Cycle 2: Why Does It Matter?

Lesson. It s All Relative 9 Part Cycle : Why Does It Matter? Lesson. It s All Relative. 5 5.. a. Negative; $0,000 Negative; 400 4. a. Loss of 0 yards Loss of 0.6 points for the day 5. 6. a. 6 6 4 4 c. 5 5. d. 5 4.8 7. The student s analogy with money is actually correct. However, that does not mean that -5 is bigger than - 4. The bigger the debt, the more the person is in the hole and the less money he or she has. Therefore, -5 is less than -4. 8. a. 7 7 c. 7, 7 d. a Lesson. Sign and Size, Part. a. 5. (.) 5.. 6. ( ) ( ) 44

0 Cycle, Part. Answers will vary depending on the year. Possible answer: 0 ( 500) 0 500 5 years. $0 $5 ($0 $5) $5 She will owe him $5 in total. 4. $,500 $0,000 $0,000 $,500 $8,500 She will have $8,500 after the debt is paid. 5. 65 The team has already gained yards on the first three downs. They will need 0 7 more yards on the fourth down to have a 0-yard gain (and get a first down). Lesson.4 Sign and Size, Part. a. (0)( 5) 0 5 5. ( )( )( 7) ()( 7) 4. a. Multiplying two negative numbers results in a positive number. For example, ( 4)( 4) 6. Adding two negative numbers results in a negative number. For example, ( 4) ( 4) 8 No. Two negative numbers can result in a positive, negative, or zero result depending on the operation. 4. Answers will vary. Possible answer: Two wrongs don t make a right. 5. Problem with parentheses Answer Problem without Parentheses New Result a. ( 4) 8 4 ( ) ( 4) 6 4 6 c. (4) 4 d. ( )( 4) 8 4 6 Do parentheses matter? Yes No No Yes 6. a. 6? 0 06 0? 0 0? 6 6 0 is undefined no such number

Lesson.4 Sign and Size, Part c. Zero divided by a number (other than zero) is zero. A number cannot be divided by zero. 7. a. Positive It depends. Possible examples: ( 7) 4, 7 4 c. Negative d. Positive e. Positive f. Negative g. Negative 8. Answers will vary. Possible calculation with a negative number: 4 4 6 6 6 ( ) Yes. The result is always. Possible calculation with a fraction: 65 5 5 5 Lesson.5 An Ounce of Prevention. 784 ( ) ( 0) 7 7 7. Current average: 75 74 7 0 7. To average an 80, point total 80. So her point total must be 0 points. She needs 0 0 00 on 4 the last test to have a B average.. Answers will vary. Any scores, between 0 and 0, which add to will work. Possible answers: 7, 7, 7; 6, 7, 8; 0, 0, ; 9, 9, ; 5, 6, 0 4. Answers will vary. It s often easy to gain weight, but hard to lose it. It is easy to get in debt, but hard to get out of it. Addition is usually easier than subtraction. Multiplication is easier than division. 5. To split the check evenly, the total bill must be divided by five. You would each pay $55 $. This amount also represents the average of the individual bills. 5

Cycle, Part 6. a. 6 8 9 4 4 44 47 49 50 40 4 0 0 The mean income for Group is $4,000/year 6 8 9 4 4 44 47 49 50 50 5 0 0 The mean income for Group is $5,000/year The mean is affected by one extreme value in a data set. When the last income was changed from $50,000 to $50,000, the mean increased by $0,000. c. No. For Group, a mean of $5,000/year is not very representative of the typical income since only one of the ten incomes is higher than that. 7. a. Answers will vary. Possible answer: 40, 45,50,55,60 50,55, 60, 65, 70 c. New mean: 60 d. When 0 was added to each score, the mean also increased by 0. e. To increase the mean by 8 points, the teacher should add 8 points to each student s test score. 8. a. 50 00 50 75 75 75 50 75 c. The mean does not tell us whether the two test scores are close to the mean of 75 or more spread out. The mean only tells us about the center, not the spread of the data. 9. a. 76455 60 0 6 6 The mean is the balancing point for the data. If we think of the number line as a teeter-totter, the mean of 0 is where the data values will balance. 0. a. The average height should be in inches, not square inches. Height is a linear measurement, not an area. The average square footage should be in square feet, not feet. Square footage is a measurement of area, not length. Lesson.6 Measure Up. a. c. (4 ab) ( ab) 4 a b ( ab),744a 4 4,744ab 0 6 5 ( x ) x 7x 9 9 6 4m 4m m 8mn 8n n b

Lesson.6 Measure Up. a. C r (4 in.) 5 in. Ar ( ft) ft. m V d g g / c g g c c g g c The volume will be in cubic centimeters. 4. v D 0.87 g (m/sec) m/sec m /sec m/sec m m sec sec m sec sec m m The distance will be in meters.

4 Cycle, Part 5. T L g ft ft/sec ft ft sec sec ft ft sec sec The time will be in seconds. Lesson.7 Count Up. No, they are not like terms. They do not have the same exponents on the x factors and they do not have the same exponents on the y factors... x xyxyx () xy( ) x xy x ( x )( xy)( xy)( x )() 6,888x y 6 6,888x y 4. The polynomial is a trinomial since it has three terms. 5. a. The student appears to have added 5x 6y to get xy, but they are not like terms and should not be combined. Instead, the xy terms should be combined and the x term and y term should be left separate. xy5x6y8 xy ( xy8 xy) 5x6y xy 5x 6y The expression still has like terms ( 5 x and 8x ) that should be combined. 5x 0xx8 x (5 8) x (0 ) x x x 6. Answers will vary. Possible answer: Expression that contains like terms: 5x 6yx

Lesson.7 Count Up 5 Expression that does not contain like terms: 5x 6y xy 7. S r rl ( in.) ( in.)(4 in.) = (4 in. ) (8 in. ) (4 8) in. in. 7.7in. Cycle Part Recap. a. 5 5 0 (5 x ) 5 ( x ) 5x 50x 50 5 5x 505 45 0x 0x c. d. e. f. 4 4 4 5x 5x 0x 4 4 44 8 5x 5x 5x 5x 0 0 5 5 0 ( 5) Concepts and Applications. a. Yes; The sum of negative numbers will always be negative. The average will then be a negative sum divided by a positive number, which will always be negative. Answers will vary. Possible answer: 0,0,0,0,0,,,,,. $50(4) $5() $00 $75() $00 $75 $00 $50 $75 $50 $5 Your net debt is $5.

6 Cycle, Part Part Cycle : Why Does It Matter? Lesson.9 Order Up. a. 5 5 (4 ) ( 8) 9 9 5 8 9 0 9 0 9 0 5 5 4 6 68. a. 5x 5( ) y ( 4) 0 4 4 0 6 5 8 5x 5(0) y ( 4) 0 ( 4) 0 5x 5( 4) y (0) 0 0 undefined (weight in pounds)(70) BMI= (height in inches) (50 lb)(70) (7 in.) 05,450 lb 5,84 in. 0.4 lb/in.. Answers will vary based on the calculator used. 4. a. Knee height = 6 in..54 cm/in. 40.64 cm Height in centimeters 84.88 (0.480) (.840.64) 40.05 cm Height in inches 40.05 cm.54 cm/in. 55.4 in.

Lesson.9 Order Up 7 Answers will vary. 5. Expression d best illustrates the process to make cupcakes. The dry ingredients are combined and the wet ingredients are combined. And then the total is divided into 4 cupcakes. 6. a. The multiplier is.097..0975($45) $49.9 The multiplier is 0.80. 0.80($75) $60 Lesson.0 Does Order Matter?. a. ( yx) z or z( x y) x ( y z). a. ( yx) z or zxy ( ) x( yz ). a. Answers will vary. Possible answer: 6 but 6 Divide 5 by 5 implies 5 5 when the intended problem is 5 5. You could say, Divide 5 by 5 or Divide 5 into 5. How you verbalize the problem matters since it indicates the order of the numbers in the division problem. 4. a. 6 Switch the 8 and the. 8888 6 c. 8 is easier to do with repeated addition because there are fewer numbers to add and keep track of in your head. 5. a. Your friend s grade would be represented by x 5 since it is 5 points lower than yours. Your grade would be represented by x 5 or 5 x since it is 5 points higher than his. 6. Group all the positive numbers together and all the negative numbers together and then combine those results. 7. a. Commutative Property Associative Property

8 Cycle, Part 8. a. 5 0 0 6 5 4 4 4 6 7 5 5 5 4 4 7 5 7 c. The two operations commute. You can multiply first then simplify or simplify first then multiply. You will get the same answer either way. Lesson. Fair Share. a. 4( x 6 x ) 4x46x 4x 64x (x)( x) x( x) ( x) x xx x x. 77(0) 707 40 4 54. a. 5 55 6 9 6

Lesson. Fair Share 9 4 6 5 6 0 9 6 6 6 9 6 6 6 6 9 6 Yes c. No 4. a. Distributive Property Commutative Property c. 0 0 0 6 d. Method (student s nontraditional way): 4 4 4 4 4 4 Method (traditional approach): 9 7 4 4 4 4 4

40 Cycle, Part 5. a. c. ( ab)( ab) (ab)a (ab)b a ba ab b a b 495 (50 )(50 ) 50,500,499 768 (70 )(70 ) (70 )(70 ) 70 4,900 4 4,896 Commutative Property for Multiplication 6. Each output is one less than three times the input value. Input Output 0-5 5 4 00 99 n n Lesson. Seat Yourself. 5( x8) 4( x) 6 5x40 4x4 6 x 50. 8( n) 786n7n 5. a.

Lesson. Seat Yourself 4 One way to think about this is that there are 4 corner tables that each seat people. All the other tables ( n 4 of them) seat person. So there are 4 ( n4) 8n4n 4 seats for n tables. c. Number of tables Number of chairs 4 8 8 6 6 0 n n 4 You get the same general result, n 4, as was achieved by using the physical situation in part d. Yes. The number of tables in this arrangement must be a multiple of 4, and 0 is a multiple of 4. If 0 tables were used, 4 people could be seated. e. n 46 n tables would be needed to seat 6 people. 4. a. 9 You will meet at mile marker. c. B A d. e. f. g. ( ) B A A ( BA) A B A A B B ( BA) B B A A B A B ( A B) A B Yes, they are equivalent.

4 Cycle, Part Lesson. Punt, Pass, Kick.. (5 in.) (5 in.) hyp 5 in. 5 in. hyp 50 in. hyp hyp 50 in. hyp 7. in. (4 ft) (leg) (0 ft) 6 ft (leg ) 00 ft (leg ) 84 ft leg 9. ft. a. All are whole numbers and 8 5 89 7 All are whole numbers and 6 0,56 4 4, 45, 5 c. d. The triangles are similar. 4. a. No. Since4 7, the lengths do not make a triangle. In order to make a triangle the sum of the lengths of the shorter sides must be greater than the length of the longest side.

Lesson. Punt, Pass, Kick 4 Yes. Since4 6, the lengths do make a triangle. No. Since 4 6, the lengths do not make a right triangle. c. Answers will vary. Some possible answers: 4 ft,5 ft, 6 ft ; in.,8 in., 0 in. 5. If the area of the square is 00 square inches, then each side is 0 inches long. (0 in.) (0 in.) hyp 00 in. 00 in. hyp 00 in. hyp hyp 00 in. hyp 4. in. So the diagonal of the square is about 4. inches long. 6. a. 847 446 distance 77, 409 98,96 distance 96,5 distance 96,5 distance distance 957ft Yes. The crime was committed about 957 feet from a church, making it a felony.

44 Cycle, Part Lesson.4 Ramp Up. m 8 8 4 0 7 5. The graph goes through the points (0,) and (5,0) 0 m 5 0 5 5. Every time x increases by, y decreases by 5. So change in y 5 m 5. change in x 4. a. c. rise 70 inches 7 m run 0 inches rise 7 inches 7 m run inches (70 in.) (0 in.) hyp 4,900 in.,00 in. hyp 7,000 in. hyp hyp 7,000 in. hyp 0 in. 0 ft, 0 in. 5. For fractions with a numerator of, the larger the denominator, the smaller the fraction. So. The ramp with a slope of : is the steepest because it has the largest slope of the three 0 6 and the shortest run for the rise of. The ramp with a slope of :0 is the flattest because it has the smallest slope of the three and the longest run for a rise of.

Lesson.4 Ramp Up 45 6. a. 50 ft m 50 ft/mi mi 500 ft m 50 ft/mi mi,000 ft m 50 ft/mi 4 mi We get the same slope regardless of which triangle is used. c. In order to write the slope as a percent, the units on the rise and run need to be the same. 50 ft m 0.047 4.7% 5,80 ft Yes. The truck can drive safely on these roads since the slope is less than 6%, 7. a. The student put the change in x on the top of the slope fraction and the change in y on the bottom of the slope fraction. The student subtracted in a different order in the numerator vs. the denominator of the slope fraction. c. The student did not simplify the final fraction to. 8. First triangle: (50 ft) (5, 80 ft) hyp 6,500 ft 7,878, 400 ft hyp 7,940,900 ft hyp Second triangle: miles 5,80 ft/mi 0,560 ft hyp 7,940,900 ft hyp 5,85.9 ft

46 Cycle, Part (500 ft) (0,560 ft) hyp 50,000 ft,5,600 ft hyp,76,600 ft hyp hyp,76,600 ft hyp 0,57.8 ft Third triangle: 4 miles 5, 80 ft/mi,0 ft (,000 ft) (,0 ft) hyp,000,000 ft 446, 054, 400 ft hyp 447,054,400 ft hyp Lesson.5 Shortest Distance hyp 447,054,400 ft hyp,4.7 ft. d ( x x ) ( y y ) () (8 6) 44 45.0. d ( x x ) ( y y ) ( ) ( ) ( ) ( ) ( 4) 4 576 580 4.

Lesson.5 Shortest Distance 47. Student #: This student forgot the square root. Student #: This student did not handle the negative number correctly when he subtracted the x-values. It should be ( ), not. Student #: This student did not do the operations in the correct order. He needed to square and add before taking the square root. d ( x x ) ( y y ) ( ) (68) ( ) ( ) 94.6 4. a. d ( x x ) ( y y ) (4 ) ( 5) 6 8 6 64 00 0 d ( x x ) ( y y ) ( 4) (5 ) ( 6) ( 8) 6 64 00 0 c. When you square the differences in the distance formula, you get the same positive number regardless of the order of the subtraction. Changing the order of the subtraction changes the sign on the difference but does not affect the square of the difference. 5. The order of the subtraction does not matter in the distance formula, and the same answer will be obtained regardless of the order of the subtraction. The x-values do not have to be subtracted in the same order as the y-values to obtain the correct answer. The order of the subtraction does not matter in the slope formula, but the same order must be used in the numerator and denominator of the slope fraction.

48 Cycle, Part Cycle Part Recap. d ( x x ) ( y y ) ( ) (0 ) ( ) (0 ) 5 5 44 69 y y m x x 0 ( ) ( ) 0 5 Concepts and Applications.. vertical distance slope horizontal distance distance (horizontal distance) (vertical distance)

Cycle, Part Recap 49 rise grade run 00 ft,640 ft 0.076 7.6% length of road (horizontal distance) (vertical distance) 00,640 40,000 6,969,600 7,009,600,647.6 ft

50 Cycle, Part Part Cycle : Why Does It Matter? Lesson.7 Parts of Speech. The multiplication in the numerator of the fraction should be performed first.. The squaring of the x-value under the square root should be done first.. a. 4 4( x y) c. x d. x 4 4. Jake is incorrect. The addition must be performed before the square root since the square root functions as a grouping symbol. The expression is correctly simplified as follows: 5. a. 5 9 4 5.8 x y x y x y Are the expressions equal? 0 0 0 0 Yes.4 No 0 Yes 6 4 4.47 6 No 4 0 Yes 5 9 5.8 8 No Yes. Since there is at least one example in the table for which the expressions are not equal, that is enough to disprove the claim that x y x y. c. No. Even though there is at least one example in the table for which the expressions are equal, that is not enough to prove the claim that x y x y. d. No. There are examples in the table for which the two expressions are not equal. e. No. There are examples in the table for which the two expressions are not equal. 6. No. The expressions are not equivalent since, for example, (5 ) 9 but 5 54. This one counterexample shows that the expressions are not equivalent.

Lesson.7 Parts of Speech 5 ( ab) ( ab)( ab) aa ( b) ba ( b) a ababb a abb 7. a. n 5 5 75 7 square, multiply by, subtract 8. a. ( x ) (0 ) (9) 8 6 subtract, multiply by, divide by or subtract, divide by, multiply by. Lesson.8 In the Swing of Things. G x 7 9. a. square, then multiply by r (inches) A (square inches) 45.9 50.9 4 65.75

5 Cycle, Part 5 706.86 6 804.5 c. square inches d. The formula represents a function since each value of the radius is matched with exactly one value of the area. The formula does not represent a linear function since the area does not increase by the same amount each time the radius goes up by one inch. e. The radius would need to be a little less than 5 inches to yield an area of 700 square inches. f. g. The data points might look like they fall in a line, depending on the scale used for the graph. We can be sure the data is not linear, however, by analyzing the table and noting that the area values do not increase by the same amount each time the radius increases by one inch. Lesson.9 Error and Estimation: Rounding. r APY n 0.04 0.0407 4.07% n. a. divide i by n, add, raise to the n p power, subtract i p r n n 0.075 0.00655 0.655%

Lesson.9 Error and Estimation: Rounding 5 c. i p r n n 4 0.085 0.004 0.4% Cycle Wrap-Up: Cycle Profile Write the slope of the line in two ways: a m ; b y y m x x Write two statements about the value of d: d a b ; d ( x x ) (y y ) Snapshots: Commutative Associative Distributive + Positive Negative Positive + depends Negative depends x x x a b ab ( x ) x a b a b Positive Negative Positive + Negative + Cycle Wrap-Up: Vocabulary Check. negative number. integers. opposite 4. absolute value 5. real numbers 6. mean 7. exponent, base 8. like terms, coefficients 9. polynomial 0. monomial, binomial, trinomial. degree. PEMDAS. commutative property 4. associative property 5. distributive property

54 Cycle, Part 6. Pythagorean theorem 7. hypotenuse, legs 8. Pythagorean triple 9. slope 0. distance formula. operation, objects. radical function Cycle Wrap-Up: Concepts and Applications Review. 8 ( 8) 6 8 ( 8) 880 8 ( 8) 64 8 ( 8). a. 9 96 ( ) 9 6 9 9 9 ( x ) 6 x c. ( x) 6 x 6 x x x x xx. a. Friday: $00 $00 $50 $5 $0 $0 $05 ; $05 $0 (overdraft fees of $5 per item) $5 Saturday: $5 $0 $5 $70 ; $70 $0 (overdraft fees of $5 per item) $5 (overdraft fee of $5 per day) $85

Cycle Wrap-up 55 Sunday: $85 $85 70 $70 $5 (overdraft fees of $5 per item) $5 (overdraft fee of $5 per day) $80 It would be better if the smaller charges cleared first on Friday. Since there is a charge based on the number of items once the account is overdrawn, the fee will be less if most of the smaller items clear first. If the book store charge was processed last, it would be the only one to incur a fee on Friday since the other items total to $55 and would be covered by the $00 balance. c. No. If you transfer $50 first thing Saturday, the balance will only be $5 and insufficient to cover both charges on Saturday. Transferring $50 on Saturday morning would avoid one of the overdraft fees on Saturday though. 4. a. c. 7 78 75 8 58.5 4 7 78 75 8 76.75 4 58.5 76.75 0.4 4% 76.75 5. S r rh ( ft) ( ft)(0 ft) 8 ft 40 ft 48 ft 50.8 ft 6. a. Yes, but with more than one variable. c. x and x are like terms. 4 x,5 x, and 9x are like terms. x 4xy y5xx 9xx y y 7. a. (5 4)( 4) ( 9)( ) 8 (5 4)( 4) ( 9)( 4) 9 94 8 6 8

56 Cycle, Part c. (5 4)( 4) 5( 4) 4( 4) 5 544 44 0 0 8 56 8 8. 9. 99(5) (00 )(5) 005 5,500 5,475 8 8 hyp 4 4 hyp 648 hyp 648 hyp hyp 5.5 in. 0. 5 b 0 5 b 00 b 75 b 75 b 8.7 ft. a. 7 chairs n c. There is one seat for each table, plus two more seats on the ends.

. First, find the difference of the x-coordinates and the difference of the y-coordinates. Second, square each of those differences. Third, add the squared differences. Fourth, take the square root of that sum.. First, subtract the y-values. Second, subtract the x-values in the same order. Third, divide the difference in the y-values by the difference in the x-values. 4. Expression Translation x Take the square root of x and then add. x Take the square root of the quantity x plus. 5 Divide 5 by the sum of x and. x 5 x Divide 5 by x and then add. x Square x and then multiply the result by ( x ) Square the product of and x. x Take the absolute value of the difference of x and. x Take the absolute value of x and then subtract. Cycle Wrap-up 57 5. Answers will vary. One possible answer is ( (,6) and (5,). 6. a. 57 65.5 7 764.5 69.5 58.75 57.75 69.5 64 649 64.9 inches 0 0 C r d 64.9 in. d 64.9 in. d 0.7 in. c. Take each circumference and divide by to produce the following set of diameters (each rounded to the nearest hundredth)

58 Cycle, Part 8.4 0.85.9.60 0.45. 8.70 8.8.04 0.7 Average diameter: 8.4 0.85.9... 0.8 06.66 0.7 in. 0 0 This produces the same result. 7. A steep line would have a large, positive value for the slope. Since there is no largest positive number, there is no steepest line. For any line, you can always draw one that is a little steeper because for any positive slope you can always find a slightly larger positive number. A vertical line may seem to be the steepest line, but by definition, it does not have a slope.