Mathematics • Year 8 • Unit 2 • Lesson 4
Tables — Mixed Challenge
Pull together everything from Lesson 4: building tables from rules, testing for linearity, reading coordinate pairs, and finding the rule from a table. Six mixed problems, one "find the mistake", and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 4. Decide what's being asked BEFORE writing. 3 marks each
1.1 Build a table for y = 4x − 1 using x = 0, 1, 2, 3, 4. State the constant rate of change.
1.2 Apply the Linear Test to: x: 0, 1, 2, 3, 4 y: 1, 3, 7, 13, 21. Show first differences and state your conclusion.
1.3 A table shows x: 0, 1, 2, 3, 4 and y: 12, 9, 6, 3, 0. (a) Show it's linear. (b) Write the rule.
1.4 A linear table has x = 0 → y = 4 and x = 1 → y = 9. Find the rule, then predict y when x = 7.
1.5 A table comes from the rule y = 2x + 3. List the four (x, y) coordinate pairs for x = −1, 0, 1, 2.
1.6 Sienna's table records x: 0, 1, 2, 3, 4 with y: 5, 7, 9, 11, 13. (a) Confirm it's linear. (b) Write the rule. (c) Use the rule to find the value of x that gives y = 99.
2. Find the mistake
Another student applied the Linear Test to this table:
x: 0 1 2 3 4
y: 1 3 6 10 15
Their working:
Line 1: First differences: 3 − 1 = 2, 6 − 3 = 3, 10 − 6 = 4, 15 − 10 = 5.
Line 2: The first differences (2, 3, 4, 5) are all different.
Line 3: But they all INCREASE by 1, so they form a pattern.
Line 4: Therefore the relationship IS linear because the differences have a pattern.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong (i.e. what the Linear Test actually requires).
(c) Write the correct conclusion: is the table linear or non-linear, and why?
Stuck? The Linear Test requires first differences to be EQUAL, not "in a pattern". "Differences themselves growing" → curved relationship.3. Open-ended challenge — design two tables
This question has more than one valid answer. 4 marks
3.1 Both tables below have x = 0, 1, 2, 3, 4 and start at y = 2.
(i) Design a LINEAR table starting (0, 2) whose final value is (4, 22). Show first differences to confirm it's linear, and write the rule.
(ii) Design a NON-LINEAR table that ALSO starts (0, 2) and ends (4, 22). Show that the first differences are NOT all equal.
Bonus: for the non-linear table, make every y-value still increase.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — y = 4x − 1
x: 0, 1, 2, 3, 4 → y: −1, 3, 7, 11, 15. Constant rate of change is 4 (the coefficient of x).
1.2 — Linear Test on y: 1, 3, 7, 13, 21
First differences: 2, 4, 6, 8 — NOT all equal → non-linear.
1.3 — y: 12, 9, 6, 3, 0
(a) Differences: −3, −3, −3, −3 → all equal → linear.
(b) Rate −3, y-intercept 12 → y = −3x + 12 (also written 12 − 3x).
1.4 — Linear table x = 0 → 4, x = 1 → 9
Difference 5 per step → y = 5x + c. At x = 0, y = 4 → c = 4. Rule: y = 5x + 4. At x = 7: 5(7) + 4 = 39.
1.5 — Pairs for y = 2x + 3
x = −1: y = 1. x = 0: y = 3. x = 1: y = 5. x = 2: y = 7. Pairs: (−1, 1), (0, 3), (1, 5), (2, 7).
1.6 — y: 5, 7, 9, 11, 13
(a) Differences: 2, 2, 2, 2 → linear.
(b) Rate 2, y-intercept 5 → y = 2x + 5.
(c) 2x + 5 = 99 → 2x = 94 → x = 47.
2 — Find the mistake
(a) The mistake is on Line 4.
(b) The Linear Test requires first differences to be EQUAL, not just "patterned". Differences 2, 3, 4, 5 are all different, so the relationship is NOT linear — it's a curve (in fact y = (x² + x)/2 + 1, a quadratic-style relationship).
(c) Correct conclusion: the relationship is non-linear, because the first differences are not all equal. If plotted, the (x, y) points would form a curve, not a straight line.
3 — Two tables, same endpoints (sample solution)
(i) Linear: from (0, 2) to (4, 22) means total change +20 over 4 steps = constant +5 per step.
x: 0, 1, 2, 3, 4 → y: 2, 7, 12, 17, 22. Differences: 5, 5, 5, 5 (all equal) → linear. Rule: y = 5x + 2.
(ii) Non-linear: e.g. x: 0, 1, 2, 3, 4 → y: 2, 3, 5, 9, 22. Differences: 1, 2, 4, 13 — clearly not equal → non-linear. (Endpoints still (0, 2) and (4, 22). ✓)
Bonus: the table above also has every y-value strictly increasing (2 < 3 < 5 < 9 < 22). ✓
Other valid examples (non-linear): 2, 3, 7, 13, 22 (diff 1, 4, 6, 9); 2, 4, 8, 14, 22 (diff 2, 4, 6, 8); anything starting at 2, ending at 22, with non-constant first differences.
Marking: 1 mark for correct linear table; 1 mark for the linear rule y = 5x + 2; 1 mark for a valid non-linear table with shown unequal differences; 1 mark for the bonus (all y-values strictly increasing).