How To Understand And Understand Fractions

Transcription

1 CARPENTRY MATH ASSESSMENT REVIEW This material is intended as a review. The following Learning Centres have more resources available to help you prepare for your assessment Nanaimo ABE Learning Centre: Building 05. Phone to confirm hours of operation. No booking is required. Cowichan Campus Learning Centre: Call to schedule an appointment. Powell River Campus Learning Centre: Call to confirm hours of operation. In order to work through this booklet, you need to know how to add, subtract, multiply and divide: Whole numbers Fractions Decimals Basic math review, is available at

2 The Imperial System of Measurement 1 Canada uses the metric system most of the time! However, there are still places and occasions where the imperial system of measurement is used. People often talk about their height in feet and inches or their weight in pounds. Many recipes measure in cups and teaspoons. Another example is the term two by four when talking about lumber. That term means that a plank is roughly two inches thick and four inches wide. Another place where the imperial system of measurement is often seen is in the grocery store, especially in the meat/fish and fresh produce sections. Prices and weights are often given in both metric and imperial units of measurement. For example, you may see a sign advertising Potatoes 89 a pound (lb.) or $1.96 per kilogram (kg). The USA uses only a system of measurement related to the imperial one, so items imported from there often do not have a metric equivalent given. Cookbooks frequently use one or the other system of measurement. For all these reasons, it is important to understand both systems and be able to convert one into the other. The most common imperial units of measurement are: Quantity Unit Symbol length foot ft. weight pound lb. volume gallon gal. Here are the most common conversions of imperial units of measurement: Length Weight Volume 1 foot (ft. ) = 1 inches (in.) 1 pound (lb.) = ounces (oz.) 1 pint (pt.) = cups 1 yard (yd.) = 3 feet 1 ton = 000 pounds (lbs.) 1 quart (qt.) = pints 1 mile (mi.) = 580 feet or 1760 yards 1 gallon (gal.) = 4 quarts When we convert units in the imperial system, we use a familiar rule: When we convert a larger unit to a smaller unit, we multiply by the conversion factor. When we convert a smaller unit to a larger unit, we divide by the conversion factor. Example 1 A two by four measures 6 feet 8 inches. How long is it in inches? Solution: As 1 foot = 1 inches, the conversion factor is 1. Since we are converting a larger unit (feet) to a smaller unit (inches), we multiply by 1. 6 feet = 6 x 1 = 7 inches. Then we need to add the 8 inches to the 7 inches. 6 feet 8 inches = = 80 inches Or we can write it in a shortened way, using quotation marks: represents feet and represent inches. So 6 8 = 80 Example How many pints or cups are in 1 gallon of milk? 1 gallon = 4 quarts and 1 quart = pints So 1 gal. = 4 quarts x the conversion factor of = 8 pints. 1 pint = cups so 8 pints x the conversion factor of = cups. VIU/CAP/Z:\Master Files\Math Reviews\Imperial System of Measurement.doc/Jan 6 010

3 Exercise: 1. Circle the larger of the two units: a) in. ft. b) mi. yd c) pt. cup d) lb. oz. e) qt. gal. f) ton lb.. Convert the following: a) 5 lbs = oz. b) mi. = yds. c) 106 ins. = d) 3 gals. = pints e) 60 ins. = f) 100 lbs. = tons g) 7 ft. = yds. h) 7 ft = ins. i) 3 yds. = j) 3 qts. = cups k) pts. = gals. l) 41 oz. = lbs oz. 3 Convert the following weights as shown: a) The baby weighs 10 oz. = lbs oz. b) Mary wants to lose 48 oz through her new diet = lbs oz. c) The Olympic weightlifter was able to lift a quarter of a ton = lbs. 4 Convert the following heights as shown: a) The doorway is 6 high = ins. b) Sally is 64 tall = c) Fred has grown to 77 = 5 Joe can reach 6 11 above the ground. He wants to make the basketball team so he has to be able to reach the rim of the basketball hoop. The hoop is 9 10 above the ground. How far will Joe have to leap to reach that rim? 6 Tim plans to make a birthday cake for his girlfriend. He needs half a pound of butter and one and half pounds of flour. How many ounces of each does he need? oz. butter and oz. flour. Answers: 1 a) ft. b) mi. c) pt. d) lb e) gal. f) ton a) 80 oz. b) 350 yds. c) 8 10 d) 4 pts. e) 5 0 f) 0.6 tons g) 9 yds. h) 84 ins. i) 36 j) 1 cups k) gals. l) lbs. 9 oz. 3 a) 7 lbs. 8 oz. b) 3 lbs. 0 oz. c) 500 lbs. 4 a) 74 ins. b) 5 4 c) 6 ft. 5 ins. 5 Joe needs to leap ft. 11 ins. 6 Tim needs 8 oz. of butter and 4 oz. of flour.

4 1 LINEAR MEASURES : THE METRIC SYSTEM The metric system is used in most countries of the world, and the United States is now making greater use of it as well. The metric system does not use inches, feet, pounds, and so on, although units for time and electricity are the same as those you use now. An advantage of the metric system is that it is easier to convert from one unit to another. That is because the metric system is based on the number 10. The basic unit of length is the metre. It is just over a yard. In fact, 1 metre 1.1yd. (comparative sizes are shown) 1 Metre 1 Yard The other units of length are multiples of the length of a metre: 10 times a metre, 100 times a metre, 1000 times a metre, and so on, or fractions of a metre: of a metre, of a metre, of a metre, and so on. Metric Units of Length 1 kilometre (km) = 1000 metres (m) 1 hectometre (hm) = 100 metres (m) 1 dekametre (dam) = 10 metres (m) 1 metre (m) 1 1 decimetre (dm) = metre (m) 10 dam and dm are not used much. 1 centimetre (cm) = 1 metre (m) millimetre (mm) = 1 metre (m) 10 You should memorize these names and abbreviations. Think of kilo- for 1000, hecto- for 100, and so on. We will use these prefixes when considering units of area, capacity, and mass (weight). THINKING METRIC To familiarize yourself with metric units, consider the following. 1 kilometre (1000 metres) is slightly more than 1 mile (0.6 mi). 1 metre is just over a yard (1.1 yd). 1 centimetre (0.01 metre) is a little more than the width of a paper-clip (about 0.4 inch). 1 cm VIU/CAP/Z:\ABE_LC\Master Files\Math Reviews\Linear Measures The Metric System.doc/April 8 010

5 1 inch is about.54 centimetres 1 millimetre is about the width of a dime The millimetre (mm) is used to measure small distances, especially in industry. mm 3 mm The centimetre (cm) is used for body dimensions and clothing sizes, mostly in places where inches were previously used. 10 cm (47. in.) 3 ft 11in. 53 cm (0.9 in.) 39 cm (15.3 in.) The metre (m) is used to measure larger objects (for example, the height of a building) and for shorter distances (for example, the length of a rug) 5 m (8 ft) 3.7 m (1 ft).8 m (9 ft) The kilometre (km) is used to measure longer distances, mostly in situations in which miles were previously used.

6 MENTAL CONVERSION AMONG METRIC UNITS 3 When you change from one unit to another you can move only the decimal point, because the metric system is based on 10. Look at the table below: Units km hm dam m dm cm mm Example: Complete: 8.4 mm = cm Think: To go from mm to cm will mean I will have fewer cm than mm because cm are larger than mm. So I move the decimal point one place to the left. 8.4 mm 0.84 so, 8.4 mm = 0.84 cm Example: Complete: km = cm Think: To go from km to cm means that there will be many more cm than there were km because cm are smaller than km. So I move the decimal place to the right 5 places km so, km = cm Example: Complete: 1 m = cm Think: To go from m to cm m are bigger, and cm smaller so, there will be more cm than I started with. I can move the decimal place to the right places. 1 m = 1.00 m 1.00 cm so, 1m = 100 cm Make metric conversions mentally as much as possible. The most commonly used units of metric measurement are: km m cm mm

7 4 PRACTICE : CONVERTING BETWEEN METRIC UNITS Complete. Do as much as possible mentally. Avoid using a calculator! 1. a) 1 km = m. a) 1 hm = m 3. a) 1 dam = m b) 1 m = km b) 1 m = hm b) 1 m = dam 4. a) 1 dm = m 5. a) 1 cm = m 6. a) 1 mm = m b) 1 m = dm b) 1 m = cm b) 1 m = mm km = m 8. 9 km = m cm = m cm = m m = km m = km m = km m = km m = cm. 435 m = cm cm = m mm = m m = cm m = dm 1. 1 mm = cm. 1 cm = km 3. 1 km = cm 4. km = cm ANSWERS 1. a) a) a) b) b) b) a) a) a) b) b) b)

8 USING IMPERIAL AND METRIC RULERS We have probably all heard the old saying: Measure twice; cut once. Using a ruler efficiently to measure materials and construction is an essential aspect of carpentry. In the previous sections, you have already studied both metric and imperial measurement. In this section, you will put that information into practical use for carpentry. Many trades use both imperial and metric systems of measurement, so you need to know how to read and use both types of rulers and tapes. Often both systems are on the same ruler / tape. This is convenient, but beware of reading the numbers for one system and using the units of measurement for the other. An Imperial ruler, usually 1 foot (ft or ) long, is divided into inches ( ) and parts of inches. An Imperial tape is similarly divided, but is much longer. Many Imperial measures of length divide inches into halves, quarters, eighths, sixteenths - and even thirty-secondths. Metric measures are divided into multiples of 10, starting with millimetres (mms) and centimetres (cms). A 1 metre (m) rule is divided into 100 centimetres or millimetres. On your carpentry assessment, you will not be given how many inches are in a foot, millimetres and centimetres in a metre, so you need to know those. Here is a quick review of past learning, plus an exercise, using that knowledge, with a ruler and a tape. Practice Exercise 1 How many of these fractions of an inch are there in one inch? a) quarters b) sixteenths c) halves d) eighths Find the answers to how many a) cms in 3 m b) mms in 55 cms c) mms in m 5 cms d) cms in 350 mms 3 In the boxes below, label the measurements shown on the ruler. Write both numbers and units of either inches (ins or ) or millimetres / centimetres (mms / cms), whichever is appropriate. You will notice that sometimes the arrow is not exactly on the line of measurement. It is as close as possible. a) b) c) d) e) f) g) h) VIU/CAP/Z: \Master Files\Math Reviews\Carpentry\Using a Ruler.doc/January 6, 010

9 P b) P d) e) Practice Exercise Answers 1 a) 4 b) c) d) 8 a) 300 cms b) 550 mms c) 050 mms d) 35 cms 3 a) = 1 9" b) = 14" = 7" 8 c) = 4 1" d) = 6 15" e) =.6 cms f) = 7 cms g) = 13. cms h) = 18.9 cms or 6 mms or 13 mms or 189 mms Exercise 1 Label this tape with the measurement points below. Use an arrow and the letter of the measurement point to show your accurate reading. inches mms and cms a) 3 1 " 11 cms c) 5 3 " 8 f).6 cms g) 1.5 cms h) 7 1 " 100 mms P i) 8 1" " j) 18 cm 5 mm Convert these fractions of inches into sixteenths. Use a ruler or tape to help you. a) 3" 8 b) 3" 4 c) 1" d) 1 1" 4 e) 7" 8 f) 1 5" 8 3 Convert the fractions below into equivalent fractions. You are given the denominators. All can be read as simple fractions [e.g. the questions below] or as fractions of an inch [e.g. the answers at the end]. If you are working with measurement, the units of inch/inches are often understood. You can see that metric measurement does not have fractions; it uses decimals to show parts. a) 7 = b) = = c) 1 1 = d) 5 = e) 10 = f) 3 = Follow this example to see how to add one measurement to another. Then do the problems below. e.g. a) Add 1" 3" 1" to 5. [Hint: change fractions into equivalent forms e.g. = 4".] 4 4

10 3 5 3" + + 4" = 7 7" (the arrow is as close as possible) (8.5 cms) + 5.5cms = (14 cms) b) Add 5.5 cms to 85 mms. (Change both measurements to the same unit before adding: 85 mms = 8.5 cms.) Now try these by following the same method and similar marking: c) Add 3 5 " to 4 1 " 8 d) Take / Subtract 4.5 cms from 1 mms. [Hint: what do you need to do with cms or mms?]

11 4 Exercise Answers: 1) Arrows are as close as possible. a) e) c) h) i) d) b) f) j) g) a) 6" b) 1" c) 8" d) 0" e) 14" f) 6" 3 a) 14" 3 b) 1" = 4" 4 c) 1" 8 d) 0" 3 e) 5" 8 f) 1" 4 c) 4 1" 8 5" +3 = 4 " + 3 5" = 7 7" d) 1 mms = 1. cms 4.5 cms =.7 cms or 1mms 45 mms = 7 mms

12 GEOMETRY Includes Perimeter of Polygons Perimeter of Circles (Circumference) Area of Polygons Area of Circles Note: This study pack is intended for students who have previously studied this material, but wish to review prior to assessment. VIU/CAP/Q:\Study Packages\Master Files\Math Reviews checked March \Geometry.doc/Jan 6 010

13 PERIMETER OF POLYGONS A polygon is a geometric figure with 3 or more sides. The perimeter of a polygon is the distance around the outside of the figure, or the sum of the length of each of its sides. Sometimes formulae are used in calculating the perimeter to make things easier. The most common formulae used are as follows: 1 Perimeter of a Rectangle or a Parallelogram: P = (l + w) or P = l + w Perimeter of a Square: P = 4 s Example 1: Find the perimeter of a rectangle that is 6 mm by 9 mm P = 6 mm + 6 mm + 9 mm + 9mm = 30 mm or P = l + w = (9 mm) + (6 mm) = 18 mm + 1 mm = 30 mm or P = (l + w) = (6 mm + 9 mm) = (15 mm) = 30 mm Example : Find the perimeter of a square with a side that is 10 cm long P = 10 cm + 10 cm + 10 cm + 10 cm = 40 cm or P = 4s = 4 (10 cm) = 40 cm Example 3: Find the perimeter of a parallelogram that has a length of 1m and a width of 5m P = (l + w) = (1 m + 5 m) = (17 m) = 34 m or P = / + w = (1 m) + (5 m) = 4 m + 10 m = 34 m or P = 5 m + 1 m + 5 m + 1 m = 34 m Example 4: Find the perimeter of a triangle that has the sides 3 mm, 6.5 mm, and 8.6 mm P = 3 mm mm mm = 18.1 mm

14 PRACTICE Find the perimeter of the following shapes: ANSWERS a) 30 m b) 17 cm c) 0 m d) 8 mm e) 54.8 in f) 44 cm g) 67. ft h) 3.4 m i) 48 ft j) 47 ft k) 1 km l) 66.6 ft m) cm n) 510 m

15 CIRCLE GEOMETRY CIRCUMFERENCE Circumference is the name for the perimeter (or distance around the outside) of a circle. In this circle, the centre is Z. A, B, and C are points on the circle. Radius: The distance from the centre of the circle to any point on the circle is called the radius (r). (ZA is a radius. ZB and ZC are too). Diameter: The distance from any point on the circle, passing through the centrepoint and continuing on to the outer edge of the circle (d). (AB is the diameter of the circle to the right.) 3 To find the circumference (or perimeter) of the circle, use one of the following formulae: (1) C = π d OR () C = π r π is called pi and is about 3.14 or 7 Example A: If the circle has a radius of 5 cm, then (1) C =π d () C = π r The diameter would be twice the radius (or 5 cm x = 10 cm) C = (3.14) (10 cm) C = 31.4 cm. The radius is 5 cm So C = () (3.14) (5) C = 31.4 cm Both formulae work equally well. You may choose either one. Now let's practice: 1. A circle has a diameter of 0 m. What is the circumference?. A circle has a radius of 7 km. What is the circumference? 3. Find the circumference for the following circles: a) radius(r) = 14 cm b) diameter (d) = 60 mm c) radius (r) = 15 m (use π = ) 7 ANSWERS 1) C = π d ; C = (3.14)(0); C = 6.8 m ) C = π r; C = ()(3.14)(7); C = km b) C = π d ; C = 3.14(60) = mm 3) a) C = π r; C = () (14) = 88 cm 7 c) C = π r ; C = ()(3.14)(15); C = 94. m AREA OF POLYGONS

16 The area of a polygon is the number of squares (of a particular unit) that it takes to cover the surface of the polygon. Formulae are used to calculate the area. The most common formulae are Area of a square = s 1 Area of a triangle = b h Area of a rectangle = l w Area of a parallelogram = b h or b h 4 Area of a trapezoid = 1 h (a + b) Example 1: Find the area of a triangle which has a base of 10 mm and a height of 9 mm. 1 b x h A = b x h A = 1 10 mm x 9 mm A = (10 mm x 9 mm) A = or 1 A = (90 mm 90 mm ) A = A = 45 mm A = 45 mm Example : Find the area of a square that has a side with a length of 6 cm. A = s A = (6 cm) A = 36 cm Example 3: Find the area of a rectangle that has a length of 3.7 m and a width of.4 m. A = l x w A = (3.7 m) x (.4 m) A = 8.88 m Example 4: Find the area of a parallelogram that has a base of 1 cm and a height of 13 cm. A = b x h A = (1 cm) x (13 cm) A = 73 cm Example 5: Find the area of the trapezoid below A = 1 h (a + b) A = 1 (9 m) (66 m + 63 m) 1 A = (9 m) (19 m) A = m *Notice that units in the answers are units (squared) PRACTICE Find the area of the following shapes:

17 a) b) c) 5 5 cm 10 cm 1.6 m 5 cm 3 cm d) e) f) 14 mm 1 cm 1. km 4 mm 11 cm 8.6 km g) h) i).3 ft 1.3 yd 10 ft 1 ft 4 ft Note: There are no diagrams for j) to q). j) A square 35 ft on a side. k) A parallelogram with height of 14 in. and base 3 in. l) A rectangle with length of 8.8 m and width of 4. m. m) A triangle with height of 9 km and base of 5. km. n) 3 A rectangle 4 7 mile by 8 mile. o) A triangle with a base of 8 3 yard and height of 1 yd. p) A parallelogram with base of 9 1 ft and height of ft. q) A trapezoid with one base of 6 m and one base of 4 m. The height is 8 m. ANSWERS a) 15 cm b) 5 cm c) m d) 336 mm e) 11 cm f) 5.46 km g) 5.9 ft h) 44 ft i) yd j) 15 ft k) 3 in l) m m) 3.4 km n) o) 15 yd or 0.47 yd p) 3 1 mi or 0.66 mi ft or ft q) 40 m 8

18 CIRCLE GEOMETRY - AREA To find the area of a circle, use the formula A =π r Where A = area of the circle: π = pi 3.14 or r = the radius of the circle 7 6 so A = π r A = 3.14 x (10) A = 3.14 x 100 A = 314 cm r = the radius of the circle = 10 cm *notice: answers are units Now let s practice: A: Find the circumference and the area of the following circles: (1) the radius = 4 km () the diameter = 10 m B: Find the circumference and the area of the following circles: (1) the radius = 14 in. () the diameter = 0 mm ANSWERS A: 1) C = π r; C = (3.14)(4): C = 5.1 km A = π r A = (3.14)( 4) = 50.4 km ) C= π d ; C = (3.14)(10); C = 31.4m A = π r A = (3.14)(5) = 78.5 m B: 3) C = π r; C = ( 3.14)( 14) = 87.9 in A = π r A = (3.14)(14) = in 4) C= π d ; C = (3.14)(0) = 6.8 mm A = π r A = (3.14)(10) = 314 mm

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21 1 THE PYTHAGOREAN THEOREM The Pythagorean Theorem is a mathematical rule which gives a quick and efficient method of finding the third side in any right-angled triangle. The name looks and sounds scary, but the use is straightforward and quick to learn. Pronounce the word like this [pie-thag-or-ee-an]. If you know the length of any two sides in any triangle that has a 90 angle, shown by a small square in that angle, then you can work out the length of the missing side. In your carpentry assessment, you will NOT be given this rule, so you need to know it well. Side A + Side B = Side C and Side B + Side A = Side C C so B Side C Side B = Side A Side C Side A = Side B and A as there is a 90 angle between sides A and B. The rule is The square of the hypotenuse is the sum of the squares of the other two sides. Can you tell which side is the [hi-pot-a-newss]? It is Side C opposite the right angle. Here is a diagram to prove that is true: 5 x 5 = 5 3 x 3 = 9 4 x 4 = If you square each side of a right-angled triangle, you can see what happens. The square of a side with length 5 has 5 squares. It is opposite the right angle so it is the Hypotenuse. The side with length 4 has squares; it is sometimes called the Altitude. The side with length 3 can be called the Base and has 9 squares. Their squares ( + 9) add up to 5 squares = the same as the number of squares off the hypotenuse. In the construction trades, other terms are often used: Hypotenuse = travel Altitude Base = rise = run VIU/CAP/Z:\Master Files\Math Reviews\Pythagorean Theorem.doc/Jan 6 010

22 The Method is a popular term for this rule, as it is an easy way to show how the rule works. However, you can see that other numbers can be used instead of 3, 4 and = 10 so = If you don t know the length of the long side opposite the right angle, you can work it out: = 100 = 10 so the length of the third side is the square root of 100 = 10. Practice Exercise 1 A B C If the line AB is 1 cms long and the line BC is 9 cms long, what is the length of the line AC? Ans. AB + BC = AC = AC = 5 cms so AC= 15 cms Practice Exercise X Z If the line XZ is 0 inches long and the line ZY is 5 inches long, how long is line XY? Ans. ZY - XZ = XY 5-0 = XY = 5 Y XY = 5 = 15 inches

23 3 Ex. 1: Find the length of the missing dimension in each of the right-angled triangles in the following problems, based on the information given: Hint: Always draw a diagram and label it with the information you are given and mark the unknown part as x or y or your favourite letter. 1. Base / Run = 9 cms and Altitude / Rise = 1 cms. What is the length of the Hypotenuse / Travel?. Two shorter sides of a right-angled triangle are 1 and. How long is the other side? 3. With an altitude of 15 metres and a hypotenuse of 5 metres, what is the distance of a gate from a house? 4. The hypotenuse is 15 meters. One side is 10 meters. How long is the third side? (Use a calculator. Don t worry if you do not find an even number.) 5. The base of a single stair is 9, while the hypotenuse from one stair tread to another is 15. What is the height (altitude / rise) of a single stair? 6. A triangular roof system needs to be constructed for a renovation. The length of the bottom of the triangle on one side of the roof is 1, while the rise of the rafter (the perpendicular dimension) is 5. How long should the rafter (the hypotenuse) be? Construction workers use the Pythagorean Theorem in many situations e.g. ensuring square corners in buildings, calculating measurements for stairs, laying out perimeters (outlines) of buildings, working out the length of rafters, etc. = you must know how to use it! Answers to Exercise: 1) 15 cms () 0 (3) 0 (4) m (5) 1 (6) 13