Write the equivalent fractions

What this sequence teaches

To write equivalent fractions, count in the top number along the top and the bottom number along the bottom, until there are six fractions: 2/3 = 4/6 = 6/9 = 8/12 = 10/15 = 12/18. Every fraction in the list is the same amount. This is Dylan Kane’s method for adding fractions: list the equivalent fractions of each, then pick the pair with the same denominator. Skip-counting replaces “what do I multiply 4 by to get 12?”, a missing-factor problem many students can’t do even when they know their tables.

This atom is an exception to “set up the working, never evaluate it”. The counting is the atom, and writing it unevaluated would mean writing (2 + 2)/(3 + 3) — “add the tops, add the bottoms”, the very misconception the rule for adding fractions has to fight.

The teaching sequence

Example 1 — 2/3. The full reveal. The 2/3 drops down from the question, then five empty fractions appear, so students can see where to stop. The 2 lights up, and the top row is counted: 2, 4, 6, 8, 10, 12. Pause. The 3 lights up, and the bottom row is counted: 3, 6, 9, 12, 15, 18.

Example 2 — 2/5. Only the bottom number changes. Example 1’s answer stays on screen, dimmed. On Reveal, the whole top row stays where it is and brightens; only the bottom row is counted again, in 5s.

Example 3 — 3/5. Only the top number changes, so now the bottom row stays and only the top row is counted again. Without this example, the top number is 2 in every teaching example, and a student could decide the top always counts in 2s.

Together, Examples 2 and 3 show the two rows are counted separately. That is the point of the sequence: nothing is “added to the top and the bottom”.

Running the sequence

• Keep it uninterrupted. The testing sequence is where students take over.
• Reveal is slow on purpose: about fifteen seconds for a full reveal. Each new number waits before it appears, and it is blue for a moment as it arrives, along with the number it counts in. Ask “what comes next?” before each one if you like.
• Moving to Example 2 is slow too. Example 1’s answer dims and is labelled “Previous answer”, the page holds still for a second, then only the 3 in the question changes to a 5. Ask: “How much of the answer will change?”
• Replay plays everything since the last example: the change to the question, then (after Reveal) the change to the answer.

The rule and the script

The rule, exactly as Show rule displays it: Count in the top number along the top, and the bottom number along the bottom, until there are six fractions.

• Example 1 — “I counted in 2s along the top and 3s along the bottom. Every fraction is 2 parts for every 3, so they’re all the same amount.”
• Example 2 — “The top is still 2, so the top row stays the same. Only the bottom row changes: now it counts in 5s.”
• Example 3 — “Now only the top changes, to 3. The top row counts in 3s; the bottom row stays the same.”

When Why? is open, the page lights up the top row and the top number, then the bottom row and the bottom number, then both: colour, underline and a pulse, so it works on a washed-out projector. A fraction wall or a clock face (2/3 of an hour is 40 minutes; so is 4/6, and 8/12) is a good way to show why the fractions are equal.

Why always six?

A fixed stop means no judgement about when to stop, and it is the length the next atom needs: two lists of six, from which students pick the pair with the same denominator. Including the first fraction in the six keeps the count honest.

Where this atom sits

Before it: “Are these fractions ready to be added or subtracted?” — only the denominators matter. After it: “Pick the pair that’s ready to be added or subtracted”, then the rule for adding and subtracting fractions.

About the testing sequence

Six items in a designed order. The first is a reset: a fraction that shares no number with the teaching examples, so students have to use the rule rather than adapt a teaching answer. Some items then change exactly one number 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, and Reveal keeps the row that hasn’t changed and counts only the other row again. After a reset, the previous answer is wiped.

What each item is diagnosing

• 3/10. Reset, with easy tables so the class gets off to a good start. Watch for 3/10, 4/11, 5/12 — adding 1 to the top and the bottom — and for 3/10, 6/11, counting on by 1 from 10 instead of counting in 10s. The 3 was the top number in Example 3, so the top row matches Example 3’s; the bottom row is new.
• 1/10. Only the top changes. The bottom row should be identical to the last board. A changed bottom row means the student thinks both rows must change. A student who doubles writes 1/10, 2/20, 4/40 and goes wrong at the third fraction.
• 1/4. Only the bottom changes. Watch for 1/4, 2/5, 3/6: adding 1 to the top and the bottom. This is also check question 1 from the routine plan.
• 5/9. Reset, in the 9s table. A board that looks like the last one was edited, not counted.
• 5/8. Only the bottom changes. A student counting the top in 8s has swapped the roles.
• 4/7. Reset, in the 7s table, now that the rule is secure. A student who doubles (4/7, 8/14, 16/28) gets the first two fractions right and goes wrong at the third.

Common confusions

Times-table slips show up plainly on the board; they are a fluency gap, not a misunderstanding of the rule. Five or seven fractions instead of six usually means the student didn’t count the first one.

Discussion prompts

• “Say the rule, then use it on this one.”
• “Which row stayed the same as last time? Why?”
• “Is 2/5 really the same amount as 3/6? How do you know?”

About the expansion sequence

Six items that show how far the rule reaches. Each changes one number from the item before, and each uses the same rule on something new. The order does the teaching: don’t skip items or run them out of order.

What each step adds

1. 4/3. An improper fraction, one change from the last test item (4/7). The rule doesn’t care which number is bigger.
2. 11/3. Counting in 11s along the top.
3. 11/12. Counting in 12s along the bottom — past the times tables the routine needs.
4. 11/b. A letter on the bottom: b, 2b, 3b …
5. a/b. A letter on top too. The list a/b = 2a/2b = 3a/3b … is the rule itself: counting in the top and the bottom is multiplying both by the same number.
6. 2a/b. Counting in 2a’s: 2a, 4a, 6a …

Where classes stall

• Counting in a’s. We write a, not 1a, in the same way we write x, not 1x. A student who writes 1a is right, just not how we usually write it.
• 4/3. Some students feel a fraction “should” be smaller than 1. The list still works.

Discussion prompts

• “What’s new in this one? Did the rule change?”
• “Look at a/b = 2a/2b = 3a/3b. What is the rule really doing?”