MATH CHAPTER 10 EXAMPLES AND DEFINITIONS

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1 MATH CHAPTER 10 EXAMPLES AND DEFINITIONS Section 10.1 Systems of Measurement It can be helpful to have an everyday approximate equivalent to units of measure. These are called benchmarks. Some benchmarks for customary units: 1 inch diameter of a quarter, length of a small paperclip 2 inches width of a credit card 6 inches length of a dollar bill 1 yard height from floor to doorknob Volume is the amount of space occupied by an object. Capacity is the amount that can be contained by an object. Liquid volume: ounce, cup, pint, quart, gallon, barrel, hogshead 8 oz = 1 cup 2 c = 1 pint 2 pt = 1 quart 4 qt = 1 gallon 63 gal = I hogshead (beer) Dry volume: pint, quart, gallon, peck, bushel 8 oz = 1 cup 2 c = 1 pint 2 pt = 1 quart 4 qt = 1 gallon 2 (dry) gallons = 1 peck 4 pk = 1 bushel (bu) Cooking measures: 1 Tablespoon = 3 teaspoons 16 T = 1 cup Mass is a measure of the quantity of matter. Weight is a measure of how heavy something is. Weight is caused by the force of gravity pulling down upon an object. An object's weight depends on what planet or moon it's on (unlike mass, which is constant). Ounce, pound, stone, hundredweight, ton 16 oz = 1 pound (A pint s a pound, the world around) 14 lbs = 1 stone 2000 lbs = 1 ton Ex. A) Using appropriate units which is the best unit to describe the amount of water in a swimming pool? (a) cup (b) pint (c) quart (d) gallon a bicycle's length? (a) inches (b) feet (c) miles (d) yards weight of a whale? (a) ounces (b) pounds (c) tons (d) too heavy to weigh weight of a pencil? (a) ounces (b) pounds (c) tons (d) too light to weigh can of soda (liquid measure)? (a) pints (b) quarts (c) gallons (d) ounces 1

2 CUSTOMARY/METRIC CONVERSIONS KNOW: 1 inch 2.54 centimeters 1 mile 1.6 kilometers 1 quart 1 Liter 2.2 pounds 1 kilogram A few mnemonics for metric prefixes: kilometer hectometer dekameter meter decimeter centimeter millimeter King Henry Danced Merrily Down Center Main King Henry Died Monday Drinking Chocolate Milk kilo hecto deka meter/liter/gram deci centi milli Kathy Hall Drinks Milk/Lemonade/Gatorade During Class Mondays Incidental Note: The abbreviation K is now commonly used for 1000 of something dollars, meters, or bytes, for example. Some metric benchmarks: Length: millimeter thickness of a dime centimeter width of index fingernail meter doorknob to floor kilometer 9 football fields Volume: 1 cubic centimeter = 1 milliliter 1 teaspoon 5 milliliter 1000 ml = 1 L Weight/mass: 1 cubic centimeter of water masses 1 gram. 1 gram a paperclip 5 g a nickel Ex. B) Use dimensional analysis to convert each of the following: (a) 6 mm = (b) 125 cm = (c) 5 inches = cm m cm 2

3 Ex. C) Choose an appropriate unit of measure for each measurement. Use customary units, metric units, and (if possible) nonstandard units. (a) distance from home to a grocery store (b) weight of a refrigerator (c) water in a fish tank (d) height of a door (e) perfume in a bottle of perfume Temperature: C = 5 9 ( F! 32) F = C Ex. D) My Celsius thermometer says that I have a fever of 40 C. Should I be alarmed? Celsius temperature jingle: 30 is hot, 20 is nice, 10 is chilly, 0 is ice! ***Always remember to use units of measure! Practice Problems for Section How might you answer these questions from your students: what measurement tool would you use to find the (a) mass of a worm? (b) length of a bean? (c) circumference of your wrist? (d) volume of an apple? 2. Perform the following operations. Express your answer in centimeters. 4.2 m + 53 cm 2846 mm 3. A third grader measures her height in both centimeters and inches. Which measurement will have more units? Why? 4. It is a cold day. Which would be a greater increase in temperature, an increase of 10 F or an increase of 10 C? Why? 3

4 Section 10.2 Perimeter and Area The perimeter of a simple closed curve is the distance around the curve. The circumference of a circle is its perimeter. The ratio of the circumference of a circle to its diameter is always pi. C = 2πr Ex. E) Determine the circumference of a circle with a diameter measuring 5.8 centimeters. Use as an approximation for pi. Area is the measure of a region i.e. the measure of the interior of a closed curve. Ex. F) True/False counterexample: (a) Two figures that are congruent will have the same perimeter. (b) Two figures that have the same perimeter are congruent. (c) Two figures that are congruent will have the same area. (d) Two figures that have the same area are congruent. Areas of Circles, Triangles, and Quadrilaterals The altitude or height of a triangle is the perpendicular segment from any vertex to the line that contains the opposite side. Area Formulas 1. The area A of a circle that has radius r: A = πr The area A of a rectangle that has length l and width w: A = lw. 3. The area A of a triangle that has base b and height h: A = ½bh. 4. The area A of a parallelogram that has base b and height h: A = bh. 5. The area A of a trapezoid that has parallel sides of lengths b and c and height h: A = ½( b + c)h. The quantity ½( b + c) is referred to as the average base. 4

5 Ex. G) Find the area of a trapezoid with a base lengths of 5 inches and 9 inches and a height of 4 inches. The Pythagorean Theorem If a right triangle has legs of lengths a and b and a hypotenuse of length c, then c 2 = a 2 + b 2. Ex. H) A triangle has side lengths 8, 8, and 10 inches. Find the area of the triangle to the nearest tenth of an inch. Ex. I) The radius of the larger circle below is 12 units. The intersection point of the two smaller circles is the center of the larger circle. (a) Find the combined area of the two smaller circles in the figure below. (b) Find that part of the area of the outer circle not included within the two smaller circles. Give all answers as exact values (in other words, in terms of π). Section 10.2 Bonus Problem (2 points) A square has an area of 16 square meters. The square has a circle inscribed within it, and another circle circumscribing it. Find the areas of these two circles. Give your answer to the nearest hundredth of a meter. 5