Math: the Lay of the Land

First, the test itself, plainly. Mathematical Reasoning runs 115 minutes on a computer. A calculator is on the screen for all but the first few questions, and so is the formula sheet, the whole way through — you are not expected to memorize the area of a circle. Read that again, because it changes what studying means: the test checks whether you have a working toolkit, not whether you're "a math person." Nobody is born one; there are only people who've met the tools and people who haven't yet.

One more fact about the test, because it decides how this page is arranged. A little under half the questions are arithmetic — numbers, fractions, decimals, percents, measurement. A little over half are algebra, in one form or another. So the families below come in three groups: the number families, which most people have met before; the shape, data, and chance families, which sit in the middle; and the algebra families, which are where the test spends most of its time and where most students have the least ground under them.

This page's whole job is the introductions — the name of each family and what it involves — so that when a quiz or a test question arrives, you recognize it. Nothing here asks you to solve anything. When you want to find out what you actually know, the Flyover Diagnostic asks a few questions from every family and tells you where to start.

1. Numbers and the four operations. Place value — what a digit is worth by where it stands — rounding, comparing, and adding through dividing, including the order of operations: the agreed sequence for doing a calculation so that everyone arrives at the same answer. Factors, multiples, and primes belong here too: which numbers build a number, and which it builds.

2. Signed numbers — positives and negatives. The number line continues below zero — temperatures, debts, yards lost, floors below the lobby. Signed numbers have their own small set of rules for adding, subtracting, multiplying, and dividing. The rules are short, and the test uses them constantly once algebra starts, because half the numbers in algebra are negative.

3. Fractions. Parts of a whole: what a fraction is, comparing them, adding, multiplying, dividing. On the test, fractions are the usual clothing of division and of "what part of." They also turn up inside algebra, where a slope is a fraction and a rate is a fraction, so the work you do here is never only about fractions.

4. Decimals. Fractions written in tenths and hundredths. You already practice these daily — every price is a decimal — and the skills are lining up the point, knowing what each place means, and moving between a decimal and the fraction that says the same thing.

5. Percents, ratios, and proportion. "Per hundred," and the art of scaling: discounts, tax, tips, interest, unit prices, a recipe doubled, "3 out of every 5," a map scale. This family is the test's workhorse — more everyday questions live here than anywhere outside algebra — and proportion is also the tool that solves similar triangles later, in geometry.

6. Measurement and units. Inches to feet, minutes to hours, quarts to cups, and the reason a square foot needs the conversion factor twice. Also reading a scale, and keeping units honest through a calculation so the answer comes out in the units the question asked for.

7. Geometry. The biggest family on this page, and the one most students have seen least. Perimeter and area of rectangles, triangles, and parallelograms; circles — radius, diameter, circumference, area, with π; volume of boxes, cylinders, cones, and spheres; angles on a line, angles where two lines cross, and the three angles of a triangle; the Pythagorean theorem for the missing side of a right triangle; and similar figures, where two shapes have the same shape but not the same size. Every formula is on the sheet. Recognizing which shape you are looking at is not.

8. Data, graphs, and statistics. Reading tables, bar graphs, line graphs, dot plots, and histograms without being fooled by them, and the four summary numbers — mean, median, mode, and range — that each describe a pile of data in one figure. The test also asks for a weighted mean now and then, and asks which summary number is the honest one to use.

9. Probability and counting. How likely something is, written as a fraction, a decimal, or a percent: favorable outcomes over all outcomes. Then the probability that something does not happen, the probability of two things happening together, and counting how many ways a choice can be made — three shirts and four pairs of trousers make twelve outfits. It is a small family, and it is the one most often left out of a study plan.

10. Exponents, roots, and scientific notation. Repeated multiplication and its undoing: squares and square roots, cubes and cube roots. Then the rules that let powers be combined — add the exponents when you multiply, subtract when you divide, multiply when you raise a power to a power — and what a zero, a negative, or a fractional exponent means. Scientific notation is the same idea put to work on numbers too big or too small to write out.

11. Expressions and polynomials. Writing a situation as an expression, and then handling the expression itself: gathering like terms, multiplying out parentheses, adding and subtracting expressions, multiplying two binomials together, and factoring — running the multiplication backward to find what the expression was built from. An expression is not a question; it is an amount with a letter still in it. Factoring is the piece to watch: here it is something you do to an expression, and in the next family it becomes the way you solve a quadratic equation.

12. Equations and inequalities. An equation says two amounts are equal, and solving it means finding the number the letter stands for while keeping the balance true. One step, then two, then letters on both sides, then parentheses. Inequalities are the same work with "at least," "no more than," "under" — and one rule of their own, that multiplying or dividing by a negative turns the sign around. Word problems live here too: the hard part is writing the equation, not solving it. Quadratic equations belong here as well — the ones where the letter is squared, solved either by factoring or with the quadratic formula, which the test gives you on the sheet.

13. Lines, slope, and systems. A straight line drawn on a grid, and the equation that describes it: y = mx + b, where m is the slope — how steep, and which way — and b is where the line crosses the vertical axis. Reading slope from a graph, from two points, or from a table; writing the equation from a description; and systems, which are two lines at once and one pair of numbers that satisfies both. The everyday version of a system is two prices and two quantities: adult tickets and child tickets, sold together for one total.

14. Functions and the coordinate plane. A function is a rule that takes one number in and gives one number back, written f(x). The test asks you to work a rule on a number, to find which input gives a stated output, to match a table or a graph to its rule, and to say whether a relationship is a function at all. Functions are drawn on the coordinate plane, and this family also covers the plane itself: plotting points, the four quadrants, the distance between two points, and the midpoint of a line segment.

Where a family page has an empty part — no app for this one yet — it says so rather than leaving it out, so you can see what is there and what is not. The diagnostics cover every family either way: they ask about it, explain every answer, and will make five more problems of the same kind for any family you got wrong. That is the stopgap until the quizzes and the apps catch up, and they are being written one at a time.

In what order? Down the list, if you have the time; the mathematics is built that way, and each family leans on the ones above it. If your test date is close, take the diagnostic first and start where it says. And an eighteen-lesson foundations course, built this past summer, is being moved onto this shelf — until it lands, it lives at its summer address.