The Answer Key That Marked a Correct Method Wrong
A Class 7 mathematics teacher in a 36-student class has just finished addition of rational numbers. On a five-item exit ticket, 21 students wrote −3/4 + 5/6 = 2/10, having added numerators and denominators; 8 wrote 2/24; 4 wrote 1/12; 3 left it blank. The teacher's answer key accepts only 1/12 and marks the 2/24 responses wrong. In brief interviews, several students say "a fraction is just two numbers" and cannot place 1/12 on a number line between 0 and 1. The textbook page presents the LCM method with three worked examples, and the teacher had explained it twice on the board. One 40-minute period remains before the unit test, plus two 25-minute remedial slots each week. The teacher must decide whether to re-teach the procedure or rebuild the underlying idea of a fraction as a quantity.
1. Classify the response patterns and identify which are conceptual errors, which are procedural, and which are not errors at all. What does each pattern tell you?
Read the worked answer
The 21 students who wrote 2/10 have applied a whole-number rule to fractions: add the tops and add the bottoms. This is a conceptual error, not carelessness. They are treating the fraction as two independent numerals rather than as one quantity. Their inability to place 1/12 on a number line confirms that the denominator has no meaning as "size of the piece."
The 8 students who wrote 2/24 are not wrong. 2/24 simplifies to 1/12. They used a general common-denominator method: (−3×6 + 5×4)/(4×6) = 2/24. The method is valid, though unsimplified. Marking them wrong is a scoring error and may punish a more robust strategy than the textbook's LCM shortcut. The teacher should check whether they can explain and simplify, not remediate them as if they failed.
The 4 students with 1/12 and no working are correct in answer but may be answer-getting. They need an explanation check: Can they represent the sum? Can they justify the denominator?
The 3 blanks need a diagnostic conversation; blank is not a category. The overall pattern is that most students have a procedure without a quantity concept, while a minority have a valid alternative procedure that the key does not recognise.
2. Design a 40-minute re-teaching sequence that uses a representation and a cognitive-conflict task. State what you will listen for.
Read the worked answer
0–7 min: estimate first. Ask students to locate −3/4 and 5/6 on a number line. Then ask: is the sum closer to 0, 1/2 or 1? Students who wrote 2/10 must confront that 2/10 is 0.2, while −0.75 + 0.83 is about 0.08. The conflict is between their answer and the estimate.
7–20 min: same-size pieces. Use folded paper strips or a bar model. Fold one strip into fourths and another into sixths. Shade −3/4 and 5/6. Ask: can we add the shaded amounts while the pieces are different sizes? Students physically cut and re-fold both into twelfths. The denominator becomes the size of the piece; the numerator becomes the count.
20–32 min: return to the algorithm. Connect the strips to −9/12 + 10/12 = 1/12. For the 2/24 group, ask them to simplify and explain why their method also gives equal-sized pieces. For the 1/12 group, ask for a number-line justification.
32–40 min: exit ticket. Give one new sum and one number-line placement.
Listen for: "You can't add different-sized pieces"; "the bottom tells the size"; "I estimated first"; and whether students can explain 2/24 = 1/12. Do not listen only for the correct answer.
3. How would you assess whether the repair is durable, and what would you do with the students who already answered correctly?
Read the worked answer
Durability requires delayed and transfer tasks, not a repeat of the same exit ticket. One week later, give a short quiz with: one addition with unlike denominators; one subtraction; one number-line placement; one error-analysis item ("A student wrote 2/5 + 1/3 = 3/8. What did they do wrong?"); and one word problem. Include a "explain your denominator" prompt. Compare pre- and post-responses for the 21 students; the target is fewer whole-number-rule responses and better justification, not just a correct answer.
The 4 students with 1/12 and no working should not simply be given harder sums. Ask them to represent the sum on a number line and write a sentence explaining why twelfths are needed. If they cannot, they belong in the same conceptual repair group despite the correct answer.
The 8 students with 2/24 need extension, not remediation: simplify, explain why their general method works, and compare its efficiency with the LCM method. They can also test whether the method works for subtraction. This keeps them engaged and gives the teacher a chance to check whether their success is procedural or conceptual. The 3 blank responses need individual diagnosis before grouping.