Add and subtract fractions

What this routine teaches

Every atom of adding and subtracting fractions, run together, one click per line of working. It assumes students have met each atom on its own first:

1. Are these fractions ready to be added or subtracted? Each denominator lights up in turn, then “Ready” or “Not ready” is written. The denominators stay orange as the record of the decision.
2. Write the equivalent fractions, once for each fraction (not ready only). Each list is its own click.
3. Pick the pair that’s ready to be added or subtracted (not ready only). Each denominator in both lists lights up in turn, smallest first; any larger ones are crossed out; then the pair is written with the sign from the question.
4. Recall the rule for adding and subtracting fractions. Ask students to say it before you click. It appears instantly, in the approved wording.
5. Add or subtract fractions with the same denominator. The denominators merge into one; the numerators drop onto the bar and become one number.

The button never says “Show answer”. The number of steps depends on the decision (three when the fractions are ready, six when they aren’t), so a label that counted down would give the decision away.

The two examples

Example 1 — 7/9 + 1/9 = 8/9. Ready: three clicks (decide, the rule, the answer).

Example 2 — 7/9 + 1/6 = 17/18. Only the second denominator changes, from 9 to 6. On Next the working fades, 8/9 dims and is labelled, the page holds still, and only then does the 9 change to a 6. Ask: “Will the answer change?” Now the fractions aren’t ready, so the lists and the pick come in: 18 and 36 are in both lists, and the smallest is 18. Then the rule, and 14 + 3 over 18.

Running the sequence

• Ask for each line on whiteboards before you click it. Nothing moves until you click.
• Replay plays the last step again (or, before step 1, the change to the question).
• Why? gives the reason for the step on screen and lights up what it refers to.
• Answers are left exactly as the rule gives them. Simplifying and mixed numbers are separate routines.

About the testing sequence

Eight questions in a fixed order. Students write every line on their whiteboards. There is no scoring; the whiteboards are the data. Early on, check boards after each line; later, ask for the whole working and click through to compare.

What each item catches

• 1/2 + 1/3 = 5/6. Easy tables. The tops match, so it can look “ready already”; only the denominators matter.
• 1/2 − 1/3 = 1/6. Only the sign changes, so subtraction arrives as one change. The decision, the lists and the denominator are all the same; only the numerator of the answer changes.
• 3/8 + 1/8 = 4/8. Ready. The answer could be simplified; it is left as the rule gives it.
• 3/8 + 1/4 = 5/8. One change, and now not ready. 3/8 is already at the start of its list, so it is copied as it is.
• 3/8 + 3/4 = 9/8. One change. The numerator is bigger than the denominator, and that’s fine: the answer stays as 9/8.
• 3/5 − 2/5 = 1/5. Ready subtraction: keep the 5, subtract the numerators.
• 1/3 − 1/9 = 2/9. 1/9 is copied as it is. Keep the question’s order: 3/9 − 1/9, not 1/9 − 3/9.
• 1/300 − 1/900 = 2/900. Looks hard, but it is the item before with every denominator 100 times bigger. Counting in 300s is no harder than counting in 3s. Students who do it feel smart.

Reading the boards

• A denominator of 9 + 9 or 18 on a ready question: they added the denominators. Go back to the rule.
• A list is wrong: go back to writing the equivalent fractions.
• The lists are right but the pair is wrong: go back to picking the pair.
• The working is right but they simplified: fine as maths, but the routine stops at the rule’s answer.

About the expansion sequence

Seven questions that take the routine further: first to three fractions, then into algebra, which is the heart of this card. Run them in order. After the first item, every new thing arrives as one change from the question before, so students can see what changed and predict what it will do. Before each click, ask: “What’s changed? What will that do to the working?” The last question is the hardest on the page.

1. 1/6 + 4/9 − 1/3 = 5/18. Three fractions, so three lists. The pick looks for the smallest denominator in all three lists: only 18 is. Work along the numerators from left to right: 3 + 8 − 6.
2. 1/6 + 4/15 − 1/3 = 3/30. One change: the 9 becomes 15. Now no denominator is in all three lists after six fractions, so there is an extra step: keep counting the list with the smallest denominator until one is. The 3s list carries on to 30. Ask what will happen before you click.
3. 3/5 + 1/5 = 4/5. Familiar, so that the algebra can arrive as one change.
4. 3/x + 1/x = 4/x. One change: the 5s become x. The denominators are still the same, so nothing else changes.
5. 3/x + 1/2x = 7/2x. One change: the second denominator becomes 2x, so the fractions aren’t ready. Count in x along the bottom: x, 2x, 3x …
6. 3/(x + 2) + 1/(2x + 4) = 7/(2x + 4). One change: x becomes x + 2 everywhere. Count in (x + 2): x + 2, 2x + 4, 3x + 6 … Everything works exactly as before, and the numerator is still 7.
7. 3a/(x + 2) + 1/(2x + 4) = (6a + 1)/(2x + 4). One change: the 3 becomes 3a. Count in 3a along the top. 6a and 1 can’t be put together, so the numerator stays as 6a + 1.

A good question for the class

“Could we add 1/x + 1/(x + 2) this way?” No: count in x and count in (x + 2), and the lists never meet. That’s why the denominators here are an expression and a multiple of it. Fractions like 1/x + 1/(x + 2) need a different method, which comes later.