Chapter 11

11 NCM7 2nd ed SB TXT.fm Page 378 Saturday, June 7, 2008 6:17 PM

MEASUREMENT

Measuring is important in our lives: ‘How long until my birthday?’ ‘How heavy is my school bag?’ ‘How much water to fill the swimming pool?’ If you worked in a kitchen, you would be continually measuring: ‘How much flour is needed for a cake?’ ‘What amount of water is needed to cook spaghetti?’ ‘For how long do we roast a chicken?’ In this chapter, we look at how to measure volume, capacity, mass and time.

11

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11 NCM7 2nd ed SB TXT.fm Page 379 Saturday, June 7, 2008 6:17 PM

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11 NCM7 2nd ed SB TXT.fm Page 380 Saturday, June 7, 2008 6:17 PM

Start up Worksheet 11-01 Brainstarters 11

Skillsheet 11-01

1 Each cube in these drawings represents one cubic centimetre (1 cm3). Find the volume of each figure. a b c d

e

f

g

h

i

j

k

l

2 Write the time shown on each of these clocks. a b 11 12

Units of time

Skillsheet 11-02

11 12

1

10

2

8

4

3

9

Telling the time

7

d

6

5

11 12

1

8

4

Skillsheet 11-03

8

4

3

9

3

9

6

2

e 2

7

10

7

10

c

1

6

5

11 12

1 2

8

4 6

8

4

3

9

3 7

2

f

10

6

5

11 12

1

10

2

8

4

3

9 7

5

6

3 Write the times shown on these watches using 12-hour time (am or pm). a

24-hour time

Worksheet 11-02

b 04:15

TV times

380

NEW CENTURY MATHS 7

c

13:20

1

10

7

9

5

11 12

20 :17

5

11 NCM7 2nd ed SB TXT.fm Page 381 Saturday, June 7, 2008 6:17 PM

4 Write these as 24-hour times. a 4:00pm b 1:00am d 5:15am e 6:38pm g 8:46am h 9:30pm

c 3:30am f 12:30pm i 10:17pm

5 Write these 24-hour times as 12-hour times. a 1800 hours b 0400 hours d 0530 hours e 1330 hours g 1930 hours h 2005 hours j 0630 hours k 1015 hours

c f i l

2200 hours 1915 hours 2145 hours 1140 hours

6 Test your general knowledge by answering these questions. a What is the meaning of BC and AD? b How many years in a century? c How many months in a year? d How many hours in a day? e How many minutes in an hour? f How many days in a year? g How many days in a month? h How many weeks in a year? i What is a leap year? Why are leap years necessary? 7 A leap year occurs when the year can be evenly divided by 4, except for years ending in 00 that are not exactly divisible by 400. The year 2000 was a leap year because it is divisible by 400. The year 2100 is not a leap year because it is not divisible by 400. a Make a list of all the leap years there are between 1949 and 1983. b How many leap years are there between 1991 and 2021? 8 Calculate the following. a 5 × 100 d 7000 ÷ 1000 g 6.01 ÷ 10

b 26 × 1000 e 350 × 100 h 4.05 ÷ 100

Working mathematically

Skillsheet 8-01

c 1800 ÷ 10 f 2.4 × 100 i 13.71 × 1000

Multiplying by 10, 100, 1000

Applying strategies

Comparing volumes To complete this activity you will need measuring equipment such as a measuring cylinder, cup or jug. 1 Bring to school as many different containers as you can find. As a group, arrange them in order, from smallest volume (occupying the least space) to largest volume (occupying the most space). 2 Write how the order was decided. 3 Check your estimates by filling the containers with either water or sand and comparing results. 4 Discuss your results with your teacher.

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11-01 Volume

!

The volume of a solid is the amount of space occupied by the solid.

Often, informal (everyday) units are used to refer to volume. For example: • a cup of flour • a cup of milk

Standard units of volume 3

1 cm A cubic centimetre is the amount of space that a cube with each side measuring 1 cm would occupy. The volume of the cube is one 1 cm cubic centimetre, or 1 cm3. A cubic millimetre is the amount of space that a cube with each 1 cm 1 cm side measuring 1 mm would occupy. The volume of the red cube is one cubic millimetre, or 1 mm3. There are 1000 cubic millimetres in one cubic centimetre. 1 cubic centimetre

1 cubic millimetre 1 cm

1000 mm3 1 cm3 = 10 mm × 10 mm × 10 mm = 1000 mm3

1 cm

10 mm

1 cm

TLF

L 164

Inside a cubic metre

10 mm 10 mm

A cubic metre is the amount of space that a cube with each side 1 m would occupy, that is, 1 m3. A washing machine is about half a cubic metre. There are 1 000 000 cubic centimetres in one cubic metre. The diagram below illustrates this. 100 cm (or 1 m) 100 cm (or 1 m)

100 cm (or 1 m)

1 m3 = 100 cm × 100 cm × 100 cm = 1 000 000 cm3

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Unit cubic millimetre cubic centimetre cubic metre

Abbreviation mm3 cm3 m3

Conversion

!

1 cm3 = 1000 mm3 1 m3 = 1 000 000 cm3

The diagram below will help you convert units. × 1 000 000 m3

× 1000 cm3

mm3

÷ 1 000 000

÷ 1000

Example 1 1 Convert 12 000 mm3 into cm3. Solution 12 000 mm3 = (12 000 ÷ 1000) cm3 = 12 cm3

cm3 ÷ 1000

2 Convert 48 m3 into cm3. Solution 48 m3 = (48 × 1 000 000) cm3 = 48 000 000 cm3

mm3

× 1 000 000 m3

cm3

Exercise 11-01 1 Write an example of the items that could be measured by each informal unit of volume. a cup(s) b box(es) c handful d pinch e bucket(s) f packet g capsule(s) h can(s) i teaspoon j wheelbarrow k carton l capful 2 What unit of volume would you use when measuring the volume of: a a textbook? b a backpack? c the carton for a large TV? d a large suitcase? e a match box? f a room? 3 Copy and complete the following. a 3 cm3 = mm3 3 c 2.6 cm = mm3 e 7.2 m3 = cm3 g 1 m3 = mm3 i 126 000 000 cm3 = m3 k 25 m3 = mm3 m 63 000 cm3 = m3

Ex 1

b d f h j l n

5 m3 = cm3 3 4000 mm = cm3 66 000 mm3 = cm3 2300 cm3 = m3 3450 mm3 = cm3 78 000 mm3 = m3 1.4 mm3 = cm3

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4 Use any types of cubes to complete these constructions. a Build as many different solids as you can with a volume of 3 cubes (that is using 3 cubes). Sketch each one. b Build as many different solids as you can with a volume of 4 cubes (that is using 4 cubes). Sketch each one. c Build as many different solids as you can with a volume of 5 cubes. Sketch each one. 5 What is the approximate volume of a brick? Select from A, B, C or D. A 1000 cm2 B 20 cm2 C 1600 cm3 D 2100 cm3 6 Match the correct volume (A to G) with each of the items (a to g) listed. a a bottle of liquid paper A 200 m3 b a box of tissues B 3890 m3 c a glass of water C 1250 cm3 d a bottle of lemonade D 5000 cm3 e a classroom E 20 000 mm3 f a school hall F 250 cm3 g a cereal package G 2200 cm3

Working mathematically

Applying strategies

Building a cubic metre As a group activity, construct your own cubic metre. Write a short report on how you did this.

Volumes of rectangular prisms 1 The rectangular prisms at the top of the next page are made up of 1 cm cubes. Copy and complete the following table. Shape

Number of cubes in one layer

a b c d e f

384

NEW CENTURY MATHS 7

Number of layers

Volume (cm3)

11 NCM7 2nd ed SB TXT.fm Page 385 Saturday, June 7, 2008 6:17 PM

a

b

c

d

e

f

2 Copy and complete this table for the rectangular prisms in Question 1.

height

length

breadth

Shape

Length (cm)

Breadth (cm)

Height (cm)

Volume (cm3)

a

4

4

1

16

b c d e f 3 What is the relationship between the length, breadth and height of a rectangular prism and its volume? 4 Write the relationship as a rule: Volume of a rectangular prism =

×

×

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11-02 Volume of a rectangular prism The volume of the rectangular prism is V = length × breadth × height V=l×b×h

!

Example 2 Worksheet 11-03

Find the volume of the rectangular prism on the right. 8 cm

Solution V=l×b×h = 18 × 12 × 8 = 1728 The volume is 1728 cm3.

Volume

12 cm 18 cm

Exercise 11-02 Ex 2

1 Find the volume of each of these rectangular prisms. a b

4c

c

m

4 cm 9 cm

21 cm

5 cm 5 cm

17 cm 17

d

36 cm

3 cm

e

cm

3 cm

f

15 cm 9 cm

3 cm 3 cm 11 m

15 cm

g

3m

15 cm

m 0c

18

1.2 m

h

2.4 m 1.8 m 33.5 m

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NEW CENTURY MATHS 7

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2 Find the missing measurements for the rectangular prisms in the table. Prism

Length

Breadth

Height

Volume

a

50 cm

50 cm

50 cm

b

5 cm

10 cm

18 cm

c

4m

2.5 m

1.4 m

d

24 mm

16 mm

11 mm

10 cm

10 cm

e f

5 mm

100 mm3

2 mm

g

1.5 m

h

22 cm

i

70 mm

j

1.8 m

2000 cm3

3m

27 m3

5 cm

880 cm3 70 000 mm3

10 mm

9 m3

10 m

3 Find the volume of each of these solids. (Hint: You will need to find the volume of two rectangular prisms each time.) a

3 cm

2 cm

b

1 cm

5 mm 30

3 mm

cm

cm

d

4 mm

12

6m

c

3 mm 2 mm

2 mm

7 cm

8 mm

4 cm

10 cm 8m 3m

cm

10 m

8m 8m

e

28

3m 2m

10 cm m 0c

1

8m

f

20 cm

cm

16 m

1 cm

24

45 m

32 m

25

cm

20 mm

g

10 mm 50 mm

50 45 mm

cm

14 mm CHAPTER 11 VOLUME, MASS AND TIME

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4 Find the volume of each of these rectangular prisms. (Hint: make sure that all the measurements you use are in the same units.) a b c 5 cm 15 mm

40 mm

2.5 cm

10 cm

0.5 m

15 mm

d

e

500 mm

10

cm

10 cm

80 cm

2m

50 cm 3m

1.5 m

Working mathematically

Applying strategies and communicating

What is your volume? Imagine that you are made up of rectangular prisms.

head neck

arms

torso

legs

feet

1 With the help of a partner, make measurements of your body. Use them to find dimensions (to the nearest centimetre) for each of the prism body parts. 2 Sketch each body part prism and label its dimensions. 3 Use the prisms to find your volume, in cm3. 4 Write a report of what you did, showing all diagrams and calculations. Explain how you found the dimensions (length, breadth and height) for the prisms. Do you believe you found a good approximation of your volume? Why?

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11-03 Capacity ‘What is the capacity of the water tank?’ Capacity is the amount of fluid (liquid or gas) in a container. The standard units of capacity are the litre (L), the millilitre (mL) and the kilolitre (kL). The same units are used to describe the volume of any liquid. A teaspoon holds about 5 mL. A tall standard carton of milk holds 1 L. Unit millilitre litre kilolitre

Abbreviation mL L kL

Conversion

!

1 L = 1000 mL 1 kL = 1000 L

The diagram below will help you convert capacity units. × 1000 1 kL

× 1000 1L

1 mL

÷ 1000

÷ 1000

It is also useful to know the relationship between volume and capacity.

1 cm3 contains 1 mL

!

1 m3 contains 1 kL

This means that a cubic centimetre can hold 1 mL of liquid, while a cubic metre can hold 1000 L of liquid.

1 mL

1 cm3

1 m3 = 1 kL

× 1 000 000 =

Exercise 11-03 1 Find the capacity of: a a variety of milk containers c a standard soft drink can e the petrol tanks of a variety of cars g a petrol tanker

b d f h

four different-sized soft drink bottles a standard cup your local swimming pool a small fruit juice pack

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2 State what unit of capacity you would use when measuring: a a glass of milk b a dam c a petrol tank d a bottle of medicine e the amount of soft drink consumed in a week 3 Copy and complete the following. a 7000 mL = L c e g i k m o

3 1--2

L=

mL

2500 mL = L 4000 mL = L 6.2 L = mL 5 kL = L 25 000 kL = L 2.3 mL = L

b 2L=

mL

d 10 000 mL = f h j l n p

1.5 L = 8.5 L = 1750 mL = 9000 L = 520 mL = 6 mL =

L mL mL L kL L kL

4 Select A, B, C or D to complete this statement. The capacity of a bottle of cough medicine is approximately equal to: A 200 mL B 500 mL C 1500 mL D 2000 mL 5 Match the correct capacity (A to J) with the items (a to j) listed. a car petrol tank A 200 mL b liquid paper B 23 kL c bath tub C 5 mL d bucket of water D 70 L e can of drink E 1250 mL f glass of water F 1875 kL g Olympic swimming pool G 20 mL h bottle of lemonade H 9L i teaspoon I 375 mL j water storage tank J 180 L 6 A jug holds 2 L of water. How many 250 mL glasses could be filled from it? 7 James is inviting 30 friends to a party. He calculates that each person will drink 1800 mL of soft drink. a How many litres of soft drink must he buy? b James intends to buy large 2 L bottles of drink, how many bottles must he buy? 8 A bottle of medicine holds 100 mL. Tara was told to take 5 mL twice a day. For how many days can Tara take the medicine before it runs out? 9 A tap leaks 10 mL of water every 50 seconds. How much water will be lost in: a 1 second? b 1 minute? c 3 hours? d 1 day? 10 Your skin releases moisture as a way of controlling body temperature. On average 200 mL is released per hour. If all this moisture was captured, how long would it take to fill a 1.25 L soft drink bottle? 11 A lunch box is made in the shape of a rectangular prism. Its dimensions are 20 cm, 15 cm and 9 cm. a Find the volume of the lunch box, in cm3. b How many mL of water would fit in the lunch box?

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12 Gina’s swimming pool is a rectangular prism 8 m long, 4 m wide and 1.5 m deep. a Find the volume of the swimming pool. b How many litres of water would be needed to fill the pool? (Hint: 1 m3 holds 1 kL.) 13 A fish tank is a rectangular prism 60 cm long, 40 cm high and 30 cm wide. a Find the volume of the tank. b How many litres of water will it hold?

Just for the record

Water, water, everywhere To help you better understand the size of a litre and a kilolitre, here are some examples of water use in and around the home: • Washing your hands/face uses 5 L • Brushing your teeth (tap running) uses 5 L • Brushing your teeth (tap not running) uses 1 L • Cooking and making coffee/tea uses 8 L per day • Flushing the toilet uses 9 L to 13 L • Flushing the toilet (half flush) uses 4.5 L to 6 L • Household tap uses 18 L per minute • Washing the dishes (hand) uses 18 L • Washing the dishes (dishwasher) uses 25 L per cycle • Bath uses 85 L to 150 L • Shower (8 minutes) uses 80 L to 120 L • Washing machine (front loading) uses 120 L per cycle • Washing machine (top loading) uses 180 L per cycle • Washing the car (with hose) uses 100 L to 300 L • Garden sprinkler uses 1 kL to 1.5 kL per hour • Garden hose uses 1.8 kL per hour • Swimming pool (backyard) uses 20 kL to 55 kL • Campbelltown swimming pool (Olympic 50 m) uses 1870 kL On average, a four-person Sydney house (with garden) uses 936 litres of water per day. Half of it is used by outside taps or is flushed in a toilet. How much water does your household use each day? Find out by asking your parents to show you the water bill.

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Working mathematically

Applying strategies and reasoning

Volume by displacement Archimedes, a mathematician and inventor from ancient Greece, discovered that the volume of an object fully immersed in a fluid equals the volume of the displaced fluid. (‘Displaced’ means moved from its position.) 1 Fill a measuring jug with 500 mL of water. 2 Choose at least five objects that can be safely immersed in the jug of water. 3 Copy and complete the following table for each object. (Remember: 1 mL takes up the same space as 1 cm3). Name of object

Original water level

Water level after putting object in

Difference in water level

Volume of object in cm3

500 mL 500 mL 500 mL 500 mL 500 mL

4 By placing a 1 cm cube in a medicine cup with water, show that a cubic centimetre displaces 1 mL of water. 5 By placing a cube with edges measuring 10 cm in a large measuring container, show that 1 L of water is displaced by the cube.

Using technology

Comparing volume and capacity of two dams Brogo Dam is situated near Bega on the south coast of NSW. Windamere Dam, which is larger in comparison, is situated between Dubbo and Newcastle. Some data for the dams is shown below.

Source: www.waterinfo.nsw.gov.au

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1 Copy the table on the previous page into a spreadsheet. (Note: ML = Megalitres.) 2 In the spreadsheet, ‘Capacity’ refers to the total capacity of a dam. ‘Volume’ means the amount of water in the dam and ‘% Capacity’ calculates that volume as a percentage of the total capacity. To find the percentage capacity on 11 July 2006, enter =B4/$C$1 in cell C4 and click %. 3 Fill Down the formula from cell C4 to C16. Notice that, by using an absolute cell reference ($C$1), each volume is divided by the same number (the total capacity).

4 Repeat the steps above to calculate Windamere Dam’s percentage capacity, in cells H4 to H16. 5 Answer the following questions in the given cells: a In cell A18, explain (in one sentence) why it is inappropriate to graph the data for both dams on a single graph. b State the driest month for: i Brogo Dam (answer in cell A19) ii Windamere Dam (B19). c State the wettest month for: i Brogo Dam (A20) ii Windamere Dam (B20). d Use the ‘% Capacity’ column to find the month which shows the lowest dam level for each dam. Answer in cell A21 (Brogo) and B21 (Windamere). e Which area is more suitable for the development of a new town? Begin your answer in cell A22. Give reasons to support your answer. f Which dam is located in an area more likely to receive rain? Give reasons to support your answer. Answer in cell A25.

11-04 Mass You are asked to pick up: • a cubic metre of feathers

• a cubic metre of cement

You can lift the feathers but not the cement! The volume is the same but the mass is different. Even though they each take up the same amount of space, one is much heavier. CHAPTER 11 VOLUME, MASS AND TIME

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Mass is the amount of matter in an object. The standard unit of mass is one kilogram (kg). Other units used are the milligram (mg), the gram (g) and the tonne (t). A drawing pin has a mass of about 1 g. An egg has a mass of about 60 g. A litre of water has a mass of exactly 1 kg. A medium-sized car has a mass of about 1.5 t. Unit milligram gram kilogram tonne

Abbreviation mg g kg t

Conversion 1 g = 1000 mg 1 kg = 1000 g 1 t = 1000 kg

The diagram below will help you convert units. × 1000

t

× 1000

× 1000

kg

÷ 1000

Working mathematically

g

÷ 1000

mg

÷ 1000

Reflecting

Mass of household objects Each member of the group must find the mass of eight household objects. Taking it in turns, each person names the object and the rest of the group guesses its mass. Use a table like the one below. Object

My estimate

Actual mass

Difference

Check each guess against the actual mass and work out the difference between them. Did you get better at estimating by the end of the exercise? Why?

Exercise 11-04 Note: You will need a variety of weighing scales. 1 What unit of mass would you use when measuring: a a piece of fruit? b an elephant? d a car? e a television?

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c a schoolbag?

!

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2 Measure the mass of: a this textbook d a shoe g a jumper

b your lunchbox e a pencil case h a brick

c your schoolbag f yourself i an apple

3 Copy and complete the following. a 3000 g = kg b 2t= kg d 9000 kg = t e 7.5 t = kg g 1.5 kg = g h 3800 kg = t

c 4 kg = g f 10 000 mg = i 3g= mg

4 Select A, B, C or D to complete the following statement. A tub of margarine weighing 500 g has a mass greater than: A 2.5 kg B 0.01 tonnes C 60 000 mg

D 0.8 kg

5 Copy and complete the following, using a ,  or = sign. a 700 g 0.6 kg b 0.8 g 95 mg c 3500 kg d 1.7 kg g 4000 mg

1700 g 0.04 kg

e 0.007 t

7 kg

h 0.03 kg

3g

g

f 640 mg

3.5 t 0.7 g

6 Match the items given (a to j) with the masses listed (A to J). a an egg A 400 g b an elephant B 16 g c a house brick C 25 kg d a medium-sized car D 80 kg e an adult E 6t f a can of soft drink F 500 g g a 50c piece G 10 kg h a 7-year-old child H 50 g i a tub of margarine I 3 kg j a large watermelon J 1t 7 Measure the mass of 1 L of water. Write a report on how you did it.

Working mathematically

Applying strategies

Investigating mass 1 Investigate the sport of weight-lifting. 2 a Obtain a schedule of postal charges from the post office. Imagine that you have five pen-friends in different countries (you choose the countries) and want to send a present to each. Choose the presents, work out the mass of each when wrapped to send, and calculate the cost of sending each one by airmail and by sea. b Work out how much you will save by sending the presents early by sea. 3 Library or Internet research a Choose 10 animals and estimate their masses. Then compare your estimates with data you find at the library or on the Internet. b Find 10 world records that have something to do with mass; for example, the heaviest man, the lightest baby, etc. CHAPTER 11 VOLUME, MASS AND TIME

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TLF

L 347

Mental skills 11

Maths without calculators

Rainforest: book a flight Skillsheet 11-03

24-hour time

24-hour time

24-hour time

12-hour time

24-hour time

12-hour time

0000 hours

12:00 midnight

1200 hours

12:00 midday

0100 hours

1:00am

1300 hours

1:00pm

0200 hours

2:00am

1400 hours

2:00pm

0300 hours

3:00am

1500 hours

3:00pm

0400 hours

4:00am

1600 hours

4:00pm

0500 hours

5:00am

1700 hours

5:00pm

0600 hours

6:00am

1800 hours

6:00pm

0700 hours

7:00am

1900 hours

7:00pm

0800 hours

8:00am

2000 hours

8:00pm

0900 hours

9:00am

2100 hours

9:00pm

1000 hours

10:00am

2200 hours

10:00pm

1100 hours

11:00am

2300 hours

11:00pm

To convert from 24-hour time to 12-hour (am/pm) time, look at the first two digits. • If they are 00, then it is just after 12:00 midnight. • If they are less than 12, then it is ‘am’ (morning) time. Write ‘am’. • If they are 12 or more, it is ‘pm’ (afternoon/evening) time. Subtract 12 and write ‘pm’. Then insert a colon (:) before the last two digits. 1 Consider these examples. a Convert 1850 hours to 12-hour time. The first two digits are 18. 18 is more than 12, so it is ‘pm’ time, and we need to subtract 12. 18 − 12 = 6 ∴ 1850 hours = 6:50pm b Convert 0430 hours to 12-hour time. The first two digits are 04. 04 is less than 12, so it is ‘am’ time. ∴ 0430 hours = 4:30am c Convert 0015 hours to 12-hour time. The first two digits are 00. 00 is 12:00 midnight, so it is just after midnight ∴ 0015 hours = 12:15am d Convert 2223 hours to 12-hour time. The first two digits are 22. 22 is more than 12, so it is ‘pm’ time, and we need to subtract 12. 22 − 12 = 10 ∴ 2223 hours = 10:23pm

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2 Now convert these examples to 12-hour time. a 0845 hours b 1320 hours c 1750 hours e 2105 hours f 1832 hours g 1115 hours i 1440 hours j 0320 hours k 1655 hours m 0108 hours n 1018 hours o 2000 hours

d h l p

0017 hours 0238 hours 2331 hours 0643 hours

We can also convert from 12-hour time to 24-hour time: • If it is 12:00 midnight, change the 12 to 00. • If it is ‘am’ time or 12:00 midday, then keep the hour as it is, but make sure it has two digits (for example 02, 09). • If it is 1:00pm or later, then add 12 to the hour. Then remove the colon (:) before the last two digits. 3 Consider these examples: a Convert 4:10am to 24-hour time. It is ‘am’ time, so keep the hour (4), then insert a ‘0’ so it has two digits, (04). ∴ 4:10am = 0410 hours. b Convert 4:10pm to 24-hour time. It is ‘pm’ time, so add 12 to the hour. 4 + 12 = 16 ∴ 4:10pm = 1610 hours. c Convert 12:47am to 24-hour time. It is just after 12:00 midnight, so change the 12 to 00. ∴ 12:47am = 0047 hours. d Convert 12:47pm to 24-hour time. It is just after 12:00 midday, so leave the 12 as it is. ∴ 12:47pm = 1247 hours. 4 Now convert each of these to 24-hour time. a 6:35pm b 8:05am c e 2:21am f 12:30pm g i 9:08am j 9:50pm k m 1:59am n 10:18pm o

11:45am 3:48pm 12:42am 10:46am

d h l p

11:20pm 7:11pm 7:39am 5:23pm

11-05 Timelines

Worksheet 11-04

Timelines record events in the order in which they happen, along a regular scale. A timeline for a puppy’s first 32 weeks could look like this: opened eyes

left mother

learnt to dug up play fetch new plants made a mess on the carpet

0

8

ate a slipper

16

ate cake from table

chased first cat

24

32 Weeks

You need to work out the scale used on the timeline before you can get information from it. On this timeline there are eight major divisions between 0 and 32, so each interval (unit) represents 4 weeks. CHAPTER 11 VOLUME, MASS AND TIME

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History of the calendar

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From the timeline, you can see that, at 24 weeks, the puppy chased its first cat. It left its mother at about 6 weeks and at 20 weeks it started digging up the garden.

Exercise 11-05 1 a Copy this timeline.

3000 BC

2000 BC

1000 BC

1000 AD

b How many years does each interval (unit) on the timeline represent? (This is called the scale of the timeline.) c Write the following dates on the timeline in the correct boxes. AD 1 The birth of Christ 753 BC The founding of the city of Rome About 1600 BC Introduction of the current Chinese year system 3111 BC Start of the Mayan ‘Long Count’ 544 BC Date recorded as the birth of Buddha AD 1792 Declaration of the 1st French Republic AD 622 Traditional date for the flight of Muhammad 2

G

A

1770

C

1870 E

F

B

1970 D

H

This timeline shows events from the first 200 years of white settlement in Australia. a What is the scale of this timeline? b Match the letters on the timeline with these events. 1851 Gold was discovered at Warrandyte, Victoria 1932 Sydney Harbour Bridge was opened 1974 Darwin was devastated by Cyclone Tracy 1956 Melbourne hosted the Olympic Games 1813 The explorers Blaxland, Wentworth and Lawson crossed the Blue Mountains 1788 The First Fleet arrived in Port Jackson 1982 Brisbane hosted the Commonwealth Games 1901 The Federation of the Australian States to form the Commonwealth of Australia

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3 The table below shows the names of Australia’s Governors-General and the year they each took office, from 1960 to 2003. Name

Year

A

Viscount Dunrossil

1960

B

Lord Casey

1965

C

Sir Zelman Cowen

1977

D

Viscount De L’Isle

1961

E

Right Reverend Dr Peter Hollingworh

2001

F

Sir William Deane

1996

G

Sir Paul Hasluck

1969

H

Sir John Kerr

1974

I

William Hayden

1989

J

Sir Ninian Stephen

1982

K

Major-General Michael Jeffery

2003

a Copy the timeline below and complete it by writing in the letters to indicate when each Governor-General took office. (Two have been done for you.) A 1960

C 1972

1984

1996

2008

b What is the scale of this timeline? c Which Governor-General was in office for the longest period of time? d Which Governor-General was in office for the shortest time? 4 Draw a timeline to show these events for the period between 1945 and 2010. 1969 Astronauts first walked on the moon 1945 World War II ended 1989 Wayne Gardner won his first Australian 500 cc Motorcycle Grand Prix 19__ The year you were born 1985 The Aboriginal people were granted land rights to Uluru (Ayers Rock) 1964 The Beatles toured Australia 1983 Australia II won the America’s Cup 1956 The first television transmission in Australia occurred 2000 The Olympic Games were held in Sydney 2002 Steve Fossett flew solo around the world in a balloon 2006 The Crocodile Hunter, Steve Irwin, died 20__ (Enter your own important event.) 5 Draw a timeline to display these famous Australian inventions and discoveries. 1906 The surf-lifesaving reel for use at Bondi Beach was invented 1919 The preferential voting system was first used for the House of Representatives 1922 Vegemite was developed by Dr Cyril Callister 1930 The world’s first mechanised letter-sorter was installed in the Sydney GPO, built by A. B. Corbett 1945 The Hills rotary clothes line was invented by Lance Hill 1952 The Victa rotary lawnmower was developed by Mervyn Victor Richardson CHAPTER 11 VOLUME, MASS AND TIME

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1979 1983 1988

Race-cam was first used by Channel Seven at the Bathurst 1000 car races The ‘Bionic ear’ cochlear implant came on the market Plastic banknotes, developed by the CSIRO, were first released

6 a Work with a partner or in a small group to write a list of important events that have occurred in your lifetime. Try to make a personal list. b Draw a timeline to show these events.

Working mathematically

Communicating

Timeline display By yourself, or with a partner, create a timeline for one of the following: • Major disasters of the world • Historical events of another country • Achievements in science • Achievements in sport • Wars of the last 150 years • Women in history • Prime Ministers of Australia • A topic approved by your teacher

Worksheet 11-05

11-06 Converting units of time

Metric units

Unit second minute hour day

!

Abbreviation s min h day

Conversion 1 min = 60 s 1 h = 60 min = 3600 s 1day = 24 h × 3600

× 24 day

h ÷ 24

× 60

× 60 min

s ÷ 60

÷ 60 ÷ 3600

Example 3 1 Round each of these amounts of time to the nearest hour. a 7.83 hours b 12 hours 19 minutes c 2 hours 43 minutes 6 seconds Solution a 7.83 h ≈ 8 h When rounding hours and minutes to the nearest hour, we use 30 minutes as the halfway mark because there are 60 minutes in 1 hour. b 12 h 19 min ≈ 12 h (round down because 19 min  30 min) c 2 h 43 min 6 s ≈ 3 h (round up because 43 min  30 min)

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2 Round each of these amounts of time to the nearest minute. a 11.4 minutes b 25 minutes 37 seconds c 3 hours 6 minutes 30 seconds Solution a 11.4 min ≈ 11 min When rounding minutes and seconds to the nearest minute, we use 30 seconds as the halfway mark because there are 60 seconds in a minute. b 25 min 37 s ≈ 26 min (because 37 s  30 s) c 3 h 6 min 30 s ≈ 3 h 7 min (because we round 30 s up)

Example 4 1 Convert 7 minutes into seconds. Solution 7 minutes = 7 × 60 seconds = 420 seconds

2 Convert 91 days into weeks. Solution 91 days = 91 ÷ 7 weeks = 13 weeks

Example 5 Convert 275 minutes into hours and minutes. Solution There are 60 minutes in 1 hour. 275 ÷ 60 = 4 remainder 35 275 minutes = 4 h 35 min

Most scientific calculators have a degrees-minutes-seconds key, ° ' " or DMS , that is useful for calculations involving minutes and seconds (base 60). This key can be used to convert decimal answers for time to hours-and-minutes or minutes-and-seconds. Calculating the answer to Example 5 in this way: 275 minutes = 275 ÷ 60 h = 4.583 333 3… h Press ° ' " to get 4° 35′ 0″ on the calculator display, which means 4 h 35 min.

Exercise 11-06 1 State which unit of time (hours, minutes, or days) would be used to measure each event. a snapping your fingers five times, as fast as possible b a day-night cricket match c running once around the school oval d building a house e flying from Sydney to Broken Hill f watching a DVD from beginning to end g the life span of a grasshopper CHAPTER 11 VOLUME, MASS AND TIME

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Ex 3

Ex 4

Ex 5

2 Write these times correct to the nearest hour. a 4 h 14 min b 11.5 h d 7 h 48 min 19 s e 3.42 h

c 6 h 27 min f 2 h 30 min

3 Write these times correct to the nearest minute. a 17 min 51 s b 8.8 min d 4 h 20 min 19 s e 12.31 min

c 4 min 7 s f 1 h 28 min 40 s

4 Convert: a 6 hours to minutes c 9 weeks to days e 3 days to hours g 2 weeks to hours

b d f h

15 minutes to seconds 2.5 years to weeks 2 years to days 4.25 hours to minutes

i 8.5 days to hours

j 10 1--- minutes to seconds

k 7.2 centuries to years

l 3 fortnights to days

5 Convert: a 480 seconds to minutes c 96 hours to days e 468 weeks to years g 60 hours to days i 330 seconds to minutes and seconds k 135 minutes to hours and minutes m 405 minutes to hours and minutes 6 Find the number of seconds in: a 1 hour b 1 day

2

b d f h j l n

70 days to weeks 200 minutes to hours and minutes 560 seconds to minutes and seconds 126 days to weeks 24 weeks to fortnights 470 years to centuries 167 minutes to hours and minutes c 1 year

7 Are you over a million seconds old? Find your age in seconds to answer this question.

Just for the record

Minutes and seconds In Chapter 2, you learned that there are 360° in a revolution because the ancient Babylonians used a base 60 number system and believed that a year lasted 360 days. (How many days is a year actually?) The Babylonians, who lived where Iraq is today in 2000 BC, invented the units for measuring angles and time. That is why there are 60 minutes in an hour and 60 seconds in a minute. The word ‘minute’ has another meaning. When pronounced ‘my-newt’, it means tiny, but this meaning is also related to the minute as a unit of time. A minute is a tiny fraction of an hour, and comes from the Latin ‘pars minuta prima’, meaning the first division (or part) of an hour. The word ‘second’ also means coming after first, and this meaning is also related to the second as a unit of time. Find out how.

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11-07 Time calculations

Worksheet 11-06 Time calculations

Example 6 What is the time 7 hours 40 minutes after 11:52pm? Solution 7 hours after 11:52pm is 6:52am. 40 minutes after 6:52am is 7:32am.

+7h 11:52pm

+ 8 min + 32 min 6:52am 7am 7:32am

Example 7 What is the difference in time between 8:35am and 3:10pm? Solution +6h + 10 min 25 min From 8:35am to 9:00am = 25 minutes From 9:00am to 3:00pm = 6 hours 3pm 3:10pm 8:35am 9am From 3:00pm to 3:10pm = 10 minutes Total time difference = 25 min + 6 h + 10 min = 6 h 35 min OR Convert to 24-hour time first. Then use the calculator’s ° ' " or DMS key to enter hours and minutes, and subtract the times.

Example 8 Find 7 h 5 min − 3 h 24 min. Solution 7 h 5 min − 3 h 24 min = 6 h 65 min − 3 h 24 min = (6 − 3) h + (65 − 24) min = 3 h 41 min OR Use the calculator’s ° ' " or DMS key to enter hours and minutes, and subtract the times.

Exercise 11-07 1 What time will it be: a 5 hours after 3:00pm? c 28 minutes after 7:15pm? e 3 hours 19 minutes after 10:49pm? g 9 hours after 5:14pm? i 2 1--- hours after 4:02am? 4

Ex 06

b d f h j

8 hours after 11:00am? 3 hours 32 minutes after 9:45am? 4 hours after 9:32am? 45 minutes after 3:30pm? 12 hours 40 minutes after 2:45am?

CHAPTER 11 VOLUME, MASS AND TIME

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2 A marathon began at 10:20am. Here are some of the competitors and the times they ran. Write the runners in their order of finishing and the time each crossed the finishing line. Mike 3:11 (3 h 11 min) Joe 2:23 Anna 2:54 Pathena 3:01 Ken 2:59 Gail 3:42 3 What is the difference in time between 10:42am and 2:13pm? Select A, B, C or D. A 3 h 31 min B 4 h 55 min C 8 h 29 min D 12 h 55 min Ex 7

Ex 8

Worksheet 11-07 World time zones

4 What is the difference in time between: a 7:15pm and 8:20pm? c 4:09am and 9:53am? e 7:27am and 1:12pm? g 7:45pm and 10:10pm? i 4:15pm and 6:02pm? k 8:40am and 4:19pm?

b d f h j l

5 Find: a 2 h 15 min + 4 h 32 min c 7 h 12 min + 5 h 18 min e 9 h 37 min + 2 h 52 min

b 3 h 25 min + 8 h 27 min d 1 h 42 min + 6 h 27 min f 4 h 49 min + 7 h 18 min

6 Find: a 6 h 42 min − 3 h 13 min c 15 h 57 min − 9 h 48 min e 8 h 18 min − 3 h 27 min

b 12 h 37 min − 5 h 6 min d 6 h 2 min − 4 h 17 min f 5 h 31 min − 3 h 48 min

11-08 Standard time World time zones The world is divided into 24 main time zones. Time is the same throughout each zone. The centre of each time zone is a meridian of longitude (an imaginary line running from the North Pole to the South Pole). The meridians are 15° apart. The system used to divide the world was first suggested by Sir Sanford Fleming (1827–1915), a Canadian civil engineer and scientist. In 1884, scientists from 27 nations met in Washington and devised the time system we now use.

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10:16am and 12:06pm? 11:15pm and 3:08am? 9:36pm and 9:14am? 2:24am and 3:07am? 10:25am and 2:33pm? 6:45am and 8:10pm?

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The map below shows how times around the world are related. All time is measured in relation to the time at Greenwich (in London), either ahead or behind Greenwich Mean Time (GMT), also known as UTC (Coordinated Universal Time). Australia’s time is ahead of Greenwich Mean Time since Australia is east of Greenwich. America’s time is behind Greenwich Mean Time since America is west of Greenwich. 180°W 150°W 120°W 90°W

60°W

30°W

0°

20°E

West of Greenwich (behind GMT)

60°E

90°E

120°E

150°E 180°E

East of Greenwich (ahead of GMT)

N

Greenwich Meridian Athens

International Date Line

New York

Beijing Hong Kong

Honolulu Equator

Rio de Janeiro

Greenwich Meridian

International Date Line

Helsinki Moscow Greenwich Geneva

Ottawa San Francisco

80°

12:00 2:00am 4:00am 6:00am 8:00am 10:00am 12:00 midnight noon

Perth

Sydney

60°

40° 20° 0° 20° 40°

60° 2:00pm 4:00pm 6:00pm 8:00pm 10:00pm 12:00 midnight

Exercise 11-08 1 State whether each of these cities is ahead of or behind Greenwich Mean Time. a Sydney b Auckland c Rio de Janeiro d Perth e Beijing f Honolulu g Moscow h Athens i Hong Kong j Helsinki k New York l Ottawa 2 From the given map, find the time in each of these cities when it is noon in Greenwich. a Sydney b Perth c New York d Beijing e San Francisco f Honolulu g Moscow h Geneva 3 What is the time difference between: a Sydney and Perth? c Sydney and Honolulu? e Sydney and New York? g San Francisco and New York? i Geneva and Perth?

b d f h j

Sydney and Beijing? Sydney and Moscow? Perth and Beijing? Honolulu and Moscow? San Francisco and Geneva?

CHAPTER 11 VOLUME, MASS AND TIME

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4 If it is 2:00pm in Sydney, what is the time in: a Greenwich? b Perth? c New York? e San Francisco? f Honolulu? g Moscow?

d Beijing? h Geneva?

5 A cricket match being played in India is telecast live at 7:00pm Sydney time. What is the local time of the cricket match if Sydney’s time is 4 1--- hours ahead of India’s? 2

6 Simone, in Newcastle, wants to use the Internet to chat with her cousin Zac in Vancouver, Canada. The time in Vancouver is 18 hours behind the time in Newcastle. At what time should Simone log on to the Internet to catch Zac when it is 3:00pm in Vancouver? 7 Brisbane is 2 hours behind New Zealand. A plane leaves New Zealand at midday and takes 3 hours to fly to Brisbane. What is the local time in Brisbane when the plane lands? Select A, B, C or D. A 11am B 1pm C 3pm D 5pm 8 Find out what happens if you cross the International Date Line (IDL). Why isn’t the IDL straight?

11-09 Australian standard time This map shows the three time zones for Australia. Australian Western Standard Time (AWST)

Australian Central Standard Time (ACST)

Australian Eastern Standard Time (AEST)

Northern Territory Queensland Western Australia

South Australia New South Wales Victoria Tasmania

−2 hours 10am

1 −2

hour 11:30am

Zero 12noon

Note: During daylight saving periods, add 1 hour.

Exercise 11-09 1 State whether each location is ahead of, behind or has the same time as Adelaide. a Sydney b Melbourne c Darwin d Perth e Mt Isa (Qld) f Geraldton (WA) g Cobar (NSW) h Ceduna (SA) i Cairns (Qld)

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2 What is the time difference between: a Sydney and Adelaide? b Melbourne and Perth? d Hobart and Darwin? e Canberra and Perth?

c Adelaide and Melbourne? f Brisbane and Canberra?

3 If it is 11:00pm in Sydney, what time is it in: a Melbourne? b Adelaide? d Darwin? e Hobart?

c Perth? f Canberra?

4 If it is 11:30pm in Adelaide, what time is it in: a Melbourne? b Sydney? c Perth? d Darwin? e Hobart? f Brisbane? 5 a Joe flies from Sydney to Perth, taking 4 hours. If he leaves Sydney at 2pm, what time does he land in Perth? Give your answer as Perth local time. b When Joe flies home, he leaves Perth at 9am. What time does he land in Sydney? Give your answer as Sydney local time. 6 a Find out when daylight saving begins and ends. b Why do we have daylight saving? c How does daylight saving affect the different time zones? d If it is 12:30pm in Western Australia (not on daylight saving), what time is it in New South Wales on Eastern Standard Daylight Saving Time?

11-10 Timetables

Worksheet 11-08 Tide chart

Exercise 11-10 1 Airline timetable Daniel and his volleyball team need to fly from Sydney to Brisbane for a championship tournament. Daniel logged on to the Internet site for Thomson Airways and found the following flight schedule for 12 October. Flight number

Sydney departure time

Brisbane arrival time

TH503

0905

1030

TH511

0935

1100

TH038

1005

1130

TH114

1040

1210

TH514

1105

1230

TH051

1135

1300

a How long does flight TH503 take from Sydney to Brisbane? b The team plans to meet at Sydney airport at 10:45am. How long will they need to wait for the next available flight? c The team needs to be at the hotel in Brisbane by 12:30pm. If it takes 30 minutes to drive from the airport to the hotel, what is the latest flight the team can catch from Sydney? d What is the flight number of the flight that takes longer to reach Brisbane than the others? Give one reason why it might take longer. CHAPTER 11 VOLUME, MASS AND TIME

407

TLF

L 764

Journey planner: quickest route 1

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2 Bus service timetable Forward: Sydney to Wagga Wagga

Return: Wagga Wagga to Sydney

Sydney Strathfield Yagoona Liverpool Mittagong Goulburn* Yass Jugiong Gundagai Wagga

Wagga Gundagai Jugiong Yass Goulburn* Mittagong Liverpool Yagoona Strathfield Sydney

2:30pm 3:00pm 3:20pm 3:45pm 4:40pm 5:40pm 7:10pm 7:55pm 8:20pm 9:30pm

7:15am 8:25am 8:54am 9:41am 10:41am 12:10pm 1:05pm 1:20pm 1:35pm 2:05pm

* 30 minute meal stop at Goulburn

a b c d

How long does the trip from Sydney to Wagga take? How long would the trip take without a meal break? Ali joins the return bus at Jugiong and gets off at Liverpool. How long is his trip? Find the time taken from Liverpool to Sydney and from Sydney to Liverpool. Suggest a reason for the difference.

3 Countrylink train timetable Goulburn to Sydney — Monday to Friday GOULBURN MARULAN TALLONG WINGELLO PENROSE BUNDANOON EXETER MOSS VALE BURRADOO BOWRAL MITTAGONG YERRINBOOL BARGO TAHMOOR PICTON CAMPBELLTOWN STRATHFIELD SYDNEY

am

am

am

pm

pm

pm

pm

pm

5:08 5:26 5:32

7:27 7:45 7:51

8:17

1:47

4:26

6:47

Bookings

Bookings

Bookings

Bookings

5:39

7:58

essential

essential

essential

essential

5:44 5:50 5:55 6:05 6:10 6:13 6:17 6:30 6:41 6:48 6:56 7:23

8:03 8:09 8:14 8:24

8:52

2:22

9:05

2:35

9:11 9:16

2:41 2:46

10:12

3:41 4:17 4:29

7:45 8:03 8:09 8:16 8:21 8:27 8:32 8:42 8:47 8:50 8:54 9:07 9:18 9:25 9:33 10:00

8:12

10:11 10:42 10:54

2:45 3:03 3:09 3:16 3:21 3:27 3:32 3:42 3:47 3:50 3:54 4:07 4:18 4:25 4:33 5:00

8:30 8:34

9:08 9:30

6:20

7:22 5:13

7:35 7:41 7:46

6:13 7:00 7:13

8:42 9:12 9:24

11:04

a Michael has an interview in Sydney on Tuesday at 10:45am. At what time must he catch the train in Goulburn? b What is the difference in the time taken to travel from Goulburn to Sydney on the 5:08am train and the 8:17am train? c Georgina travels from Penrose to Yerrinbool, arriving at 4:07pm. How long did the trip take? d You have been visiting friends in Moss Vale and are returning to Sydney. Decide which train you would catch and explain why. e A new train is added to the timetable, leaving Goulburn at 11:12am. Write out a timetable for this train if it stops at the same stations as the 6:47pm train.

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4 The Explorer Bus The Explorer Bus operates in Sydney, Canberra and Melbourne. It takes tourists on a tour of the city and allows them to visit places of interest. Below is a winter timetable for an Explorer Bus in a capital city. Depart Explorer depot City cathedral Railway station Parliament Museum City square Zoo Dockland shops Arts centre Water gardens Hall of fame

10:00 10:08 10:15 10:24 10:35 10:45 11:00 11:12 11:19 11:30 11:38

10:25 10:33 10:40 10:49 11:00 11:10 11:25 11:37 11:44 11:55 12:03

10:50 10:58 11:05 11:14 11:25 11:35 11:50 12:02 12:09 12:20 12:28

11:15 11:23 11:30 11:39 11:50 12:00 12:15 12:27 12:34 12:45 12:53

11:45 11:53 12:00 12:09 12:20 12:30 12:45 12:57 1:04 1:15 1:23

12:00 12:08 12:15 12:24 12:35 12:45 1:00 1:12 1:19 1:30 1:38

12:25 12:33 12:40 12:49 1:00 1:10 1:25 1:37 1:44 1:55 2:03

12:50 12:58 1:05 1:14 1:25 1:35 1:50 2:02 2:09 2:20 2:28

1:15 1:23 1:30 1:39 1:50 2:00 2:15 2:27 2:34 2:45 2:53

11:50

12:15

12:40

1:05

1:35

1:50

2:15

2:40

3:05

Arrive Explorer depot

a How many buses are needed to meet the winter Explorer Bus timetable? Explain how you arrived at your answer. b Vo, Binh and Vicki came to the city by train, arriving at the station at 11:42am. They caught the Explorer Bus to the zoo. What is the earliest time they could expect to arrive at the zoo? Explain your answer. c Manuel and Sofia are dropped off by car at the ‘City cathedral’ at 10:25am. They arrange to meet their hosts at the ‘Hall of fame’ at 2:45pm. They want to spend at least half an hour at the museum, photograph the ‘City square’ and do some souvenir shopping at the Dockland shops. Plan a list of times for them to catch the Explorer Bus to do these things and meet their hosts on time. d In summer, extra Explorer tours leave the depot at 11:30am, 1:30pm and 2:30pm. Make a list of departure times that appear in the timetable for each of these tours.

Working mathematically

Applying strategies and reasoning

Round trip Plan a trip around the world with at least three stopovers (for example, Berlin, London, New York). Obtain some airline timetables so you can give details of departures and arrivals. Work out how much time is actually spent flying. Does it matter if you head east or west when you start? What effect does the International Date Line have on your trip?

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Using technology

Graphing winning times The table below shows the gold medal winning time of the women’s 400-metre track event in the Olympic Games held from 1972 to 2004. Year

1972

1976

1980

1984

1988

1992

1996

2000

2004

Time (s)

51.08

49.28

48.88

48.83

48.65

48.83

48.25

49.11

49.41

1 Copy the data, as shown in the table above, into a spreadsheet. 2 Highlight the data, open Chart Wizard (by clicking on or choosing Insert Chart) and select XY (Scatter). Click on the option that shows points joined with straight lines

.

3 Give the graph an appropriate title and axes labels. Click ‘Next’. Save the graph ‘As new sheet’. 4 On the graph, position the mouse over a data point. (Do not click on it.) You can view the specific details of the Olympic year and winning time.

5 Use your spreadsheet, graph and the formulas below to answer these questions. a In what year was the fastest gold medal winning time run? In cell A5, enter =min(B2:J2). In cell B5, enter the year that corresponds to this time. b In cell A6, type the label ‘Average’. In cell B6, use the formula =average(B2:J2) to calculate the average winning time for this event, from 1972 to 2004. c In cell A7, enter =max(B2:J2) to find the slowest winning time in this event. In cell B7, enter the year that corresponds to this time. d In cell A8, enter a formula to find the difference between the fastest and slowest winning times. e Predict the gold medal time at the 2008 Olympic Games in this event. Justify your answer. f In cell A9, enter a formula to calculate the speed, in metres per second, of the fastest women’s 400-m runner, from 1972 to 2004. g Starting in cell A10, write a paragraph describing the changes in winning times for this event between 1972 and 2004. h In cell A15, suggest reasons why the pattern of gold medal times has changed between 1972 and 2004.

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Working mathematically

Applying strategies and reasoning

Time puzzlers Try to solve as many of the following puzzles as you can, on your own or in a group. Record your solution and how you solved the puzzle each time. Try the puzzles out on your family and friends. Puzzler 1

Puzzler 3

If it takes 3 1--- minutes to soft boil 1 egg, 2

how long will it take to soft boil 3 eggs? Puzzler 2 Here is a way to find someone’s age. Give them the following instructions. • Think of any number between 1 and 10. • Square it. • Subtract 1. • Multiply the result by the original number. • Multiply that by 3. • Add the digits of the answer. • Add your age in years and tell me the result. Now comes the trick: • First you need to guess the first digit of their age (that is, are they in their teens, 20s, 50s, etc.?). • Add the digits of the result you have been given. • Subtract the first digit of their age from this sum to get the second digit of their age.

The floral clock shown above gains half a minute during the day due to the warmth of the sun, and loses one-third of a minute during the cool of the night. If the clock was set to the correct time on 1 January, when will it be 5 minutes fast? Puzzler 4 A doctor prescribed 15 pills and told his patient to take one every half-hour. How long would it take the patient to finish the course of pills? (Note: The answer is not 7 1--- hours.) 2

Puzzler 5 Some months have 31 days, some have 30 days. How many months have 28 days? Puzzler 6 How long in our time is a metric hour if: 1 minute = 100 seconds and: 1 hour = 100 minutes?

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Power plus 1 The diagram on the right shows a tank. The tank is half-filled with water. Find the amount of water in the tank. 2 A cube has a volume of 512 cm3. Find the length of each side of the cube.

14 cm 15 cm 30 cm

3 A children’s pool is in the shape of a cross as shown on the right. Each side is 3 m long. The pool is filled with water to a depth of 300 mm. a Find the area of the pool surface. b Calculate the volume of water, in cubic metres (m3). c If water is charged for at $0.80 per kL, how much does it cost to fill the pool?

3m 3m

3m 3m

4 A doctor orders 5.2 litres of fluid each day to be given to a patient in drops. Each 1 mL of fluid is equivalent to 15 drops. How many drops of fluid per minute are needed for the patient to receive the required dose? 5 The diagram on the right shows a container in the shape of a rectangular prism. a How many cubes of side length 60 cm could be stacked in the container? b If each cube has a mass of 25 kg, how many tonnes would the container carry?

3m 3m

12 m

6 Calculate the volume of each solid below. a b 16 cm

8 cm

100 cm 2 cm 13 cm

13 cm 20 cm

8 cm

16 cm

16 cm

30 cm

2 cm

7 A rectangular box 40 cm long and 12 cm wide contains 2880 cm3 of sugar. How deep is the sugar in the box if it is spread evenly? 8 South Australia is 1 1--- hours ahead of Western Australia. Anna is flying from Perth to 2 Port Augusta. If the flight takes 2 1--- hours and the flight leaves Perth at 10:00am on 2 Sunday, at what time will the plane land in Port Augusta? 9 What happens if you travel east across the International Date Line? 10 If a 1 cm3 container can hold 1 mL, explain why a 1 m3 container can hold 1 kL.

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Chapter 11 review Language of maths capacity cubic millimetre gram kilolitre milligram timetable 24-hour time

Worksheet 11-09

cubic centimetre Central Standard Time Greenwich Mean Time litre millilitre time zone volume

cubic metre Eastern Standard Time kilogram mass timeline tonne Western Standard Time

Measurement crossword

1 What is the difference between volume and capacity? 2 Look up the different meanings of ‘capacity’ in the dictionary. How are these related to its mathematical meaning? 3 Find out the difference between a tonne and a ton. 4 What is a megalitre (ML)? 5 The word ‘minute’ can be pronounced differently and has different meanings. Find how the other meanings relate to a ‘minute’ meaning a fraction of an hour. 6 In Summer, the eastern states of Australia use AEDST instead of AEST. Explain.

Topic overview • Write in your own words what you have learnt about volume, mass and time. • What parts of this topic were new to you? • What parts of this topic did you have difficulty with? Discuss them with a friend or your teacher. • Give some examples of situations where you would use what you know about volume, mass and time. • Copy this summary into your workbook and complete it. Use colour to help you remember your summary. Check it with other students and your teacher. V _____ l×b×h

M _____ • mg •g • kg •t

cm3 m3

VOLUME, MASS and TIME 11

C _______ • mL •L • kL

12

1 2 3

10 9 T ____

4

8 7

6

5

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Topic test 11

Exercise 11-01

Chapter revision 1 Copy and complete the following. a 20 cm3 = mm3 c 7500 mm3 = cm3

b 0.5 m3 = cm3 b 230 000 mm3 =

Exercise 11-02

2 Count the cubes in this solid to find its volume. Each cube equals 1 cm3.

Exercise 11-02

3 Find the volume of each of these prisms. a

b

4 cm

2m

m3

4 cm

5m 20 cm

8m

c

10 m

d 7 mm

6m 8 mm 12 mm

e

6m

f

15 cm 20 cm

6m

7m

10 cm 15 m 5m 15 m

Exercise 11-02

4 The biggest iceberg on record was called B9. It had the same volume as a rectangular prism with dimensions 160 km long, 50 km wide and 250 metres high. When B9 melted, how many litres of water was contained in B9? (1 kL of water will occupy 1 m3.)

Exercise 11-03

5 Copy and complete the following. a 2000 mL = L c 7L= mL e 1750 mL = L

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b 3 kL = d 3300 L = f 2.5 L =

L kL mL

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6 Select A, B, C or D to complete the following. The mass of an egg is closest to: A 5g

B 50 g

C 500 g

Exercise 11-04

D 5 kg

7 Eighteen trucks, each carrying 12 000 kg of debris, were required to clear a building site. How many tonnes of debris were cleared altogether?

Exercise 11-04

8 Copy and complete the following. a 5000 g = kg

Exercise 11-04

c 1 1--- t = 2

e 4000 mg =

b 2g=

mg

d 6500 kg =

kg g

f 1.5 kg =

t g

9 Make up a timeline for your life from age 0 to 12.

Exercise 11-05

10 Write each of these amounts of time correct to the nearest hour. a 9 h 50 min b 3.2 h c 4 h 12 min 49 s

Exercise 11-06

11 Write each of these amounts of time correct to the nearest minute. a 2 min 36 s b 10.5 min c 3 h 23 min 40 s

Exercise 11-06

12 Copy and complete the following. a 56 days = weeks c 960 s = min e 7 days = h

Exercise 11-06

b 4h= d 5 years = f 750 min =

13 What is the time: a 5 hours after 10:42pm? c 55 minutes before 7:15pm?

b 2 hours 28 minutes after 5:23am? d 7 hours 36 minutes before 1:19am?

e 15 hours 34 minutes after 7:00am? 14 What is the difference in time between: a 5.26am and 9:45am? c 1316 hours and 2003 hours? e 2347 hours and 0006 hours? 15 Find: a 6 h 45 min + 3 h 20 min c 4 h 33 min + 2 h 24 min

min weeks h Exercise 11-07

f 3 1--- hours after 3:40pm? 4

Exercise 11-07

b 11:56pm and 7:30am? d 0750 hours and 1425 hours? f 1529 hours and 3:28pm? Exercise 11-07

b 3 h 16 min − 1 h 26 min d 4 h 19 min − 2 h 50 min

16 If it is 10:00am in Sydney, use the maps on pages 405 and 406 to help you work out the time in: a Perth b Rio de Janeiro c Adelaide d Moscow e Hong Kong f San Francisco

Exercise 11-08

17 Use the information on page 406 to answer the following question. When it’s 2am in Sydney, what time is it in: a Melbourne? b Perth? c Darwin? d Canberra?

Exercise 11-09

18 Use the Countrylink train timetable on page 408 to answer the following. a How long does the 7.45 pm train take to get to Sydney? b What time do you need to catch the train at Moss Vale to be in Sydney by 11 am?

Exercise 11-10

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