3/4 + 1/2 = 4/6: Twenty-Seven Students, One Coherent Rule
Ms. Sana marks 42 Grade 6 scripts on fractions. Twenty-seven students wrote 3/4 + 1/2 = 4/6. Eleven of those also wrote 5/6 − 1/2 = 4/4, and one of them wrote 2/3 × 3/5 = 6/15, explaining 'you just do the same thing to the top and the bottom'. Eight students gave correct answers but could not say why the denominators have to match. Seven students applied the algorithm correctly, finished in five minutes and then sat idle. The unit test is in ten days and the textbook moves on to decimals next week.
1. Name the misconception, account for its origin, and explain why telling students to 'find the LCM' will not remove it.
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The misconception. Fraction notation is being read as two independent whole numbers - a count on top and a label underneath - so any operation is applied to both, exactly as it would be to whole numbers.
Origin. Fractions are usually introduced as shaded parts of a shape, where the denominator functions as a name ('these are quarters') rather than a number with a position on the number line. Under that reading, 3/4 and 1/2 are two objects, not two magnitudes, and 'add' means combine the counts on top and the counts below.
The trap in the scripts. 2/3 × 3/5 = 6/15 is numerically correct as an unsimplified product. The student's rule accidentally works for multiplication, which confirms the rule and makes it much harder to dislodge.
Why 'find the LCM' fails. It is a competing rule with no reason attached. The student now has two rules and no basis for preferring yours, so under time pressure and unfamiliar items they revert. Nothing in their mental model has changed; only a procedure has been added on top of it.
The repair that is needed: fractions must become single numbers with magnitude, and the wrong answer's absurdity must become visible to the student rather than asserted by the teacher.
2. Choose one representation and design a single task that forces the student to confront the error themselves.
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Representation: the number line. Not pie diagrams - circles do not add across different denominators easily, and they reinforce the 'parts of a whole' reading that caused the problem.
The confrontation task (5 minutes, before any teaching):
Without calculating, decide whether 3/4 + 1/2 is more or less than 3/4. Then mark 3/4, 1/2 and 4/6 on this number line.
Students see 4/6 sitting below 3/4. Adding a positive amount made the number smaller. The conflict belongs to the student, not to the teacher's correction - that is the difference between this and marking the script wrong.
Then build the correct idea: fraction strips. 3/4 = 6/8; 1/2 = 4/8; total 10/8. Students write the sentence: I can add eighths because the pieces are the same size. That is the reason the LCM rule never gave them.
Make the habit structural: for two weeks every item requires estimate → represent → compute → check against estimate. Marked, so it is not skipped. Also flag that multiplication does not need this step, and say why - that is what protected the error.
3. Plan the 40-minute lesson for these three groups and specify what the unit test must contain so the fix is actually tested.
Read the worked answer
Group C - 27 with the misconception (teacher-led, 15 min). The number-line sequence above, then paired practice with a mandatory estimate column. These students need the teacher, not a worksheet.
Group B - 8 correct but unexplained (independent, 15 min). Write an explanation for a Grade 4 child of why 3/4 + 1/2 is not 4/6, then a bridging task with unlike denominators requiring simplification. In the last 10 minutes they act as checkers for Group C - articulating the justification is precisely their gap, so the pairing serves both.
Group A - 7 fast finishers. Not more of the same. Task: 'Find two fractions that add to 4/6 and explain why your method is valid,' then write a wrong worked example with the error explained in words. This produces board material for Friday and is harder than the algorithm.
What the unit test must contain. At least one item that cannot be answered by the LCM procedure: place four fractions on a number line; judge whether a given sum is reasonable and say why. Without this, a student who has merely memorised the new rule scores full marks and you have no evidence the model changed. Ten days is enough for one re-check, not for decimals and a re-check - tell the coordinator that.