never asked about. The algebraic spine is common to every version of the course; the ordering is not.