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MEASUREMENT

MEASUREMENT. Time. - Morning = am. e.g. 6:30 am. - Evening = pm. e.g. 2:45 pm. e.g . Add 2 ½ hours to 7:55 pm. 7:55 + 2 hours =. 9:55 pm. Useful to add hours and minutes separately. 9:55 + 30 min =. 10:25 pm. 24 Hour Clock. - Morning = 0001 – 1159. 1200 = midday.

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MEASUREMENT

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  1. MEASUREMENT

  2. Time - Morning = am e.g. 6:30 am - Evening = pm e.g. 2:45 pm e.g. Add 2 ½ hours to 7:55 pm 7:55 + 2 hours = 9:55 pm Useful to add hours and minutes separately 9:55 + 30 min = 10:25 pm 24 Hour Clock - Morning = 0001 – 1159 1200 = midday - Evening = 1201 – 2359 2400/0000 = midnight e.g. Change to 24 hour time If past midday, add 12 hours to the time a) 11:15 am 1115 b) 4:15 pm 4:15 + 12 hours = 1615 For am times, leave unchanged. For times 12:00 - 12:59 am subtract 12 hours e.g. Change to 12 hour time a) 1020 10:20 am b) 1950 1950 - 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

  3. 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) A = 2 - 0.2 B = 10 + 2.5 = 1.8 = 12.5 Each gap = (2 – 1) ÷ 5 Each gap = (20 – 10) ÷ 4 = 0.2 = 2.5

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

  5. Use of Lengths - Millimetres (mm) Very accurate measurements e.g. Width of toenail WHEN GIVING AN TO A MEASUREMENT QUESTION, ALWAYS STATE THE UNIT! - 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 ÷10 ÷100 ÷1000 To convert to bigger units, we divide To convert to smaller units, we multiply base unit mm cm m km ×10 ×100 ×1000

  6. e.g. Convert a) 45 mm to cm = 45 ÷ 10 b) 8 cm to mm = 8 ×10 = 4.5 cm = 80 mm c) 3.8 km to m = 3.8 × 1000 d) 1600 m to km = 1600 ÷ 1000 = 3800 m = 1.6 km e) 120 cm to m = 120 ÷ 100 f) 1.82 m to cm = 1.82 ×100 = 1.2 m = 182 cm g) 850 mm to m = 850 ÷ 10 ÷ 100 h) 43 m to mm = 43 ×100 ×10 = 0.85 m = 43000 mm - 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 or 0.52 m + 3.6 m 26 mm – 17 mm or 2.6 cm – 1.7 cm = 412 cm = 4.12 m = 9 mm = 0.9 cm e.g. If Paula swims 80 lengths of 50 m each, how many km does she swim? 80 × 50 = 4000 m 4000 ÷ 1000 = 4 km

  7. 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 cm 800000 ÷ 100 ÷ 1000 = 8 km Speed D - How fast an object is travelling SPEED = DISTANCE ÷ TIME S T e.g. What is the speed of a bus travelling 300 km in 4 hours? 300 ÷ 4 = 75 km 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 = 8 km 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 = 2.5 hours

  8. 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 ÷1000 ÷1000 ÷1000 base unit mg g kg t ×1000 ×1000 ×1000

  9. e.g. Convert a) 6000 mg to g = 6000 ÷ 1000 b) 8.5 g to mg = 8.5 ×1000 = 6 g = 8500 mg c) 3500 g to kg = 3500 ÷ 1000 d) 4 kg to g = 4 × 1000 = 3.5 kg = 4000 g e) 1200 kg to t = 1200 ÷ 1000 f) 9.6 t to kg = 9.6 ×1000 = 1.2 t = 9600 kg - 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 or 6.42 kg + 0.32 kg 450 kg – 120 kg or 0.45 t – 0.12 t = 6740 g = 6.74 kg = 330 kg = 0.33 t e.g. A bookshop posts 5 books, each weighing 850 g. What is the total weight in kg? 850 × 5 = 4250 g 4250 ÷ 1000 = 4.25 kg

  10. Liquid Volume (Capacity) Conversions ÷1000 e.g. Convert a) 200 mL to L = 200 ÷ 1000 base unit = 0.2 L mL L b) 1.5 L to mL = 1.5 ×1000 = 1500 mL ×1000 - 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 or 0.26 L + 1.2 L = 2800 mL – 1430 mL or 2.8 L – 1.43 L = 1460 mL = 1.46 L = 1370 mL = 1.37 L e.g. 200 mL is poured from a 1 L container. How much is left in the container? 1000 – 200 = 800 mL or 1 – 0.2 = 0.8 L

  11. 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 = milli = 1 . 1000 e.g. mm, mg, mL c = centi = 1. 100 e.g. cm k = kilo = 1000× e.g. km, kg

  12. Perimeter Always add in missing lengths - The total distance around an object (total length of ALL its sides) e.g. Calculate the perimeter of the following: 10 m ALWAYS remember to add in the UNIT a) b) 7 cm 8 m Perimeter = 7 + 7 + 8 Perimeter = 8 + 10 + 8 + 10 = 22 cm = 36 m Area - Uses squared units such as cm2 and m2 - Can be estimated by counting the squares of a grid e.g. ALWAYS remember to add in the UNIT Area = 9 cm2

  13. Squares and Rectangles - Area = length × width (A = l × w) e.g. Calculate the following areas: a) b) Area = 9 × 9 Area = 6 × 3 9 cm = 81 cm2 = 18 m2 ALWAYS remember to add in the UNIT

  14. Triangles - Area = ½ × base × height (A = ½ × b × h) e.g. Calculate the following areas: a) b) Area = ½ × 10 × 7 Area = ½ × 8 × 5 ALWAYS remember to add in the UNIT ALWAYS use the VERTICAL height = 35 cm2 = 20 m2

  15. 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) Area = 5 × 4 Area = 6 × 3.5 ALWAYS remember to add in the UNIT ALWAYS use the VERTICAL height = 20 cm2 = 21 m2

  16. 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 = 5 × (6 + 12) 2 = 5 × 9 = 45 m2

  17. 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 = 8 × 8 Area 2 = ½ × (11 – 8) × 8 = 64 = 12 1 2 ALWAYS remember to add in the UNIT Total Area = Area 1 + Area 2 = 64 + 12 = 76 cm2 1. By subtracting areas e.g. Calculate the following area: Area 1 = ½ × 10 × 9 Area 2 = 4 × 4 = 45 = 16 2 Total Area = Area 1 – Area 2 1 = 45 – 16 = 29 cm2

  18. Land Areas - 1 Hectare (ha) = 10,000 m2 e.g. Calculate the area of the paddock in hectares: Area = 360 × 210 = 75600 m2 Area (in hectares) = 75600 ÷ 10000 = 7.56 ha Circles The diameter is the longest CHORD of a circle since it has to pass through the centre. Radius (r) Centre Diameter (d)

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

  20. Area of a Circle Remember if you are given the diameter, you must halve it to find the radius. - To calculate the area of a circle, use the following formula: - Area = π × r2 e.g. Calculate the area of the following circles: a) b) c) r d Area = ½ × π × 62 = 56.55 m2 (2 d.p.) Area = π × 2.52 Radius = 3 cm = 19.63 cm2 (2 d.p.) Area = π × 32 = 28.27 cm2 (2 d.p.) ALWAYS remember to add in the UNIT As we are dealing with a semi-circle, we multiply by ½

  21. Surface Area ALWAYS remember to add in the UNIT - Find area of each face and add them together e.g. Calculate the surface area of the following: b) a) Diameter = 20 m Height = 25 m 2 × π × 102 Area of circles = Area of one face = 6 × 6 = 628.32 = 36 Curved area = π × 20 × 25 Surface Area = 36 × 6 = 1570.80 = 216 m2 Surface Area = 628.32 + 1570.80 = 2199.12 m2 (2 d.p.)

  22. Area Conversions - Square the unit conversion number when changing area units e.g. b) Convert 35 mm2 to cm2 a) Convert 42 m2 to cm2 = 42 × 1002 = 35 ÷ 102 = 420000cm2 = 0.35cm2

  23. Volume ALWAYS remember to add in the UNIT - The amount of space an object take up - Measured using cubic units i.e. cm3, mm3 - 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 cm3 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 = ½ × 4 × 5 × 5 = 50 cm3 Volume = 5 × 3 × 6 = 90 m3

  24. ALWAYS remember to add in the UNIT Volume of Pyramids - Volume = 1/3 × base (end) area × vertical height e.g. Calculate the volume of the following shapes: a) b) × 6 Volume = 1/3 × 4 × 5 Volume = 1/3 × π × 9.852 × 11.65 cm3 (2d.p.) cm3 = 40 = 1183.66 Composite Figures - Split into regular shapes and add/subtract volumes e.g. Calculate the volume of the following: Volume of sphere = 4/3 × π × r3 Volume of hemisphere = ½ × 4/3 × π × 2.53 = 32.72 32.72 + 55.63 Volume of cone = 1/3 × π × 2.52 × 8.5 Total Volume = = 88.35 cm2 (2 d.p.) = 55.63

  25. Volume Conversions - Cube the unit conversion number when changing volume units e.g. b) Convert 965 mm2 to cm2 a) Convert 0.65 m3 to cm3 = 0.65 × 1003 = 965 ÷ 103 = 650000cm3 = 0.965cm3

  26. Liquid Volume Everyday measures: 1 cup = 250 mL (water = 250g) 1 tablespoon = 15 mL (water = 15g) 1 teaspoon = 5 mL (water = 5g) 1 cm3 = 1 mL and 1000 cm3 = 1 litre e.g. Change 600 cm3 into litres: 600 cm3 = 600 mL Remember: 1 L = 1000 mL 600 ÷ 1000 = 0.6 L e.g. How much water (in L) can fit into the following tank? Volume = 40 × 20 × 30 1 cm3 = 1 mL = 24000 cm3 1 L = 1000 mL = 24000 mL = 24000 ÷ 1000 = 24 L 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 and tank = 49 kg Weight of water = 24 kg

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