Add or subtract fractions with the same denominator

What this sequence teaches

Two fractions with the same denominator are added or subtracted by keeping the denominator and adding or subtracting the numerators. This is the last step of the routine for adding and subtracting fractions: once the fractions are ready, students say the rule (the fact “To add or subtract fractions, keep the denominator, and add or subtract the numerators.”), then use it here.

Students write the answer, not a calculation. Most transformation atoms on this site set up the working and never work it out. This one works it out, for two reasons: adding the numerators is the rule, not an extra step; and every sum in every sequence is one students can recall (2 + 5, never 17 + 26). The arithmetic never gets in the way of the idea.

Nothing is ever simplified or converted. 9/12, 9/9 and 9/7 are finished answers on this page. Simplifying and mixed numbers are separate routines; you can mention them if you like, but the page never does.

The teaching sequence

Each change keeps one half of the answer and changes the other, so students see each half respond only to its own part of the question.

Example 1 — 2/9 + 5/9 → 7/9. On Reveal, the two 9s pulse, then fly down and land on top of each other: one 9. Then the 2, the plus and the 5 drop onto the bar, sit there for a beat as “2 + 5”, and collapse into 7. The contrast is the lesson: the 9s merge into the same number; the numerators make a new number.

Example 2 — 2/9 + 6/9 → 8/9. Only the 5 changes, to a 6. The 9 in the answer brightens where it is; only the numerator is worked again.

Example 3 — 2/11 + 6/11 → 8/11. Only the denominators change. The 8 brightens where it is; the 11s merge into one 11. It is 11, not 22: a different denominator is kept, never added.

Running the sequence

• Keep it uninterrupted and quick. The testing sequence is where students take over.
• Moving to Examples 2 and 3 is slow on purpose: the answer dims and is labelled “Previous answer”, the page holds still for a second, then only the changed numbers change. Ask: “What will change in the answer?”
• Replay plays everything since the last example.

The rule and the script

The rule, exactly as Show rule displays it: Keep the denominator. Add or subtract the numerators. It is the fact students have already learned, without its first words, and in the same order: the denominator first.

• Example 1 — “I kept the denominator: 9. I added the numerators: 2 + 5 = 7.”
• Example 2 — “I kept the denominator: 9. One numerator changed, so the new numerator is 2 + 6 = 8.”
• Example 3 — “The numerators are the same, so the numerator is still 8. The denominator is now 11, so I kept 11.”

When Why? is open, the page lights up the denominators (in the question and the answer), then the numerators and signs and the answer’s numerator, then everything. Signs take colour and a pulse, never an underline.

The word “numerator” may be new to some students; the fact atom introduces it. We avoid “tops” because “add the tops” is half of the chant behind the commonest error, 2/9 + 5/9 = 7/18.

Where this atom sits

Before it: “Are these fractions ready to be added or subtracted?”, the lists and the pick, and the fact “Recall the rule for adding and subtracting fractions”. After it: the full routine, “Add and subtract fractions”.

About the testing sequence

Six items in a designed order. The first is a reset, sharing nothing with the teaching examples, on easy numbers. After that, most items change one thing from the item before, so a wrong whiteboard points at one thing. Show the item, students write on mini-whiteboards, then reveal. There is no scoring: the whiteboards are the data.

What each item is diagnosing

• 4/10 + 3/10 → 7/10. Reset. A board with 7/20 added the denominators too.
• 4/10 − 3/10 → 1/10. Only the sign changes: this is where subtraction arrives. A board with 7/10 always adds; 1/0 subtracted the denominators.
• 4/9 − 3/9 → 1/9. Only the denominators change. The 1 stays; a board still showing 10 edited the last answer without reading the question.
• 3/7 + 2/7 → 5/7. Reset: no last answer to edit.
• 3/7 + 6/7 → 9/7. One numerator changes, and the answer comes out with a numerator bigger than the denominator. That is fine, and it is the answer, just as it is. Students who have never seen this often think they’ve gone wrong; this item is there so they meet it here first.
• 3/25847 + 6/25847 → 9/25847. Only the denominators change, to a five-digit number. It looks hard and isn’t: the denominator is only kept, so its size costs nothing. The 9 stays where it was. This is the item where students feel smart.

Common confusions

• Adding the denominators (7/20). Point back to the reveal: the denominators land on top of each other.
• Converting 9/7 to a mixed number, or simplifying. Not wrong maths, but not this job. On this page, the answer is the fraction the rule gives.

Discussion prompts

• “Say the rule, then use it on this one.”
• “What changed in the question? So what will change in the answer?”
• “Does a big denominator make this harder? Why not?”

About the expansion sequence

Eight items that show how far the rule reaches. Each adds one new thing, and the arithmetic never gets harder: every sum is still one students can recall. What changes is how the question and the answer look. The order does the teaching: don’t skip items or run them out of order.

What each step adds

1. 3/12 + 6/12 → 9/12. The answer could be simplified. It stays 9/12: on this page the answer is what the rule gives.
2. 3/9 + 6/9 → 9/9. Only the denominators change, and the answer is a whole one, written 9/9, just as it is.
3. 3/9 + 6/9 − 2/9 → 7/9. A third fraction, taken away, with numbers only. Work along the numerators left to right. Boards with 7/27 added the denominators; 11/9 added all three.
4. 2a/7 + 4a/7 → 6a/7. Letters on top. The rule doesn’t change: add the numerators, 2a + 4a = 6a.
5. 2a/7 + 4/7 → (2a + 4)/7. One term loses its letter. 2a and 4 can’t be put together, so the numerator is 2a + 4, just as it is. Boards with 6a/7 combined unlike terms.
6. 5/2x + 3/2x → 8/2x. A letter in the denominator. It is kept, like any denominator: not 4x, not 2x².
7. 5/2x − 1/2x + 3/2x → 7/2x. A third fraction, taken away, in the middle. Left to right: 5 − 1 + 3 = 7. The minus only takes away the 1; a board with 1/2x took away the 1 and the 3.
8. a/c − b/c → (a − b)/c. The general form: the rule itself, written in letters.

Where classes stall

• 9/9. “Isn’t that just 1?” Yes, and the page leaves it as 9/9. Simplifying is another job.
• 2a + 4. Students expect one number on top. An answer that is still a sum is finished.
• Three fractions. Read the signs one at a time, left to right.

Discussion prompts

• “What’s new in this one? Did the rule change?”
• “Why can’t we write 6a here?”