Mathematics • Year 9 • Unit 2 • Lesson 1
Linear vs Non-Linear — Mixed Challenge
Pull together every classification tool from Lesson 1: spot the family from the equation, the table, or the description. Catch one tricky mistake, then tackle an open-ended challenge that lets you invent your own data.
1. Mixed problems — equation, table or description
Each question needs you to choose the right detection tool: scan the equation, check first differences, or read a verbal clue. Show your working. 3 marks each
1.1 Classify each equation as linear or non-linear, and where non-linear name the family: (a) $y = -2x + 9$ (b) $y = x^2 - 7$ (c) $y = 12/x$ (d) $y = 4^x$ (e) $y = 3$.
1.2 A relationship gives $x: 0, 1, 2, 3, 4$ and $y: 1, 3, 9, 27, 81$. Show using first differences that it is non-linear. Then identify which family it belongs to (with one-sentence reasoning).
1.3 A relationship gives $x: 0, 1, 2, 3, 4$ and $y: -1, 2, 5, 8, 11$. Decide linear or non-linear using first differences, and write a rule of the form $y = mx + c$ that fits the data.
1.4 Sketch (in the margin or your book) a rough graph of $y = x^2$ for $x = -3$ to $3$. Then sketch $y = x$ on the same axes. State which is linear and which is non-linear, and give the $x$-value(s) where the two graphs cross.
1.5 A scientist measures bacteria in a dish each hour: $h: 0, 1, 2, 3$ gives $N: 50, 100, 200, 400$. Show that the relationship is non-linear, identify the family, and find a rule of the form $N = N_0 \times 2^h$ that matches the data.
1.6 Sort these into linear and non-linear groups, naming the family for each non-linear one: $y = x^2$, $y = 4x + 1$, $y = 1/x$, $y = 2^x$, $y = -3x$, $y = x^2 + 5$, $y = 10/x$.
2. Find the mistake
Another student has tried to classify five equations. Their working is shown below. Exactly two of their answers are wrong. Spot them, explain why, and fix them. 3 marks
Student's classifications:
A: $y = 3x - 2$ → Linear ✓
B: $y = x^2 + 5$ → Linear (because there's an $x$ in it)
C: $y = 7/x$ → Non-linear, hyperbola ✓
D: $y = 2^x$ → Non-linear, parabola (because it has a power)
E: $y = -x + 4$ → Linear ✓
(a) Which two classifications are wrong?
(b) For each wrong one, explain in one sentence why the student's reasoning is mistaken.
(c) Write out the corrected classification (including family name) for each wrong one.
Stuck? Compare each equation to the three signature patterns: $x^2$, $1/x$, $a^x$. Then check whether the student has matched the right family or the wrong one.3. Open-ended challenge — invent the data
This question has many valid answers. Be creative. 4 marks
3.1 Invent three tables of values, each with $x = 0, 1, 2, 3, 4$. Make one linear, one a parabola, and one an exponential.
For each table:
(i) Write the $y$-values you chose.
(ii) Show the first differences.
(iii) State which family it represents and give the rule (e.g. $y = 2x + 1$, $y = x^2$, $y = 3^x$).
Bonus: Your three tables must NOT use the rules $y = x$, $y = x^2$ or $y = 2^x$ (the lesson's defaults).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Mixed classify
(a) Linear. (b) Non-linear, parabola. (c) Non-linear, hyperbola. (d) Non-linear, exponential. (e) Linear ($y = 0 \cdot x + 3$ — a horizontal straight line).
1.2 — $y: 1, 3, 9, 27, 81$
First differences: $2, 6, 18, 54$ — NOT constant, so non-linear. Each $y$ is $3$ times the last ($1, 3, 9, 27, 81$), so the rule is $y = 3^x$, which is an exponential.
1.3 — $y: -1, 2, 5, 8, 11$
First differences: $3, 3, 3, 3$ — constant, so the relationship is linear. The constant difference is the gradient $m = 3$. At $x = 0$, $y = -1$, so $c = -1$. Rule: $y = 3x - 1$.
1.4 — $y = x^2$ vs $y = x$
$y = x$ is the linear straight line through the origin with gradient $1$. $y = x^2$ is the non-linear U-shaped parabola, also through the origin but symmetric across the $y$-axis. They cross where $x^2 = x$, i.e. $x(x - 1) = 0$, so $x = 0$ and $x = 1$. Crossing points: $(0, 0)$ and $(1, 1)$.
1.5 — Bacteria
First differences in $N$: $50, 100, 200$ — not constant, so non-linear. Each $N$ is double the previous, so the family is exponential. Test $N = 50 \times 2^h$: at $h = 0, 1, 2, 3$ we get $50, 100, 200, 400$ — all match. Rule: $N = 50 \times 2^h$.
1.6 — Sorting task
Linear: $y = 4x + 1$, $y = -3x$ (both fit $y = mx + c$).
Non-linear — parabolas: $y = x^2$, $y = x^2 + 5$.
Non-linear — hyperbolas: $y = 1/x$, $y = 10/x$.
Non-linear — exponentials: $y = 2^x$.
2 — Find the mistake
(a) The two wrong classifications are B and D.
(b) B: "$y = x^2 + 5$ is linear because there's an $x$ in it" — having an $x$ is not enough to be linear. The $x^2$ takes it out of $y = mx + c$ form. D: "$y = 2^x$ is a parabola because it has a power" — the power is on the $2$, not on the $x$. Parabolas come from $x^2$. Equations with $x$ as the exponent are exponentials, not parabolas.
(c) B: Non-linear, parabola (the $x^2$ signature). D: Non-linear, exponential ($x$ is the exponent on base $2$).
This is exactly the trap the lesson flags — mistaking "any equation with $x$" for linear, and confusing "power" location.
3 — Open-ended challenge (sample solutions)
Linear example: rule $y = 4x + 2$. Values: $2, 6, 10, 14, 18$. First differences: $4, 4, 4, 4$ — constant, confirming linear.
Parabola example: rule $y = x^2 + 3$. Values: $3, 4, 7, 12, 19$. First differences: $1, 3, 5, 7$ — NOT constant, but the second differences are constant at $2$, confirming parabola.
Exponential example: rule $y = 3^x$. Values: $1, 3, 9, 27, 81$. First differences: $2, 6, 18, 54$ — each is $3$ times the previous, confirming exponential.
Marking: 1 mark per valid table with correct first differences and named family, plus 1 mark for clarity. Award full marks for any three distinct rules that follow the brief (no defaults).