Mathematics • Year 9 • Unit 2 • Lesson 20
Unit Synthesis in Context
Five real-world scenarios spanning every family in Unit 2: a fountain, a thrown javelin, a screen-time decay, a coin-flip pattern, and a circular pizza. Identify, model, solve, and interpret — units included, impossibles rejected.
1. Word problems
For each: identify the family, do the calculation, interpret with units. 3 marks each
1.1 — Park fountain. A circular fountain has equation $x^2 + y^2 = 64$ (metres), with the centre of the fountain at the origin.
(a) State the radius.
(b) Name the family.
(c) The fountain edge is to be lined with tiles. Estimate roughly how many metres of tile-edge are needed (use circumference $= 2\pi r$, take $\pi \approx 3.14$).
1.2 — Thrown javelin. A javelin is thrown so its height (m) after $t$ seconds is $h = -5t^2 + 20t + 2$.
(a) State the launch height (sub $t = 0$).
(b) Identify the family.
(c) Find the time of maximum height (axis of symmetry: half-way between the roots, but a quick formula is $t = \dfrac{-b}{2a}$ — for $a = -5, b = 20$ this is $t = 2$ s). Then state the max height with units.
1.3 — Screen-time decay (exponential). A school's average daily screen-time was $4$ hours and falls by factor $0.8$ each week of a "screen-down" campaign: $S = 4 \cdot 0.8^w$ (hours, weeks).
(a) Compute $S$ after $1$ week and $4$ weeks.
(b) Identify the family.
(c) State the equation of the horizontal asymptote and what it means in context.
1.4 — Coin-flip doubling. A class tracks how many "heads in a row" a coin flips. The probability of $n$ consecutive heads is $P = \dfrac{1}{2^n}$.
(a) Compute $P$ for $n = 1, 2, 3, 4, 5$ (as fractions).
(b) Name the family (look at the form — does it match parabola, hyperbola or exponential?).
(c) State $P$ at $n = 10$ as a fraction.
1.5 — Circular pizza shared. A circular pizza with area $A = \pi r^2$ ($\pi \approx 3.14$) of radius $r$ cm is split equally between $n$ people; each person's share (in cm²) is $S = \dfrac{\pi r^2}{n}$.
(a) For a pizza of $r = 15$ cm, write the share formula $S$ in terms of $n$ (compute the constant).
(b) Compute $S$ for $n = 2, 5, 10$ (round to nearest cm²).
(c) Name the family (in $n$).
2. Explain your thinking
Use full sentences. 4 marks
2.1 A classmate looks at the equation $y = (x + 3)^2 - 2$ and reads the vertex as $(3, -2)$ instead of $(-3, -2)$. In your own words, explain (i) why the sign of $h$ flips when reading vertex form $y = a(x - h)^2 + k$, (ii) write out the rewriting trick that makes the sign obvious ($(x + 3) = (x - (-3))$), and (iii) give a similar example of a vertex-form equation and its CORRECT vertex.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Park fountain
(a) Radius $= \sqrt{64} = 8$ m.
(b) Family: circle.
(c) Circumference $\approx 2 \times 3.14 \times 8 = 50.24$ m. Approximately $50$ metres of tile-edge needed.
1.2 — Thrown javelin
(a) Launch height: $h(0) = -5(0) + 20(0) + 2 = 2$ m.
(b) Family: parabola (projectile under gravity).
(c) Time of max: $t = \dfrac{-20}{2 \times (-5)} = \dfrac{-20}{-10} = 2$ s. Sub: $h(2) = -5(4) + 20(2) + 2 = -20 + 40 + 2 = 22$ m. Max height: $22$ metres at $t = 2$ seconds.
1.3 — Screen-time decay
(a) $S(1) = 4 \times 0.8 = 3.2$ hours. $S(4) = 4 \times 0.8^4 = 4 \times 0.4096 \approx 1.64$ hours.
(b) Family: exponential decay (base $0.8$ is between $0$ and $1$).
(c) Horizontal asymptote: $S = 0$. In context: the average daily screen-time approaches zero hours but never quite reaches it (the curve gets infinitely close without touching $0$).
1.4 — Coin flips
(a) $n = 1$: $\tfrac{1}{2}$. $n = 2$: $\tfrac{1}{4}$. $n = 3$: $\tfrac{1}{8}$. $n = 4$: $\tfrac{1}{16}$. $n = 5$: $\tfrac{1}{32}$.
(b) Family: exponential decay ($P = (\tfrac{1}{2})^n$, base $\tfrac{1}{2} < 1$). NOTE: it's NOT a hyperbola, even though $\dfrac{1}{2^n}$ has a denominator — the $n$ is in the EXPONENT, which is the exponential signature, not the hyperbola one.
(c) $P(10) = \dfrac{1}{2^{10}} = \dfrac{1}{1024}$ — about $0.1\%$, so getting $10$ heads in a row is rare.
1.5 — Circular pizza shared
(a) For $r = 15$ cm: $A = 3.14 \times 15^2 = 3.14 \times 225 = 706.5$ cm². Share formula: $S = \dfrac{706.5}{n}$.
(b) $S(2) = 353$ cm² (each person). $S(5) = 141$ cm². $S(10) \approx 71$ cm².
(c) Family (in $n$): hyperbola ($S \cdot n = 706.5$, the constant-product signature of inverse proportion).
2.1 — Explain your thinking (sample response)
In vertex form $y = a(x - h)^2 + k$, the vertex is at $(h, k)$ — but the bracket contains $(x - h)$, with a MINUS sign. When the equation actually says $(x + 3)$, we have to rewrite the plus as a minus before reading $h$: $(x + 3) = (x - (-3))$, which means $h = -3$ (not $+3$). The vertex is therefore $(-3, -2)$, not $(3, -2)$. The trick is to ALWAYS check the sign by rewriting the bracket in $(x - h)$ form: if it's already $(x - h)$, then $h$ is positive; if it's $(x + h)$, then $h$ is actually negative. Another example: for $y = (x - 5)^2 + 7$, we read $h = 5$ and $k = 7$, so the vertex is $(5, 7)$ — here the sign matches directly because the bracket already shows $(x - 5)$.
Marking: 1 mark for "minus sign in the standard form $(x - h)$"; 1 mark for the rewriting trick $(x + 3) = (x - (-3))$; 1 mark for the correct vertex $(-3, -2)$; 1 mark for a correctly worked example.