APES Math Instructional Guide

Transcription

1 APES Math Instructional Guide Your Name PURPOSE OF THIS GUIDE: In this class, you will be asked to do math <GASP!!!>. And now for the double whammy you will need to do the majority of math in the class WITHOUT a calculator <DOUBLE GASP!!>. That s right WITHOUT A CALCULATOR! The reason for this is because the AP exam has a free-response question that has you solve a realistic math problem associated with environmental science. And you cannot use a calculator on the exam. So, this means we need to refresh you on how to do math without your calculator. That is the bad news. The good news is that you have already learned all of this math earlier in your schooling many times before. We just need to refresh you! The following packet is a guide of instructions to how to solve various math skills or problems you will be asked to know for this course. This packet is set up to show you the steps of how to solve a particular type of problem and then give you a chance to practice. We will assign each module as we go along. There is no need to do the whole packet at once. Please only work on the part that you are assigned at the time. The packet will review the following math topics: - Unit Conversion (converting units of different unit measurement systems) - Metric Unit Conversion - Computing Volume - Computing Percent Change - Setting Up and Reducing Ratios

2 MODULE 1 UNIT CONVERSION Date When Due: Many of the problems you will encounter will ask you to convert from one unit to another. This may require many steps. This seems overwhelming. However, it is not, if you know how to properly set up the problem. There are no hard 100% solid rules to solving these. However, the best piece of advice I can offer is this: WORK WITH THE UNITS FIRST. THEN WORK WITH THE NUMBERS LAST!!!! The units are FAR more important. Set those up first. They guide you through the rest of the problem. EXAMPLE PROBLEM: Here is a problem that might be seen and then a proper set up of the problem: You are working with a household cleaner that is toxic if consumed in a large enough dose. A child in your care has consumed the 500 ml bottle of the chemical. It has a toxicity rating of 25 ml / 1 kg of body mass. At what weight (in lbs) would a child need to have at minimum in order to avoid a severe toxicity reaction? (1 kg = 2.2 lbs) Step 1: Identify the starting value of the problem. In this problem, the problem revolves around a household cleaner that was consumed. The value that starts this is the 500 ml consumed. Step 2: Identify the ending value of the problem. The problem wants to solve the weight in lbs. of a child. So, the unit of your answer is lbs. Step 3: Set up the start and end and fill in the units in between to get from start to end. NO NUMBERS!! 500 ml x x = lbs (In order to convert from unit to unit, you multiply your starting number by fractions.) Step 4: Fill in the fractions with units. If you start with ml, then to get out of ml you put it in the opposite position of the next fraction. In this problem, since ml starts on the top, then you put it in the bottom of the following fraction. Then you fill in the other part of the fraction with a unit you can convert into, based on the other information given in the problem (HINT if the starting number is not a fraction, then set it up like a fraction with the denominator being the number 1 without a unit): 500 ml x kg x = lbs 1 ml 500 ml x kg x lb = lbs 1 ml kg (NOTICE no numbers have gone into the fractions just units in opposite places! Remember, the problem gave me a conversion for kg to ml and also gave me one for lb to kg. The last unit left is the unit I want. So I am done and can stop setting up fractions)

3 Step 5: Now fill in the numbers that go with the conversions. 500 ml x 1 kg x 2.2 lb = lbs 1 25 ml 1 kg Step 6: Any numbers on the top of the fractions get multiplied together. Then divide that answer by any numbers on the bottom of the fractions. Now you have the answer. 500 ml x 1 kg x 2.2 lb = 1100/25 = 44 lbs 1 25 ml 1 kg [Units in opposite places in these fractions cancel out. So ml are set up in opposite places they cancel. The kg also are set up that way. They cancel. All that is left is lb] PRACTICE PROBLEMS: Please set up the conversion like I have shown, even if you don t have to and understand how to do it otherwise. You will get the chance later to do other ways. However, this assignment requires that you show all the steps, then show the final answer! #1) Your community is installing a windmill to supplement its energy needs. It is rated to produce 6 MW (megawatts). How many homes can this supply energy for if 1 home uses 1200 W? (1 MW = 1,000,000 W) Show your set up of the problem, WITHOUT numbers in the conversion: Now fill in the numbers and compute the final answer: #2) The BP oil spill in the Gulf of Mexico took 90 days in length. It is estimated that the spill released 50,000 barrels of oil/day. How many gallons was spilled into the Gulf in total during this time? (1 barrel = 42 gallons) Show your set up of the problem, WITHOUT numbers in the conversion: Now fill in the numbers and compute the final answer:

4 #3) You own a farm with 400 cows. You grow corn to feed the cows. An average cow eats 25 kg of corn each day. How much corn do you need to grow to feed all the cows on the farm for a day? Then calculate how much corn you need for the year for all cows. (You feed corn to the cows for 300 days/year, while you feed them other things during the rest of the year). Show your set up of the problem, WITHOUT numbers in the conversion: Now fill in the numbers and compute the final answer: #4) Most energy used in the U.S. is produced from coal. There are 300,000,000 people in the U.S. It is estimated that one person is responsible for the burning of 1 ton of coal per year to do what they do in their daily life. For every 100 lbs of coal burned, 2.5 lbs of sulfur go into the atmosphere. How much sulfur is put into the air by coal burning each year for all the people in the U.S.? [1 ton = 2000 lb] Show your set up of the problem, WITHOUT numbers in the conversion: Now fill in the numbers and compute the final answer:

5 #5) Your family uses a hot water heater to heat water for showers. It uses 0.2 kwh of energy/gallon of water used (kwh stands for kilowatt-hour, which is the way the electric company measures your energy use). The cost of a kwh is $0.10/kWh. Your family uses gallons of hot water per year for showers. How much money will you spend on electric energy to heat the water for your family showers for a DAY? (You do not use the hot water heater each day of the year. So, only calculate for 300 days in the year). Show your set up of the problem, WITHOUT numbers in the conversion: Now fill in the numbers and compute the final answer: 6) Your community has a landfill to place its trash. The landfill will fill with rain water over the year which collects toxic materials from the trash. Then the water will try to drain down into the soil, taking the toxic materials with it into the ground and groundwater. In order to prevent this contamination, the landfill is designed to catch this drainage water and treat it to remove the toxins. Your landfill collects 900 m 3 of rain water each year. As that water goes down through the landfill, it collects a toxic metal called cadmium at a concentration of 2 g/m 3 of water. The cost of filtering the cadmium out of the water for the community is $5000/kg of cadmium removed. How much money does your community need to spend on cadmium removal of landfill water each year? (Remember, 1 kg = 1000 g) Show your set up of the problem, WITHOUT numbers in the conversion: Now fill in the numbers and compute the final answer:

6 MODULE 2 COMPUTING PERCENT CHANGE Date When Due: When asked to compute percent change, you will be asked to compare values between two different time periods. You will then calculate the percent increase or decrease in that value over time. This formula is used if you are asked to compute percent change, percent increase, or percent decrease. To compute percent change, you must know this equation: (New Value Old Value) * 100 OR (N O) * 100 Old Value O New value means the value that is the latest chronologically in the problem Old value means the value that is the earliest chronologically in the problem Remember this: NOO! (This is the order of how to set up the variables in the problem from top to bottom) To solve the equation, determine the new and old values of the given problem. Substitute these values into the equation and solve. Sample Problem: You are studying a population of deer in your local area. 10 years ago, there were 90 deer in the population. The current population estimate is 126 deer. What is the percent change in the deer population? First, determine the new and old values: New Value = 126 deer Old Value = 90 deer Then substitute these values in and solve: Try these samples! (126 90) * 100 = 36 * 100 = 3600 = 40% ) Before the 2004 tsunami, a coastal region in Thailand had an average water depth of 5 m. Right after the tsunami, the water depth was 50 m. Compute the percent change in water depth. 2) The amount of arable land in the US in the year 2000 was 20% of total land area. Currently, the amount of arable land in the US is 17.5% of our total land area. Compute the percent change in arable land area in the US over this time period.

7 MODULE 3 RATIOS Date When Due: A ratio is a comparison of two different values. Ratios can be written in the following forms: 3:1 3 to 1 3/1 This ratio is saying that 3 parts is compared to 1 part of something. Steps to Writing Ratios: To solve a ratio, set it up like a fraction. Put the first number asked for in the ratio in the numerator. Put the second number asked for in the denominator. Then simplify the fraction! - Do not reduce any number lower than the number 1! - Do not leave a decimal in any portion of the fraction! Example 1: Set up a ratio of $40,000,000 to $6,000,000 $ = $ = $40 = 20 The answer is 20:3 or 20 to 3 or 20/3 $ $ $6 3 Example 2: In a 50 ml sample of soil, there is 27 ml of sand and 1.2 ml of clay. What is the ratio of sand to clay? 27 ml = 270 ml 3 = 90 ml = 90 2 = 45 The answer is 45:2 or 45 to 2 or 45/2 1.2 ml 12 ml 3 ml Try these samples! 1) Create a ratio of 7500 km to 500 km. 2) What is the ratio of the earthquake magnitudes of earthquakes A to B, if earthquake A was measured at 4.8 on the Richter scale while earthquake B was measured at 8.0 on the Richter scale? 3) What is the ratio of erosion from wind to the erosion from water of soil on croplands if the wind removes 0.6 tons/acre and water removes 2.6 tons/acre? 4) Set of a ratio of monthly precipitation of the tundra to the rainforest if the tundra had 12 mm of precipitation while the rainforest had 15 cm of precipitation.

8 MODULE 4 - METRIC SYSTEM REVIEW Date When Due: This is a review of how to convert units in the metric system. Since science uses the metric system, the College Board and AP exam assume you know all about metric conversion, even if you probably do not. This is showing you how to convert units WITHIN the metric system (Example: Converting types of liters to other types of liters). However, if you need to turn a liter into a gallon for example, these are two very different units and you use the method from Module 1. The following method is my own way of doing metric system conversions. Use this method if you want or your own method. However, I think you will like this method. Part 1: Learn and MEMORIZE Metric Prefixes First thing you need to know is what the metric prefixes mean. Each prefix relates to a value of 10 to a certain power. You must know these. Here are the ones we are likely to see on our AP exam: Tera- (T) = Giga- (G) = 10 9 Mega- (M) = 10 6 Kilo- (k) = 10 3 No Prefix = 10 0 Centi- (c) = 10-2 Milli- (m) = 10-3 Micro- (µ) = 10-6 Nano- (n) = 10-9 Pico- (p) = These prefixes will be attached to metric units. Here are some commonly used metric units (although there may be more than these): o Meter (unit of length) o Liter (unit of volume) o Gram (unit of mass) o Second (unit of time) o Watt (unit of power) o Joule (unit of work/energy) o Volt (unit of voltage) o Byte (unit of data storage)

9 Part 2: Simple Metric Conversion Metric system conversion involves different powers of 10. This means, that in order to convert units in the metric system, you will either divide or multiply by powers of 10. This is good news for you, because dividing or multiplying by powers of 10 is the same as moving the decimal point of a number. So, when doing metric conversions, there are only 2 things you need to do. They are: o Determine how many places you will move the decimal point of your number. o Determine if you will move the decimal point right or left. This is how I determine these things. Here is an example: o Sample Problem: Convert 7500 megawatts to watts 1) You name the powers of ten for each unit in the problem. Convert 7500 megawatts to watts (no prefix on watts) ) Figure out how many spaces on a number line would you have to shift to get from the value of the one power to the value of the other. Convert 7500 megawatts to watts (no prefix on watts) (to get from 6 to 0 on a number line, you would move 6 spots) 3) This amount you would shift is the amount of places you will shift the decimal place in your original number to get your newly converted answer. We will ultimately shift the decimal point 6 places in this problem. 4) Now go back to the problem and select the power that has the largest value. Draw an arrow from that larger power to the small power with the arrow starting at the larger and going to the smaller. Convert 7500 megawatts to watts (no prefix on watts) (6 is larger that 0. So I start the arrow at 6 and draw to 0) 5) Now you know which way to move the decimal point. You will shift to the right.

10 6) So, now you have all the information you need. You will shift right 6 places. Convert 7500 megawatts to watts (no prefix on watts) converts to ) Your final answer is 7,500,000,000 watts Another Example: Convert 600 mg to kg places 600. converts to after shifting 6 places to the right Final answer: kg PRACTICE PROBLEMS: Please convert these quantities. For this, please show the set up of work. 1) Convert 895 centiliters into liters 2) Convert 4.66 megabytes into kilobytes 3) Convert meters into millimeters 4) Convert 85 TW into GW

11 5) Convert 810 ng into g 6) Convert kj into MJ Part 3: More Complex Metric Conversion (Squared and Cubic Units!) Sometimes, problems will ask you to convert squared metric units into other squared metric units (in cases where you are talking about areas). Or you might have to convert cubed metric units into other cubed metric units (in cases you are talking about volumes). These require an extra step to the conversion to ensure you have the correct answer! Here is a sample. o Sample Problem: Convert 7500 m 2 to Mm 2 1) Do the normal conversion process from before. Mainly, determine the number of places to shift and which direction to shift. Assume for this stage that the cubed on the unit does not exist. Convert 7500 m 2 to Mm places 2) Normally, you would shift the decimal to the left 6 places. However, there is one more step for dealing with squared or cubic units. You must multiply the number of shift places by the value of the exponent on the unit. In this case, we have a squared unit. So we shift 6 places times 2 (for the squared) for a total of 12 places! Convert 7500 m 2 to Mm places x 2 (for square units) = 12 places

12 3) So now shift places to the left! 4) Your final answer is Mm *** NOTE if you a cubic unit, multiply your shift value by three, since a cubic is something raised to the third power! PRACTICE PROBLEMS: Convert these values. Show set up and work!! 7) Convert km 2 to m 2 8) Convert 35 cm 3 to m 3 9) Convert 0.09 mm 3 to cm 3 10) Convert 225 cm 2 to km 2 11) Convert Gm 3 to km 3

13 MODULE 5 COMPUTING VOLUME Date When Due: In terms of water measurements (and some land based measurements), you will need to know how to compute how much volume a certain amount of water or chunk of land takes up. Volume measures the 3 dimensional space that something fills up. Volume is measured in cubic units. So, some examples of volume measurements in the metric system would be: m 3, cm 3, km 3, mm 3, and many others. Notice that all the examples have a cubed exponent, meaning that this is measuring volume! Here is an example of a typical problem that could be asked of you in regards to computing volume: The Gulf of Mexico has an average depth of 1500 m. It has an estimated surface area of 1,500,000 km 2. What is the volume of the Gulf of Mexico? Express answer in km 3, m 3, and L. (Note that 1000 L = 1 m 3 ) So you need to solve for volume! Here is the general equation for solving the volume of anything: VOLUME = AREA x DEPTH/HEIGHT So, all you need is a measure of the area and a measure of depth or height. Multiply together. HOWEVER, the area units and the depth/height units MUST be in the same size. This means they must both possess the same type of prefix. For example, if the area is expressed in km 2, then the depth/height must be in km. They must both be scaled with the prefix of kilo-. Or they could both be expressed in regular meters. Whatever scale you choose, you must keep it consistent before multiplying! Remember: Anything with a square unit = AREA measurement Anything with no exponent with the unit = DEPTH/HEIGHT measurement Anything with a cube unit = VOLUME measurement Let s solve this example: 1) First, define the information you have been given: 2) It appears that my area measurement is rated in the kilo- prefix, while my depth is rated in normal meters (no prefix). I CANNOT JUST MULTIPLY THEM TOGETHER! I must change one of them to make the prefixes consistent. 3) I will change the depth from 1500 m to km. Here is the work for that:

14 4) Now I have area in km 2 and my depth in km. Now I can multiply. The result is volume. And what unit is volume expressed in? Look to see! 5) So I have just solved the problem and computed the volume in km 3. But I also need to compute the number of m 3. This is simple. I can use simple metric system conversion. Look: 6) Now that I did this, I now have the volume in km 3 and m 3. The last requirement of the problem was to compute the volume in L. I was told that 1000 L = 1 m 3. Use unit conversion and convert the m 3 to L. Here is how I solved it: 7) That s it. Your official answers are: Remember, to compute volume, multiply the area and depth together. BUT make sure all the units are the same size before you do that!

15 PRACTICE PROBLEMS Show all work!!!! 1) Hennepin County has done an audit of how much concrete covers the land. This is often called impervious surface, since it prevents the percolation of water, which means more runoff will occur. There was 350,000,000 m 2 of impervious surface in the county. Each year, 80 cm of precipitation falls on this impervious surface. Compute the volume of water that is unable to percolate in Hennepin County on its impervious surfaces. Please compute this in km 3, m 3, and L. (1 m 3 = 1000 L) 2) The area of land that the Dust Bowl impacted in the US was 400,000 km 2. The average topsoil erosion on this land was 120 mm. Compute the volume of soil lost during the Dust Bowl from American lands. Provide the answer in km 3 and m 3.

16 3) A mining company needs to remove the topsoil of a plot of land to get to gold deposits below. The company removes 12 m of topsoil across the plot. The plot is 900 km 2. Compute the volume of topsoil that is removed over the whole plot and must be replaced after mining is complete. Provide your answer in km 3 and m 3. 4) The Great Barrier Reef is struggling to survive due to climate change and ocean acidification. The size of the reef was 400,000 km 2. Under non-stressed conditions, the coral grows 3 mm/year. Compute the total volume of new coral that should be grown in the reef per year when not stressed. Place final answer in km 3 and m 3.

17 MORE PRACTICE PROBLEMS!!!!! Module 1: Unit Conversion Problem Practice 1) Module 2: Percent Change Module 3: Ratios Module 4: Metric System Conversion Module 5: Volume Calculations