Using Properties of Equality to Solve Equations

Transcription

1 LESSON 9 Using Properties of Equality to Solve Equations California Mathematics Content Standards AF.0,. Write and solve one-step linear equations in one variable. MR.0,. Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques. MR.0,. Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. MR.0,.7 Make precise calculations and check the validity of the results from the context of the problem. Power Up facts mental math Power Up M a. Powers/Roots: + 00 b. Number Sense: c. Geometry: What is the perimeter of this rectangle? Dimensions are in centimeters. 6 cm d. Money: Double $.00. $0.00 e. Statistics: The attendance figures for the first three games were 0,, and 88. What is the range of those figures? 68 f. Number Sense: 0 80 g. Measurement: How many pints are in a quart? pt h. Calculation:,,, +,, +,, 7 problem solving Choose an appropriate problem-solving strategy to solve this problem. If Sam can read 0 pages in 0 minutes, how long will it take Sam to read 00 pages? hours New Concept We have practiced solving equations by inspection using logical reasoning. In this lesson we will use inverse operations to solve equations. Inverse operations undo each other. Addition and subtraction are inverse operations. A number plus five, minus five equals the number. Multiplication and division are inverse operations. A number times five, divided by five equals the number. Lesson 9 6

2 We use inverse operations to isolate the variable in an equation. We can illustrate the process with a balance-scale model: x 7 The balanced scale shows that the weight on the left side equals the weight on the right side. We can represent the relationship with this equation: x + = To isolate x we need to remove the. Since is added to x, we remove the by subtracting (which is the inverse operation of addition). However, if we subtract from the left side, we must also subtract from the right side to keep the scale balanced. x + = x = 7 x 7 We find that x equals 7. Isolating a variable means to get the variable, like x, alone on one side of the equal sign. Notice these two aspects of isolating a variable.. We choose the operation that isolates the variable.. We perform the operation on both sides of the equals sign to keep the equation balanced. Performing the same operation on both sides of an equation preserves the equality of the equation. Below are four properties of equality we can use to manipulate equations. Operation Properties of Equality Addition: If a = b, then a + c = b + c. Subtraction: If a = b, then a c = b c. Multiplication: If a = b, then ac = bc. Division: If a = b, then a c b c if c is not 0. Formulate Use numbers to give an example of each property. See student work. 66 Saxon Math Intermediate 6

3 Example After spending $.0, Rondall had $.70. How much money did he have before he spent $.0? Solve this equation to find the answer. Then check your answer. x.0 =.70 We see that.0 is subtracted from x. To isolate x we add.0 to the left side to undo the subtraction, and we add.0 to the right side to keep the equation balanced. The result is a simpler equivalent equation. Step: Justification: x.0 =.70 Given equation x = Added.0 to both sides x = 8.00 Simplified We find that Rondall had $8.00. To check our answer, we substitute 8.00 for x in the equation. x.0 =.70 Equation =.70 Replaced x with =.70 Simplified Since both sides of the equation equal.70, the solution is correct. Example If a -month subscription to a magazine costs $8.0, what is the average cost per month? Solve this equation to find the answer. Then check your answer. m = 8.0 The variable is multiplied by. To isolate the variable we divide the left side by. To keep the equation balanced we divide the right side by. Step: Justification: m = 8.0 Given equation m 8. m = 0.70 m = 0.70 Divided both sides by Simplified Simplified Explain In the context of the problem, what does m = 0.70 mean? The average cost per month for the subscription is $0.70. We check the answer by substituting 0.70 for m. m = 8.0 Equation (0.70) = 8.0 Replaced m with = 8.0 Simplified Our answer is correct. Lesson 9 67

4 Example Thinking Skills Connect When the variable is multiplied by a fraction, why do we multiply by the reciprocal to help us isolate the variable? The product of a number and its reciprocal is always, so we multiply by the reciprocal to get x, or x. Solve and check: w = The variable is multiplied by. We can divide by, or multiply both sides by the multiplicative inverse (reciprocal) of. Step: Justification: w Given equation w Multiplied both sides by w 9 8 w = 9 8 Simplified Simplified We substitute 9 for w in the equation to check. 8 w = equation 9 8 = 9 = 8 = Our answer is correct. replaced w with 9 8 reduced simplified It is customary to express a solution with the variable on the left side of the equal sign. Since the quantities on either side of an equal sign are equal, we can reverse everything on one side of an equation with everything on the other side of the equation. This property of equality is called the symmetric property. Symmetric Property of Equality If a = b, then b = a. Example Solve and check:. = x. We may apply the Symmetric Property at any step. Step: Justification:. = x. Given equation. +. = x. +. Added. to both sides.8 = x Simplified x =.8 Symmetric Property 68 Saxon Math Intermediate 6

5 We check our answer.. = x. equation. =.8. replaced x with.8. =. simplified Our answer is correct. We have used cross products to solve proportions. = w = w Cross products are equal because they are numerators of equal fractions with a common denominator. = w = w = w Another way to understand why cross products are equal is to multiply both sides of the proportion by the denominators. Consider the proportion again. = w 70 If we multiply both sides by, the denominator on the left-hand side cancels. Since and are multiplicative inverses, their product is. = w 70 = w 70 Then we multiply both sides by 70, and the other denominator cancels. 70 = w We end up with equal cross products. 70 = w Discuss In the first step of this method, how do we know that the equality statement is still true once we have multiplied both sides of the equation by? If two quantities are equal, then if they are each multiplied by the same factor, they will still be equal. Lesson 9 69

6 Example Solve this proportion using two different methods. = x a. Multiply both sides of the equation by the denominators. b. Use equal cross products. a. We will multiply both sides of the equation by and by. = x Original equation = x Multiplied both sides by = x Canceled the denominator = x Multiplied both sides by = x Canceled the denominator 8 = x Simplified 8 = x Divided both side by 8 = x Simplified b. we will solve by using equal cross products = x Original equation = x Cross products are equal 8 = x Simplified 8 = x Divided both sides by 8 = x Simplified Analyze Explain why using cross-multiplication is a reasonable method for solving proportion problems. Equal cross products are the result of multiplying both sides of the equation by the denominators and canceling the denominators. Cross multiplication skips these steps, so it is faster. Lesson Practice a. What does isolate the variable mean? Manipulate the equation so that the variable is alone on one side of the equation. b. Which operation is the inverse of subtraction? Addition c. Which operation is the inverse of multiplication? Division Justify Solve each equation. Justify the steps. Then check your answer. d. x x e. x. =. x =. 70 Saxon Math Intermediate 6 f.. x x =.8 g. x x h. Peggy bought pounds of oranges for $.0. Solve the equation x = $.0 to find the price per pound. $0.90 per pound i. From 8 a.m. to noon, the temperature rose degrees to C. Solve the equation t + = to find the temperature at 8 a.m. C

7 Written Practice Distributed and Integrated *. (86) *. (Inv. 7) The ratio of the length to the width of the rectangular lot was to. If the lot was 60 ft wide, how long was the lot? 0 ft Keenan does not know the correct answer to two multiple-choice questions. The choices are A, B, C, and D. If Keenan just guesses, what is the probability that Keenan will guess both answers correctly? 6. Ratio Length Width A. C. l 60 l 60. (0). (RF9). (69, 7) 6. (69) 7. () 8. (, 7) 0. (80). (). (6) If the sum of four numbers is, what is the mean of the four numbers? 6 The rectangular prism shown below is constructed of -cubic-centimeter blocks. What is the volume of the prism? 6 cm Write 9 as a decimal number and as a percent. 0.6; 6% Write as a decimal number and add it to.. What is the sum?. +. = 6.7 What number is % of 80? 6 (0.) ( (6, 7) ) With one toss of a pair of number cubes, what is the probability that the total rolled will be a prime number? (Add the probability for each prime number total.) 6 = Twenty of the two dozen members voted yes. What fraction of the members voted yes? 6 Analyze If the rest of the members in problem voted no, then what was the ratio of no votes to yes votes? 6 Find each unknown number:. (9, 8) *. (90) 6. () 7. (88) 8. (RF) 9. (8) w 9 7 *. (8) 6 m 0 6 Conclude In what type of triangle are all three sides the same length? Equilateral triangle What mixed number is 8 of 00? 7 Which property is illustrated by the equation Distributive Property 6 ( ) = A triangular prism has how many faces? faces How many pints of milk is quarts of milk? a. Create a function table showing the number of pints in,, and quarts. Graph the ordered pairs in your table and estimate the answer to the question. See student graph; 0 b. Find the rule for the function and use it to find the number of pints in quarts. Multiply number of quarts by for number of pints; 0 Lesson 9 7

8 * 0. (6, 7). (6) *. (8). (77) Represent Use a factor tree to find the prime factors of 800. Then write the prime factorization of 800 using exponents. Round the decimal number one hundred twenty-five thousandths to the nearest tenth n = $.0 $.00 The diagonal segment through this rectangle divides the rectangle into two congruent right triangles. What is the area of one of the triangles? mm 0. One possibility: mm. (7) *. (9) 6. (89) 7. () 8. (RF7, 9) Write 7 0 as a percent. 8% 7 x = x = 8 6 mm Multiply hours times 7 dollars per hour. $ hours $7 hour = a. What is the probability of rolling a 6 with a single roll of a number cube? b. What is the probability of rolling a number less than 6 with a single roll of a number cube? c. Name the event and its complement. Then describe the relationship between the two probabilities. Represent The coordinates of the four vertices of a quadrilateral are (, ), (0, ), (, ), and (, ). What is the name for this type of quadrilateral? Trapezoid c. Sample: The event is rolling 6 and its complement is not rolling 6. The relationship between the two probabilities is that their sum is : =. * 9. (87) 0. (, ) Explain The formula for the area of a triangle is A bh If the base measures 0 cm and the height measures cm, then what is the area of the triangle? Explain your thinking. Generalize Write the rule for this sequence. Then write the next four numbers. 6, 8, 6,, 6, 8, 6,, 6, 8,... Sample: The rule is add 6 to the preceding number cm ; Sample: A bh A, (0 cm) ( cm) 00 cm 0 cm 7 Saxon Math Intermediate 6

9 LESSON 9 Writing Fractions and Decimals as Percents, Part California Mathematics Content Standards NS.0,. Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips. MR.0,. Determine when and how to break a problem into simpler parts. MR.0,. Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques. Power Up facts mental math problem solving Power Up L a. Number Sense: b. Percent: 0% of c. Decimals: 00. d. Money: Shelby purchased a DVD for $.7. If she paid with a $0 bill, how much change should she receive? $. e. Probability: Kody flipped a quarter and it landed heads up. If he flips the quarter again, what is the probability it will land tails up? f. Algebra: If b = 600, what is b 0? 0 g. Algebra: If n = 8, what does n equal? 0 h. Calculation: 6 +,, +,, +, Choose an appropriate problem-solving strategy to solve this problem. The basketball team s points-per-game average is 88 after its first four games. How many points does the team need to score during its fifth game to have a points-per-game average of 90? 98 New Concept Since Lesson 7 we have practiced changing a fraction or decimal to a percent by writing an equivalent fraction with a denominator of % % 00 Read as find the square root. Lesson 9 7

10 In this lesson we will practice another method of changing a fraction to a percent. Since 00% equals, we can multiply a fraction by 00% to form an equivalent number. Here we multiply by 00%: Thinking Skills Justify How can you use mental math to change a decimal to a percent? Move the decimal point two places to the right and write a percent sign. 00% 00% Then we simplify and find that equals 60%. 00% 60% We can use the same procedure to change decimals to percents. Here we multiply 0.7 by 00% % = 7.% To change a number to a percent, multiply the number by 00%. Example Change to a percent. We multiply by 00%. 00% 00% To simplify, we divide 00% by and write the quotient as a mixed number. % 00% Example Write. as a percent. We multiply. by 00%.. 00% = 0% In some applications a percent may be greater than 00%. If the number we are changing to a percent is greater than, then the percent is greater than 00%. Example Write as a percent. We show two methods below. Method : We split the whole number and fraction. The mixed number means. We change each part to a percent and then add. 7 Saxon Math Intermediate 6

11 00% + % = % Method : We change the mixed number to an improper fraction. The mixed number equals the improper fraction 9. We then change 9 to a percent % % Example Write 6 as a percent. Method shown in Example is quick, if we can recall the percent equivalent of a fraction. Method is easier if the percent equivalent does not readily come to mind. We will use method in this example. We write the mixed number 6 as the improper fraction 6 and multiply by 00%. 6 00% 00% 6 Now we divide 00% by 6 and write the quotient as a mixed number. 00% 6 6 % Example Twenty of the thirty students on the bus were girls. What percent of the students on the bus were girls? We first find the fraction of the students that were girls. Then we convert the fraction to a percent. Girls were 0 0 Qor R of the students on the bus. Now we multiply the fraction by 00%, which is the percent name for. We can use either 0 0 or, as we show below % 00 % 66 0 % or 00% 00% 66 % We find that 66 % of the students on the bus were girls. Lesson Practice Change each decimal number to a percent by multiplying by 00%: a. 0. 0% b % c. 0..% d. 0. % e.. 0% f. 0.0.% g % h.. % i % Lesson 9 7

12 Change each fraction or mixed number to a percent by multiplying by 00%: j. 66 % k. 6 6 % l. 8 % m. % n. 80% o. % p. What percent of this rectangle is shaded? 8 % q. Connect What percent of a yard is a foot? % Written Practice Distributed and Integrated *. (9). (0). (RF9) Analyze Ten of the thirty students on the bus were boys. What percent of the students on the bus were boys? % Connect On the Celsius scale water freezes at 0 C and boils at 00 C. What temperature is halfway between the freezing and boiling temperatures of water? 0 C Connect If the length of segment AB is the length of segment AC, and if segment AC is cm long, then how long is segment BC? 8 cm A B C. (9) *. (9) What percent of this group is shaded? 0% Change to a percent by multiplying by 00%. 66 % * 6. (9) 7. (69) 8. (9) Change. to a percent by multiplying. by 00%. 0% 6. 6 (Begin by writing 6 as a decimal number.) 0. x = 9. (6) How much is of 60? 70 Tommy placed a cylindrical can of spaghetti sauce on the counter. He measured the diameter of the can and found that it was about 8 cm. Use this information to answer problems 0 and. 0. () *. (8). (6). (89) The label wraps around the circumference of the can. How long does the label need to be?. cm Analyze How many square centimeters of countertop does the can occupy? 0. cm (67) Multiply 6 inches by foot per inches. ft 6 in. ft in. = 76 Saxon Math Intermediate 6

13 . (RF) 7. (8) 9. (7) 0. (RF9). (7). (RF). (66) $ $ $. 6. () 7 x 9 8. (6) If ninety percent of the answers were correct, then what percent were incorrect? 0% Write the decimal number one hundred twenty and three hundredths. 0.0 Arrange these numbers in order from least to greatest:.,.,,.,,,. Conclude A pyramid with a square base has how many edges? 8 edges What is the area of this parallelogram? 80 in. 8 in. 0 in. *. (90). (9) Multiple Choice The parallelogram in problem is divided into two congruent triangles. Both triangles may be described as which of the following? C A acute B right C obtuse During the year, the temperature ranged from 7 F in winter to 0 F in summer. How many degrees was the range of temperature for the year? 0 F * 6. (RF7, 77) Model The coordinates of the three vertices of a triangle are (0, 0), (0, ), and (, 0). Graph the triangle and find its area. 8 sq. units 6. y 7. (Inv. ) Latondra s first nine test scores are shown below.,,, 9,,, 0,, a. What is the mode of these scores? b. What is the median of these scores? x 8. (88) 9. (Inv., 8) 0. (, 69) Evaluate a(b + c) if a =, b =, and c =. Sandra filled the aquarium with 6 quarts of water. How many gallons of water did Sandra pour into the aquarium? a. Create a function table showing the number of quarts in,, and gallons. Graph the ordered pairs in the function table and estimate the answer. See student graph; gallons b. Determine a rule for the function table and use it to write and solve an algebraic equation to answer the question. Rule: number of quarts are the number of gallons multiplied by ; x = 6; x = gallons A bag contains lettered tiles, two for each letter of the alphabet. What is the probability of drawing a tile with the letter A? Express the probability ratio as a fraction and as a decimal rounded to the nearest hundredth. 6 ; 0.0 Lesson 9 77

14 LESSON 9 California Mathematics Content Standards AF.0,. Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches). AF.0,. Solve problems involving rates, average speed, distance, and time. Formulas and Substitution Power Up facts mental math problem solving Power Up K a. Number Sense: 0 8 b. Percent: % of 6 c. Money: Jerome bought paint for $7.99 and a paintbrush for $.7. What was the total cost? $0.6 d. Decimals: e. Estimation: Estimate the area of this rectangle. cm f. Number Sense: 0 70 g. Algebra: If x =, what does x equal? h. Calculation: 8 9, +,,, 6, +,, 0 Choose an appropriate problem-solving strategy to solve this problem. Denzel put purple marbles, 7 red marbles, and brown marble in a bag and shook the bag. If he reaches in and chooses a marble without looking, what is the probability that he chooses a red marble? A purple or brown marble? What is the probability of not choosing a red marble? If Denzel does choose a red marble, but gives it away, what is the 7 probability he will choose another red marble? 0 ; 0 ; 0 ; New Concept A formula is a literal equation that describes a relationship between two or more variables. Formulas are used in mathematics, science, economics, the construction industry, food preparation anywhere that measurement is used. To use a formula, we replace the letters in the formula with measures that are known. Then we solve the equation for the measure we wish to find. 78 Saxon Math Intermediate 6

15 Example Thinking Skills Analyze What measures do you know? What unknown measure do you need to find? Use the formula d = r t to find t when d is 6 and r is 9. This formula describes the relationship between distance (d), rate (r), and time (t). We replace d with 6 and r with 9. Then we solve the equation for t. d = r t formula 6 = 9t substituted t = divided by 9 Example Distance (d ) and the rate (r ); time (t ). Use the formula d = rt to find the distance Anthony traveled after driving for hours at 60 miles per hour. Explain why your answer is reasonable. We are given the time and the rate. We replace t with hr and r with 60 mi hr. d = rt Formula d = ( 60 mi hr ) ( hr ) Substituted d = ( 60 mi hr ) ( hr ) Rewrote numbers as improper fractions d = ( 0 60 mi hr ) ( hr ) Cancelled d =0 mi Multiplied Anthony traveled 0 mi. This answer is reasonable because for each hour that Anthony drove, he traveled 60 miles. In two hours, he drove 0 miles, and in a half hour he drove another 0 miles. Together this is 0 miles. Example Alexandria biked 6 miles in hours. What was her average speed? During her bike ride, Alexandria may not have traveled at a constant speed. Some stretches may have been fast and others slow. Her average speed is the rate at which she would have completed the trip if she had ridden at a constant rate. We can find her average speed with the distance formula. We are told the distance and time. We replace d with 6 miles and t with hours. d = rt Formula 6 mi = r hr Substituted 6 mi = r Divided both sides by hr hr r = 6 mi Simplified hr Alexandria s average speed was 6 miles per hour. The rate at which she rode may have been faster at some points and slower at others, but averaging 6 miles per hour for hours, she completed the 6-mile trip. Lesson 9 79

16 Example Use the formula F =.8C + to find F when C is 7. This formula is used to convert measurements of temperature from degrees Celsius to degrees Fahrenheit. We replace C with 7 and simplify. F =.8C + formula F =.8(7) + F = F = 98.6 substituted multiplied added Thus, 7 degrees Celsius equals 98.6 degrees Fahrenheit. Discuss How could we use the formula to convert from degrees Fahrenheit to degrees Celsius? Replace F with the number of degrees Fahrenheit and solve for C. Lesson Practice a. Use the formula A = bh to find b when A is 0 and h is. b. Use the formula A = bh to find b when A is 0 and h is. 0 c. Use the formula d = rt to find t when d is 00 and r is 60. d. Use the formula F =.8C + to find F when C is 0. 0 e. Connect The formula for converting from Fahrenheit to Celsius is often given as F = 9 C +. How are 9 and.8 related?.8 is the decimal equivalent of 9 Written Practice Distributed and Integrated. () *. (89) *. (7). (78) *. (9) * 6. (9) 7. () What is the total price of a $.79 item when 7% sales tax is added to the price? $9.00 Jeff is.67 meters tall. How many centimeters tall is Jeff? (Multiply.67 meters by 00 centimeters per meter.) 67 centimeters Analyze If of the 0 seeds sprouted, how many seeds did not sprout? seeds 8 The ratio of wigglers to sliders was 7 to. If there were sliders, how many wigglers were there? 9 Change 6 to its percent equivalent by multiplying 6 by 00%. 6 % Analyze What is the percent equivalent of.? 0% How much money is 0% of $.00? $ Saxon Math Intermediate 6

17 8. (80) 9. (RF6, 8) There are markers in a box: blue, red, and yellow. Without looking, Lee gives one marker to Keondra and takes one for himself. What is the probability that both markers will be blue? = Evaluate The circumference of the front tire on Elizabeth s bike is 6 feet. How many complete turns does the front wheel make as Elizabeth rides down her 0-foot driveway? turns 0. (88). (). (67). (6) 6. () 8. (69) 9. (6) * 0. (8) Multiple Choice The expression ( + ) equals which of the following? B A ( ) + B ( ) + ( ) C + 7 D (8) 0. (7) How many fourths are in? (, ) (decimal answer) (.). Estimate Find the sum of 6,.9, and 8. by rounding each number to the nearest whole number before adding. Explain how you arrived at your answer. 9; Round 6 to 6, round.9 to, and round 8. to 8. Then add 6,, and 8. Analyze The diameter of a round tabletop is 60 inches. a. What is the radius of the tabletop? 0 inches 8. (7) b. What is the area of the tabletop? (Use. for π.) 86 square inches Arrange these numbers in order from least to greatest: %,, 0. Find each unknown number.. (9). (8) *. (RF9), %, 0. y +. =.6. (8) x = 8 x 6 Connect AB is mm long. AC is mm long. How long is BC? 8 mm A B C 6. (9) 7. (8) * 8. (87) The formula c =.n is used to convert inches (n) to cm (c). Find c when n is. 0.8 cm What is the ratio of a pint of water to a quart of water? The formula for the area of a parallelogram is A = bh. If the base of a parallelogram is. m and the height is 0.9 m, what is the area of the parallelogram? How can estimation help you check your answer? m ; Sample:. m rounds to m and 0.9 m rounds to m. A ( m)( m) m..08 m is close to so I know my answer is reasonable. Lesson 9 8

18 * 9. (89) Multiply. liters by 000 milliliters per liter. 00 milliliters. liters 000 milliliters liter 0. (, 69) If this spinner is spun once, what is the probability that the arrow will end up pointing to an even number? Express the probability ratio as a fraction and as a decimal. ; 0. Early Finishers Real-World Connection The local university football stadium seats 60,000 fans, and average attendance at home games is 8,00. It has been determined that an average fan consumes. beverages per game. a. If each beverage is served in a cup, about how many cups are used during an average game? Express your answer in scientific notation cups b. Next week is the homecoming game, which is always sold out. A box of cups contains 0 cups. How many boxes of cups will be needed for the game? boxes 8 Saxon Math Intermediate 6

19 LESSON 9 Transversals California Mathematics Content Standards MG.0,. Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms. MR.0,. Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Power Up facts mental math problem solving Power Up M a. Number Sense: b. Number Sense: c. Percent: 0% of 80 0 d. Decimals: e. Statistics: Zach plays soccer. He scored goals in 0 games. Find the mean number of goals Zach scored per game. 0. goal f. Number Sense: 60 0 g. Algebra: If r = 8, what is r + r? 7 h. Calculation:, +,, 8,,,,, Choose an appropriate problem-solving strategy to solve this problem. What are the next four numbers in this sequence:, 6,,,...,, 7, New Concept A line that intersects two or more other lines is a transversal. In this drawing, line r is a transversal of lines s and t. r s t Math Language Parallel lines are lines in the same plane that do not intersect and are always the same distance apart. In the drawing, lines s and t are not parallel. However, in this lesson we will focus on the effects of a transversal intersecting parallel lines. Lesson 9 8

20 Below we show parallel lines m and n intersected by transversal p. Notice that eight angles are formed. In this figure there are four obtuse angles (numbered,,, and 7) and four acute angles (numbered,, 6, and 8). p exterior interior exterior m n Thinking Skills Verify Why does every pair of supplementary angles in the diagram contain one obtuse and one acute angle? If the sum of the angles is 80, and one angle is less than 90 (acute), then the other must be greater than 90 (obtuse). Notice that obtuse angle, and acute angle, together form a straight line. These angles are supplementary, which means their measures total 80. So if measures 0, then measures 70. Also notice that and are supplementary. If measures 70, then measures 0. Likewise, and are supplementary, so would measure 70. There are names to describe some of the angle pairs. For example, we say that and are corresponding angles because they are in the same relative positions. Notice that is the upper left angle from line m, while is the upper left angle from line n. exterior interior exterior p m n Angle and 8 are also alternate exterior angles. Which angle corresponds to? Angle 6 corresponds to. Which angle corresponds to 7? Angle corresponds to 7. Since lines m and n are parallel, line p intersects line m at the same angle as it intersects line n. So the corresponding angles are congruent. Thus, if we know that measures 0, we can conclude that also measures 0. The angles between the parallel lines (numbered,,, and 6 in the figure on previous page) are interior angles. Angle and are on opposite sides of the transversal and are called alternate interior angles. Name another pair of alternate interior angles. Angle and 6 Alternate interior angles are congruent if the lines intersected by the transversal are parallel. So if measures 0, then also measures 0. Angles not between the parallel lines are exterior angles. Angle and 7, which are on opposite sides of the transversal, are alternate exterior angles. Name another pair of alternate exterior angles. Alternate exterior angles formed by a transversal intersecting parallel lines are congruent. So if the measure of is 0, then the measure of 7 is also 0. 8 Saxon Math Intermediate 6

21 While we practice the terms for describing angle pairs, it is useful to remember the following. When a transversal intersects parallel lines, all acute angles formed are equal in measure, and all obtuse angles formed are equal in measure. Thus any acute angle formed will be supplementary to any obtuse angle formed. Example Transversal w intersects parallel lines x and y. w x y a. Name the pairs of corresponding angles. b. Name the pairs of alternate interior angles. c. Name the pairs of alternate exterior angles. d. If the measure of is, then what are the measures of and 6? a. and, and 6, and 7, and 8 b. and 6, and c. and 7, and 8 d. If measures, then also measures and 6 measures 6. Example Refer to the diagram from Example to answer the following. Select all correct answers. a. and are complementary angles supplementary angles adjacent angles vertical angles b. and are complementary angles supplementary angles adjacent angles vertical angles c. What angle is supplementary to but not adjacent to? 6 Lesson 9 8

22 a. Angles and are supplementary (together measure 80 ) and adjacent (share a common side). b. Angles and are vertical angles. c. Angle is obtuse and 6 is acute. Since lines x and y are parallel, and 6 are supplementary. Also, 6 is not adjacent to since they do not share a common side. From this list, only 6 fits the criteria. Angles and are adjacent to. Angle is not supplementary to since it is also obtuse. Example Lesson Practice e. Angle,, and 7 each measure 0. Angle,, 6, and 8 each measure 7. Draw a pair of parallel lines intersected by a transversal so that all angles formed are right angles. Which angles are equal in measure? Which angles are supplementary? All of the right angles formed are equal in measure, and any pair of right angles is a pair of supplementary angles. a. Which line in the figure at right is a transversal? Line c b. Which angle is an alternate interior angle to? c. Which angle corresponds to 8? d. Which angle is an alternate exterior angle to 7? e. Conclude If the measure of is 0, what is the measure of each of the other angles in the figure? c f g Written Practice Distributed and Integrated *. (, ). () *. (0). (9) *. (9) When the sum of.0 and.0 is subtracted from the product of.0 and.0, what is the difference? 0 A.-kilogram object weighs the same as how many objects that each weigh 0. kilogram? 0 objects If the mean of 8 numbers is, what is the sum of the 8 numbers? 96 Conclude What is the name of a quadrilateral that has one pair of sides that are parallel and one pair of sides that are not parallel? Trapezoid a. Write 0. as a percent. % b. Write. as a percent. 0% 86 Saxon Math Intermediate 6

23 * 6. (9) * 7. (9) 8. (7) * 0. (88). (9). (9). (6). () 6. (6) 8. () 0. (69) *. (6) Write 6 as a percent. 8 % Multiple Choice Three of the numbers below are equivalent. Which one is not equivalent to the others? C A B 00% C 0. D How much is of 60? 00 (6) 6 Evaluate x(y + z) when x = 0., y = 0., and z = x 7 = The formula below may be used to convert temperature measurements from degrees Celsius (C) to degrees Fahrenheit (F). Find F to the nearest degree when C is 7. 6 F F =.8C + Factor and reduce: ()() (6) (67) of $.00 $ (). (decimal answer) $6.0 $0.78 What is the ratio of the number of cents in a dime to the number of cents in a quarter? Find each unknown number: *. (8). (RF9) *. (8) * 6. (89) * 7. (RF7, 66) 8. (8) n = 6 *. (8) 0.n = 0 Model Draw a segment inches long. Label the endpoints R and T. Then find and mark the midpoint of RT. Label the midpoint S. What are the lengths of RS and ST? See student work. RS = ST = 7 8 inch Solve this proportion: w Multiply hours by 6 dollars per hour: dollars hours 6 dollars hours Connect The coordinates of the vertices of a parallelogram are (0, 0), (6, 0), (, ), and (, ). What is the area of the parallelogram? units Estimate The saying A pint s a pound the world around refers to the fact that a pint of water weighs about one pound. About how many pounds does a gallon of water weigh? About 8 pounds Lesson 9 87

24 9. (9) l m. Use the figure to answer the following: a. Which line is a transversal? n b. is adjacent to which two angles? and 0. (. 69) What is the probability of rolling a prime number with one roll of a number cube? Express the ratio as a fraction and as a decimal., Saxon Math Intermediate 6

25 LESSON 9 Sum of the Angle Measures of Triangles and Quadrilaterals California Mathematics Content Standards MG.0,. Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms. MG.0,. Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle. MG.0,. Draw quadrilaterals and triangles from given information about them (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle). Power Up facts mental math problem solving Power Up L a. Number Sense: b. Percent: % of 80 0 c. Decimals: d. Geometry: What is the perimeter of a regular pentagon whose sides are each. cm long? 6.0 cm e. Measurement: Valerie jogged. kilometers and then walked 00 meters. Altogether, how many meters did she jog and walk? 800 m f. Number Sense: 0 60 g. Algebra: If w = 0, what does 7w equal? 70 h. Calculation: 6 8, +,,, +,,,, Choose an appropriate problem-solving strategy to solve this problem. Carlos reads pages in minutes, and Malik reads pages in minutes. If they both begin reading 00-page books at the same time and do not stop until they are done, how many minutes before Malik finishes will Carlos finish? 90 minutes New Concept If we extend a side of a polygon, we form an exterior angle. In this figure is an exterior angle, and is an interior angle. Notice that these angles are supplementary. That is, the sum of their measures is 80º. Lesson 9 89

26 Thinking Skills Verify Act out the turns Elizabeth made to verify the number of degrees. Recall that a full turn measures 60º. So if Elizabeth makes three turns to get around a park, she has turned a total of 60º. Likewise, if she makes four turns to get around a park, she has also turned 60º. The sum of the measures of angles,, and is 60. The sum of the measures of angles,,, and is 60. If Elizabeth makes three turns to get around the park, then each turn averages 0º per turn turns If she makes four turns to get around the park, then each turn averages 90º per turn turns Recall that these turns correspond to exterior angles of the polygons and that the exterior and interior angles at a turn are supplementary. Since the exterior angles of a triangle average 0º, the interior angles must average 60º. A triangle has three interior angles, so the sum of the interior angles is 80º ( 60º = 80º). The sum of the interior angles of a triangle is 80º. The sum of angles,, and is 80. Since the exterior angles of a quadrilateral average 90º, the interior angles must average 90º. So the sum of the four interior angles of a quadrilateral is 60º ( 90º = 60º). The sum of the interior angles of a quadrilateral is 60º. The sum of angles,,, and is Saxon Math Intermediate 6

27 Example What is the unknown angle measure x in ABC? The letter x is used to represent the unknown angle measure. The measures of the interior angles of a triangle total 80º. x + 60º + 70º = 80º Since the measures of B and C total 0º, m x = is 0º. A x B C Example Find the unknown angle measures a and b. The angle which measures a is supplementary to the angle which measures º. a + º = 80º a = º The measures of the interior angles of the triangle total 80º. The right angle measures 90º and a is º. 90º + º + b = 80º º + b = 80º b = º Example Find x and y. Since A is a right angle, the angle which measures x is complementary to the angle that measures 0º. x + 0º = 90º x = 60º The unknown angles are part of a right triangle. The three angle measures total 80º. We found that x is 60º, and the right angle measures 90º. 60º + 90º + y = 80º 0º + y = 80º y = 0º Lesson 9 9

28 Example What is mt in quadrilateral QRST? The measures of the interior angles of a quadrilateral total 60º. m T + 80º + 80º + 0º = 60º The measures of Q, R, and S total 70º. So m T is 90º. R S T Q Lesson Practice Quadrilateral ABCD is divided into two triangles by segment AC. Use for problems a c. B 6 A C D f a. What is the sum of m, m, and m? 80º b. What is the sum of m, m, and m 6? 80º c. Generalize Draw a quadrilateral. What is the sum of the measures of the four interior angles of the quadrilateral? See student work; 60º d. Draw a scalene right triangle. If one of the acute angles measures 0º, what are the measures of the other two angles. See student work; 90º, 60º e. Draw a regular quadrilateral. What is the measure of each interior angle of a regular quadrilateral? See student work; 90º f. Model Nashawn made five left turns as she ran around the park. Draw a sketch that shows the turns in her run around the park. Then find the mean number of degrees in each turn. Written Practice Distributed and Integrated. () *. (0). (88). (7) How many quarter-pound hamburgers can be made from 00 pounds of ground beef? 00 hamburgers Connect On the Fahrenheit scale water freezes at F and boils at F. What temperature is halfway between the freezing and boiling temperatures of water? F Estimate the value of πd when π.9 and d is 9.87 meters. 0 m Compare: 8 < Saxon Math Intermediate 6

29 *. (9) * 7. (9) 9. (7) 0. (9). (9). (). (RF) 7. (69) 8. (68) 9. () Write as a percent. * 6. (9) % Write 0.7 as a percent. * 8. (9) 70% Write as a percent. 0% Write 7 8 as a percent. 87 % Use division by primes to find the prime factors of 0. Then write the prime factorization of 0 using exponents. 6 x = 6 a. Solve for h: A = bh h = A b b. Use the solution from part a to find h when A = 6 and b = (8) (). 7 (decimal answer) (6) ( 0.)( 0.) Jovita bought cubic yards of mulch for the garden. She will need. cubic yards for the flowerbeds. How much mulch is left for Jovita to use for her vegetable garden? Write your answer as a fraction. 6 cubic yards Analyze If 80% of the 0 students passed the test, how many students did not pass? 6 students * 0. () *. () Compare: > Predict What is the next number in this sequence? 0. (or 0 ) Find each unknown number:. () *. (77)..., 000, 00, 0,,... a = (8) 7 w 77 The perimeter of this square is 8 in. What is the area of one of the triangles? 7 in. Refer to the table below to answer problems 7. Mark s Personal Running Records Distance Time (minutes:seconds) mile 0:8 mile : mile :00. (RF) 6. (RF) If Mark set his -mile record by keeping a steady pace, then what was his -mile time during the -mile run? :0 Multiple Choice What is a reasonable expectation for the time it would take Mark to run miles? B A 9:0 B :00 C :00 Lesson 9 9

30 * 7. (RF) * 8. (9) Formulate Write a question that relates to this table and answer the question. See student work. Transversal t intersects parallel lines r and s. Angle measures 78. t r s 9. (9) a. Analyze Which angle corresponds to? 6 b. Find the measures of and 8. m = 0 ; m 8 = 78 a. Find m A. 9º b. Find x (, 6) What is the probability of rolling a composite number with one roll of a number cube? 9 Saxon Math Intermediate 6

31 LESSON 96 Fraction-Decimal-Percent Equivalents California Mathematics Content Standards NS.0,. Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips. MR.0,. Develop generalizations of the results obtained and the strategies used and apply them in new problem situations. Power Up facts mental math Power Up M a. Number Sense: b. Money: A bag of sunflower seeds costs $0.89. Keiko paid for a bag with a $ bill. How much change should she receive? $. c. Decimals: d. Percent: 0% of one yard is how many feet? ft e. Measurement: How many pints are in quarts? pt f. Estimation: Estimate 8 6 by rounding each factor and then multiplying. 000 g. Algebra: If h 9 =, what is h? 00 h. Calculation: 6 6,,, 8,,, 8,, +, 9 problem solving Choose an appropriate problem-solving strategy to solve this problem. Copy this factor tree and fill in the missing numbers: New Concept Fractions, decimals, and percents are three ways to express parts of a whole. An important skill is being able to change from one form to another. This lesson asks you to complete tables that show equivalent fractions, decimals, and percents. Lesson 96 9

32 Example Complete the table. Fraction Decimal Percent a. b. c. 0. d. e. f. 0% The numbers in each row should be equivalent. For we write a decimal and a percent. For 0. we write a fraction and a percent. For 0% we write a fraction and a decimal. a b. 00% 0% c. 0. d % = 0% 0 e. 0% 0 00 f. 0% = 0.0 = 0. Lesson Practice Connect Complete the table. Fraction Decimal Percent c. e. i. k. 0 a. 0.6 b. 60% 0.8 d. 80% f. 0. 0% g. 0.7 h. 7% 0. j. % l. 0.0 % Written Practice Distributed and Integrated. (, 0) When the sum of and is divided by the product of and, what is the quotient? 6 *. (89). (7) Analyze Tzara is feet tall. She is how many inches tall? 66 inches If of the 00 runners finished the race, how many runners did not finish the race? 0 runners 96 Saxon Math Intermediate 6

33 *. (9) Lines p and q are parallel. p q m a. Which angle is an alternate interior angle to? 8. (9) * 6. (, ) b. If measures 8º, what are the measures of 6 and 7? m 6 = 8 ; m 7 = 9 x = 6 Estimate Use a ruler to measure the diameter of a quarter to the nearest sixteenth of an inch. How can you use that information to find the radius and the circumference of the quarter? 7. () * 8. () 9. () * 0. (9) *. (8). (). (9). (69) 7. () * 8. (89) 9. (7) 0. (86) Multiple Choice Connect Which of these bicycle wheel parts is the best model of the circumference of the wheel? C A spoke B axle C tire Predict As this sequence continues, each term equals the sum of the two previous terms. What is the next term in this sequence?,,,,, 8,,... If there is a 0% chance of rain, what is the probability that it will not rain? 0.8 or Write as a percent. % Analyze 0.08w = $0.60 $ (6) 00 The following formula can be used to find the area, A, of a trapezoid. The lengths of the parallel sides are a and b, and the height, h, is the perpendicular distance between the parallel sides. A = (a + b)h Use this formula to find the area of the trapezoid. cm 6.9 (decimal). 6. (68) 0 6. (fraction) Explain If a shirt costs $9.79 and the sales-tax rate is 6%, what is the total price including tax? Explain how you can check your answer using estimation. What fraction of a foot is inches? What percent of a meter is centimeters? % The ratio of children to adults in the theater was to. If there were children, how many adults were there? 7 adults inch; sample: Divide by to find the radius; multiply by π to find the circumference $0.98; sample: Round $9.79 to $0 and 6% to 0%. 0% of $0 is $. $0 + $ = $. Since both numbers were rounded up, the actual answer should be a little less. Lesson 96 97

34 . (9, ) Arrange these numbers in order from least to greatest:,, 0,,,, 0,,. (9) Classify These two triangles together form a quadrilateral with only one pair of parallel sides. What type of quadrilateral is formed? Trapezoid *. (RF) Conclude Do the triangles in this quadrilateral appear to be congruent or not congruent? Not congruent *. (9) a. Analyze What is the measure of A in ABC? 0º b. Analyze What is the measure of the exterior angle marked x? 70º A. (7, 69) Write 0% as a a. simplified fraction. C 0 0 x B b. simplified decimal number (8) 7. (96) The diameter of this circle is 0 mm. What is the area of the circle? (Use. for π.) mm Complete the table: 0 mm Fraction Decimal Percent a. e b. 0% c. 0. d. % f. 0. 0% * 8. (89) 9. (6, ) Multiply 0 inches by foot per inches. 0 ft 0 in. ft in. A bag contains 0 red marbles and blue marbles. a. What is the ratio of red marbles to blue marbles? b. If one marble is drawn from the bag, what is the probability that the marble will be blue? 7 * 0. (90) Conclude An architect drew a set of plans for a house. In the plans, the roof is supported by a triangular framework. When the house is built, two sides of the framework will be 9 feet long and the base will be feet long. Classified by side length, what type of triangle will be formed? Isosceles triangle 98 Saxon Math Intermediate 6

35 LESSON 97 Algebraic Addition Activity California Mathematics Content Standards NS.0,. Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations. MR.0,. Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Power Up facts mental math problem solving Power Up K a. Number Sense: b. Percent: How many seconds is 0% of one minute? 0 sec c. Money: $ $7.8 $.7 d. Geometry: Angles A and B each measure 6. What is m C? e. Decimals: f. Measurement: The top of the desk is feet inches above the floor. How many inches is that? 9 in. g. Algebra: If y =.7, what is y + y? 7. h. Calculation: 9,,,,, 0, +, 9,, Choose an appropriate problem-solving strategy to solve this problem. Cezar and his friends played three games that are scored from 00. His lowest score was 70 and his highest score is 00. What is Cezar s lowest possible three-game average? What is his highest possible three-game average? 80; 90 New Concept Thinking Skills Discuss What addition equation shows that a positive charge and a negative charge neutralize each other? One model for the addition of signed numbers is the electrical-charge model, which is used in the Sign Game. In this model, signed numbers are represented by positive and negative charges that can neutralize each other when they are added. The game is played with sketches, as shown here. The first two levels may be played with two color counters. + + = 0 Lesson 97 99

36 Activity Sign Game In the Sign Game pairs of positive and negative charges become neutral. After determining the neutral pairs we count the signs that remain and then write our answer. There are four skill levels to the game. Be sure you are successful at one level before moving to the next level. level Positive and negative signs are placed randomly on a screen. When the game begins positive and negative pairs are neutralized so we cross out the signs as shown. (Appropriate sound effects strengthen the experience!) Use counters to act out the game. Choose one color to represent positive charges and another to represent negative charges. Remove all pairs of counters that have different colors. These are neutralized charges. Before After Two positives remain. After marking positive-negative pairs we count the remaining positives or negatives. In the example shown above, two positives remain. With counters, two counters of one color remain. See whether you can determine what will remain on the three practice screens below: One negative Zero remain. All 7 negatives remain. level Positives and negatives are displayed in counted clusters or stacked counters. The suggested strategy is to combine the same signs first. So + combines with + to form +, and combines with to form 7. Then determine how many of which charge (sign) remain. 600 Saxon Math Intermediate 6

37 There were three more negatives than positives, so remain. With counters, stacks of equal height and different colors are removed until only one color (or no counters) remains. See whether you can determine how many of which charge will remain for the three practice screens below: level Reading Math A negative sign indicates the opposite of a number. means the opposite of. Likewise, ( ) means the opposite of, which is. (+) means the opposite of +, which is. Positive and negative clusters can be displayed with two signs, one sign, or no sign. The first step is to change a double sign to one sign. A cluster with no sign, with, or with + + is a positive cluster. A cluster with + or with + is a negative cluster. If a cluster has parentheses, look through the parentheses to see the sign. With counters, for invert a negative to a positive, and for + invert a positive to a negative. Examples of Positives ( ) = + No sign or double sign () Examples of Negatives (+) = = + +( ) = = + + = ++= + + = Change to one sign () remain See whether you can determine how many of which charge remain for the following practice screens: level Extend Level to a line of clusters without using a screen. + ( ) ( ) (+) + (+6) Use the following steps to find the answer: Step : Change to single signs: Step : Combine same signs: 9 + Step : Find what remains: + Lesson 97 60

38 Lesson Practice Simplify: a b. + (+) (+) ( 6) 0 c d. + ( ) ( 9) (+7) + (+) + e f. ( 0) (+0) ( 0) + ( 0) 0 Written Practice Distributed and Integrated. (6). (78) *. (7) *. (, 7). (88) 6. (9) 7. (7) * 8. (9) 9. (69) 0. (). (). (97) *. (8). (67) 7. () Analyze A foot-long ribbon can be cut into how many -inch lengths? 8 lengths Twenty paintings were sold. If the ratio of sold to unsold was to 9, how many were unsold? Analyze If 8 of the group voted yes and 8 voted no, then what fraction of the group did not vote? Connect Nine months is a. what fraction of a year? b. what percent of a year? 7% a. Compare: (0.)(0.) + (0.)(0.) = 0.( ) b. What property is illustrated in a? Distributive Property x = x = If of the pie was eaten, what percent of the pie was left? 80% Write the percent form of 7. 7 % 6 6. (decimal answer) (97) Analyze Solve this proportion: 0 n () (. 0.0) If the sales-tax rate is 7%, what is the tax on a $. purchase? $ Saxon Math Intermediate 6

39 8. (Inv. ) Analyze The table shows the percent of the population aged 6 with some senior high school education. The figures are for the year 00. Use the table to answer a c. Country Percent 8. c. the median; Sample: The Peru % mode is the number that Iceland 7% appears most Poland 6% often. The median is the number in Italy % the middle. The median may be Greece % the same as the Chile 6% mode, but it will not always be the Luxembourg % same. a. Find the mode of the data. 6% b. If the data were arranged from least to greatest, which country or countries would have the middle score? Poland and Chile c. Explain What is the term used for the answer to problem b? Will this quantity always be the same as the mode in every set of data? Explain. 9. (7) 0. (). (76, 87) *. (77, 90) Write the prime factorization of 900 using exponents. Think of two different prime numbers, and write them on your paper. Then write the greatest common factor (GCF) of the two prime numbers. The GCF is. Explain The perimeter of a square is meters. How many centimeters long is each side? Explain your thinking. 0 centimeters; P = s, meters = 00 cm, 00 cm = s, 00 = s, s = 0 cm a. What is the area of this triangle? 0 cm 0 cm cm 8 cm b. Classify Is this an acute, right, or obtuse triangle? Obtuse triangle *. (9) a. What is the measure of B in quadrilateral ABCD? 8 b. What is the measure of the exterior angle at D? 70 D 0 90 A C 7 B Lesson 97 60

40 Complete the table to answer problems 6. Fraction Decimal Percent *. (96) *. (96) * 6. (96) * 7. (RF9, ) 8. () 9. (8) 0. (9) a. 0.6 b. 60% a. 0 0 b. 0. % a. 0. b. 0% Model Draw AC inches long. Find and mark the midpoint of AC, and label the midpoint B. What are the lengths of AB and BC? AB = BC = 8 inch A B C There are cards in a bag. Eight of the cards have letters written on them. What is the chance of drawing a card with a letter written on it? % Compare: gallon < liters a. Solve for d: C = πd d = C π b. Use the solution from part a to find d when C = 6.8 (Use. for π). 0 Early Finishers Real-World Connection Jesse displays trophies on shelves in the family room. Two of the 6 trophies on each shelf are for soccer. How many trophies are NOT for soccer? Write one equation and use it to solve the problem. Sample: (6 ) = 6 60 Saxon Math Intermediate 6

41 LESSON 98 Addition of Integers California Mathematics Content Standards NS.0,. Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations. MR.0,. Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Power Up facts mental math problem solving Power Up L a. Number Sense: b. Percent: 0% of 8 c. Money: Gwen had $0. She spent $8.7 on groceries. How much money did she have left over? $.8 d. Decimals: e. Number Sense: f. Measurement: How many cups are in pints? cups g. Algebra: If g = 6, what is g? 6 h. Calculation: 8 8,, 9,, +,, +,, Choose an appropriate problem-solving strategy to solve this problem. If two people shake hands, there is one handshake. If three people shake hands, there are three handshakes. If four people shake hands with one another, we can picture the number of handshakes by drawing four dots (for people) and connecting the dots with segments (for handshakes). Then we count the segments (six). Use this method to count the number of handshakes that will take place between Bill, Phil, Jill, Lil, and Wil. 0 handshakes New Concept Math Language Integers consist of the counting numbers (,,,...), the negative counting numbers (,,,...), and 0. In this lesson we will practice adding integers. The dots on this number line mark the integers from negative five to positive five ( to +). 0 If we consider a rise in temperature of five degrees as a positive five (+) and a fall in temperature of five degrees as a negative five ( ), we can use the scale on a thermometer to keep track of the addition. Lesson 98 60

42 Thinking Skills Analyze How is a thermometer like a number line? How is it different? Sample: Both can contain positive and negative integers at even intervals. A number line is usually drawn horizontally. Thermometers are usually vertical. Imagine that the temperature is 0 F. If the temperature falls five degrees ( ) and then falls another five degrees ( ), the resulting temperature is ten degrees below zero ( 0 F). When we add two negative numbers, the sum is negative. + = 0 Imagine a different situation. We will again start with a temperature of 0 F. First the temperature falls five degrees ( ). Then the temperature rises five degrees (+). This brings the temperature back to 0 F. The numbers and + are opposites. When we add opposites, the sum is zero. 0 0 F Starting from 0 F, if the temperature rises five degrees (+) and then falls ten degrees ( 0), the temperature will fall through zero to F. The sum is less than zero because the temperature fell more than it rose = Example Math Language Opposites are numbers that can be written with the same digits but with opposite signs. They are the same distance, in opposite directions, from zero on the number line. Add: +8 + We will illustrate this addition on a number line. We begin at zero and move eight units in the positive direction (to the right). From +8 we move five units in the negative direction (to the left) to The sum is +, which we write as = Saxon Math Intermediate 6

43 Example Thinking Skills Generalize When two negative integers are added, is the sum negative or positive? Add: + Again using a number line, we start at zero and move in the negative direction, or to the left, five units to. From we continue moving left three units to 8. Negative = 8 The sum is 8. Example Add: We start at zero and move six units to the left. Then we move six units to the right, returning to zero = 0 Example Add: (+6) + ( 6) Sometimes positive and negative numbers are written with parentheses. The parentheses help us see that the positive or negative sign is the sign of the number and not an addition or subtraction operation. (+6) + ( 6) = 0 Negative 6 and positive 6 are opposites. Opposites are numbers that can be written with the same digits but with opposite signs. The opposite of is, and the opposite of is (which can be written as +). On a number line, we can see that any two opposites lie equal distances from zero. However, they lie on opposite sides of zero from each other. opposites 0 opposites If opposites are added, the sum is zero. + + = = 0 Lesson

44 Instead of subtracting 6 from 0, we can add the opposite of 6 to 0. The opposite of 6 is In both problems the answer is. Adding the opposite of a number to subtract is called algebraic addition. We change subtraction to addition by adding the opposite of the subtrahend. Subtraction: minuend subtrahend = difference (sign change) Addition: addend + opposite of = sum subtrahend a. b. Lesson Practice Model Find each sum. Draw a number line to show the addition for problems a and b. Solve problems c h mentally a b. + 7 c d e. (+) + ( ) f. (+0) + ( ) + g. ( 0) + ( ) h. ( 0) + (+) Find the opposite of each number: i j. k. 0 0 Written Practice Distributed and Integrated. (, ). (). (0, ) *. (98) *. (98) If 0.6 is the divisor and. is the quotient, what is the dividend? 0.7 If a number is twelve less than fifty, then it is how much more than twenty? 8 If the sum of four numbers is.8, what is the mean of the four numbers?.7 Model Illustrate this problem on a number line: + + Find each sum mentally: a b c. + + d * 6. (98) Solve each subtraction problem using algebraic addition: a. + b. 0 c. + + d. + * 7. (90, 9) Analyze What is the measure of each angle of an equilateral triangle? Saxon Math Intermediate 6

45 8. (66) * 9. () 0. (RF9) Quadrilateral ABCD is a parallelogram. If angle A measures 70, what are the measures of angles B, C, and D? m B = 0 ; m C = 70 ; m D = 0 a. If the spinner is spun once, what is the probability that it will stop in a sector with a number? How do you know your answer is correct? b. Estimate If the spinner is spun 0 times, about how many times would it be expected to stop in the sector with the number? About times Find the volume of the rectangular prism at right. 0 in. C D B A 9. a. ; Sample: Two thirds of the top half of the circle is labeled. 6 in.. (6) Twelve of the 7 students in the class are boys. What is the ratio of girls to boys in the class? 7 in. in.. (Inv. 8) *. (9). (7). (9) 8. (7) 9. (8) Thirty number cubes are rolled simultaneously. Predict the number of s that will result. The fraction is equal to what percent? 66 % If 0% of the students brought their lunch to school, then what fraction of the students did not bring their lunch to school? x = 9 6. () (. + 0.) (6) Estimate If the diameter of a circular plastic swimming pool is 6 feet, then the area of the bottom of the pool is about how many square feet? Round to the nearest square foot. (Use. for π.) About 8 square feet * 0. (9). (8) Use the formula b = A to find b when A is and h is 6. 8 h Solve this proportion: 9 0 x Rectangle ABCD is 8 cm long and 6 cm wide. Segment AC is 0 cm long. Use this information to answer problems and. D A. (77) What is the area of triangle ABC? cm C B. (RF7) What is the perimeter of triangle ABC? cm. (RF6) Measure the diameter of a nickel to the nearest millimeter. millimeters *. () Estimate Calculate the circumference of a nickel. Round to the nearest millimeter. (Use. for π.) 66 millimeters Lesson

46 6. (, 69) A bag contains marbles. Eight of the marbles are red and are blue. If you draw a marble from the bag without looking, what is the probability that the marble will be blue? Express the probability ratio as a fraction and as a decimal rounded to the nearest hundredth., 0. Connect Complete the table to answer problems 7 9. Fraction Decimal Percent * 7. (96) * 8. (96) * 9. (96) 9 0 a. a. a. 0.9 b. 90%. b. 0% b. 0.0 % 0. (8) A full one-gallon container of milk was used to fill two one-pint containers. How many quarts of milk were left in the one-gallon container? quarts Early Finishers Real-World Connection These three prime factorizations represent numbers that are powers of 0. Simplify each prime factorization. 00 0,000 00,000 Use exponents to write the prime factorization of another number that is a power of 0. Sample: 000 = 60 Saxon Math Intermediate 6

47 FOCUS ON CONCEPTS P What Adding Two Negative Integers Means California Mathematics Content Standards NS.0,. Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations. MR.0,. Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. We know that a negative integer plus a negative integer equals a negative integer. For example, ( 0) + ( 0) = ( 0). Why is this true? First we will look at a real world example. If Sam owes his sister $0, Sam has 0 dollars. If he asks her for another $0 loan, he now owes her another $0. ( $0) + ( $0) = ( $0) If Sam wants to owe his sister $0, he must pay back $0 and $0, or $0. Now think about what a negative number is. For every number a, there is a number a, such that a + ( a) = 0. For example, 0 + ( 0) = 0. You have seen that this is true on a number line in earlier lessons. We will use this idea and the addition properties to see why it is true that [( 0) + ( 0)] = [0+0]. We are saying that [( 0) + ( 0)] is the opposite of [0+0]. If this is true, their sum should be zero. We will begin by finding the sum of [(-0)+(-0)] and [0+0] [( 0) + ( 0)] + [0 + 0] ( 0) + ( 0) Used Associative Property to ungroup the addends ( 0) ( 0) + 0 Used Commutative Property to reorder the addends [( 0) + 0] + [( 0) + 0] Used Associative Property to regroup the addends The sum of a number and its opposite is zero. 0 Simplified The sum of 0 implies that [( 0) + ( 0)] is the opposite of [0 + 0]. This means that [0 + 0] is the opposite of [( 0) + ( 0)]. So it is true that [( 0) + ( 0)] = [0 + 0] Predict Will the sum of any two negative integers always be a negative integer? Try finding the sums below to help you answer this question. a. ( ) + ( 6) ( 9) b. ( ) + ( ) ( ) c. ( ) + ( ) ( ) Possible response: Yes, the sum will be negative. Focus on Concepts P 6

48 LESSON 99 Subtraction of Integers California Mathematics Content Standards NS.0,. Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations. MR.0,. Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Power Up facts mental math problem solving Power Up M a. Number Sense: b. Percent: % of 8 c. Decimals: d. Time: Goldie spoke on the phone for 8 minutes 0 seconds. How many seconds is that? 0 sec e. Geometry: The perimeter of this equilateral triangle is 9.6 centimeters. What is the length of each side?. cm f. Number Sense: 0 0 g. Fractions: h. Calculation: 7 7, +,,,,,,, +, Choose an appropriate problem-solving strategy to solve this problem. If the last page of a section of large newspaper is page 6, what is the fewest number of sheets of paper that could be in that section? 9 sheets New Concept Math Language A positive number and a negative number whose absolute values are equal are opposites. Recall that the graphs of and are the same distance from zero on the number line. The graphs are on the opposite sides of zero. 0 This is why we say that and are the opposites of each other. is the opposite of is the opposite of 6 Saxon Math Intermediate 6

49 We can read as the opposite of. Furthermore, ( ) can be read as the opposite of the opposite of. This means that ( ) is another way to write. () () 0 There are two ways to simplify the expression 7. The first way is to let the minus sign signify subtraction. When we subtract from 7, the answer is. 7 = The second way is to use the thought process of algebraic addition. To use algebraic addition, we let the minus sign mean that is a negative number and we treat the problem as an addition problem. 7 + ( ) = Notice that we get the same answer both ways. The only difference is in the way we think about the problem. We can also use algebraic addition to simplify this expression: 7 ( ) We use an addition thought and think that 7 is added to ( ). This is what we think: 7 + [ ( )] Notice that we include ( ) in brackets [ ], a symbol of inclusion like parentheses, in order to group the symbols and number. The opposite of is, so we can write 7 + [] = 0 We will practice using the thought process of algebraic addition because algebraic addition can be used to simplify expressions that would be very difficult to simplify if we used the thought process of subtraction. Example 0 Thinking Skills Generalize What is the sum of +7 and 7? Simplify: ( ) We think addition. We think we are to add and ( ). This is what we think: ( ) + [ ( )] The opposite of is itself. So we have ( ) + [] = Example Simplify: ( ) (+6) We see three numbers. We think addition, so we have [ ( )] + ( ) + [ (+6)] Lesson 99 6

50 We simplify the first and third numbers and get [+] + ( ) + [ 6] = 9 Note that this time we write as +. Either or + may be used. Example Simplify: 0 6 This problem directs us to subtract a negative six from negative ten. Instead, we may add the opposite of negative six to negative ten = Example Simplify: ( ) (+) Instead of subtracting a positive five, we add a negative five. ( ) (+) ( ) + ( ) = 8 Lesson Practice Generalize Use algebraic addition to simplify each expression. a. ( ) (+) b. ( ) ( ) c. (+) () d. ( ) (+) ( ) e. ( 8) + ( ) (+) f. ( 8) (+) + ( ) g. Which is greater, + ( 6) or ( 6)? Explain. ( 6), Subtracting 6 is the same as adding +6 while adding 6 is the same as subtracting +6. Written Practice Distributed and Integrated *. (78). (RF). (0) For every three daisies there were a dozen dandelions. If there were daisies, how many dandelions were there? 0 Connect A shoe box is the shape of what geometric solid? Rectangular prism Analyze If the mean of six numbers is, what is the sum of the six numbers? 7. (RF6, 6) *. (89) If the diameter of a circle is inches, what is the radius of the circle? inch What is the cost of.6 pounds of meat priced at $.6 per pound? $.9 6 Saxon Math Intermediate 6

51 6. (RF9) Suppose AC is cm long. If AB is the length of AC, then how long is BC? 9 cm A B C * 7. (98) Find each sum mentally: a. + 7 b c d * 8. (98) * 9. (Inv. 6) Solve each subtraction problem using algebraic addition: a. + b c d e. Generalize Describe how to change a subtraction problem into an addition problem. Sample: Change the sign of the subtrahend and add. Explain Two coins are tossed. a. What is the probability that both coins will land heads up b. What is the probability that one of the coins will be heads and the other tails? Complete the table to answer problems 0. Fraction Decimal Percent 9. There are four equally likely outcomes. H T H T H T a. One of the four outcomes is HH, so the probability is. HH HT TH TT * 0. (96) *. (96) *. (96) a. a. 0 a. 0.7 b. 7%.6 b. 60% b. 0.0 % b. Two of the four outcomes are HT and TH, so the probability is.. (6) 6. (6) Find each unknown number: 6. (9, 8) 8. (8) 0. (RF9). () x 7. (7) 0.06n = $0. $.0 9. (8) 6. (9) 8 0 x 6n = Connect Nia s garage is 0 feet long, 0 feet wide, and 8 feet high. a. How many -by--by--foot boxes can she fit on the floor (bottom layer) of her garage? 00 boxes b. Altogether, how many boxes can Nia fit in her garage if she stacks the boxes 8 feet high? 00 boxes x + = Multiple Choice Estimate If a roll of tape has a diameter of inches, then removing one full turn of tape yields about how many inches? Choose the closest answer. C A in. B in. C 7 in. D 9 in. 0 ft 0 ft 8 ft. (9) Use the formula h = A b to find h when A is. m and b is.6 m. h =.8 m Lesson 99 6

52 Use the figure to answer problems and.. (RF) *. (9) *. (98) 6. (8) Together, these three triangles form what kind of polygon? Pentagon Generalize What is the sum of the measures of the angles of each triangle? 80 At 6 a.m. the temperature was 8 F. By noon the temperature was F. The temperature had risen how many degrees? F Connect To what decimal number is the arrow pointing on the number line below? (, ) What is the probability of rolling a perfect square with one roll of a number cube? The probability of rolling or is. * 8. (RF7, 77) Connect What is the area of a triangle with vertices located at (, 0), (0, ), and (0, 0)? 6 units * 9. (89) 0. (8) Explain How can you convert 8 feet to yards? If a gallon of milk costs $.80, what is the cost per quart? $0.9 per quart Early Finishers Real-World Connection The surface of the Dead Sea is approximately 08 meters below sea level. Its greatest depth is 0 meters. In contrast, Mt. Everest reaches a height of 8,80 meters. What is the difference in elevation between the summit of Mt. Everest and the bottom of the Dead Sea? Show your work. 08 m Dead Sea 0 m 988 m; 08 m 0 m = 78 m; 880 m ( 78 m) = 988 m 9. Sample: We set up 8 feet as a fraction and multiply by the number of feet in a yard: 8 feet yard. Then we simplify by canceling: 8 feet feet yard 8 yd 6 yd feet 66 Saxon Math Intermediate 6

53 LESSON 00 Ratio Problems Involving Totals California Mathematics Content Standards NS.0,. Use proportions to solve problems (e.g., determine the value of N if _ 7 = N, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. MR.0,. Determine when and how to break a problem into simpler parts. Power Up facts mental math Power Up K a. Number Sense: b. Number Sense: c. Estimation: A rectangle has a length of.9 cm and a width of.0 cm. Estimate the perimeter. 0 cm d. Functions: Tim earned $7.0 for each hour of work. He made this table to show the rate of pay. How much money would Tim earn for hours of work? $0.00 Hours Pay $7.0 $.00 $.0 e. Decimals: f. Algebra: If t = 8, what is t? 6 g. Measurement: How many cups are in a quart? cups h. Calculation: +, 0,, 9, +,, 7, +, problem solving Choose an appropriate problem-solving strategy to solve this problem. How many different triangles of any size are in this figure? 0 Lesson 00 67

54 New Concept Thinking Skills Model Use red and yellow counters or buttons to model the problem. Math Language A proportion is a statement that shows two ratios are equal. In some ratio problems a total is used as part of the calculation. Consider this problem: The ratio of boys to girls in a class was to. If there were 7 students in the class, how many girls were there? We begin by drawing a ratio box. In addition to the categories of boys and girls, we make a third row for the total number of students. We will use the letters b and g to represent the actual counts of boys and girls. Ratio Actual Count Boys b Girls g Total 9 7 In the ratio column we add the ratio numbers for boys and girls and get the ratio number 9 for the total. We were given 7 as the actual count of students. We will use two of the three rows from the ratio box to write a proportion. We use the row we want to complete and the row that is already complete. Since we are asked to find the actual number of girls, we will use the girls row. And since we know both total numbers, we will also use the total row. We solve the proportion below. Ratio Actual Count Boys b Girls g Total g 7 9g = 7 g = We find that there were girls in the class. If we had wanted to find the number of boys, we would have used the boys row along with the total row to write a proportion. 68 Saxon Math Intermediate 6

55 Example The ratio of football players to band members on the football field was to. Altogether, there were 7 football players and band members on the football field. How many football players were on the field? We use the information in the problem to make a table. We include a row for the total. The ratio number for the total is 7. Ratio Actual Count Football Players f Band Members b Total 7 7 Next we write a proportion using two rows of the table. We are asked to find the number of football players, so we use the football players row. We know both totals, so we also use the total row. Then we solve the proportion. Ratio Actual Count Football Players f Band Members b Total 7 7 We find that there were 0 football players on the field. 7 f 7 7f = 7 f = 0 Example The ratio of basketball players to soccer players in the room was to 7. If the basketball players and the soccer players in the room totaled 8, how many were basketball players? We use the information in the problem to form a table. We include a row for the total number of players. Basketball Players Soccer Players Total Ratio Actual Count b 7 s 8 = b 8 b = 8 b = 0 To find the number of basketball players, we write a proportion from the basketball players row and the total row. We solve the proportion to find that there were 0 basketball players in the room. Lesson 00 69

56 a. Lesson Practice Represent Use ratio boxes to solve problems a and b. S C T b. R N T Ratio A. C. s c 8 7 Ratio A. C. r n 60 a. Sparrows and crows perched on the wire in the ratio of to. If the total number of sparrows and crows on the wire was 7, how many were crows? 7 crows b. Raisins and nuts were mixed by weight in a ratio of to. If 60 ounces of mix were prepared, how many ounces of raisins were used? ounces c. Model Using 0 red and 0 yellow color tiles (or 0 shaded and unshaded circles) create a ratio of to. How many of each color (or shading) do you have? See student work. Sample: and 8 Written Practice Distributed and Integrated On his first six tests, Cleavon had scores of 90%, 9%, 96%, 9%, 8%, and 9%. Use this information to answer problems and.. (Inv. ). (0) a. Which score occurred most frequently? That is, what is the mode of the scores? 9% b. The difference between Cleavon s highest score and his lowest score is how many percentage points? That is, what is the range of the scores? % What was Cleavon s average score for the six tests? That is, what is the mean of the scores? 9% *. (, 8) *. (8) *. (8) 6. (8) 7. (99) 8. (8) In basketball there are one-point baskets, two-point baskets, and three-point baskets. If a team scored 96 points and made 8 one-point baskets and 6 three-point baskets, how many two-point baskets did the team make? Explain how you found your answer. Multiple Choice Analyze Which ratio forms a proportion with 7? C A 7 B 7 C D Complete this proportion: Four is to five as what number is to twenty? 6 Arrange these numbers in order from least to greatest:, 0., 0, 0.,,, 0., 0., 0 a. ( ) ( 6) b. ( ) (+6) 0 c. ( 6) ( ) d. (+6) ( ) 0 The area of the square in this figure is 00 mm. a. What is the radius of the circle? 0 mm b. What is the diameter of the circle? 0 mm c. What is the area of the circle? (Use. for π.) mm. 0 two-point baskets; Sample: (8 ) + (6 ) + (b ) = 96, b = 96, 6 + b = 96, b = 60, b = 0 60 Saxon Math Intermediate 6

57 Connect Complete the table to answer problems 9. * 9. (96) * 0. (99) *. (99). (6). (6) 6. (6, 69) 7. (RF) 8. (9) * 9. (Inv. 6) Fraction Decimal Percent a. 00 a. 9 0 a. 0.6 b. 6% 0.0 b. % b % 6 9. (67) () Estimate Convert 7 to a decimal number by dividing by 7. Stop dividing after three decimal places, and round your answer to two decimal places. 0. An octagon has how many more sides than a pentagon? more sides x = 7 x = 6 7 Analyze Sector on this spinner is a 90 sector. If the spinner is spun twice, what is the probability that it will stop in sector both times? 6 0. () If the spinner is spun 00 times, about how many times would it be expected to stop in sector? About 7 times. (00) Draw a ratio box for this problem. Then solve the problem using a proportion. The ratio of boys to girls in the class was to. If there were 0 students in the class, how many were girls? girls. (0) The average of four numbers is. What is their sum? 0 *. (RF0) Connect When Andy was born, he weighed 8 pounds ounces. Three weeks later he weighed 0 pounds ounce. How many pounds and ounces had he gained in three weeks? pound ounces *. (9). (88) 6. (Inv. 8) Lines s and t are parallel. a. Which angle is an alternate interior angle to? b. If the measure of is 76, what are the measures of and? m = 76 ; m = 0 Which property is illustrated by the equation x(y + z) = xy + xz? Distributive Property A spinner has equal spaces marked one through. If it is spun 0 times, how many times can we predict it will stop on? r s t 7. (66, 77) a. What is the area of the parallelogram at right? in. b. What is the area of the triangle? 6 in. c. What is the combined area of the parallelogram and triangle? 0 in. 6 in. in. 6 in. in. in. Lesson 00 6

58 * 8. (76) 9. (RF7) 0. () How many milligrams is half of a gram? 00 milligrams Model The coordinates of the endpoints of a line segment are (, ) and (, ). The midpoint of the segment is the point halfway between the endpoints. What are the coordinates of the midpoint? (, ) Estimate Tania took 0 steps to walk across the tetherball circle and steps to walk around the tetherball circle. Use this information to find the approximate number of diameters in the circumference of the tetherball circle. About. diameters 6 Saxon Math Intermediate 6

59 INVESTIGATION Focus on Similar Figures 0 California Mathematics Content Standards NS.0,. Use proportions to solve problems (e.g., determine the value of N if /7 = N/, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. MG.0,. Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle. Triangles that are the same shape and size are congruent. The two triangles below are congruent. Each triangle has three angles and three sides. The angles and sides of triangle ABC correspond to the angles and sides of triangle XYZ. X C A By rotating, translating, and reflecting triangle ABC, we could position it on top of triangle XYZ. Then their corresponding parts would be in the same place. B Z Y A corresponds to X. B corresponds to Y. C corresponds to Z. AB corresponds to XY. BC corresponds to YZ. AC corresponds to XZ. If two figures are congruent, their corresponding parts are congruent. So the measures of the corresponding parts are equal. Example These triangles are congruent. What is the perimeter of each? in. in. in. in. We will rotate the triangle on the left. in. in. in. in. Now we can more easily see that the unmarked side on the left-hand triangle corresponds to the -inch side on the right-hand triangle. Since the triangles are congruent, the measures of the corresponding parts are equal. So each triangle has sides that measure inches, inches, and inches. Adding, we find that the perimeter of each triangle is inches. in. + in. + in. = in. Investigation 0 6