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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. Natural Numbers: {1, 2, 3, …..} Whole Numbers {0, 1, 2, 3, ….} Integers: {…-2, -1, 0, 1, 2...} Rational Numbers Ex: 5, 0, -5/3, 7/9, 0.5, 4/1, 1.3333…. (Includes repeating decimals and terminating decimals) Irrational Numbers: Ex: 𝟐, π, -2.5613402817….. (Includes non-terminating decimals) 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. Tools and Rules: Vocabulary: Uniform Motion- motion at a constant speed Isosceles triangle-two sides, two angles are the same. Examples: D = RT See Examples on pg. 43-45 and Pg. 49-52 Consecutive numbers: 5 less than a number n, n + 1, n + 2,… n, n + 2, n + 4...(for odd/evens) Area of a Triangle: A = ½bh Area of a square or rectangle: A=bh 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. 2) 3) 4) 5) Write down given info. Choose a variable to replace unknowns. Write an equation using variable. Solve equation to find variable. 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. Tools and Rules: Will do on the board Vocabulary: Absolute Value= the distance between a number and the origin (0). 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. Tools and Rules: Vocabulary: Absolute Value= the distance between a number and the origin (0). 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 – y1 = m(x - x1) 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. Tools and Rules: See next slide… Vocabulary: System of equations- a set of linear equations with the same two variables. 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) < or > boundary line will be dotted < or > boundary line will be solid Examples: Next slide… 1) Graph 3x – 2y < 8 2) Graph x – 3 < 0 1) Graph 3x – 2y < 8 2) 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 Vocabulary: Function- A relation between two sets (say D and R) that assigns to each member of D exactly one member of R. Objectives: Understanding and graphing functions Domain = Input values Range = Output values Value (of a function) = members of its range Tools and Rules: Ways to test a function: Mapping Vertical Line Test (of a graph) Function notation: f(x) = 3x + 2 (Read as: “f of x is 3x + 2) Two limitations of domain: When denominator = 0 −# Book: f, g, F, G, and phi (see pg. 142) = used to name functions tells you the “rule” for x Examples: Next slide… 1) f(x) = 8 – ½ x ; D = {-4, -2, 0, 2, 4) 2) If f(x) = 1 𝑥 2 +5𝑥+6 , what is the domain? (Hint: Think what can x NOT b?) 1) f(x) = 8 – ½ x ; D = {-4, -2, 0, 2, 4) R = {6, 7, 8, 9, 10} 2) If f(x) = 1 𝑥 2 +5𝑥+6 , what is the domain? (Hint: Think what can x NOT b?) D = {x: x ≠ -2 and x ≠ -3) 3) Pg. 145 #35 Vocabulary: Objectives: Recognizing Linear equations have a constant change. and using linear functions Tools and Rules: Examples: Next slide… f(x) = mx + b Rate of change m = 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒇(𝒙) 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙 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. x f(x) 0 1 1 4 2 3 Vocabulary: Relation- any set of ordered pairs. Objectives: Know the Function- a relation in which each input value is linked difference between a relation and a function. Tools and Rules: Ways to test a function: Mapping Vertical Line Test (of a graph) to a specific output value. 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… Examples: Next slide… Tools and Rules: When you subtract polynomials, remember to distribute the negative sign!! 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, Vocabulary: and subtract polynomials. See Homework or pg. 167 Examples: Tools and Rules: Simplify polynomials by combining like terms. When you subtract polynomials, remember to distribute the negative sign!! 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 Examples: Tools and Rules: (a and b must be real numbers, m and n must be positive integers) 1. aman = am+n 2. (ab)m = ambm 3. (am)n = amn (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. Examples: Tools and Rules: FOIL: First, Outside, Inside Last 3 special cases: (a + b)2 = a2 + 2ab + b2 (a- b)2 = a2 -2ab + b2 (a+ b)(a – b) = a2 - b2 Next slide Multiply 1) (2a – 5)(3a + 4) 2) (3c + 2)2 3) (4k + 7)(4k – 7) 4) (3x + 1)(x2 + 2x – 6) Objectives: Use the distributive property to multiply polynomials. Vocabulary: binomial- a polynomial that has 2 terms. trinomial- a polynomial with 3 terms. Examples: Tools and Rules: FOIL: First, Outside, Inside Last 3 special cases: (a + b)2 = a2 + 2ab + b2 (a- b)2 = a2 -2ab + b2 (a+ b)(a – b) = a2 - b2 Next slide Multiply 1) (2a – 5)(3a + 4) = 6a^2 -7a - 20 2) (3c + 2)2 3) (4k + 7)(4k – 7) 4) (3x + 1)(x2 + 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: 48u2v2 and 60uv3w Find the prime factorization of 98. Find the GCF and the LCM for 12a2s3 and 8a2r2s Find the prime factorization of 98. Find the GCF and the LCM for 12a2s3 and 8a2r2s 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 Area? = X2 x 1 1 =X =1 Objectives: Factor quadratic Polynomials. Vocabulary: Quadratic Polynomial= ax2 + bx + c (a ≠ 0) Tools and Rules: Examples: Next slide T-chart: (see board) *Use different method if the x2 term has a coefficient of 1 or another number. Factor x2 + 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 x2 – 2x - 8 Factor 15t2 – 16t + 4. Use a different method for factoring when the leading term (one with degree 2) has a coefficient > 1. Practice: Factor 2m2 – m - 1 Objectives: Factor polynomials using GCF, special products, and grouping. Tools and Rules: Perfect Square Trinomials: 1) a2 +2ab + b2 = (a + b)2 2) a2 –2ab + b2 = (a - b)2 Difference of Squares: 3) a2 – b2 = (a + b)(a – b) Vocabulary: None Examples: Next slide Type 1: Two variables Type 2: Negative degree 2 term Pg. 190 Example 6 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) a2 +2ab + b2 = (a + b)2 2) a2 –2ab + b2 = (a - b)2 Difference of Squares: 3) a2 – b2 = (a + b)(a – b) Sum and Difference of Cubes: a3 + b3 = (a + b)(a2 – ab + b2 ) a3 – b3 = (a - b)(a2 + ab + b2 ) 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 Tools and Rules: Vocabulary: (Leave a space, wait for later) Examples: (Leave a space, wait for (Leave a space, wait for later) 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. 1st: Write the equation with 0 as one side 2nd: Factor the polynomial side of the equation. 3rd: 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) 12 + 4m = m^2 {6, -2}