MATH ALGEBRA AND FUNCTIONS

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1 Students: 1. Students express quantitative relationships using algebraic terminology, expressions, equations, inequalities and their graphs. 1. Use variables and appropriate operations to write an expression, equation, inequality, or system of equations or inequalities which represent a verbal description. Expression Write expressions for the following: a) Three less than a number b) as large as area A c) Three less than four times the number of eggs required for a chocolate cake d) decreased by the product of 3 and a number. e) more than three times a number Write the following verbal statements as algebraic expressions: a) The square of a is increased by the sum of twice a and 3. b) The product of 3 and 1/2 of a is decreased by the quotient, a divided by (-4). ( FW) Equation Five less than a number is equal to 15. What is the number? 12

2 Sarah bought 12 feet of framing material to make a rectangular frame for her oil painting. She uses all the framing material with no waste and the length is twice the width. Write an equation that represents this situation. The sum of two numbers is 24. One number is two more than the other. Find the numbers. Inequality The sum of three times a number and seven is greater than 13. A number n is less than or equal to 5. Jarvis wants to earn at least $23 this week. His father has agreed to pay him $2.25 an hour to weed the garden and $5.00 to mow the lawn. Write the inequality where h is the number of hours he spends weeding the garden and n is the number of times he mows the lawn. Choose two pairs of numbers that add up to less than 5 and differ by less than 4. 13

3 2. Use order of operations correctly to evaluate algebraic expressions, such as 3(2x + 5) 2. Evaluate. 5(2x + 7) with x = 4 3(x + 5) with x = 5 (-x) 2, -x 2 with x = 2 y 2 /3x with y = 6 and x = 3 h + k 2 /3h with h =2 and k = 4 y 2 - (x 3-4) with x =3 and y = 5 Given x = (-2) and y = 5, evaluate. a. x 2 +2x - 3 b. y(xy - 7) / 10 ( FW) *3. Simplify numerical expressions by applying properties of rational numbers (identity, inverse, distributive, associative, commutative), and justify the process used. Name the property illustrated. a) x(y + -y) = x (0) b) x(y + -y) = xy +x(-y) c) x(x + -y) = (y + -y)(x) d) x (y = -y) = x(-y +y) e) x(y(l/y)) = x(1) ( FW) Solve: = -3/8 x -8/3 = 8 x 23 = 8(20 + 3) = 6 x 99 = 6(100-1) = 4. Use algebraic terminology correctly (e.g., variable, equation, term, coefficient, inequality, expression, constant). Name the variable(s), coefficient(s) and constant(s) in each expression. 3x 2 - x + 5 3x 4 + 3x 2 -x 10 14

4 How many terms are in the expressions? 3x + y + 3 3x 2 Name the like terms. 3x -2x 4x 2 x What is the difference between an equation and an expression? 5. Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in terms of the situation represented by the graph. Write a description for what this graph might represent. y 2. Students interpret and evaluate expressions involving integer powers and simple roots. Students: 1. Interpret positive whole number powers as repeated multiplication and negative whole numbers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. altitude 10, Time/minutes) Write as the product of equal factors. 2 3 = a 4 = 2 3 = a -4 = Write without using exponents. 3y 4 Write using exponents. aabbcc 2mmnn mmm/nn 15

5 Simplify and evaluate expressions that include exponents Evaluate for a = 3, b = 5 a 7 b 5 a -2 b (a 2 )(a 4 ) (a -2 )(a -5 ) a 5 /a 4 a 3 /a -2 a -3 /a Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials, when the latter results in a monomial with an integer exponent. Multiply monomials x 4 x 2 n n 2-2x x 5 -c 5 c x 2 (x 3 ) 3x 2 (5x 3 ) Divide monomials x 4 /x a 5 /a 3 2x 3 /x 2-4x 2 /2x 2 Power of monomials (2/3) 4 (-2x) 2 (x 4 ) 3 Extracting roots 4 4x 2 Simplify as a non-fraction. x 5 /x 3 x 3 /x 5 x 5 /x 5, (42a 5 b 3 )/(14a 2 b 9 ) x -8 /x -7, (a 7 b -3 c 9 )/(a 4 b -3 c 10 ) ( FW) 3. Students graph and interpret linear and some non-linear functions. Students: 1. Graph linear, quadratic and cubic functions. 16 Graph y = 2x, y = 2x 2, y = 2x 3. What are the similarities and differences between the graphs?

6 Use the table to write the equation and graph the linear function. Explain the relationship between the table, graph and equation. x y x The area of a square is found by using the formula A = s 2. Graph this function to estimate the area of the square whose side is 3.5 units long. The Red fir is 8 inches tall and grows 6 inches per year. The Monterey pine is 12 inches tall and grows 5 inches per year. Use a graph to show when they are approximately the same height? 2. Plot the values from the volumes of a 3-D shape for various values of its edge lengths. Graph using the length of the side of a cube for x, and its volume for y. 17

7 Write down the equation of the surface area of a cube of side length x. Graph the surface area as a function of x. ( FW) * 3. Graph linear functions, noting that the vertical change (change in y-vary) per unit horizontal change (change in x- value) is always the same and know that the ratio (rise over run) is called the slope of a graph. Does the following chart represent linear(constant) growth? x y Find the slope of a line On a 5% grade, how many units of altitude do you gain for every 200 units you move in the horizontal direction? 25 units? 1 unit? Graph y = 2x and y = 4x. Find the slope of each line. Which one is steeper? Graph y = 3x and y = -3x. Which one has a negative slope? Graph the pair of points and determine the slope of the line through them. (3,1), (1,3) 18

8 Find the slope of a line that contains the points (-2,5) and (4,-5) and graph the line. Graph the equation y = 2x +2. Where does the graph cross the y axis? If the equation was changed to y = 2x - 2, where would the graph cross the y axis? Identify and use the slope intercept for to graph a linear function Graph y = 2/3x - 3 using the slope intercept form. *4. Plot values of the quantities whose ratio is always the same. Fit a line to the plot and understand that the slope of the line equals the quantities. Measure several different size cans and plot the relationship between the circumference and diameter. 12 Circumference Diameter 19

9 Use trend lines to interpret data Which package is the best buy? Draw a trend line to represent the relationship. 12 D Number of E cassettes C per A F package B 0 0 Cost Per Package $15 *4. Students solve simple equations and inequalities over the rational number. Students: * 1. Solve two-step equations and inequalities in one variable over the rational numbers, interpret the solution(s) in terms of the context from which they arose and verify the reasonableness of the results. Interprets the solution of two-step problems in terms of the context from which they arose and verify the reasonableness of the results 20 Solve two step equations. 15 = 3x + 2 c/4-19 = 17 Solve each inequality and graph their solutions on a number line. 9 5x - 3 m/3-7 > 11 Dr. Herrera recommended that Mitchell take eight tablets on the first day and then 4 tablets each day until his prescription was used. How many days will Mitchell be taking the pills after the first day?

10 You earn $2.00 for every magazine subscription you sell plus a salary of $10 a week. How many subscriptions do you need to sell each week to earn at least $40 a week? *2. Solve multi-step problems involving rate, average speed, distance and time, or direct variation. Quintan and Mark leave Elk Grove going in opposite directions. Quintan drives 10 miles per hour faster than Mark. In 4 hours they are 476 miles apart. How fast is each person traveling? The Daytona race is 500 miles long. If the winner finished the race in 3 hours, 11 minutes, and 10 seconds, what was the average speed in mile per hour? The Smiths drove to a friends house at 45 km/h. The return trip took 1 hour less because they traveled at 60 km/h. How long did it take them to get home? A car goes 45 mph and travels 200 miles. How many hours will it take? ( FW) 21

11 A plane leaves San Francisco flying at 450 mph. 1/2 hour later a second plane tries to catch the first, flying at 600 mph. How long will it take the second plane to catch the first? How far from San Francisco will this happen? ( FW) 22

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