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MEASUREMENT

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Time - Morning = ame.g. 6:30 am - Evening = pme.g. 2:45 pm e.g. Add 2 ½ hours to 7:55 pm 7:55 + 2 hours =9:55 pm 9:55 + 30 min = Useful to add hours and minutes separately 24 Hour Clock - Morning = 0001 – 1159 - Evening = 1201 – 2359 1200 = midday 2400/0000 = midnight e.g. Change to 24 hour time a) 11:15 amb) 4:15 pm 10:25 pm 11154:15 + 12 hours =1615 If past midday, add 12 hours to the time e.g. Change to 12 hour time a) 1020b) 195010:20 am1950 - 12 hours =7:50 pm For times 1300 up to 2359, subtract 12 hours and make time pm For times 0000 up to 0059, add 12 hours and make time am For am times, leave unchanged. For times 12:00 - 12:59 am subtract 12 hours

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Money - Dollars = number before the decimal point - Cents = number after the decimal point e.g. $2.76 Rounding Money - For answers dealing in money, always leave answers rounded to 2 d.p. e.g. Leave $2.76 as is, DO NOT round it up to $2.80 Scales (Uniform) - Make sure you know what even division represents e.g. What value does each letter represent? a)b) Each gap= (2 – 1) ÷ 5 = 0.2 A = 2 - 0.2 = 1.8 Each gap= (20 – 10) ÷ 4 = 2.5 B = 10 + 2.5 = 12.5

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Scales (Non-Uniform) - Scales where the gaps are not equal e.g. Radio Frequencies Temperature - 0°C is freezing - 100°C is boiling For WATER e.g. Overnight the temperature drops 9°C from 7°C. What is the new temperature? - Everyday unit is generally degrees Celsius (°C) A good grasp of Integers is important when dealing with temperature! New temperature= 7 – 9 = -2 °C

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Use of Lengths - Millimetres(mm)Very accurate measurements e.g. Width of toenail - Centimetres(cm)Small object measurements e.g. Student heights - Metres(m)Buildings, sports etc e.g. Length of a Basketball court - Kilometres(km)Distances e.g. Distance of Hamilton to Cambridge Length Conversions mmcmmkm ÷10÷100÷1000 To convert to bigger units, we divide To convert to smaller units, we multiply ×10×100×1000 WHEN GIVING AN TO A MEASUREMENT QUESTION, ALWAYS STATE THE UNIT! base unit

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e.g. Convert a) 45 mm to cmb) 8 cm to mm g) 850 mm to mh) 43 m to mm e) 120 cm to mf) 1.82 m to cm c) 3.8 km to md) 1600 m to km = 45 ÷ 10 = 4.5 cm = 8 ×10 = 80 mm = 850 ÷ 10 ÷ 100 = 0.85 m = 43 ×100 ×10 = 43000 mm = 120 ÷ 100 = 1.2 m = 1.82 ×100 = 182 cm = 3.8 × 1000 = 3800 m = 1600 ÷ 1000 = 1.6 km - When performing calculations involving lengths, first convert all measurements to the same unit e.g.a) 0.52 m + 360 cm b) 2.6 cm – 17 mm 52 cm + 360 cm = 412 cm or 0.52 m + 3.6 m = 4.12 m 26 mm – 17 mm = 9 mm or 2.6 cm – 1.7 cm = 0.9 cm e.g. If Paula swims 80 lengths of 50 m each, how many km does she swim? 80 × 50 = 4000 m4000 ÷ 1000 = 4 km

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Scale Diagrams - Drawings representing real life situations - We use a scale to determine real life sizes of a drawing e.g. If a map has a scale of 1:200000, how much would 4 cm on the map equate to in real life? 4 × 200000 = 800000 cm800000 ÷ 100 ÷ 1000 = 8 km Speed - How fast an object is travelling SPEED = DISTANCE ÷ TIME e.g. What is the speed of a bus travelling 300 km in 4 hours? 300 ÷ 4 = 75km hr -1 DISTANCE = SPEED × TIME e.g. How far does Jenny walk if she walks at a speed of 4 km hr -1 for 2 hours? 4 × 2 = 8km TIME = DISTANCE ÷ SPEED e.g. Paul cycles 80 km at a speed of 32 km hr -1. How long does he bike for? 80 ÷ 32 = 2.5hours S D T

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Use of Weights - Milligrams(mg)Very accurate measuring e.g. Weight of an eyelash - Grams(g)Accurate measuring e.g. Weights of cooking ingredients - Kilograms(kg)People, objects that can be carried e.g. Student weights - Tonnes(t)Very heavy objects e.g. Shipping containers, elephants Weight Conversions mggkgt ÷1000 ×1000 base unit

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e.g. Convert a) 6000 mg to gb) 8.5 g to mg e) 1200 kg to tf) 9.6 t to kg c) 3500 g to kgd) 4 kg to g = 6000 ÷ 1000 = 6 g = 8.5 ×1000 = 8500 mg = 1200 ÷ 1000 = 1.2 t = 9.6 ×1000 = 9600 kg = 3500 ÷ 1000 = 3.5 kg = 4 × 1000 = 4000 g - When performing calculations involving weights, first convert all measurements to the same unit e.g.a) 6.42 kg + 320 g b) 0.45 t – 120 kg 6420 g + 320 g = 6740 g or 6.42 kg + 0.32 kg = 6.74 kg 450 kg – 120 kg = 330 kg or 0.45 t – 0.12 t = 0.33 t e.g. A bookshop posts 5 books, each weighing 850 g. What is the total weight in kg? 850 × 5 = 4250 g4250 ÷ 1000 = 4.25 kg

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Liquid Volume (Capacity) Conversions mLL ÷1000 ×1000 base unit e.g. Convert a) 200 mL to L b) 1.5 L to mL = 200 ÷ 1000 = 0.2 L = 1.5 ×1000 = 1500 mL - When performing calculations involving capacity, first convert all measurements to the same unit e.g.a) 260 mL + 1.2 L b) 2.8 L – 1430 mL 260 mL + 1200 mL = 1460 mL or 0.26 L + 1.2 L = 1.46 L = 2800 mL – 1430 mL = 1370 mL or 2.8 L – 1.43 L = 1.37 L e.g. 200 mL is poured from a 1 L container. How much is left in the container? 1000 – 200 = 800 mLor 1 – 0.2 = 0.8 L

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Prefixes - The prefix (first letter if there are two) of a unit, gives the size - The second letter gives the base unit of what you are measuring m c k = milli= 1. 1000 e.g. mm, mg, mL = centi= 1. 100 e.g. cm = kilo= 1000×e.g. km, kg

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Perimeter - The total distance around an object (total length of ALL its sides) e.g. Calculate the perimeter of the following: a)b) Always add in missing lengths 7 cm Perimeter = = 22cm ALWAYS remember to add in the UNIT 10 m 8 m Perimeter = = 36 m Area - Uses squared units such as cm 2 and m 2 - Can be estimated by counting the squares of a grid e.g. Area =cm 2 ALWAYS remember to add in the UNIT 7 + 7 + 88 + 10 + 8 + 10 9

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Squares and Rectangles - Area = length × width (A = l × w) e.g. Calculate the following areas: a)b) 9 cm Area = = 81 ALWAYS remember to add in the UNIT cm 2 Area = = 18 m 2 9 × 9 6 × 3

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ALWAYS remember to add in the UNIT Triangles - Area = ½ × base × height (A = ½ × b × h) e.g. Calculate the following areas: a)b) Area = = 35cm 2 Area = = 20 m 2 ALWAYS use the VERTICAL height ½ × 10 × 7½ × 8 × 5

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Parallelogram - Both pairs of opposite sides are parallel and equal in length - Area = base × height (A = b × h) e.g. Calculate the following areas: a)b) ALWAYS remember to add in the UNIT Area = = 20cm 2 5 × 4Area = = 21 m 2 ALWAYS use the VERTICAL height 6 × 3.5

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Trapezium - One pair of opposite sides are parallel - Area = height × average of parallel sides -Area = h × (a + b) 2 e.g. Calculate the following area: ALWAYS remember to add in the UNIT Area = = 45m2m2 5 × (6 + 12) 2 = 5 × 9

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Compound Areas - Complex shapes made up of 2 or more regular shapes - Areas can be calculated in 2 ways 1. By adding areas e.g. Calculate the following area: Area 1 = = 64 8 × 8 12 Area 2 =½ × (11 – 8) × 8 = 12 Total Area = Area 1 + Area 2 = 64 + 12 = 76 ALWAYS remember to add in the UNIT cm 2 1. By subtracting areas e.g. Calculate the following area: 1 Area 1 = = 45 4 × 4Area 2 =½ × 10 × 9 = 16 Total Area = Area 1 – Area 2 = 45 – 16 = 29cm 2 2

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Land Areas - 1 Hectare (ha) = 10,000 m 2 e.g. Calculate the area of the paddock in hectares: Area = = 75600m2m2 360 × 210 Area (in hectares) =75600 ÷ 10000 = 7.56ha Circles Centre Radius (r) Diameter (d) The diameter is the longest CHORD of a circle since it has to pass through the centre.

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Circumference (Perimeter of a Circle) - To calculate the circumference, use one of these two formula: 1. Circumference = π × diameter (d) 2. Circumference = 2 × π × radius (r) π (pi) is a special number 3.141.... Whose decimal part never repeats and is infinite in length e.g. Calculate the circumference of the following circles: a)b) d r Circumference =π × 8.2 = 25.76 ALWAYS remember to add in the UNIT cm (2 d.p.) Circumference =2 × π × 3.5 = 21.99m (2 d.p.)

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Area of a Circle - To calculate the area of a circle, use the following formula: - Area = π × r 2 Remember if you are given the diameter, you must halve it to find the radius. e.g. Calculate the area of the following circles: a)b)c) r Area =π × 2.5 2 = 19.63cm 2 (2 d.p.) ALWAYS remember to add in the UNIT d Radius =3 cm Area =π × 3 2 = 28.27cm 2 (2 d.p.) Area =½ × π × 6 2 = 56.55m 2 (2 d.p.) As we are dealing with a semi-circle, we multiply by ½

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Surface Area - Find area of each face and add them together e.g. Calculate the surface area of the following: a) b) Area of one face =6 × 6 = 36 Surface Area = 36 × 6 = 216 m2m2 Diameter = 20 m Height = 25 m ALWAYS remember to add in the UNIT Area of circles = 2 × π × 10 2 = 628.32 Curved area =π × 20 × 25 = 1570.80 Surface Area =628.32 + 1570.80 = 2199.12 m 2 (2 d.p.)

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Area Conversions - Square the unit conversion number when changing area units e.g. a) Convert 42 m 2 to cm 2 b) Convert 35 mm 2 to cm 2 = 42 × 100 2 = 420000 cm 2 = 35 ÷ 10 2 = 0.35 cm 2

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Volume - The amount of space an object take up - Measured using cubic units i.e. cm 3, mm 3 - Volume can be determined by counting 1 cm cubes e.g. The volume of the following shape made up of 1 cm cubes is? Volume =5 ALWAYS remember to add in the UNIT cm 3 Volume of Prisms - Prisms are 3D shapes with two identical and parallel end faces - Volume = end area × length (depth) e.g. Calculate the volume of the following shapes: a)b) Volume =5 × 3× 6 = 90m3m3 Volume =½ × 4 × 5× 5 = 50cm 3

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Volume of Pyramids - Volume = 1/3 × base (end) area × vertical height e.g. Calculate the volume of the following shapes: a)b) Volume = 1/3 × 4 × 5 × 6 = 40 cm 3 ALWAYS remember to add in the UNIT Volume = 1/3 × π × 9.85 2 × 11.65 = 1183.66 cm 3 (2d.p.) Composite Figures - Split into regular shapes and add/subtract volumes e.g. Calculate the volume of the following: Volume of sphere = 4/3 × π × r 3 Volume of hemisphere =½ × 4/3 × π × 2.5 3 = 32.72 Volume of cone =1/3 × π × 2.5 2 × 8.5 = 55.63 Total Volume = 32.72 + 55.63 = 88.35cm 2 (2 d.p.)

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Volume Conversions - Cube the unit conversion number when changing volume units e.g. a) Convert 0.65 m 3 to cm 3 b) Convert 965 mm 2 to cm 2 = 0.65 × 100 3 = 650000 cm 3 = 965 ÷ 10 3 = 0.965 cm 3

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Liquid Volume 1 cm 3 = 1 mL and 1000 cm 3 = 1 litre e.g. Change 600 cm 3 into litres: 600 cm 3 =600 mL 600 ÷ 1000= 0.6 L Remember: 1 L = 1000 mL e.g. How much water (in L) can fit into the following tank? Volume =40 × 20× 30 = 24000cm 3 1 cm 3 = 1 mL = 24000 mL 1 L = 1000 mL = 24000 ÷ 1000 = 24 L Everyday measures: 1 cup = 250 mL (water = 250g) 1 tablespoon = 15 mL (water = 15g) 1 teaspoon = 5 mL (water = 5g) e.g. If the tank weight 25 kg, how much will the tank, full of water weigh? 1 litre of water = 1 kg = 1000g (and 1 mL of water = 1 g) Weight of water = 24 kg Weight of water and tank = 49 kg

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G K Powers 2013 Cambridge University Press 8. Applications of perimeter, area and volume Study guide 1.

G K Powers 2013 Cambridge University Press 8. Applications of perimeter, area and volume Study guide 1.

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