Write the lists, then pick the pair

What this chain teaches

Two atoms of adding and subtracting fractions, run together: Write the equivalent fractions (once for each fraction) and Pick the pair that’s ready to be added or subtracted. It assumes students have met each one on its own. On its own, the picking atom hands students the lists; here they make the lists and then use them, which is the hardest stretch of the routine. Nothing is added up: the chain stops at the pair.

The atom names never appear on screen. The line at the top, “Get these fractions ready”, echoes the question students have already met: “Are these fractions ready to be added or subtracted?”

The steps

1. List the first fraction. It drops from the question, five empty fractions appear, then the tops are counted, a pause, then the bottoms, until there are six.
2. List the second fraction, in the same way.
3. Pick the pair. Each denominator that is in both lists lights up in turn, smallest first. Any larger ones are crossed out together, and then the two fractions are written with the sign from the question.

Nothing moves between steps until you click, so you can check boards after each list.

The two examples

Example 1 — 4/9 + 1/6 → 8/18 + 3/18. 18 and 36 are both in both lists; take the smallest.

Example 2 — 4/9 + 1/3 → 4/9 + 3/9. Only the 6 changes, to a 3. On Next the lists fade, the previous answer dims and is labelled, the page holds still, and only then does the 6 change. Ask: “Which list will change?” Only the second. The smallest denominator in both lists is now 9, which is the first fraction in the first list, so 4/9 is copied as it is. The previous answer stays at the bottom until the new one is written.

Running the sequence

• Replay plays the last step again (or the change to the question, before step 1).
• Why? gives the reason for the step on screen and lights up what it refers to.

About the testing sequence

Five questions in a fixed order, starting with easy tables. Students write both lists and the pair on their whiteboards. There is no scoring; the whiteboards are the data. Early on, check boards after each list. Later, ask for all three lines at once and click through to compare.

What each item catches

• 1/2 + 3/5 → 5/10 + 6/10. A first item in the 2s and 5s. The only match is the product of the denominators.
• 1/3 + 1/4 → 4/12 + 3/12. The tops match, so 1/3 + 1/4 can look “ready already”. Only the denominators matter.
• 1/3 + 1/6 → 2/6 + 1/6. One change, the 4 to a 6. Three denominators are in both lists; the smallest is 6, and 1/6 is copied as it is. A board where both fractions changed took “both must change” as the rule.
• 3/8 + 5/6 → 9/24 + 20/24. Harder tables, with tops above 1, so the counting itself can go wrong.
• 7/10 + 5/6 → 21/30 + 25/30. The 30 is 3rd in one list and 5th in the other: a match is not in the same place in both.

Reading the boards

• A list is wrong: go back to writing the equivalent fractions (count the top number along the top and the bottom number along the bottom).
• The lists are right but the pair is wrong: go back to picking the pair (the smallest denominator in both lists).
• Students added them up: right, but ask them to stop at the pair, so a wrong board shows which step went wrong.

About the expansion sequence

Four questions that take the chain further. Run them in order.

1. 5/6 − 1/4 → 10/12 − 3/12. Subtraction. The lists and the pick work exactly as before; the sign comes from the question.
2. 4/9 − 1/6 → 8/18 − 3/18. Keep the question’s order: 3/18 − 8/18 is a different calculation.
3. 4/9 − 1/5 → 20/45 − 9/45. One change, the 6 to a 5, and now no denominator is in both lists after six fractions. This item has an extra step: if no denominator is in both lists, keep counting the list with the smaller denominator until one is. Step 3 continues the 5s list (35, 40, 45) on the line below; step 4 picks the pair. Ask students what to do before you click step 3.
4. 2/1 − 3/4 → 8/4 − 3/4. 2/1 is the whole number 2, written as a fraction. Its list counts in 1s along the bottom, so it meets any denominator. Here the whole number is written as 2/1; a question that just says “2 − 3/4” needs that step first, and comes later.

Discussion prompts

• “Which list will change?” (before item 3)
• “No denominator is in both lists. What now?”
• “How would you write 2 − 3/4 so that you could make its list?”