The miskeyed system: a correct student against a printed answer key
You teach Grade 8 algebra in a gymnasium in Karaganda. Today's objective is 'solve systems of linear equations by substitution'. The task on the board, taken from the department's question bank, is: 2x + 3y = 12 and 4x + 6y = 20. The printed answer key gives x = 3, y = 2. Twelve minutes into the lesson Aisha says the problem is 'broken': when she substitutes, she gets 0 = -4. Damir, working by elimination, also wrote 0 = -4 and concluded 'no solution'. Two other students stopped working and asked whether they had made a mistake. There are 20 minutes left; you still have to introduce the graph of a linear equation in two variables. The same item appears as item 9 on the department's term test next week, and the answer key has already been printed and handed to all four Grade 8 teachers.
1. Aisha asks: 'So is the answer no solution, or is the book wrong?' What mathematics must you establish, and how do you spend the remaining 20 minutes?
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Establish the mathematics, not the authority.
- From 2x + 3y = 12, x = 6 - 1.5y. Substituting into 4x + 6y = 20 gives 24 - 6y + 6y = 20, i.e. 24 = 20. Elimination gives the same 0 = -4. The system is inconsistent.
- Generalise with the ratio test: a1/a2 = 2/4 = 1/2, b1/b2 = 3/6 = 1/2, c1/c2 = 12/20 = 3/5. Equal coefficient ratios with an unequal constant ratio means parallel, distinct lines: no solution.
- Aisha and Damir are right and the key is wrong. Say so plainly.
Use the 20 minutes. Roughly 8 minutes making the contradiction visible: both methods side by side on the board, then a quick sketch from intercepts (0,4) and (6,0) against (0,10/3) and (5,0), so students see the parallelism rather than being told about it. Then 7 minutes on pairs choosing k and c in 2x + 3y = 12, 4x + ky = c to produce one, none and infinitely many solutions. The last 5 minutes become the planned introduction to graphs, now with the best possible lead-in already on the board.
What students should leave with. Not 'the book is wrong' but 'an answer key is a claim, not evidence; we check by substitution, by elimination and geometrically'. Tell them the key will be corrected. Hiding the error teaches students that the printed answer outranks their own valid reasoning, which is precisely the belief that later produces exam failures.
2. Damir's elimination solution is fully correct but the department test will mark it wrong. What do you do before the test, and how do you mark?
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Before the test: written, specific, private.
- Send the head of department a short note today: the item, the printed key, the substitution and elimination working, and the parallel-line graph. Attach one student script.
- Propose a replacement item testing the same objective, so the blueprint is not left with a hole and the department is not simply asked to delete an item.
- If nothing changes, do not announce the answer to a live test - that is unfair to the other three classes and indefensible if challenged. Instead, instruct your students to show substitution working in the margin for any inconsistent system, and raise a formal post-marking review if they lose marks.
- Keep a dated record of the note and the response. That record is what makes correction possible afterwards.
When marking. Damir's '0 = -4, therefore no solution' is a complete solution for this system. Mark the mathematics. Applying a wrong key silently would give a correct student a lower mark and teach him that being right is not the point.
Why not stay quiet. Four classes sit this item. Silence protects a working relationship for one week and tells roughly 120 students that not making trouble matters more than being correct. Raising it privately, with evidence and a proposed fix, is the version of collegiality that survives an inspection.
3. Design a 12-minute follow-up task that uses this error and gives you evidence of who actually understands consistency of a system.
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The task. Pairs receive 2x + 3y = 12 with 4x + ky = c, and three cards: ONE SOLUTION / NO SOLUTION / INFINITELY MANY.
- 3 min: choose k and c for each case; write the reason, not only the numbers.
- 5 min: swap with another pair, who checks by substitution and writes 'confirmed' or the correction.
- 4 min: harvest four pairs, one case each, and write the ratio rule on the board.
Why this works. Students must vary the parameters themselves, which is the difference between recognising a no-solution system and being able to construct one. It also makes the standard misconception visible immediately: many pairs will choose k = 6 with c = 20 and label it 'infinitely many', reproducing exactly the confusion the miskeyed item created. The only prompt they need is: 'what does 4x + 6y equal when you double the first equation?'
What the evidence tells you.
- Correct k, c and a stated reason for all three cases: secure.
- Correct answers with vague reasoning: partially secure; ask for the substitution in writing.
- k = 6 offered for all three cases: not yet; reteach through the graph.
Collect the cards and sort them into three piles. The pile names are tomorrow's grouping, and the sorted cards are your documented formative evidence for the term.