Math 6, Unit 9 Notes: Measurement and Geometry Customary and Metric Units of Measure Objective: (6.3)The student will estimate corresponding units of measure between customary and metric systems for temperature, length, and weight/mass. (6.4)The student will compare corresponding units of measure between customary and metric systems for temperature, length, and weight/mass. The customary system is the measurement system we use in the United States. The metric system is used almost everywhere else in the world. Measurement the metric system in particular is embedded in the science program at the middle school level. Measurement is an objective is to be addressed repeatedly throughout the science course. Be sure to collaborate with your science department! If you do not have a measurement tool, like a ruler, measuring cup or a scale, it is good to have a benchmark or estimate that you can use. It will also help you to choose the appropriate measurement when asked to measure an object, as well as make a comparison between the two systems of measurement. You may want to make an exercise having students determine their own benchmarks. Customary Units: LENGTH Unit Abbr. Benchmark inch in width of your thumb foot ft spread fingers-touch thumbs** yard yd length of baseball bat mile mi 12 city blocks or 20 football fields ** Metric Units: LENGTH Unit Abbr. Benchmark millimeter mm thickness of a CD (or dime) centimeter cm width of your pinky finger meter m about the distance from the floor to your belly button (for an average 6 th grader) kilometer km about half a mile Customary Units: WEIGHT/MASS Unit Abbr. Benchmark ounce oz a slice of bread pound lb a loaf of bread ton T a small car Metric Units: WEIGHT/MASS Unit Abbr. Benchmark gram g large paper clip kilogram kg hardcover textbook Math 6 Notes Unit 9: Measurement and Geometry Page 1 of 12
Customary Units: CAPACITY Unit Abbr. Benchmark fluid ounce fl oz a tablespoonful quart qt large bottle of water gallon gal large plastic milk jug Metric Units: CAPACITY Unit Abbr. Benchmark gram g large paper clip kilogram kg hardcover textbook Listed below are a few websites with more information: http://www.aaamath.com/mea.html http://www.nist.gov/public_affairs/kids/metric.htm http://www.edhelper.com/metric_system.htm Precision Objectives: (6.1)The student will be able to select the more precise unit given two measures. (6.2)The student will explain how the size of the unit measure used effects precision. We often use numbers that are not exact. Measurements are approximate there is no such thing as a perfect measurement. The precision of a number refers to its exactness to the level of detail to which the tool can measure. Measurements cannot be more precise than the measuring tool. This is very important in science! Example: To what degree of precision can you measure a length using this ruler? To the nearest 1 of an inch 8 To the nearest mm The smaller the unit of measurement, the more precise the measure. Consider some measures of time, such as 15 seconds and 15 hours. A measure of 15 seconds implies it is precise to the nearest second, or a time interval between 14.5 and 15.5 seconds. The time of 15 hours is far less precise: it suggests a time between 14.5 and 15.5 hours. The potential error in the first interval is 0.5 seconds; the potential error in the 15 hours scenario is 0.5 hours or 1800 seconds. Because the potential for error is greater, the 15-hour-measure is less precise. Math 6 Notes Unit 9: Measurement and Geometry Page 2 of 12
Example: Choose the more precise measurement in the given pair. (a) 3 m, 35 km (b) 12 inches, 1 foot (c) 1 pound, 1 ounce 3 m is more precise (meters are smaller than km) 12 inches (inches are smaller than a foot) 1 ounce (an ounce is smaller than a pound) The number of decimal places in a measurement can also affect precision. Using time again, a measure of 5.1 seconds is more precise than 5 seconds. The 5.1 measurement implies a measure precise to the nearest tenth of a second. The potential error in 5.1 seconds is 0.05 seconds, compared to the potential error of 0.5 seconds with the measure of 5 seconds. Example: Choose the more precise measurement in the given pair. (a) 5.4 m, 5.67 m (b) 3 yards, 3.6 yards 5.67 is more precise (hundredths of a meter smaller than tenths) 3.6 is more precise (tenths of a yard more precise than yards) Have students actually measure items. They can reinforce this idea of precision by measuring objects to the nearest inch, then nearest one-half inch and finally one-fourth inch; measuring to the nearest cm, then nearest mm. CRT Example: CRT Example: Math 6 Notes Unit 9: Measurement and Geometry Page 3 of 12
Converting Units: Customary (U.S) System As a review: a proportion shows that two ratios are equivalent. Using proportions is an easy way to convert units of measure. Customary (U.S) Measurements Length Weight/Mass Capacity 1 foot = 12 inches 1 pound = 16 ounces 1 cup = 8 fluid ounces 1 yard = 3 feet 1 ton = 200 pounds 1 pints = 2 cups 1 mile = 5,280 feet 1 quart = 2 pints 1 gallon = 4 quarts Example: Convert 36 quarts to gallons. We know that 1 gallon = 4 quarts. We can set up the proportion quarts/gallons or gallons/quarts. We will choose quarts/gallons. 36 quarts 4 quarts = x gallons 1 gallon ( )( ) 36 1 = x 4 36 = 4x 9 = x Therefore, 36 quarts = 9 gallons Example: Convert 10 gallons to quarts. We know that 1 gallon = 4 quarts. 10 gallons 1 gallon = x quarts 4 quarts 10 4 = x 1 40 = x Therefore, 10 gallons = 40 quarts Example: Convert 90 inches to yards. 90 inches 3 feet ; x yards 1 yard if we do not know how many inches there are in 1 yard, we need to determine that first. 3 feet 1 foot = x inches 12 inches 3 12 = x 1 36 = x There are 36 inches in one yard. 90 inches 36 inches = x yards 1 yard 90 1 = x 36 90 = 36x 2.5 = x Therefore, 90 inches = 2.5 yards Math 6 Notes Unit 9: Measurement and Geometry Page 4 of 12
Another way to convert units is to use what is called conversions factors and unit analysis. Let s look at the same example given above, converting 90 inches to yards. We need to consider the units of inches, feet and yards. We can set up conversion factors that are equal to 1, ie. 1 ft 1 yd and 12 in 3 ft We arrange them this way so that the labels of ft cancel. So our problem can be written: 90 in 1 1 90 ft yd = yd or 2.5 yards 1 12 in 3 ft 36 Converting Units: Metric System You can use both conversion factors and proportions to convert metric units. However, since the system is based on factors of 10, you are multiplying or dividing by powers of 10. All we need to know is how to move the decimal point what direction and how many places. We need to start with the meaning of the metric prefixes. Again, note that to move from one unit to another is simply multiplying or dividing by 10. kilo- 1000 hecto- 100 deka- or deca- 10 base or unit (meter, liter, gram) 1 deci-.1 centi-.01 milli-.001 A way to remember the order of these units is to think: King Henry Doesn't (Usually) Drink Chocolate Milk Or King Henry Died (By) Drinking Chocolate Milk where the first letter matches the first letter of the prefix, and the U refers to the unit or the B refers to the base (meter, liter or gram). Math 6 Notes Unit 9: Measurement and Geometry Page 5 of 12
If you memorize them in this order, you will know the answer to our questions of how to move the decimal point. For example, let s convert 24 hectometers to centimeters. We would list: km, hm, dkm, m, dm, cm and mm. Then we need to determine, how many jumps I would make from hectometers to get to centimeters. km hm dkm m dm cm mm I would jump four places, to the right. I will move my decimal that way, filling in zeros as place holders: 24 24.0 24.0 240000 Therefore, 24 hm = 24,000 cm. Example: Convert 54,653 m to km. Let s determine the number of jumps and the direction to move. km hm dkm m dm cm mm We need to move the decimal 3 places to the left. 54,653 54653.0 5 4 6 5 3. 0 54.653 Therefore, 54,653 m = 54.653 km. CRT Example: CRT Example: Math 6 Notes Unit 9: Measurement and Geometry Page 6 of 12
Time and Temperature Students must be able to convert one unit of time to another. Information students should know: Time 1 year = 365 days 1 day = 24 hours 1 year = 12 months 1 hour = 60 minutes 1 year = 52 weeks 1 minute = 60 seconds 1 week = 7 days Example: Convert 84 hours to days. 84 hours 1 day 84 hours = = 3.5 days 1 24 hours 24 Example: Convert 8100 seconds to hours. 8100 seconds 1 minute 1 hour 8100 hours = = 2.25 hour 1 60 seconds 60 minutes 3600 CRT Example: Scales to measure temperature are Celsius and Fahrenheit. Not only should you learn to use the formulas, but also learn to estimate the conversion. For instance, to estimate a conversion from C to F, you would double the degrees C and add 30. To estimate a conversion from F to C, you would subtract 30 and take one-half. Math 6 Notes Unit 9: Measurement and Geometry Page 7 of 12
Formulas Estimation Temperature Conversion 9 5 F = C+ 32 C = F 32 5 9 1 F = 2C+ 30 C = F 30 2 ( ) ( ) Example: Estimate 25 C in F; then use the formula. Estimation: F F ( )( ) 2 25 + 30 80 F F 9 = ( 25 ) + 32 5 = 77 Formula: Example: Estimate 40 F in C; then use the formula. Estimation: C 1 40 30 2 C 5 F ( ) Formula: 5 C = 40 32 9 C = 4.4 F ( ) Perimeter Objectives: (5.5)The student will model formulas to find the perimeter of plane figures. (5.6)The student will apply formulas to find the perimeter of plane figures. The perimeter of a polygon is the sum of the lengths of the segments that make up the sides of the polygon. 2 m Example: Find the perimeter of the regular pentagon. Since the pentagon is regular, we know all five sides have a measurement of 2 meters. So we simply multiply 5 2for an answer of 10 meters for the perimeter. Example: Find the perimeter for a rectangle with length 7 feet and width 2 feet. Math 6 Notes Unit 9: Measurement and Geometry Page 8 of 12
Since opposite sides of a rectangle are equal in length, P = 27 ( ) + 22 ( ) or 18 feet. Example: Find the length of the missing side if you know the perimeter is 67 meters. Perimeter is equal to the sum of the sides. 67 = 13 + 15 + 13 + k 67 = k + 41 26 = k The length of the missing side is 26 meters. k 13 m 13 m 15 m Circles: Circumference Objectives: (5.5)The student will model formulas to find the circumference of plane figures. (5.6)The student will apply formulas to find the circumference of plane figures. (6.11)The student will use the concept of central angles to represent a fraction of a circle. A circle is defined as all points in a plane that are equal distance (called the radius) from a fixed point (called the center of the circle). The distance across the circle, through the center, is called the diameter. Therefore, a diameter is twice the length of the radius, or d = 2r. We called the distance around a polygon the perimeter. The distance around a circle is called the circumference. There is a special relationship between the circumference and the diameter of a circle. Let s get a visual to approximate that relationship. Take a can with 3 tennis balls in it. Wrap a string around the can to approximate the circumference of a tennis ball. Then compare that measurement with the height of the can (which represents three diameters). You will discover that the circumference of the can is greater than the three diameters (height of the can). You can make an exercise for students to discover an approximation for this circumference/diameter relationship which we call π. Have students take several circular objects, measure the circumference (C) and the diameter (d). Have students determine C d for each object; have groups average their results. Again, they should arrive at answers a little bigger than 3. This should help convince students that this ratio will be the same for every circle. C We can then introduce that π d = or C = πd. Since d = 2r, we can also write C = 2π r. Please note that π is an irrational number (never ends or repeats). Mathematicians use to represent the exact value of the circumference/diameter ratio. Math 6 Notes Unit 9: Measurement and Geometry Page 9 of 12
Example: If a circle has a diameter of 4 m, what is the circumference? Use 3.14 to approximate π. State your answer to the nearest 0.1 meter. Using the formula: C = πd C (3.14)(4) C 12.56 The circumference is about 12.6 meters. Many standardized tests (including the CRT and the district common exams) ask students to leave their answers in terms of π. Be sure to practice this! Example: If a circle has a radius of 5 feet, find its circumference. Do not use an approximation for π. C = 2πr Using the formula: C = 2π 5 The circumference is about 10π feet. C = 10π CRT Example: Example: A circle has a diameter of 24 m. Using π 3.14, find the circumference. Round your answer to the nearest whole number. Using the formula: C = πd ( ) C (3.14) 24 C 75.36 The circumference is about 75 meters. Constructing a circle graph will show how parts relate to a whole. We need to think of the circle as representing the whole or 100%. Example: Make a circle graph to represent the results of a survey regarding favorite color. Math 6 Notes Unit 9: Measurement and Geometry Page 10 of 12
Favorite Color Color Number of Votes Blue 140 Green 105 Red 84 Orange 21 1. Find the total number of students surveyed: 140 + 105 + 84 + 21 = 350. 2. Find the % of the total (whole) represented by each group of students. Color Number of Votes % Blue 140 140 40% 350 = Green 105 105 30% 350 = Red 84 84 24% 350 = Orange 21 21 6% 350 = 3. There are 360 in a circle. We can determine the measure of the angles we need to create by multiplying 360 by each %. Color Number of Votes % % of 360.40 360 = 144 Blue 140 40% ( )( ) Green 105 30% (.30)( 360 ) = 108 Red 84 24% (.24)( 360 ) = 86.4 86 Orange 21 6% (.06)( 360 ) = 21.6 22 4. Use a compass and construct a circle. Construct a radius. Math 6 Notes Unit 9: Measurement and Geometry Page 11 of 12
5. Use a protractor to draw an angle measuring 144 : make the vertex of the angle at the center of the circle, one side the radius you already drew, and the other side another radius, forming the 144 angle. This angle is called a central angle. 144 6. Repeat for the other sections. Label. Favorite Color blue orange red green Math 6 Notes Unit 9: Measurement and Geometry Page 12 of 12