Equivalent fractions: find the operation
A transformation atom from mrbartonmaths.com
What this sequence teaches
Students name the one operation that takes one fraction to an equivalent one. They read it across, from the left fraction to the right one, and the bottom always does the same as the top. The whiteboard answer is a single operation: ×4 or ÷4.
The teaching sequence
Example 1 — 3/7 and 12/28 (×4). Reveal fills the top arrow, pauses, fills the bottom arrow, then both labels fly down and merge into one answer below the line. Two arrows, one operation.
Example 1 → Example 2. The answer dims and is labelled “Previous answer”; the arrows go back to question marks; then the two fractions swap places, one arcing over the other. Ask: “Same two fractions. Is it still ×4?”
Example 2 — 12/28 and 3/7 (÷4). Reveal fills the arrows again, then the answer changes as little as it can: the 4 stays and brightens, and only the sign changes. Going from the bigger fraction to the smaller one divides.
Two examples are enough. Top-heavy fractions, powers of ten and new numbers are left to the testing sequence, where students produce the answer rather than watch it.
Running the sequence
- Keep it fast: about thirty seconds, with no practice between examples. Pause for predictions before each Reveal.
- The pauses inside the reveal are thinking time: “What’s the top been multiplied by?” before the top fills, “And the bottom?” before the bottom does.
- Replay plays everything since the last example: on Example 2, the swap (and, after Reveal, the answer changing); on Example 1, the reveal.
The rule and the script
The rule, exactly as Show rule displays it: For the following examples: go from the left fraction to the right one. What is the top multiplied or divided by? That’s the answer, and the bottom is the same.
- Example 1 — “3 to 12 is ×4, and 7 to 28 is ×4 too. The answer is ×4.”
- Example 2 — “The same two fractions, but now the bigger one comes first. 12 to 3 is ÷4, and 28 to 7 is ÷4. Going the other way turns ×4 into ÷4.”
Why? steps through the answer: the two tops and the top arrow turn blue and pulse, then the two bottoms and the bottom arrow, then both, with the answer. Whether students see the rule and the reasons is your call. The page says “top” and “bottom”; later: “Their proper names are the numerator and the denominator.”
Why these conventions?
One operation, not a calculation. Every operation on the teaching and testing cards is a times-table fact or a power of ten, so it is recalled rather than worked out. For a student without secure tables this atom is a calculation, and a wrong board may mean a missing fact rather than a missing idea.
÷4, not ×¼. Accept ×¼ if a student writes it, but ÷ is what’s taught here.
Why fill both arrows? The answer only needs one, but filling both shows the bottom agrees — which is what makes the fractions equivalent.
Where this atom sits
Before it: “Are these fractions equivalent?”, with the same arrows. After it: completing a fraction (3/7 = ?/28): find the operation from the numbers you have, then apply it to the missing one. This atom is the first of those two stages, on a complete pair.
About the testing sequence
Six items in a designed order, answered on mini-whiteboards; reveal each before moving on. There is no scoring — the whiteboards are the data. The first item is a reset, unlike either teaching example, so students use the rule rather than adapt Example 2’s answer. After that, small changes alternate with fresh starts.
What each item is diagnosing
- 5/6 and 15/18 (×3). Reset. ×4 or ÷4 means a teaching answer was reused.
- 2/10 and 6/30 (×3). New fractions, same operation. ×5 means reading down (2 to 10, inside one fraction). A changed answer means “the numbers changed, so my answer must”: the same ×3 is the point of the item. Ask: “Did you change your answer? Why not?”
- 6/30 and 2/10 (÷3). The two fractions swap. ×3 means × and ÷ backwards, or always multiplying.
- 9/4 and 45/20 (×5). Reset. Top-heavy fractions. ÷3 or ×2 means the last board was edited.
- 9/4 and 18/8 (×2). Only the operation changes. +9 or +4 means using the difference. Doubling is the most tempting case, because doubling is adding a number to itself — and +9 on top, +4 underneath don’t match.
- 700/900 and 7/9 (÷100). Reset. A power of ten, bigger fraction first. ×100, or ÷10 from counting zeros in one number, are the usual slips.
Discussion prompts
- “Say the rule. Now use it on 6/30 and 2/10.”
- “Why isn’t 9/4 and 18/8 ‘+9’? Check the bottom.”
- “Write a pair of fractions whose answer is ÷4.”
About the expansion sequence
Six items beyond the testing card, in a fixed order, to see how far the idea stretches. The first is one change from the last test item (only the operation changes), but no previous answer is carried across the card boundary. 2 and 18/9 is always last.
Two kinds of item
The rule still works, on harder numbers: 56/72 and 7/9 (÷8), 4/9 and 28/63 (×7), 0.6/0.5 and 6/5 (×10: decimals inside a fraction), and 2 and 18/9 (×9). For the last, Reveal first writes 2 as 2/1, then fills the arrows.
The answer isn’t a whole number: 8/6 and 12/9 (×1½) and 10/15 and 6/9 (×3/5). A student who thinks the operation must be a whole number searches, finds nothing, and gets stuck. Accept ×3/2 for the first, and “÷5, then ×3” for the second. The easier route is to spot the simpler fraction both come from (2/3).
Who it’s for
Students secure on the testing card. A miss on ÷8 or ×7 may be times tables rather than equivalence, and that’s acceptable here.
Discussion prompts
- “What do you multiply 8 by to get 12?”
- “Find the simpler fraction that 10/15 and 6/9 both come from.”
- “Write 2 as a fraction three different ways.”