Kronos wants to travel from Rigel Kentaurus to Earth. The distance is 4.3 light years. A light year is 5.9 x 0 2 miles. This is the distance light travels in one year. Centaurus Centaurus Rigel Kentaurus Rigel Kentaurus Kronos can travel at,000,000 miles per hour. If he is 500 years old when he leaves Rigel Kentaurus, how old will he be when he reaches Earth? Centaurus Centaurus Rigel Kentaurus Rigel Kentaurus Classroom Strategies Blackline Master I - 33 Page 03
There are 2.6 x 0 8 people in the USA. The average amount of of soft drinks consumed per person in in the U.S. United is States 44. gallons. is 44. gallons. China has.3 x 0 9 people. If If we gave all our our yearly soft drink consumption for one of soft year drinks to China, to China, how much how much would there be be for each person? Page 04 Classroom Strategies Blackline Master I - 34
Name Date The Last Digit From the Lane County Mathematics Project; Grade 8 5 000 is a huge number. In fact, the answer has close to 700 digits. Do you think it is possible to determine the last digit of 5 000? Or 2 000? Find the last digit (units digit) of each of these numbers.. 000 0. 0 000 2. 2 000. 000 3. 3 000 2. 0 000 4. 4 000 3. 3 000 5. 5 000 4. 4 000 6. 6 000 5. 5 000 7. 7 000 6. 79 000 8. 8 000 7. 273 000 9. 9 000 8. 000 000 Bonus : What are the last two digits of 6 000? Classroom Strategies Blackline Master Page 05 I - 35
Name Date Tower of Hanoi There is an ancient legend that in the great tower of Hanoi there stand three diamond spindles. On the middle one there is a stack of 64 golden disks of different sizes, each one smaller than the one below it. Monks in the temple have the task of moving the disks from one spindle to another, but they can move only one disk at a time, and they can never place a larger disk on top of a smaller one. The legend says that when this task is complete, the temple will disappear in a clap of thunder and the world will end. If the monks are very efficient and move these disks in the quickest way possible with each move lasting only one second, how long do we have until the world ends? To find out how long we have until the world ends, start with a smaller problem and search for a pattern. If there was only one disk, how many moves would it take to transfer that? What if there were only two disks? Fill in the table below. Number of disks Minimum number of moves 2 3 4 5 6 7 Can you find a pattern? According to the legend, how long do we have until the world ends? Page 06 Classroom Strategies Blackline Master I - 36
Classroom Strategies Blackline Master Page 07 I - 37
Page 08 Classroom Strategies Blackline Master I - 38
Absolute Value Triangle Puzzle Absolute Value Triangle Puzzle -5 3-3 - 7 3 30-38 -3-6 -5 + 2 6-5 -6-3 8-4 - 2 7-3-0 6 + -2 2 - -8 7-9 - -3 3-2 2-3 0 6 6-0 - 4(-3) 2-4 -4-7 5-3.4-2.4-8 3-4 - 5 2 4 7 7-20 -2-5 - -6 0.47 - -5.47 Cut the triangles apart. Reassemble the puzzle so that touching edges have equivalent expressions. The result should be the shape shown in miniature below. Classroom Strategies Blackline Master Page 09 I - 39
3 + 5 + 5 + 2 8 9 8 9 3 + 5 5 + 2 + 8 8 9 9 8 5 5 6 8 5 6 5 7 / 9 Associative/Commutative Properties for Addition 6 Associative/Commutative Properties for Multiplication + 8 6 24 7 8 6 24 + 24 24 599 5942 24 599 5942 Distributive Property 5942 Substitution Page 0 Classroom Strategies Blackline Master I - 40
24 + 7 + 95 + 3 + 26 + 5 80 95 + 5 + 24 + 26 + 7 + 3 Associative/Commutative Properties for Addition 25 22 3 4 6600 25 4 22 3 Associative/Commutative Properties for Multiplication 6 42 + 54 6 6 6 42 + 54 24 599 0 278 92 Distributive Property 0 0 24 599 278 92 Associative/Commutative Properties for Multiplication Classroom Strategies Blackline Master Page I - 4
+ + 2 + 6 7 9 9 7 + 6 + + 2 7 7 9 9 2 7 3 9 50 7 2 3 7 9 50 7 29 74 + 29 26 / 3 Associative/Commutative Properties for Addition 300 Associative/Commutative Properties for Multiplication 2900 29(74 + 26) Distributive Property 9(2) 399 9(20 + ) Distributive Property Page 2 Classroom Strategies Blackline Master I - 42
75 + (-6) + 25 + (-4) 80 75 + 25 + -6 + (-4) Associative/Commutative Properties for Addition 3 0 4 25 6 0 3 25 4 6 2 4 9 36 ( + ) 36 + 2 36 4 9 5% of $40 20 Associative/Commutative Properties for Multiplication 7 Distributive Property Commutative Property for Multiplication 6 (0% + 5%)40 Substitution Classroom Strategies Blackline Master Page 3 I - 43
399 + 407 + 20 + 393 400 399+20 + 407 + 393 Associative/Commutative Properties for Addition / 9 45 2 0. 6 6 45 2 / 9 0. 6 4 36 ( 70 ) 36 70 4 Associative/Commutative Properties for Multiplication 630 Associative /Commutative Properties for Multiplication 2(49) 588 2(50 ) Substitution Page 4 Classroom Strategies Blackline Master I - 44
3x + 2y + 4x + 9y 7x + y 3x + 4x + 2y + 9y Associative/Commutative Properties for Addition 4x 2y 8xy 4 2 xy Associative/Commutative Properties for Multiplication 36 ( x + 2y) 36x + 72y 36 x + 36 2y Distributive Property 75x + 25x 00x x(75 + 25) Distributive Property Commutative Property for Multiplication Classroom Strategies Blackline Master Page 5 I - 45
5x + 2x 2 + 3x + 5x 2 8x + 7x 2 5x + 3x + 2x 2 + 5x 2 Associative/Commutative Properties for Addition / 2 (4x y) 2xy / 2 4 xy Associative Property for Multiplication / 2 ( 4 + 2y) 2 + y / 2 4 + / 2 2y Distributive Property 7x + (-5x) + 3x + (-5x) 0 (7x + 3x) + (-5x +( -5x)) Associative /Commutative Properties for Addition Page 6 Classroom Strategies Blackline Master I - 46
Alien Math + $ * # @! $ @ # $! * * #! * $ @ # $ * # @! @! $ @ * #! * @! # $ X $ * # @! $ @! # * $ *! @ # $ * # # # # # # @ * $ #! @! $ * # @! Wow! A family of aliens just had lunch in your back yard, and one of their kids left his homework for you to find. It seems they use the same symbols for addition and multiplication that we do, but different symbols for the numbers. Use the addition and multiplication tables above to discover something about the alien math. Is there an additive identity? If so, what is it? How do you know? Is there a multiplicative identity? If so, what is it? How do you know? Is addition commutative? How do you know? Is multiplication commutative? How do you know? Make up some alien addition and multiplication problems to see whether the associative properties hold for this number system. Make up some problems to see whether the distributive property works in this system. What is the additive inverse of each number? Do all the numbers have a reciprocal? What is the reciprocal of each number? Can you figure out the Earth number that matches each symbol? How do these aliens do math? Classroom Strategies Blackline Master Page 7 I - 47
Associative Property for Addition (4 + 3 / 5 ) + 2 / 5 = 4 + ( 3 / 5 + 2 / 5 ) (a + b) + c = a + (b + c) Associative Property for Multiplication (52 x 4) x 25 = 52 x (4 x 25) a(bc) = (ab)c Additive Identity 0 + 56 = 56 a + 0 = a Multiplicative Identity 75 = 75 a = a Additive Inverse 3 + (-3) = 0 -a + a = 0 Page 8 Classroom Strategies Blackline Master I - 48
Multiplicative Inverse 3 ( / 3 ) = ( / a ) a = for a 0 Distributive Property ( / 3 )(6 + 300) = ( / 3 )6 + ( / 3 )(300) a(b+c) = ab + ac Commutative Property for Addition 25 + 62 = 62 + 25 a + b = b + a Commutative Property for Multiplication 4 5 = 5 4 ab = ba Multiplicative Property of Zero 7564 0 = 0 a 0 = 0 Classroom Strategies Blackline Master Page 9 I - 49
Associative Property for Addition (7 + 89) + = 7 + (89 + ) (x + 2y) +3y = x + (2y + 3y) Distributive Property 24( / 3 + / 8 ) = 24( / 3 ) + 24( / 8 ) 3(x+y) = 3x + 3y Additive Identity 95 + 0 = 95-4 + 0 = -4 Multiplicative Identity 6 = 6 x = x Additive Inverse 5 + (-5) = 0 -a + -(-a) = 0 Page 20 Classroom Strategies Blackline Master I - 50
Multiplicative Inverse 7 ( / 7 ) = ( / x ) x = for x 0 Associative Property of Multiplication / 2 (4 / 3 ) = ( / 2 4) / 3 / 5 (0x) = ( / 5 0)x Commutative Property of Addition 6 + 40 = 40 + 6 x + x 2 = x 2 + x Commutative Property of Multiplication 7 22 = 22 7 x(2) = 2x Multiplicative Property of Zero 9 0 = 0 x 0 = 0 Classroom Strategies Blackline Master Page 2 I - 5
Heaps and Holes II Modeling Variables Name The giant's wife from the story of Jack and the Beanstalk likes to make surprises for the boys who venture to climb to the top of the beanstalk. Each morning she wraps up several boxes with magic beans inside. The beans are invisible, so no one can look inside and see how many are there. Some days she puts hundreds of beans in each box. Some days she puts only 4 beans in each box. Some days she will put only /2 a bean in each box. The boxes she wraps up each day all have the same number of beans. So she can remember how many are inside, she labels each box with a secret code letter. Each box marked with the same letter has the same number of beans inside. Write a variable expression for the number of beans pictured here. x x x x. Do you think that 3x + 2 and 5x are the same? Draw a diagram to represent each expression. 3x + 2 5x 2. Is 4x + 0 different from 4x? Draw a diagram to represent each of these expressions. 4x + 0 4x 3. Is (2x+3) different from 2x+3? Draw a diagram to represent each of these expressions. (2x+3) 2x+3 Page 22 Classroom Strategies Blackline Master I - 52
4. Is 3(x + ) the same as (3x + )? Draw a diagram to represent each of these expressions. 3(x + ) (3x + ) 5. Draw a diagram to represent (2x + ) and (3x + ). 2x + 3x + What is (2x + ) + (3x + )? We can use the heaps and holes drawings to add expressions involving negative values. x -x - 6. Draw a diagram to show the result of adding (3x + 2) and (2x - 4). 7. Draw a diagram to show (3x - 2) + (5 - x). 8. Draw a diagram to show (x + 2) + 2(32 (3-- x). x). (x + 2) 2(3 - x) Classroom Strategies Blackline Master I - 53 Page 23
What about subtraction? (3x - 4) - (2x + ) Start with... (3x - 4). (3x - 4 ) You could take away 2x, but there is not a (+) to take away. We fix this by adding a zero. Now we can take away 2x and. The answer is x - 5. 9. Draw a diagram to illustrate (x - 4) - (2x + 3). 0. Draw a diagram to illustrate 2(x + 2) (3 - x). Page 24 Classroom Strategies Blackline Master I - 54
Algebraic Expressions Puzzle 2x+3 2-x -x 2-3x 3-x 3x+5 3(x-2) 3x-6 (x+7)-2(x+5) -(x+) -x-3 4x-(3x+2) -3x 6-3(x+5) -x- 3x-5-2(3-x) x+3 4x-5 (3x-8) -6+2x 3x+3-2x-3 2x-4 (4x-) x-3 0 +(x-3) 2x+ +(x-6) x-2 4-x-(2x) 4-3x 4x+ -3-2x -3x- -(3+2x) 5x-2-(8x-) 7-(5-x) x+2 6+2x - / 2 (6+8x) -3-4x 2(3+x) -(2-x) x- 3x-2 3-2x x+ (2x+3) /2 (2x+2) -6-2x 3x+4 -x-2 x+ (x-0)+(4-3x) 2x++(x+3) 3x+ -3x-6-3(x+2) -3(x-2) 6-3x 3x- 3(x+4)-3 3x+6 x-4 4-x 2x-5 3x-4 Cut out the squares above. Fit the squares together so that touching edges are equivalent. Classroom Strategies Blackline Master I - 55 Page 25
D Hamburgers $2.00 Fries $0.40 Drink $.20 C Hamburgers $.60 Fries $0.80 Drink $.00 A Hamburgers $.20 Fries $0.80 Drink $0.80 Bottom Line Cards B Hamburgers $.00 Fries $0.40 Drink $.20 D You buy one of each 2 C 2 hamburgers, 2 drinks, and one fry are bought. A 3 hamburgers and a drink are purchased B You buy 2 fries, one burger, and one drink. D You have a 0% off coupon and you pay 5% tax after the discount. C You pay 5% tax. A You add a 5% tip. B You have a 25% off coupon D You have a $2.50 off coupon. How much do you pay? C You split the cost with a friend. How much do you pay? 3 4 A You have only $3, how much more do you need? B What is your change from $0? Page 26 Classroom Strategies Blackline Master I - 56
qqq3 qqq qqq5 qqq 4 8 8 0 qqq qqq2 qqq qqq 5 3 4 3 qqq 2 qqq 2 Classroom Strategies Blackline Master I - 57 Page 27
qqq7 8 qqq 6 qqq2 5 Page 28 Classroom Strategies Blackline Master I - 58
qqq26 qqq6 qqq36 qqq66 qqq8 qqq96 qqq2 qqq24 Classroom Strategies Blackline Master I - 59 Page 29
qqq26 qqq8 qqq26 qqq6 qqq36 qqq2 qqq66 qqq2 qqq46 qqq56 qqq2 qqq0 qqq46 qqq6 qqq26 qqq0 qqq86 qqq2 qqq46 qqq20 Page 30 Classroom Strategies Blackline Master I - 60
qqq26 qqq36 qqq24 qqq36 qqq26 qqq20 qqq36 qqq30 qqq5 6 qqq24 qqq0 6 qqq6 qqq2 6 qqq24 qqq4 6 qqq6 qqq26 qqq2 qqq36 qqq8 Classroom Strategies Blackline Master I - 6 Page 3
0 % 25 % qqq46 qqq0 qqq66 qqq5 60 % 30 % 75 % 40 % Page 32 Classroom Strategies Blackline Master I - 62
70 % 80 % 33 % 3 33 %. %.0 %.40 % 66 % 66 2 % 3.50 % Classroom Strategies Blackline Master I - 63 Page 33
I have.5 x 0 2. I have 2. I have 2 x 0 2. Who has (2 - ) -? Who has 200 written in scientific notation? Who has an irrational number between 3 and 4? I have 5. I have -2. I have 9. Who has the additive inverse of 2? Who has 20% of 45? Who has 2,000 written in scientific notation? I have 2 x 0 3. I have.5 x 0-2. I have 2 / 3. Who has 0.05 written in scientific notation? Who has the absolute value of 2 / 3? Who has the multiplicative inverse of - / 5? I have -5. I have 2 x 0-2. I have 2 8. Who has 0.02 written in scientific notation? Who has 2 5 2 3? Who has - (- / 5 )? Page 34 Classroom Strategies Blackline Master I - 64
I have / 5. I have / 2. I have 3 / 2. Who has the multiplicative inverse of 2? Who has the reciprocal of 2 / 3? Who has the absolute value of 4? I have 4. I have / 4. I have.3. Who has the multiplicative inverse of 4? Who has the absolute value of.3? Who has the additive inverse of 4? I have -4. I have 64. I have - 3 / 2. Who has the square of 8? Who has the reciprocal of - 2 / 3? Who has 50% of 36? I have 8. I have. I have -. Who has the square root of? Who has the only negative number which is its own reciprocal? Who has 3,600 written in scientific notation? Classroom Strategies Blackline Master I - 65 Page 35
I have 3.6 x 0 3. I have 0. I have 3.6 x 0 4. Who has only number which is its own additive inverse? Who has 36,000 written in scientific notation? Who has the reciprocal of - 4 / 3? I have - 3 / 4. I have 90. I have 4 / 3. Who has 0% of 900? Who has the additive inverse of - 4 / 3? Who has the multiplicative inverse of / 5? I have 5. I have 3 / 4. I have 3.6 x 0-3. Who has the absolute value of - 3 / 4? Who has 0.0036 written in scientific notation? Who has the additive inverse of / 5? I have - / 5. I have 3. I have 0.9. Who has an irrational number between 5 and 6? Who has 0% of 9? Who has 50 written in scientific notation? Page 36 Classroom Strategies Blackline Master I - 66
NAME DATE Solve each problem below. Show your work on a separate sheet of paper. Do not use a calculator.. Find the fraction which would be closest to the sum shown below. 7.52 + 8.99 + 0.23 + 0.0004 A) 6 B) 6 C) 6 3 D) 6 4 2 4 2 2. Find the fraction which would be closest to the difference shown below. 8.90-0.30007 A) 8 2 B) 8 3 C) 8 D) 8 3 5 4 2 5 3. Mr. Jones promises to pay 0% interest on a loan. If the loan is for $2000, how much interest must he pay? 4. A shirt cost $9 and the tax on the shirt is 6%. How much tax will be paid? 5. A theater owner plans to give every of his third customers a a free free movie poster. If If 659 659 customers come to the come theater, to how the theater, many posters how many should posters he have should to give he away? have to give away? 6. A movie star is planning to sign photographs for his fans. He knows from experience that all of the fans will bring at least one picture to sign, and about of them will bring two. If he is told 4 there are 598 fans, about how many pictures should he expect to sign? 7. Mark now makes $6.80 per hour. He is told he will get a 25% raise. How much will he make per hour after the raise? 8. Mary just spent $2 for dinner at a restaurant. If she wants to leave a 5% tip, how much should she leave? 9. Susan has 9.7 meters of ribbon. She wants to make lots of pillows. Each one needs 30 cm of ribbon. How many pillows can Susan make with the ribbon she has? 0. Tom has a board 5 feet long. He wants to cut the board into pieces that are each 4 3 inches long. How many pieces can he cut from his board? 4 Classroom Strategies Blackline Master Page 37 I - 67
NAME DATE Complete the chart below. Express answers in simplest form.. 2. 3. Fraction Decimal Percent 2 0.6 0% 4. 5. 3 4 0.33 6. 7. 8. 9. 0. 3 3 000.5 30% 32% Simplify each of the following. Show your work.. 3 + 2. + 3 3. + 8 4 5 0 3 2 9 4. 6 + 3 5. 2 4-3 6. 5-2 3 2 3 5 8 8 7. 5 x 2 8. 5 3 9. 2 x 5 5 20. 2 3 2. 5 + 2 4 3 22, 4 + 3 4 Write each fraction in decimal form. 3 4 2 23. 3 24. 7 25. 8 9 2 7 Page 38 Classroom Strategies Blackline Master I - 68
NAME DATE Algebraic Expressions. There may be more than one correct answer for each of the following.. Circle any of the following which are equivalent to the algebraic expression : x. A) x B) 0 + x C) 0 x D) 2(x) - 2 2. Circle any of the following which are equivalent to the number 0. A) 0 + x B) 0 x C) 0 x - x D) x + (-x) 3. Circle any of the following which are equivalent to the number. A) x B) x - x C) x D) x 4. Circle any of the following which are equivalent to the expression : 3(x + y). A) 3xy B) 3x + y C) 3x + 3y D) (x + y) 3 5. Circle any of the following which are equivalent to the expression: (x + y) + 2. A) x + y + 2 B) x + (y + 2) C) xy + 2 D) 2xy x x Evaluate the following 6. 8 - (7-3) 7. 4 (8-2) 8. 7 + 2 5 9. 6 + 3 4-2 0. 6 4 x 2. 5 + 2 3 2 2. 6-2 + 4 3. 5 + 6-6 + 7 4. 7 + 3 (5-3) 4 5. 6 - + 8(5-3) 2 Classroom Strategies Blackline Master Page 39 I - 69
NAME DATE Exponents and Scientific Notation - No Calculators Allowed Write each number in scientific notation. 8,400 2. 0.320 3. 0.002 4. 0.0000024 Write each number in standard form. 5..7 x 0-2 6. 2.25 x 0-5 Simplify each of the following. You may write your answer as an exponential expression. 7. 2 3 2 4 = 8. x 5 x 3 = 9. (3 3 ) 2 = 0. (x 2 ) 4 = 5 5 x. = 2. 4 = 5 2 x 3 Solve each problem below. 3. A space ship will be traveling 3.6 x 0 8 miles, and a fuel chip will be used for every.2 x 0 5 miles. How many chips will be needed? 4. A rectangular box has a length of x 2, a width of x, and a height of x. Determine the volume of the box. Evaluate each expression below if A = -, B = - 4, C = 2, and D = 3. 5. -5A 6. A + B + C + D 7. B - A 8. C - A 9. AD 20. A + D 2. B + C Page 40 Classroom Strategies Blackline Master I - 70
NAME DATE. A shoe store pays $24 for a pair of sneakers. The manager plans to mark the shoes up 30% to determine the selling price. What will be the selling price of the shoes? 2. The price of popcorn at a movie theatre was $2.00 one week, and $2.75 the next week. What was the percent of increase in the price of the popcorn? 3. A tennis racket normally sells for $48 but is on sale for 30% off. What is the sale price of the racket? 4. Gasoline was priced at $.50 per gallon one week, and the next week the price fell to $.40. What was the percent of decrease in the price? 2 5. John makes $4.50 per hour at his job, but of his pay is deducted for taxes and insurance. 5 How much will he take home for each hour worked? 6. Sue makes $4.80 per hour and will receive a 5% raise at the end of the year. What will her new hourly rate be? 7 7. A worm was 8 of an inch long when it was first measured. By the end of the next week, its length had increased by 25%. How long was the worm at that time? 8. During one week a plant grew 3 x 0 - meters. The following week a scientist recorded that the plant grew 4 x 0 2 millimeters. In all, how many centimeters did the plant grow in those two weeks? 9. The perimeter of a certain trapezoid can be found by the formula: P = 3s + (s + 4), where s is the length of the shortest side. What is the perimeter of this trapezoid if the shortest side has a length of 6 cm? 0. If a ball is thrown upwards, its height at any time can by found by the formula, h = -6 (t - ) 2 + 46 where h is the height in feet, and t is the time in seconds. How high is the ball after it has been in the air two seconds? Classroom Strategies Blackline Master Page 4 I - 7
NAME DATE For each of the following, determine an algebraic expression that describes y in terms of x.. 2. 3. x y x y x y 4 - -4 2 8 0 0 x -4 2-3 6 0 5 x 2 3 4 7 0 9 x 4. Each month, a club records a positive number if they make a profit and a negative number if they owe money at the end of the month. The results for five months are given below. January +25 February +5 March -0 April -5 May +7 What is the net result for all five months? 5. James Bond is in a car that is flying at an altitude of,200 feet. Suddenly he dives into the ocean to a depth of 840 feet. How far did he dive? 6. The beginning balance for Mary s checking account was $45 on Tuesday. Mary wrote three checks for $2.50 each on Wednesday and one check for $24.95 on Thursday. Friday she deposited the baby-sitting money she earned ($4.25 an hour for 4 hours of work). How much does she have in her account after these transactions? Page 42 Classroom Strategies Blackline Master I - 72