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Math · Stage 6

Fraction telescoping (splitting into differences)

A sum like 1/(1·2)+1/(2·3)+1/(3·4)+… looks endless — until you split every piece into a difference and watch almost all of it cancel, leaving only the two ends.

✦ Math📘 Stage 6 · S6⏱ ~14 min🧒 Ages 10–12★ Free to start
I can sum a telescoping fraction series by splitting terms.
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Kavi the owl for Math
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Kavi and the Collapsing Chain 🔭

Kavi met a scary-looking sum: 1/2 + 1/6 + 1/12 + 1/20 + 1/30. “Too many fractions!” 😵 Then Kavi spotted that each piece is a tiny gap: 1/6 = 1/2 − 1/3, 1/12 = 1/3 − 1/4, 1/20 = 1/4 − 1/5 … “If I write every term as a difference, the inside bits knock each other out like dominoes!” 🁢 Only the very first and the very last piece were left standing: 1 − 1/6 = 5/6. That domino trick has a name — telescoping.

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What your child will learn

1Which splits are correct?The whole trick rests on one move: 1/(n(n+1)) = 1/n − 1/(n+1). The little piece is the gap between two neighbouring unit fractions. Sort each line — is it a correct split or a wrong one?
21/(2·3) is the gap between 1/2 and 1/3Cut a bar into 2·3 = 6 equal parts. One part is 1/6. That same 1/6 is exactly how much bigger 1/2 is than 1/3 — the gap 1/2 − 1/3.
3Don't add — telescope!Tap each card. One way grinds through messy fractions; the other splits every term and lets the middle cancel.
4Wider gaps? Pull out a factorWhen the two numbers on the bottom are k apart (not 1 apart), the difference is k times too big — so you pull a 1/k out in front. The inside differences still telescope; the 1/k just scales the final answer.
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A peek inside

Find 1/(1·2) + 1/(2·3) + 1/(3·4).
split each term: (1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4). The 1/2 and the 1/3 each cancel → 1 − 1/4

Try this at home: Split into a difference: 1/(7·8) = __

Ready to play and learn?

Start the free interactive lesson, or print the handout for screen-free play.

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Part of Math · Stage 6 · S6 on KAVI Academy — a secular, whole-child learning platform for ages 5–12.