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Mastering Algebraic Competencies: Modeling, Rational Numbers, and Polynomial Operations

This comprehensive math module focuses on essential algebraic skills including data analysis (minimum, quartiles, median), understanding rational and irrational numbers, and mastering polynomial operations. Students will learn to add, subtract, multiply, and divide polynomials while adhering to exponent laws. Additionally, the module covers combinations and probability, along with long and synthetic division methods. It emphasizes practical exercises and warm-up problems to reinforce learning and build a solid foundation in algebra.

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Mastering Algebraic Competencies: Modeling, Rational Numbers, and Polynomial Operations

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  1. Common Core math 3 Unit 2-Modelling Algebraic Competency Ms. C. Taylor

  2. Warm-Up • Using the data given: 23, 45, 23, 45, 67, 54, 34, 89, 56, 76, 12, 76 • Give the Minimum, Lower Quartile, Median, Upper Quartile, Maximum, and standard deviation. (Use Calculator)

  3. Rational & Irrational Numbers • Rational Numbers repeated in a pattern, terminate, or can be expressed as a ratio of two integers. • Irrational Numbers don’t repeat in a pattern, never terminate, and cannot be expressed as a ratio of two integers. • Examples of Rational Numbers: 8/9, 2.66666666, 3.0 • Examples of Irrational Numbers: , √2

  4. Exponent Laws And the law about Fractional Exponents:

  5. Polynomial Operations • We can add, subtract, multiply, & divide polynomials. • Polynomials consist of 2 or more terms. • Examples: • When we add or subtract polynomials we DO NOT mess with the exponents! • Add ) + ( • Subtract (

  6. Polynomial Operations • When we multiply or divide polynomials then we have to use Exponent Laws to deal with the exponents! • What do you do when you multiply like bases of exponents? • What do you do when you divide like bases of exponents? • Examples: • Multiply • Multiply • Multiply • Divide • Divide

  7. Warm-Up • What is the probability of driving a Honda or a Nissan, or a Ford. The probability of a Honda is 0.05, probability of a Nissan is 0.35 and the probability of a Ford is 0.67.

  8. Factoring • ALWAYS look for a GCF (Greatest Common Factor) • If a = 1, then find the factors that multiply together to get “c” and add to get “b”. • If a ≠ 1, follow the steps for a = 1, then you have to group the first two and the last two. • Difference of Squares: • Sum & Difference of Cubes: S.O.A.P (Same Opposite Always Positive) • (

  9. Factoring Examples

  10. Factoring Examples

  11. Warm-Up • I want to choose a first, second, and third place winner. I have 25 students to choose from, how many combinations are possible? • Mr. Jones would like to choose 7 students for a history project. There are 89 students that he can choose from, how many combinations are possible?

  12. Long Division • Divide the same way as you would with regular integers.

  13. Synthetic Division • With this division method you only need the coefficients and the only thing to remember is if the divisor is NOT in (x –r) form then you have to use a “–r”.

  14. Warm-Up • Classify the following numbers as irrational or rational: • √2 • 2.4 •  • 5.99999999999999…….

  15. Simplifying Rational Expressions

  16. Warm-Up • Add the polynomials: • (2x2 + 5x – 2) + (-3x2 – 6x + 5) • (-x3 – 2x2 + 3x + 5) + (6x2 – 8x – 9)

  17. Add & Subtract Rational Expressions • More examples in class.

  18. Warm-Up • Add the fractions + . • Subtract the fractions - . • Multiply the fractions * . • NO CALCULATOR!

  19. Multiply & Divide Rational Expressions • More examples in class.

  20. Warm-Up • Simplify the rational expression and state the restrictions:

  21. Inverses of Functions • The relations formed when the independent variable(x) is exchanged with the dependent variable(y) in a given relation. • Given f(x)={(3,4),(1,-2),(5,-1),(0,2)} give the inverse. • To solve algebraically do the following 3 steps: • Set the function equal to y. • Swap the x and y variables. • Solve for y. • Solve the following for its’ inverse f(x)=x – 4 • Solve the following for its’ inverse f(x) =

  22. Warm-Up • Add the following rational expression: . • Subtract the following rational expression:

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