directed numbers - hkep · 2 summer adventure in mathematics 1 (third dition) 3 1. number line –5...
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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