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Multiplying Polynomials: The Area Model & FOIL

Students learn to multiply binomials using an area model — a grid that organizes every product — connecting the visual method to the FOIL shortcut and building toward factoring and quadratics.

Grade 9Polynomials50 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Multiply two binomials.
  • ✓Use an area model.
  • ✓Connect it to FOIL.
  • ✓Combine like terms.
Essential Question

How do you multiply (x + 2)(x + 3)? A simple grid turns a confusing expression into four easy pieces.

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Model · Interactive Tool

The Area Model

Project this and change the numbers. The grid shows every product — combine the like terms to get the expanded polynomial.

✖️ Multiply the binomials — fill the area gridTry it
a: 2
b: 3

The Lesson · Gradual Release (I Do · We Do · You Do)

50 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — Four Pieces

5 min

(x + 2)(x + 3) — how do you multiply this out?

👩‍🏫 Teacher Moves

  • Show a binomial product.
  • Ask how to expand it.
  • Set the goal: multiplying polynomials.

🎒 Student Actions

  • Read the product.
  • Wonder how.
  • Predict the method.
2

I Do — The Area Grid

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Set up a 2×2 area grid.
  • Fill in each product.
  • Combine like terms.

🎒 Student Actions

  • Set up the grid.
  • Fill the cells.
  • Combine terms.
3

We Do — Model Together

13 min

Class uses the Area Model.

👩‍🏫 Teacher Moves

  • Change the numbers; read the grid.
  • Connect grid cells to FOIL.
  • Write the expanded form.

🎒 Student Actions

  • Set the values.
  • Read the grid.
  • Write the answer.
4

You Do — On Your Own

12 min

Students multiply binomials.

👩‍🏫 Teacher Moves

  • Assign binomial products.
  • Use the area model or FOIL.
  • Combine like terms.

🎒 Student Actions

  • Multiply binomials.
  • Show the work.
  • Combine terms.
5

Close — Close

5 min

One product.

👩‍🏫 Teacher Moves

  • Give a binomial product.
  • Expand it.
  • Hand out the exit ticket.

🎒 Student Actions

  • Expand it.
  • Show the method.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School: Algebra I).

CCSS
A-APR.1

Add, subtract, and multiply polynomials.

CCSS
A-SSE.2

Use the structure of an expression to identify ways to rewrite it.

CCSS
A-SSE.1

Interpret expressions that represent a quantity in terms of its context.

CCSS
A-APR.A

Perform arithmetic operations on polynomials.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Label the area grid clearly.
  • Sentence frame: “First, Outer, Inner, Last.”
  • Use color for like terms.

Support & Access

IEP / 504
  • Use the area grid every time.
  • Start with (x + a)(x + b).
  • Combine terms with support.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Multiply (ax + b)(cx + d).
  • Square a binomial.
  • Connect back to factoring.
Materials

What to gather

  • 📽️Projector / board
  • 📓Math notebooks
  • 💻The Area Model
  • 📐Grid paper
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

polynomialan expression of terms with variables and exponentsbinomiala polynomial with two termsterma part of an expressionarea modela grid used to multiplyFOILFirst, Outer, Inner, Lastlike termsterms with the same variable partcoefficientthe number multiplying a variableexpandto multiply out
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
Multiply: (x + 2)(x + 3).
x² + 5x + 6.
QUESTION 2
In the area model for (x + 4)(x + 1), what goes in the top-left cell?
x² (from x times x).
QUESTION 3
What does FOIL stand for?
First, Outer, Inner, Last — the four products of two binomials.

Going deeper? Square a binomial.

Have students expand (x + a)² with the area model and discover the pattern x² + 2ax + a². A printable polynomial sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Multiplying PolynomialsFlip
Card 1
Term
Tap to flip →
Meaning
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Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Multiplying PolynomialsQuiz
Score: 0
1 / 6
Question 1
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Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Multiplying PolynomialsPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: area model, binomial, coefficient, expand, FOIL, like terms, polynomial, term
  1. a grid used to multiply
  2. the number multiplying a variable
  3. a polynomial with two terms
  4. terms with the same variable part
  5. an expression of terms with variables and exponents
  6. to multiply out
  7. First, Outer, Inner, Last
  8. a part of an expression

Part B · Show what you learned

  1. Multiply: (x + 2)(x + 3).
  2. In the area model for (x + 4)(x + 1), what goes in the top-left cell?
  3. What does FOIL stand for?
Answer key — Part A: 1) area model · 2) coefficient · 3) binomial · 4) like terms · 5) polynomial · 6) expand · 7) FOIL · 8) term
Part B: 1) x² + 5x + 6. 2) x² (from x times x). 3) First, Outer, Inner, Last — the four products of two binomials.