The Revision Book II

⚹   Gradient = Rise ÷ Run

A line's gradient measures how steep it is. Gradient = rise ÷ run — the change in y divided by the change in x between any two points. Practise on six lines: first two guide you step by step; the rest ask you to predict, including negative and fractional gradients.

⚹   The Y-Intercept

You just worked out gradient = rise ÷ run. Written formally:

m  =  (y₂ − y₁) ÷ (x₂ − x₁)

Now we'll build the second piece of a line's equation — the y-intercept — one step at a time.

⚹   Find the Line

For each pair of points, find the gradient m and the y-intercept c, then write the line as y = mx + c.

⚹   Parallel Lines — Same Gradient

Parallel lines have the same gradient. If L has gradient m, any line parallel to L also has gradient m. To find the y-intercept of a specific parallel line, substitute its point into y = mx + c and solve for c.

⚹   The Point Between

Given two points, the midpoint is the point exactly halfway between them. We’ll work through four pairs — first two guide you, last two ask you to predict. By the end you’ll have worked out the formula yourself.

⚹   Distance Between Two Points

You just worked out the midpoint formula:

midpoint  =  ((x₁ + x₂) / 2,  (y₁ + y₂) / 2)

Now we'll build the other essential coordinate-geometry tool — the distance between two points — one piece at a time.

⚹   Find Both

For each pair of points, find the midpoint, then the distance. Distances are exact integers for these questions — you should always get a whole number out.

⚹   Reverse Midpoint — Find the Other Endpoint

If M is the midpoint of segment AB and you know A, you can find B. Each coordinate of B satisfies B = 2M − A. That is: Bx = 2·Mx − Ax and By = 2·My − Ay. (Rearranged from the midpoint formula M = (A+B)/2.)

⚹   The Multiplier Behind Interest

Most people think of interest as something you add to your balance. There’s a faster way to see it: as a multiplier. We’ll work through four accounts — the first two guide you step by step; the last two ask you to predict. By the end you’ll know why compound interest is exponential.

⚹   Crunch the Numbers

For each account, use A = P(1 + r)ⁿ to find the final amount. Round to the nearest cent. You’ll need a calculator.

⚹   Beyond the Formula

The formula changes form when you want the interest amount, when compounding is monthly, when you need to work backwards to find the principal, or when you want to know how long something takes to double.

⚹   Two Things at Once

When you do two independent random things (like toss a coin and roll a die), how likely is a particular pair of outcomes? We’ll work through four scenarios — first two guide you step by step, last two ask you to predict.

Answers can be entered as fractions (e.g. 1/12) or decimals (e.g. 0.0833).

⚹   Rules of Probability

You just discovered the multiplication rule for independent events (events where one doesn't affect the other):

P(A and B) = P(A) × P(B)    [when A and B are independent]

Now we'll add four more tools: the complement, "at least one" problems, the union rule, and conditional probability — one atom at a time.

⚹   Apply the Rules

Six questions mixing the multiplication rule, the complement, and dependent events.

⚹   Venn Diagrams — Inclusion-Exclusion

Inclusion-exclusion for two overlapping sets A and B: |A ∪ B| = |A| + |B| − |A ∩ B|. Subtract the overlap once — otherwise members of both sets get counted twice. Then P(neither) = 1 − P(A ∪ B).

⚹   The One Move That Differs

Solving an inequality is almost the same as solving an equation. Almost. There’s exactly one move where the rules change. We’ll work through four problems — first two guide you, last two ask you to predict. By the end you’ll have found the move.

⚹   Inequality Moves

You discovered in Stage 1 that dividing by a negative flips the inequality sign. Now we'll work through all the moves — from single-step adds and subtracts, up through two-step solves with and without the flip.

⚹   Solve

Six inequalities. Some require the flip, some don’t. Watch your signs.

⚹   Translate the Phrase to a Symbol

Word problems live or die on the phrasing. At least X → ≥ X (inclusive). At most X → ≤ X (inclusive). More than X → > X (strict). Fewer / less than X → < X (strict). No more than = ≤. No less than = ≥. Pick direction first, then the boundary value.

⚹   The Structure of a Table

A two-way table sorts a group two ways at once — dogs or not, cats or not. Every row sums to its row total; every column sums to its column total; those totals all add up to one grand total. When a cell is missing, you can always find it by subtraction. We’ll work through four tables — first two guide you, last two ask you to predict.

⚹   Two-Way Tables & Venn Diagrams

You discovered how to find a missing cell in a two-way table. Now we'll build every other skill — reading totals, computing probabilities, working with Venn diagrams, and using inclusion-exclusion — one concept at a time.

⚹   Fill, Convert, Read

Six mixed questions: filling tables, reading Venn regions, and extracting probabilities.

⚹   Inclusion-Exclusion — Forward & Reverse

The core identity: |A ∪ B| = |A| + |B| − |A ∩ B|. And the total-splits-into-two rule: Total = |A ∪ B| + Neither. Any Venn word problem with just two sets is a rearrangement of these two equations. Forward: find neither from knowing both. Reverse: find both from knowing neither.