Percent Change FINDING SCALE FACTOR In Go With the Floe on page 4, you learned how to find the percent change of areas of sea ice. You can also apply percent change in problems involving similar figures. Similar figures have the same shape, but they are not necessarily the same size. They differ in size by a certain scale factor. The scale factor is the number by which a figure s side or base length can be multiplied in order to yield the length of a second figure s corresponding side or base. You can use scale factor to determine the percent change between the two figures. 4 cm Figure 1 3.5 cm Figure 2 8 cm EXAMPLE: Find the scale factor of the figures above and determine the percent change from Figure 1 to Figure 2. To find the scale factor, divide either the length of the base or side of Figure 2 by the length of the base or side of Figure 1. Here we used the base: base of Fig. 2 7 base of Fig. 1 = 8 = 0.875 The scale factor of these two figures is 0.875: 7 cm To convert the scale factor to a percentage, move the decimal point two places to the right and add a % sign: 0.875 087.5 87.5% The scale factor is equivalent to 87.5%, which means that Figure 2 is 87.5% the size of Figure 1. To find the percent change between these two figures, subtract your scale factor from 100%: 100% 87.5% = 12.5% So the percent change is 12.5%. Since Figure 2 is smaller than figure 1, this is a percent decrease. Use this information to answer the questions below. 1 10 in. 24 in. Figure 1 A. What is the scale factor of the figures above? B. Express the scale factor as a percentage. C. What is the percent change between these two figures? Is it a percent increase or decrease? 8 in. 2 3 mm 19.2 in. Figure 1 5 mm Figure 2 Figure 2 3 mm 1.5 mm 5 mm 2.5 mm A. What is the scale factor of the figures above? B. Express the scale factor as a percentage. C. What is the percent change between these two figures? Is it a percent increase or decrease?
Percent Change SEA ICE SHRINKS & GROWS In Go With the Floe on page 4, you practiced calculating percent change to determine the extent by which sea ice is changing in the Antarctic and the Arctic. Use what you learned to answer five more questions about changes in sea ice at both poles. Round all of your answers to the nearest percent. Antarctica South Pole 1On average, there are 5.8 million square miles of Arctic sea ice during winter, and only 2.7 million square miles at the end of the summer-melt season. What is the percent decrease in Arctic sea ice between winter and the end of summer? 2In Antarctica, there are 6.9 million square miles of sea ice during the winter, but only 1.1 million square miles at the end of the summer. What is the percent decrease in Antarctic sea ice between winter and the end of summer? 3In Antarctica, the average maximum extent of sea ice is about 7 million square miles, and the average minimum is about 1.16 million square miles. What is the percent decrease from average maximum to average minimum? 4In September 2013, the average maximum extent of Arctic sea ice was about 5.4 million square miles. In March 2014, the extent was about 14.8 million square miles. What was the percent increase during this time? 5In September 1999, the extent of Antarctic sea ice was about 19 million square kilometers. In September 2013, the extent was up to 19.8 million square kilometers. What was the percent increase in sea-ice extent during this time period?
MEASURING SECTOR SIZE In Head Protector on page 6, you practiced making a circle graph based on a set of data. Before making a circle graph, you need to calculate the size of each sector, or wedge, in terms of its number of degrees out of 360 (the total number of degrees in a circle). One way to find the size of a sector is to set up a proportion that compares the measured data with the degrees of a circle. EXAMPLE: The table to the right shows the birth months of players on the L.A. Lakers. Find the size of the circle-graph sector for the number of players born in August. STEP 1: Set up a proportion: players born in August total players = degrees of the sector 2 total degrees of a circle 15 = x 360 STEP 2: Scale up the left side of the proportion to find the unknown number. In this example, because 15 multiplied by 24 is 360, multiply the top and bottom of the left side of your proportion by 24: 2 5 24 15 5 24 Circle Graphs x = 360 STEP 3: Solve to find the unknown number: x = 2 5 24 = 48 So the sector representing players born in August should be 48 degrees out of a total of 360. Birth Months of the LA Lakers Birth Month Number of Players January 0 February 1 March 0 April 1 May 0 June 5 July 2 August 2 September 0 October 0 November 4 December 0 TOTAL 15 Use this information and the data in the table to answer the following questions. 1Find the sector size for players born in February. 2Which month(s) will have a sector the same size as February s? 3 Find the sector size for players born in July. 4 Find the sector size for players born in November. A. Which month would have the largest segment of 5 a circle graph based on the data? B. What would be the size of that sector?
Circle Graphs graphing girls concussions In Head Protector on page 6, you made a circle graph by calculating the percent of total concussions reported for each sport. Use what you learned and the information in the chart below to create a second circle graph showing the distribution of concussions among female high school athletes in three sports. Round all percent calculations to the nearest whole percent. Circle Graph TITLE: Concussions in Girls High School Sports, 2012 Sport Number of Concussions Soccer 159 Lacrosse 60 Basketball 107 Total 326 Source: Epidemiology of Concussions Among United States High School Athletes in 20 Sports, Marar et. al., The American Journal of Sports Medicine, 2012. 1A. What percent of girls concussions occurred in soccer? B. What percent occurred in lacrosse? C. What percent occurred in basketball? 2Use the information provided in the table to complete the circle graph above and give it a title. Label each wedge you make with the sport it represents and each sport s percent of total concussions. (Hint: The circle above is divided into 24 slices of 15-degree increments, for a total of 360 degrees. Remember to convert your percent calculations into degrees to determine the number of wedges to allot to each sport.) 3Female athletes playing which sport experienced the most concussions? 4If you combined the number of lacrosse and basketball concussions, would it account for a higher or lower percent of total concussions than soccer-related concussions? 5What do you think accounts for the higher percentage of concussions in soccer than in either lacrosse or basketball? Explain your thinking.
Volume triangular prisms In The Math of Minecraft on page 8, you practiced finding the volume of rectangular prisms using the formula for volume: length 5 width 5 height. In this formula, length 5 width represents the base of the rectangular prism you re working with. This is the same as the formula for the area of a square or rectangle. When you find the volume of a prism, you are multiplying the area of its base by its height. The same applies to triangular prisms. To find the volume of a triangular prism, first find the area of its triangular base (b). Then multiply the area of the base by the length of the prism (l). EXAMPLE: Find the volume of the triangular prism to the right. First, find the area of the triangular base. The formula for area of a triangle is: 1 2 bh = 1 2 5 8 cm 5 3 cm = 12 cm2 Next, multiply the area of the base by the length of the prism: 12 cm 2 5 12 cm = 144 cm 3 So the volume of the prism is 144 cm 3. Use this information to answer the following questions. 3 cm 8 cm 12 cm A. What is the area of the 1 triangular base of the prism to the right? B. What is the volume of the prism? 2 A. What is the area of the triangular base of the prism to the right? B. What is the volume of the prism? 16 cm 18 cm 12 cm 10 cm 12 cm 9 cm 3 A. What type of prism is this? B. Without knowing the formula for the area of this prism s base, can you think of another way to find the volume of the prism? C. Find the volume of the prism above, showing your work. 10 cm 4 cm 7 cm 4 cm
Volume BUILDING BLOCKS In The Math of Minecraft on page 8, you calculated the volume of structures in Minecraft using the formula for the volume of a rectangular prism: V = length 5 width 5 height Use what you learned and the Minecraft block diagram below to answer five more volume questions about structures you could build in the game. h = 1 meter l = 1 meter w = 1 meter 1A shelter built in Minecraft has a three-layer foundation of stone blocks. If the foundation is 20 blocks long and 10 blocks wide, what is its volume in cubic meters? 2A square, human-made lake is 5 blocks long, 5 blocks wide, and 4 blocks deep. What is the volume of the lake? 3A sugarcane farm gets its water from 3 irrigation canals that border the crops. Each canal is 2 blocks wide and 2 blocks deep. One canal is 15 blocks long, while the other two are both 12 blocks long. What is the combined volume of all 3 canals? 4A square tower has a height of 15 blocks and its sides are each 3 blocks long. It sits on a rectangular pedestal that s 2 blocks high, 4 blocks long, and 6 blocks wide. What is the approximate combined volume of the tower and its pedestal? Round your answer to the nearest whole number. 5A structure 30 blocks long and 5 blocks high has a volume of 450 m 3. What is its width?
Probability THEORETICAL PROBABILITY In Against the Odds on page 14, you learned how to make predictions using experimental probability. Experimental probability is the likelihood that an event will happen based on data from prior events. You can also make predictions based on theoretical probability. While experimental probability is based on what happened in the past, theoretical probability is based on what should happen in the future. The formula for finding the theoretical probability of an event is: P = number of ways the event can occur total number of equally likely outcomes EXAMPLE: A standard die has six sides, which are numbered 1 to 6. On a single roll, what is the probability of rolling a 6? P = 1 6 number of times that 6 appears on the die total number of possible outcomes So the probability of rolling a 6 is 1 6. This is a theoretical probability, because it is not based on experimental data. Instead, it s based on the likelihood of an event happening out of a distribution of equally likely outcomes. Use this information to answer the following questions. Express your answers in simplest form. 1On a single roll of an ordinary die, what are the possible outcomes? 2What is the probability of rolling an odd number? 3If you were to roll the die 48 times, how many times would you expect to roll a 2? 4If you were to roll the die 70 times, how many times would you expect to roll an even number? A. Are your predictions for No.4 based on theoretical 5 probability or experimental probability? Explain. B. Do you think that theoretical probability can differ from experimental probability in certain cases? Why or why not?
Probability PANDA PREDICTIONS In Against the Odds on page 14, you practiced using experimental probabilities to make predictions about the births of rare animals like male calico cats and albino lobsters. Use what you learned and the probability chart below to answer five more questions about the odds of certain animal births. Animal Odds Animal Probability Panda twins 1 in 2 Panda triplets 1 in 100 Male calico cat 1 in 3,000 Blue lobster Albino lobster 1 in 2 million 1 in 100 million 1Out of 500 panda births, how many would likely be sets of twins? 2Out of 24,000 calico cat births, how many would likely be males? 3A. Out of 300,000,000 lobster births, how many lobsters would likely be blue? B. How many would likely be albino? 4Out of 200 panda births, how many more sets of twins are likely to be born than sets of triplets? 5If there are 100 panda births in a year, how many pandas in total do you predict will have been born? Explain your reasoning.
Practice Test 1Between spring and fall of 2013, Arctic sea ice dropped from 15.1 million square km. to 5.4 million square km. What was the percent decrease during this time period, rounded to the nearest percent? ISSUE SKILLS REVIEW 6A Minecraft shelter that is 10 blocks wide, 7 blocks long, and 8 blocks high has a 3-layer stone foundation. What is the combined volume of the shelter and its foundation? 2Antarctic sea ice increased from an average of 18.7 million square km. in September 2000 to an average of 19.2 million square km. in September 2010. What was the percent increase between these two periods, rounded to the nearest percent? 7Out of a total of 1,651 concussions reported among high school athletes in 2012, 159 occurred in girls soccer. What percent of concussions occurred in girls soccer? Round to the nearest percent. 3The probability of a giant panda giving birth to twins is 1 in 2. If there are 16 panda births in one year, how many sets of twins do you predict there will be? 4Using the probability given in question 3, predict the total number of pandas likely to result from 10 births. 5Each block in Minecraft has a volume of 1 m 3. What is the volume of a shelter that is 16 blocks long, 12 blocks wide, and 20 blocks high? 8A total of 262 concussions occurred in boys and girls soccer combined, 103 of which happened in boys soccer. What percent of those 262 concussions occurred in boys soccer? Round to the nearest percent. 9Write the following operation as an algebraic expression: Subtract 7 from x. Write the following operation as an algebraic 10 expression: Divide 12 by the sum of 2 and a.