Unlocking the Secrets of Factoring Cubes

Algebra 2 Grades High School 6:44 Video
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Lesson Description

This lesson explores the techniques for factoring the sum and difference of cubes, building upon previous knowledge of factoring differences of squares. It covers identifying cubic terms, applying the factoring formulas, and simplifying expressions.

Video Resource

Factoring Cubes

Kevinmathscience

Duration: 6:44
Watch on YouTube

Key Concepts

  • Identifying perfect cubes
  • Sum of cubes factoring formula: a³ + b³ = (a + b)(a² - ab + b²)
  • Difference of cubes factoring formula: a³ - b³ = (a - b)(a² + ab + b²)
  • Factoring out common factors before applying cube factoring formulas

Learning Objectives

  • Students will be able to identify expressions that can be factored as the sum or difference of cubes.
  • Students will be able to apply the appropriate factoring formula (sum or difference of cubes) to factor expressions.
  • Students will be able to simplify factored expressions.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of factoring and the difference of squares. Introduce the idea of factoring cubes and explain what a 'cube' means in mathematical terms. Briefly mention the real-world applications of factoring, such as in engineering and physics.
  • Video Presentation (10 mins)
    Play the "Factoring Cubes" video by Kevinmathscience. Instruct students to take notes on the formulas and steps involved in factoring the sum and difference of cubes.
  • Formula and Example Review (10 mins)
    Write the sum and difference of cubes formulas on the board. Work through the examples from the video, emphasizing each step. Encourage students to ask questions and clarify any confusion.
  • Guided Practice (15 mins)
    Provide students with practice problems, starting with simpler examples and gradually increasing the complexity. Work through the first few problems as a class, then have students work independently or in pairs. Provide assistance as needed.
  • Independent Practice (10 mins)
    Assign a set of practice problems for students to complete independently. This can be done in class or as homework. Encourage students to check their answers with each other and discuss any discrepancies.
  • Wrap-up and Assessment (5 mins)
    Summarize the key concepts covered in the lesson. Answer any remaining questions. Administer the multiple choice and fill in the blank quizzes to assess student understanding.

Interactive Exercises

  • Cube Identifier
    Present students with a list of numbers and algebraic terms. Have them identify which are perfect cubes.
  • Factoring Challenge
    Divide students into small groups and give each group a challenging factoring problem involving the sum or difference of cubes. Have them work together to solve the problem and present their solution to the class.

Discussion Questions

  • How does factoring the sum/difference of cubes relate to factoring the difference of squares?
  • What are some common mistakes students make when factoring cubes, and how can we avoid them?
  • Can all polynomial expressions be factored? Why or why not?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Pattern recognition
  • Critical thinking

Multiple Choice Questions

Question 1:

Which of the following is the correct formula for factoring the sum of cubes, a³ + b³?

Correct Answer: (a + b)(a² - ab + b²)

Question 2:

Which of the following is the correct formula for factoring the difference of cubes, a³ - b³?

Correct Answer: (a - b)(a² + ab + b²)

Question 3:

What is the first step in factoring an expression like 8x³ - 27?

Correct Answer: Find the cube root of each term.

Question 4:

Factor the expression: x³ + 8

Correct Answer: (x + 2)(x² - 2x + 4)

Question 5:

Factor the expression: 27y³ - 1

Correct Answer: (3y - 1)(9y² + 3y + 1)

Question 6:

What is the result of squaring the first term in the small bracket when factoring the sum or difference of cubes?

Correct Answer: The first term in the large bracket

Question 7:

What sign do you use to combine the numbers in the small bracket?

Correct Answer: Dependant on original question

Question 8:

What sign is always used as the last sign in the large bracket?

Correct Answer: Always +

Question 9:

What is the middle term of the large bracket equal to?

Correct Answer: The first term multiplied by the second term

Question 10:

Which of the following represents a cube number?

Correct Answer: 8

Fill in the Blank Questions

Question 1:

The formula for the sum of cubes is a³ + b³ = (a + b)(a² - __ + b²).

Correct Answer: ab

Question 2:

The formula for the difference of cubes is a³ - b³ = (a - b)(a² + __ + b²).

Correct Answer: ab

Question 3:

Before factoring cubes, always check for a common __.

Correct Answer: factor

Question 4:

When factoring the difference of cubes, the binomial factor (a - b) has a __ sign.

Correct Answer: negative

Question 5:

In the factored form of a sum or difference of cubes, the trinomial factor (a² ± ab + b²) cannot be factored further using __ numbers.

Correct Answer: real

Question 6:

A cube is when a number is multiplied by itself _____ times.

Correct Answer: three

Question 7:

The small bracket in the factoring formula always contains _____ terms.

Correct Answer: two

Question 8:

The large bracket in the factoring formula always contains _____ terms.

Correct Answer: three

Question 9:

The small bracket and large bracket are joined together through ______.

Correct Answer: multiplication

Question 10:

x³ is an example of a _____.

Correct Answer: cube

Educational Standards

A2.A.2.1:Factor polynomial expressions including, but not limited to, trinomials, differences of squares, sum and difference of cubes, and factoring by grouping, using a variety of tools and strategies. A2.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies.

Teaching Materials

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