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An Afternoon with Algebra

An Afternoon with Algebra. Claudia Hart, Northwest Catholic H.S. chart@nwcath.org Kathleen Reilly, St. James School koneill2@cox.net. Outline of Afternoons with Algebra. Session 1: Functions and equations Session 2: Graphing; prep for midterm exam

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An Afternoon with Algebra

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  1. An Afternoon with Algebra Claudia Hart, Northwest Catholic H.S. chart@nwcath.org Kathleen Reilly, St. James School koneill2@cox.net

  2. Outline of Afternoons with Algebra Session 1: Functions and equations Session 2: Graphing; prep for midterm exam Session 3: Systems, Exponents, Polynomials, and Quadratics Session 4: Quadratic Formula, Radicals, Distance/Midpoint, Rational Expression, Rational Equations; Prep for final exam All sessions include explanations, including possible pitfalls and common mistakes; take-away worksheets; discussion and sharing; and snacks!

  3. Agenda for 4/25/12 Discussion of current topics (from Session 3) Quadratic Formula Radical Expressions and equations Distance and Midpoint Formulas Rational Expressions and equations 6.  Preparing for the End-of-course exam 7. Discussion and sharing: Summer math maintenance. 8. Conclusion

  4. Discussion of current topics Quadratics - multiplying and factoring Systems  Study Guide End-of-course Exam - available to principals

  5. Quadratic Formula Review solving by factoring:  x2 - 6x - 16 = 0 Show x2 - 16 = 0 and x2 = 16 Possible Pitfall: Forgetting ± Show x2 - 6x - 16 = 0 with completing the square If time, do more of Completing the Square Complete the square on ax2 - bx + c = 0 to develop the quadratic formula

  6. Quadratic Formula (continued) ax2 + bx + c = 0

  7. Quadratic Formula continued • Use quadratic formula to solve x2 - 6x - 16 = 0. • Remember to solve for zero and put the left side in descending order. • You can solve some that students have already solved by factoring. • Then try x2 - 6x - 10 = 0, using a calculator for square root of 76.  If possible, show that solution (x-intercepts) on a graphing calculator. • Show one example with a negative discriminant, such as x2+ 4x + 5 = 0 and tell students this is an Algebra II topic.  You can also look at the graph to see it never crosses the x-axis, so the roots are imaginary.

  8. Quadratic Formula (continued) Quadratic Rap on youtube.com (words provided) - to the tune of "Do your ears hang low?" and "Do your chain hang low?" http://www.youtube.com/watch?v=dBtUetKJzOU You can filter it through safeshare.tv to avoid comments, video suggestions, etc. http://www.safeshare.tv/w/VqEFUKdATH The video also has a verse about adding rational expressions.

  9. Radical Expressions Have students write square numbers (perfect squares) as far as they can.  The symbol       indicates the principal root and is always positive. Start with Then:   Check on a calculator to see they are the same. Then: Possible Pitfall: Writing 3 * 4 for 12 instead of 4 * 3

  10. Radical Expressions continued • Adding/subtracting:  review with 3x2 + 8x2 and 1/5 + 3/5. • When adding, we need like whatever and the like part gets carried to the answer. • EX:   = •   = •   = Possible Pitfall: Adding the radicands • Again, check on calculator.

  11. Radical Expressions continued • Multiplying:  Start with • You can multiply rational parts by rational parts and irrational parts by irrational parts. • And the granddaddy of them all:

  12. Radical Expressions continued • Distribute:  and • Sometimes you can simplify after multiplying: • Possible Pitfall: Students want to change any 4 to a 2 even if the square root symbol is “used up” • And the granddaddy of them all:

  13. Radical Equations     5, so x = ?   5, so x = ? 5, so x = ? Steps:  "Clean" radical term Square both sides Solve Be sure to check for extraneous roots

  14. Radical Equations continued • An extraneous root is a solution to the transformed equation that does not satisfy the original equation.  Therefore it is not part of the solution. • Extraneous roots can show up in equations involving radicals and equations involving rational expressions. • In equations involving radicals, extraneous roots can show up if you have variables outside the radical as well as inside, such as    = x. • You can demonstrate extraneous roots by looking at the graph of the original and the graph of the transformed equation.

  15. Radical Equations continued No extraneous roots:   y1 =   (blue) and y2 = x2 - 6x + 5 (red)

  16. Radical Equations continued Extraneous roots:    y1 =   (blue) and y2 = x2 - x – 6 (red).

  17. Distance Formula • The distance formula is derived from the Pythagorean Theorem -- demonstrate the distance between (2,5) and         (-2,3) by drawing the triangle. •   c2 = (xA - xB)2 + (yA - yB)2 keeps Pythagorean variables. • http://www.mathopenref.com/coorddist.html has a demonstration. • Leave answer in simplified radical form or use calculator. • Use points on vertical and horizontal lines first so students can count, such as A(-3, 4) B(-3, 1)  C(2,1)  D(2, 4).

  18. Distance Formula continued • Then  E(0,0)  F(1,3)  G(6,4)   H(8,1) • Then L(-4,-1)  M(-3,4)  N(0,2)  P(1,0) • With all examples, show how you can subtract in different directions and still get the same answer. • Cut strips of graph paper to use as rulers.

  19. Midpoint Formula  Using the same points as for the distance formula, ask students to find the midpoint by folding the paper or some other method. Then develop the Midpoint Theorem. http://www.mathopenref.com/coordmidpoint.html shows a demonstration. Include examples with fractional answers. 

  20. Distance and Midpoint Formulas • Investigate different shapes using the distance and midpoint formulas, as well as slope of parallel and perpendicular lines.  Some examples. • Right triangle has 2 perpendicular lines • Square has adjacent sides perpendicular, opposite sides parallel and equal in length. • Trapezoid has one pair of opposite sides parallel and one pair of opposite sides not parallel. • A midsegment joins the midpoints of the legs of a trapezoid.  It is parallel to the bases and its length is half the sum of the lengths of the 2 bases. • See project on Coordinate Geometry (Handout p. 31-37).

  21. Rational Expressions Review basic fractions  Do not assume they will recall the following without a review: Remind  them:    anything        = 1    itself(except 0) ______0_______  = 0    anything (except 0) anything     is undefined.....causes "black holes." 0

  22. Rational Expressions Review simplifying   Simplify.20 = 4 x 5   =   4                  255 x 5        5   =  Tricky 2x + 3y - 2x + 2y                            15 y2

  23. Simplifying algebraic fractions   x + 5   common pitfall:  They will want to cancel x's         x + 3 To prove this is not possible, Let x =2.  or talk about "cookies". Discuss Restrctions:  You must restrict the variables in a denominator by excluding any value that makes the denominator equal to zero. "prevent black holes" Rational Expressions

  24. Simplify. 3x + 6   = 3(x+ 2)     3x + 9      3(x + 3) It is ok to cancel the 3's but not the x's......why? Common pitfall: They think that the restrictions are both -3 and -2.  Remind them that it is ok for a numerator to be zero. Simplify. x2 - 9         = (x + 3)(x - 3)x≠ -1 or -3     (x +1)(3+x)       (x +1)(3 + x) Common pitfall:  They do not write the restriction of the cancelled factor.  Prove it by plugging them both into the original problem. Rational Expressions

  25. Rational Expressions    Simplify.  2x2 + x - 3    = (x - 1)(2x + 3)      2 - x - x2         (1 - x)(2 + x) Common pitfall:  They forget that x-1 and 1-x are opposites and they must apply the -1  Review that x+4 and 4+x can cancel without a negative one.

  26. Rational Expressions Review multiplying fractions. 2  .   5restrictions?      5      8 Reminder:  You can multiply first and then simpliy or simplify and then multiply. Common pitfall:  For some strange reason they want to cross multiply. Multiply.  6x   .   y2                                              restrictions?     2y3       15 Common pitfall:  They think they can only cross simplify.  Remind them that any numerator can simplify into any denominator.

  27. Rational Expressions Multiply.  x2 - x - 12  .  x2 - 25      x2 - 5x          x + 3 Restrictions?

  28. Rational Expressions Review Simple Division. To divide by a fraction, you must multiply by its reciprocal.

  29. Rational Expressions "Smartie Problem" or "No homework problem"

  30. Rational Expression Review Simple adding and subtracting of fractions(same denominator). Common Pitfall:  believe it or not....they may add the denominators! Most common pitfall of all!!!!!!!: They may forget to subtract the second term. It helps to tell them to add in ( ).

  31. Rational Expressions Restrictions? Common pitfall:  Some will still cross out the 4s.

  32. Rational Expressions Add and Subtract Fractions (unlike denominators) Restrictions?

  33. Mixed Expressions     Great discussion on how we change a mixed number to an improper fraction.

  34. Equations with Fractional Coefficients

  35. Fractional Equations An equation with a variable in the denominator of one or more terms is called a fractional equation.  Note:  there may be times when there are no solutions. Solve.

  36. Fractional Equations  Solve.

  37. Discussion Graphing Calculator Skills Graph an equation Intersection of 2 lines Solving an equation by finding zeros and by finding intersection of the 2 sides of the equation Finding a specific function value Use Insert and Delete buttons Find an appropriate window Set up a table and view a table Enter an absolute value equation (and solve) Raise to a power other than 2 or 3 Zoom In and Zoom Out (Store values in a variable)

  38. Discussion How are things going in your classes?

  39. Discussion How to maintain math skills over the summer:

  40. Planning for 2012-2013

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