Semester A Notes

 The Real Number System  Objective: Review the real number system.  Tools and Rules

 Real Numbers {R}: Any number that can be shown on the number line.  Rational Numbers{Q}: Any number that can be written in the form a/b (where b ≠ 0, and a and b are integers).  Irrational Numbers{Q’}: Any number that cannot be written in the form a/b where a and b are integers.  Integer {Z}: A “whole” number (not a fraction or decimal) including positives, negatives, and 0.  Whole numbers: Same as set of integers, but not including Negative Numbers  Natural Numbers{N}: Same as integers, but not including Negatives or 0.

 Why do we classify numbers?  Have a 1-2 minute discussion with the person next to you, and write down why you think having a system of classification is useful for math.  Be prepared to discuss with the class.

 Objectives: To review the methods used to simplify numerical expressions and to evaluate algebraic expressions.  Tools and Rules  PEMDAS  |a| =  a if a > 0 (+ #)  0 if a = 0  a if a < 0 (- #)  Vocabulary  Sum  Difference  Product  Quotient  Power  Base  Exponent  Variable  Expression

 (A) Pg. 11, #24  (B) Pg. 11 #40

 Objectives: To review properties of equality of real numbers and properties for adding and multiplying real numbers.  Tools and Rules/Vocabulary  See properties of equality on pg. 14  See properties of real numbers on pg. 15  Commutative Property (x and +)  Associative Property(x and +)  Identity Properties(x and +)  Inverse Properties (x and +)  Opposite  Reciprocal  Distributive Property  Combines x and + or –  Multiplicative Property of 0  Multiplicative Property of -1

 Pg. 18 #17

(+)(+) = (+)(-)= (-)(-)= (-)(+)= (+)/(+)= (+)/(-) = (-)/(-) = (-)/(+) =  (5) + (-9) =  (5) – (9) =  (-5) +(9) =  (-5) + (-9)=  (-5) – (-9) =  (-5) – (9)=  To subtract any number, add the opposite (-)(-)(-) =? (-)^15 =? (-) ^20 = What’s the rule?

 Why is it important to have an order of operations when simplifying or solving problems?  Why do we need properties in math? How are they useful? (Extra: Can you think of an analogy?)  Have a 1-2 minute discussion with the person next to you, and write down your conclusions.  Be prepared to discuss with the class.

 Objectives: To review properties of equality and properties for adding and multiplying real numbers.  Vocabulary:  Field- A set of numbers that has all the properties listed  (The set of Real Numbers is a field.)  Tools and Rules/Vocabulary  See properties of equality on pg. 14  See properties of real numbers on pg. 15  Commutative Prop (x and +)  Associative Prop (x and +)  Identity Prop (x and +)  Inverse Prop (x and +)  Opposite  Reciprocal  Distributive Prop  Combines x and + or –  Multiplicative Prop of 0  Multiplicative Prop of -1

 Pg. 18 #17

(+)(+) = (+)(-)= (-)(-)= (-)(+)= (+)/(+)= (+)/(-) = (-)/(-) = (-)/(+) =  (5) + (-9) =  (5) – (9) =  (-5) +(9) =  (-5) + (-9)=  (-5) – (-9) =  (-5) – (9)=  To subtract any number, add the opposite (-)(-)(-) =? (-)^15 =? (-) ^20 = What’s the rule?

 Objectives: Solve equations with one variable.  Vocabulary:  Root = Answer  Formula = an equation showing a relationship between two or more variables.  Constant = Number that doesn’t change in a formula  Tools and Rules: One Variable equation answers: -1 solution (most cases) -No solutions (“empty” set or “null” set) -All values are solution (Identity)  Examples:  (1) -7.5(x – 3) = -30  (2) 3p – (p – 9) = 2(p + 5)  (3) 3(2s – 3) = 6(s – 1) – 15

 Objectives: Translating word phrases into equations and expressions.  Vocabulary:  Uniform Motion- motion at a constant speed  Isosceles triangle-two sides, two angles are the same.  Tools and Rules:  D = RT  Consecutive numbers:  n, n + 1, n + 2,…  n, n + 2, n (for odd/evens)  Area of a Triangle: A = ½bh  Area of a square or rectangle: A=bh   Examples:  See Examples on pg and Pg  5 less than a number  What is the sum of 3 consecutive even numbers if the middle one is x?

 Objectives: Persevere in mapping out and solving word problems.  Vocabulary:  Patience: Be patient when solving these problems! Don’t give up!!!  Tools and Rules: Plan for Solving a Word Problem: 1) Read problem carefully. Draw a picture. Write down given info. 2) Choose a variable to replace unknowns. 3) Write an equation using variable. 4) Solve equation to find variable. 5) Reread problem to check results.  Examples:  Group Activity

 Objectives: Solving simple and combined inequalities.  Vocabulary:  Conjunction- Inequality with AND. (True = Both True)  Disjunction- Inequality with OR. (True = One is true)  Tools and Rules:  <less than  >greater than  <less than or equal to  >greater than or equal to  ≠NOT equal to  When you divide or multiply by a -#, FLIP the inequality sign.  Examples:  Next page

 Solve, then graph: 3x + 16 < 7

 Solve, then graph: -2 < -x + 3 < 4

 Solve, then graph: c + 7 < 4 OR 7 – c < 1

 Objectives: Solve word problems using inequalities  Vocabulary:  None  Tools and Rules: Pg. 70 in textbook (phrase to translation)  Examples:  Next slide…

The Granada Hills football team is ordering sweatshirts for a fundraiser. The team is charged $5.50 for each sweatshirt and there is a delivery charge of $25 for the order. If the team sells the sweatshirts for $8 each, how many sweatshirts must be ordered and sold to produce a profit of at least $80?

 Objectives: Solving and graphing absolute value sentences.  Vocabulary:  Absolute Value= the distance between a number and the origin (0).  Tools and Rules: Will do on the board  Examples:  Next page

 |x| = 1  |x| > 1  |x| < 1

1) Solve |3x – 2| = 8 2) Solve |3 – 2t| < 5 3) Solve |2z – 1| + 3 > 8

 Objectives: Solve word problems using inequalities  Vocabulary:  None  Tools and Rules: Pg. 70 in textbook (phrase to translation)  Examples:  Next slide…

The Granada Hills football team is ordering sweatshirts for a fundraiser. The team is charged $5.50 for each sweatshirt and there is a delivery charge of $25 for the order. If the team sells the sweatshirts for $8 each, how many sweatshirts must be ordered and sold to produce a profit of at least $80?

 Objectives: Solving and graphing absolute value sentences.  Vocabulary:  Absolute Value= the distance between a number and the origin (0).  Tools and Rules:  Examples:  Next page

1) Solve |3x – 2| = 8 2) Solve |3 – 2t| < 5 3) Solve |2z – 1| + 3 > 8

 Objectives: Graphing a linear equation in two variables and finding the slope of a line.  Vocabulary:  (Referring to Pg. 107, draw a picture and label each of the following)  X-axis  Y-axis  Origin  Quadrants I, II, III, IV  Slope- The steepness of a line; change in y over change in x.  Tools and Rules: AX + BY = C (where A and B are not 0) is a linear equation in two variables. (Standard Form) One way to graph is find random values of x and solve for y. (create a table).  Examples:  Next slide…

 1) Graph:  (a) (0, -2)  (b) (1, -4)  (c) 2x + y = 5  (d) x = -3  2) Find the slope of a line containing the points (3, 5) and (4, 2).

1) Graph: (a) (0, -2) (b) (1, -4) (c) 2x + y = 5 (d) x = -3 2) Find the slope of a line containing the points (3, 5) and (4, 2). m = -3/1 or -3

 y = mx + b (m = slope)  Write each line in slope-intercept form:  x – y = 2  y + 3x = 1  2x + 3y = 5  Do you notice a pattern?  What is the “rule” when finding the slope from Standard form?  The slope of the line Ax + By = C (where B is not 0) is –A/B

 Objectives: Find the equation of a line given limited information.  Vocabulary:  Y-intercept: where graph crosses the y-axis  X-intercept: where graph crosses the x-axis  Tools and Rules: Point-Slope Form y – y 1 = m(x - x 1 ) Slope-Intercept Form y = mx + b Standard Form Ax + By = C Parallel lines = same slope Perpendicular lines = product of slopes = -1  Examples:  Next slide…

1) Give the standard form of the equation that goes through the points (1, 3) and (3, 7). 2) Give the standard form of the equation of the line through point P that is: (a) Parallel to L (b) Perpendicular to L P: (1,2) L: 3x + 2y = 4 (*Hint: first find the slope of L, then use point-slope form to find the line and convert it to Standard Form)

 Objectives: Solving systems of linear equations in two variables.  Vocabulary:  System of equations- a set of linear equations with the same two variables.  Tools and Rules:  See next slide…  Examples:  Next slide…

 When solving a system of equations, there are three possibilities:  1) The lines intersect  One solution (a point) (x, y)  2) The lines never intersect and are parallel  No solution  3) The lines are the same (“coinciding”)  Infinitely many solutions But wait, there’s more…

 There are two ways to solve systems of equations:  1) Substitution  2) Elimination (or, “linear-combination”)  *In both cases, the goal is to get rid of one of the variables so that we can combine the two equations into one to solve.

1) Solve using substitution: 2x + 3y = 13 x – y = 9 2) Solve using elimination (linear combination): 3x + 2y = 4 2x – 5y = -29

 Solve the system of equations: 4x - y = 7 3x – 5y = 11 Tell whether you used substitution or elimination and why.

Solve the systems: 1) 2x – y = 7 4x – 2y = 9 2) 3x – 2y = x + 8 y = x – 4

 Objectives:Graph single linear inequalities and systems of inequalities.  Vocabulary: None  Tools and Rules:  > or > shade above the line (or to the right)  < or < shade below the line (or to the left)  boundary line will be dotted  boundary line will be solid  Examples:  Next slide…

1) Graph 3x – 2y < 8 2) Graph x – 3 < 0

1) Graph 3x – 2y < 82) Graph x – 3 < 0

Solve 3x – 2y < 6

1) Graph the system: 2x + y > 1 x – y < 3

 Graph the system of inequalities: x > 0; y < 0; x + y > -1

 Objectives: Understanding and graphing functions  Vocabulary: Function- A relation between two sets (say D and R) that assigns to each member of D exactly one member of R. Domain = Input values Range = Output values Value (of a function) = members of its range  Examples:  Next slide…

 3) Pg. 145 #35

 Objectives: Recognizing and using linear functions  Vocabulary: Linear equations have a constant change.  Examples:  Next slide…

 The table shows a linear function. Fill in the missing information and state what its rate of change is. Then, plot the points and graph the line.  xf(x)

 Objectives: Know the difference between a relation and a function.  Vocabulary:  Relation- any set of ordered pairs.  Function- a relation in which each input value is linked to a specific output value.  Tools and Rules:  Ways to test a function:  Mapping  Vertical Line Test (of a graph)  Examples:  Next slide…

Use mapping to test whether the relation is a function. Then, check your work by graphing and using the vertical line test: (a) {(2, 1), (1, -2), (1, 2)} (a) {(3, 2), (1, 4), (5, 2), (7, 6)}

 Objectives: To simplify, add, and subtract polynomials.  Vocabulary: See next page (leave lots of room for this section…  Tools and Rules:  When you subtract polynomials, remember to distribute the negative sign!!  Examples:  Next slide…

 Working with a partner, open up to pg. 167 in your textbook.  Individually, write an example for each of the bold vocabulary words.  Then, trade with your partner, and write down their example also on your paper. Constant Monomial (Term) Coefficient Degree of a monomial Like Terms (like monomials) Polynomial Simplified Polynomial (Polynomial in Standard Form) Degree of a polynomial

 Simplify (Combine like terms and put in Standard Form). Then, give the degree of the polynomial: 1) 3x – x^2 – 9 + 2x^2 + 2 – 5x 2) Add -3x^2 – 4x + 9 and 2x^3 – 4x^2 – 5 3) Subtract 3x^2 + 4x – 5 from 2x^2 – 9x + 7

 Objectives: To simplify, add, and subtract polynomials.  Vocabulary: See Homework or pg. 167  Tools and Rules:  Simplify polynomials by combining like terms.  When you subtract polynomials, remember to distribute the negative sign!!  Examples:  Next slide…

 Simplify (Combine like terms and put in Standard Form). Then, give the degree of the polynomial: 1) 3x – x^2 – 9 + 2x^2 + 2 – 5x 2) Add -3x^2 – 4x + 9 and 2x^3 – 4x^2 – 5 3) Subtract 3x^2 + 4x – 5 from 2x^2 – 9x + 7

 Objectives: To use laws of exponents to multiply a polynomial by a monomial.  Vocabulary: None  Tools and Rules:  (a and b must be real numbers, m and n must be positive integers)  1. a m a n = a m+n  2. (ab) m = a m b m  3. (a m ) n = a mn  Examples:  (I will write on the whiteboard.)  Pg. 172

 Objectives: Use the distributive property to multiply polynomials.  Vocabulary: binomial- a polynomial that has 2 terms. trinomial- a polynomial with 3 terms.  Tools and Rules:  FOIL: First, Outside, Inside Last 3 special cases:  (a + b) 2 = a 2 + 2ab + b 2  (a- b) 2 = a 2 -2ab + b 2  (a+ b)(a – b) = a 2 - b 2  Examples:  Next slide

Multiply 1) (2a – 5)(3a + 4) 2) (3c + 2) 2 3) (4k + 7)(4k – 7) 4) (3x + 1)(x 2 + 2x – 6)

 Objectives: Use the distributive property to multiply polynomials.  Vocabulary: binomial- a polynomial that has 2 terms. trinomial- a polynomial with 3 terms.  Tools and Rules:  FOIL: First, Outside, Inside Last 3 special cases:  (a + b) 2 = a 2 + 2ab + b 2  (a- b) 2 = a 2 -2ab + b 2  (a+ b)(a – b) = a 2 - b 2  Examples:  Next slide

Multiply 1) (2a – 5)(3a + 4) = 6a^2 -7a ) (3c + 2) 2 3) (4k + 7)(4k – 7) 4) (3x + 1)(x 2 + 2x – 6)

 Objectives: Find the GCF and LCM of integers and monomials.  Vocabulary: A factor (noun)- a smaller unit of a number or polynomial found by division. To Factor (verb)- To write a number or polynomial as the product of its individual factors. Prime #- An integer that is only divisible by itself and 1.  Tools and Rules:  Prime Factorization: Writing a number as the product of only prime numbers.  (Trick: Divide the number by the smallest prime number until you have only prime numbers).  GCF-Greatest Common Factor (Hint: Factor is smaller than the number).  LCM-Least Common Multiple (Hint: Multiple is larger than the number).  Examples:  Next slide

Find the GCF and the LCM of the following numbers: 20, 28. With your seat partner, please solve this problem. Be prepared to discuss. Next, solve 84, 56, 140

 To find the GCF, take the least power of each common prime factor.  To find the LCM, take the greatest power of each prime factor.  Example:  Find the GCF and the LCM of the following numbers: 72, 108, 126.  GCF = 18  LCM = 1512

 Find the GCF and the LCM of the following monomials:  48u 2 v 2 and 60uv 3 w

 Find the prime factorization of 98.  Find the GCF and the LCM for 12a 2 s 3 and 8a 2 r 2 s

 Find the prime factorization of 98.  Find the GCF and the LCM for 12a 2 s 3 and 8a 2 r 2 s

 If we think of degree-2 polynomials as rectangles, we can model them with tiles:

 Each group will get a box or bag of tiles.  Each bag will have the following pieces (colors will vary):  Red = Positive Blue (or Purple) = Negative. x x 1 x 1 = X 2 Area? = X = 1

 Objectives: Factor quadratic Polynomials.  Vocabulary: Quadratic Polynomial= ax 2 + bx + c (a ≠ 0)  Tools and Rules:  T-chart: (see board) *Use different method if the x 2 term has a coefficient of 1 or another number.  Examples:  Next slide

 Factor x 2 + 2x -15  What is the form of our answer going to be?  What two numbers are we looking for?  Hint: Look at the second and last terms. (Reverse “FOIL” in your mind).  To give us -15, we must get a ____ of two numbers.  To give us 2x, we must get a _____ of the same two numbers.

 Factor x 2 – 2x - 8

 Factor 15t 2 – 16t + 4.  Use a different method for factoring when the leading term (one with degree 2) has a coefficient > 1.  Practice:  Factor 2m 2 – m - 1

 Objectives: Factor polynomials using GCF, special products, and grouping.  Vocabulary: None  Tools and Rules: Perfect Square Trinomials: 1) a 2 +2ab + b 2 = (a + b) 2 2) a 2 –2ab + b 2 = (a - b) 2 Difference of Squares: 3) a 2 – b 2 = (a + b)(a – b)  Examples:  Next slide

 Type 1: Two variables  Pg. 190 Example 6  Type 2: Negative degree 2 term  Pg. 189 Example 3

 Factor the polynomials:  1) 3x^3 – 15x^2 Method: Find GCF  2) y^2 – 49 Method: Use difference of two squares.  3) xy + 5x + 3y + 15 Method: Use grouping.

 1) 24x^2y – 40xy  2) x^2 + 14x + 49  3) 81 – 4a^2  4) 10q^2 – 5q + 2qt - t

 Objectives: Factor polynomials using GCF, special products, and grouping.  Vocabulary: None  Tools and Rules: Perfect Square Trinomials: 1) a 2 +2ab + b 2 = (a + b) 2 2) a 2 –2ab + b 2 = (a - b) 2 Difference of Squares: 3) a 2 – b 2 = (a + b)(a – b) Sum and Difference of Cubes: a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) a 3 – b 3 = (a - b)(a 2 + ab + b 2 )  Examples:  Next slide

 x^3 – 8  (x – 8)(x^2 + 8x + 64)  2g^4 + 54g  (g + 3)(g^2 – 3g + 3^2)

 t^3 – 64  (t – 4)(t^2 + 4t + 16)  a^6 + b^3  (a^2 + b)(a^4 – a^2b + b^2)

 Objectives: Solve Polynomial Equations  Vocabulary: (Leave a space, wait for later)  Tools and Rules: (Leave a space, wait for later)  Examples:  (Leave a space, wait for later)

 Is your polynomial an expression or an equation? Discuss with your partner.  Set your polynomial equal to zero.  Now it is known as a polynomial equation. (Write this in the vocabulary section).  Polynomial equation- A polynomial that equals zero, or its equivalent*.  *It could be rearranged.  If we wanted to solve the equation, we would need to find what value of the variable makes the equation a true statement.  The roots of a polynomial are the values of x (or variable) that satisfies the equation. (Write this in vocabulary).

 Factor the equation  Use the zero-product property*  Zero-Product Property: ab = 0 if and only if a = 0 or b = 0.  1 st : Write the equation with 0 as one side  2 nd : Factor the polynomial side of the equation.  3 rd : Solve the equation by setting each factor equal to 0.  * Write in Tools and Rules section

 Solve (x – 5)(x + 2) = 0  Solution set = {5, -2}  Solve x^2 = x + 30  Solution set = {-5, 6}  Solve 3x^2 = 4x(2x – 1)  Solution set = {0, 2/3, 2}

 1) q(2q + 6) = 0  {0, -3}  2) t^2 + 7t + 6 = 0  {-1, -6}  3) m = m^2  {6, -2}