NSM 8 Chapter 1

1 Review of last year’s work I’d better work through this chapter, just to make sure.

I hope you remember last year’s work!

Chapter Contents 1:01 1:02 1:03 1:04 1:05 1:06 1:07

Beginnings in number Number: Its order and structure Fractions Decimals Percentages Angles Plane shapes

NS4·1 NS4·1 NS4·3 NS4·3 NS4·3 SGS4·2

1:08 Solid shapes 1:09 Measurement 1:10 Directed numbers 1:11 The number plane 1:12 Algebra Working Mathematically

SGS4·1 MS4·1, MS4·2, MS4·3 NS4·2 PAS4·5 PAS4·1–4

SGS4·3

Learning Outcomes As this is a review chapter many outcomes are addressed. These include: NS4·1, NS4·2, NS4·3, PAS4·1, PAS4·2, PAS4·3, PAS4·4, PAS4·5, MS4·1, MS4·2, MS4·3, SGS4·1, SGS4·2, SGS4·3 Working Mathematically Stage 4 1 Questioning, 2 Applying Strategies, 3 Communicating, 4 Reasoning, 5 Reflecting Click here

Note: A complete review of Year 7 content is found in Appendix A located on the Interactive Student CD. Appendix A

1

This is a summary of the work covered in New Signpost Mathematics 7. For an explanation of the work, refer to the cross-reference on the right-hand side of the page which will direct you to the Appendixes on the Interactive Student CD.

1:01 | Beginnings in Number

Outcome NS4·1

Exercise 1:01 1

2

3

4 5 6 7

CD Appendix

Write these Roman numerals as basic numerals in our number system. a LX b XL c XXXIV d CXVIII e MDCCLXXXVIII f MCMLXXXVIII g VCCCXXI h MDCXV Write these numerals as Roman numerals. a 630 b 847 c 1308 d 3240 e 390 f 199 g 10 000 h 1773 Write the basic numeral for: ■ ‘Basic’ a six million, ninety thousand means b one hundred and forty thousand, six hundred ‘simple’. c (8 × 10 000) + (4 × 1000) + (7 × 100) + (0 × 10) + (5 × 1) d (7 × 104) + (4 × 103) + (3 × 102) + (9 × 10) + (8 × 1) Write each of these in expanded form and write the basic numeral. a 52 b 104 c 23 d 25 Write 6 × 6 × 6 × 6 as a power of 6. Write the basic numeral for: a 8 × 104 b 6 × 103 c 9 × 105 d 2 × 102 Use leading digit estimation to find an estimate for: a 618 + 337 + 159 b 38 346 − 16 097 c 3250 × 11·4 d 1987 ÷ 4 e 38·6 × 19·5 f 84 963 ÷ 3·8

1:02 | Number: Its Order and

A:01A

A:01A

A:01B

A:01D A:01D A:01E A:01F

Outcome NS4·1

Structure

Exercise 1:02 1

2

3

2

Simplify: a 6×2+4×5 b d (6 + 7 + 2) × 4 e Simplify: a 347 × 1 b d 3842 + 0 e Write true or false for: a 879 + 463 = 463 + 879 c 4 + 169 + 96 = (4 + 96) + 169 e 8 × (17 + 3) = 8 × 17 + 8 × 3 g 7 × 99 = 7 × 100 − 7 × 1

NEW SIGNPOST MATHEMATICS 8

CD Appendix

A:02A 12 − 6 × 2 50 − (25 − 5)

c 4 + 20 ÷ (4 + 1) f 50 − (25 − [3 + 19])

84 × 0 1 × 30 406

c 36 + 0 f 864 × 17 × 0

A:02B

A:02B b d f h

76 × 9 = 9 × 76 4 × 83 × 25 = (4 × 25) × 83 4 × (100 − 3) = 4 × 100 − 4 × 3 17 × 102 = 17 × 100 + 17 × 2

4

List the set of numbers graphed on each of the number lines below. a –1

0

1

2

3

4

5

6

7

–1

0

1

2

3

4

5

6

7

15

16

17

18

19

20

21

22

23

6

6·5

7

7·5

8

8·5

9

9·5

10

0

0·1

0·2

0·3

0·4

0·5

0·6 0·7

0·8

0

1 4

1 2

3 4

1

1 14

1 12

A:02C

b c d e f 5

1 43

2

Use the number lines in Question 4 to decide true or false for: a 34 e 7·5 < 9 f 0 > 0·7 g 1--- < 1---

A:02C d 0 > −1 h 1 1--- > 3--4

2

4

4

6

Which of the numbers in the set {0, 3, 4, 6, 11, 16, 19, 20} are: a cardinal numbers? b counting numbers? c even numbers? d odd numbers? e square numbers? f triangular numbers?

A:02D

7

List all factors of: a 12

A:02E

8

9

10

b 102

List the first four multiples of: a 7 b 5

c 64

d

140

c 12

d

13

A:02E

Find the highest common factor (HCF) of: a 10 and 15 b 102 and 153 c 64 and 144

A:02E d

Find the lowest common multiple (LCM) of: a 6 and 8 b 15 and 9 c 25 and 20

294 and 210 A:02E

d

36 and 24

11

a List all of the prime numbers that are less than 30. b List all of the composite numbers that are between 30 and 40.

A:02F

12

a b c d e

A:02G

13

Find the smallest number that is greater than 2000 and: a is divisible by 2 b is divisible by 3 d is divisible by 5 e is divisible by 6 g is divisible by 9 h is divisible by 10 j is divisible by 25 k is divisible by 100

14

Use a factor tree to write 252 as a product of prime factors. Write 400 as a product of prime factors. Write 1080 as a product of prime factors. Find the HCF of 400 and 1080. Find the LCM of 400 and 1080.

Complete the following: a If 152 = 225, then 225 = . . . c If 132 = 169, then 169 = . . .

A:02H c f i l

is divisible by 4 is divisible by 8 is divisible by 11 is divisible by 12 A:02I

83

3

b If = 512, then 512 = . . . d If 43 = 64, then 3 64 = . . .

CHAPTER 1 REVIEW OF LAST YEAR’S WORK

3

1:03 | Fractions

Outcome NS4·3

Exercise 1:03 1

CD Appendix

Change each fraction to a whole or mixed numeral. ----------a 10 b 9--c 87 2

2

4

Simplify each fraction. 8 a ----b

10

100

b

19 --------100

−

b

9 -----10

a

3 -----10

6

a

3 --5

7

a Which fraction is the smaller:

+

4 -----10

4

5

+

3 -----10

d

3 1--7

d

18 -----24

A:03D 20 -----50

Complete the following equivalent fractions. a 2--- = -----b 3--- = --------5

d

A:03C

4

10

10

4

10

Change each mixed numeral to an improper fraction. 3 a 3 1--b 5 ----c 1 3--2

3

A:03B 11 -----8

6 --------100

3 --4

−

3 --4

2 --5

b Which fraction is the larger:

or

or

c

15 --------100

c

4 --1

=

-----10

d

1 --3

c

3 --8

+

7 --8

d

5 -----12

c

3 --4

+

2 --5

d

31 --------100

A:03E =

--------120

−

1 -----12

A:03F

1 --5

A:03G

−

6 ------ ? 10

A:03H

1 --- ? 3

8 -, c Arrange in order, from smallest to largest: { -----

d Arrange in order, from largest to smallest: 8

a 3 2--- + 1 3---

9

a

×

3 --4

b

10

a 4×

3 --5

b 1 7--- × 3

11

a

÷

1 --2

12

a Find

3 --5

5

2 --5

9 -----10

4

b 10 7--- − 3 1--8

3 -----10

c 5 − 2 3---

2

×

3 --8

÷

d 10 1--- − 1 1---

5

7 -----10

c

3 --5

×

5

2 --3

d

c 3 3--- × 1 1---

8

b

17 3 ------ , --- }. 10 20 4 { 1--- , 3--- , 1--- , 2--- }. 2 5 4 3

4

3 --5

c 4÷

3

3 --5

9 -----10

A:03K

d 1 1--- ÷ 3 3---

A:03L

2

A:03M

CD Appendix 1 ------ ) 10

1 --------- ) + (7 × 100 1 - ) + (2 × × ----10

+ (3 ×

b Write (6 × 100) + (8 × 10) + (4 × 1) + (0 NEW SIGNPOST MATHEMATICS 8

4

Outcome NS4·3

Exercise 1:04

4

A:03J

5

2

b What fraction of 2 m is 40 cm?

of 2 km.

a Write (1 × 10) + (7 × 1) + (5 ×

15 -----16

A:03I

d 2 1--- × 1 4---

1:04 | Decimals 1

×

2

1 ------------ ) as a decimal. 1000 1 --------- ) as a decimal. 100

A:04A A:04A

2

Change each decimal to a fraction or mixed numeral in simplest form. a 0·7 b 2·13 c 0·009 d 5·3 e 0·85 f 0·025 g 1·8 h 9·04

A:04B

3

Change each fraction or mixed numeral to a decimal. 9 13 a ----b -------c 1 1---

A:04B

10 3 --5

e 4

f

100 33 --------200

99 d 2 --------

2

g

5 --8

100 3 -----11

h

Write in ascending order (smallest to largest): a {0·3, 0·33, 0·303} b {2, 0·5, 3·1}

A:04C c {0·505, 0·055, 5·5}

Do not use a calculator to do these. 5

a 3·7 + 1·52

b 63·85 − 2·5

c 8 + 1·625

d 8 − 1·625

A:04D

6

a 0·006 × 0·5

b 38·2 × 0·11

c (0·05)2

d 1·3 × 19·1

A:04E

7

a 0·6 × 100

b 0·075 × 10

c 81·6 ÷ 100

d 0·045 ÷ 10

A:04F

8

a 48·9 ÷ 3 e 3·8 ÷ 0·2

b 1·5 ÷ 5 f 0·8136 ÷ 0·04

c 8·304 ÷ 8 g 875 ÷ 0·05

d 0·123 ÷ 4 h 3·612 ÷ 1·2

A:04G

9

a $362 + $3.42

b $100 − $41.63

c $8.37 × 8

d $90 ÷ 8

A:04I

10

a b c d

■ When you round off,

A:04J

Round off 96 700 000 to the nearest million. Round off 0·085 to the nearest hundredth. Round off 86·149 correct to one decimal place. . Write 0.6 , rounded off to two decimal places.

you are making an approximation.

1:05 | Percentages

Outcome NS4·3

Exercise 1:05

CD Appendix

1

Write each percentage as a fraction or mixed numeral in simplest form. a 9% b 64% c 125% d 14 1--- %

A:05A

2

Write each fraction or mixed numeral as a percentage. 37 a 3--b 1 3--c --------

A:05B d

Write each percentage as a decimal. a 47% b 4% e 50% f 104%

c 325% g 12·7%

d 300% h 0·3%

Change each decimal to a percentage. a 0·87 b 1·3

c 5

d 0·825

2

4

3

4

5

8

300

4 3--5

A:05C

A:05D

a 8% of 560 L b 70% of 680 g c 5% of $800 d 10% of 17·9 m e Joan scored 24 marks out of 32. What is this as a percentage? f 250 g of sugar is mixed with 750 g of salt. What percentage of the mixture is sugar.

A:05F

CHAPTER 1 REVIEW OF LAST YEAR’S WORK

5

1:06 | Angles

Outcome SGS4·2

Exercise 1:06 1

CD Appendix

A:06A

Name each angle marked with a dot, using the letters on the figures. a b S c d D P A

B

Q

B

B

C

2

T

E

3

5

A:06B b

A

C B

Classify each angle using one of these terms: acute, right, obtuse, straight, reflex, revolution. a b c d

f

g

A:06D c

d d°

a°

e

h

b vertically opposite angles d adjacent supplementary angles.

Find the value of the pronumeral in each. a b

47°

e°

f

f°

b°

72° 48° c °

g

NEW SIGNPOST MATHEMATICS 8

105°

h

120°

g° 73°

A:06C

A:06D

Draw a pair of: a adjacent complementary angles c alternate angles

58°

6

D

N

C

e

4

M

Use a protractor to measure ∠ABC. a A B

C

A

68°

h°

122°

1:07 | Plane Shapes

Outcome SGS4·3

Exercise 1:07

CD Appendix

1

a b c d e

2

Choose two of these names for each triangle and find the vale of the pronumeral: equilateral, isosceles, scalene, acute-angled, right-angled, obtuse-angled a b c d

What is the name of the shape on the right. How many vertices has this shape? How many sides has this shape? How many angles has this shape? How many diagonals has this shape?

b°

60°

A:07A

A:07D

55°

40° 110°

70°

a°

60°

e

f e°

d°

c°

70°

g

h 75°

100°

h° g°

f°

40°

45°

3

65°

Calculate the value of the pronumeral in each quadrilateral. a a° b b° c d c°

A:07G 120°

60°

100°

60° 120°

d°

80°

4

a Give the special name of each figure in Question 3. A:07F b Which of the shapes in Question 3 have: i opposite sides equal? ii all sides equal? iii two pairs of parallel sides? iv only one pair of parallel sides? v diagonals meeting at right angles?

5

Use a ruler, a pair of compasses and a protractor to construct each of these figures. a b c cm

m

3·2

4c

m

60° 6 cm

3·8 cm

cm

4c

5

5 cm

A:07B A:07E

60°

80° 5 cm

CHAPTER 1 REVIEW OF LAST YEAR’S WORK

7

1:08 | Solid Shapes

Outcome SGS4·1

Exercise 1:08

A

B

C

D

E

1

a Give the name of each solid above. b Which of these solids have curved surfaces? c For solid B, find: i the number of faces (F) ii the number of vertices (V) iii the number of edges (E) iv number of edges + 2 (ie E + 2) v number of faces + number of vertices (ie F + V)

A:08A A:08B A:08C

2

Name the solid corresponding to each net. a b

A:08D c

a Draw the front view of this prism. b Draw the side view of this prism. c Draw the top view of this prism.

3

Front

8

CD Appendix

NEW SIGNPOST MATHEMATICS 8

A:08E

1:09 | Measurement

Outcome MS4·1, MS4·2, MS4·3

Exercise 1:09 1

Write down each measurement in centimetres, giving answers correct to 1 decimal place. a

b

0 mm 10

2

CD Appendix

20

c 30

d

40

Complete each of these. a 3000 mm = . . . . . . . cm d 7 km = . . . . . . . m

50

60

70

A:09B

e 80

90

100

110

120

A:09C b 2500 mL = . . . . . . . L e 7·8 kg = . . . . . . . g

c 630 mg = . . . . . . . g f 2·5 m = . . . . . . . cm

3

An interval is 8·4 cm long. It must be divided into 12 equal parts. How many millimetres would be in each part?

A:09C

4

Find the perimeter of each of these figures. a b 9m

A:09D c 8·2 m

8m

3·8 cm

5m 6·2 cm 10 m

5

Write the time on each clock in both conventional and digital time. a b c d 12

9

12

3

9

12

3

9

A:09E 12

3

3

6

6

6

Write each of these as a 24-hour time. a 20 minutes past 5 (before noon) c 12 noon

A:09E b 30 minutes past 5 (after noon) d 57 minutes past 11 (after noon)

7

a Rajiv ran at a speed of 5 m/s for 20 s. How far did he run? b Taya walked at a constant speed for 50 s. During this time she travelled 150 m. What was her speed? c A train travelling with a speed of 30 km/h travelled a distance of 120 km. How long did it take?

A:09F

8

Find the area of each figure below. a b

A:09G

3 cm

4·5 cm

c

1m

7 mm

30 cm

50 cm 30 cm CHAPTER 1 REVIEW OF LAST YEAR’S WORK

9

d

e

f 5 cm

5 cm

8m

12 cm

7 cm

6m

9

Find the volume of each prism. a b 3 cm

3 cm

3 cm

A:09H c

d

2 cm 7 cm

3 cm

3 cm

20 m 2 cm 5 cm 9m

10

11

Select the most likely answer. a A teaspoon of water would contain: b A milk carton would contain: c 1 mL is the same volume as: d A small packet of peas has a mass of: e A bag of potatoes has a mass of:

11 m

A:09I A A A A A

5L 1 mL 1 m3 5 kg 20 kg

B B B B B

5 mL 1 kL 1 cm3 50 g 20 g

C C C C C

The gross mass, net mass and container’s mass of a product are required. Find the missing mass. a Gross mass = 500 g, net mass = 400 g, container’s mass = g. b Net mass = 800 g, container’s mass = 170 g, gross mass = g.

5 kL 1L 1 mm3 5g 20 mg A:09J

• List the 3D shapes that you can find in this picture.

10

NEW SIGNPOST MATHEMATICS 8

1:10 | Directed Numbers

Outcome NS4·2

Exercise 1:10 1 2

3

4

CD Appendix

Which members of the following set are integers: {−3, 1--- , −1·5, 4, 0, −10}? 2 Give the basic numeral for each of the following. a −7 + 11 b −3 + 15 c −9 + 2 d −25 + 5 e 2 − 13 f 7 − 10 g −7 − 5 h −10 − 3 i 6 − (−10) j 14 − (−1) k 3 + (+7) l 15 + (+1) m 10 − (3 − 9) n 15 − (2 − 5) o 3 + (−7 + 11) p 11 + (−5 + 18) Simplify: a −4 × −3 b −8 × −2 c −0·2 × −3 d −0·1 × −15 e −4 × 14 f −5 × 8 g 7 × (−1·1) h 6 × (−12) i −35 ÷ (−5) j (−40) ÷ (−10) k 60 ÷ −6 l 14 ÷ −7 – 21 – 24 – 48 – 1·8 m --------n --------o --------p ----------–3 –4 6 2 Write down the basic numeral for: a −3 + 6 × 2 b −4 − 8 × 2 c 6−4×4 d −30 + 2 × 10 e −8 + 6 × −3 f 10 + 5 × −2 g (2 − 20) ÷ 3 h (8 − 38) ÷ −3 i 8 × $1.15 − 18 × $1.15 j 35° + 2 × 15° − 4 × 20°

1:11 | The Number Plane

A:11B

y B

3 2

–2

H –1

A C

0

1

2

■ The negatives are on the left on the x-axis. The negatives are at the bottom on the y-axis.

x

3

–1

I

E

–2 J

2

(0, 0) is the origin

D

1

–3

A:10

CD Appendix

Find the coordinates of each of the points A to J.

G

A:10D

Outcome PAS4·5

Exercise 1:11 1

A:10A A:10B A:10C

–3

F

On a number plane like the one in Question 1, plot the following points. Join them in the order in which they are given, to draw a picture. (2, 0) (3, 0) (3, −1) (1, −1) (1, −1·5) (2, −1·5) (2, −1) (−2, −1) (−2, −1·5) (−1, −1·5) (−1, −1) (−3, −1) (−3, 0) (−2, 0) (−1, 1) (1, 1) (2, 0) (−2, 0)

A:11C

CHAPTER 1 REVIEW OF LAST YEAR’S WORK

11

1:12 | Algebra

Outcome PAS4·1–4

Exercise 1:12 1

CD Appendix

If s represents the number of squares formed and m is the number of matches used, find a rule to describe each pattern and use it to complete the table given. a

,

,

A:12A

, ...

m=.......... s

1

2

3

4

10

20

30

100

m b

,

,

, ... m=..........

s

1

2

3

4

10

20

30

100

m 2

3

4

5

Rewrite each of these without the use of × or ÷ signs. a 5×h+2 b a+3×y c 6 × (a + 7)

Rewrite each of these, showing all multiplication and division signs. a 3a + 8 b 5p − 6q c 4(x + 2) d Given that x = 3, find the value of: a 6x b 2(x + 5)

5×a÷7 a+7 -----------3

A:12B A:12C

c 5x2

If a = 2 and b = 5, find the value of: 10a a 3a + 7b b --------b

d

10 − 3x A:12C

c 4a(b − a)

d

a 2 + b2

6

If m = 2t + 1, find the value of m when t = 100.

A:12C

7

Discover the rule connecting x and y in each table.

A:12D

a

8

12

A:12B d

x

1

2

3

4

5

y

7

10

13

16

19

Simplify: a 1m e f×5 i 8x × 0

NEW SIGNPOST MATHEMATICS 8

b

x

1

2

3

4

5

y

11

15

19

23

27 A:12E

b 1×a f a×b j 4y × 0

c 4×y g 5×k k 6m + 0

d h l

y+y+y+y 5×a×b 3a × 1

9

10

11

12

13

m 7a + 5a q m − 3m

n 10a + a r 4b − 6b

o 7b − b s 4x2 + 3x2

p t

114a − 64a 6ab − 5ab

Simplify: a 3 × 5a e a × 4b i 12t ÷ 3 m 15r ÷ 10

b f j n

c g k o

d h l p

8x × 4y −6y × 3 10a ÷ 5b 5ab × 4b

A:12E

Simplify: a 5m + 7m − 10m d 12a + 3b − 2a g 6a + 7b − 2a + 5b j 7a2 − 4a + 2a2

6 × 10b 6m × 5 30t ÷ 3 8m ÷ 6

7m × 3p −3k × −5 6m ÷ 2a 3ab × 7a

A:12E

b e h k

c f i l

8x − 6x − x 7p + 2q + 3p + q 4m + 3 − 2m + 1 2x2 + 3x + 2x

Expand, by removing grouping symbols: a 3(a + 9) b 5(x + 2) d 9(2a − 3) e 6(4t + 3) g m(m + 7) h a(a − 3) Solve these equations. a x+5=9 b x + 4 = 28 e 6m = 42 f 5m = 100

5x + 2y + 7y 3r + 2A + 3A + 5r 8m + 2a − 2m − 8a 2x2 + 3x + 2x + 3 A:12F

c 10(m − 4) f 5(2 + 4x) i a(6 + a) A:12H

c 12 − a = 5 g m+7=2

d h

6 − a = −1 m − 1 = −5

The sum of two consecutive numbers is 91. What are the numbers?

A:12I A:11A A:11B

14

,

,

, ...

This pattern of triangles formed from matches gives the following table. Number of triangles (t)

1

2

3

4

Number of matches (m)

3

5

7

9

Plot these ordered pairs on a number plane like the one to the right.

m 9

7

5

3

1 0

1

2

3

4

5

t

1:12 Fraction, decimal and percentage equivalents

Challenge worksheet 1:12 The bridges of Königsberg

CHAPTER 1 REVIEW OF LAST YEAR’S WORK

13

assi

men gn t

1A

Chapter 1 | Working Mathematically 1 Use ID Card 7 on page xx to identify: a 5 b 7 c 8 d 17 e 18 f 19 g 20 h 21 i 22 j 23 Consecutive numbers follow one after the other.

a the number of days that passed until all debris was removed. b the average mass of debris that was removed in each hour of labour. c the number of tonnes of debris removed if 1 ton = 907 kilograms. (Remember: 1 tonne = 1000 kg.) 5 This graph appeared in the Sunday Telegraph where a survey carried out by the Commonwealth Bank of Australia suggested that Australians are not saving enough of their income. Savings 12·7%

Mortgage/rent 17·6%

2 Use a calculator to find: a three consecutive numbers that have a sum of 822. b three consecutive numbers that have a sum of 1998. c three consecutive numbers that have a sum of 24 852. d three consecutive numbers that have a product of 336. e three consecutive numbers that have a product of 2184. f three consecutive numbers that have a product of 15 600. 3 Which counting number, when squared, is closest to: a 210? b 3187? c 2·6? 4 After the twin towers of the World Trade Centre fell in New York on 11 September 2001, it took 3.1 million hours of labour to remove 1 590 227 tons of debris. If this was completed on 10 May 2002, find:

1 Roman numerals 2 Multiplication tables A 3 Multiplication tables B 4 Order of operations 5 Operations with decimals 6 Equivalent fractions

14

NEW SIGNPOST MATHEMATICS 8

Entertainment 10·3% Credit card 8·2% Motoring 6·4% Household 26·4%

Other 18·4%

a If the wage represented here is $960 per week, how much is allocated to: i savings? ii mortgage/rent? iii credit card? iv motoring? b What do you think was the most popular reason for saving? c The three top reasons for saving were given in the article. Try to guess these three reasons in just five guesses.