What this is for. Every question on the Mathematical Reasoning test belongs to one of fourteen families. This asks one question from each, so you can find out in a quarter of an hour which families you already have and which you do not. It is not a test and nothing is recorded. Flyover I asks the same fourteen families in far more depth, and takes about an hour.
One question from each of the fourteen families, in order. About fifteen minutes. Type an answer and press Check — you get the reasoning for every one, and a list of the families worth your time.
- Family 1 · Numbers and the four operations 9 + 6 × 2 = 21, not 30. Multiplication happens before addition, whatever order they are written in, so this is 9 + 12. The order of operations is not about importance; it is an agreement, so that everyone doing this calculation lands on the same number. Family 1 has the rest of it.
- Family 2 · Signed Numbers (positives & negatives) −7 + 3 = −4. Start at −7 and walk three steps toward zero. You do not reach it, so you are still on the cold side, four steps short. Owing seven and paying back three still leaves you owing four. Family 2 has the rest of it.
- Family 3 · Fractions 1⁄2 + 1⁄3 = 5⁄6. You can only add pieces of the same size, so re-cut both into sixths: 3⁄6 + 2⁄6. Then add the count and keep the name. The bottom number is never added. Family 3 has the rest of it.
- Family 4 · Decimals 0.8 × 5 = 4. Multiply as though the point were not there — 8 × 5 = 40 — then count the decimal places in the question. There is one, so the answer gets one: 4.0, which is 4. Family 4 has the rest of it.
- Family 5 · Percents, ratios, and proportion What is 20% of 150? 30. Percent means per hundred, so 20% is 0.20. “Of” means multiply: 0.20 × 150 = 30. Every percent question becomes ordinary multiplication once the percent is written as a decimal. Family 5 has the rest of it.
- Family 6 · Measurement and units How many inches are in 5 feet? 60. Twelve inches to a foot, and you are going from a big unit to a small one, so the number grows: 5 × 12 = 60. Asking first whether the number should grow or shrink catches a wrong-way answer before you write it. Family 6 has the rest of it.
- Family 7 · Geometry A rectangle is 7 cm by 6 cm. What is its area? 42 square cm. Area is length times width. The unit changes too: you multiplied cm by cm, so the answer is in square cm. An area answer in plain cm means something went into the wrong slot. Family 7 has the rest of it.
- Family 8 · Data, graphs, and statistics Find the mean of 4, 8 and 6. 6. The mean is the fair share: add them and divide by how many there are. 4 + 8 + 6 = 18, and 18 ÷ 3 = 6. It is what each value would be if the total were shared out evenly. Family 8 has the rest of it.
- Family 9 · Probability and counting A bag holds 2 red marbles and 3 blue. You draw one without looking. What is the probability it is red? 2⁄5. Probability is the favorable outcomes over all of them: 2 red out of 5 marbles. The test will take it as a fraction, a decimal or a percent, because they are the same number. Family 9 has the rest of it.
- Family 10 · Exponents, roots, and scientific notation 33 = 27. The raised number counts how many times the base is used as a factor, not what to multiply by: 3 × 3 × 3 = 27, not 3 × 3 = 9. That misreading is the most common error in this family. Family 10 has the rest of it.
- Family 11 · Expressions and polynomials Simplify: 4x + 2x 6x. Four of something plus two more of the same something is six of them. Like terms gather; the variable and its exponent stay as they are. Adding the variables to make 6x² is the slip to avoid. Family 11 has the rest of it.
- Family 12 · Equations and inequalities Solve for x: 2x + 5 = 13 x = 4. Undo what was done, in reverse. The 5 was added, so take it from both sides: 2x = 8. The x was doubled, so halve both sides: x = 4. Whatever you do to one side you do to the other, and the balance stays true. Family 12 has the rest of it.
- Family 13 · Lines, slope, and systems What is the slope of the line through (0, 0) and (2, 6)? 3. Slope is the rise over the run: the line climbs 6 while it runs 2, and 6 ÷ 2 = 3. Take the two points in the same order top and bottom or the sign comes out wrong. Family 13 has the rest of it.
- Family 14 · Functions and the coordinate plane f(x) = x + 7. What is f(3)? 10. f(3) means put 3 in wherever x appears: 3 + 7 = 10. The notation looks like multiplication and is not — f(3) is not f times 3. It is a machine with a number going in and a number coming out. Family 14 has the rest of it.