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Thinking Mathematically By: Mac Sinsky and Matt Hipp

Thinking Mathematically By: Mac Sinsky and Matt Hipp.

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Thinking Mathematically By: Mac Sinsky and Matt Hipp

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  1. Thinking MathematicallyBy: Mac Sinsky and Matt Hipp

  2. Solving 1st power equations in one variableA. Special cases where variables cancel to get all realsExample: 2x+6=2(x+3)-Distribute the 2 2x+6=2x+6Cancel out to equal no setB. Equations containing fractional coefficientsExample: x+ = 5 =2, so x=3Equations with variables in the denominatorExample:28 divided by 7 =4, so x=4

  3. In the next slides you will review:All the Properties, and then take a Quiz on identifying the Property Names

  4. Addition Property (of Equality) Definition: If one number is added to two sides of an equationExample: If 3=3. then 2 + 3=3 + 2 Multiplication Property (of Equality) Definition: If one number is multiplied to two sides of an equationExample: If 2=2, then 2 x 5=5 x 2

  5. Reflexive Property (of Equality) Definition: One number equals the same numberExample: 10=10 Symmetric Property (of Equality) Definiton: One number equals the same number, even if it’s in a different order Example: If 6=8, then 8=6 Transitive Property (of Equality) Definition: One number equals another number, and that number equals another numberExample: If 12=15, and 16=12. then 15=16

  6. Associative Property of Addition Definition: When three numbers are added, and changing the order has the same end resultExample: (5 + 6)+ 4=5+(4 + 6) Associative Property of Multiplication Definition: When three numbers are multiplied, and changing the order has the same end resultExample: (4 x 7)x 2=4 x(2 x 7)

  7. Commutative Property of Addition Definition: When two numbers are added, and, if the order is changed, the result is the same Example: 7 + 6=6 + 7 Commutative Property of Multiplication Definition: When two numbers are multiplied, and, if the order is changed, the result is the sameExample: 9 x 8=8 x 9

  8. Distributive Property (of Multiplication over Addition) Definition: When the distributive property equals the distributive property broken downExample: 4x(5+8)=4x5 +4 x 8

  9. Prop of Opposites or Inverse Property of Addition Definition: When a number added to its opposite equals zeroExample: 7 +(-7)=0 Prop of Reciprocals or Inverse Prop. of Multiplication Definition: when two numbers are multiplied, and the answer is 1Example: 4 x ¼=1

  10. Identity Property of Addition Definition: When 0 plus a number equals the number that was added to 0Example: 0 + 5=5 Identity Property of Multiplication Definition: When 1 times a number equals the number that was multiplied by the 1Example: 1 x 4=4

  11. Multiplicative Property of Zero Definition: When 0 times a number equals 0 Example: 0 x 3=0 Closure Property of Addition Definition: When two numbers are added to equal a different number Example: 5 + 3=8 Closure Property of Multiplication Definition: When two numbers are multiplied to equal a different number Example: 4 x 6=24

  12. Product of Powers Property Definition: When two powers and their exponents are added Example: 22 x 25=4 x 32=128 is the same as 22 + 5 =27=128 Power of a Product Property Definition: When you want to find the power of a productExample: (3 × 4)2 = 122 = 144 is the same as 32 × 42 = 9 × 16 = 144 Power of a Power Property Definition: When you want to find the power of a powerExample: (22)3 = 43 = 64 is the same as 22×3 = 26 = 64

  13. Quotient of Powers Property Definition: When dividing two powers that have the same base, subtract the exponents Example: = Power of a Quotient Property Definition: When dividing two bases with the same exponents, cancel out the common factors Example: is the same as

  14. Zero Power Property Definition:When zero is the exponent the answer is 1Example: Negative Power Property Definition: When the power is a negative, the answer is 1 over that power. Moving the power to the denominator takes away the negative.Example:

  15. Zero Product Property Definition: When two numbers equal zero, one of the numbers must be equal to zero Example: If ab=0, then a=0 or b=0

  16. Product of Roots PropertyDefinition: When two numbers, both inside the root, equal both numbers in separate rootsExample:= x Quotient of Roots PropertyDefinition: When a fraction is in a root, simplify the numbers.Example:

  17. Root of a Power Definition: When an exponent is in a square root Example: -The power and the root cancel, so final answer is 2

  18. Power of a root Definition: When the square root is powered Example: - Multiply with the power and the number, then use the root

  19. Now you will take a quiz!Look at the sample problem and give the name of the property illustrated. Click when you’re ready to see the answer. 1. 30 + 2 = 2 + 30 Answer: Commutative Property (of Addition)

  20. Now you will take a quiz!Look at the sample problem and give the name of the property illustrated. Click when you’re ready to see the answer. 2. If 65=54, and 25=65, then 25=54 Answer: Transitive Property (of Equality)

  21. Solving 1st power inequalities in one variableA. With only one inequality signExample:2+x > 1516>15, so 2+14>15B. ConjunctionExample:3 < x+15 < 2018 is greater than 3, but less than 20, so 3 < 3+15 < 20C. DisjunctionExample:x+2 > 4045 > 40, so 43+2 > 40

  22. slopes of all types of lines: equations of all types of linesExample: y=3x+5standard/general form: Ax+By=Cpoint-slope form: (y-y1)=m(x-x1)how to graph: Use slope-intercept form (y=mx+b)Example: y=3x+5(5,0) is y-intercept, and 3 is slopeup (3,1) from y-intercept Linear Equations in Two Variables

  23. Linear Equations in Two Variables (cont.) how to find intercepts: find intercepts by using slope-intercept form: y=mx+b how and when to use the point-slope formula: The point-slope formula=

  24. Substitution MethodDefinition: When you subtract one side of an equation with another to get an answerExample: 5x+y=10, x+4y=10--Get y alone, solve for x1. y=10-5x, x+4(10-5x)=102. x+40-20x=10, -19x=10-403. y=10-5(30)y=-140 Linear Systems

  25. Linear Systems (cont.) Addition/Subtraction Method (Elimination)Definition: When you add two equations to find the variable. Example: 5x+4y=4, 4x-4y=5 -Add these two equations • 9x=9 or x=1 -Plug in x to first equation to find y 2. 5(1)+4y=4, 4y=-1 Y=-1/4

  26. Linear Systems (cont.) Check for understanding of the terms dependent, inconsistent and consistent Dependent: When two lines are on top of each other Example: Consistent: When two lines cross once Example:

  27. Linear Systems (cont.) Inconsistent: When two lines are parallel Example:

  28. Sum and Difference of Cubes: Break the factors down: Difference of Squares:Break the factors down:Grouping 3x1:Break the factors down:GCF: Find GCF:Solve: Factoring

  29. Factoring (cont.) • Grouping 2x2 • Example: • -Multiply with the power(2) • Multiply 9 with the number of the power(2) • Solve: Reverse FOIL Example: (x+5)(5+x) Solve: PST -PST stands for perfect square trinomial Example: -Multiply using the power • Multiply the number 9 times the number of the power(2) Solve:

  30. A. Simplify by factor and cancelExample: -factor, then cancel out common factors Rational Expressions

  31. Rational Expressions (cont.) Addition and subtraction of rational expressionsExample: -Add numerator, and cancel

  32. Rational Expressions (cont.) Multiplication of rational expressionsExample: -Multiply, and simplify

  33. Quadratic equations in one variable Factoring Example for any terms: x=3 Example for binomials: X=5 Example for trinomials: -Put zero on one side, then find PST PST=-4

  34. Quadratic equations in one variable (cont.) Square root of both sides Example: X=8 Complete the square Example: • Get zero on its own side, then simplify

  35. A. What does f(x) mean?: f(x)=yB. Find the domain and range of a functionDomain: to see how much of x is coveredRange: to see how much of y is coveredExample:(1,1)(4,4)(5,3)Domain is 4- From 1 to 5Range is 3- From 1 to 4 Functions

  36. Functions (cont.) Given two ordered pairs of data, find a linear function that contains those pointsExample: (3,5)(7,10) -Find the slope -Put slope in point-slope formula

  37. Functions (cont.) Quadratic functionsFormula: Example: Because a=1, graph opens up To find x-intercept, set y to 0 0=(x-4)(x+2)=(4,0)(-2,0) To find y-intercept, set y to 0 Vertex: formula= Axis of symmetry: x=1 The discriminant ( ) tells you the number of x-intercepts.

  38. When two exponents are multiplied, addExample:When an exponent is multiplied by a power, multiply the powersExample:If an exponent has a 0 power, the answer is 1Example:When an exponent has a negative power, the answer is a fractionExample: Simplifying Expressions with Exponents

  39. When square root has a second power, the square root and power cancel outExample: x+2=225x=223 Simplifying Expressions with Radicals

  40. Word Problems Mike spent $95 on Packer tickets. This was $25 less than twice of what John spent. How much were John’s tickets? 2x-25=95 2x=70 X=35 John’s Packer tickets cost $35.

  41. Word Problems (cont.) There are n number of players on a basketball team. This is four less than two times the amount of coaches. How many coaches are on the team? 2x-4=n 2x=b+4 X=2 There are two coaches.

  42. Word Problems (cont.) • The sum of two consecutive numbers is 21. What are the numbers? 2x=21-1 2x=20 X=10 The numbers are 10 and 11

  43. Word Problems (cont.) Two students are running for class president. One student got 30 more votes than the other student. If the total amount of votes is 45, how much votes did each person get? x+x+30=35 One x is bigger than the other x, so the first x is 10, and the second x is 5

  44. When do you use this?: Linear regression on the TI-84 calculator is used when you want to find the line of best fit, or when you want to see the stat plots. You can also see both of these together.How does your calculator help?: Your TI-84 calculator can find out and graph your line of best fitExample: (-18,4)(-11,7)(-2,12.5)(1,14)(6,16) Line of Best Fit or Regression Line

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