Illustrative Math Alignment: Grade 6 Unit 7

Topic

Polynomials

Description

Factored Quadratic: Expressing a polynomial of degree 2 as the product of one or two linear terms. Examples: x2 + 2x + 1 = (x + 1)2, x2 - 7x + 12 = (x - 3)(x - 4), x2 - 2x = x(x - 2). Highlights the use of factorial in binomial expansions and permutations.

Topic

Polynomials

Description

Factored Cubic: Expressing a polynomial of degree 3 as the product of linear and/or quadratic terms. Examples: x3 + 3x2 + 3x + 1 = (x + 1)3, x3 + 6x2 + 11x + 6 = (x + 1)(x + 2)(x + 3). Illustrates how polynomials can be decomposed into simpler factors for solving equations and simplifications.

Topic

Polynomials

Description

Factored Cubic: Expressing a polynomial of degree 3 as the product of linear and/or quadratic terms. Examples: x3 + 3x2 + 3x + 1 = (x + 1)3, x3 + 6x2 + 11x + 6 = (x + 1)(x + 2)(x + 3). Illustrates how polynomials can be decomposed into simpler factors for solving equations and simplifications.

Topic

Prime and Composite Numbers

Description

Factor: A number or expression that divides another number or expression evenly, with no remainder. For example, factors of 24 include 1, 2, 3, 4, 6, 8, 12, and 24. Algebraically, x2 - 1 has factors (x + 1) and (x - 1). The concept of factors is fundamental to breaking down numbers into primes and understanding their composition.

Topic

Fractions

Description

The Greatest Common Factor (GCF) is the largest number that divides two numbers evenly, used to simplify fractions. For example, the GCF of 24 and 36 is 12, enabling simplification of 24/36 to 2/3. This ensures fractions are in their simplest form. It supports simplifying fractions and ensures efficient fraction operations.

Topic

Prime and Composite Numbers

Description

Factor Tree: A visualization tool to break a number into its factors, which can be used to create a prime factorization. For example, 24 = 23 * 3. This tool helps students visually decompose numbers into their prime components.

Topic

Prime and Composite Numbers

Description

Common Factors: The factors that two or more numbers have in common. For example, 15 and 24 have the common factors 1 and 3. Similarly, 9 and 36 have the common factors 1, 3, and 9. The concept of common factors extends the understanding of divisibility and primes, allowing learners to identify shared properties between numbers, which is foundational for further topics like greatest common factor.

Topic

Prime and Composite Numbers

Description

Greatest Common Factor (GCF): Of the common factors of two numbers, the one with the greatest value. For example, the GCF of 24 and 36 is 12. The GCF is crucial for comparing numbers and simplifying fractions.

Topic

Prime and Composite Numbers

Description

Prime Factors: The factors of a number that are also prime numbers. For example, the prime factors of 24 are 2 and 3. This concept links the idea of primes to factorization, essential for understanding number composition.

Topic

Quadratics

Description

Factoring: Writing a quadratic expression as the product of linear factors. This method is a fundamental algebraic process for solving quadratics and simplifying expressions. A core algebraic technique for quadratic problem-solving.

Relevance to Topic: This video series provides a deep dive into the topic of Quadratics. It explores mathematical concepts and demonstrates real-world applications to help solidify understanding.

Topic

Prime and Composite Numbers

Description

Prime Factorization: Expressing a number as the product of its prime factors. For example, 24 = 23 * 3. Prime factorization is critical for many areas of math, including solving problems in algebra and number theory.

Topic

Quadratics

Description

Quadratic Function in Factored Form: A quadratic function written so as to indicate the root(s) of the quadratic. Not all quadratics can be written this way. This connects directly to finding roots and understanding intercepts graphically. Relates to factoring methods and their graphical implications in identifying roots.

Topic

Polynomials

Description

Factor Theorem: For a polynomial P(x), if P(a) = 0 for some number a, then (x - a) is a linear factor of P(x). Example: P(x) = x2 - 7x + 12, P(3) = 0 implies (x - 3) is a factor of P(x). Provides a systematic approach to expand binomials, foundational in algebra.

Topic

Polynomials

Description

Factor Theorem: For a polynomial P(x), if P(a) = 0 for some number a, then (x - a) is a linear factor of P(x). Examples: P(x) = x2 - 7x + 12, P(3) = 0 implies (x - 3) is a factor of P(x). Provides a systematic approach to expand binomials, foundational in algebra.

Topic

Prime and Composite Numbers

Description

Proper Factor: For any number n, the proper factors are the factors of n, not including n itself. For example, the proper factors of 12 are 1, 2, 3, 4, and 6. This concept connects to divisors and emphasizes the exclusion of the number itself.

Topic

Polynomials

Description

Binomial Factor: A polynomial P(x), of degree n, is said to have a binomial factor (x - a) if it can be written as the product of this factor and another polynomial of degree n - 1. Formula: P(x) = (x - a)(an-1 * x(n-1) + ... + a0). Provides a method for expanding binomials systematically, linking combinatorics and algebra.

Topic

Polynomials

Description

Binomial Factor: A polynomial P(x), of degree n, is said to have a binomial factor (x - a) if it can be written as the product of this factor and another polynomial of degree n - 1. Formula: P(x) = (x - a)(an-1 * x(n-1) + ... + a0). Provides a method for expanding binomials systematically, linking combinatorics and algebra.

Topic

Rationals and Radicals

Description

Factoring a radical expression involves extracting whole number factors. For example, √(36x2) = 6x or solving x = (-6 ± √(36 - 16)) / 4, which simplifies to -3 ± √5 / 2. Factoring radicals is a key step in simplifying expressions and solving equations with square roots.

Topic

Rationals and Radicals

Description

Factoring a radical expression involves extracting whole number factors. For example, √36x2 = 6x or solving x = (-6 ± √(36 - 16)) / 4, which simplifies to -3 ± √5 / 2. Factoring radicals is a key step in simplifying expressions and solving equations with square roots.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Polynomials

Description

Factorial: For a positive integer n, the product of all integers in the range 1 ≤ n. Written n! and pronounced "n factorial". Examples: 3! = 3 × 2 × 1 = 6, 5! = 5 × 4 × 3 × 2 × 1 = 120. Links the concept of polynomial roots to their factors, aiding in factorization.

Topic

Polynomials

Description

Factorial: For a positive integer n, the product of all integers in the range 1 ≤ n. Written n! and pronounced 'n factorial'. Examples: 3! = 3 × 2 × 1 = 6, 5! = 5 × 4 × 3 × 2 × 1 = 120. Links the concept of polynomial roots to their factors, aiding in factorization.

This is the transcript that goes with the video segment entitled Video Tutorial: Quadratics: Solving Quadratic Equations in Standard Form by Factoring.

To see the complete collection of video transcriptsy, click on this link.

This is the transcript that goes with the video segment entitled Video Tutorial: Quadratics: Solving Quadratic Equations in Standard Form by Factoring.

To see the complete collection of video transcriptsy, click on this link.

This is the transcript for the TI-Nspire Mini-Tutorial entitled, Testing If One Polynomial Is a Factor of Another.

To see the complete collection of video transcriptsy, click on this link.

Topic

Polynomials

Description

This segment focuses on factoring techniques for polynomials, including using patterns like the difference of squares and sum-product relationships. Applications include solving equations and finding polynomial roots. Key vocabulary includes factoring, roots, quadratic trinomial, and binomial product.

This video is essential to understand the topic of Polynomials. It provides insights into mathematical concepts and their applications, with a focus on building foundational understanding.

Topic

Polynomials

Description

This segment focuses on factoring techniques for polynomials, including using patterns like the difference of squares and sum-product relationships. Applications include solving equations and finding polynomial roots. Key vocabulary includes factoring, roots, quadratic trinomial, and binomial product.

This video is essential to understand the topic of Polynomials. It provides insights into mathematical concepts and their applications, with a focus on building foundational understanding.

Topic

Polynomials

Description

This segment focuses on factoring techniques for polynomials, including using patterns like the difference of squares and sum-product relationships. Applications include solving equations and finding polynomial roots. Key vocabulary includes factoring, roots, quadratic trinomial, and binomial product.

This video is essential to understand the topic of Polynomials. It provides insights into mathematical concepts and their applications, with a focus on building foundational understanding.

Topic

Polynomials

Description

This video discusses factorials and their applications in counting, permutations, and combinations. It explains the growth of factorials, illustrating how they are used for calculating arrangements and combinations. Examples cover permutations of objects, arranging items, and deriving combinations for groups. Key terms include factorials, permutations, combinations, and the fundamental counting principle. Applications extend to password generation and probability calculations. The segment bridges counting techniques with practical problem-solving.

Topic

Polynomials

Description

This video discusses factorials and their applications in counting, permutations, and combinations. It explains the growth of factorials, illustrating how they are used for calculating arrangements and combinations. Examples cover permutations of objects, arranging items, and deriving combinations for groups. Key terms include factorials, permutations, combinations, and the fundamental counting principle. Applications extend to password generation and probability calculations. The segment bridges counting techniques with practical problem-solving.

This worksheet covers the topic of factors and multiples.

Related Resources

Worksheet Library

This worksheet covers the topic of factors and multiples.

Related Resources

Worksheet Library

In this worksheet students practice the skill of simplifying fractions by finding the GCF of the numerator and denominator. This is a companion to a tutorial on the same topic.

Related Resources

Worksheet Library

This is part of a collection of math worksheets on the use of the TI-Nspire graphing calculator. Each worksheet supports a companion TI-Nspire Mini-Tutorial video. It provides all the keystrokes for the activity.

Related Resources

Worksheet Library

This is part of a collection of math worksheets on the use of the TI-Nspire graphing calculator. Each worksheet supports a companion TI-Nspire Mini-Tutorial video. It provides all the keystrokes for the activity.

Related Resources

Worksheet Library