Table of Contents Introduction...v About the CD-ROM...vi Standards Correlations... vii Ratios and Proportional Relationships... 1 The Number System... 14 Expressions and Equations... 28 Geometry... 42 Statistics and Probability... 49 Answer Key... 55 iii Mathematics Warm-Ups for CCSS, Grade 6
Standards Correlations Mathematics Warm-Ups for Common Core State Standards, Grade 6 is correlated to five domains of CCSS Grade 6 mathematics. The page numbers, titles, and standard numbers are included in the table that follows. The full text of the CCSS mathematics standards for Grade 6 can be found in the Common Core State Standards PDF at http://www.walch.com/ccss/00001. Page number Title CCSS addressed Ratios and Proportional Relationships 1 Jumping Jellies! 6.RP.1 2 Solving Proportions 6.RP.2 3 Money, Money, Money 6.RP.3a 4 Birthday Roses 6.RP.3b 5 Paolo s Pizza Pricing 6.RP.3b 6 Stamping Around 6.RP.3b 7 Faster Than a Speeding Bullet 6.RP.3b 8 Let It Snow 6.RP.3b 9 Square Pizza 6.RP.3c 10 Playing with Proportions 6.RP.3c 11 How Sweet It Is 6.RP.3d 12 Oil and Vinegar 6.RP.3d 13 Find the Distance 6.RP.3d The Number System 14 Ribbon and Bows 6.NS.1 15 Baking Blueberry Pies 6.NS.1 16 Super Sub Sandwich 6.NS.1 17 Fun with Fractions 6.NS.1 (continued) vii Mathematics Warm-Ups for CCSS, Grade 6
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 6.RP.1 Jumping Jellies! Use the information from the scenario below to answer the questions. Annie has a big bowl of jelly beans. She has the following number of each flavor: 25 strawberry 25 lemon 50 pineapple 25 blueberry 75 lime 1. How many strawberry jelly beans are there? 2. How many blueberry jelly beans are there? 3. How does the number of strawberry jelly beans compare to the number of blueberry jelly beans? 4. How many pineapple jelly beans are there? 5. How many jelly beans are there altogether? 6. How does the number of pineapple jelly beans compare to the total number of jelly beans? What fraction of the jelly beans are pineapple? 1 Mathematics Warm-Ups for CCSS, Grade 6
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 6.RP.2 Solving Proportions Remember that a proportion is two equal ratios. a c = b d To solve a proportion, cross-multiply. a b = c d ad = bc Solve the following proportions for x. 1. x 3 = 6 9 2. 1 = x 2 5 3. 3 4 8 = x 4. x 3 x = 4 8 2 Mathematics Warm-Ups for CCSS, Grade 6
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 6.RP.3a Money, Money, Money You got a great job babysitting for $8 an hour. Fill out the table below to figure out how much you would make if you worked a full 8-hour day. Number of hours Amount earned 1 $8 2 3 $24 $32 5 6 $56 8 3 Mathematics Warm-Ups for CCSS, Grade 6
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 6.RP.3b Birthday Roses Read the scenario and answer the question that follows. Alonzo is planning to purchase roses for his mother on her birthday. He has seen them advertised at 12 roses for $15.00 and 20 roses for $23.00. Which is the better buy? Show your work and explain your reasoning. 4 Mathematics Warm-Ups for CCSS, Grade 6
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 6.RP.3b Paolo s Pizza Pricing Paolo has just started working at his Uncle Antonio s pizza parlor. He is trying to figure out which size meat lover s pizza provides the best value. Meat Lover s Special Size Price 9-inch round pizza $10.50 12-inch round pizza $15.00 18-inch round pizza $19.00 Which pizza listed on the menu provides the best value? Write a few sentences that explain your reasoning. 5 Mathematics Warm-Ups for CCSS, Grade 6
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 6.RP.3b Stamping Around In 2012, a book of 20 first class stamps cost $9. How much would one stamp cost? Show your work and write your answer below. 6 Mathematics Warm-Ups for CCSS, Grade 6
Answer Key Ratios and Proportional Relationships Jumping Jellies!, p. 1 1. 25 2. 25 3. There are the same number of each. 4. 50 5. 200 6. There are 50 pineapple jelly beans and 200 total. 1/4 of the jelly beans are pineapple. Solving Proportions, p. 2 1. 2 2. 2.5 Money, Money, Money, p. 3 3. 10 2/3 4. 6 Number of hours Amount earned 1 $8 2 $16 3 $24 4 $32 5 $40 6 $48 7 $56 8 $64 Birthday Roses, p. 4 Students could scale to 60 roses, which yields $75 and $69 respectively, or find the price per rose, which yields $1.25 and $1.15 respectively. Thus, 20 roses for $23.00 is the better buy per rose. Paolo s Pizza Pricing, p. 5 The 18-inch pizza provides the best value. 9-inch pizza: 63.6 square inches/10.5 = 6.06 square inches per $1.00; 12-inch pizza: 113.1 square inches/15 = 7.54 square inches per $1.00; 18-inch pizza: 254.5 square inches/19 = 13.39 square inches per $1.00 Stamping Around, p. 6 Divide $9 by 20 to get $0.45. In 2012, one stamp cost $0.45. Faster Than a Speeding Bullet, p. 7 Distance equals rate times time, so 240 = 4r. In this example, r = 60, so Anna s mother drove 60 mph. Let It Snow, p. 8 275 25 = 10; therefore, Estaban has shoveled 10 walkways. Square Pizza, p. 9 There will be no remainder. The friends predict that they will eat 110% of the pizza (10% + 50% + 35% + 15% = 110%), so there won t be enough. Playing with Proportions, p. 10 16/20 simplifies to 4/5. 80% = 80/100, which also simplifies to 4/5. Therefore, both fractions are equivalent, and 16/20 is 80%. How Sweet It Is, p. 11 To compare numbers given in various forms, it is often easiest to put them into the same form. In this example, rewriting all of the numbers in decimal form is most efficient. 30% = 0.3, 23% = 0.23, 1/5 = 0.2, and 2/9 = 0.2. Therefore, the numbers in order from least to greatest are 0.18, 1/5, 2/9, 23%, 0.25, and 30%. Oil and Vinegar, p. 12 You can set up and solve a proportion: 3/2 = 12/x. 3x = 24, and x = 8. You will need 8 tablespoons of vinegar. You can also encourage students to come up with the denominator of a fraction that is equivalent to 3/2 which has 12 in the numerator. Find the Distance, p. 13 Since there are 3 feet in a yard, divide 15 by 3 to get 5. Or, you could set up and solve a proportion: 3 feet/1 yard = 15 feet/x yards = 3x and then x = 5. Students may also use repeated addition or may share another method. Support all ways. The Number System Ribbon and Bows, p. 14 Veronica can make 9 bows, with 1 3 yard remaining. Baking Blueberry Pies, p. 15 45 pies (or 44, depending on rounding) Super Sub Sandwich, p. 16 1. 25 students would get a portion with 1 4 remaining. 2. 17 students would get a portion. 3. Students should use drawings of the sub divided into appropriate length portions. Fun with Fractions, p. 17 10; student drawings will vary. Using 10 10 Grids, p. 18 0.3 0.4 = 0.12; 3 10 4 10 = 12 100 ; 30% of 40% is 12% What Day of the Week?, p. 19 Anatole was born on a Saturday. 759/7 = 108 R3 Greatest Common Factor, p. 20 The GCF is 12. Find the prime factorization of each number and compare. Then take the product of the common factors. 55 Mathematics Warm-Ups for CCSS, Grade 6
Table of Contents Introduction...v About the CD-ROM...vi Standards Correlations... vii Ratios and Proportional Relationships... 1 The Number System... 10 Expressions and Equations... 23 Geometry... 27 Statistics and Probability... 45 Answer Key... 55 iii Mathematics Warm-Ups for CCSS, Grade 7
Standards Correlations Mathematics Warm-Ups for Common Core State Standards, Grade 7 is correlated to five domains of CCSS Grade 7 mathematics. The page numbers, titles, and standard numbers are included in the table that follows. The full text of the CCSS mathematics standards for Grade 7 can be found in the Common Core State Standards PDF at http://www.walch.com/ccss/00001. Page number Title CCSS addressed Ratios and Proportional Relationships 1 Balancing a Milk Bottle 7.RP.1 2 Check It Out 7.RP.2a 3 Comparing Rectangles 7.RP.2b 4 Input and Output 7.RP.2b 5 Pizza Pizza Pizza 7.RP.2b 6 Tree Height 7.RP.2b 7 Vanishing Wetlands 7.RP.3 8 Percent Increase or Decrease 7.RP.3 9 Bread Prices 7.RP.3 The Number System 10 Chip-Board Integers I 7.NS.1c 11 Working with Integers I 7.NS.1d 12 Working with Integers II 7.NS.1d 13 Chip-Board Integers II 7.NS.1d 14 Chip-Board Integers III 7.NS.1d 15 Integer Practice 7.NS.1d 16 Make a True Sentence 7.NS.1d, 7.NS.2c 17 10 10 Grids 7.NS.2d (continued) vii Mathematics Warm-Ups for CCSS, Grade 7
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 7.RP.2a Check It Out The graph below illustrates the distance a car traveled at 25 miles per hour. Examine the graph closely and describe how it looks to a partner. y x What information lies on the x-axis? What information lies on the y-axis? Does it make sense that the graph looks as it does? Explain. Write your observations below. 2 Mathematics Warm-Ups for CCSS, Grade 7
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 7.RP.2b Comparing Rectangles Examine the two rectangles below. Notice that the larger rectangle s length and width are three times the smaller rectangle s length and width. 2 cm 4 cm 12 cm 6 cm Answer the following questions. 1. What is the area of the smaller rectangle? 2. What is the area of the larger rectangle? 3. How many times larger is the area of the larger rectangle than the smaller rectangle? Is this what you expected? Explain. 3 Mathematics Warm-Ups for CCSS, Grade 7
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 7.RP.2b Input and Output Consider the tables below. Figure out the rule that takes the input value and gives the corresponding output value. Describe each rule in words. Example: Input 1 2 3 4 5 Output 4 5 6 7 8 Rule: Add 3 to the input to get the output. 1. Input 5 7 9 11 13 Output 10 14 18 22 26 2. Input 10 20 30 40 50 Output 6 16 26 36 46 3. Input 24 28 32 36 40 Output 12 14 16 18 20 4. Input 0 2 4 6 8 Output 3 7 11 14 19 4 Mathematics Warm-Ups for CCSS, Grade 7
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 7.RP.2b Pizza Pizza Pizza At Suki s Pizza Parlor, each slice of pizza costs $1.50. Aaron had been studying adding decimal numbers in math class, and so while waiting for his pizza, he made the chart below. Number of slices Price 1 $1.50 2 $3.00 3 $4.50 4 $6.00 5 $7.50 Use Aaron s chart to answer the following questions. 1. How would you describe how the numbers in the first column change? 2. How would you describe how the numbers in the second column change? 5 Mathematics Warm-Ups for CCSS, Grade 7
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 7.RP.2b Tree Height Use what you know about proportional relationships to answer the questions that follow. Show your work for each problem. 1. Justine, who is 60 inches tall, casts an 8-foot shadow. At the same time and place, how long a shadow would a 35-foot tree cast? 2. If another tree nearby casts a 40-foot shadow, how tall is this tree? 6 Mathematics Warm-Ups for CCSS, Grade 7
RATIOS AND PROPORTIONAL RELATIONSHIPS CCSS 7.RP.3 Vanishing Wetlands Bonita works for the Desoto Park Service. The Little Otter Wetland Area that she monitors has had drought conditions recently. She is preparing a report on the drought for the park service. The grid below represents a model for the change in area of the wetland. What was the percent change in wetland area from August 2010 to August 2011? Explain your thinking. August 2010 model August 2011 model 7 Mathematics Warm-Ups for CCSS, Grade 7
Answer Key Ratios and Proportional Relationships Balancing a Milk Bottle, p. 1 3.44 miles per hour Check It Out, p. 2 Answers will vary. The graph is a straight line at an angle. The information on the x-axis could be the time the car has traveled. The information on the y-axis could be the distance covered. However, the information could be reversed on either axis. Students may say that the graph makes sense because the car is moving at a constant rate. Comparing Rectangles, p. 3 1. 8 cm 2 2. 72 cm 2 3. 9 times larger (this is found by finding the ratio of the two areas [72/8]). Students may say that they expected the area to be three times larger (since the length and width are three times larger). Encourage them to realize that since both the length and the width are three times larger (and they are multiplying these larger numbers), the area will end up being 3 3 or 9 times larger. Input and Output, p. 4 1. Multiply the input by 2 to get the output. 2. Subtract 4 from the input to get the output. 3. Divide the input by 2 to get the output. 4. Multiply the input by 2 and then add 3 to get the output. Pizza Pizza Pizza, p. 5 1. The numbers increase by 1. 2. The numbers increase by 1.50. Tree Height, p. 6 1. The tree s shadow is 56 feet long. 2. The tree is 25 feet tall. Vanishing Wetlands, p. 7 The original area on the grid is approximately 52 square units. The new area is approximately 34 square units. This represents a 34.6% decrease in area. Percent Increase or Decrease, p. 8 1. growth; ratio = 3; 300% increase 2. growth; ratio = 1.1; 110% increase 3. decay; ratio =.6; 60% decrease 4. decay; ratio =.2; 20% decrease 5. decay; ratio =.38; 38% decrease Bread Prices, p. 9 1. 32% 2. $1.37/loaf 3. The 2010 price is 32% greater. If the trend were continuing, one might assume that the percent increase would be closer to 16%. The Number System Chip-Board Integers I, p. 10 Chip board 1 should have 6 black chips. Chip board 2 could show the 6 black chips with 6 zeros included. Chip board 3 would show 6 white chips remain after 12 black chips are removed. So, ( 6) ( 12) = +6. Working with Integers I, p. 11 1. 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 2. 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 3. 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Working with Integers II, p. 12 1. ( 9) + 7 = ( 2) 2. ( 4) + ( 6) = ( 10) 3. 7 + ( 14) = 7 Chip-Board Integers II, p. 13 1. 5 + ( 8) = ( 3) 2. ( 6) + 10 = 4 3. ( 8) + 3 = ( 5) Chip-Board Integers III, p. 14 1. Chip board 1: 5 black chips; Chip board 2: 5 black chips and 9 white chips; Chip board 3: 4 white chips 2. Chip board 1: 12 white chips; Chip board 2: 12 white chips and 7 black chips; Chip board 3: 5 white chips 3. Chip board 1: 8 black chips; Chip board 2: 5 black chips crossed out or removed; Chip board 3 (or 2): 3 black chips 55 Mathematics Warm-Ups for CCSS, Grade 7
Table of Contents Introduction...v About the CD-ROM...vi Standards Correlations... vii The Number System... 1 Expressions and Equations... 3 Functions... 12 Geometry... 38 Statistics and Probability... 52 Answer Key... 55 iii Mathematics Warm-Ups for CCSS, Grade 8
Standards Correlations Mathematics Warm-Ups for Common Core State Standards, Grade 8 is correlated to five domains of CCSS Grade 8 mathematics. The page numbers, titles, and standard numbers are included in the table that follows. The full text of the CCSS mathematics standards for Grade 8 can be found in the Common Core State Standards PDF at http://www.walch.com/ccss/00001. Page number Title CCSS addressed The Number System 1 Irrational Numbers 8.NS.1 2 Where Do They Go? 8.NS.2 Expressions and Equations 3 Tearing and Stacking Paper 8.EE.4 4 Linear vs. Nonlinear 8.EE.5 5 Graphing Linear Functions 8.EE.5 6 The Seesaw Problem 8.EE.7b 7 Systems of Linear Equations 8.EE.8a, 8.EE.8b 8 Exploring Systems of Equations 8.EE.8b 9 Nathan s Number Puzzles I 8.EE.8b 10 Understanding Systems of Linear Equations 8.EE.8b 11 Nathan s Number Puzzles II 8.EE.8c Functions 12 Contaminated Drinking Water 8.F.1 13 Cars and Drivers 8.F.2 14 Slope-Intercept Form 8.F.3 15 Slope-Intercept Equations 8.F.3 (continued) vii Mathematics Warm-Ups for CCSS, Grade 8
THE NUMBER SYSTEM CCSS 8.NS.1 Irrational Numbers Use the information below and what you know about irrational numbers to answer the question that follows. An irrational number is a number that cannot be expressed as a fraction. Any decimals that are not terminating and do not repeat are irrational numbers. More technically, a rational number is a number that can be expressed in the form x, where x and y are y integers and y is not 0. Is 2 an irrational number? Why or why not? Explain your thinking. 1 Mathematics Warm-Ups for CCSS, Grade 8
THE NUMBER SYSTEM CCSS 8.NS.2 Where Do They Go? Using the number line below, show approximately where each number would fall. Explain your thinking. 0 1 2 3 4 5 6 7 8 9 10 1. 96 2. 35 3. 24 4. 17 2 Mathematics Warm-Ups for CCSS, Grade 8
EXPRESSIONS AND EQUATIONS CCSS 8.EE.4 Tearing and Stacking Paper Read the scenario that follows, and then answer the questions. Mr. Andres poses the following problem to his math class: Take a large sheet of paper and tear it exactly in half. Then you have 2 sheets of paper. Put those 2 sheets together and tear them exactly in half. Then you have 4 sheets of paper. Continue this process of tearing and putting together for a total of 50 tears. If the paper is only 1 1,000 of an inch thick, how many sheets of paper would there be? How thick or tall would the stack of paper be? 3 Mathematics Warm-Ups for CCSS, Grade 8
EXPRESSIONS AND EQUATIONS CCSS 8.EE.5 Linear vs. Nonlinear A linear equation is an equation that can be graphed by a straight line. A nonlinear equation is an equation that cannot be represented by a line. Determine whether the following graphs are linear or nonlinear. Write your answer on the line below each graph. 1. 2. 3. 4. 5. 6. 4 Mathematics Warm-Ups for CCSS, Grade 8
EXPRESSIONS AND EQUATIONS CCSS 8.EE.5 Graphing Linear Functions Using a function to generate output will lead to the production of a set of ordered pairs. We use ordered pairs when plotting points on a coordinate plane. For example, look at the chart below. The output can be used as the y-coordinate. Input (x) Function (2x + 1) Output (y) 1 2(1) + 1 3 2 2(2) + 1 5 3 2(3) + 1 7 4 2(4) + 1 9 The ordered pairs created are (1, 3), (2, 5), (3, 7), and (4, 9). They all lie on a straight line. You can connect the points to see all the other points on the line. Graph the line generated by each function below. Use x-values from at least 0 to 5. 1. y = 2x 2 2. y = x + 1 5 Mathematics Warm-Ups for CCSS, Grade 8
EXPRESSIONS AND EQUATIONS CCSS 8.EE.7b The Seesaw Problem Read the information in each problem, and then answer the questions. 1. Caleb and his friend Alex are playing on the seesaw at the playground. Alex weighs 70 pounds. Caleb and Alex balance perfectly when Alex sits about 3 feet from the center and Caleb sits about 2 1 feet from the center. About how much does Caleb weigh? 2 2. Alex s identical twin brother Samuel joins them and sits next to Alex. Can Caleb balance the seesaw with both Alex and Samuel on one side, if Samuel weighs about the same as Alex? If so, where should Caleb sit? If not, why not? 6 Mathematics Warm-Ups for CCSS, Grade 8
Answer Key The Number System Irrational Numbers, p. 1 Yes, it s irrational. It s a decimal that continues without repeating. Where Do They Go?, p. 2 17 24 35 96 0 1 2 3 4 5 6 7 8 9 10 Expressions and Equations Tearing and Stacking Paper, p. 3 2 50 = 1.125899907 10 15 sheets of paper /1,000 = 1.125899907 10 12 inches /12 = 9.382499224 10 10 feet /5,280 = 17,769,884.89 miles Linear vs. Nonlinear, p. 4 1. nonlinear 2. linear 3. nonlinear 4. nonlinear 5. linear 6. nonlinear Graphing Linear Functions, p. 5 1. graph of ordered pairs (0, 2), (1, 0), (2, 2), (3, 4), (4, 6), (5, 8) 2. graph of ordered pairs (0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6) The Seesaw Problem, p. 6 1. Caleb weighs about 84 pounds. 2. If Alex and Samuel continue to sit at 3 feet, then 140 3 = 84 D, and D = 5 feet. If the seesaw is long enough, then Caleb can balance them. Otherwise, he cannot. Systems of Linear Equations, p. 7 Answers will vary. 1. Sample answer: to find an ordered pair that satisfies both equations 2. Sample answer: Graph both lines on the same set of axes, and locate the intersection point of the two lines. 3. Sample answer: Substitute values of x into both equations. Look for identical y values for the same value of x. 4. Sample answer: solve algebraically by substitution or elimination 5. Sample answer: If the values in the table have the same constant rate of change, if the lines are parallel, or if the slopes are the same but have different y-intercepts, then there is no solution. Exploring Systems of Equations, p. 8 1. a. Check graphs for accuracy. b. x 0 1 2 3 4 5 y 3 1 1 3 5 7 2. a. Check graphs for accuracy. b. x 0 1 2 3 4 5 y 12 9 6 3 0 3 3. (3, 3) 4. (3, 3) 5. The point satisfies both equations. 6. The point only works in the first equation. 7. Any other point chosen will only work in one of the equations, not in both equations. Nathan s Number Puzzles I, p. 9 Word puzzles will vary. 1. (2, 1) 2. ( 2, 4) 3. not possible Understanding Systems of Linear Equations, p. 10 Students choices and rationales may vary. 1. ( 4, 5) 3. ( 20, 10) 2. ( 6, 4) 4. (3.6, 2.4) Nathan s Number Puzzles II, p. 11 1. x + y = 10; x + 2y = 8; (12, 2) 2. x + 2y; x + y = 15; (10, 5) 3. x y = 2; 2x 2y = 4; Any values for x and y that differ by 2 will work in this puzzle. There are infinite solutions. Functions Contaminated Drinking Water, p. 12 1. 24,975; 20,350; 15,725 55 Mathematics Warm-Ups for CCSS, Grade 8