Mississippi Math A3.A.11

MathGrades 9–12Algebra (A) Analyze and Manipulate Functions

The Standard

Verify the Binomial Theorem by mathematical induction or by a combinatorial argument.

2025 Mississippi College- and Career-Readiness Standards for Mathematics · Use Polynomials Identities to Solve Problems

Teacher's field guide

What Students Need to Do: Students show why the binomial expansion works for any nonnegative whole-number exponent. They either count choices from the factors or prove the pattern with a base case and an induction step.
What Mastery Looks Like: A student can explain why the coefficient of a^(n-k)b^k counts the ways to choose b from k of n factors. They account for every value of k from 0 to n, not just a few examples.
Common Misconceptions: Checking n = 2 and n = 3 does not prove the pattern for every n. Students may count choices for a when k is the exponent of b. In induction, they may skip the step that combines coefficients using Pascal's rule.
How to Assess It: Exit ticket: In (a+b)^n, why is the coefficient of a^(n-k)b^k the number of ways to choose k of n factors? Explain why your answer accounts for every term.

Lesson moves

Give pairs four cards labeled a on one side and b on the other; list all 16 arrangements and group matching products.
Ask, 'Why does choosing two b's from five factors give the coefficient of a^3b^2 in (a+b)^5?'
Give groups scrambled proof cards for the base case and induction step; have them order the cards and mark where Pascal's rule applies.
List five optional pizza toppings; count pizzas with exactly two toppings, then find that count as the coefficient of x^2 in (1+x)^5.

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