Mathematics 20-2

A rate compares two quantities with DIFFERENT units — $\frac{\$1.42}{1\text{ L}}$, $\frac{72\text{ beats}}{1\text{ min}}$ — and a UNIT rate has denominator 1.

Similar shapes have equal angles and proportional sides; if the length scale factor is $k$, the AREA scale factor is $k^2$ (area is two lengths multiplied).

Set up a proportion and keep units aligned: to convert a rate, multiply by a well-chosen $\frac{\text{unit}}{\text{unit}} = 1$ so the unwanted units cancel.

Reverse it: $k = \sqrt{\dfrac{A_2}{A_1}}$ — take the square root of the area ratio to recover the length scale factor.

A scale factor is a ratio of SAME-unit lengths: $\text{scale factor}$$= \frac{\text{diagram length}}{\text{actual length}}$ — a 1:50 drawing means 1 unit on paper is 50 real units.

A transversal across parallel lines makes CORRESPONDING angles equal, ALTERNATE angles equal, and CO-INTERIOR angles supplementary ($\text{sum} = 180°$).

Chain the parallel-line properties: label what you know, find one new angle at a time, and cite the property (corresponding, alternate, co-interior, supplementary) for each.

Sine law: $\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$ — each angle's sine over its OPPOSITE side (or the reciprocal form for finding a side).

A proof is a chain of justified statements: each line states a fact AND the reason (given, definition, or a proven property) — no measuring allowed.

Angles on a straight line sum to $180°$ and angles around a point sum to $360°$ — often the last step that closes a problem.

Inductive reasoning generalizes from specific cases to a conjecture — 'the pattern I keep seeing probably always holds'.

Match a strategy to the puzzle: WORK BACKWARD from the goal, LOOK FOR A PATTERN across small cases, or SIMPLIFY to a smaller version and scale up.

Extract perfect-square factors: $\sqrt{50} = \sqrt{25}\cdot\sqrt{2} = 5\sqrt{2}$; for variables $\sqrt{x^3} = x\sqrt{x}$ with the restriction $x \ge 0$ so the root is defined.

Isolate the radical FIRST, then square both sides: $\sqrt{x+3} = 5 \Rightarrow x + 3 = 25$.

A conjecture is only a PLAUSIBLE claim: one COUNTEREXAMPLE disproves it outright, while more supporting cases only raise confidence, never certainty.

Standard deviation $\sigma$ measures SPREAD about the mean — small $\sigma$ = data clustered tightly, large $\sigma$ = data spread out; it says nothing about where the centre is.

A confidence INTERVAL is a range of plausible values for the true population value; the confidence LEVEL (e.g. 95%) is how often such an interval would capture the truth over many samples.

For a normal distribution the 68–95–99.7 rule holds: about 68% of data lie within $1\sigma$ of the mean, 95% within $2\sigma$, 99.7% within $3\sigma$.

Higher confidence widens the interval (more certain of capture, but less precise) — there is a trade-off between confidence and precision.

A z-score $z = \dfrac{x - \mu}{\sigma}$ counts how many standard deviations $x$ is from the mean — positive above, negative below.

In vertex form $y = a(x - p)^2 + q$ the vertex is $(p, q)$ and the axis of symmetry is the vertical line $x = p$ — mind the sign: $(x+2)$ means $p = -2$.

Set the equation to zero, factor, then apply the ZERO-PRODUCT property: if $(x-r)(x-s)=0$ then $x = r$ or $x = s$.

$a$ sets the opening: $a > 0$ opens up (vertex is the minimum), $a < 0$ opens down (vertex is the maximum).

The roots ARE the x-intercepts of $y = ax^2 + bx + c$ — solving and graphing tell one story.

The y-intercept comes from $x = 0$; x-intercepts (zeros) come from solving $y = 0$ — there can be 0, 1 or 2.

PLAN: choose a focused topic with a genuine mathematical core, write a guiding question, and identify credible sources (books, peer-reviewed or reputable sites) — noting each for citation.

CONDUCT then PRESENT: gather and organize evidence, distinguish fact from opinion, and communicate findings clearly with visuals, correct mathematical notation, and cited sources.