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1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Remember to grab a green Homework check slip! Assignment #: 3.2.1 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) What’s wrong with this picture? 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Assignment #: 3.2.1 Write your answers for the following numbers: 2. 3. 4. 5. 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Assignment #: 3.2.1 1-6 Write your answers for the following numbers: 2. No real solution 3. +/- 3 4. x=-2 x=-4 5. No real solution 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Review Ratio and Proportions 5 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Notes Ratio: Number of specific things Total number of items or Part Whole 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) What does a ratio mean? In Orem, The male to female ratio is 100:101 This means for every 100 males there are 101 females in Orem At Mt. View High School the student to teacher ratio in 28:1 This means for every 28 students there is 1 teacher at Mt. View 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) To change a ratio from a fraction to a percent, divide then move the decimal two to the right. Move two to the right and then round to the tenth. 7/9 = .77777 = 77.8% 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Use the information to write a ratio in lowest terms. a. Hamburgers to veggie burgers The number of people who prefer the following: b. Hot dogs to total Hamburger: 32 Hot dogs: 26 c. Hot dogs to hamburgers Veggie burger: 9 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) In a survey of 40 people the results are: regular skittles: 24 sour skittles: 6 double flavor: 10 What is the proportion of sour to total? 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Notes A proportion is an equation written in the form stating that two ratios are equivalent. To solve proportions, cross multiply. 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Solving Proportions Cross Multiplying 7 x 12 36 7(36) = 12x 252 = 12x 21 = x 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Solve for x. 3 x 5 15 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Solve for x. 5 x 20 4 1 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Setting Up Proportions If you are comparing apples to oranges, you could set the equation up two ways: Apples 1 Apples 2 = Oranges 1 Oranges 2 Remember: KEEP RATIOS or CONSISTANT!! Apples 1 Oranges 1 = Apples 2 Oranges 2 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) A dance number needs 4 guys for every 3 girls. If there are 42 girls, how many guys are there? 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) A dance number needs 4 guys for every 3 girls. If there are 56 dancers, how many guys are there? 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) A dance number needs 4 guys for every 7 girls. If there are 42 girls, how many guys are there? 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) How to use the BLUE paper: Example: Word Their Definition My Definition Variable Represents a value or The letters in the unknown quantity equations 19 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Introduction Algebraic expressions are mathematical statements that include numbers, operations, and variables to represent a number or quantity. We know that a variable is a letter used to represent a value or unknown quantity that can change or vary. We have seen several linear expressions such as 2x + 1. In this example, the highest power of the variable x is the first power. In this lesson, we will look at expressions where the highest power of the variable is 2. 20 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Notes • A quadratic expression (ax2 + bx + c) is an expression where the highest power of the variable is the second power. • The quadratic expression 4x2 + 6x – 2 is made up of many component parts: terms, factors, coefficients, and constants. • A factor is one of two or more numbers or expressions that when multiplied produce a given product. In the given expression, the factors of 4x2 are 4 and x2 and the factors of 6x are 6 and x. 21 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Notes • A coefficient is the number multiplied by (in front of) a variable. • Example: In the given expression, the coefficient of the term 4x2 is 4 and the coefficient of the term 6x is 6. • A constant term that does not contain a variable. The value of the term does not change. • Example: –2 is a constant. 22 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Pg. 253 in your Folder In order to pay for upkeep of a local highway, the transportation department has set up tollbooths at each of the highway’s exits. Drivers are charged a toll of $1.20 for use of the highway, and are then charged an additional $0.04 per mile driven. 1. Write an algebraic expression that can be used to represent the total toll charged if m represents the number of miles driven. 2. What is the toll charged to drive 14 miles on this highway? 3.1.1: Identifying Terms, Factors, and Coefficients 23 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) 2. What is the toll charged to drive 14 miles on this highway? • Substitute 14 for m and evaluate the expression. 1.20 + 0.04(14) = 1.20 + 0.56 = 1.76 • The toll charged for driving 14 miles on the highway is $1.76. 24 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Notes • A monomial is a number, a variable, or the product of a number and variable(s). (Example: 5x2, 4, y) • A polynomial is a monomial or the sum of monomials. A polynomial can have any number of terms. • A binomial is a polynomial with two terms. (Example: 6x + 9) • A trinomial is a polynomial with three terms. (Example: 4x2 + 6x – 2) • Like terms (same variables raised to the same power) can be combined by adding. 25 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice Example 1 – with steps Identify each term, coefficient, and constant of 6(x – 1) – x(3 – 2x) + 12. Classify the expression as a monomial, binomial, or trinomial. Determine whether it is a quadratic expression. 26 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 1, continued 1. Simplify the expression. The expression can be simplified by following the order of operations and combining like terms. 6(x – 1) – x(3 – 2x) + 12 Original expression 6x – 6 – x(3 – 2x) + 12 Distribute 6 over x – 1. 6x – 6 – 3x + 2x2 + 12 Distribute –x over 3 – 2x. 3x + 6 + 2x2 Combine like terms: 6x and –3x; –6 and 12. 2x2 + 3x + 6 Rearrange terms so the powers are in descending order. 27 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 1, continued 2. Identify all terms. There are three terms in the expression: 2x2, 3x, and 6. 28 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 1, continued 3. Identify all coefficients. The number multiplied by a variable in the term 2x2 is 2; the number multiplied by a variable in the term 3x is 3; therefore, the coefficients are 2 and 3. 29 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 1, continued 4. Identify any constants. The quantity that does not change (is not multiplied by a variable) in the expression is 6; therefore, 6 is a constant. 30 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 1, continued 5. Classify the expression as a monomial, binomial, or trinomial. The polynomial is a trinomial because it has three terms. 31 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 1, continued 6. Determine whether the expression is a quadratic expression. It is a quadratic expression because it can be written in the form ax2 + bx + c, where a = 2, b = 3, and c = 6. ✔ 32 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Addition Increased by More than Combined Total of Sum Added to Multiplication Of Times Product of Increased by a factor of Subtraction Decreased by Minus, less Difference between Less than, fewer than Division per, Out of Ratio of, quotient of Percent (divide by 100) 33 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice Example 2 Translate the verbal expression ‘take triple the difference of 12 and the square of x, then increase the result by the sum of 3 and x” into an algebraic expression. Identify the terms, coefficients, and constants of the given expression. Is the expression quadratic? 34 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice Example 3 – with steps A fence surrounds a park in the shape of a pentagon. The side lengths of the park in feet are given by the expressions 2x2, 3x + 1, 3x + 2, 4x, and 5x – 3. Find an expression for the perimeter of the park. Identify the terms, coefficients, and constant in your expression. Is the expression quadratic? 35 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 3, continued 1. Find an expression for the perimeter of the park. Add like terms to find the perimeter, P. P = 2x2 + (3x + 1) + (3x + 2) + 4x + (5x – 3) Set up the equation using the given expressions. P = 2x2 + 3x + 3x + 4x + 5x + 1+2–3 Reorder like terms. P = 2x2 + 15x Combine like terms. The expression for the park’s perimeter is 2x2 + 15x. 36 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 3, continued 2. Identify all terms. There are two terms in this expression: 2x2 and 15x. 37 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 3, continued 3. Identify all coefficients. The number multiplied by a variable in the term 2x2 is 2; the number multiplied by a variable in the term 15x is 15; therefore, 2 and 15 are coefficients. 38 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 3, continued 4. Identify any constants. Every number in the expression is multiplied by a variable; therefore, there is no constant. 39 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Guided Practice: Example 3, continued 5. Determine whether the expression is a quadratic expression. It is a quadratic expression because it can be written in the form ax2 + bx + c, where a = 2, b = 15, and c = 0. ✔ 40 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Shanna wants to decorate the triangular deck behind her house. The base of the triangle is 10 meters shorter than the altitude. What are the terms, factors, and coefficients of the quadratic expression that represents the area of the deck to be decorated? 41 3.1.1: Identifying Terms, Factors, and Coefficients 1-5 (Ratio, Proportions) Identify terms, factors, coefficients (3.1.1) Assignment Practice 3.1.1 Pg. 259-260 (odds) R1-5 Ratio and Proportions (odds) 42