Are these fractions ready to be added or subtracted?
A categorical atom from mrbartonmaths.com
What this sequence teaches
Fractions are ready to be added or subtracted when their denominators are exactly the same. Nothing else matters: not the numerators, not the sign, not the size of the numbers, not whether a fraction is top-heavy.
The teaching sequence
Example 1 — 5/7 + 5/9 (not ready). The numerators match and the denominators don’t. Students who look at the tops say yes.
Example 1 → 2. The verdict clears and the stage holds still for a second. Then the left fraction flips, its two numbers arcing past each other; a pause; then the right fraction flips the same way. The same three numbers are on screen before and after. Only their places change.
Example 2 — 7/5 + 9/5 (ready). Now the two 5s are on the bottom. Top-heavy fractions are fine. The flip is a change of example, not a method: 5/7 is not the same as 7/5. It is worth asking “Is 5 sevenths the same as 7 fifths?” here.
Example 2 → 3. The shared 5 becomes 3048 in both fractions. The numerators never move.
Example 3 — 7/3048 + 9/3048 (ready). Nobody can calculate this in their head, and nobody needs to. Students decide by reading the denominators. A good moment for them to feel clever.
Example 3 → 4. A clean fade: a new example.
Example 4 — 5/12 − 1/12 (ready). Subtraction, proper fractions, a numerator of 1. Ready can look like anything.
Example 4 → 5. Only the second denominator changes, 12 to 6.
Example 5 — 5/12 − 1/6 (not ready). 6 goes into 12, but “related” is not “the same”. This is the boundary students should leave with. The minus sign has already been seen on a ready pair, so it can’t be the reason.
What this sequence doesn’t address
Left to the testing sequence: matching numerators on a ready pair; the same numbers appearing in both fractions; fractions that could be simplified; look-alike denominators such as 17 and 71; equivalent fractions with different denominators; taking a bigger fraction from a smaller one.
Left to the expansion sequence: letters in the denominators, three fractions, and a whole number with a fraction.
Not covered: mixed numbers.
The rule and the script
The rule, exactly as the Show rule button displays it: For the following examples: if the denominators are exactly the same, the fractions are ready to be added or subtracted.
- Example 1 — “These are not ready to be added or subtracted. How do I know? Because the denominators are not the same: one is 7 and the other is 9.”
- Examples 2, 3 and 4 — “These are ready to be added or subtracted. How do I know? Because the denominators are exactly the same: both are 5” (then 3048, then 12).
- Example 5 — “These are not ready to be added or subtracted. How do I know? Because the denominators are not the same: one is 12 and the other is 6. 6 goes into 12, but that doesn’t make them the same.”
Whether students see the words is your call. Nothing appears unless you click. Show rule puts the rule above the stage until you turn it off; it works before Example 1 too. Why? appears after each reveal: it shows the justification and highlights both denominators, never the numerators or the sign. None of the reasons mentions the flip.
Where this atom sits
This is the decision at the start of adding or subtracting two fractions. After a yes comes the fact: keep the denominator, add or subtract the numerators. After a no comes making equivalent fractions with the same denominator.
Running the sequence
Pause before each reveal. The pause is the prediction moment.
What Reveal does. Before the tick or cross appears, each denominator pulses in turn, left to right. Students’ eyes go to exactly what decides it, even if nobody says a word.
Replay appears on Examples 2, 3 and 5 and replays the change from the example before, with the same pauses.
About the testing sequence
Ten items in a new random order each time: five are ready, five are not. The rule decides every one. There is no Show rule button here. Why? works as in the teaching sequence, after each answer.
The items themselves are still: the flip in the teaching sequence marks a change of example, not what “ready” means. After each answer, as on Reveal in the teaching sequence, each denominator pulses in turn before the feedback appears.
What each item is diagnosing
6/25 + 6/25 (ready). Matching numerators on a ready pair. It closes the route “matching tops means not ready”, which Example 1 could otherwise leave behind.
3/5 + 5/3 (not ready). The same two numbers appear in both fractions. That they swap places is deliberate: it is the trap.
2/6 + 3/6 (ready). “You have to simplify first.” As written, the denominators are the same.
4/17 − 9/71 and 254/1003 − 9/1030 (not ready). Denominators that look alike at a glance, small and big. These reward a careful read.
2/7 + 3/14 (not ready). “One goes into the other.” The smaller denominator comes first here; in Example 5 it came second, so position can’t be what was learned.
17/4 − 3/4 (ready). “Top-heavy fractions must be converted first.”
13/20 − 19/20 (ready). “You can’t take a bigger fraction from a smaller one” leads to a no for the wrong reason.
3/4 − 6/8 (not ready). The two fractions are worth the same, yet the denominators are different.
248/365 − 1/365 (ready). “They’re too different in size to go together.”
Discussion prompts
- “Say the rule. Now use it on the one with 3 fifths and 5 thirds.”
- “Which number decided it?”
- On the 3 quarters and 6 eighths: “They’re worth the same. Why isn’t that enough?”
Reading the summary
Each cell has two channels: a green or red tint shows whether the student answered correctly; the ✓ or ✗ in the corner shows whether the fractions are ready.
Saying yes to the 2 sevenths and 3 fourteenths, or to the 3 quarters and 6 eighths, suggests “close enough” is still being accepted. Saying no to the 6 twenty-fifths pair or the 2 sixths and 3 sixths suggests students are looking at the numerators or simplifying first. Missing the 17 and 71 or the 1003 and 1030 suggests reading too quickly.
About the expansion sequence
Six items a little beyond the examples the rule was built on, to see how far students’ understanding transfers. The rule began “For the following examples”; this is where students meet its edge. It works exactly like the testing sequence.
The two kinds of item
The rule still works on an unfamiliar surface.
- 3/x + 7/x (ready): letters on the bottom.
- x/9 − 4/9 (ready): a letter on top changes nothing. Mainly for students who think “letters, so it can’t be ready”.
- 2/(x+1) − 9/(x+1) (ready): a whole expression on the bottom, exactly the same in both.
- 6/11 + 2/11 + 5/9 (not ready): three fractions. The rule never said “both”, so it still applies, and “most of them match” is not enough.
Blind matching, or the bare rule, isn’t enough.
- 8/x + 1/x² (not ready): “they both have x” is the letter version of “one goes into the other”.
- 4 + 2/5 (not ready): the 4 shows no denominator, so there is nothing to compare. After an answer the 4 pulses, then the 5. The way through is the meaning of a whole number: 4 is 4/1, and 1 isn’t 5.
Who it’s for
Students who have met letters standing for numbers. Algebraic fractions themselves are not needed.
Discussion prompts
- On 4 + 2 fifths: “Where’s the 4’s denominator?”
- On the three fractions: “Which one stops them being ready?”
Reading the summary
The same two channels as the testing sequence. Saying no to the items with letters suggests the rule is being tied to numbers rather than to the denominators. Saying yes to the x and x² item suggests “close enough” has returned on new ground.