Pick the pair that’s ready to be added or subtracted

What this sequence teaches

Two fractions that aren’t ready to be added or subtracted have been written out as lists of equivalent fractions (Dylan Kane’s method). This atom picks the pair to use: find the smallest denominator that is in both lists, and write those two fractions, in the question’s order, with the question’s sign. Nothing is added up yet: “keep the denominator and add the numerators” is the next atom.

The teaching sequence

Only the second fraction changes, so the first list, 5/6 = 10/12 = … = 30/36, sits still for the whole card.

Example 1 — 5/6 + 1/8. One denominator, 24, is in both lists. On Reveal the two 24s pulse and turn blue, then 20/24 drops from the first list, the plus drops from the question, and 3/24 drops from the second list. The 24 is 4th in one list and 3rd in the other: a match is never in the same place in both lists.

Example 2 — 5/6 + 1/9. One number changes, 8 to 9, and now two denominators are in both lists: 18 and 36. Both pulse; the 18s turn blue and the 36s grey out and are struck through. Take the smallest. (36 would work, but the answer would need simplifying at the end.)

Example 3 — 5/6 + 1/2. 9 becomes 2. The smallest denominator in both lists is 6, which is the first fraction in the first list, so 5/6 is copied as it is. Without this example, both fractions change every time, and “both must change” looks like the rule.

Running the sequence

• Keep it uninterrupted. The testing sequence is where students take over.
• Reveal has two stages, in the order the rule says them: find the denominator (every match pulses; the smallest stays blue, the rest are passed over), a pause, then write the two fractions with the sign.
• Moving to Example 2 is slow on purpose. Example 1’s answer dims and is labelled “Previous answer”, the page holds still for a second, then only the 8 changes to a 9 — in the question and along the second list. Ask: “Will the answer change much?” Here it changes almost completely, so Reveal plays the whole find-and-write again.
• Replay plays everything since the last example.

The rule and the script

The rule, exactly as Show rule displays it: Find the smallest denominator in both lists, and write those two fractions with the sign from the question.

• Example 1 — “24 is the denominator in both lists. I wrote those two fractions, 20/24 and 3/24, with the plus from the question.”
• Example 2 — “Now two denominators are in both lists: 18 and 36. I took the smallest, 18. The first list didn’t change.”
• Example 3 — “6 and 12 are in both lists. The smallest is 6, and that’s the first fraction in the first list, so 5/6 is copied as it is.”

When Why? is open, the page lights up the shared denominator in both lists and in the answer, then the two fractions picked and where they went, then the sign. Colour, underline and a pulse, so it works on a washed-out projector. The sign takes colour and a pulse, never an underline.

The convention

Write the two fractions exactly as they appear in the lists, in the order of the question, with the question’s sign — and stop there. For adding, a pair written the other way round isn’t wrong maths, but keeping the question’s order means nothing changes when subtraction arrives.

Where this atom sits

Before it: “Are these fractions ready to be added or subtracted?” and “Write the equivalent fractions”. After it: the chain of writing the lists and picking the pair, then the rule for adding and subtracting fractions.

About the testing sequence

Six items in a designed order. The first is a reset: a question that shares nothing with the teaching examples, so students have to use the rule rather than adapt a teaching answer. Some items then change one thing from the item before, and two more resets start again from nothing. Show the item, students write on mini-whiteboards, then reveal. There is no scoring — the whiteboards are the data.

After a one-change item, the previous answer stays on screen, dimmed and labelled. Where most of the answer survives, Reveal keeps what survives and brings in only what’s new; where almost everything changes, it plays the whole reveal again.

What each item is diagnosing

• 3/4 + 1/6 → 9/12 + 2/12. Reset. Two denominators are in both lists, 12 and 24. A board with 18/24 + 4/24 took a match that wasn’t the smallest — or multiplied the denominators, since 24 is 4 × 6.
• 1/6 + 3/4 → 2/12 + 9/12. The same two fractions, swapped. A board with 9/12 first was copied from the last answer, or sorted by denominator. On Reveal the two numbers arc past each other.
• 1/6 + 1/4 → 2/12 + 3/12. The tops now match all along both lists, so 3/18 + 3/12 is sitting right there. Matching tops is the misconception. This is the routine’s check question for this atom, the other way round.
• 2/3 + 4/9 → 6/9 + 4/9. Reset. 4/9 is the pick, at the start of its list. A board with 8/18 or 12/27 skipped the first fraction — “both must change”.
• 2/3 + 4/5 → 10/15 + 12/15. Only one match, and this time it is the product of the denominators. A student who has decided “never the product” goes wrong here.
• 3/10 + 1/4 → 6/20 + 5/20. Reset, bigger denominator first. The 20 is 2nd in one list and 5th in the other, so “the same place in both lists” is plainly wrong.

Common confusions

Students who go on and add (9/12 + 2/12 = 11/12) have done the next two atoms as well: right, but not what was asked. Ask them to stop at the pair, so a wrong board shows which step went wrong.

Discussion prompts

• “Say the rule, then use it on this one.”
• “Which denominators are in both lists? Which one do we take?”
• “Why does the answer start with 2/12 this time?”

About the expansion sequence

Seven items that show how far the rule reaches. Each adds one new thing, and each uses the same rule. The order does the teaching: don’t skip items or run them out of order.

What each step adds

1. 3/10 − 1/4 → 6/20 − 5/20. Subtraction, one change from the last test item. The sign comes from the question, so the answer takes away too. Keeping the question’s order now matters: 5/20 − 6/20 is a different calculation.
2. 3/10 − 1/5 → 3/10 − 2/10. Three denominators in both lists — 10, 20 and 30 — and the smallest is the first fraction itself.
3. 13/10 − 1/5 → 13/10 − 2/10. A top bigger than the bottom. The rule doesn’t care.
4. 13/10 − 1/15 → 39/30 − 2/30. A denominator past 10, with two matches, 30 and 60.
5. 13/10a − 1/15a → 39/30a − 2/30a. A letter on every denominator. The lists are the same with the a carried along, and the answer is the last one with an a on each bottom.
6. 3/5 − 2/5 → 3/5 − 2/5. A new start: the denominators are already the same. Every denominator is in both lists, the smallest is 5, and the pair is the question itself. This is why, in the full routine, fractions that are already ready skip this step.
7. 3/1 − 2/5 → 15/5 − 2/5. One change: the 5 under the 3 becomes a 1. 3/1 is the whole number 3, and its list counts 1, 2, 3, 4, 5, 6 along the bottom, so it meets any small denominator. The 2/5 stays put; only the first fraction moves along its list.

Where classes stall

• 3/1. Some students don’t accept 6/2 or 15/5 as “3”. The list is the rule doing its job: every fraction in it is the same amount.
• The letters. 30a and 30 are different denominators. Every denominator in both lists has the a.
• 3/5 − 2/5. “There’s nothing to do” is the right instinct. The rule agrees: it hands back the question.

Discussion prompts

• “What’s new in this one? Did the rule change?”
• “How would you write 3 − 2/5 so that the rule can work on it?”