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UNIT ONE

UNIT ONE. MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques.

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UNIT ONE

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  1. UNIT ONE • MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques. • a. Represent functions using function notation.b. Graph the basic functions f(x) = x(n), where n = 1 to 3, f(x) = √x, f(x) = |x|, and f(x) = 1/x.c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes.d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior.e. Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior.f. Recognize sequences as functions with domains that are whole numbers.g. Explore rates of change, comparing constant rates of change (i.e., slope) versus variable rates of change. Compare rates of change of linear, quadratic, square root, and other function families. • MM1G2. Students will understand and use the language of mathematical argument and justification. • a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples, and indirect proof as appropriate.b. Understand and use the relationships among a statement and its converse, inverse, and contrapositive.

  2. UNIT TWO • MM1A2 Students will simplify and operate with radical expressions, polynomials, and rational expressions. • a. Simplify algebraic and numeric expressions involving square root.b. Perform operations with square roots.c. Add, subtract, multiply, and divide polynomials.d. Expand binomials using the Binomial Theoreme. Add, subtract, multiply, and divide rational expressions.f. Factor expressions by greatest common factor, grouping, trial and error, and special products limited to the formulas below. • (x + y)2 = x2 + 2xy + y2(x - y)2 = x2 - 2xy + y2(x + y)(x - y) = x2 - y2(x + a)(x + b) = x2 + (a + b)x + ab(x + y)3 = x3 + 3 x2y + 3xy2 + y3(x - y) 3 = x3 - 3 x2y + 3xy2 – y3 • g. Use area and volume models for polynomial arithmetic. • MM1A3. Students will solve simple equations. • a. Solve quadratic equations in the form ax² + bx + c = 0, where a = 1, by using factorization and finding square roots where applicable.

  3. UNIT THREE • MM1G3. Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons. • a. Determine the sum of interior and exterior angles in a polygon.b. Understand and use the triangle inequality, the side-angle inequality, and the exterior-angle inequality.c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, HL).d. Understand, use, and prove properties of and relationships among special quadrilaterals: parallelogram, rectangle, rhombus, square, trapezoid, and kite. e. Find and use points of concurrency in triangles: incenter, orthocenter, circumcenter, and centroid.

  4. Start-up Activity The slope of a line passing through point (1,3) is (-3). Write the rule (or equation) of this line. m = y2 – y1 x2 – x1 -3 = y – 3 x - 1 y – 3 = (-3)(x – 1) y – 3 = -3x + 3 y = - 3x + 6 y = -3(x – 2)

  5. Important Properties True for all Real Numbers a, b, c

  6. Adding & Subtracting Expressions Adding Expressions Subtracting expressions • Remove the parentheses • Use Commutative & Associative Properties to Rearrange and Group “like terms” • Simplify by combining “like terms” • Change the subtracted expression to its opposite • Add the expressions (4d – 2) – (5d – 3) (4d – 2) + (-5d + 3) 4d – 2 + 3 – 5d 4d – 5d – 2 + 3 - d + 1 (3x + 4) + (2x - 1) 3x + 4 + 2x – 1 3x + 2x +4 -1 5x +3

  7. Multiplying & Dividing Expressions Multiplying Expressions Dividing Expressions • ab = ba • (ab)c = a(bc) • a(b + c) = ab+ ac

  8. Dividing by Fractions - NASCAR • Distance = Rate * Time • Distance /Rate = Rate *Time/Rate • Time = Distance/Rate • Time = 1 Miles/ x Miles/Minute • Time = 1 Miles * Minute/ x Miles • f(x) = (1/x ) Minutes

  9. Ticket Out The Door 9/17 • Write a Paragraph containing at least 3 complete sentences: • The most interesting thing I learned in Math since school started. What was it about this topic that you liked most? • What I hope to learn more about this week.

  10. Typical Binomial Squares • (a + b)2 = a2 + 2ab + b2 • (a - b)2 = a2 – 2ab + b2 • Prove using F.O.I.L. • (a + b) (a + b) = a2 + ab + ab + b2 • = a2 + 2ab + b2 • Sum and Difference Pattern (Conjugates) • (a + b) (a - b) = a2 - ab + ab - b2 • = a2 – b2

  11. Binomial Theorem & Pascal’s Triangle • (a + b)0 = 1 • (a + b)1 = 1a + 1b • (a + b)2 = 1a2 + 2ab + 1b2 • (a + b)3 = 1a3 + 3a2b +3ab2 + 1b3 1 1 1 1 2 1 1 3 3 1

  12. Solving Polynomial EquationsIn Factored Form • Factoring and the Zero-Product Property • Ifab = 0, then: • a = 0, or • b = 0, or • Both a and b = 0 • If (x + 1) (x – 2) = 0, then • (x + 1) = 0, or • (x - 2) = 0, or • Both are zero

  13. Solving Polynomial Equations(cont.) • Factor x2 + bx + c • Factor ax2 + bx + c • Factor Special Products • a2 – b2 • ( a + b) 2 • (a - b) 2 • Factor Polynomials Completely • Writtenas a product of unfactorable polynomials with integer coefficients

  14. Mid-Term Assessment • How do you convert Miles per Hour to Miles per Minute? • In the NASCAR problem, it was discovered that the Rational Function, f(x) = 1/x, was a good fit to reflect the relationship described in the problem. Identify f(x), 1, and x in this problem. • In what units was time to be measured in #2, above? • After an accident on the track which caused a 5-minute delay, (a) reflect this delay in a new f(x) relationship. (b) What impact will this have on the original graph? • The problem then changed the venue to Daytona, which has a 2.5 mile track. (a) Write a new f(x) which reflects this change. (b) What impact does this distance change have on the original graph? • In the Tiling Task, what is the relationship between Mario’s Figure No. and (a) number of rows? (b) Number of tiles in each figure? • Mario has 5,000 tiles and wants to know how many rows he needs to use them. How should he proceed? • How many rows will he use to place the maximum number of available tiles? • How were Triangular Numbers defined in the problem? • What do you get when you add two Triangular Numbers?

  15. NASCAR Task

  16. Mario’s Tiling Vision

  17. Latasha’s Tiling Vision

  18. Triangular Numbers Sum of the Numbers 1 through n is:

  19. YE OLDE VILLAGE SHOPPE • Define the Variables • Draw a Picture • Define Relationships • Construct a Table • Graph the Results • Draw Conclusions w w x

  20. Ladder Length Learning Task

  21. Warm-up Simplify using the Distributive Property. • 1. 4(x – 3) • 2. -2(x + 5) • 3. x(x + 10) • 4. -x(x – 7)

  22. Warm-up • Simplify using the Distributive Property. • 1. 4(x – 3) • A) 4x – 3 B) 4x + 12 C) x – 12 D) 4x – 7 E) 4x - 12 • 2. -2(x + 5) • A) -2x + 3 B) -2x – 10 C) 2x – 10 D) -2x + 5 E) x - 10 • 3. x(x + 10) • A) 2x + 10x B) 2x + 10x C) x2 + 10 D) x2 + 10x E) 10x2 • 4. -x(x – 7) • A) -7 B) 7x2 C) -x2 + 7x D) x2 - 7x E) 7x

  23. Number Magic • Think of a number less than 10 • Double it • Add 10 • Divide it in half • Subtract the 1st number you thought of from the answer • Your final answer is 5! • (or you did something wrong!!!)

  24. I’ve Got Your Number • Pattern 1: (x+a)(x+b) = x2 + (a + b)x + ab • (72)(76) • A. (70 + 1)(75 + 1) B. (70 +2)(70 + 6) C. (68 +3)(70 + 6) D. (70 + 2)(72 + 3) • (31)(39) • A. (30 + 1)(30 + 9) B. (27 + 5)(35 + 3) C. (29 + 4)(30 + 9) D. (30 + 1)(32 + 5) • (52)(48) • A. (43 + 8)(40 + 8) B. (50 +2)(50 - 2) C. (43 + 9)(43 + 6) D. (40 + 12)(40 + 8)

  25. I’ve Got Your NumberPattern 1 HW • (x+a)(x+b) = x2 + (a+b)x + ab • (x + 3)(x + 4) • x2 + 7x + 12 • (x + 5)(x + 2) • x2 + 7x + 10 • (x + 9)(x + 1) • x2 + 10x + 9 • (x + 7)(x + 6) • x2 +13x + 42

  26. I’ve Got Your NumberPattern 1 HW #5 x 2 2x x2 (x+2)(x+1) = x2 + 3x + 2 x 1x 2 1

  27. I’ve Got Your Number Problem #1, a. b. c. • Case 1: a positive, b positive: Figure A • Case 2: a positive, b negative: Figure D • Case 3: a negative, b positive: Figure B • Case 4: a negative,b negative Figure C

  28. I’ve Got Your Number Problem #1, a. b. c. • Case 1: a positive, b positive: Figure A • Case 2: a positive, b negative: Figure D • Case 3: a negative, b positive: Figure B • Case 4: a negative,b negative Figure C

  29. I’ve Got Your Number #2. Use Pattern 1 to calculate each of the following products. • (a) (52)(57) = • (50 + 2 ) (50 + 7) = 502 + (2 + 7) (50) + (2)(7) = 2500 + 450 + 14 = 2964 • (b) (16)(13) = • (10 + 6)(10 + 3) = 102 + (6 + 3)(10) + (6)(3) = 100 + 90 + 18 = 208 • (c) (48)(42) = • (40 + 8)(40 + 2) = 402 + (8 + 2)(40) + (8)(2) = 1600 + 400 + 16 = 2016 • (d) (72)(75) = • (70 + 2)(70 + 5) = 702 + (2 + 5)(70) + (2)(5) = 4900 + 490 + 10 = 5400

  30. I’ve Got Your Number #3. • (a) (34)(36) = • 1224 • (b) (63)(67) = • 4221 • (c) (81)(89) = • 7209 • (d) (95)(95) = • 9025 Verify with Pattern 1 • = 900 + 10(30) + 24 = 1224 Verify with Pattern 1 • = 3600 + 10(60) + 21 = 4221 Verify with Pattern 1 • = 6400 + 10(80) + 9 = 7209 Verify with Pattern 1 • = 8100 + 10(90) + 25 = 9025

  31. I’ve Got Your Number #3a.-f. • To represent two-digit numbers with the same ten’s digit • start by using n to represent the ten’s digit. • So, n is 3 for part (a) What is n for parts (b), (c), and (d)? • Next, represent the first two-digit number as 10 n + a and the second one as 10 n + b. • In part (a): (32)(38) = (30 + 2)(30 + 8) = (10n + a)( 10n + b) for n = 3, a = 2, and b = 8. • Now figure (b), (c), & (d)

  32. I’ve Got Your Number #3h • (32)(38) = (30 + 2)(30 + 8) = • (10n + a)( 10n + b) for n = 3, a = 2, and b = 8 • Since a + b = 10, • = 100n2 + (10)(10n) + ab • = 100n2 + 100n + ab • = 100(n2+ n) + ab • Let k = (n2 + n) • = 100k + ab • Therefore, (32)(38) = 100(3)(4) + 16 = 1216 n2 + n = n(n+1)

  33. I’ve Got Your Number #4 How much Bigger? • How much bigger is the area of a square if you add 4 to its length and width? • (x+4)(x+4) • = x2 + 8x + 16 • How much bigger if you add y to length and width? • (x + y)(x + y) • x2 + 2xy + y2 • Pattern 2: Square of a Sum x 4 x2 x x + 4 4 x + 4

  34. I’ve Got Your Number #5 • 3022 = • (300 + 2)2 = (300)2 + 2(300)(2) + 22 = 90000 + 1200 + 4 = 91204 • 542 = • (50 + 4)2 = (50)2 + 2(50)(4) + (4)2 = 2500 + 400 + 16 = 2916 • 852= • (80 + 5)2 = (80)2 + (80)(5) + (5)2 = 6400 + 800 + 25 = 7225 • 2.12= • (2 + 0.1)2 = (2)2 + 2(2)(0.1) + (0.1)2 = 4 + 0.4 + 0.01 = 4.41

  35. I’ve Got Your Number #6 What is the Volume? • If Length = Width = Height = x • How much larger if sides increase by 4? • (x + 4)(x2 + 8x + 16) • = x3 + 12x2 + 48x + 64 • How much larger is sides increase by y? • (x+y)3 = x3+3x2y+3xy2+y3 • Pattern 3: Cube of a Sum x3 x x x

  36. I’ve Got Your Number #6e • 113 = • (10 + 1)3 = 103 + 3(10)2(1) + 3(10)(1)2 + 13 = 1000 + 300 + 30 +1 = 1331 •  233 = • (20 + 3)3 = 203 + 3(20)2(3) + 3(20)(3)2 + 33 = 4000 + 3600 + 180 + 27 = 7807 •  1013 = • (100 + 1)3 = 1003 + 3(100)2(1) + 3(100)(1)2 + 13= 1000000 + 30000 + 300 + 1 = 1030301

  37. I’ve Got Your Number #6f • Use the cube of the sum pattern to simplify the following expressions. • (t + 5)3 = • t3 + 3(t2)(5) + 3(t)(52) + 53 = t3 + 15t2 + 75t + 125 • (w + 2) 3 = • w3 + 3(w2)(2) + 3(w)(22) + 23 = w3 + 6w2 + 12w + 8

  38. I’ve Got Your Number #7-9 • Pattern 2: Square of a Sum • (x + y) 2 = x2 + 2xy + y2 • Pattern 4: Square of a Difference • (x – y) 2 = x2- 2xy + y2 • Pattern 5: Cube of a Difference • (x - y)3 = x3 - 3x2y + 3xy2 - y3 • Pattern 6: Conjugates • (x + y)(x – y) = x2 – y2

  39. I’ve Got Your Number #7-10 • (101)(99) = • (100 + 1)(100 – 1) = 1002 – 12 = 10000 – 1 = 9999 • (22)(18) = • (20 + 2)(20 – 2) = 202 – 22 = 400 – 4 = 396 • (45)(35) = • (40 + 5)(40 – 5) = 402 – 52 = 1600 – 25 = 1575

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