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Building Content Knowledge for Common Core Algebra I

Join us for an engaging Algebra I workshop where we delve into essential mathematical concepts. Explore new concepts in Algebra I, focusing on arithmetic and geometric sequences, including explicit and recursive formulas, and the complete square method for solving quadratics. Enjoy a collaborative environment featuring presentations and lesson development. By the end of the day, you will master how to derive the quadratic formula and apply various sequence equations to real-world problems. Perfect for educators and students alike!

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Building Content Knowledge for Common Core Algebra I

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  1. Building Content Knowledge for Common Core Algebra I Kimberly Louttit & Tricia Profic

  2. AGENDA • Explore Concepts New to Algebra I (8:30 – 11:45) • Arithmetic and Geometric Sequences • Complete the Square & Derive Quadratic Formula • Residuals • Lunch (11:45 – 12:30) • Modules 3-5 Analysis (12:30 – 3:00) • Lesson Development • Presentations and Overview of Designed Lessons

  3. SEQUENCES • Notation • f(n) • A(n) • an • Explicit vs. Recursive • Explicit Formula: a formula that allows a direct computation of any term for a sequence • Recursive Formula: a formula that requires the computation of all previous terms in order to find the value of the specific term for a sequence • Arithmetic vs. Geometric • Common difference • Common ratio

  4. ARITHMETIC SEQUENCE • Recursive Formula • List the first 5 terms of the sequence: • an = an – 1 – 5, where a1 = 12 and n ≥ 1 • 12, 7, 2, -3, -8 • Find a6 • a6 = -13 • Find a100 • a100 = -483 • How else could a recursive formula look for this question? • an + 1= an– 5, where a1 = 12 and n ≥ 1 • f(n) = f(n – 1) – 5, where f(1) = 12 and n ≥ 1 • f(n + 1) = f(n) – 5, where f(1) = 12 and n ≥ 1

  5. ARITHMETIC SEQUENCE • Explicit Formula • an = a1 + (n – 1)d, where d is the common difference • Write an explicit formula for the given recursive formula an= an – 1 – 5, where a1 = 12 and n ≥ 1 • d = -5 • an = 12 + (n – 1)(-5) • an = 12 – 5n + 5 • an = 17 – 5n

  6. GEOMTRIC SEQUENCE • Recursive Formula • List the first 5 terms of the sequence: • f(n + 1) = 10f(n), where f(1) = 4 and n ≥ 1 • 4, 40, 400, 4,000, 40,000 • Find a6 • a6 = 400,000 • How else could a recursive formula look for this question? • an = 10an – 1 , where a1 = 4 and n ≥ 1 • an + 1 = 10an, where a1 = 4and n ≥ 1 • f(n) = 10f(n – 1), where f(1) = 4 and n ≥ 1

  7. GEOMTRIC SEQUENCE • Explicit Formula • an = a1rn – 1 , where r is the common ratio • Write an explicit formula for the given recursive formula f(n + 1) = 10f(n), where f(1) = 4 and n ≥ 1 • r = 10 • an= 4(10n – 1)

  8. SEQUENCES • Determine the recursive and explicit formulas for the following sequences: • 14, 21, 28, 35 • 8, 2, , , • Word Problem: • The local football team won the championship several years ago, and since then, ticket prices have been increasing $20 per year. The year they won the championship, tickets were $50. Write a recursive formula and an explicit formula for a sequence that will model ticket prices. Is the sequence arithmetic or geometric?

  9. SEQUENCES • Consider the sequence 13, 24, 35. If the nth term is 299, find the value of n. • f(n) = 13 + (n – 1)(11) • f(n) = 13 + 11n – 11 • f(n) = 2 + 11n • 299 = 2 + 11n, n = 27 • If 2, x, y, -54 forms a geometric sequence, find the values of x and y. • an = 2rn – 1 • a4= 2r 4 – 1 • -54 = 2r3 , r = -3 • x = 2(-3) = -6 and y = 2(-3)(-3) = 18

  10. COMPLETE THE SQUARE • To Solve a Quadratic Equation by Completing the Square • Move the constant to the opposite side (if necessary) • Divide through by the coefficient of x2 • Take half of the coefficient of the x term and square it. Add this to both sides of the equation. • Convert the one side of the equation to a squared binomial (because it should now be a perfect square trinomial) and simplify the other side of the equation to a single constant. • Square root both sides, remembering the in front of the constant that was square rooted. • Solve for x.

  11. COMPLETE THE SQUARE • Solve x2 + 6x – 12 = 0 by completing the square.

  12. COMPLETE THE SQUARE • Solve x2 – 6x = 2 by completing the square.

  13. COMPLETE THE SQUARE • -4x2 = 24x + 11 • x2 – 3x – 8 = 0

  14. DERIVE THE QUADRATIC FORMULA

  15. RESIDUALS • Refer to Hand Out ☺

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