Building Content Knowledge for Common Core Algebra I

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 ☺