Review of last
year’s work
1
I hope you remember last year’s work!
I’d better work through this chapter, just to
make sure.
Chapter Contents
1:01 Beginnings in number NS4·1
1:02 Number: Its order and structure NS4·1
1:03 Fractions NS4·3 1:04 Decimals NS4·3 1:05 Percentages NS4·3 1:06 Angles SGS4·2 1:07 Plane shapes SGS4·3 1:08 Solid shapes SGS4·1 1:09Measurement MS4·1,MS4·2,MS4·3 1:10 Directed numbers NS4·2
1:11 The number plane PAS4·5
1:12 Algebra PAS4·1–4
Working Mathematically
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
Note: A complete review of Year 7 content is found in Appendix A located on the Interactive Student CD.
Click here
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 NS4·1Write 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:
a six million, ninety thousand
b one hundred and forty thousand, six hundred
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
Outcome NS4·1Structure
Simplify: a 6 × 2 + 4 × 5 b 12 − 6 × 2 c 4 + 20 ÷ (4 + 1) d (6 + 7 + 2) × 4 e 50 − (25 − 5) f 50 − (25 − [3 + 19]) Simplify: a 347 × 1 b 84 × 0 c 36 + 0 d 3842 + 0 e 1 × 30 406 f 864 × 17 × 0 Write true or false for:a 879 + 463 = 463 + 879 b 76 × 9 = 9 × 76 c 4 + 169 + 96 = (4 + 96) + 169 d 4 × 83 × 25 = (4 × 25) × 83 e 8 × (17 + 3) = 8 × 17 + 8 × 3 f 4 × (100 − 3) = 4 × 100 − 4 × 3 g 7 × 99 = 7 × 100 − 7 × 1 h 17 × 102 = 17 × 100 + 17 × 2
Exercise 1:01
■ ‘Basic’ means ‘simple’. A:01A A:01A A:01B A:01D A:01D A:01E A:01F CD Appendix 1 2 3 4 5 6 7Exercise 1:02
A:02A A:02B A:02B CD Appendix 1 2 3List the set of numbers graphed on each of the number lines below. a b c d e f
Use the number lines in Question 4 to decide true or false for:
a 3 < 5 b −1 < 0 c 7 > 4 d 0 > −1 e 7·5 < 9 f 0 > 0·7 g < h 1 >
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? List all factors of:
a 12 b 102 c 64 d 140 List the first four multiples of:
a 7 b 5 c 12 d 13 Find the highest common factor (HCF) of:
a 10 and 15 b 102 and 153 c 64 and 144 d 294 and 210 Find the lowest common multiple (LCM) of:
a 6 and 8 b 15 and 9 c 25 and 20 d 36 and 24
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 Use a factor tree to write 252 as a product of prime factors. b Write 400 as a product of prime factors.
c Write 1080 as a product of prime factors. d Find the HCF of 400 and 1080.
e Find the LCM of 400 and 1080.
Find the smallest number that is greater than 2000 and:
a is divisible by 2 b is divisible by 3 c is divisible by 4 d is divisible by 5 e is divisible by 6 f is divisible by 8
g is divisible by 9 h is divisible by 10 i is divisible by 11
j is divisible by 25 k is divisible by 100 l is divisible by 12 Complete the following:
a If 152 = 225, then b If 83 = 512, then c If 132 = 169, then d If 43 = 64, then A:02C A:02C A:02D A:02E A:02E A:02E A:02E A:02F A:02G A:02H A:02I 4 6 7 5 4 3 2 0 1 –1 6 7 5 4 3 2 0 1 –1 22 23 21 20 19 18 16 17 15 9·5 10 9 8·5 8 7·5 6·5 7 6 0·7 0·8 0·6 0·5 0·4 0·3 0·1 0·2 0 0 1 4 1 2 3 4 1 4 1 2 3 4 1 1 1 1 2 5 1 4 --- 1 2 --- 1 4 --- 3 4 ---6 7 8 9 10 11 12 13 14 225 = . . . 3 512 = . . . 169 = . . . 3 64 = . . .
1:03
|
Fractions
Outcome NS4·3Change each fraction to a whole or mixed numeral.
a b c d
Change each mixed numeral to an improper fraction.
a 3 b 5 c 1 d 3
Simplify each fraction.
a b c d
Complete the following equivalent fractions.
a = b = c = d =
a + b − c + d −
a + b − c + d −
a Which fraction is the smaller: or ?
b Which fraction is the larger: or ?
c Arrange in order, from smallest to largest: { , , }.
d Arrange in order, from largest to smallest: { , , , }.
a 3 + 1 b 10 − 3 c 5 − 2 d 10 − 1 a × b × c × d ×
a 4 × b 1 × 3 c 3 × 1 d 2 × 1 a ÷ b ÷ c 4 ÷ d 1 ÷ 3 a Find of 2 km. b What fraction of 2 m is 40 cm?
1:04
|
Decimals
Outcome NS4·3 a Write (1 × 10) + (7 × 1) + (5 × ) + (3 × ) + (7 × ) as a decimal. A:04Ab Write (6 × 100) + (8 × 10) + (4 × 1) + (0 × ) + (2 × ) as a decimal.
Exercise 1:03
A:03B A:03C A:03D A:03E A:03F A:03G A:03H A:03I A:03J A:03K A:03L A:03M CD Appendix 1 10 2 --- 9 4 --- 87 10 --- 11 8 ---2 1 2 --- 3 10 --- 3 4 --- 1 7 ---3 8 10 --- 20 50 --- 15 100 --- 18 24 ---4 2 5 ---10 --- 3 4 ---100 --- 4 1 ---10 --- 1 3 ---120 ---5 3 10 --- 4 10 --- 19 100 --- 6 100 --- 3 8 --- 7 8 --- 5 12 --- 1 12 ---6 3 5 --- 3 10 --- 9 10 --- 3 4 --- 3 4 --- 2 5 --- 31 100 --- 1 5 ---7 3 4 --- 6 10 ---2 5 --- 1 3 ---8 10 --- 17 20 --- 3 4 ---1 2 --- 3 5 --- 1 4 --- 2 3 ---8 2 5 --- 3 4 --- 7 8 --- 1 2 --- 3 5 --- 1 5 --- 1 2 ---9 2 5 --- 3 4 --- 3 10 --- 7 10 --- 3 5 --- 2 3 --- 9 10 --- 15 16 ---10 3 5 --- 7 8 --- 3 4 --- 1 3 --- 1 2 --- 4 5 ---11 9 10 --- 1 2 --- 3 8 --- 3 5 --- 3 5 --- 1 2 --- 3 4 ---12 3 5---Exercise 1:04
A:04A CD Appendix 1 1 10 --- 1 100 --- 1 1000 ---1 10 --- 1 100---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
Change each fraction or mixed numeral to a decimal.
a b c 1 d 2
e f g h
Write in ascending order (smallest to largest):
a {0·3, 0·33, 0·303} b {2, 0·5, 3·1} c {0·505, 0·055, 5·5}
Do not use a calculator to do these.
a 3·7 + 1·52 b 63·85 − 2·5 c 8 + 1·625 d 8 − 1·625 a 0·006 × 0·5 b 38·2 × 0·11 c (0·05)2 d 1·3 × 19·1 a 0·6 × 100 b 0·075 × 10 c 81·6 ÷ 100 d 0·045 ÷ 10 a 48·9 ÷ 3 b 1·5 ÷ 5 c 8·304 ÷ 8 d 0·123 ÷ 4 e 3·8 ÷ 0·2 f 0·8136 ÷ 0·04 g 875 ÷ 0·05 h 3·612 ÷ 1·2 a $362 + $3.42 b $100 − $41.63 c $8.37 × 8 d $90 ÷ 8 a Round off 96 700 000 to the nearest million.
b Round off 0·085 to the nearest hundredth. c Round off 86·149 correct to one decimal place. d Write , rounded off to two decimal places.
1:05
|
Percentages
Outcome NS4·3Write each percentage as a fraction or mixed numeral in simplest form.
a 9% b 64% c 125% d 14 %
Write each fraction or mixed numeral as a percentage.
a b 1 c d 4
Write each percentage as a decimal.
a 47% b 4% c 325% d 300% e 50% f 104% g 12·7% h 0·3%
Change each decimal to a percentage.
a 0·87 b 1·3 c 5 d 0·825 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:04B A:04C A:04B A:04D A:04E A:04F A:04G A:04I A:04J 2 3 9 10 --- 13 100 --- 1 2 --- 99 100 ---3 5 --- 33 200 --- 5 8 --- 3 11 ---4 5 6 7 8 9
■ When you round off, you are making an approximation. 10 0.6.
Exercise 1:05
A:05A A:05B A:05C A:05D A:05F CD Appendix 1 1 2 ---2 3 4 --- 3 8 --- 37 300 --- 3 5 ---3 4 51:06
|
Angles
Outcome SGS4·2Name each angle marked with a dot, using the letters on the figures.
a b c d
Use a protractor to measure ∠ABC.
a b
Classify each angle using one of these terms: acute, right, obtuse, straight, reflex, revolution.
a b c d
e f g h
Draw a pair of:
a adjacent complementary angles b vertically opposite angles c alternate angles d adjacent supplementary angles.
Find the value of the pronumeral in each.
a b c d e f g h
Exercise 1:06
A:06A A:06B A:06C A:06D A:06D A B C T E B S D N M P Q D C B A B C A C B A 58° a° 47° b° 48° c° 72° 105° d° 73° e° 120° f° 68° g° h° 122° CD Appendix 1 2 3 4 51:07
|
Plane Shapes
Outcome SGS4·3 a What is the name of the shape on the right.b How many vertices has this shape? c How many sides has this shape? d How many angles has this shape? e How many diagonals has this shape?
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
e f g h
Calculate the value of the pronumeral in each quadrilateral.
a b c d
a Give the special name of each figure in Question 3. 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?
Use a ruler, a pair of compasses and a protractor to construct each of these figures.
a b c
Exercise 1:07
A:07A A:07D A:07G A:07F A:07B A:07E a° 60° 60° b° 70° 70° c° 40° 110° d° 55° e° 45° f ° 100° g° 75° 40° h° 65° a° b° c° 60° 60° 120° d° 80° 100° 120° 5 cm 4 cm 6 cm 4 cm 5 cm 60° 3·2 cm 5 cm 60° 80° 3 ·8 cm CD Appendix 1 2 3 4 51:08
|
Solid Shapes
Outcome SGS4·1a 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)
Name the solid corresponding to each net.
a b 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.
Exercise 1:08
A:08A A:08E A:08B A:08C A:08D A B C D E Front CD Appendix 1 2 31:09
|
Measurement
Outcome MS4·1, MS4·2, MS4·3Write down each measurement in centimetres, giving answers correct to 1 decimal place.
a b c d e
Complete each of these.
a 3000 mm = . . . cm b 2500 mL = . . . L c 630 mg = . . . g d 7 km = . . . m e 7·8 kg = . . . g f 2·5 m = . . . cm An interval is 8·4 cm long. It must be divided into 12 equal parts. How many millimetres would be in each part?
Find the perimeter of each of these figures.
a b c
Write the time on each clock in both conventional and digital time.
a b c d
Write each of these as a 24-hour time.
a 20 minutes past 5 (before noon) b 30 minutes past 5 (after noon) c 12 noon d 57 minutes past 11 (after noon) 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?
Find the area of each figure below.
a b c
Exercise 1:09
A:09B A:09E A:09C A:09C A:09D A:09E A:09F A:09G 0 mm 10 20 30 40 50 60 70 80 90 100 110 120 9 m 8 m 10 m 5 m 8·2 m 3·8 cm 6·2 cm 12 3 9 6 12 3 9 12 3 9 12 3 6 4·5 cm 3 cm 30 cm 1 m 50 cm 30 cm 7 mm CD Appendix 1 2 3 4 5 6 7 8d e f
Find the volume of each prism.
a b c d
Select the most likely answer.
a A teaspoon of water would contain: A 5 L B 5 mL C 5 kL b A milk carton would contain: A 1 mL B 1 kL C 1 L c 1 mL is the same volume as: A 1 m3 B 1 cm3 C 1 mm3
d A small packet of peas has a mass of: A 5 kg B 50 g C 5 g e A bag of potatoes has a mass of: A 20 kg B 20 g C 20 mg
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.
A:09I A:09H A:09J 6 m 8 m 7 cm 5 cm 12 cm 5 cm 9 3 cm 3 cm 3 cm 3 cm 2 cm 7 cm 2 cm 3 cm 5 cm 20 m 9 m 11 m 10 11
• List the 3D shapes that you can find in this picture.
1:10
|
Directed Numbers
Outcome NS4·2Which members of the following set are integers: {−3, , −1·5, 4, 0, −10}? A:10A
Give the basic numeral for each of the following. A:10B
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 m n o p
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
Outcome PAS4·5Find the coordinates of each of the points A to J. A:11B
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)
Exercise 1:10
A:10A A:10B A:10D A:10 A:10C CD Appendix 1 1 2 ---2 3 21 – 3 – --- –24 4 – --- –48 6 --- –1·8 2 ---4Exercise 1:11
A:11B A:11C y x 3 2 1 0 –1 –2 –3 –1 –2 –3 1 2 3 G D B A C E F J H I (0, 0) is the origin ■ The negatives are on the left on the x-axis. The negatives are at the bottom on the y-axis. CD Appendix 1 21:12
|
Algebra
Outcome PAS4·1–4If 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
m = . . . .
b
m = . . . .
Rewrite each of these without the use of × or ÷ signs.
a 5 × h + 2 b a + 3 × y c 6 × (a + 7) d 5 × 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) c 5x2 d 10 − 3x
If a = 2 and b = 5, find the value of:
a 3a + 7b b c 4a(b − a) d a2 + b2
If m = 2t + 1, find the value of m when t = 100. Discover the rule connecting x and y in each table.
Simplify: a 1m b 1 × a c 4 × y d y + y + y + y e f × 5 f a × b g 5 × k h 5 × a × b i 8x × 0 j 4y × 0 k 6m + 0 l 3a × 1 s 1 2 3 4 10 20 30 100 m s 1 2 3 4 10 20 30 100 m a x 1 2 3 4 5 b x 1 2 3 4 5 y 7 10 13 16 19 y 11 15 19 23 27
Exercise 1:12
A:12A A:12B A:12B A:12C A:12E A:12C A:12C A:12D, ...
,
,
, ...
,
,
CD Appendix 1 2 3 a+7 3 ---4 5 10a b ---6 7 8m 7a + 5a n 10a + a o 7b − b p 114a − 64a q m − 3m r 4b − 6b s 4x2 + 3x2 t 6ab − 5ab Simplify: a 3 × 5a b 6 × 10b c 7m × 3p d 8x × 4y e a × 4b f 6m × 5 g −3k × −5 h −6y × 3 i 12t ÷ 3 j 30t ÷ 3 k 6m ÷ 2a l 10a ÷ 5b m 15r ÷ 10 n 8m ÷ 6 o 3ab × 7a p 5ab × 4b Simplify: a 5m + 7m − 10m b 8x − 6x − x c 5x + 2y + 7y d 12a + 3b − 2a e 7p + 2q + 3p + q f 3r + 2A + 3A + 5r g 6a + 7b − 2a + 5b h 4m + 3 − 2m + 1 i 8m + 2a − 2m − 8a j 7a2 − 4a + 2a2 k 2x2 + 3x + 2x l 2x2 + 3x + 2x + 3
Expand, by removing grouping symbols:
a 3(a + 9) b 5(x + 2) c 10(m − 4) d 9(2a − 3) e 6(4t + 3) f 5(2 + 4x)
g m(m + 7) h a(a − 3) i a(6 + a)
Solve these equations.
a x + 5 = 9 b x + 4 = 28 c 12 − a = 5 d 6 − a = −1
e 6m = 42 f 5m = 100 g m + 7 = 2 h m − 1 = −5
The sum of two consecutive numbers is 91. What are the numbers?
1:12 Fraction, decimal and percentage equivalents
Challenge worksheet 1:12 The bridges of Königsberg
A:12H A:12E A:12E A:12F A:12I A:11A A:11B 9 10 11 12 13 14
, ...
,
,
0 1 2 3 4 5 1 3 5 7 9 m t This pattern of triangles formed from matchesgives the following table.
Plot these ordered pairs on a number plane like the one to the right.
Number of triangles (t) 1 2 3 4