Summer Adventure in Mathematics 1 (Third Edition)2 3

1. Number line

–4 0–2–3–5

negative numbers(less than 0)

increasing in value

positive numbers(greater than 0)

–1 42 31 5

2. Operations among directed numbers

(a) Addition

(i) (+a) + (+b) = +a + b

(ii) (-a) + (+b) = -a + b

(iii) (+a) + (-b) = +a - b

(iv) (-a) + (-b) = -a - b

(b) Subtraction

(i) (+a) - (+b) = +a - b

(ii) (-a) - (+b) = -a - b

(iii) (+a) - (-b) = +a + b

(iv) (-a) - (-b) = -a + b

(c) Multiplication

(i) (+a)(+b) = +ab (ii) (-a)(+b) = -ab

(iii) (+a)(-b) = -ab (iv) (-a)(-b) = ab

(d) Division

(i) ++

= +ab

ab

(ii) −+

= −ab

ab

(iii) +−

= −ab

ab

(iv) −−

= +ab

ab

–5 0–2 52

5 > 2 but –5 < –2

Mathematics Tips

E.g. (–7) + (–9) = –7 – 9 = –16

-9 can be interpreted as ‘move 9 units to the left’ from the point -7 along the number line.

–16 –14–15 –12–13 –10–11 –8–9 –7

Mathematics Tips

E.g. (–9) – (–7) = –9 + 7 = –2

+7 can be interpreted as ‘move 7 units to the right’ from the point -9 along the number line.

–9 –7–8 –5–6 –3–4 –2

Mathematics Tips

For multiplication or division between two numbers:

(i) I f the signs of the two numbers are different, the sign of the product/quotient is ‘negative’.

(ii) If the signs of the two numbers are the same, the sign of the product/quotient is ‘positive’.

Mathematics Tips

Directed Numbers1

Directed Numbers2 3

Matching

1. -5 - (-8) • • -2

2. 2 + (-4) • • 42

3. (-3) × 6 • • 2

4. (-8) ÷ (-4) • • 3

5. (-6) × (-7) • • -18

Fill in the Blanks6. -9 + (-4) - (-6) 7. -2 - (-3) × 4

= -9 - + = -2 - ( × )

= = -2 - ( )

= -2 +

=

True or False8. The sum of two negative numbers must be negative.

9. When a negative number is divided by a positive number, the quotient must be negative.

10. The product of two negative numbers must be positive.

11. 0, -1, -2, -3 are in ascending order.

Multiple-choice Questions12. Which of the following expressions is equal to 3 + (-2) × (-6)?

A. (3 - 2) × (-6) B. (3 - 2) × 6

C. 3 - 2 × 6 D. 3 + 2 × 6

13. Which of the following numbers is the greatest?

A. – 12

B. – 13

C. – 14

D. – 15

+(-a) = -a

-(-a) = +a

➤ multiplication first

➤

a

60°

A

C

B

D

52°

A

C

B

D

x

63°E

F

O

A

C

B

Dy + 10°

65°

Geometry28 29

True or False10. A pentagonal pyramid has 7 faces.

11. The sum of two acute angles must be an obtuse angle.

12. In the figure, if AEC and BED are straight lines, then b = 35°.

Multiple-choice Questions13. Which of the following sets of angles can be the interior angles of a triangle?

A. 18°, 56°, 104° B. 90°, 35°, 55°

C. 55°, 65°, 65° D. 100°, 80°, 0.1°

14. In the figure, find a.

A. 30° B. 60°

C. 90° D. 120°

Find the unknowns in the following questions (15 – 17).

15. 16. 17.

18. In the figure, AOD, BOE and COF are straight lines. Find x.

19. In the figure, ABCD is a trapezium with AB // DC. Find y.

35°b

A

EB

D

C

c

130°d

30° 70°e

60°

25°

35°

A

CDB

xy

y 60°60°

A

CB D

A

C

B

D65°

45°

70°

E

A

CB

KJ

D

108°

EG

FH

LI

A

C

B

D

55°

E24°

x + 15°

A

CB

D

z

55°

E

Summer Adventure in Mathematics 1 (Third Edition)30 31

20. In the figure, BA // CD and BC // DE. Find z.

21. In the figure, AEC is a straight line. Show that AB // DC.

22. In the figure, BCD is a straight line. Find ∠ACD.

23. In the figure, BCD is a straight line. Find x and y.

24. In the figure, AILB, CJKD, EIJF and GLKH are straight lines. BA // DC and EF // GH. If ∠BLG = 108°, find ∠CJF.

25. In the figure, BA // DE, ∠ABC = 55° and ∠CDE = 24°. Find x.

➤

find ∠EIB or ∠GKD first

Coordinate System56 57

21. In the figure, BCDE is a trapezium. A is a point on DE. Find the area of ∆ABC.

22. In the figure, point A(3, 150°) is rotated clockwise about the pole O through 90° to point B. Point B is then rotated clockwise about the pole through 90° again to point C.

(a) Find the polar coordinates of point B and point C.

(b) Draw the square ABCD.

23. In the figure, ∆PQR is an isosceles triangle in a rectangular coordinate plane. It is first rotated clockwise about the origin through 90°, then reflected about a line passing through the origin and the point (7, 7) to become the image ∆P′Q′R′.

(a) Draw the image ∆P′Q′R′.

(b) Tommy claims that if ∆PQR is rotated anticlockwise about the origin through 180°, the image can be reflected about the x-axis to overlap ∆P′Q′R′. Do you agree? Explain your answer.

EYA

A(8, 4)A(8, 4)

C(10, 2)C(10, 2)

B(2, 0)B(2, 0)

E D

x

y

O

90°60°

30°

330°

300°270°

240°

210°

180°

150°

120°

1 20°

3O

A

y

0x

−1

−2−1

−3

−2−3 1

12

2 3 4 5 6 7−4−5−6−7

−4−5−6−7

34567

P

R

Q

Summer Adventure in Mathematics 1 (Third Edition)58 PB

In a polar coordinate system, the polar coordinates of the points A, B and C are (5, 120°), (4, 210°) and (3, 300°) respectively. Find the area of ∆ABC. Ref: DSE Practice Paper Paper 1, Q6

Solution:Note that AOC is a straight line.

120°

150°

180°

210°

240°270°

300°

330°

90°60°

30°

0°1 2 3 4 5 6O

CB

A

X

Area of ∆ABC = AC × OB2

= (5 + 3) × 42

sq. units

= 16 sq. units

Follow-up24. In a polar coordinate system, the polar coordinates of the points D, E and F are (4, 60°), (7, 240°) and (6, 330°)

respectively. Find the area of ∆DEF.

Identify the base and the height of the triangle.

Scoring Tips

1. coordinates 坐標 2. polar coordinate system 極坐標系統

3 rectangular coordinate system 直角坐標系統 4. x-axis x 軸

5. y-axis y 軸

Revision TestPB 67

Section A (16 marks, each answer carries 1 mark)

1. − −− −

=12 ( 18)( 2)( 3)

A. -6 B. -5

C. 5 D. 6

2. Consider the polynomial 4x3 + 5x + 6. Which of the following is the coefficient of x?

A. 4 B. 5

C. 6 D. -6

3. Which of the following is the reasonable estimate of 153 + 149 + 151 + 152 + 146?

A. 100×5 B. 150 × 5

C. 200 × 5 D. 250 × 5

4. The marked price of a product is $360. If it is sold at 15% off, find the selling price.

A. $54 B. $306

C. $345 D. $414

5. In the figure, AB // CD. Find x.

A

C

B

D

x

65°

A. 15° B. 65°

C. 115° D. 155°

6.

Find the image of the above figure after rotating about O through 90° in a clockwise direction.

A. O B. O

C.

O

D.

O

7. What is the condition for ΔABC ≅ ΔDEF?

A

C

B D

F

E

A. SSS B. SAS

C. AAS D. RHS

8. In the figure, the volume of the prism is

Base area = 25 cm2

8 cm

A. 50 cm3. B. 100 cm3.

C. 150 cm3. D. 200 cm3.

O

Marks: 64Revision TestHKEP

Bridging Task72 7372 73

Short Questions1. Expand (3 + 4u)2 by using (a + b)2 = a2 + 2ab + b2.

(3 + 4u)2 =

2. It is given that the sum of the interior angles of an n-sided polygon is (n - 2) × 180°.

Find the sum of the interior angles of a pentagon.

The sum of the interior angles of a pentagon =

3. It is given that the area of a circle is πr2, where r is a radius of the circle.

In the figure, find the area of the shaded region. (Take π = 3.14.)

The area of the shaded region =

4. In ΔABC, it is known that if AB = AC, then ∠ABC = ∠ACB.

In ΔDEF, DE = DF and ∠DEF = 68°. Find ∠EDF.

∠EDF =

10 cm

10 cm

CB

A

FE

D

68°

Bridging Task