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 pm9:55 + 30 min =

Useful to add hours and minutes separately

24 Hour Clock

- Morning = 0001 – 1159 - Evening = 1201 – 2359

1200 = midday2400/0000 = midnight

e.g. Change to 24 hour time

a) 11:15 am b) 4:15 pm

10:25 pm

1115 4:15 + 12 hours = 1615

If past midday, add 12 hours to the time

e.g. Change to 12 hour timea) 1020 b) 195010:20 am 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

For am times, leave unchanged. For times 12:00 - 12:59 am subtract 12 hours

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

1 2

B

10 20

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

Scales (Non-Uniform)

- Scales where the gaps are not equal

e.g. Radio Frequencies

700 800 900 1000 1100 1300 kHz

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

Use of Lengths

- Millimetres (mm) Very accurate measurementse.g. Width of toenail- Centimetres (cm) Small object measurementse.g. Student heights- Metres (m) Buildings, sports etce.g. Length of a Basketball court- Kilometres (km) Distancese.g. Distance of Hamilton to Cambridge

Length Conversions

mm cm m km

÷10 ÷100 ÷1000To 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

e.g. Convert

a) 45 mm to cm b) 8 cm to mm

g) 850 mm to m h) 43 m to mm

e) 120 cm to m f) 1.82 m to cm

c) 3.8 km to m d) 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 mm52 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 m 4000 ÷ 1000 = 4 km

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

- How fast an object is travellingSPEED = DISTANCE ÷ TIME

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

S

D

T

Use of Weights

- Milligrams (mg) Very accurate measuringe.g. Weight of an eyelash- Grams (g) Accurate measuringe.g. Weights of cooking ingredients- Kilograms (kg) People, objects that can be carriede.g. Student weights- Tonnes (t) Very heavy objectse.g. Shipping containers, elephants

Weight Conversions

mg g kg t

÷1000 ÷1000 ÷1000

×1000 ×1000 ×1000

base unit

e.g. Convert

a) 6000 mg to g b) 8.5 g to mg

e) 1200 kg to t f) 9.6 t to kg

c) 3500 g to kg d) 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 kg6420 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 g 4250 ÷ 1000 = 4.25 kg

Liquid Volume (Capacity) Conversions

mL L

÷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 mL260 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 mL or 1 – 0.2 = 0.8 L

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

Perimeter

- The total distance around an object (total length of ALL its sides) e.g. Calculate the perimeter of the following:

a) b)

8 cm

7 cm 8 m

10 m

Always add in missing lengths

7 cm

Perimeter = = 22 cm

ALWAYS remember to add in the UNIT

10 m

8 m

Perimeter = = 36 m

Area

- Uses squared units such as cm2 and m2

- Can be estimated by counting the squares of a gride.g.

= 1 cm2Area = cm2

ALWAYS remember to add in the UNIT

7 + 7 + 8 8 + 10 + 8 + 10

9

Squares and Rectangles

l

w- Area = length × width (A = l × w)

e.g. Calculate the following areas:

a) b)

9 cm

9 cm

6 m

3 mArea = = 81

ALWAYS remember to add in the UNIT

cm2Area = = 18 m2

9 × 9 6 × 3

ALWAYS remember to add in the UNIT

Triangles

h

b

- Area = ½ × base × height (A = ½ × b × h)

e.g. Calculate the following areas:

a) b)

7 cm

10 cm

5 m

8 m

12 m

Area = = 35 cm2

Area = = 20 m2

ALWAYS use the VERTICAL height

½ × 10 × 7 ½ × 8 × 5

Parallelogram

- Both pairs of opposite sides are parallel and equal in length

h

b

- Area = base × height (A = b × h)

e.g. Calculate the following areas:

a) b)

5 cm

4 cm

6 m

3 .5 m 4 m

ALWAYS remember to add in the UNIT

Area = = 20 cm2

5 × 4 Area = = 21 m2

ALWAYS use the VERTICAL height

6 × 3.5

Trapezium

- One pair of opposite sides are parallel

a

b

h- Area = height × average of parallel sides

- Area = h × (a + b) 2

e.g. Calculate the following area:

7m

12 m

6 m

5 m

ALWAYS remember to add in the UNIT

Area =

= 45 m2

5 × (6 + 12) 2

= 5 × 9

Compound Areas

- Complex shapes made up of 2 or more regular shapes- Areas can be calculated in 2 ways

1. By adding arease.g. Calculate the following area:

8 cm

8 cm

11 cm

Area 1 = = 64

8 × 8

1 2

Area 2 = ½ × (11 – 8) × 8 = 12

Total Area = Area 1 + Area 2 = 64 + 12

= 76

ALWAYS remember to add in the UNIT

cm2

1. By subtracting arease.g. Calculate the following area:

10 cm

4 cm x 4 cm square

9 cm

1

Area 1 = = 45

4 × 4 Area 2 = ½ × 10 × 9 = 16

Total Area = Area 1 – Area 2= 45 – 16

= 29 cm2

2

Land Areas

- 1 Hectare (ha) = 10,000 m2

e.g. Calculate the area of the paddock in hectares:

360 m

210 m

Area = = 75600 m2

360 × 210

Area (in hectares) = 75600 ÷ 10000 = 7.56 ha

Circles

Centre

Radius (r)

Diameter (d)

The diameter is the longest CHORD of a circle since it has to pass through the centre.

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)

8.2 cm3.5 m

dr

Circumference = π × 8.2 = 25.76

ALWAYS remember to add in the UNIT

cm (2 d.p.)Circumference = 2 × π × 3.5

= 21.99 m (2 d.p.)

Area of a Circle

- To calculate the area of a circle, use the following formula:

- Area = π × r2

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)

2.5 cm

6 cmradius = 6 mr

Area = π × 2.52 = 19.63 cm2 (2 d.p.)

ALWAYS remember to add in the UNIT

d

Radius = 3 cmArea = π × 32

= 28.27 cm2 (2 d.p.)

Area = ½ × π × 62 = 56.55 m2 (2 d.p.)

As we are dealing with a semi-circle, we multiply by ½

Surface Area

- Find area of each face and add them together

e.g. Calculate the surface area of the following:

a) b)

6 m6 m

6 m

Area of one face = 6 × 6

= 36

Surface Area = 36 × 6

= 216 m2

Diameter = 20 m

Height = 25 m

ALWAYS remember to add in the UNIT

Area of circles = 2 × π × 102

= 628.32

Curved area = π × 20 × 25

= 1570.80

Surface Area = 628.32 + 1570.80

= 2199.12 m2 (2 d.p.)

Area Conversions

- Square the unit conversion number when changing area units

e.g.

a) Convert 42 m2 to cm2 b) Convert 35 mm2 to cm2

= 42 × 1002

= 420000 cm2

= 35 ÷ 102

= 0.35 cm2

Volume

- 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

ALWAYS remember to add in the UNIT

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)

5 m6 m

3 m

4 cm

height = 5 cm

5 cm

Volume = 5 × 3 × 6 = 90 m3

Volume = ½ × 4 × 5 × 5 = 50 cm3

Volume of Pyramids

- Volume = 1/3 × base (end) area × vertical height

e.g. Calculate the volume of the following shapes:

a) b)height = 6 cm

4 cm

5 cm

height = 11.65 cm

diameter = 19.7 cm

Volume = 1/3 × 4 × 5 × 6

= 40 cm3

ALWAYS remember to add in the UNIT

Volume = 1/3 × π × 9.852 × 11.65

= 1183.66 cm3 (2d.p.)Composite Figures

- Split into regular shapes and add/subtract volumes

e.g. Calculate the volume of the following:

r = 2.5 cm

8.5 cmVolume of sphere = 4/3 × π × r3

Volume of hemisphere = ½ × 4/3 × π × 2.53 = 32.72

Volume of cone = 1/3 × π × 2.52 × 8.5

= 55.63

Total Volume = 32.72 + 55.63

= 88.35 cm2 (2 d.p.)

Volume Conversions

- Cube the unit conversion number when changing volume units

e.g.

a) Convert 0.65 m3 to cm3 b) Convert 965 mm2 to cm2

= 0.65 × 1003

= 650000 cm3

= 965 ÷ 103

= 0.965 cm3

Liquid Volume

1 cm3 = 1 mL and 1000 cm3 = 1 litre

e.g. Change 600 cm3 into litres:

600 cm3 = 600 mL 600 ÷ 1000

= 0.6 LRemember: 1 L = 1000 mL

e.g. How much water (in L) can fit into the following tank?

40 cm

30 cm

20 cm

Volume = 40 × 20 × 30 = 24000 cm3

1 cm3 = 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