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Algebra II

Algebra II. Lessons 1-5. LESSON 1. Polygons- Triangles Transversals Proportional Segments. Polygon. A polygon is a closed geometric figure located in one plane (flat surface) whose sides are: Straight line segments Intersect only at endpoints. Convex vs. concave polygons.

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Algebra II

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  1. Algebra II Lessons 1-5

  2. LESSON 1 Polygons- Triangles Transversals Proportional Segments

  3. Polygon • A polygon is a closed geometric figure located in one plane (flat surface) whose sides are: • Straight line segments • Intersect only at endpoints

  4. Convex vs. concave polygons Convex polygons can have all interior points connected without ever going outside the shape. In concave polygons it is possible to have to go outside the shape to connect interior points.

  5. NAMES OF POLYGONS

  6. REGULAR POLYGONS • A regular polygon is defined as a polygon which has all sides congruent and all angles congruent. • SQUARE • EQUILATERAL TRIANGLE

  7. A DIAGONAL of any polygon is defined as a line through the interior of the polygon which connects two of the vertices. How many other diagonals does this pentagon have?

  8. TRIANGLES By angle:

  9. By sides:

  10. Angle Sum Theory For a convex polygon, the sum of the interior angles equals the following: Where n-equals the number of sides of the polygon Sum of interior angles in degrees = (180) X (n-2)

  11. EQUILATERAL TRIANGLE All the sides are congruent Therefore All the angles are congruent (60 degrees each)

  12. ISOSCELES TRIANGLE In a triangle – if two sides are congruent then the angles opposite those side must also be congruent. 20 20 50 50

  13. Solve for the value of x and y.

  14. Solve for the value of x and y.

  15. TRANSVERSALS AND PARALLEL LINES If these lines are parallel they will never intersect. This is a transversal through the parallel lines. Let’s see what happens when there are parallel lines cut by a transversal.

  16. PROPORTIONAL SEGMENTS 3 2 9/2 3 When 3 or more parallel lines are cut by two transversals, the lengths of corresponding segments of the transversals are proportional.

  17. ALGEBRA II Lesson 2 Negative Exponents Product and Power Theorems for Exponents Circle Relationships

  18. ZERO EXPONENT THEOREM • Any non-zero number (or variable) taken to the zero power is equal to - - - 1

  19. NEGATIVE EXPONENTS • Definition of • If n is any real number and x is any real number that is NOT zero:

  20. Negative Exponents (cont) • When we write an exponential expression in reciprocalform, the sign of the exponent must be changed. • If exponent is negative, it is changed to positive in the reciprocal form • If exponent if positive, it is changed to negative in the reciprocal form.

  21. Product theorem for Exponents • If m and n and x are real numbers and

  22. Why???? Wow, that was easy!

  23. Power Theorem for Exponents • If m and n and x are real numbers. EXTENSION OF POWER THEOREM:

  24. Now - PROVE IT!

  25. Wow, that was easy! MORE

  26. ALGEBRA II Lesson 3 Evaluation of Expressions Adding Like Terms

  27. EXPRESSIONS • Numerical expressions: an arrangement of numerals and symbols • Algebraic expressions: can contain variables that represent unknown numbers.

  28. Evaluating Expressions EVALUATE: Let x = -2 and y = -4

  29. Let a = -2 and b = 4

  30. Adding Like Terms LIKE terms: terms whose literal components represent the same number regardless of the numbers used to replace the variables. What???? Simply put: terms whose variables are alike.

  31. 3xyz -2zyx LIKE TERMS SIMPLIFY by adding LIKE terms: 3xy – 2x + 4 – 6yx + 3x

  32. ALGEBRA II Lesson 4 Distributive Property Solutions of Equations Change Sides – Change Signs

  33. DISTRIBUTIVE PROPERTY • Definition: • a(b+c) = ab + ac Demonstrate.

  34. Equations • Demonstration.

  35. Solutions of Equations • From Algebra I the five steps to solving an equation are • Eliminate the parenthesis. • Add like terms on both sides. • Eliminate the variable on one side or the other. (I don’t like to deal with negatives) • Eliminate the constant term on the side with the variable. • Eliminate the coefficient of the variable.

  36. 12 – (2x + 5) = -2 + (x – 3)

  37. Change Sides – Change Signs • Transposition: Change sides – change signs. Change sides – change signs

  38. Use transposition to solve for p: P – 3x + 4 = 7y

  39. Solve for y:

  40. ALGEBRA II Lesson 5 Word Problems Fractional Parts of a number

  41. Example 5.1: Twice a number is decreased by 7, and this quantity is multiplied by 3. The result is 9 less than 10 times the number. What is the number? Highlight the IMPORTANT stuff. Twice decreased by quantity multiplied result less than 10 times Define the variable (unknown). N= number Write in algebraic terms 2n -7 3( ) = -9 10( ) Write the equation and solve: 3(2n-7) = 10n-9

  42. Example 5.2: The number of ducks on the pond was doubled when the new flock landed. Then, 7 more ducks came. The resulting number was 13 less than 3 times the original number. How many ducks were there to begin with? Highlight: Define: Algebraic terms: Equation & Solve:

  43. Fractional Parts of a Number Finding a part (fraction) of a number means to multiply the number times the fraction. (A lot like percent).

  44. Example 5.4: One fifth of the clowns had red noses. If 30 clowns had red noses, how many clowns were there in all? Highlight: Define: Algebraic terms: Equation & Solve:

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