COSMO · STUDY + TEACHING DECK

AIME Algebra

May 30, 2026

A question worth sharing

Telescoping Products

Factor each term before multiplying; a long chain can collapse to two endpoints.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use telescoping, factorization, endpoint to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

Evaluate the product (1 - $\frac{1}{4}$ )(1 - $\frac{1}{9}$ )(1 - $\frac{1}{16}$ ) ... (1 - $\frac{1}{100}$ ).

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Factor each difference of squares before multiplying.
Write neighboring factors on separate lines so cancellation is visible.
The first and last uncancelled factors determine the answer.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

telescoping

a chain whose interior terms cancel

factorization

rewriting one expression as a useful product

endpoint

a term that survives after the interior cancels

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

\prod_{k=m}^{n}\left(1-\frac{1}{k^2}\right)=\frac{(m-1)(n+1)}{mn},\qquad 2\le m\le n

The idea

Factor each term before multiplying; a long chain can collapse to two endpoints.

What to remember

Rewrite the chain until the interior has nothing left to contribute.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Evaluate the product from k = 2 to 6 of ( $k^{2}$ - 1)/ $k^{2}$ .

Predict each next step

Rewrite each factor: (k - 1)(k + 1)/ $k^{2}$ .
The product becomes ( $\frac{1}{2}$ )( $\frac{3}{2}$ )( $\frac{2}{3}$ )( $\frac{4}{3}$ )( $\frac{3}{4}$ )( $\frac{5}{4}$ )( $\frac{4}{5}$ )( $\frac{6}{5}$ )( $\frac{5}{6}$ )( $\frac{7}{6}$ ).
Cancel every interior numerator with its neighboring denominator.
The survivors are $\frac{1}{2}$ and $\frac{7}{6}$ , so the value is $\frac{7}{12}$ .

Which step needs a reason, not just a calculation?

See the structure

The middle disappears

\prod_{k=m}^{n}\left(1-\frac{1}{k^2}\right)=\frac{(m-1)(n+1)}{mn},\qquad 2\le m\le n

Separate the two ratio chains and keep their endpoints; every interior factor cancels.

Read the representation

A telescoping product is like a row of linked tiles: each interior numerator cancels a later denominator, leaving only the first and last edges.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

A telescoping form

Factor before expanding or approximating.
Line terms up so cancellation is paired.
Check only the surviving endpoints.

A brute-force form

Multiply decimals and lose exact structure.
Expand every factor unnecessarily.
Miss a denominator that should cancel.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

Multiply decimals and lose exact structure.
Expand every factor unnecessarily.
Miss a denominator that should cancel.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Make the cancellation ledger

Your shared task

Write a product or sum on a long strip of paper. Mark each interior term with the exact term that cancels it.

Explore

4 min - factor each term

Compare

6 min - draw cancellation pairs

Defend

5 min - state the endpoint rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

Cancellation is a proof step only when the factors are written exactly.
Endpoint bookkeeping is often faster and safer than evaluating every term.
The same structure appears in sums after partial fractions split one fraction into two.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Find the sum S = 1/(1*2) + 1/(2*3) + ... + 1/(8*9).

Predict each next step

Use 1/(k(k+1)) = 1/k - 1/(k+1).
The sum is (1 - $\frac{1}{2}$ ) + ( $\frac{1}{2}$ - $\frac{1}{3}$ ) + ... + ( $\frac{1}{8}$ - $\frac{1}{9}$ ).
Every interior fraction cancels with the next term.
Only 1 and - $\frac{1}{9}$ survive, so S = $\frac{8}{9}$ .

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A telescoping expression is a long chain with only its endpoints exposed.

Build the argument

For sums, write each term as a difference; for products, write each nonzero factor as a ratio.
Align the chain: interior summands cancel with opposite signs, or interior numerator factors cancel with denominators.
Only the boundary terms or factors remain; record the starting and ending indices before canceling.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

Evaluate the product from k = 3 to 12 of (1 - 1/ $k^{2}$ ).

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Telescoping appears in probability, finite differences, amortized algorithm analysis, and numerical formulas. The recurring question is: can a complicated total be rewritten as changes between consecutive states?

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

What makes a product telescope rather than merely simplify?

Question 2

How does partial fraction decomposition create a telescoping sum?

Question 3

Which endpoints survive in a finite chain, and why?

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

Rewrite the chain until the interior has nothing left to contribute.

On the practice set, mark the two endpoint terms before performing any cancellation.

A good question belongs to everyone in the room.