1 Concepts of Measurement CHAPTER 11

Slide 2 Chapter 11.1Linear Measure 11.2Areas of Polygons and Circles 11.3The Pythagorean Theorem and the Distance Formula 11.4Surface Areas 11.5Volume, Mass, and Temperature 11 Concepts of Measurement

Slide 3 NCTM Standard: Measurement All students will recognize that objects have attributes that are measurable explore length, weight, time, area, and volume (grades 3-5) learn about area, perimeter, volume, temperature, and angle measure learn both customary and metric systems know rough equivalences between the metric and customary systems.

Slide Linear Measurement  The English System  Dimensional Analysis (Unit Analysis)  The Metric System  Distance Properties  Distance Around a Plane Figure  Circumference of a Circle  Arc Length

Slide 5 The English System Unit Equivalent in Other Units yard (yd)3 feet foot (ft)12 inches mile (mi)1760 yd or 5280 ft

Slide 6 Dimensional Analysis (Unit Analysis) Dimensional Analysis: a process to convert from one unit of measurement to another. Works with unit ratios (ratios equivalent to 1)

Slide 7 Example: Which is faster, 50 miles per hour or 50 feet per second? Answer: 50 mph = 73.3 fps 73.3 fps > 50 fps Therefore, 50 mph is faster than 50 fps

Slide 8 Example: Convert: 219 ft = __________ yd Answer: 73

Slide 9 Example: Convert: 8432 yd = __________ mi Answer: 4.79

Slide 10 Example: Convert: 0.2 mi = __________ ft Answer: 1056

Slide 11 Example: Convert: 64 in = __________ yd Answer: 1.78

Slide 12 PrefixSymbolFactor kiloh1000 hectoh100 dekada10 decid0.1 centic0.01 millim0.001 The Metric System

Slide 13 UnitSymbol Relationship to Base Unit kilometerkm1000 m hectometerhm100 m dekameterdam10 m metermbase unit decimeterdm0.1 m centimetercm0.01 m millimetermm0.001 m Different units of length in the metric system are obtained by multiplying a power of 10 times the base unit.

Slide 14 Benchmarks for Metric Units can be used to estimate a meter, decimeter, centimeter, and a millimeter. Kilometer is commonly used for measuring longer distances: 1 km = 1000 m or nine football fields, including end zones, laid end to end.

Slide 15 Converting Metric Units: are accomplished by multiplying or dividing by power of 10. We simply move the decimal point to the left or right depending on the units.

Slide 16 Now Try This 11-2Page 740 If our money system used metric prefixes and the base unit was a dollar, give metric names to each of the following: a) dime b) penny c) $10 bill d) $100 bill e) $1000 bill a) dime - decidollar b) penny - centidollar c) $10 bill - dekadollar d) $100 bill - hectodollar e) $1000 bill - kilodollar Answer:

Slide 17 Example Convert: 278 km = _________ m 278,000 or move the decimal place 3 places to the right. Answer:

Slide 18 Example Convert: 278 m = _________ cm 2.78 or move the decimal place 2 places to the left. Answer:

Slide 19 Example Convert each of the following: 278 mm = ________m Answer: or move the decimal place to the left 3 places.

Slide 20 Distance Properties 1.The distance between any two points A and B is greater than or equal to 0, written (AB  0). 2.The distance between any two point A and B is the same as the distance between B and A, written (AB = BA). 3.For any three points, A, B, and C, the distance between A and B plus the distance between B and C is greater than or equal to the distance between A and C, written (AB + BC  AC).

Slide 21 Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC Can a triangle be made with sides that are 15 cm, 18 cm, and 37 cm? No: = is less than 37

Slide 22 Now Try This 11.3Page 742 If two sides of a triangle are 31 cm and 85 cm long and the measure of the third side must be a whole number of centimeters, a) What is the longest the third side can be? Answer: = 116 – 1 = 115 cm because (116 – 31) = 85 b) What is the shortest the third side can be? Answer: 85 – 31 = = 55 cm because ( = 86)

Slide 23 Distance Around a Plane Figure Perimeter – the length of a simple closed curve, or the sum of the lengths of the sides of a polygon. Perimeter has linear measure.

Slide 24 Example: How many feet of molding are needed to go around the entire room? Answer: = 47 feet (10 + 7) + (18 – 12) = = 11 feet = 58 feet

Slide 25 Circumference of a Circle Circle – the set of all points in a plane that are the same distance from a given point, the center. Circumference – the perimeter of a circle. Pi – (π) the ratio between the circumference of a circle and the length of its diameter. d = diameter r = radius p = π = pi C = circumference

Slide 26 Example Find: a.The circumference of a circle with radius 10 m. Answer: b.The radius of a circle with circumference 18π ft. Answer: C = πd or C = 2 π r C = (2)(π)(10) C = 20 π m C ≈ 62.8 m The length of the diameter (d) is twice the radius (r)

Slide 27 Arc Length r = radius p = π = pi q = 0°

Slide 28 Example Find: a.The length of a 36° arc of a circle with diameter 5 inches. Answer: b.The radius of an arc whose central angle is 54 ° and whose arc length is 15 cm. Answer: Radius is ½ the diameter

Slide 29 HOMEWORK 11-1 Pages 746 – 748 # 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27