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Wicomico High School Mrs. J. Austin 2009-2010 Chapter 7 : System of Equations.

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Presentation on theme: "Wicomico High School Mrs. J. Austin 2009-2010 Chapter 7 : System of Equations."— Presentation transcript:

1 Wicomico High School Mrs. J. Austin Chapter 7 : System of Equations

2 Class Format MiniQuiz: 10 minutes Cognitive Tutor Program on Computer GROUP A DAY 1 GROUP B DAY 2 Textbook Lesson / Practice at Desk GROUP B DAY 1 GROUP A DAY 2

3 Cognitive Tutor Program Self-pacing program with immediate feedback. Log-in using your first name and last name The lesson will open and to begin you will click on the yellow box in the upper right-hand corner. The first lesson has eight sections of Solving Equations On the right-hand side of the screen, there are two drop- down menus. Transformations and Applications Use the choices to solve the problem Click Done and the next problem will appear RED means incorrect. Use the UNDO button and try again.

4 Systems of Linear Equations Two linear equations graphed on the SAME coordinate plane. What are the THREE things that could happen? They could CROSS or INTERSECT They could NEVER CROSS or be PARALLEL They could be ON TOP OF EACH OTHER or COINCIDE

5 Solving a System of Linear Equations By Solving a System of Linear Equations, we are asking: Are there any Values for x and y that will “satisfy” or make BOTH equations TRUE? Is there a POINT that will make BOTH equations TRUE? What is the POINT OF INTERSECTION of these two lines? Find the values for x and y that will make BOTH equations TRUE.

6 Solving a System By Graphing 7.1 Transform each equation to Slope-Intercept Form For EACH of the TWO equations: PLOT the y – intercept, b COUNT, rise over run using the Slope, m. DRAW the straight line.

7 Solving a System By Graphing 7.1 Transform each equation into Slope-Intercept Form.

8 Solving a System By Graphing 7.1 GROUP A: Work Session Solving Systems using the Graphing Method. Textbook: Pg. 431 (1-28) GROUP B: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 1

9 Solving a System By Graphing 7.1 GROUP B: Work Session Solving Systems using the Graphing Method. Textbook: Pg. 431 (1-28) GROUP A: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 2

10 Solving a System By Substitution 7.2 Transitive Property A variable can be REPLACED with its equivalent. If two equations equal the SAME thing, they must then EQUAL each other. AND THEN The two equations are SET equal to each other

11 Solving a System By Substitution EXAMPLE: Now we know y=2. Substitute this into an equations to find x. Solve the first equation for x. Substitute the expression in for x in the second equation. Now SOLVE for y.

12 Solving a System By Substitution GROUP A: Work Session Solving Systems using the Substitution Method. Textbook: Pg. 439 (1-28) GROUP B: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 1

13 Solving a System By Substitution GROUP B: Work Session Solving Systems using the Substitution Method. Textbook: Pg. 439 (1-28) GROUP A: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 2

14 Solving a System By Elimination 7.3 Transitive Property Replace a variable with its equivalent. Set two expressions equal to each other, when they BOTH equal the same thing! Property of Equality Add or Subtract two equations to create a new equivalent equation. Transform the look of an equation by multiplication.

15 Solving a System By Elimination 7.3 Using Addition: ___________ Write the equations one above the other. Be sure the variables are lined up. Draw a line under them. Combine the Like-Terms to create a NEW equation. Solve for the variable. Substitute your answer into one of the equations to find the other variable.

16 Solving a System By Elimination 7.3 Using Subtraction _____________ Write the equations one above the other. Be sure the variables are lined up. Draw a line under them. Subtract the Like-Terms to create a NEW equation. Solve for the variable. Substitute your answer into one of the equations to find the other variable.

17 Solving a System By Elimination 7.3 GROUP A: Work Session Solving Systems using the Graphing Method. Textbook: Pg. 431 (1- 28) GROUP B: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 1

18 Solving a System By Elimination 7.3 GROUP B: Work Session Solving Systems using the Graphing Method. Textbook: Pg. 443 – 444 (1-35) GROUP A: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 2

19 Solving a System By Elimination 7.4 Using Multiplication Before Adding or Subtracting: ___________________ __________________ Write the equations one above the other. Be sure the variables are lined up. Multiply the top equation by 4. Multiply the lower equation by 7. Draw a line under them. Distribute through each equation. Combine or Subtract the Like- Terms to create a NEW equation.

20 Solving a System By Elimination 7.4 Using Multiplication Before Adding or Subtracting. What would you MULTIPLY by? Solve the System. Did you get the solution: Solve the System. Did you get the solution:

21 Solving a System By Elimination 7.4 GROUP A: Work Session Solving Systems using the Graphing Method. Textbook: Pg (1-34) GROUP B: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 1

22 Solving a System By Elimination 7.4 GROUP B: Work Session Solving Systems using the Graphing Method. Textbook: Pg (1-34) GROUP A: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 2

23 Special Types of Systems 7.5 One Solution: Lines Intersect at one point. Lines have different slopes. Lines may be PERPENDICULAR if they cross at 90⁰ angles. No Solution: Lines do not intersect. Lines have the SAME slope and DIFFERENT y-intercepts. Lines are PARALLEL. Many Solutions: Lines touch on every point. Lines have the SAME slope and SAME y-intercepts. Line COINCIDE.

24 Special Types of Systems 7.5 How Many Solutions Does the System Have?. System: Answer Choices: One Intersecting Lines None Parallel Lines Many Coinciding Lines

25 Special Types of Systems 7.5 GROUP A: Work Session Solving Systems using the Graphing Method. Textbook: Pg. 462 – 463 (1-31) GROUP B: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 1

26 Special Types of Systems 7.5 GROUP B: Work Session Solving Systems using the Graphing Method. Textbook: Pg. 462 – 463 (1-31) GROUP A: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 2

27 Writing and Solving Systems Slope –Intercept Form: Total Cost scenarios with given rates of change. The movie theater charges $8 per a ticket for its general customers. It offers a movie club discount of $5 per ticket if you join the club for a one-time fee of $15. How many movies would you have to go see to make joining the club beneficial? Write a system. Let x = the number of movie tickets y = total cost

28 Writing and Solving Systems Standard or General Form: Two different Items are given. At a grocery store, a customer pays a total of $9.70 for 1.8 pounds of potato salad and 1.4 pounds of coleslaw. Another customer pays a total of $6.55 for 1 pound of potato salad and 1.2 pounds of coleslaw. How much do 2 pounds of potato salad and 2 pounds of coleslaw cost? Write the system. Let: x = cost of potato salad y = cost of coleslaw.

29 Writing and Solving Systems High School Assessment Practice Questions: READ the question ALL the way through. RE-READ and define the variables. RE-READ and WRITE two equations to model the scenario. DECIDE which METHOD you will use to solve the System of Equations. SOLVE the System. RE-READ the question. Use YOUR SOLUTION to CONSTRUCT a written ANSWER to the question.

30 Solving a System By Writing 7.5 GROUP A: Work Session Solving Systems using the Graphing Method. Textbook: Pg. 458 (1-6) Pg. 464 ( 36-40) GROUP B: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 1

31 Solving a System By Writing 7.5 GROUP B: Work Session Solving Systems using the Graphing Method. Textbook: Pg. 458 (1-6) Pg. 464 ( 36-40) GROUP A: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 2

32 Solving Systems of Linear Inequalities 7.6 Graphing Linear Inequalities: Graph the first line. Shade the area defined by the first line. Graph the second line. Shade the area defined by the second line. The SOLUTUION to the System of Linear Inequalities is the AREA OF INTERSECTION. Re-shade the section of the graph that has been shaded by both of the equations.

33 Solving Systems of Linear Inequalities 7.6 GROUP A: Work Session Solving Systems using the Graphing Method. Textbook: Pg. 458 (1-6) Pg. 464 ( 36-40) GROUP B: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 1

34 Solving Systems of Linear Inequalities 7.6 GROUP B: Work Session Solving Systems using the Graphing Method. Textbook: Pg. 458 (1-6) Pg. 464 ( 36-40) GROUP A: Computer Session Cognitive Tutor Unit 19 Solving Equations DAY 2

35 Solving Systems of Linear Inequalities 7.6

36 Chp 7 Review Two linear functions graphed on a coordinate plane. They could CROSS, INTERSECT They could NEVER CROSS, PARALLEL They could be ON TOP OF EACH OTHER, COINCIDE Solving a System By Graphing 7.1 Solving a System By Substitution 7.2 Solving a System By Elimination 7.3 Identifying the Point of Intersection Testing a Solution to a System of Equations Special Types of Systems Writing and Solving Systems of Equations

37 Surveys and Sampling TAKE A SAMPLE:

38 Types of Sampling Simple Random Sampling Every person has an “equally likely chance of being chosen” and each is independent of the other. Stratified Random Sampling The population is divided into groups, then by groups every person has an “equally likely chance of being chosen” and each is independent of the other. Convenient Sampling Only people in “your area” get to participate Self-Selected Sampling Only people who volunteer to answer your survey get to participate. Representative Sampling The people who get to answer your survey match the gender and ethnicity of the population.

39 Survey Project STEP 1: Create 5 questions about the given topic. STEP 2: Create two questionnaires on a half- sheet of paper each. STEP 3: Decide on your Sampling Technique and how your are going to conduct your survey. STEP 4: Conduct your Survey of the school population. STEP 5: You must have a minimum of 30 responses to make the survey VALID. STEP 6: Compile your results in a table and with a Box-and-Whisker Plot. STEP 7: Create a POSTER to display your survey, graphics, and findings. STEP 8: Write a summary and conclusion based on your survey results.

40 Exponential Functions Chp 8 intro

41 Graphing Exponential Functions 8.5 Growth Functions:

42 Graphing Exponential Functions 8.6 Decay Functions:

43 The Laws of Exponents 8.1

44 The Laws of Exponents 8.2

45 Negative Exponents 8.3

46 Exponents and Scientific Notation 8.4

47 Chapter 8 Review

48 Simulations

49 Types of Devices Dice or Number Cube Six numbers; evens/odds Spinners Any number of sections of EQUAL size can be made Coins Two sides, heads/tails Deck of Playing Cards Thirteen cards; Four Suits; Two Colors Colored Chips in a Bag Any number of chips and colors can be used Random Number Generator (TI-84 calculator) Ten digits, any number of choices can be made. Random Number Table Ten digits, any number of choices can be made.

50 Setting Up Simulations Decide which device will FIT your numbers. PercentageFractionSimple Fraction DevicesDesign 10% Spinner 10 Chips in Bag 10 sections; 1 red chip, others blue 25% Two coins Spinner 4 sections; 4 chips 30% 10Chips in a Bag Spinner 3 of one color, rest another color 35% noneNumber Generator35 choices out of % Coins, Cards, Spinner, Die,etc. 2 sections, colors, sides, 66.6% Dice, Spinner, Chips in a Bag 1 die, 4 numbers 3 chips, 2 one color 75% Two Coins, Spinner, 4 Chips, Cards, … 4 sections, 3 one outcome, 1 the other

51 Polynomial Functions Chp 9. Constant Function NOT a POLYNOMIAL Linear Function 1 st degree polynomial Quadratic Function 2 nd degree polynomial Cubic Function 3 rd degree polynomial Quartic Function 4 th degree polynomial Fifth Degree Function Activity: Pg. 560 Graphing Polynomial Functions

52 Types of Functions Even Functions degree is even tails go in the same direction Odd Functions Degree is odd Tails go in opposite directions One-to-One Functions Every x and be paired with only one y Continuous Functions No breaks in the curve. No jumps, skips, holes, oscillations,…. Discrete Functions Fractions are not possible as outcomes. A series of points that are NOT connected. Piece-wise Functions The function is defined by two or more equations. Each piece of the function has a different equation. Step Functions A discrete function with horizontal lines that appear like steps going up or down the y axis. Finite Functions The function is BOUNDED The function has a distinct domain and/or range Infinite Functions The function has NO BOUNDS The function has an unrestricted Domain and Range, -∞ to ∞

53 Polynomial Vocabulary 9.1 Classifying Polynomials: Monomial one term Binomial two terms Trinomial three terms Polynomial many terms Degree of a Polynomial

54 Adding Polynomials 9.1 Two Step Process: Align the like-terms Combine the coefficients Remember: The exponents DO NOT CHANGE

55 Subtracting Polynomials 9.1 Three Step Process: 1. Distributing the negative sign 2. Align the like-terms 3. Combining the coefficients Remember: The exponents DO NOT CHANGE

56 Work Session 9.1 Adding and Subtracting Polynomials Pg 558 (38 – 42) evens Problem Solving

57 Algebra Tiles 9.2 Activity: Page 561

58 Multiplying Polynomials 9.2 Distributive Method

59 Multiplying Polynomials 9.3 Square a Binomial Difference of Squares

60 Multiplying Polynomials 9.3 FOIL Method Work Session Page 572 (1-37)

61 Multiplying Polynomials 9.3 Punnett Square Page 573 #40 and #42

62 The Zero Product Property 9.4 Finding the Zeros of a Function

63 The Zero Product Property 9.4 Vertical Motion Formula Rocket Project Begins

64 Factoring Polynomials 9.5 GCF Factoring out the Greatest Common Factor

65 Factoring Polynomials 9.5 F O I L (first, outer, inner, last) Multiply to get C and add to get B Multiply to get C and Subtract to get B

66 Factoring Polynomials 9.5 Work Session Page 587 (42 – 55)

67 Factoring Polynomials 9.5 Work Session Page (59 – 81)

68 Factoring Polynomials 9.6 Slip- Slide-Divide Method Using Algebra Tiles Pg. 592 (1-8)

69 Factoring Polynomials 9.6 Slip- Slide-Divide Method Work Session Page 596 ( 1-21)

70 Factoring Polynomials 9.6 Slip- Slide-Divide Method Work Session Page 597 ( 22-37)

71 Factoring Polynomials 9.6 Slip- Slide-Divide Method Work Session Page 597 ( 38-57)

72 Factoring Polynomials 9.6 Slip- Slide-Divide Method Work Session Page 598 – 599 (58-81)

73 QUIZ Factoring Polynomials 9.6 Page 599 (1-25)

74 Factor Special Products 9.7 Difference of Squares Perfect Square Trinomials

75 Factor Special Products 9.7 Work Session: Page 603 (1-39)

76 Factor Special Products 9.7 Work Session: Page (46-76)

77 Factor Polynomials Completely Factoring Out a Common Binomial Factoring By Grouping 2 by 2 Factoring By Grouping 3 by 1

78 Factor Polynomials Completely Factoring Out a Common Binomial Factoring By Grouping 2 by 2 Factoring By Grouping 3 by 1 Work Session: Pg. 610 (1-42)

79 Factor Polynomials Completely Factoring Out a Common Binomial Factoring By Grouping 2 by 2 Factoring By Grouping 3 by 1 Work Session: Pg. 611 (43-66)

80 Factor Polynomials Completely Factoring Out a Common Binomial Factoring By Grouping 2 by 2 Factoring By Grouping 3 by 1 Work Session: Pg. 612 (68-89)

81 QUIZ Factor Polynomials 9.8 Page 613 (1-19)

82 Chapter 9 Review

83 Graphing Quadratic Equations 10.1 General Form of a Quadratic Function: A indicates the width of the parabola B is used to find the Axis of Symmetry C is the y-intercept Parabola Shape: Positive “holds water” concave up Negative “doesn’t hold water” concave down Shifting of the Parabola: Vertical Shift Horizontal Shift Skinny Wide

84 Graphing Quadratic Equations 10.2 Finding the Axis of Symmetry Finding the Vertex (maximum or minimum) Graphing a Parabola: 1. Dot in the vertical line for the Axis of Symmetry 2. Plot the Vertex 3. Plot the y-intercept and its reflected point. 4. Plot another point and its reflection. 5. Draw a smooth curve connecting the points.

85 Graphing Quadratic Equations 10.2 Work Session: Page 638 – 639 (1-37)

86 Conic Form of a Parabola Page 669 Reference Green Globs Activity:

87 Solve Quadratic By Graphing 10.3

88 Solve Quadratics With Square Roots 10.4

89 Solve Quadratic By: Completing the Square 10.5

90 Solve Quadratics By: the Quadratic Formula 10.6 Quadratic Formula:

91 The Discriminate of the Formula 10.7

92 Comparing Linear, Exponential, and Quadratic Functions 10.8

93 Performing Regression Work Session with Graphing Calculators: Page (1-3)

94 Chapter 10 Review

95 Benchmark Review # 3


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