Lesson #1 Using Variables to Create Models of the Real World 1. Write the algebraic expression that is represented by these algebra tiles. Answer: x + 0. Identify each constant, term and expression for x x + Constant(s) Term(s) Expression(s) Constant(s) Term(s) z and x Expression(s) x x +. What is the coefficient of the x term for 8x + 7y =? Answer The coefficient of the x term is 8. 4. The area of a rectangle is length times width. The perimeter is found by adding all the sides. Write the formula that would help calculate the amount of fencing needed for the playground shown in the diagram. Answer Calculate perimeter to find amount of fencing. Perimeter = x + 4 + 17 + x + x + 4 + 17 x (your terms may be in a different order) Combine like terms Perimeter = 4x + 4 Your final answer should look like this. Lesson # Classifying and Comparing Numbers. List the absolute value and opposite of -0. Answer Absolute value -0 = 0 The opposite of -0 is 0. 6. List the absolute value and opposite of - 6 Answer Absolute value = - 6 Since the negative sign is outside the absolute value brackets, it stays. Opposite of -6 is 6. 1
Directions for questions 7 1 List all of the number classifications that apply to the given number. The classifications are: natural, whole, integer, rational, irrational, real. 7. - Answer integer, rational, real 8. 0 Answer whole, integer, rational, real 9. ¾ Answer rational, real 10. 16/4 Answer whole, integer, rational, real 16/4 simplifies to 4 which is a whole number 11. - 64 Answer integer, rational real - 64 = - 8 which is an integer 1. Answer irrational, real 1. A number that is a repeating decimal Answer rational, real Terminating decimals and repeating decimals are rational. 14. 14 0 = Answer Undefined Any number divided by zero is undefined. Lessons # and #4 Evaluating Algebraic Expression Containing Exponents 1. Write ()()(x)(x)(x)(x) in exponential form and standard form. Exponential form Standard form Answer Exponential form x 4 Standard form 9x 4 16. Which of the following is true? Why? a) -11 = (-11) b) -11 > (-11) c) -11 < (-11) Answer -11 < (-11) is true because the exponent only applies to the number to which it is attached or to the numbers inside the parenthesis to which the exponent is attached. -11 = - 11 (-11) = (-11)(-11) = +11 17. = Write the answer in exponential form and list the property of exponents Answer 7 Product of Powers
18. ( ) = Write the answer in exponential form and list the property of exponents Answer 6 Power of a Power x 19. = Write the answer in exponential form and list the property of exponents x Answer x Quotient of Powers 0. (14 x ) 0 = Write the answer in standard form and list the property of exponents Answer (14 x ) 0 =1 Zero Exponent Property 1. x Write the answer in exponential form and list the property of exponents Answer Power of Quotient x x 6. Write the answer in exponential form and list the properties of exponents 8 6 6 1 Answer 6 or Either is correct. Quotient of Powers and Negative Exponent Property 8 6 6 Lesson # How are Subtraction and Division Related to Addition and Multiplication. - + 4 0 + = Answer - + 4 0 + = - 4. 4x + 7 (-6x) = Answer 4x + 7 (-6x) =10x + 7 1 4 6. 1 4 6 4 Answer 6. 7 1 Find the common denominator first then solve. 4 8 4 Answer 7 1 6 7 1 18 1 1 8 4 8 4 4 6 8 4 4 4 4 4 7. What is the inverse operation of subtraction?
Answer Addition is the inverse of subtraction. 8. What is the inverse operation of multiplication? Answer Division is the inverse of multiplication. When dividing by a fraction, always invert the fraction then multiply. 9. A ship left the dock and sailed miles west. Then the ship sailed 0 miles east. Represent this motion on the number line. Write a mathematical expression that models the ship s motion relative to the ship s dock. Consider eastward motion as positive and westward motion as negative. Answer The ship s dock is at zero on the number line. The ship moves west (left) units then east (right) 0 units. The ship ends up 1 miles east of the dock. The mathematical expression is - + 0 = 1 0. A box contains 1 marbles. A scoop can hold x marbles. John removed scoops of marbles from the box then added 10 marbles back into the box. a) Write an algebraic expression that models, John removed scoops of marbles from the box then added 10 marbles back into the box. b) Use the algebraic expression from part A to write a new algebraic expression that models the number of marbles remaining in the box. Answer a) -x + 10 John removed scoops of marbles, so x is negative. He returned 10 marbles back to the box, so 10 is positive. b) Marbles remaining in the box 1 x + 10 = 14 x There are 14 x marbles left in the box. 1. (x)(-x ) = Answer (x)(-x ) = -1x 4 ac. 6 6ac ac 1 Answer 6a c a c 6 4 Lesson #6 (Evaluating Expressions by Adding, Subtracting, Multiplying and Dividing Real Numbers). List the correct order of operations used to simplify - + 4*-7. Answer Exponent 7 = 49 so -+4*-49 Multiply 4* so - + 0 49 Add and subtract - + 0 49 = - 4
4. List the correct order of operations used to simplify (8 ) + 7 Answer Parenthesis (8 ) = 6 so (6) + 7 Exponent 6 = 6 (6) + 7 Multiply (6) = 108 108 + 7 Add and subtract 108 + 7 = -96. List the correct order of operations used to simplify 6*9 Answer The fraction bar is a grouping symbol. Simplify the numerator. Exponent = 4 so 6*9 4 4 4 Multiply 6*9 = 4 Subtract (We are still simplifying the numerator.) 0 Now we can Divide 0 =10 6. Use the correct order of operations to simplify [4 + 6 (9 11)] Answer [4 + 6 (9 11)] = [4 + 6 (-)] = [4 + 6(-)] = [4 7] = [-68] = + 68 = 7 Lesson #7 Using the Distributive Property 7. Simplify 7(a 4b + ) Answer 7(a 4b + ) = 14a 8b + 8. Volume equals length times width times height. Write and simplify the expression that models the volume of a box with a length of, a width of x and a height of. Answer V = ()(x )() = (x 1)() = 10x 0 or V = ()()(x ) = 10(x ) = 10x 0 9. Use the distributive property to simplify this expression. x 4 8 7 Answer We added extra steps so you could see why the 7 s cancel. 4 4(8 x ) 4( ) 4(4* 7 x 8 ) 4( x * 7) 16 x ( 0) 16x 0 7 7 7 7 7 40. Simplify x(-x + 6) Answer x(-x + 6) = (-x)(-x) + (-x)(6) = x + (-6x) = x 6x Lesson #8 Combining Like Terms 41. Simplify + 8(x+) Answer + 8(x+) = + 8x + 16 = + 16 + 8x = 1 + 8x Distribute the 8, Commute the 16, Add + 16 4. Simplify and list the properties used for each step (-)(1)(x + ) + (x 1)
Answer (-)(1)(x + ) + (x 1) Multiply from left to right so (-)(1) -4(x + ) + (x-1) Distribute (-4)(x) + (-4)() + ()(x) ()(1) Multiply from left to right Find the additive inverse of - -1x + (-90) + 44x + (-) Commute 44x left -1x + 44x + (-90) + (-) Associate (-1x + 44x) + [(-90) + (-)] Combine like terms -91x - 11 4. Tania works 10 hours per week. Represent Tania s salary per hour as T. James makes dollars an hour more than Tania and works 1 hours per week. Bernie makes times 4 dollars per hour less than Tania (T-4) and works 1 hours per week. What is the expression that represents the total salary? List the property used in each step. Answer Tania = 10T James = 1(T + ) Bernie =1[(T - 4)] 10T + 1(T + ) + 1[(T 4)] Distribute within the brackets 10T + 1(T + ) + 1[T 1] Distribute the 1 across T + and 1 across T - 1 10T + 1T + 60 + 4T 180 Commute the 60 to the right 10T + 1T + 4T + 60 180 Associate terms with T and the terms that are constants (10T + 1T + 4T) + (60 180) Combine like terms 67T - 10 44. 1 8 1 1 m m g 4 4 Answer 4. (x + y) 4(x + y) = Answer (x + y) 4(x + y) Change the sign on the 4 and add (x + y) + (-4)(x + y) Distribute the. Distribute the -4. 10x + y + (-4x) + (-8y) Rearrange using the commutative property 10x + (-4x) + y + (-8y) Combine like terms. 6x + (-6y) or 6x 6y Lesson #9 Working with Ratios 46. The grocery store stocks shelves of store brand cereal for every shelves of name brand cereal. What is the ratio of name brand cereal to store brand cereal? Write this as a ratio and as a fraction. 6
Answer The ratio of name brand cereal to store brand cereal is :. In fraction form 47. A truck can travel 96 miles on gallons of gas. Express this as a rate, then calculate the unit rate as miles per gallon 96 miles 18 miles Answer The rate is The unit rate is gallons gallon. 48. What is the ratio of rabbits to foxes in a certain nature reserve if the biologist counted 10 rabbits and 6 foxes? Answer The ratio of rabbits to foxes is :1. If you answered 10:6, you need to reduce by dividing both by 6. If you answered 1:, you wrote the ratio of foxes to rabbits. Keep the ratio in the correct order. Lesson #10 Using Unit Analysis to Guide Problem Solving 49. Convert miles per gallon to kilometers per liter. The unit rates are 1mile:1.609 kilometers and 1 gallon:.79 liters. Answer: You can do this in one step or two steps. Do you see how the units cancel? one step method two step method 0. The pet store sells two brands of dog food. Brand A costs $8.0 for. pounds. Brand B costs $8. for 40 ounces. Which brand is more expensive? There are 16 ounces in a pound. Show your work. Answer: Brand B is more expensive. You can convert Brand A to ounces or you can convert Brand B to pounds. Here are both answers. Either method is correct. $8.0 1 lb $0.1 $8. $0.1 Brand A to ounces Brand B. lb 16 oz oz 40 oz oz 7
Brand A $8.0 $.4. lb lb $8. 16 0z $.0 Brand B to pounds 40 oz 1 lb lb 8