Lesson 1 Reteach. A Plan for Problem Solving. Example. Exercises

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2 Lesson Reteach A Plan for Problem Solving Four-Step Plan When solving problems, it is helpful to have an organized plan to solve the problem. The following four steps can be used to solve any math problem.. Understand get a general understanding of the problem. Plan make a plan to solve the problem and estimate the solution. Solve use your plan to solve the problem. Check check the reasonableness of your solution Example Understand Plan According to a recent study, out of every 0 people is left-handed. If there are 7 people in the eighth grade, predict the number of students who are left-handed. You know that out of 0 people is left-handed. You also know that there are 7 people in the eighth grade. You need to predict how many of the students are left-handed. Make a table to organize the information and look for a pattern. Number of people Number who are left-handed Solve Check By extending the pattern, you can predict that 7 students will be left-handed. For every 0 students in the class, is left-handed. There are 7 groups of 0 in a class of 7 and 7 = 7. The answer is correct. Use the four-step plan to solve each problem.. James needs to buy one can of orange soda for every three cans of cola. If James buys cans of cola, how many cans of orange soda should he buy?. Bob s Video Venue has a membership fee of $.00 and DVD rentals are $.0 each. Video Heaven has no membership fee and DVD rentals are $.00 each. How many DVDs must be rented in order for Bob s Video Venue to be more economical?. A cookie shop offers 6 varieties of cookies and bakes dozen of each kind every day, from Monday through Friday. How many cookies are baked in four weeks?. Find the next term in, 6, 8,, 6,.... Jeremy needs to type a 00-word report for science class. He knows he can type about 9 words per minute. About how long will it take Jeremy to type his report? Math Accelerated Chapter The Language of Algebra

3 Lesson Reteach Words and Expressions A numerical expression contains a combination of numbers and operations such as addition, subtraction, multiplication, and division. Verbal phrases can be translated into numerical expressions by replacing words with operations and numbers. + - plus the sum of increased by more than minus the difference of decreased by less than times the product of of divide the quotient of divided by among Example Write a numerical expression for the verbal phrase. the product of seventeen and three Phrase the product of seventeen and three Expression 7 Evaluate, or find the numerical value of, expressions with more than one operation by following the order of operations. Step Step Step Evaluate the expressions inside grouping symbols. Multiply and/or divide from left to right. Add and/or subtract from left to right. Example Evaluate the expression. ( + 6) + ( + 6) + = (9) + Evaluate ( + 6). = 6 + Multiply and 9, and and. = 8 Add 6 and. Write a numerical expression for each verbal phrase.. eleven less than twenty. the product of seven and twelve. the quotient of forty and eight. sixteen more than fifty-four. the sum of thirteen and eighteen 6. three times seventeen Evaluate each expression (6 + ) -. (6 + ) -. [( - ) + ()]. [( + 7) 9] -. (8 7) -. (8) (9) Math Accelerated Chapter The Language of Algebra

4 Lesson Reteach Variables and Expressions An algebraic expression is a combination of variables, numbers, and at least one operation. A variable is a letter or symbol used to represent an unknown value. To translate verbal phrases with an unknown quantity into algebraic expressions, first define the variable. Algebraic Expressions The letter x is most often used as a variable. b 7d means 7 d. mn means m n. x+ 7d - means b. mn b To evaluate an algebraic expression, replace the variable(s) with known values and follow the order of operations. Substitution Property of Equality Words Symbols Example If two quantities are equal, then one quantity can be replaced by the other. For all numbers a and b, if a = b, then a may be replaced by b. Evaluate the expression if r = 6 and s =. 8s - r 8s - r = 8() - (6) = 6 - or Replace r with 6 and s with. Multiply. Then subtract. Translate each phrase into an algebraic expression.. twelve more than four times a number. the difference of sixty and a number. the quotient of the number of chairs and four. a number of books less than twenty-three. twenty dollars divided among a number of friends minus three Evaluate each expression if r =, s =, and t =. 6. t - rs 9. s + r 0t. s t 7. rs 8. t( + r) t 0.. (t - s)7. (t + r) - (r + s). st - rs (r + ) Math Accelerated Chapter The Language of Algebra

5 Lesson Reteach Properties of Numbers In algebra, there are certain statements called properties that are true for any numbers. Property Explanations Example Commutative Property of Addition a+b=b+a 6+=+6 9=9 Commutative Property of Multiplication a b=b a = 0 = 0 Associative Property of Addition (a + b) + c = a + (b + c) ( + ) + 7 = + ( + 7) = Associative Property of Multiplication (a b) c = a (b c) ( ) 8 = ( 8) 80 = 80 a+0=0+a=a = = 0 Multiplicative Identity a = a=a = = Multiplicative Property of Zero a 0=0 a=0 0 = 0 = 0 Additive Identity To simplify an algebraic expression, perform all operations. Properties can be used to help simplify an expression with variables. Example Simplify the expression. Commutative Property of Addition Associative Property of Addition Add 9 and 7. Name the property shown by each statement = + 7. ( ) = ( ). =. p 0 = 0 Simplify each expression.. + (x + 6) 6. (a) 7. ( + f ) (n + 7) + 9. (7 x) 8 0. (s 0) Math Accelerated Chapter The Language of Algebra (9 + r) + 7 (9 + r) + 7 = (r + 9) + 7 = r + (9 + 7) = r + 6

6 Lesson Reteach Problem-Solving Strategies Below are some examples of problem-solving strategies that can be used to solve a problem. Selecting and applying the appropriate strategy is significant in solving problems. Look for a Pattern Make a Table Guess, Check, and Revise Work Backwards Example Melanie is training for an upcoming marathon. In Week, Melanie runs miles. In Week, she runs. miles. In Week, she runs miles. Each week she increases the number of miles she runs by 0. mile. During which week will Melanie s running distance be 6 miles? One way to solve the problem is to make a table of values, starting with the given data, and extending the values until a distance of 6 miles is reached. Week Miles Melanie will be run a distance of 6 miles during Week 7 of her training. Use a strategy to solve each problem.. An aquarium contains 0 gallons of water. When the plug is pulled, water drains from the aquarium at a rate of gallons per minute. How many gallons of water still remain in the aquarium after 8 minutes?. The sum of three consecutive even numbers is 8. What are the numbers?. After a shopping trip to the mall, Ashley saw $6.0 in her purse. She spent $.80 on a pair of shoes, $9. on a necklace, and $8.8 on a belt. How much money did Ashley bring to the mall?. How many ways can you make change for $ using only dimes and nickels? Math Accelerated Chapter The Language of Algebra

7 Lesson 6 Reteach Ordered Pairs and Relations In mathematics, a coordinate system or coordinate plane is used to locate points. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. The point where the two axes intersect is the origin (0, 0). An ordered pair of numbers is used to locate points in the coordinate plane. The point (, ) has an x-coordinate of and a y-coordinate of. A relation is a set of ordered pairs, such as {(0, ), (, ), (, 6), (7, )}. A relation can also be shown in a table or a graph. The set of x-coordinates is the domain of the relation, while the set of y-coordinates is the range of the relation. Example Express the relation {(0, 0), (, ), (, ), (, )} as a table and as a graph. Then determine the domain and range. y 8 x y O x The domain is {0,,, }, and the range is {0,,, }. Refer to the coordinate plane shown at the right. Write the ordered pair that names each point.. P. Q. R. S Express the relation as a table and as a graph. Determine the domain and range.. {(, 6), (0, ), (, )} x y 6 y O 6 x O y R P Q S x Math Accelerated Chapter The Language of Algebra

8 Lesson 7 Reteach Words, Equations, Tables, and Graphs Relations can be described as words, equations, tables, and graphs. Words The distance biked is equal to miles per hour times the number of hours. Equation d = t Table Graph Time (h) Distance (mi) 6 8 Distance (mi) Time (h) Example Tori s computer backs up the file she is working on every minutes. Make a table to find the time for, 6, 9, and backups. Then graph the ordered pairs. Let m represent the number of minutes and b represent the number of backups. So, the rule is m = b. Exercise Input (b) b Output (m) () 6 (6) 0 9 (9) () 60 Minutes Number of Backups. Viktor s heart beats 7 times a minute. a. Write an equation to find the number of times Viktor s heart beats for any number of minutes. b. Make a table to find the number of times Viktor s heart beats in, 0,, and 0 minutes. c. Graph the ordered Heartbeats pairs for the relation. 00 m b Minutes Math Accelerated Chapter The Language of Algebra

9 Lesson Reteach Integers and Absolute Value A negative number is a number less than zero. A positive number is a number greater than zero. The set of integers can be written {, -, -, -, 0,,,, } where means continues indefinitely. Two integers can be compared using an inequality, which is a mathematical sentence containing < or >. Example Write an integer for each situation. Then identify its opposite and describe what it means. a. 6 feet below the surface b. strokes over par The integer is -6. The opposite is 6. It means 6 feet above the surface. The integer is + or. The opposite is -. It means strokes below par. Numbers on opposite sides of zero and the same distance from zero have the same absolute value. The symbol for absolute value is two vertical bars on either side of the number. = and - = Example Evaluate each expression. a. - b units = = + 6 =9 6 - =, 6 = 6 Simplify. On the number line, - is units from 0. Write an integer for each situation. Then identify its opposite and describe what it means.. inches less than normal. F above average. a deposit of $0. a loss of 8 yards Evaluate each expression if x = 8 and y = y 6. x - y 8. x + y 9. 6 y Math Accelerated Chapter Operations with Integers 7. x + y 0. x - y

10 Lesson Reteach Adding Integers Add Integers with the Same Sign Add their absolute values. The sum is: positive if both integers are positive. negative if both integers are negative. Example Find the sum - + (-). - + (-) = -7 Add - and -. The sum is negative. Add Integers with Different Signs Subtract their absolute values. The sum is: positive if the positive integer s absolute value is greater. negative if the negative integer s absolute value is greater. Example Find the sum = - - Subtract from -. = - or Simplify. = - The sum is negative because - >. Two numbers with the same absolute value but different signs are opposites. An integer and its opposite are also called additive inverses. This property is useful when adding or more integers. Example Find the sum + (- ) (-7). + (- ) (-7) = (- ) + (-7) Commutative Property = ( + 9) + [- + (-7)] Associative Property = + (-) or 0 Simplify. Find each sum (-). - + (-). 7 + (-). - + (-) (-0) (-8) (-) (-) (-). - + (-0) (-) (-) + (-) + (-6) (-7) (-) (- 0) (-7) (-) + + (-) Math Accelerated Chapter Operations with Integers

11 Lesson Reteach Subtracting Integers Subtract Integers To subtract an integer, add its additive inverse. Example Find each difference. a. 9-7 b = 9 + (-7) To subtract 7, add = -7 + (-) To subtract, add -. = -8 Simplify. = -0 Simplify. Example Find each difference. a. - (-) b (-) - (-) = + To subtract -, add (-) = -6 + To subtract -, add. = 9 Simplify. = - Simplify. To find the distance between two integers on a number line, you can count the units on the number line or use absolute value. Example Find the distance between and -9 on a number line. - (-9) = Find the absolute value of the difference of and -9. = Simplify. Find each difference (-9) (-6). - - (-). 9 - (-6) Find the distance between the integers on a number line.. - and and and and 7. - and and -6 Math Accelerated Chapter Operations with Integers

12 Lesson Reteach Multiplying Integers Multiply Integers with Different Signs Example The product of two integers with different signs is negative. Find each product. b. -8() -8() = -0 a. (-) (-) = - Multiply Integers with the Same Sign Example The product of two integers with the same sign is positive. Find each product. b. -7(-) -7(-) = 8 a. 6(6) 6(6) = 6 Example Evaluate xy if x = and y = -. Replace x with and y with -. Associative Property of Multiplication The product of and is positive. The product of and - is negative. Find each product.. -(7). 6(-9) (-) 6. -(-) 7. ()(-) 8. -(-6)(7) 9. -(-)(-9) Evaluate each expression if x = - and y = x. y. -x. -xy. xy. -x(-y) Math Accelerated Chapter Operations with Integers xy = ()(-) = [()](-) = (-) = -60

13 Lesson Reteach Dividing Integers Divide Integers with Different Signs The quotient of two integers with different signs is negative. Example Find each quotient. a. 6 (-) The signs are different. 6 (-) = -9 The quotient is negative. b. - 6 The signs are different. - = -7 6 The quotient is negative. Divide Integers with the Same Sign The quotient of two integers with the same sign is positive. Example Find each quotient. a. The signs are the same. = 7 The quotient is positive. b = - (-) The signs are the same. = The quotient is positive. To find the mean, or average, of a set of numbers, find the sum of the numbers and then divide by the number of items in the set. Example The diving depths in feet of 7 scuba divers studying schools of fish were -, -9, -, -8, -0, -7, and -0. Find the mean diving depth. - + (-9) + (-) + (-8) + (-0) + (-7) + (-0) 7 = -9 7 = - Simplify. The mean diving depth is - feet, or feet below sea level. Find each quotient. Find the sum of the diving depths. Divide by the number of divers.. 0 (-). -8 (-) The low temperatures in degrees Fahrenheit for a week were -,, -9,, 6, -, and -. Find the mean temperature. 8. During rounds of golf, James had scores of, -, 0, -, and -. Find the mean of his golf scores. Math Accelerated Chapter Operations with Integers

14 Lesson 6 Reteach Graphing in Four Quadrants The coordinates are (negative, positive). (-, +) The coordinates are (negative, negative). Example (-, -) - - y The coordinates are (positive, positive). (+, +) O x (+, -) The coordinates are (positive, negative). Graph and label each point on a coordinate plane. Name the quadrant in which each point lies. a. M(-, ) Start at the origin. Move units left. Then move units up and draw a dot. Point M(-, ) is in Quadrant II. M b. N(, -) Start at the origin. Move units right. Then move units down and draw a dot. Point N(, -) is in Quadrant IV. O x N Graph and label each point on the coordinate plane. Name the quadrant in which each point is located.. A(, 6). B(-, ). C(0, -). D(-, -). E(, 0) 6. F(, -) 7. G(-, ) 8. H(, -) 9. I(6, ). K(, -) 0. J(-, -8). L(-7, -) y O 6 7 8x Math Accelerated Chapter Operations with Integers y

15 Lesson Reteach Fractions and Decimals Some fractions can be written as decimals by making equivalent fractions with denominators of 0, 00, or,000. All fractions can be written as decimals by dividing the numerator by the denominator. Repeating decimals have a pattern in their digits that repeats without ending. If the repeating digit is zero, then the decimal is a terminating decimal. Example Write as a decimal is a terminating decimal. Example Write as a decimal is a repeating decimal. You can indicate that a decimal repeats by writing a bar or line over the repeating digit(s): 9 = 0.. It may be easier to compare numbers when they are written as decimals. Example < Replace with <, >, or = to make a true sentence. 6 Write as a decimal is to the left of on the number line, so -0.7 < Write each fraction as a decimal. Use a bar to show a repeating decimal Replace each with <, >, or = to make a true sentence Math Accelerated Chapter Operations with Rational Numbers

16 Lesson Reteach Rational Numbers A number that can be written as a fraction is a rational number. Mixed numbers, integers, terminating decimals, and repeating decimals can all be written as fractions. Any number that can be expressed as a b, where a and b are integers and b 0 is a rational number. Example Write each number as a fraction. a. = 7 Write the mixed number as an improper fraction. b is hundredths. 0. = 00 or 7 0 Simplify. Numbers can be classified into a variety of different sets. The diagram at the right illustrates the relationships among the sets of natural numbers, whole numbers, integers, and rational numbers. Decimal numbers such as π =.9... and are infinite and nonrepeating. They are called irrational numbers. Rational Numbers Integers Whole 0 Natural Example Identify all sets to which each number belongs. a This is neither a whole number nor an integer. Since can be written as - 8, it is rational. 00 b This is a nonterminating and nonrepeating decimal. So, it is irrational. Write each number as a fraction Identify all sets to which each number belongs Math Accelerated Chapter Operations with Rational Numbers

17 Lesson Reteach Multiplying Rational Numbers a c a c To multiply fractions, multiply the numerators and multiply the denominators: =, where b d b d b, d 0. Fractions may be simplified before or after multiplying. For negative fractions, assign the negative sign to the numerator. Example Find 7. Write in simplest form. 8 = 7 Rename mixed numbers as improper fractions. 8 = = 0 = or 0 Divide and by, and 8 and by. Multiply. Simplify. Algebraic expressions are expressions which contain one or more variables. Variables can represent fractions in algebraic expressions. Example ab = 7 Evaluate ab if a = and b = -. Write the product in simplest form. - ( )( ) - = ( 7 ) ( ) - = ( 7 ) ( ) 7 Replace a with and b with -. 7 Rename as = - or - The GCF of and is. Simplify. Find each product. Write in simplest form ( ) ( 7 ) ( ) 7, y = -, and z = -. Write the product in Evaluate each expression if x = 7 0 simplest form. 7. xy xz yz ( ). -x Math Accelerated Chapter Operations with Rational Numbers 9. xyz 9. y 0

18 Lesson Reteach Dividing Rational Numbers Two numbers whose product is are called multiplicative inverses or reciprocals. a b a b For any fraction, where a, b 0, a is the multiplicative inverse and a =. b b and are multiplicative inverses because =. This means that a a d ad c To divide by a fraction, multiply by its multiplicative inverse: = c =, where b, c, d 0. b b bc Find. Write in simplest form. Example d Multiply by the multiplicative inverse of,. = = Divide and 8 by their GCF,. 6 = or Simplify. Algebraic fractions are fractions which contain one or more variables. You can divide algebraic fractions just as you would divide numerical fractions. 0 Find qrs qs. Write in simplest form. Example 0 qs qrs qs = qrs 0 qs 0 0 Multiply by the reciprocal of qs,. qs = qrs 0 Divide out common factors. = Simplify. r Find each quotient. Write in simplest form. ( ) x 7. y y c 8. a ac d mn m 0. 9 yz 6x 0z. x 8d ( b b bc ab. st t. q q e de. 6i 8i. 9 0f f gh ) h Math Accelerated Chapter Operations with Rational Numbers

19 Lesson Reteach Adding and Subtracting Like Fractions To add fractions with the same denominators, called like denominators, add the numerators and write the sum over the denominator. So, a c + b c = a + c b, where c 0. Example Find 8 + (- 7. Write in simplest form. 8) 8 + (- 7 = 8) + (-7) 8 = - 8 or - Simplify. Example Find 9 +. Write in simplest form = ( + ) + ( 9 + = = 6 9 or The denominators are the same. Add the numerators. 9) Add the whole numbers and fractions separately or write as improper fractions. Add the numerators. Simplify. To subtract fractions with like denominators, subtract the numerators and write the difference over the denominator. So, a c - b c = a - c b, where c 0. Example 8-8 = - 8 = - 8 or - Find 8 -. Write in simplest form. 8 The denominators are the same. Subtract the numerators. Simplify. Find each sum. Write in simplest form ( ) + Find each difference. Write in simplest form (- ) Math Accelerated Chapter Operations with Rational Numbers

20 Lesson 6 Reteach Adding and Subtracting Unlike Fractions Fractions with different denominators are called unlike fractions. To add fractions with unlike denominators, rename the fractions with a common denominator. Then add and simplify. Example Find 7 +. Write in simplest form. 7 + = = + 7 = 9 Use 7 or as the common denominator. Rename each fraction with the common denominator. Add the numerators. To subtract fractions with unlike denominators, rename the fractions with a common denominator. Then subtract and simplify. Example Find Write in simplest form = 8 6 = = = 7 8 Write the mixed numbers as improper fractions. Rename fractions using the LCD, 8. Simplify. Subtract the numerators. Find each sum. Write in simplest form Find each difference. Write in simplest form ( ) (- 8) (-. 0 ) 6 - (- ) Math Accelerated Chapter Operations with Rational Numbers

21 Lesson Reteach Powers and Exponents { A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as a factor. So, has a base of and an exponent of, and = = 6. base exponent powerabo Example Write each expression using exponents. a The base is 0. It is a factor times, so the exponent is = 0 b. (p + )(p + )(p + ) The base is p +. It is a factor times, so the exponent is. (p + )(p + )(p + ) = (p + ) When evaluating expressions with exponents, follow the order of operations. Example Evaluate x - if x = -6. x - = (-6) - = (-6)(-6) - = 6 - = Replace x with is a factor times. Multiply. Subtract. Write each expression using exponents... (-7)(-7)(-7).. x x y y. (z - )(z - ) 6. (-t)(-t)(-t) ()()()() Evaluate each expression if g =, h = -, and m = g 0. (m - ) 8. g. -(g + ) Math Accelerated Chapter Powers and Roots 9. g - m. (h - m)

22 Lesson Reteach Negative Exponents A negative exponent is the result of repeated division. Extending the pattern below shows that - = or. = 6 = 0 = - = This suggests the following definition. a -n = n for a 0 and any whole number n. Example: 6 - = For a 0, a 0 =. Example: 9 0 = a Example a. - - = Write each expression using a positive exponent. b. y - Definition of negative exponent y Definition of negative exponent Write each fraction as an expression using a negative exponent other than -. a. 6 = 6-6 y - = b. Definition of negative exponent 8 = 8 9 = 9 - Definition of exponent Definition of negative exponent Write each expression using a positive exponent (-7) -8. b -6. n -. (-) j a - Write each fraction as an expression using a negative exponent other than Math Accelerated Chapter Powers and Roots Example 6

23 Lesson Reteach Multiplying and Dividing Monomials When multiplying powers with the same base, add the exponents. Example 7 = 7 = 7 + = 8 Example Symbols am an = am + n Example = + or 7 Find the product 7. = Product of Powers Property; the common base is. Add the exponents. Find the product a a. a - a = a - a = a - + = a - = 6a - Commutative Property of Multiplication Product of Powers Property; the common base is a. Add the exponents. Multiply. When dividing powers with the same base, subtract the exponents. Example m Symbols a = am - n, where a 0 an Example = 6 - or 6 (-8) (-8) Find the quotient. (-8) (-8) = (-8) - = (-8) Quotient of Powers Property; the common base is ( 8). Subtract the exponents. Find each product. Express using positive exponents v v. (f )(f 9). (- )(- -). (-cr -)(-r ) 6. 9z z u 6 (-6u ) 9. -m (m 6) Find each quotient. Express using positive exponents (-p 8) (-p ). (-) (-) w. w Math Accelerated Chapter Powers and Roots 0 e. - e

24 Lesson Reteach Scientific Notation Numbers like,000,000 and are in standard form because they do not contain exponents. A number is expressed in scientific notation when it is written as a product of a factor and a power of 0. The factor must be greater than or equal to and less than 0. By definition, a number in scientific notation is written as a 0 n, where a < 0 and n is an integer. Example Express the number in standard form = = = Move the decimal point 6 places to the left. Example Express the number 6,000,000 in scientific notation. 6,000,000 = 6. 0,000,000 The decimal point moves 7 places. = The exponent is positive. To compare numbers in scientific notation, compare the exponents. If the exponents are positive, the number with the greatest exponent is the greatest. If the exponents are negative, the number with the least exponent is the least. If the exponents are the same, compare the factors. Example Compare each set of numbers using <, > or =. a Compare the exponents: >. So, >. 0. b The exponents are the same, so compare the So, < factors: < Express each number in standard form Express each number in scientific notation. 7.,000,000, ,989,000, ,000,000 Order each set of numbers from least to greatest , , 7. 0, ,. 0 -,. 0, Math Accelerated Chapter Powers and Roots

25 Lesson Reteach Compute with Scientific Notation When you multiply and divide with numbers in scientific notation, multiply or divide the leading numbers first, then use the Product of Powers or Quotient of Powers properties to multiply or divide the powers of 0. Example Evaluate (.9 0 ) ( 0 ). Express the result in scientific notation. = (.9 ) (0 0 ) Commutative and Associative Properties. = (9.8) (0 0 ) Multiply.9 by. = Product of Powers = Add the exponents. When you add and subtract with numbers in scientific notation, the exponents must be the same. Sometimes you need to rewrite one of the numbers so it has the same exponent as the other. Example Evaluate (.68 0 ) + ( ). Express the result in scientific notation. = (.68 0 ) + (7 0 ) Write as 7 0. = ( ) 0 Distributive Property = Add.68 and 7. = Write in scientific notation. Evaluate each expression. Express the result in scientific notation.. (. 0 )(7. 0 ). (.07 0 )( ). (. 0 - )(7,000). ( )(8 0-7 ) (. 0 ) + (.77 0 ) 8. ( ) + ( ) 9. ( ) + (. 0 8 ) 0. ( ) - (8.8 0 ). ( ) - ( ). ( ) - (.7 0 ) Math Accelerated Chapter Powers and Roots

26 Lesson 6 Reteach Square Roots and Cube Roots A square root of a number is one of two equal factors of the number. A radical sign,, is used to indicate a positive square root. Every positive number has a positive square root and a negative square root. The square root of a negative number, such as - 6, is not real because the square of a number cannot be negative. Example Find each square root. a. - b. ± 9 Find the negative square root of ; =. - = - ± 9 = ±7 Find both square roots of 9; 7 = 9. A cube root of a number is one of three equal factors of the number. The symbol is used to indicate the cube root of a number. The cube root of a positive number is positive. The cube root of a negative number is negative. Example Find each cube root. a. 79 b. 79 = 9 Find each square root. - - = Find each cube root = or 79 (-) = (-) (-) (-) or Math Accelerated Chapter Powers and Roots

27 Lesson 7 Reteach The Real Number System The set of real numbers consists of all natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Rational numbers can be written as fractions. Rational Numbers 8 Irrational numbers are numbers that Integers - - Whole Numbers Irrational Numbers π 0... Natural Numbers Example Name all sets of numbers to which each real number belongs. Write natural, whole, integer, rational, or irrational. a. 7 This number is a natural number, a whole number, an integer, and a rational number. b. 0.6 This repeating decimal is a rational number because it is equivalent to. c. 7 It is not the square root of a perfect square so it is an irrational number. If x = y, then x = ± y. If x = y, then x = y. Example Solve the equation b =. Write the equation. b = b = ± Definition of square root b = and - Check = and (-) (-) = The solutions are and -. Name all sets of numbers to which each real number belongs. Write natural, whole, integer, rational, or irrational Solve each equation. Round to the nearest tenth, if necessary. 0. x = 9. h = 86. 6t = 78. s = 76. a =. m = -66 Math Accelerated Chapter Powers and Roots

28 Lesson Reteach Ratios A ratio is a way to compare two quantities using division. Ratios can be written in a number of ways. The ratio representing 7 out of can be written as: 7 to, 7:, and 7. Ratios are usually written as fractions in simplest form when the first number being compared is less than the second number being compared. Example correct answers number of questions = 6 0 = Express the ratio 6 correct answers out of 0 questions as a fraction in simplest form. Explain its meaning. Divide the numerator and denominator by the GCF,. The ratio of correct answers to questions is to. This means that for every questions, were answered correctly. Also, of the questions were answered correctly. When a ratio involves measurements, both quantities should have the same unit of measure. When the quantities have different units of measure, you must convert one unit to the other. It is usually easiest to convert the larger unit to the smaller unit. Example 6 feet inches = 7 inches inches = Express the ratio 6 feet to inches as a fraction in simplest form. Write the ratio as a fraction. Convert 6 feet to 7 inches. Divide the numerator and denominator by the GCF,. Written in simplest form, the ratio is to. Express each ratio as a fraction in simplest form.. weeks to plan events. 9 children to adults. 8 teaspoons to forks. 6 cups to 0 servings. 7 shelves to 8 books 6. 6 teachers to 6 students 7. inches to feet 8. 0 inches to yards 9. 9 feet to inches 0. gallons to quarts. pints to quarts. ounces to pounds Math Accelerated Chapter Ratio, Proportion, and Similar Figures

29 Lesson Reteach Unit Rates A rate is a ratio of two quantities having different kinds of units. different kinds of units 0 miles hours A unit rate is a rate with a denominator of. To change a rate to a unit rate, divide the numerator by the denominator. 0 miles miles = hours hour Example Express the rate $0 for 8 fish as a unit rate. 0 dollars 8 fish Write the ratio as a fraction. 8 0 dollars. dollars = 8 fish fish Divide the numerator and denominator by 8 to get a denominator of. 8 Express each rate as a unit rate. Round to the nearest tenth or nearest cent, if necessary.. $8 for tickets. $.9 for cans of soup. $7.90 for 6 people. 6 miles in hours. 7 pages in 8 days 6. $0 dollars over hours words in minutes 8. $6.99 for cans 9. Shawna strung necklaces in hours. How many necklaces could she string in 7 hours? 0. At Funtimes Gym, eight -hour classes cost $96. At Fitness Place, twelve -hour classes cost $. Which gym offers the best rate per hour?. Jamie downloaded 8 songs in minutes. At this rate, how many songs could he download in 0 minutes? Math Accelerated Chapter Ratio, Proportion, and Similar Figures The unit rate is $. per fish.

30 Lesson Reteach Complex Fractions and Unit Rates Complex fractions are fractions with a numerator that is a fraction, a denominator that is a fraction, or both that are fractions. Example = = = 8 or Simplify. Write the complex fraction as a division problem. Multiply by the reciprocal of, which is. Simplify. So, Simplify. is equal to Math Accelerated Chapter Ratio, Proportion, and Similar Figures

31 Lesson Reteach Converting Rates Dimensional analysis is the process of including units of measurement as factors when you compute. Example A cheetah can run short distances at a speed of up to 7 miles per hour. How many feet per second is this? You need to convert miles per hour to feet per second. Use mile = 80 feet and hour = 600 seconds. 7 mi h = 7 mi h 80 ft mi h 600 s 7 mi 80 ft h = h mi 600 s = 0 ft s Multiply by 80 ft mi and h 600 s. Divide the common factors and units. Simplify. So, 7 miles per hour is equivalent to 0 feet per second. Example Convert gallons to liters. Round to the nearest hundredth. Use liter 0.6 gallons. L gal gal Multiply by L 0.6 gal L gal 0.6 gal L or 7.8 L Simplify. 0.6 So, gallons is approximately 7.8 liters. 0.6 gal. Divide out the common units, leaving the desired unit, liter.. Jake was in a bicycle race. His average speed was miles per hour. At this rate, how many feet per hour did Jake travel?. Giant pandas can spend up to 6 hours a day eating bamboo. How many minutes per day is this?. Karin discovered that her leaky faucet was leaking. cups of water an hour. At this rate, how many gallons a day were leaking? Complete each conversion. Round to the nearest hundredth.. 8 L qt. 6 pt ml 6. kg lb 7. m in. 8. The average American uses about 90 gallons of water per day. How many liters per year is this? Math Accelerated Chapter Ratio, Proportion, and Similar Figures

32 Lesson Reteach Proportional and Nonproportional Relationships Two quantities are proportional if they have a constant ratio or rate. If they do not have the same ratio or rate, they are said to be nonproportional. Example Determine whether the distance traveled is proportional to the time. Explain your reasoning. Write the rate of distance to time for each column. Simplify each fraction. 00 = = = = 00 Time (min) Distance (yd) Since all of the rates equal 00, the distance traveled is proportional to the time. Proportional relationships can be described using equations of the form y = kx, where k is the constant ratio. The constant ratio is the constant of proportionality. Example The perimeter of a square with a side of inches is inches. A square s perimeter is proportional to the length of one of its sides. Find the constant of proportionality. Then write an equation relating the perimeter of a square to the length of one of its sides. What would be the perimeter of a square with 9-inch sides? Find the constant of proportionality between perimeter and side length. perimeter length of sides = or Words: The perimeter is times the length of a side. Variable: Let P = perimeter and s = the length of a side. Equation: P = s P = s P = (9) P = 6 Write the equation. Replace s with the length of a side. Multiply. The perimeter of a square with a side of 9 inches is 6 inches. Determine whether the set of numbers in each table is proportional. If the relationship is proportional, identify the constant of proportionality. Explain your reasoning.. Cookies 6 9 Cupcakes Population (00,000).... Years. Gloria earned $6 for babysitting hours. Find the constant of proportionality. Then write an equation relating money earned to the number of hours. How much would Gloria earn after babysitting hours? Math Accelerated Chapter Ratio, Proportion, and Similar Figures

33 Lesson 6 Reteach Graphing Proportional Relationships If two quantities are proportional, the graph of the two quantities is a straight line through the origin. You can use a graph of the quantities to find the constant ratio between the quantities, or the constant of proportionality. Example Miranda earns $ per hour for babysitting. Is the amount of money she earns proportional to the number of hours she spends babysitting? Time (hr) 0 Money ($) Graph the ordered pairs on the coordinate plane. Then connect the ordered pairs. The graph passes through the origin and is a straight line. So, the amount of money Miranda earns babysitting is proportional to the number of hours she spends babysitting. Check Write the ratio of money to time for each ordered pair in simplest form. 0 = = 60 = The ratios are all the same. The relationship is proportional. Money ($) O y (, 0) (, 60) (, ) (, ) x Time (hr) Determine whether each relationship is proportional by graphing on the coordinate plane. Explain your reasoning.. Radius Circumference π π 6π 8π 0π. Teachers Students A recipe for chocolate chip cookies uses cups flour and sticks of butter. Is the amount of butter used proportional to the number of cups of flour used? Explain your reasoning.. What is the constant of proportionality between the perimeter of a square to its side length, s? Explain what it means. Math Accelerated Chapter Ratio, Proportion, and Similar Figures

34 Lesson 7 Reteach Solving Proportions A proportion is an equation stating that two ratios or rates are equal. a b c d = An important property of proportions is that their cross products are equal. You can use this property to solve problems involving proportions. a b c d = ad = bc Example.. Solve the proportion. c = c =. = c 6. = c 6. c = 8.8 = c Cross products Multiply. Divide. Simplify. The solution is 8.8. Solve each proportion. 6 x. = 9 w. = u = 6. = s a. = w = 8 m 7. = 8 8. = 9. g = 0 0 f 6 h 7 0. =. z = 6. k. =.6. r =. d. = = q 7. d. 6. =. 7. c = n = = p y = 98. v. = Math Accelerated Chapter Ratio, Proportion, and Similar Figures

35 Lesson 8 Reteach Scale Drawings and Models Scale drawings or scale models represent objects that are either too large or too small to be drawn or built at actual size. The lengths and widths of objects on a scale drawing or model are proportional to the corresponding lengths and widths on the actual object. The scale of a drawing or model is the ratio of a given measure on the drawing or model and the corresponding measure on the actual object. If the measurements are in the same unit, the scale can be written without units. In this case, it is called the scale factor. Example A map shows a scale of inch = 6 miles. The distance between two places on the map is. inches. What is the actual distance? Let x represent the actual distance. Write and solve a proportion. map width actual width inch 6 miles =. inches x miles map width actual width x = 6. Find the cross products. x =. Simplify. The actual distance is. miles. Example Sam made a model car that is 9 inches long. The actual car that the model is based on is. feet long. Find the scale and the scale factor of the model. Write the ratio of the model s length to the length of the actual car. Then solve a proportion in which the model s length is inch and the length of the actual car is x feet. model length actual length 9 in.. ft = in. x ft model length actual length 9 x =. Find the cross products. 9x =. Simplify. x =. Divide each side by 9. Simplify. So, the scale is inch =. feet. To change this to a scale factor with the same units, first write as a ratio. in. in. scale inch =. feet. ft 8 in. :8 scale factor. Joanna knows the distance to her grandmother s house is miles. On a map, the distance is. inches. What is the scale of the map?. Kevin drew a scale drawing of his living room. The actual room is 6 feet long. If the room is inches long in the drawing, what is the scale of the drawing?. Cindy s dad made her a dollhouse that is a scale model of their house. If their house is feet tall and the model is inches tall, what is the scale of the model? Math Accelerated Chapter Ratio, Proportion, and Similar Figures

36 Lesson 9 Reteach Similar Figures Similar figures are figures that have the same shape but not necessarily the same size. If two figures are similar, then the corresponding angles have the same measure, and the corresponding sides are proportional. Because corresponding sides are proportional, you can use proportions or the scale factor to find the measures of the sides of similar figures when some measures are known. The scale factor is the ratio of a length on a scale drawing to the corresponding length on the real object. It is also the ratio of corresponding sides in similar figures. Example If the polygons ABCD and EFGH are similar, what is the value of x? CD AD = EH GH = x 6 C 7m The corresponding sides are proportional. Write a proportion. 7 B A m D F G Replace AD with, EH with 6, CD with 7, and GH with x. x = 6 7 x = x = xm Find the cross products. Simplify. 6 m E Division Property of Equality H The figures are similar. Find each missing measure. E. R. B 7m m 0 m A C R 7 in. S in. Q T M Q H x in. I N m xm U. T X in. G S 7 m P m O F D. V xm A J 8 m gm B 8m C. The art club is painting the mural shown at the right on a wall. Triangle QRS and triangle NOP are similar. a. Find the length of NO. b. Find the length of PN. Math Accelerated Chapter Ratio, Proportion, and Similar Figures Y Z m Q ft 6 ft R S P ft ft O N

37 Lesson 0 Reteach Indirect Measurement The properties of similar triangles can be use to find measurements that are difficult to measure directly. This is called indirect measurement. One type of indirect measurement is shadow reckoning. The diagram at the right shows how two objects and their shadows form two sides of similar triangles. You can use a proportion to find measures such as the height of the flag pole. ft? ft ft shadow 8 ft shadow Example A school building casts a 0.-foot shadow at the same time a.8-foot student casts a.-foot shadow. How tall is the school building to the nearest tenth? Understand Plan Solve You know the lengths of the shadows and the height of the student. You need to find the building s height. To find the height of the building, set up a proportion comparing the student s shadow to the building s shadow. Then solve. student s height.8 h =. building s height 0. The height of the school building is about. feet. student s shadow building s shadow.8 0. = h. Find the cross products..9 =.h Multiply.. h Divide each side by... Lena s house casts a shadow that is feet long at the same time that Lena casts a shadow that is. feet long. If Lena is. feet tall, how tall is her house?. Suppose a rocket outside a science museum cast a shadow that was 76 feet. At the same time, a.7-foot-tall person standing next to the rocket casts a shadow that is 9. feet long. How tall is the rocket?. A cell phone tower casts a shadow that is 9 feet. A building next to the tower is 8 feet high and casts a shadow that is. feet long. How tall is the cell phone tower? h ft 0. ft.8 ft. ft Math Accelerated Chapter Ratio, Proportion, and Similar Figures

38 Lesson Reteach Using the Percent Proportion In a percent proportion, one ratio compares part of a quantity to the whole quantity. The other ratio is the equivalent percent, written as a fraction, with a denominator of 00. Example Find each percent. a. Twelve is what percent of 6? b. What percent of 8 is 7? a b = p 00 6 = p a Replace the variables. 00 b = p = p = p 6 Find the cross products. p 8 = = 6p Multiply. 700 = 8p 7 = p Divide. 87. = p So, twelve is 7% of 6. So, 87.% of 8 is 7. Example Find the part or the whole. a. What number is.% of? b. is 6% of what number? a b = p 00 a =. a Replace the variables. 00 b = p 00 b = 6 00 a 00 =. Find the cross products. 00 = 6 b 00a = Multiply.,00 = 6b a = 0. Divide. 6 = b So, 0. is.% of. So, is 6% of 6. Use the percent proportion to solve each problem... is what percent of 8?. 9 is what percent of 00?. What percent of is?. What percent of 0 is?. What is 80% of 80? 6. What is % of 8? 7. What is 6% of 6? 8. 8 is 80% of what number? is 0% of what number? 0.. is 90% of what number? Math Accelerated Chapter 6 Percents

39 Lesson Reteach Find Percent of a Number Mentally When working with common percents like 0%, %, 0%, and 0%, it may be helpful to use the fraction form of the percent. Percent-Fraction Equivalents % = 0% = 0 0% = % = 8 6 % = 6 0% = 0% = 0 0% = 7 % = 8 % = 7% = 70% = % = 6 % = 8 66 % = 00% = 90% = % = 87 % = % = 6 Example Find 0% of mentally. 0% of = of Think: 0% =. = 7 Think: of is 7. So, 0% of is 7. When an exact answer is not needed, estimate by rounding and using mental math to compute the answer. Example Estimate. a. % of 8 b. % of 90 % is about % or. % = % of 8 is. 90 is almost 00. So, % of 8 is about. So, % of 90 is about or.. Find the percent of each number mentally.. 0% of 6. % of % of. 7% of % of 6. 0% of 7. % of 8. 7% of 9. 0% of 0 Estimate. 0. 9% of 0. % of 9. 8% of. % of 90. 0% of 00. % of Math Accelerated Chapter 6 Percents

40 Lesson Reteach Using the Percent Equation A percent equation is an equivalent form of the percent proportion. In a percent equation, the percent is written as a decimal. Example Solve each problem using a percent equation. a. Find % of 9. n = 0.(9) n = 0.9 So, % of 9 is 0.9. b. is what percent of 7? = n(7) 0. = n So, is 0% of 7. c. 90 is 0% of what number? 90 = 0.n 0 = n So, 90 is 0% of 0. Solve each problem using a percent equation.. Find 76% of.. Find 9% of 0.. Find 0% of 7.. Find 6% of 0.. Find.% of Find 8.% of Find 07% of is what percent of 800? 9. 6 is what percent of 0? is what percent of 00?.. is what percent of?. 7 is what percent of 0?.. is what percent of 80?.. is what percent of 0?. 6 is 9% of what number? 6. 9 is 9% of what number? is 90% of what number? 8.. is % of what number? 9. is % of what number? is 7.% of what number? Math Accelerated Chapter 6 Percents

41 Lesson Reteach Percent of Change A percent of change is a ratio of the amount of change to the original amount. Example Find the percent of change from 7 yards to yards. Step Step Subtract to find the amount of change. - 7 = - final amount original amount Write a ratio that compares the amount of change to the original amount. Express the ratio as a percent. amount of change percent of change = original amount = - = -0.8 or -8% 7 A percent error is a ratio of the amount of error to the actual value of a measurement. Example Dominic estimates that the length of a ribbon is 7 centimeters. It is actually 66 centimeters long. What is the percent of error of his estimate? Step Subtract to find the amount of error = 9 Subtract the actual value from the estimate. Step Write a ratio that compares the amount of change to the original measurement. Express the ratio as a percent. amount of error percent of error = 00 = actual value 66 So, the percent error is.6% Find the percent of change. Round to the nearest tenth, if necessary. Then state whether the percent of change is an increase or decrease.. from inches to 6 inches. from 8 years to 0 years. from people to 7 people. from mi per gal to mi per gal Find the percent error.. actual cost: $8 meters, estimated cost: $7 6. estimated mass: 0 grams, actual mass: grams 7. projected time: minutes, actual time: 80 minutes 8. estimated length: meters, actual length: meters Math Accelerated Chapter 6 Percents

42 Lesson Reteach Discount and Markup A store sells items for more than it pays for those items so it can make a profit. The amount of increase is the markup. The percent of markup is a percent of increase. The amount the customer actually pays for an item is the selling price. When a store has a sale, the discount is the amount by which the regular price is reduced. The percent discount is a percent of decrease. Example Find the selling price if a store pays $67 for a set of luggage and the markup is 8%. Method Find the amount of the markup first. The whole is $67. The percent is 8. You need to find the amount of the markup, or the part. Let m represent the amount of the markup. part = percent whole m = m = 6.6 Multiply. Add the markup to the cost. So, $67 + $6.6 = $0.6. Method Find the total percent first. The customer will pay 00% of the store s price plus an extra 8%, or 8% of the store s price. Let p represent the price. p =.8(67) part = percent whole p = 0.6 Multiply. The selling price is $0.6. Example Find the sale price of a purebred German Shepherd puppy that is regularly $0 and is on sale for % off. Method Find the amount of discount first. Let d represent the amount of the discount. part = percent whole d = 0. 0 d = 7.0 Multiply. Subtract the discount from the original cost. So, $0-7.0 = $9.0 Method Find the total percent first. Let p represent the sale price. The amount of the discount is %, so the customer will pay 00% - % or 6% of the original cost. p = 0.6(0) part = percent whole p = 9.0 Multiply. The sale price is $9.0. Find the selling price for each item given the cost and the percent of the markup or discount.. guitar: $00; 60% discount. MP player: $8; 78% markup. lamp: $; 8% markup Math Accelerated Chapter 6 Percents. jeans: $6; % discount