Are these fractions equivalent?

What this sequence teaches

Two fractions are equivalent when you can get from one to the other by multiplying or dividing the top and the bottom by the same number. Adding the same number to both doesn’t work, and neither does multiplying the top and the bottom by different numbers.

Every example can show two arrows: one over the tops, one under the bottoms. Each carries what was done to that part (+2, ×2, ÷3). They don’t give the answer away: students still have to judge whether the two operations keep the fraction the same.

You control what students see

Each example arrives as its two fractions only. From there you choose:

• Fractions only. Ask “Equivalent or not?” and click Reveal. The arrows and operations then play quickly, so the reason has something to point at.
• Show operations. The arrows draw in, then a long pause (“What’s been done to the top?”), then the top operation, another long pause (“And the bottom?”), then the bottom operation. Then ask for the verdict and Reveal.

On a minimal change the arrows stay if you showed them; the operations that change clear, and an operation that doesn’t change (the top ÷3 from Example 4 to 5) stays. Show operations brings in what’s missing. Replay plays everything since the example arrived.

The teaching sequence

Example 1 — 3/7 and 5/9 (not equivalent). Add 2 to the top and 2 to the bottom: the classic error. Students who think “do the same to the top and the bottom” say yes.

Example 1 → Example 2. The right fraction changes and the labels change from +2 to ×2. The numbers on the arrows stay the same; only adding becomes multiplying.

Example 2 — 3/7 and 6/14 (equivalent). The first equivalent pair, with the easiest multiplier there is.

Example 2 → Example 3. Only the multiplier changes: from 2 to a million. 3/7 stays exactly where it is.

Example 3 — 3/7 and 3,000,000/7,000,000 (equivalent). The size of the multiplier doesn’t matter, and a million costs no arithmetic. Bigger numbers don’t make a bigger fraction.

Example 3 → Example 4. A clean fade: a new example, not a change.

Example 4 — 27/12 and 9/4 (equivalent). Dividing works too; both fractions are top-heavy; and the bigger fraction is on the left. Each of those rules out a wrong idea students might have picked up (“you only multiply”, “fractions are less than 1”, “the second one is bigger”).

Example 4 → Example 5. Only the bottom-left number changes, 12 to 8. The top arrow keeps its ÷3; only the bottom label changes, to ÷2.

Example 5 — 27/8 and 9/4 (not equivalent). Both parts are divided, and both divisions come out exactly — but not by the same number. This is the over-correction students make once they know adding is wrong: “it’s been divided, so it’s equivalent”. It is the contrast they should leave with.

Why the sequence is shaped this way

• Small multipliers, big numbers. The multipliers are 2 and 3 wherever a student has to check them, so attention goes on the idea, not on times tables. The million in Example 3 shows the size of the multiplier doesn’t matter at no arithmetic cost.
• Top first, then bottom, with thinking time. The operations arrive one at a time with a long pause before each, so students can predict what was done before they see it.
• Arrows always go left to right, so Example 4 reads as dividing by 3, which is why the rule says “multiplying or dividing”.

What this sequence doesn’t address

Taking away, an unchanged bottom, and one extra zero are in the testing card. Harder multipliers (7, 8, 9) and multipliers that aren’t whole numbers (such as 10/15 and 6/9, where the multiplier is 3/5) are in the expansion card. Negative fractions and decimals inside fractions aren’t covered.

The rule and the script

The rule, exactly as Show rule displays it: For the following examples: if you can get from one fraction to the other by multiplying or dividing the top and the bottom by the same number, the fractions are equivalent.

• 3/7 and 5/9 — “These fractions are not equivalent. How do I know? Because 2 is added to the top and the bottom, and adding doesn’t keep the fraction the same size: 3/7 is less than a half, but 5/9 is more than a half.”
• 3/7 and 6/14 — “These fractions are equivalent. How do I know? Because the top and the bottom are both multiplied by 2. Cut every piece into 2: there are twice as many pieces, and we have twice as many of them.”
• 3/7 and 3,000,000/7,000,000 — the same, with “a million”.
• 27/12 and 9/4 — “…both divided by 3. Put the pieces into groups of 3: there are a third as many pieces, and we have a third as many of them.”
• 27/8 and 9/4 — “These fractions are not equivalent. How do I know? Because the top is divided by 3 but the bottom only by 2: 27/8 is more than 3, but 9/4 is only 2 and a quarter.”

Whether students see the words is your call. Show rule puts the rule above the stage and keeps it there until you turn it off. Why? appears after each reveal: it shows the reason, and the two labels pulse. The arrows are always shown — they are the picture, not the words.

Vocabulary. The page says “top” and “bottom”. Once students are fluent: “You know how I’ve been saying the top and the bottom? Their proper names are the numerator and the denominator.”

Where this atom sits

Before it: what a fraction means — the bottom says how many equal pieces, the top says how many we have. After it: finding a missing number in a pair of equivalent fractions, and simplifying.

Running the sequence

Pause for predictions before Show operations and before Reveal. If a class struggles with Example 5, go back to Example 4 and replay: “What stayed the same? What changed?”

About the testing sequence

Ten items in a new random order each time: five equivalent, five not. Each arrives as its two fractions. Students can answer straight away, or you can click Show operations first: the arrows, then (after a pause for predictions) the top operation, then the bottom. Either way, the operations appear after the answer, so Why? has something to point at. There is no Show rule button here.

The arrows are kept on this card because they are how this atom shows the relationship. Finding the multiplier yourself is a separate skill (and the arithmetic this atom deliberately keeps out); the expansion card asks for it.

What each item is diagnosing

4/7 and 12/21 (×3). A plain equivalent pair.

50/40 and 5/4 (÷10) and 16/10 and 8/5 (÷2). Dividing, with top-heavy fractions and the bigger one on the left.

9/2 and 9,000/2,000 (×1,000). Big numbers don’t make a bigger fraction.

24/32 and 6/8 (÷4). Neither fraction is in its simplest form, and they are still equivalent.

5/7 and 2/4 (−3). Adding in a new form: taking the same away from both.

3/8 and 1,000,003/1,000,008 (+1,000,000). Adding again, in its hardest form: huge numbers that look like the million example.

7/9 and 70,000/9,000 (×10,000 and ×1,000). Different multipliers where the only difference is one zero.

5/6 and 15/6 (×3 and ×1). “Only the top needs changing.” The ×1 says the bottom stayed the same.

5/8 and 10/24 (×2 and ×3). Different multipliers, small numbers.

Common confusions to watch for

“Do the same to the top and the bottom” — shows up on 5/7 and 2/4, and on the million added.

“It’s been multiplied, so it’s equivalent” — shows up on the three pairs with different multipliers.

“Bigger numbers, bigger fraction” — shows up on 9/2 and 9,000/2,000.

Discussion prompts

1. “Say the rule. Now use it on 3/8 and 1,000,003/1,000,008.”
2. “Change one number in 5/8 and 10/24 so the fractions are equivalent.” (10/16 works; so does 15/24.)
3. “Which is bigger, 5/7 or 2/4? How do you know?”

Reading the summary

Each cell has two channels: the green or red tint shows whether the student answered correctly; the ✓ or ✗ in the corner shows whether the fractions are equivalent.

About the expansion sequence

Six items just beyond what the teaching and testing sequences covered, to see how far understanding stretches. It works exactly like the testing card: each item arrives as its two fractions, and you choose whether to click Show operations before students answer. Leaving the operations hidden asks more of students here, because the multipliers are harder to spot. The last item is always 1 and 5/5, and it arrives as 1: Show operations (or answering) writes 1 as 1/1 first, then the arrows draw in. The others are in a random order.

Two kinds of item

The rule still works, with harder numbers. 5/6 and 40/48 (×8); 2/9 and 14/72 (×7 on top, ×8 underneath — one away, and not equivalent); and 1 and 5/5, where students have to see 1 as 1/1 before the rule applies.

The multiplier isn’t a whole number. 8/6 and 12/9 (×1½) and 10/15 and 6/9 (×3/5) are equivalent. A student who thinks the “same number” must be a whole number searches, finds nothing, and says no. 6/8 and 9/14 (×1½ and ×1¾) catches the opposite over-correction.

For these, there is an easier route than finding the multiplier: see whether both fractions come from the same simpler fraction. 10/15 and 6/9 are both 2/3; 6/8 is 3/4, which is 9/12, not 9/14.

Who it’s for

Students who are secure on the testing card. A miss on 5/6 and 40/48 may be times tables rather than equivalence; on this card that’s acceptable.

Discussion prompts

1. “What could you multiply 10 by to get 6?”
2. “Find a simpler fraction that both 8/6 and 12/9 come from.”
3. “How many fifths make 1? Write 1 as a fraction three different ways.”

Reading the summary

Same two channels as the testing card. Saying no to 8/6 and 12/9 or 10/15 and 6/9 usually means “the same number” has been read as “the same whole number”.