Dear Parents and Students,

Transcription

1 Dear Parents and Students, We hope you will enjoy this Math Challenge packet and work hard to complete all problems on your own or with help from a parent or guardian. All projects in this packet are based upon the Connecticut Core Standards for Math (formally known as Common Core State Standards). To learn more about these standards, please visit: We suggest doing two to three projects each week. Once you have finished a project, try to find a way to discuss it with a friend, parent, or relative. Think about how the skills and concepts in the problem you have completed are connected to other things in your home, environment, or daily routine. Find ways to apply your new understanding to real world situations. Math is all about problem-solving. One of the best ways to learn Math is to try out problems in which you have to develop your own strategy in order to find the solution. There is usually more than one way to solve Math problems. While working on the problems in this packet, you may discover shortcuts and use your own process or set of rules to calculate and determine the appropriate solution. Make sure to keep notes and include your work so that you can explain your solutions. In other works, be sure you can answer the question How do you know? Explaining your solution to a problem immediately tells others what you re learning. Have a wonderful summer! Sincerely, Sheila M. Civale STEM Coordinator Greenwich Public Schools Adapted with permission from Chartier Valley School District

2 Project #1 Expressions and Equations (EE) I can determine the properties of integer exponents by exploring patterns and applying my understanding of properties of whole number exponents. I can use the properties of integer exponents to simplify expressions. 1. Write out the following in words. For example, 3(x+5) would be three times the quantity of x and , x /5 7:3 23 1/8 85% ,000,000, Solve the problem below. The evil ruler, Gonglo has locked the beautiful, intelligent Princess Alexa in a loft high up in the castle. To be released, she must find the first perfect square number greater than 100 whose digits do not sum to a perfect square! Can you help her find the number? 3. Create a challenge problem of your own using the formula for finding a perfect square. Page 2

3 Project #2 Functions (F) I can match a function to a given situation. 1. Use the information below to answer the riddle. Amy, Connor, Jalia, Stella and Gonzo live in apartments A, B, C, D, and E in the same building. Their ages are 9, 10, 11, 13, and 14. They each get up at a different time each morning, either 6:00, 7:00, 7:30, 8:00, or 8:15. Using the clues below, determine the apartment, age and rising time for each student. Connor gets up at 6:00 and is older than the student in Apt. B. Amy is 11 and gets up two hours later than the boy in Apt. A. The ten-year old girl lives in Apt. B and it isn't Stella. The youngest student, a girl, doesn't live in Apt. E. The person in Apt. B gets up later than everyone else. Gonzo is younger than Connor and he lives in Apt. D. The shades go up at 7:30 in Apt. C. 2. Create your own riddle for a friend to solve. Page 3

4 Project #3 Directions: The Number System (NS) I can classify a number as rational or irrational based on its decimal expansion. Pretend your 10-year-old neighbor knows nothing about integers, rational, whole, natural, and irrational numbers. Write two to three paragraphs explaining the graph below. Be sure to use language that a fourth or fifth grader can understand. Page 4

5 Project #4 Directions: The Number System (NS) I can compare two or more rational or irrational numbers. Compare each pair of numbers using <, >, or =. Write one to two sentences after each problem explaining your answer. Now, create two to three similar problems and challenge an adult or friend to solve them. Page 5

6 Project #5 Directions: The Number System (NS) I can plot rational and irrational numbers on a number line. List the numbers 2/3, -2/3, 1.2, 4/3, -4/3, -1.2, -7/4 from least to greatest and then locate the numbers on the number line. You will label the number line below to fit your purposes. Page 6

7 Project #6 Functions (F) I can define the rate of change in relation to the situation. Linda traveled 110 miles in 2 hours. If her speed remains constant, how many miles can she expect to travel in 4.5 hours? Answer the question in complete sentences and justify your answer. Then, create a problem of your own involving linear functions and constant rates of speed over a specified distance. Page 7

8 Project #7 Directions: Functions (F) I can write a linear function that models a given situation. Which of the following could be modeled by y=2x+5? Answer YES or NO for each one. 1. There are initially 5 rabbits on the farm. Each month thereafter the number of rabbits is 2 times the number in the month before. How many rabbits are there after x months? YES NO 2. Joaquin earns $2.00 for each magazine sale. Each time he sells a magazine he also gets a five dollar tip. How much money will he earn after selling x magazines? YES NO 3. Sandy charges $2.00 an hour for babysitting. Parents are charged $5.00 if they arrive home later than scheduled. Assuming the parents arrived late, how much money does she earn for x hours? YES NO 4. I have a sequence of integers. The first term of the sequence is 7 and the difference between any consecutive terms is always equal to 2. YES NO 5. Sneak Preview is a members-only video rental store. There is a $2.00 initiation fee and a $5.00 per video rental fee. How much would John owe on his first visit if he becomes a member and rents x videos? YES NO 6. Andy is saving money for a new CD player. He began saving with a $5.00 gift and will continue to save $2.00 each week. How much money will he have saved at the end of x weeks? YES NO Page 8

9 Project #8 Directions: Geometry (G) I can apply the Pythagorean Theorem to find an unknown side length of a right triangle. The Pythagorean Theorem is a 2 + b 2 = c 2. The variable C represent the hypotenuse which is the side opposite the right angle. In this particular problem C is the unknown side. Solve the problem and justify your answer. To get from point A to point B you must avoid walking through a pond. To avoid the pond, you must walk 34 meters south and 41 meters east. To the nearest meter, how many meters would be saved if it were possible to walk through the pond? Page 9

10 Project #9 The Number System (NS) I can solve real-world problems that involve the addition, subtraction, multiplication, and/or division of rational numbers. 1. The United States owes approximately $14,300,000,000, in National debt. There are approximately 300 million people in the United States. Assume the United States would like to pay off the debt in one lump sum. How much would each American have to pay to pay off the National debt? 2. If each American paid $2,000 a year towards the National debt how many years would it take for the country to pay off what it owes? (Assume that there is no interest on the debt.) Page 10

11 Project #10 Directions: Expressions and Equations (EE) I can determine the volume of three-dimensional figures. I can solve real-world problems involving area, surface area and volume. Solve the following questions using the provided formulas. My sister s birthday is in a few weeks and I would like to buy her a new vase to keep fresh flowers in her house. She often forgets to water her flowers and needs a vase that holds a lot of water. In a catalog there are three vases available and I want to purchase the one that holds the most water. The first vase is a cylinder with diameter 10 cm and height 40 cm. The second vase is a cone with base diameter 16 cm and height 45 cm. The third vase is a sphere with diameter 18 cm. Page 11

12 1. Which vase should I purchase? 2. How much more water does the largest vase hold than the smallest vase? 3. Suppose the diameter of each vase decreases by 2 cm. Which vase would hold the most water? 4. The vase company designs a new vase that is shaped like a cylinder on bottom and a cone on top. The catalog states that the width is 12 cm and the total height is 42 cm. What would the height of the cylinder part have to be in order for the total volume to be 1224πcm 3 Page 12

13 Project #11 Statistics and Probability (SP) I can recognize whether or not data plotted on a scatter plot have a linear association. I can make inferences regarding the reliability of the trend line by noting the closeness of the data points to the line. This scatter diagram shows the lengths and widths of the eggs of some American birds. 1. A biologist measured a sample of one hundred Mallard duck eggs and found they had an average length of 57.8 millimeters and average width of 41.6 millimeters. Use an X to mark a point that represents this on the scatter diagram. 2. What does the graph show about the relationship between the lengths of birds' eggs and their widths? 3. Another sample of eggs from similar birds has an average length of 35 millimeters. If these bird eggs follow the trend in the scatter plot, about what width would you expect these eggs to have, on average? 4. Describe the differences in shape of the two eggs corresponding to the data points marked C and D in the plot. 5. Which of the eggs A, B, C, D, and E has the greatest ratio of length to width? Explain how you decided. Page 13

14 Project #12 Ratios and Proportional Relationships (RP) I can compute a unit rate by multiplying or dividing both quantities by the same factor. 1. DVDs can be made in a factory in New Mexico at the rate of 20 DVDs per $3, but the factory costs $80,000 to build. If they make 1 million DVDs, what is the unit cost per DVD? 2. DVDs can be made in a factory in Colorado at the rate of 10 DVDs per $1.50, but the factory costs $100,000 to build. If they make 1 million DVDs, what is the unit cost per DVD? 3. How much can a buyer save on a million DVDs by buying DVDs from New Mexico instead of DVDs from Colorado? Challenge: 4. Find an equation for the cost of making x number of DVDs in the factory in New Mexico. 5. Find an equation for the cost of making x number of DVDs in the factory in Colorado. Page 14

15 Project #13 Statistics and Probability (SP) I can use probability to predict the number of times a particular event will occur given a specific number of trials. I can use variability to explain why the experimental probability will not always exactly equal the theoretical probability. Drawing from a set of six blue beads and three gold beads, use ratios to state the likelihood of each color being drawn; conducts experiments to test predictions. Show your work and explain your answer. Page 15

16 Project #14 Ratios and Proportional Relationships (RP) I can use proportional reasoning to solve real-world percent problems, including those with multiple steps. College costs increase at about twice the inflation rate. Current increases have averaged 5% to 8% annually. According to the College Board's and Trends in College Pricing, the average total costs (including tuition, fees, room and board) were $16,140 for students attending four-year public colleges and universities in-state and $28,130 out-of-state, and $36,993 for students at four-year private colleges and universities. You can assume an additional $4,000 for textbooks, supplies, transportation and other expenses. Use the information above to calculate the cost of your freshman year of college based on a yearly increase in costs of 5% for the next four-six years. Record your work below. Page 16

17 Project #15 The Number System (NS) I can rewrite a subtraction problem as an addition problem by using the additive inverse. I can show that the distance between two inters on a number line is the absolute value of their difference. I can describe real-world situations represented by the subtraction of integers. 1. Using what you know about number lines, complete the problem below. On the number line above, the numbers a and b are the same distance from 0. What is a+b? Explain how you know. 2. Use a picture or physical objects to illustrate and create a problem for others to solve: Example: 13 (-8) 3 7 At noon on a certain day, the temperature was 13 degrees. At 10:00 p.m. the same day, the temperature was -8 degrees. How many degrees did the temperature drop between noon and 10:00 p.m.? (-7) (-3)(-7) 21 (-3) Page 17

18 Project #16 The Number System (NS) I can solve real-world problems that involve the addition, subtraction, multiplication, and/or division of rational numbers. Ocean water freezes at about -2.5 C. Fresh water freezes at 0 C. Antifreeze, a liquid used to cool most car engines, freezes at -64 C. Imagine that the temperature is exactly at the freezing point for ocean water. How many degrees must the temperature drop for the antifreeze to turn to ice? Page 18

19 Project #17 Geometry (G) I can state the relationship between supplementary, complementary and vertical angles. I can use algebraic reasoning and angle relationships to solve multi-step problems. 1. Study the diagram and coinciding information below. 2. Find 2-3 real objects in your home or neighborhood that demonstrates one or more of the same relationships expressed in the diagram above. 3. Take pictures of each of the objects you found and either download the pictures and paste them into an electronic document(s) or create a poster and paste your pictures on the poster. *If you do not have access to a digital camera and source for printing pictures, you may draw a picture of your objects instead. 4. Finally label each line, each angle, and each corresponding relationship. Use words to describe the angles and relationships formed by the intersecting lines on your document or poster (as done in the example above). Page 19

20 Project #18 Directions: Geometry (G) I can use formulas to compute the area and circumference of a circle. The figure below is composed of eight circles. Neighboring circles only share one point, and two regions between the smaller circles have been shaded. Each small circle has a radius of 5 cm. Using the formulas provided answer the questions. a. What is the radius of the large circle? b. What is the area of the large circle? c. What is the circumference of the large circle? Page 20

21 Project #19 Geometry (G) I can solve real-world problems involving area, surface area, and volume. I can use a scale drawing to determine the actual dimensions and area of a geometric figure. Part 1: 1. Read the problem below then study the scale drawing. Kiera showed her best friend a scale drawing of a new game room her father is going to build for her and her brothers and sisters. 2. Solve: If each 2 cm on the scale drawing equals 5 ft, what are the actual dimensions of Kiera s room? 3. How much carpet needs to be purchased to cover the entire floor? Part 2: 1. Design your own game room using Kiera s scale drawing as an example. 2. Using graph paper, determine the scale and be sure to write it in the top right corner of your paper. Example: (1 cm = 3 ft). 3. Then, create a scale drawing of the game room you have designed. Include game areas, furniture, and more to give the best visual representation possible. 4. Calculate the area of the floor that is showing versus the area of the floor under the furniture. *Be sure to use a ruler to create straight lines and accurate measures. Page 21

22 Project #20 Statistics and Probability (SP) I can draw inferences about a population based on collected and analyzed data. 1. Look at the following data set. It represents the height in centimeters of a group of students: 2. Answer the following questions based on the data set above. What is the mode of the set? What is the range of the set? Whose height is closest to the median height for the set? Whose height is closest to the mean height for the set? 3. On separate paper, create a scatter plot and box and whisker plot using all of the above data. *If you need help, search on the internet to find examples of scatter plots and box and whisker plots. Page 22

23 Project #21 Expressions and Equations (EE) I can write a simple algebraic inequality to represent a real-world problem. I can solve a simple algebraic inequality and graph the solution on a number line. I can describe the solution to an inequality in relation to the problem. You have tried many ways to solve problems throughout this Math Challenge Packet. Already you know that when one strategy does not lead you to a solution, you back up and try something else. Sometimes you can find a smaller problem inside the larger one that must be solved first. Sometimes you need to think about the information that is missing rather than what is there. Sometimes you need to read the problem again and look for a different point of view. Sometimes you need to tell your brain to try to think about the problem in an entirely different way perhaps a way you have never used before. Looking for different ways to solve problems is like brainstorming. Try to solve this problem. You may need to change your point of view. Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry 1200 pounds of people and gear for safety reasons. Assume the average weight of a person is 150 pounds. Each group will require 200 lbs of gear for the boat plus 10 lbs of gear for each person. 1. Create an inequality describing the restrictions on the number of people possible in a rented boat. Solve the inequality then graph the solution set. 2. Several groups of people wish to rent a boat. Group 1 has 4 people. Group 2 has 5 people. Group 3 has 8 people. Which of the groups, if any, can safely rent a boat? What is the maximum number of people that may rent a boat? Page 23

24 Extensions The following activities are based on standards you will learn in Algebra I. They may be challenging for you. Project #22 Functions (F) I can determine the properties of a function written in algebraic form. I can determine the properties of a function when given the inputs and outputs in a table. The table shows a relation between x and y. Which of these equations expresses this relation? Explain your answer in complete sentences. A. y = x + 5 B. y = x - 5 C. y=1/3(x-1) D. y = 3x + 1 How might you use information like the above in the real world? Describe a situation in which you would need to create a table and apply your understanding of a function. Page 24

25 Project #23 Expressions and Equations (EE) I can interpret the unit rate of a proportional relationship as the slope of the graph. I can use a graph, a table, or an equation to determine the unit rate of a proportional relationship and use the unit rate to make comparisons between various proportional relationships. The figure below shows the lines l and m described by the equations 4x-y=a and y=2x+b, for some constants a and b. They intersect at the point (p, q). 1. How can you interpret a and b in terms of the graphs of the equations above? 2. Imagine you place the tip of your pencil at point (p, q) and trace line l out to the point with x -coordinate p + 2. Imagine I do the same on line m. How much greater would the y - coordinate of your ending point be than mine? Now, create a function of your own. State the coordinates of two points along the line. Challenge a friend or a parent to find the slope. *Keep in mind that you may need to teach them how to find the slope before you ask them to solve your problem. You are the expert now! Page 25

26 Project #24 Equations and Expressions (EE) I can solve real-world problems and mathematical problems dealing with systems of linear equations and interpret the solution in the context of the problem. Consider the graph below showing two lines, L1 and L2 1. Find the two corresponding linear equations. 2. Find points other than the ones given in the graph; one that lies on L1 but not on L2 and one that lies on L2 but not on L1. Page 26

27 SUMMER MATH FUN FOR STUDENTS ENTERING ALGEBRA I ANSWER KEY When completing the problems we need to show all of our work and show all of our thinking. Compare your work to ours, especially if your answer is different than our answer. If the project listed is creative in its approach and each project created would be different, no answer is provided. Page

28 Project #1 Expressions and Equations (EE) I can determine the properties of integer exponents by exploring patterns and applying my understanding of properties of whole number exponents. I can use the properties of integer exponents to simplify expressions. 1. Write out the following in words. For example, 3(x+5) would be three times the quantity of x and five. 2. Solve the problem below. The evil ruler, Gonglo has locked the beautiful, intelligent Princess Alexa in a loft high up in the castle. To be released, she must find the first perfect square number greater than 100 whose digits do not sum to a perfect square! Can you help her find the number? The perfect square number that is greater than 100 but whose digits do not add to a perfect square is 256. When you add = you get 13, which is not a perfect square. 3. Create a challenge problem of your own using the formula for finding a perfect square. Page 2

29 Project #2 Functions (F) I can match a function to a given situation. 1. Use the information below to answer the riddle. Amy, Connor, Jalia, Stella and Gonzo live in apartments A, B, C, D, and E in the same building. Their ages are 9, 10, 11, 13, and 14. They each get up at a different time each morning, either 6:00, 7:00, 7:30, 8:00, or 8:15. Using the clues below, determine the apartment, age and rising time for each student. Connor gets up at 6:00 and is older than the student in Apt. B. Amy is 11 and gets up two hours later than the boy in Apt. A. The ten-year old girl lives in Apt. B and it isn't Stella. The youngest student, a girl, doesn't live in Apt. E. The person in Apt. B gets up later than everyone else. Gonzo is younger than Connor and he lives in Apt. D. The shades go up at 7:30 in Apt. C. 2. Create your own riddle for a friend to solve. Page

30 Project #3 Directions: The Number System (NS) I can classify a number as rational or irrational based on its decimal expansion. Pretend your 10-year-old neighbor knows nothing about integers, rational, whole, natural, and irrational numbers. Write two to three paragraphs explaining the graph below. Be sure to use language that a fourth or fifth grader can understand. Page

31 Project #4 Directions: The Number System (NS) I can compare two or more rational or irrational numbers. Compare each pair of numbers using <, >, or =. Write one to two sentences after each problem explaining your answer. > < = Now, create two to three similar problems and challenge an adult or friend to solve them. Page

32 Project #5 Directions: The Number System (NS) I can plot rational and irrational numbers on a number line. List the numbers 2/3, -2/3, 1.2, 4/3, -4/3, -1.2, -7/4 from least to greatest and then locate the numbers on the number line. You will label the number line below to fit your purposes. Page

33 Project #6 Functions (F) I can define the rate of change in relation to the situation. Linda traveled 110 miles in 2 hours. If her speed remains constant, how many miles can she expect to travel in 4.5 hours? Answer the question in complete sentences and justify your answer. Step 1: = 55mph Step 2: 55 x 4.5 mph = miles Then, create a problem of your own involving linear functions and constant rates of speed over a specified distance. Page

34 Project #7 Directions: Functions (F) I can write a linear function that models a given situation. Which of the following could be modeled by y=2x+5? Answer YES or NO for each one. 1. There are initially 5 rabbits on the farm. Each month thereafter the number of rabbits is 2 times the number in the month before. How many rabbits are there after x months? YES NO 2. Joaquin earns $2.00 for each magazine sale. Each time he sells a magazine he also gets a five dollar tip. How much money will he earn after selling x magazines? YES NO 3. Sandy charges $2.00 an hour for babysitting. Parents are charged $5.00 if they arrive home later than scheduled. Assuming the parents arrived late, how much money does she earn for x hours? YES NO 4. I have a sequence of integers. The first term of the sequence is 7 and the difference between any consecutive terms is always equal to 2. YES NO 5. Sneak Preview is a members-only video rental store. There is a $2.00 initiation fee and a $5.00 per video rental fee. How much would John owe on his first visit if he becomes a member and rents x videos? YES NO 6. Andy is saving money for a new CD player. He began saving with a $5.00 gift and will continue to save $2.00 each week. How much money will he have saved at the end of x weeks? YES NO Page

35 Project #8 Directions: Geometry (G) I can apply the Pythagorean Theorem to find an unknown side length of a right triangle. The Pythagorean Theorem is a 2 + b 2 = c 2. The variable C represent the hypotenuse which is the side opposite the right angle. In this particular problem C is the unknown side. Solve the problem and justify your answer. To get from point A to point B you must avoid walking through a pond. To avoid the pond, you must walk 34 meters south and 41 meters east. To the nearest meter, how many meters would be saved if it were possible to walk through the pond? Part I: Calculate the distance around the pond = 75 meter Part II: Calculate the distance through the pond by using the Pythagorean Theorem (a 2 +b 2 =c 2 ) = c = = c 2 C = (53 nearest meter) Part II: Distance saved = 22 Answer: 22 meters Page

36 Project #9 The Number System (NS) I can solve real-world problems that involve the addition, subtraction, multiplication, and/or division of rational numbers. 1. The United States owes approximately $14,300,000,000, in National debt. There are approximately 300 million people in the United States. Assume the United States would like to pay off the debt in one lump sum. How much would each American have to pay to pay off the National debt? Divide $14,000,000,000, by 300,000,000 = $47, If each American paid $2,000 a year towards the National debt how many years would it take for the country to pay off what it owes? (Assume that there is no interest on the debt.) Divide $47, by $2,000 = , now round off to 24 years Page

37 Project #10 Directions: Expressions and Equations (EE) I can determine the volume of three-dimensional figures. I can solve real-world problems involving area, surface area and volume. Solve the following questions using the provided formulas. My sister s birthday is in a few weeks and I would like to buy her a new vase to keep fresh flowers in her house. She often forgets to water her flowers and needs a vase that holds a lot of water. In a catalog there are three vases available and I want to purchase the one that holds the most water. The first vase is a cylinder with diameter 10 cm and height 40 cm. The second vase is a cone with base diameter 16 cm and height 45 cm. The third vase is a sphere with diameter 18 cm. Page

38 1. Which vase should I purchase? You should purchase the cylinder vase. If r is the radius and h is the height, then, using the fact that the radius is half the diameter, we get Cylinder volume = 1000 π cm 3 Cone volume = 960 π cm 3 Sphere volume = 972 π cm 3 2. How much more water does the largest vase hold than the smallest vase? Cylinder volume Cone volume = 40 π cm 3 3. Suppose the diameter of each vase decreases by 2 cm. Which vase would hold the most water? Cylinder volume = 640 π cm 3 Cone volume = 735 π cm 3 Sphere volume = 682 2/3 π cm 3 Now you should buy the cone. 4. The vase company designs a new vase that is shaped like a cylinder on bottom and a cone on top. The catalog states that the width is 12 cm and the total height is 42 cm. What would the height of the cylinder part have to be in order for the total volume to be 1224πcm 3. The total volume is the volume of the cylinder plus the volume of the cone. If the cylinder has height x cm then the cone has height of 42-x cm. X =30cm Page

39 Project #11 Statistics and Probability (SP) I can recognize whether or not data plotted on a scatter plot have a linear association. I can make inferences regarding the reliability of the trend line by noting the closeness of the data points to the line. This scatter diagram shows the lengths and widths of the eggs of some American birds. 1. A biologist measured a sample of one hundred Mallard duck eggs and found they had an average length of 57.8 millimeters and average width of 41.6 millimeters. Use an X to mark a point that represents this on the scatter diagram. 2. What does the graph show about the relationship between the lengths of birds' eggs and their widths? There seems to be a positive linear relationship between the length and width of the eggs. 3. Another sample of eggs from similar birds has an average length of 35 millimeters. If these bird eggs follow the trend in the scatter plot, about what width would you expect these eggs to have, on average? The line below appears to fit the data fairly well. Since it passes through (0, 0) and (50, 36), the equation of the line is y=0.72x. If x = 35, then our line would predict that y = = So we would expect the width of these eggs to be, on average, about 25 mm. Answers using different lines can vary up to 1 mm in either direction. 4. Describe the differences in shape of the two eggs corresponding to the data points marked C and D in the plot. Without reading off precise numerical values from the plot, we can see that eggs C and D have very nearly the same width, but egg D is about 12 millimeters longer than egg C. 5. Which of the eggs A, B, C, D, and E has the greatest ratio of length to width? Explain how you decided. First we note that egg E certainly has a higher length-to-width ratio than C or D, since it is both longer and narrowed. Similarly, E has a higher ratio than B because it is significantly longer, and only a tad wider. It is harder to visually identify the difference between A and E, we compute their respective length-to-width ratios numerically, which turn out to be approximately 1.3 for A and 1.6 for E. So E has the greatest ratio of length to width. Page

40 Project #12 Ratios and Proportional Relationships (RP) I can compute a unit rate by multiplying or dividing both quantities by the same factor. 1. DVDs can be made in a factory in New Mexico at the rate of 20 DVDs per $3, but the factory costs $80,000 to build. If they make 1 million DVDs, what is the unit cost per DVD? The unit cost is 23 cents per DVD. 2. DVDs can be made in a factory in Colorado at the rate of 10 DVDs per $1.50, but the factory costs $100,000 to build. If they make 1 million DVDs, what is the unit cost per DVD? The unit cost per DVD is 25 cents per DVD. 3. How much can a buyer save on a million DVDs by buying DVDs from New Mexico instead of DVDs from Colorado? 20,000 dollars Challenge: 4. Find an equation for the cost of making x number of DVDs in the factory in New Mexico. C = (80, X ) dollars 5. Find an equation for the cost of making x number of DVDs in the factory in Colorado. C = (100, X ) dollars Page

41 Project #13 Statistics and Probability (SP) I can use probability to predict the number of times a particular event will occur given a specific number of trials. I can use variability to explain why the experimental probability will not always exactly equal the theoretical probability. Drawing from a set of six blue beads and three gold beads, use ratios to state the likelihood of each color being drawn; conducts experiments to test predictions. Show your work and explain your answer. Step 1: Add to get the total number of beads Step 2: Figure out each color beads ratios Blue = 6:9 or 2:3 Gold = 3:9 or 1:3 Page

42 Project #14 Ratios and Proportional Relationships (RP) I can use proportional reasoning to solve real-world percent problems, including those with multiple steps. College costs increase at about twice the inflation rate. Current increases have averaged 5% to 8% annually. According to the College Board's and Trends in College Pricing, the average total costs (including tuition, fees, room and board) were $16,140 for students attending four-year public colleges and universities in-state and $28,130 out-of-state, and $36,993 for students at four-year private colleges and universities. You can assume an additional $4,000 for textbooks, supplies, transportation and other expenses. Use the information above to calculate the cost of your freshman year of college based on a yearly increase in costs of 5% for the next four-six years. Record your work below. Page

43 Project #15 The Number System (NS) I can rewrite a subtraction problem as an addition problem by using the additive inverse. I can show that the distance between two inters on a number line is the absolute value of their difference. I can describe real-world situations represented by the subtraction of integers. 1. Using what you know about number lines, complete the problem below. On the number line above, the numbers a and b are the same distance from 0. What is a+b? Explain how you know. Step 1: Start at zero and move a units to the right, Step 2: Then move the same number of units to the left, which puts us back at Use a picture or physical objects to illustrate and create a problem for others to solve: Example: 13 (-8) 3 7 At noon on a certain day, the temperature was 13 degrees. At 10:00 p.m. the same day, the temperature was -8 degrees. How many degrees did the temperature drop between noon and 10:00 p.m.? (-7) (-3)(-7) 21 (-3) Page

44 Project #16 The Number System (NS) I can solve real-world problems that involve the addition, subtraction, multiplication, and/or division of rational numbers. Ocean water freezes at about -2.5 C. Fresh water freezes at 0 C. Antifreeze, a liquid used to cool most car engines, freezes at -64 C. Imagine that the temperature is exactly at the freezing point for ocean water. How many degrees must the temperature drop for the antifreeze to turn to ice? The difference between the temperature that ocean water turns to a solid and antifreeze turns to a solid is -2.5 (-64). So the temperature must drop another 61.5 C after ocean water freezes for the antifreeze to turn to ice. Page

45 Project #17 Geometry (G) I can state the relationship between supplementary, complementary and vertical angles. I can use algebraic reasoning and angle relationships to solve multi-step problems. 1. Study the diagram and coinciding information below. 2. Find 2-3 real objects in your home or neighborhood that demonstrates one or more of the same relationships expressed in the diagram above. 3. Take pictures of each of the objects you found and either download the pictures and paste them into an electronic document(s) or create a poster and paste your pictures on the poster. *If you do not have access to a digital camera and source for printing pictures, you may draw a picture of your objects instead. 4. Finally label each line, each angle, and each corresponding relationship. Use words to describe the angles and relationships formed by the intersecting lines on your document or poster (as done in the example above). Page

46 Project #18 Directions: Geometry (G) I can use formulas to compute the area and circumference of a circle. The figure below is composed of eight circles. Neighboring circles only share one point, and two regions between the smaller circles have been shaded. Each small circle has a radius of 5 cm. Using the formulas provided answer the questions. a. What is the radius of the large circle? The radius of the large circle 3 5=15 cm. b. What is the area of the large circle? The area of the large circle is π(15 cm) 2 =225π cm 2. c. What is the circumference of the large circle? The circumference of the large circle is 2 π 15 = 30π cm. Page

47 Project #19 Geometry (G) I can solve real-world problems involving area, surface area, and volume. I can use a scale drawing to determine the actual dimensions and area of a geometric figure. Part 1: 1. Read the problem below then study the scale drawing. Kiera showed her best friend a scale drawing of a new game room her father is going to build for her and her brothers and sisters. 2. Solve: If each 2 cm on the scale drawing equals 5 ft, what are the actual dimensions of Kiera s room? 3. How much carpet needs to be purchased to cover the entire floor? Part 2: 1. Design your own game room using Kiera s scale drawing as an example. 2. Using graph paper, determine the scale and be sure to write it in the top right corner of your paper. Example: (1 cm = 3 ft). 3. Then, create a scale drawing of the game room you have designed. Include game areas, furniture, and more to give the best visual representation possible. 4. Calculate the area of the floor that is showing versus the area of the floor under the furniture. *Be sure to use a ruler to create straight lines and accurate measures. Page

48 Project #20 Statistics and Probability (SP) I can draw inferences about a population based on collected and analyzed data. 1. Look at the following data set. It represents the height in centimeters of a group of students: 2. Answer the following questions based on the data set above. What is the mode of the set? 144 What is the range of the set? = 47 Whose height is closest to the median height for the set? Juan is exactly at the median Whose height is closest to the mean height for the set? The mean is so Juan is also closest to the mean. 3. On separate paper, create a scatter plot and box and whisker plot using all of the above data. *If you need help, search on the internet to find examples of scatter plots and box and whisker plots. Page

49 Project #21 Expressions and Equations (EE) I can write a simple algebraic inequality to represent a real-world problem. I can solve a simple algebraic inequality and graph the solution on a number line. I can describe the solution to an inequality in relation to the problem. Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry 1200 pounds of people and gear for safety reasons. Assume the average weight of a person is 150 pounds. Each group will require 200 lbs of gear for the boat plus 10 lbs of gear for each person. 1. Create an inequality describing the restrictions on the number of people possible in a rented boat. Solve the inequality then graph the solution set. Let p be the number of people in a group that wishes to rent a boat. Then 150p represents the total weight of the people in the boat, in pounds. Also, 10p represents the weight of the gear that is needed for each person on the boat. So the total weight in the boat that is contributed solely by the people is 150p+10p=160p. Because each group requires 200 pounds of gear regardless of how many people there are, we add this to the above amount. We also know that the total weight cannot exceed 1200 pounds. So we arrive at the following inequality: 160p+200< Several groups of people wish to rent a boat. Group 1 has 4 people. Group 2 has 5 people. Group 3 has 8 people. Which of the groups, if any, can safely rent a boat? What is the maximum number of people that may rent a boat? Extensions The following activities are based on standards you will learn in Algebra I. They may be challenging for you. Page

50 Project #22 Functions (F) I can determine the properties of a function written in algebraic form. I can determine the properties of a function when given the inputs and outputs in a table. The table shows a relation between x and y. Which of these equations expresses this relation? Explain your answer in complete sentences. A. y = x + 5 B. y = x - 5 C. y=1/3(x-1) D. y = 3x + 1 How might you use information like the above in the real world? Describe a situation in which you would need to create a table and apply your understanding of a function. Page

51 Project #23 Expressions and Equations (EE) I can interpret the unit rate of a proportional relationship as the slope of the graph. I can use a graph, a table, or an equation to determine the unit rate of a proportional relationship and use the unit rate to make comparisons between various proportional relationships. The figure below shows the lines l and m described by the equations 4x-y=a and y=2x+b, for some constants a and b. They intersect at the point (p, q). 1. How can you interpret a and b in terms of the graphs of the equations above? 2. Imagine you place the tip of your pencil at point (p, q) and trace line l out to the point with x -coordinate p + 2. Imagine I do the same on line m. How much greater would the y - coordinate of your ending point be than mine? Now, create a function of your own. State the coordinates of two points along the line. Challenge a friend or a parent to find the slope. *Keep in mind that you may need to teach them how to find the slope before you ask them to solve your problem. You are the expert now! Page

52 Project #24 Equations and Expressions (EE) I can solve real-world problems and mathematical problems dealing with systems of linear equations and interpret the solution in the context of the problem. Consider the graph below showing two lines, L1 and L2 1. Find the two corresponding linear equations. 2. Find points other than the ones given in the graph; one that lies on L1 but not on L2 and one that lies on L2 but not on L1. Page

53 Helpful Websites Visit one of the websites below and learn about one of the games. Play the game together for minutes. Give your child time to play the game independently. Then, check to ensure your child is playing the game correctly and has mastered the concept &vendor=sun_microsystems_inc. Page