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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: Rational Expression, Rational Equations, Radicals, Quadratic Formula, Distance and Midpoint; 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 2/29/12 Overview of remaining topics Systems Exponents Polynomials Quadratics  6. Discussion and sharing: Fun things to do in 5  or 10 minutes.  7. Plan for the next session  8. Conclusion

  4. Overview of Remaining Topics How to access the end of year study guide and use it as check list:  www.adh-ocs.org →  Quality Catholic Education → Resources →  under Assessment look for Algebra I Study Guide.pdf and its solutions. Update from Session 2 (index cards, etc.) Word problems throughout the year

  5. Systems Motivating the Lesson:  Big Picture:  What is a system and why is it useful to be able to present data as a system? You are a busy fashion design mogul and you have 5 minutes to choose which one of two young start up companies you are going to buy....all you have is their financial reports.

  6. Derricks Dreamwear Partial Financial Report Initial 40K Feb.  50K Apr    60K Jun    70K Sasha's Sassy Wear Financial Report Initial 20K Jan    30K Feb    40K Mar    50k Apr    60K May    70K Systems Continued

  7. Systems Continued

  8. Systems Continued Discussion Questions 1.  What does the y intercept signify?  2.  What does the intersection signify? 3. What would it signify if the lines were parallel? 3.  What does a greater slope signify? 4.  Would the health of a company be related to the y intercept or the slope? 5.  Which company would you buy?  Why?

  9. Systems Continued There are an infinite number of ordered pairs(x,y) which are solutions to x+y = 4 such as (3,1), (2,2),and (4,0).....and when plotted form a straight line. Similarly, There are an infinite number of ordered pairs (x,y) which are solutions to x-y= 2 such as (6,4),(2,0), and (3,1)...and will also form a straight line when graphed. Since the ordered pair (3,1) satisfies both equations x+y=4 and x-y=2, it is called the common solution for this pair of linear equations.

  10. Systems Continued  3 scenerios of a system 1.  Lines intersect (unique solution)...consistent & independent          Different slope     ex. y=2x+1 and y = 3x+1  2.  Lines are parallel ( no solution)....inconsistent     Same slope different intercept  ex. y = 2x + 1 and y = 2x - 1 make sure you have a clean y before comparing slopes 3.  Lines coincide (infinite solutions)....dependent      Same slope same intercept

  11. Intersecting  One common solution Coinciding All solutions in common Parallel No solutions in common Discussion Question:  What would have to change to have 2 points in common? Systems

  12. Systems Three ways to solve 1.  Graphically...graph two lines on a cartesion plane to determine if there is one, no, or infinite solutions. 2.  Substitution...isolate one variable and plug into the other equation. 3.  Linear Combination (Elimination)...set up system in such a way that one of the variables drop out when you add them together. Note:  If the variables drop out in #2 and #3 that signifies paralell(if false statement remains) or coinciding line(if true statement remains). Emphasize using the easiest method for given problem.

  13. Solve Systems Graphically y=2x + 3 y=x -1

  14. Solve System by Substitution Start easy. 2x + y = 7                    3x + y =6                y = 1                              x = 18 -3y Show y= to y= method y= 3x +5 x-y =6 

  15.     5x - y = 12     3x + y = 4     4s - 5t = 3     3s + 2t = -15     6c + 7d = -15 6c - 2d = 12 * 2x = 14y x-y = x+y      3       4 Solve System by Elimination

  16. Parallel (no solutions in common)  Substitution y = 3x - 7 6x - 2y = 12 Elimination   -4x +y = -5 8x - 2y = 20  . Coinciding(all solutions in common)  Substitution 2x + 2y = 2 x + y = 1 Elimination(watch order) Solving Systems of Parallel and Coinciding lines * 2x = 14y x-y = x+y      3       4

  17. Solving Problems with Two Variables A pet shop sold 23 frogs and beta fish one week.  They sold 9 more frogs than betas.  How many of each did they sell? Let f = # of frogs       b = # of beta fish     f + b = 23           f = b + 9          what is the best method? Substitution.     b+9+ b = 23     2b + 9 = 23           -9      -9       2b = 14          b = 7

  18. Solving Problems with Two Variables A bank teller has 112 $5-bills and $10-bills for a total of $720.  How many of each does the teller have? Let x=  $5-bills        y = $10 bills      x +   y = 112    5x +10y = 720        Best Method?     either.

  19. Solving Problems with Two Variables Val has $8000 invested in stocks and bonds.  The stocks pay 4% interest, and the bonds pay 7% interest.  If her annual income from stocks and bonds is $500, how much was invested in bonds?  Let s=stocks        b=bonds           s +        b = 8000       0.04s + 0.07b = 500

  20. <   Dotted line >   Dotted line <   Solid line >   Solid Line Use y=mx+b to graph Must have a "clean y" to graph. Note:  If you multiply or divide by a negative number remember to flip the inequality sign. y < 2x + 1 y>  x - 2 Graphing Systems of Inequalities

  21. Graphing a System of Inequalities Graph and give 2 points that are solutions to the system. x - y > 2     x + 2y > 1

  22. Exponent Rules Add 2x2 + 3x2+ xAdd the coefficients of the like terms Multiply (2x2)(3x3)  Multiply coefficients and add exponents Raising to a power (2x2)3       Apply Phantom one multiply exponents Dividing  Reduce coefficients and subtract exponents 8x3 6x2

  23. Zero Exponent  5 =1       Anything  =1  5               itself Use Division Rules x2 =1   x2 x2-2 =x0 =1 Fun Problems (x10y40z30)0 2x0 + (3y)0 (blue cow)0 + 5(green pig)0 (2x3 +4x2 + 8x3)0 Exponent Rules:Special Cases

  24. Negative Exponents Note:  Negative Exponents determine the placement on a fraction. Simplify. x2y3z5      = xy-2 (not proper) x y5z5                 =  x    (proper)                     y2 Real World:  Scientific Notation Microbiology 2.3 x 10-8 Fun Examples  Landlord of a duplex...reposition your unhappy (-) tenants. a-2b4c-1 d2e-2f5 Simplify.  x-2y3z-1  x3 y-1z-5 Note:  -2x-3 (2 negs does not = +) Exponent Rules : Special Cases

  25. Simplify. 1)  a2 + b + a2 + b2 + b 2)  (7a2b2)2 + (ab)4 - 50 3)    2x(-5y6)3 4)  (3x2)(3y2) + 3x2y - (3xy)2 Exponents Practice

  26. Polynomials Vocabulary: The degree of a polynomial is the same as the degree of the monomial that has the highest degree.  The degree of a monomial is the sum of the exponents on the variables. 3x2 has degree 2 3y4 has degree 4 3x2y4 has degree 6  (not used much) 5x3 - 2x2 + 3x - 5 has degree 3 because of 5x3 The above polynomial is in descending order.

  27. Polynomials Vocabulary by degree Constant (0)- relate to constant function or doesn't vary Linear (1) - relate to line Quadratic (2) - relate to "a square has 4 sides" * Cubic (3)- relate to x-cubed [Quartic = degree of 4, Quintic = degree of 5 – handout p. 25] *Although in Latin the prefix "quadri" means four, the word "quadrus" means a square (because it has four sides) and "quadratus" means "squared." We get several other words from this: "quadrille," meaning a square dance; "quadrature," meaning constructing a square of a certain area.  From Ask Dr. Math

  28. Polynomials Vocabulary by number of terms Monomial - mono-tone Binomial - bi-cycle Trinomial - tri-cycle Consider requiring "what what" for every answer. Look at graphs of a constant, linear, quadratic, cubic to see why the degree is important.  The degree also tells how many roots the equation will have.

  29. Polynomials Adding and Subtracting Polynomials Review like terms Include lots of practice with (3x2 - 5) - (2x2+ 3x - 2) Continue vocabulary with "what what"

  30. Polynomials Multiplying Polynomials Review exponent rules. Review multiplication facts. Review factors, such as list all the factors of 20, of 30, or 45; use a system for listing the factors. Start with multiplying by a monomial (distributing) and then "un-distributing" which is the start of factoring.

  31. Polynomials Multiplying Polynomials - p. 2 Multiply a binomial by a binomial as distributing (instead of FOIL).  Consider calculating the constant second but still writing it last.  As students get better they should try to calculate the linear term in their head. Consider vertical method. Include examples of (2x + 3)(2x - 3) here and show the reverse.

  32. Polynomials Multiplying Polynomials - p. 3 Include lots of practice with (x + 4)2. Start with (2 + 3)2. Consider teaching the shortcut.  Then x2 + 6x + ___ and x2 - 14x + ___ and have students figure out the constant.  Show the factoring of each as you go. Play Tic-Tac-Times here and after Factoring.

  33. Polynomials Multiplying Polynomials - p. 4 Multiply a binomial by a trinomial Easier with the binomial first Distribute the first term in the binomial to the trinomial, then the second term in the binomial to the trinomial. Use the vertical method to group like terms If you did use FOIL, see if students can come up with an acronym for binomial * trinomial.

  34. Factoring- p. 1 Again, review multiplication facts and factors of numbers. Review factoring out a Greatest Common Factor (GCF) as "un-distributing." Start factoring of trinomials by having students multiply (x + 3)(x + 2) (x + 3)(x - 2) (x - 3)(x - 2) (x - 3)(x + 2)     and then decide how to split them into 2 groups (last term positive and last term negative)

  35. Factoring - p. 2 Practice factoring last term positive first, then last term negative.  Emphasize looking mostly at the first and last terms (these are products). Always check by multiplying.  Again, challenge some students to combine the linear term in their heads. If you get the right number for the linear term but the wrong sign, the numbers in ( ) are right but the signs are wrong. Include trinomials with a constant GCF, such as 2x2 - 2x - 12

  36. Factoring - p. 3 For -x2 + x + 6, factor out -1:     -1(x2 - x - 6) and then   -1(x - 3)(x + 2).  This will help later when simplifying rational expressions. Include Difference of Squares (x2 - 25) and perfect square trinomials (9x2 - 24x + 16) with the other examples.  Students may not see PST as a special pattern.

  37. Factoring - p. 4 Leading coefficient ≠ 1:  6x2 - x - 12 Claudia: If there is no GCF in the original trinomial, you cannot put 2 numbers in the same ( ) that do have a GCF.  Write all the possibilities on the board and then cross off those with a GCF quickly and ask students to figure out how you knew which to cross off as impossible.  They should notice that you didn't have enough time to check. (next page) Kathleen:  Tic-Tac-Toe method. 

  38. Factoring - p. 5 6x2 - x - 12        Which are impossible? (6x - 2)(1x + 6)            (3x - 2)(2x + 6) (6x - 6)(1x + 2)            (3x - 6)(2x + 2) (6x - 3)(1x + 4)            (3x - 3)(2x + 4) (6x - 4)(1x + 3)            (3x - 4)(2x + 3) (6x - 12)(1x + 1)          (3x - 12)(2x + 1) (6x - 1)(1x + 12)          (3x - 1)(2x + 12)

  39. Factoring - p. 6 Play Tic-Tac-Times (being able to factor allows you to block). We suggest leaving 4-term factoring until Algebra II unless you have a very good group and extra time. Continually ask students "What is the first step in any factoring problem?" Give lots of practice with all the styles mixed up.

  40. Quadratic Equations • Review/introduce graphing parabolas with a chart. • y=(x - 3)2 • y = x2 - 3 • y = x2 + x - 2 • y = x2 + 2x • y = -x2 + 3x • Our suggestion:  Leave finding vertex with x = -b/2a until Algebra II. • Show zeros of the function are 2 numbers that satisfy the equation, and are the zeros of the factored form.  Quadratic equations have two roots. • zeros of a function = roots of an equation = x -intercepts of the graph = solutions

  41. Quadratic Equations • Introduce the Zero Product Property:  If ab = 0 then ... • Solve quadratic equations using the Zero Product Property by first solving for zero and then factoring and setting each factor equal to zero. • Refer to the graph of y = (x + 3)(x - 4) when solving (x + 3)(x - 4) = 0.  The solution is where y = 0 (x - axis). • Show that the following all have x = 0 and x = 3 as roots (use graphing calculator if possible): • x(x - 3) = 0 • 3x(x - 3) = 0 • 10x(x - 3) = 0 • -x(x - 3) = 0 • ?Completing the square – Algebra II • Save solving by square roots and quadratic formula until after radicals.

  42. Quadratic Equations • Word Problems:  • You get 2 answers: sometimes both are usable, sometimes only one. • Geometry: length of a rectangle is 3 cm more than the width and the area is 40 cm2.  Find length and width. • Physics:  The formula h = -16t2 + 29t + 6 gives a ball's height when thrown with an initial velocity of 29 ft/s and an initial height of 6 ft.  When will the ball hit the ground? • Numbers:  The product of 2 consecutive even integers is 120.  What are the integers?  (2 possibilities) • Show finding zeros on a graphing calculator (both rational and irrational roots).

  43. Discussion Fun things to do with 5-10 minutes left in class: Bizz-Buzz Sugar-Sugar on coolmathgames.com Inspirational inverviews Read Biography of mathematicians. SAT Skills Insight: http://sat.collegeboard.org/practice/sat-skills-insight/math/band/200 or SAT Question of the Day: http://sat.collegeboard.org/practice/sat-question-of-the-day. Logic Puzzles Lateral Thinking (20 questions)

  44. Planning for Session 4 Topics: Radicals, Quadratic Formula, Distance Formula and Midpoint Theorem, Rational expressions and rational equations Bring a tip to share about math maintenance.

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