1
Lesson (1):The set of natural numbers Lesson (2): Representing natural numbers on the . number line. Lesson (3) :Addition of natural numbers. Lesson (4) :Subtraction of natural numbers. Lesson (5): Multiplication of natural numbers. 2 Lesson (6): Division of natural numbers.
Remarks:
The set of counting numbers The set of natural numbers The set of odd numbers The set of even numbers The set of prime numbers
C = { 1 , 2 , 3 , 4 , …….. } N = { 0 , 1 , 2 , 3 , …….. } O = { 1 , 3 , 5 , 7 , …….. } E = { 0 , 2 , 4 , 6 , …….. } P = { 2 , 3 , 5 , 7, …….. }
Exercise ( 1 ) A. Complete using ∈ , ∉ , ⊂ , ⊄ : 1. { 2 } …… N 2. 0 …… C 3. 5 …… N 4.
1 5
9. 10. 11.
…… N
12.
5. 4545 …… C 6. 3.5 …… N 7. 2.2 …… C 8. 0 …… N
13. 14. 15. 16.
{0} {4,5,6} Ø
…… N …… N …… N
{1,2}
…… C
2332 Ø { 0, 1, 2 ….. } C
…… N …… C …… C …… N
B. Write ( True ) or ( False ) : 1. Set of prime numbers ⊂ N. 2.
24 36
4.
∈ N
5.
3. The set of natural numbers is a finite set.
C. Complete the following: 1. ………….. is the smallest natural number 2. …………. is the smallest counting number 3. The smallest even natural number is …….. 4. The smallest odd natural number is ……….. 5. The smallest even counting number is…… 6. The smallest odd counting number is …… 3
6.
Ø ⊂ 0
C
∈ N
9.6 ∉ N
Exercise ( 2 ) A. Write each of the following sets : 1. The set of natural numbers smaller than 5. 2. The set of natural numbers bigger than 2. 3. The set of natural numbers between origin and 4. 4. The set of natural numbers equal to or bigger than 3. 5. The set of natural numbers equal to or less than 4. 6. The set of natural numbers less than 4. 7. The set of even numbers between 3 and 11. 8. The set of odd numbers greater than 8. 9. The set of prime numbers between 0 and 12. 10. The set of odd numbers bigger than 2.
B. Make a number-line graph for each set of the following : 1. { 4 , 5 , 6 }. 2. { 0 , 1 , 2 , ……. } . 3. { 1 , 2 , ……… , 9 }. 4. The number
28 4
.
5. The number
24 6
.
6. The number
30 5
. 4
7. The set of natural numbers smaller than 4. 8. The set of even numbers. 9. The set of natural numbers bigger than 0. 10.
The set of natural numbers equal to or bigger than 2.
11.
The set of natural numbers between 2 and 4.
12.
The set of natural numbers equal to or less than 5.
C. Graph each of the following 1. The number half way between 4 and 6. 2. The number one- fourth of the way from 2 to 10. 3. The number one- third of the way from 0 to 9. 4. The number half way between 2 and 8. 5. The number half way between 3 and 4. 6. The number one-third of the way from 0 to 6. 7. The number one-fourth of the way from 1 to 9. 8. The number one- third of the way from 1 to 10.
D. Write the following sets : 1.
0
1
2
3
4
5
6
2.
0
1
2
3
4
5
6
3.
0
4. 5. 6.
1
0 0
0
0
1
0
3
2
1 1
2
0
2
2
0
4
3
0
3
3
0
5
4
4
6
5
4
0
0
0
5
7
0
8
0
9 10 0
0
11
0
12
0
13
0
14
6
0
5
0
6
0
7
0
8
0
9 10 0
6
5
0
11
0
12
0
13
0
14
No .
1
Property Closure
Description
Example
a + b= c
2+3=5
2 Commutative
a+b=b+a
2+3=3+2
3 Associative
(a+b)+c (2+3)+4 = a+(b+c) =2+(3+4) = a+b+c =9
4 Additive identity
a+0=0+a=a
6
2+0=0+2=0
Exercise ( 3 ) A. Use the number line to add the following natural numbers : 1.
2+3
2.
4+2
3.
3+1
4.
5+5
B. Complete : 1. 24 + 35 + 76 + 65 = 24 + ( …… + 76 ) + 65
Associative property
= …… + ( 76 + …… ) + 65
Commutative property
= ( …… + …… ) + ( …… + …… ) Associative property = ………. + ……….. = ………
Addition
2. 37 + 19 + 63 = 37 + (19 + 63 )
…………. Property
= 37 + (63 + 19 )
…………. Property
= ( 37 + 63 ) + ……
…………. Property
= 100 + 19 = 119
Addition
3. 44 + 67 + 56 + 33 = 44 + ( 67 + 56 ) + 33 = 44 + ( 56 + ….. ) + 33 = ( … + 56 ) + ( 67 + …. ) 7
…………. Property …………. Property ……….. Property
=100 + 100 = 200
Addition
4. ‌‌‌.. is the additive neutral element ( additive identity ).
C. Write the property of addition in each of the following : 1. 12 + 9 = 9 + 12 2. 49 + 0 = 49 3. ( 4 + 6 ) + 10 = 4 + ( 6 + 10 ) 4. 33 + 24 = 57 5. 125 + 0 = 125 6. 77 + 46 = 46 + 77 7. 11 + ( 23 + 66 ) = ( 11 + 23 ) + 66 8. 300 + 300 = 600
E. Use the commutative and associative property to find the result of the following : 1. 28 + 15 + 72 2. 54 + 72 + 46 3. 81 + 43 + 19 + 57 4. 76 + 15 + 85 + 24 5. 38 + 46 + 62 + 54 6. 53 + 75 + 25 + 47
F. Use the addition properties to find the result of each : 1.
33 + 67 + 75
2.
28 + 45 + 72 + 55
3.
41 + 65 + 59 + 35
4.
250 + 75 + 25 8
5.
88 + 56 + 12 + 44
6.
29 + 74 + 71
Exercise ( 4 ) A. Which results of the following operations belong to N ? 1. 6 + 2
6. 0 – 0
11. 3 – 4
5. 3 + 5
7. 0 – 1
12. 9 – 9
6. 6 – 4
8. ( 3 + 2 ) - 5
13. 3 – ( 9 – 2 )
4. 7 – 5
9. 7 – ( 2 + 4 )
14. ( 16 – 9 ) – 7
9. 1 – 3
10. 10 – ( 6 – 2 )
15. 6 – ( 3 + 3 )
B. Use the number line if possible to find the result of each of the following : 1. 2 + 5
3. 1 + 4
5. 3 – 2
2. 2 – 7
4. 4 – 9
6. 6 + 1
C. Write ( True ) or ( False ) : 1. 29 – 9 = 9 – 29
2.
3.
4. ( 23 – 3 ) – 5 = 23 – ( 3 – 5 )
0–5=5
77 – 7 = 70
5. Subtraction operation is always possible in N. 6.We can use the associative property in subtraction. 7. (13 – 45) belong to N 8. ( 9 – 4 ) – 1 = 9 - ( 4 – 1 )
9
No .
1
Property
Example
axb=c
2x3=6
2 Commutative
axb=bxa
2x3=3x2
3 Associative
(axb)xc = ax(bxc) = axbxc
(2x3)x4 =2x(3x4) = 24
4 Distributive
a (b+c) = (a×b) + (b×c) 2(3+4) = (2×3) + (2×4)
5
Closure
Description
Multiplication identity
Multiplication 6 by zero
a ×1=1×a=a
2×1=1×2=2
a × 0 =0 × a = 0
2×0=0×2=0
10
Exercise ( 5 ) A. Use the commutative and associative properties to simplify finding the result of each of the following : 1. 25 × 37 × 4
6. 8 × 28 × 125
2. 25 × 81 × 4
7.
5 × 37 × 2
3. 25 × 20 × 16
8.
5 × 25 × 20 × 4
4. 16 × 5 × 125
9.
8 × 10 × 125 × 5
5. 2 × 16 × 50
10. 25 × 2 × 4 × 5
B. Use the distribution property to calculate : 1. 12 × 5 + 12 × 15
4. 30 × 8 – 20 × 8
2. 20 × 12 + 50 × 12 + 30 × 12
5. 8 × 5 + 3 × 5
3. 7 × 10 – 5 × 7 + 7 × 6
6. 20 × 5 + 5 × 30 – 5 × 10
C. If the dimensions of a rectangle are 7cm and 3 cm, then its length is ……. and its width is ……..
D. If the dimensions of a rectangle are 6cm and 4cm. find its area. E. If the dimensions of a rectangle are 5cm and 2cm. Find its perimeter F. If the side length of a square is 6cm, find its area G. If the side length of a square is 5 cm, find its perimeter. 11
H. If the length of a rectangle is 8cm and its width is 4cm, answer : 1. Find its perimeter and area. 2. If its width increased by 1cm, find its perimeter. 3. If its length decreased by 2cm, find its perimeter. 4. If its dimensions increased by 1cm each , find its area.
H. Complete the following to obtain correct statements and mention the used properties : 1. 25 × 85 + 25 × 15 = ……… ( ……. + …….. )
…… Property
= …….. × 100 = ……….. 2. (76 ×123 ) + ( 76 × 12 ) – ( 76 × 35) = 76 × ( 123 + …… – ……. )
…… Property
= 76 × ………. = …………… 3. ……….. is the multiplicative neutral element ( identity ). 4. 78 x 5 = ( …… x …… ) + ( …… x …… ) = …………….. + ………….. = ……………
I.Use the distributive property to find the value of each of the following: 1.
52 × 101 =
6. 15 × 412 =
2.
74 × 99 =
7. 25 × 842 =
3.
502 × 45 =
8.
4.
103 × 25 =
9. 20 × 234 =
5.
35 x 98 =
5 × 395 =
10. 25 x 274 = 12
Exercise ( 6 ) A. State whether the division operation is "possible" or " not possible" in N 1.
54 ÷ 6
5.
7 ÷ 35
2.
5 ÷ 20
6.
30 ÷ 10
3.
1 ÷ 20
7.
12 ÷ 0
4.
0 ÷ 25
8.
48 ÷ 1
B. State whether the following statements are " true" or " false" 1. 2.
63 ÷ 9 = 9 ÷ 63 (225 ÷ 15) ÷ 5 = 225 ÷ (15 ÷ 5 )
3.
234 ÷ 3 ∈ N
4. 5. 6. 7. 8.
1225 ÷ 5 ∈ N 84 ÷ 7 = 7 ÷ 84 We can use the commutative property in division. We can use the associative property in division. Any number divided by zero equals zero.
13
Unit test A.Use the number line to add the following natural numbers : 1. 2 + 3
3. 4 + 2
2. 6 – 3
4. 5 – 1
B. Make a number-line graph for each set of the following : 1. { 1 , 3 , 5 }. 2. The number
3. { 0 , 2 , 4 , ……. } . 18 6
.
4. The set of natural numbers < 4.
C. Use the commutative and associative property to find the result of the following : 1. 18 + 40 + 32
3. 24 + 72 + 26
2. 71 + 53 + 29 + 47
4. 56 + 25 + 75 + 44
D. Write ( True ) or ( False ) : 1. 48 – 8 = 8 – 48
3.
2.
4. ( 47 – 7 ) – 7 = 47 – ( 7 – 7 )
0–9=9
88 – 8 = 8
E. Use the commutative and associative properties to simplify finding the result of each of the following : 1. 25 × 36 × 4 2.
4 × 49 × 25
3.
8 × 83 × 125
4.
5 × 56 × 2
G.Use the distribution property to calculate : 1. 15 × 5 + 15 × 15
3.
30 × 9 – 20 × 9
1. 20 × 17 + 50 × 17 + 30 × 17
4.
7×4+5×4
H. State whether " true" or " false" : 1. 54 ÷ 9 = 9 ÷ 54 2. 123 ÷ 3 ∈ N
3. (220 ÷ 20) ÷ 5 = 220 ÷ (20 ÷ 5 ) 4. 725 ÷ 5 ∈ N 14
Worksheet ( 1 ) A. Which results of the following operations belong to N ? 1.
4×2
4. 25 ÷ 100
2.
0 × 34
5. ( 6 × 3 ) ÷ 9
3.
( 5 × 6 ) ÷ 12
6. ( 2 ÷ 4 ) × 11
B. Choose the correct answer : 1.
( ∈ , ∉ , ⊂ , ⊄ )
35 ÷ 7 …….. N
2.
0 5
= …..
( 0, 1 , 5 , has no meaning )
3.
7 0
= ……
( 0, 1, 7, has no meaning )
4.
The multiplicative neutral element is ………..
(0,1)
5.
The additive identity element is ………..
(0,1)
6.
Zero …… C
7.
The distributive property is possible in ………..
( ∈ , ∉ , ⊂ , ⊄ ) ( Subtraction , Multiplication , Division )
8.
The associative property is impossible in ……….. ( Subtraction , Multiplication , Addition )
C. Use the number line to graph the following : 1. The set of natural numbers equal to or less than 4. 2. The number half way between 2 and 4. 3. The set of natural numbers bigger than 2. 4. 2 + 4. 15
D. Write each of the following sets : 1. The set of natural numbers smaller than 3. 2. The set of natural numbers bigger than 5 and smaller than 9. 3. The set of natural numbers between 5 and 6. 4. The set of natural numbers equal to or less than 3.
E. Complete: 1. The additive neutral element is ……………, while the multiplicative identity element is ………….. 2. 6 × 44 = 44 × 6
…………… Property
3. ( 6 + 5 ) + 8 = 6 + ( 5 + 8 )
…………… Property
4. Zero divided by any number is equals to ….. , while any number divided by ………… is meaningless. 5. If the side length of a square 4 cm. then its perimeter is ……….. and its area is ……………. 6. The smallest number in the set N is …………., while the smallest number in the set C is ……………….. 7. 5 × ( 4 + 1 ) = 5 × …….. + 5 × ……….. 8. ( 33 + …..) + 12 = 33 + ( 7 + ……)
I. Use the distributive property to find : 1.
46 x 4
2.
55 x 5
3.
26 × 101
4.
64 × 99
5.
23 × 35 + 46 × 35
6.
32 × 37 – 32 × 34 + 32 × 17 16
J.Use the commutative and associative property to find the result of the following : 1.
53 + 28 + 47
3.
44 + 67 + 56 + 33
2.
25 × 73 × 4
4. 10 × 4 × 7
H. If the length of a rectangle is 6cm and its width is 2cm, answer : 1. Find its perimeter and area. 2. If its length decreased by 2cm, find its perimeter. 3. If its width increased by 2 cm, find its area.
I. Write ( True ) or ( False ) : 1. 1 is the additive neutral element. 2.
8 4
∈N
3. We can use the associative property in division. 4. 27 ÷ 3 = 3 ÷ 27 5. Subtraction operation is always possible in N. 6. ( 12 + 14 ) + 13 = 12 + ( 14 + 13 ) 7. Perimeter of rectangle = ( L × W ) + 2 8. The set of natural number less than 3 = { 0, 1, 2, 3 }. 9. Zero belongs to the set of natural number. 10. Zero is the multiplicative identity element in N. 11. Perimeter of rectangle = 2L + 2w. 12. The smallest counting number is zero. 13. Perimeter of square = 4 × S. 14. The smallest even natural number is 2. 17
Lesson (1): Equation -Number sentences -Inverse operation sApplicati on and problem solving. 18
Lesson (2): Number
Exercise ( 1 ) A. Tell whether the equation is true or false for the given value of the variable : 1.
33 – z = 22 ; z = 11
2.
y + 8 = 20 ; y = 3
3.
s – 14 = 34 ; s = 48
4.
9 x n = 99 ; n = 10
5.
32 ÷ h = 4 ; h = 128
6.
x + 5.23 = 8.34 ; x = 3.11
B. Solve the equation for the given replacement set : 1. 2.
6 x b = 168 ; { 28 , 29 , 30 } 89 – a = 80 ; { 8 , 9 , 10 }
3. 144 ÷ c = 12 ; { 10 , 11 , 12 } 4.
55 + h = 100 ; { 45 , 55 , 65 }
5.
x ÷ 11 = 11 ; { 0 , 1 , 121 }
6. 5.25 – y =
1 ; { 2.25 , 3.25 , 4.25 }
C. If x = 28 , find the result of each of the following : 1.
x+2
2.
x–8
3. x x 10 19
4. x ÷ 4
D. Solve the following given the value of each variable : 1. 3 x + 4 , when x = 3. 2. 25 – 2y , when y = 6. 3. 2e + 4 , when e = 5. 4. 4b + 10 , when b = 0.5.
To find the value of an unknown number we should have an Equation to solve it.
No
Forming an Equation
Solving the Equation
.
The Statement
1
If 5 is added to the number M, the result will be 12.
M + 5 = 12
M = 12 – 5 M=7
2
If 6 is subtracted from X, the result will be 10.
X – 6 = 10
X = 10 + 6 X = 16
20
3
If 2 is multiplied by Y, the result will be 6.
2Y = 6
Y=6÷2 Y=3
4
If 14 is divided by Z, the result will be 7.
14 ÷ Z = 7
Z = 14 ÷ 7 Z=2
5
If a number is divided by 2, the quotient will be 9.
X÷2=9
X=9x2 X = 18
6
Find the value of Y , if Y + 8 = 13.
Y + 8 = 13
Y = 13 – 8 Y=5
Exercise ( 2 ) A. Use the inverse operation to solve the following equations : 1. x + 4 = 24
2.
n – 17 = 17
3 . c + 5 = 12
4.
z – 10 = 2
5.
6.
3b = 33
7. s÷6=3
8.
21
4h = 20 a÷4=7
9. 148m = 148
10. n ÷ 4.22 = 5
11.
12. 144 ÷ y = 12
7r = 49
13. z – 21 = 21
14.
b + 45 = 95
B. Find the value of each variable: 1. 3c + 4 = 16
5.
5y – 20 = 25
2. 4b + 10 = 38
6.
2h – 12 = 12
3. 7x + 7 = 7
7.
6v – 6 = 0
4. 8z + 10 = 98
8. 10e – 20 = 50
C. Put ( + , - , x or ÷ ) so that the equation has the given solution : 1. 3 …. b …. 4 = 19 ; 4
5. 6 …. c …. 9 = 4 ; 6
2. 5 …. h …. 5 = 10 ; 10
6. 28 …. y …. 11 = 15 ; 7
3. 80 …. t …. 4 = 10 ; 2
7. 7 …. s …. 2 = 7 ; 2
4. 15 …. b …. 15 = 0 ; 1
8. 24 …. g …. 1 = 1 ; 12
22
Sometimes we can symbolize an Unknown Number by an Alphabet Letter as follows : No .
The Statement
Expressing by Symbols
1
The sum of X and Y is 12
2
Add 4 to the number Z
3
The product of A and B is 18
A x B or AB
4
Double (twice) the number M
2 x M or 2M
5
Three times the number N
3 x N or 3N
6
Subtract 10 from the number X
7
One fifth of a number k
8
Eight divided by a number h
9
Number A increases (more than) B by 3
10
Number D exceeds (more than) C by 5
11
Number B decreases (less than) C by 4
12
Price of a pen 5 L.E, the cost of N pens is R.
13
Twice the difference of a number and four
2( y – 4 )
14
Three times the sum of a number and five
3( x + 5 )
15
Three is less than six times a number
X + Y = 12 Z+4
X – 10 K
Exercise ( 3 ) A. Express the following using symbols : 23
8÷h or or or
A–B=3 A=B+3 D–C=5 D=C+5 C–B=4 B=C–4 R = 5N
6f – 3
1. If we add 6 to the number B. 2. Double the number T. 3. Four times the number Z. 4. Ten less than a number q. 5. One quarter a number A. 6. Divide the number H by 6. 7. Twice the sum of a number and six. 8. Half the sum of a number and twelve. 9. We bought 2 books for y L.E each and 4 copybooks for z L.E each, the total cost will be …………… L.E. 10. A rectangle with length f cm and width 7 cm, its perimeter is ………. 11. A square with side length s cm then its perimeter is ………… 12. Sara is m years old. How old will she be in 5 years?
B. Write an equation for each word sentence : 1. The product of the numbers X and Z is 12. 2. 12 is seven more than twice the number v. 3. The difference of the number z and 6 is 48. 24
4. Four times the difference of a number and two is 20. 5. The difference between two numbers is 20 , then it will be ..... 6. Four times the number w is less than the number 20 by 4.
Exercise ( 4 ) A. Khaled is " z " years old, use z to write an expression : 1.
Khaled’s age five years ago.
2.
Khaled’s age 10 years from now.
3.
Khaled’s father age, if he is four times Khaled’s age.
4.
Khaled’s sister age, if she is half Khaled’s age.
5.
Khaled’s friend age, if he is more than Khaled’s age by two years.
6.
Khaled’s brother age, if he is younger than Khaled by three years.
B. Find the whole numbers that belong to each shape : 1.
+ 7 = 14
6.
+
= 15
2.
– 2 = 10
7.
+
=
3.
+
8.
x
=
= 26
25
4.
x
= 81
5.
–
=
9.
x
=
10.
÷
=
C. Complete in the same sequence : 1.
2 , 5 , 8 , …. , …. , ….
2.
2 , 3 , 5 , 8 , …. , …. , ….
3.
3 , 6 , 9 , …. , …. , ….
4.
7 , 77 , 777 , …. , …. , ….
5.
10 , 12 , 16 , 22 , …. , …. , ….
6.
5 , 10 , 20 , 35 , …. , …. , ….
7.
4 , 9 , 14 , 19 , …. , …. , ….
8.
1 , 5 , 25 , 125 , …. , …. , ….
9.
2 , 4 , 8 , …. , …. , ….
10. 25 , 20 , 16 , 13 , …. , …. , ….
Unit test A. Tell whether the equation is true or false for the given value of the variable : 1.
44 – z = 22 ; z = 22
3. y + 6 = 20 ; y = 15
2.
s – 12 = 24 ; s = 48
4. 7 x n = 77 ; n = 10
B. Solve the equation for the given replacement set : 26
1.
3 x b = 168 ; { 55 , 56 , 57 }
3. 55 – a = 40 ; { 10 , 15 ,
20 } 2.
121 ÷ c = 11 ; { 10 , 11 , 12 }
4. 65 + h = 100 ; { 35 , 45 ,
55 }
C. Use the inverse operation to solve the following equations : 1. x + 14 = 24
3.
n – 27 = 27
2. 3c + 26 = 56
4.
4z – 12 = 36
D. Put ( + , - , x or ÷ ) so that the equation has the given solution : 1.
3 …. b …. 5 = 28 ; 5
3.
6 …. c …. 9 = 2 ; 3
2.
5 …. h …. 5 = 50 ; 2
4.
3 …. y …. 11 = 3 ; 11
E. If z = 45 , find the result of each of the following : 1. z + 5
2. z – 5
3. z x 5
4. z ÷ 5
F. Put ( + , - , x or ÷ ) so that the equation has the given solution : 1. v …. 13 …. 9 = 12 ; the solution is 39 2. c …. ( 15 …. 5 ) …. 7 = 7 ; the solution is 10
G. Write an equation for each word sentence : 1. The product of the numbers a and b is 24. 2. 14 is eight more than twice the number w. 27
3. Five times the number y is less than the number 25 by 5. 4. The difference of the number z and 9 is 26.
28
Worksheet ( 2 ) Choose the correct answer :
A.
1. Thirteen is four less than a number is expressed by …………. ( 13 = 4 – n , 13 = n – 4 , 13 – 4 = n , 4 – 13 = n ) 2. Which of the following is equivalent to 6 x 34 ? ( ( 6 x 3 ) + ( 6 x 4 ) , ( 6 x 30 ) + ( 6 x 4 ) , ( 3 x 6 ) + ( 40 x 6 ) ) 3. 3 x 5 – 2 = 7 + ( 3 2) ( + , – , x , ÷ ) 4. 1 , 3 , 9 , 27 , 81 , ……. ( 713 , 243 , 371 ) 5. 16h = 80, then h = …… ( 3 , 4 , 5 ) 6. The additive identity element is …… ( 0 , 1 , Ø )
Complete :
B. 1. 2. 3.
v + ( h + …… ) = v + ( …… + m ) If y = 4 and z = 48 ÷ y, then z = …… 8 x 56 = ( …… x …… ) + ( …… x …… ) = …… + …… = …… 4. A square with side length 5 cm and it decreased by 1 cm, then its area was …… and now its area became …… 5. The value of 2b – 3, when b = 5 is …………..
C. Ramy is " m " years old. Use m to write an expression for
each of the following : 1. 2. 3. 4.
Ramy’s age three years ago. Ramy’s age six years from now. Ramy’s father’s age, if he is four times as old as Ramy. Ramy’s sister’s age, if she is half as old as Ramy.
D. Use the number line to graph the following : 1.
3+5 3+3
2. 6 – 3
E. Use the distributive property to find : 29
3.
1.
36 x 4
3.
22 x 6
2.
42 × 101
4. 36 × 28 + 64 × 28
F. Express the following using symbols : 1. Twice the sum of a number and six. 2. Divide the number h by 23. 3. One third a number b. 4. Twice the difference of a number and four.
G.
Solve the following :
1. x + 16 = 32, then half x = ……
2.
4. n – 64 = 36, then 4n = ……
3c + 30 = 60, then 2c = …..
3. 11b = 121, then 5b = …..
5.
z ÷ 22 = 88, then
z = ……
6.
5y – 15 = 20, then 7y = …..
H. Write an equation for each word sentence : 1. The product of the numbers a and b is 36. 2. 15 is eight more than twice the number h. 3. Five times the number n is less than the number 6 by 2. 4. The difference of the number y and 23 is 25.
I. Find the whole numbers that belong to each shape : 30
1.
+
= 72
8.
x
=
2.
x
= 35
9.
+
=
3.
–
=
10.
÷
=
J.
Put ( + , - , x or ÷ ) in each space :
1. 3 …. b …. 4 = 19 ; 4 2. 6 …. c …. 9 = 4 ; 6 3. 5 …. h …. 5 = 10 ; 10 4. 8 …. y …. 11 = 15 ; 7
K. Use the commutative and associative property to find
the result of the following : 1. 32 + 120 + 48 2. 34 + 57 + 66 + 43 3. 125 × 56 × 8 4. 16 × 5 × 7
L. If the side length of a square is 6cm, find: 1. Its perimeter and area. 2. Its perimeter if its side length decreased by 2cm. 3. Its area if its side length increased by 2 cm.
M. Write ( True ) or ( False ) : 1. 0 is the additive identity element. 31
2.
5 2
∈N
3. We can use the associative property in subtraction. 4. 241 x 32 = 32 x 241 5. Subtraction operation is always impossible in N. 6. ( 22 – 15 ) + 13 = 22 – ( 15 + 13 ) 7. Perimeter of rectangle = 2L + 2W 8. The set of natural number equal or more than 2 = { 3 , 4 , 5 , …… }. 9. Zero belongs to the set of counting number. 10.
m ÷ 3 = 9, then m = 3.
32
Lesson (1) : Graphing points and figures.
Some figures have one or more folding
Lesson (2) : Geometric transformation. Lesson (3) : Reflection. 33
line, the line in which makes it close exactly on each other and this line called a line or an axe of symmetry, and these figures will be called symmetrical figures; Some examples for symmetrical figures :
Some examples for non symmetrical figures :
Some basic figures and no of their axes of symmetry :
Square : 4
Equilateral triangle : 3
Rectangle : 2
Isosceles triangle : 1 34
Rhombus : 2
Scalene triangle : 0
Parallelogram : 0
Circle : uncountable
Trapezium : 0
Isosceles trapezium : 1
We use the graph copybook to draw the 2-coordinate plane.
35
ď&#x192;ź The two numbers which give the location of a point on a grid are called the Coordinates of the point ( ordered pair ).
36
ď ś Using the graph above: - The ordered pair of the point A = ( 2 , 2 ) - The ordered pair of the point B = ( 7 , 2 ) - The ordered pair of the point C = ( 7 , 5 ) - The ordered pair of the point D = ( 2 , 5 ) After joining the points A , B , C , D we get the rectangle ABCD. AB = DC = 5 cm BC = AD = 3 cm
Exercise ( 1 ) A.After graphing the following points and connecting them in order (2,3) , (7,3) , (2,6) , (7,6) Find the following : a.The name of the drawn figure. b The perimeter of the drawn figure. c. Draw one line of symmetry.
B.In the two coordinate plane, draw the â&#x2C6;&#x2020; ABC in which A = ( 2 , 2 ) , B= ( 6 , 2 ) and C = ( 2 , 6 ), then find the following : 1. The length of each side.
37
2.The type of the triangle according to its side length and according to the measure of its angles. 3. Draw the ∆ XYZ congruent to the ∆ ABC. 4.Write the ordered pair of each point of the ∆ XYZ .
C. In the two coordinate plane, graph the point A = ( 2 , 2 ) , B = ( 6 , 2 ) and C = ( 2 , 6 ), by connecting the points in order find the following : 1.The name of the resulted figure. 2.The length of each side. 3.The measure of angle A. 4.The type of the figure according to its side length and according to the measure of its angles. D. Graph the figure XYZL by giving the following ordered pairs to each point : X( 3 ,1 ) , Y( 7 , 1 ) , Z( 7 ,5 ) , L( 3 , 5 ) 1. What is the name of the figure XYZL ? 2. Draw all axes of symmetry if it has. 3. Find the perimeter of the figure. 4. Find the area of the figure. 5. Determine the coordinates of the midpoint of XY and the midpoint of YZ.
38
Look carefully to each kind of the transformation in the following table.
No Transformation . Type
Movement Name
1
Rotation
Turn
2
Translation
Slide
3
Reflection
Flip
39
Before / After
Exercise ( 2 ) 1. Tell whether each transformation is the result of a Flip, a Slide or a Turn : 1.
4.
2.
5.
3.
6.
2. In the two coordinate plane, draw the ∆ ABC in which A = ( 1 , 1 ), B = ( 3 , 4 ) and C = ( 1 , 6 ); in the same page draw the ∆ XYZ in which X = ( 5 , 1 ) ,Y = ( 7 , 4 ) and Z = ( 5 , 6 ) then state the type of the transformation of ∆ XYZ. 3. In the two coordinate plane, draw the ∆ DEF in which D = ( 1 , 3 ), E = ( 5 , 3 ) and F = ( 3 , 5 ); in the same page draw the ∆ LMN in which L = ( 7 , 5 ) , M = ( 7 , 1 ) and N = ( 9 , 3 ) then state the type of the transformation of ∆ LMN. 4. Graph the figures ABCD and XYZL in which the points of the first are A= ( 3 , 1 ) , B = ( 6 , 1 ) , C = ( 4 , 3 ) , D = ( 1 , 3 ) and the points 40
of the second are X = ( 1 , 4 ) , Y = ( 4 , 4 ) , Z = ( 6 , 6 ) , L = ( 3 , 6 ) and state the type of transformation and the name of the movement.
After drawing the ∆ ABC we are going to draw ∆ A'B'C' which is the reflection of ∆ ABC on the line of reflection ( L ).
41
42
The two numbers which give the location of a point on a grid are called the Coordinates of the point ( ordered pair ). In ∆ ABC ,
A=(1,2) , B=(4,2) , C=(1,6)
∆ A'B'C' the reflection of ∆ ABC should have the points A' = ( 9 , 2 ) , B' = ( 6 , 2 ) , C' = ( 9 , 6 ) If we draw ∆ A'B'C' with other ordered pairs, it will not be the reflection of ∆ ABC
Exercise ( 3 )
A. Copy the ∆XYZ in the coordinate plane, and then draw its reflection on the reflection line ( L ), and state the ordered pairs of each new point : X' = ( …… , …… ) Y' = ( …… , …… ) Z' = ( …… , …… )
43
B. In the two coordinate plane, draw the reflection of ∆ABC on the reflection line
( L ) giving the ordered pairs of the reflection points :
A' = ( …… , …… ) B' = ( …… , …… ) C' = ( …… , …… )
C. In the two coordinate plane, draw the reflection of the parallelogram LMNO on the reflection line ( L ) giving the ordered pairs of the reflection points : 44
L' = ( …… , …… ) M' = ( …… , …… ) N' = ( …… , …… ) O' = ( …… , …… )
D. Graph the figure ABCD in which the points A = ( 1 , 1 ) , B = ( 7 , 1 ) , C = ( 9 , 4 ) , D = ( 3 , 4 ) then draw its image by reflection on DC.
E. In the two coordinate plane, draw the ∆ DEF in which D = ( 4 , 0 ), E = ( 4 , 10 ) and F = ( 1 , 5 ), then draw the reflection image of the ∆ DEF on DE and write the ordered pair of the new points.
F. In the two coordinate plane, draw the figure XYZL in which X = ( 5 , 1 ), Y = ( 5 , 9 ), Z= ( 1 , 7 ) and L = ( 1 , 3 ), then draw the reflection image of the figure on XY , then draw one line of symmetry to cut both into two symmetrical figures.
G. Graph the figure ABCD in which the points A= ( 2 , 2 ) , B = ( 8 , 2 ) , C = ( 8 , 8 ) , D = ( 2 , 8 ) , then write the name of the drawn figure and draw all the lines of symmetry possible to this figure.
H. Graph the figure for the coordinates given. Then graph a congruent figure by adding the number 4 to each of the coordinates : (3,0) , (5,2) , (3,6) , (1,2)
I. Graph the figure for the coordinates given. Then graph a congruent figure by adding the number 6 to each of the coordinates : (1,3) , (8,3) , (6,6) , (3,6)
J. Graph the figure for the coordinates given. Then graph a congruent figure by adding the number 3 to each of the coordinates ( 2 , 3 ) , 45
( 4 , 2 ) , ( 4 , 6 ) , ( 2 , 5 ) after drawing the other figure state the type of transformation.
K. In the two coordinate plane, draw the figure with ordered pairs ( 5 , 2 ) , ( 5 , 4 ) and ( 9 , 4 ) then draw another figure with switching the numbers and state the kind of movement.
Worksheet ( 3 ) A. Choose the correct answer : 1. Twenty four is ten more than a number is expressed by …………. ( 24 = n + 10 , 24 + 10 = n , 24 = 10 + n , 10 + 24 = n ) 2. Which of the following is equivalent to 27 x 452 ? ( (452 x 2) + (7 x 452) , (20 x 452) + (452 x 7) , (452 x 70) + (2 x 452) ) 3. 4 x 5 + 8 = 18 + ( 20 2) ( + , – , x , ÷ ) 4. 6 , 18 , 54 , ……. ( 72 , 60 , 162 ) 5. m ÷ 3.27 = 6 , then m = …… ( 19.62 , 9.27 , 2.27 ) 6. The multiplicative neutral element is …… ( Ø , 1 , 0 )
B. Complete : 1. 2. 3. 4.
n x ( a + b ) = ( …… x …… ) + ( …… x …… ) If y = 4 and z = 48 ÷ y, then z = …… ( 73 x 17 ) + ( 17 x 27 ) = …… x ( …… + …… )= …… + …… = …… The ……… and ……… properties can be possible in addition and multiplication operations. 5. The ………. property is only possible in multiplication. 6. If we flipped an image then it is ………… transformation. 7. Turning an image is called ………… 8. The two numbers which give the location of a point are called ……… 9. The symmetrical figure has at least ……… axe ( line ) of symmetry. 10. If we turned the letter b it will be …… but if we flip it, it will be ……… 46
11. In switching the coordinates ( 2 , 4 ) & ( 5 , 0 ) they will be ….. & ….. 12. The reflection of the number 8008 is …………
C. Use the number line to graph the following : 1. 4 + 4 2. 8 – 5 3. 5 + 5 4. The set of prime numbers between 1 and 15. 5. The set of natural numbers equal to or less than 6. 6. The set of even numbers greater than zero.
L. Use the distributive property to find : 1. 5 x 202
3.
48 x 4
2. 6 x 555
4. 12 × 16 + 16 × 28 + 20 x 16
M. Express the following using symbols : 1. Four times the difference of a number and six. 2. Twice the sum of six and half the number h. 3. One third a number b. 4. Twice the difference of a number and four.
N.
Solve the following equations : 1. 2x + 10 = 32
4.
4n – 64 = 36
2. x + 15 = 45
5.
z ÷ 4 = 36
3. 10b = 120
6.
3y ÷ 2 = 12
O. Write an equation for each word sentence then solve it : 1. The sum of twice a number and four is 18. 47
2. 30 is twelve more than twice the number h. 3. 16 is less than Three times the number n by 2. 4. The difference of the number y and 25 is 75.
P. Put the parentheses to make the following sentences true : 1.
6+6÷6–6=1
2.
6+6÷2–6=0
3.
2–8÷8–0=1
4.
8x2÷4–4=0
Q. Write the following sets : 1. 2. 3.
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
R. Use the addition and multiplication properties to solve the following : 1. 312 + 120 + 80 2. 30 + 77 + 70 + 23 3. 25 × 50 × 8 48
4. 16 × 125 × 5
K. Tell whether each transformation is the result of a Flip, a Slide or a Turn : 1.
2.
3.
L. In the two coordinate plane, draw the ∆ DEF in which D = ( 1 , 3 ), E = ( 5 , 3 ) and F = ( 3 , 5 ); in the same page draw the ∆ LMN in which L = ( 7 , 5 ) , M = ( 7 , 1 ) and N = ( 9 , 3 ) then state the type of the transformation of ∆ LMN.
M. In the two coordinate plane, draw the ∆ LMN in which L = ( 5 , 1 ), M = ( 10 , 5 ) and N = ( 0 , 5 ), then draw the reflection image of the ∆ LMN on NM and write the ordered pair of the new points.
N. Graph the figure for the coordinates given. Then graph a congruent figure by adding the number 4 to each of the coordinates: (3,0) , (5,2) , (3,6) , (1,2)
49
Lesson (1) : Investigating circumference. Lesson (2) :Area of triangle. Lesson (3) :Area of parallelogram. Lesson (4) :Area of rhombus. Lesson (5) :Area of square. 50
We are going to study that the distance around the circle ( the perimeter of the circle ) is called the circumference of the circle. Note : C = the circumference /
d = the diameter
d = 2 x r = 2r
&
/ r=
r = the radius 1 2
xr = r÷2
To get the circumference we have this rule :
C= π xd =2xπxr [ where π =
22 7
≃ 3.14 ( to the nearest hundredth ) ]
From this rule we can deduce these rules to find the diameter or the radius.
d diameter
Circumference
C 2
radius r
d=
C
π Exercise ( 1 ) 51
r=
r π C 2x
π
A. Find the circumference to the nearest whole number : ( π = 3.14) 1.
Diameter = 5.4 cm.
4. Radius =
3.5 cm.
2.
Diameter = 32 mm.
5. Radius = 236 mm.
3.
Diameter = 6.6 cm.
6. Radius =
7 cm.
B. Find the circumference to the nearest whole number : (π =
)
1. d = 21 cm.
4.
r = 14 cm.
2. d = 35 cm.
5.
r =
3. d = 70 mm.
6.
r = 280 mm.
7 cm.
C. Find the circumference to the nearest whole number : ( π = 3.14) 1.
4. 7cm
2cm
2.
5 cm
3.
12 cm
D. Find the diameter and the radius from the following circumferences of some circles : 1.
C = 88 cm
(π =
)
2.
C = 44 cm
(π =
)
52
3.
C = 21.98 cm
( π = 3.14)
4.
C = 28.26 cm
( π = 3.14)
Exercise ( 2 ) A. Which is longer, the circumference of a circle of radius = 2.3 cm, or the perimeter of a square of side length 3.5 cm.
( π = 3.14)
B. Which is bigger, the perimeter of a rectangle 4.5 cm long and 3.5 cm wide or the circumference of a circle of diameter 7 cm.
(π =
)
C. Find the difference between the circumference of two circles with radii 4.6 cm and 2.6 cm. approximating the result to the nearest one decimal place. ( π = 3.14)
D. A circle of circumference 33 cm, find the length of its diameter. ( π =
)
E. A circle of circumference 628 mm, find the length of its radius.( π = 3.14 ) F. Find the sum of the perimeters of an equilateral triangle of side length 7 cm, and a circle of diameter 7 cm.
(π =
)
G. Find the difference between the perimeter of a square with side length 10 cm, and the circumference of a circle of diameter 10 cm. ( π = 3.14 )
H. The perimeter of a rectangle = the circumference of a circle of radius 7 cm, find the width of the rectangle if its length = 16 cm. 53
(π =
)
I. Which is smaller in perimeter, a rectangle with dimensions 10 cm and 4 cm or a circle with radius 3.5 cm?
(π =
)
J. Find the difference between the circumference of a circle of radius 21cm, and another circle of diameter 14 cm.
(π =
)
Exercise ( 3 ) A. The figure represents a fence surrounding a garden, the diameter is 9 m. Find the length of the fence.
( π = 3.14)
4m 9m
B. Calculate the perimeter of the figure opposite, if AC = BC = 5 cm. and the radius of the circle N = A3.5 cm . N
(π =
)
C
B
C. The figure opposite is formed of a semicircle ( half a circle ) and a line segment, then its perimeter equals ……………….. cm. 14cm 54
(π =
)
D.Find the perimeter of the figure opposite.
( π =3.14 )
8 cm 4 cm
(h) height
h base ( b )
h
b
b
Note :
A = the Area /
b = the base
/
h = the height
To get the area we have this rule :
Area of triangle =
A =
x base x height 55
x b x h
From this rule we can deduce these rules to find the base or the height.
A
b h A
b= or
h=
A
0.5 x h
0.5 x b
2A
2A
b=
Exercise ( 4 ) h =
b
h
A. Find the area of each of the following triangles : figure ( 1 )
figure ( 2 )
figure (3 )
5cm
4cm
3 cm 6 cm
8 cm
5 cm
1. Area of figure ( 1 ) = …………………………………………………….. 2. Area of figure ( 2 ) = …………………………………………….. 3. Area of figure ( 3 ) = …………………………………………………..
B. Find the length of the base of each of the following triangles : 1.
Its area = 6 cm² and its height = 4 cm.
2.
The height = 5 cm and its area = 10 cm². 56
3.
The area = 8 cm² and its height = 8 cm.
4.
The height = 5 cm and its area = 7.5 cm².
C. Find the length of the height in each of the following triangles : 1.
Its area = 12 cm² and its base = 4 cm.
2.
The base = 5 cm and its area = 15 cm².
3.
The area = 9 cm² and its base = 6 cm.
4.
The base = 4 cm and its area = 14 cm².
Exercise ( 5 ) A. Find : 1. The area of a triangle its base is 6 cm. long and its height = 3.5 cm. 2. The area of an equilateral triangle its perimeter = 24 cm & height = 7 cm. 3. The area of triangle if its height equals the side length of a square with perimeter 40 cm, the base of the triangle = 3 cm. 4. The height of a triangle if its base = 12 cm, and its area = 24 cm². 5. The length of the base of a triangle if its area = 60 cm², the height is equals to 10 cm.
B. Which is larger in area : A piece of land in the shape of triangle with base 8 cm, and height 5 cm or a piece of land of a rectangle with dimensions 6 cm. and 5 cm.? 57
C. Calculate : The perimeter of the opposite figure if its area = 24 cm.
6 cm
10 cm
D. Find the total area of the opposite figure : 7 cm 4 cm 12 cm
Remember :
The parallelogram is a quadrilateral.
Each two opposite sides are equal and parallel.
The diagonal bisects it into two triangles.
Its area = the area of these 2 triangles.
Area = 2 (
x base x height ) = base x height
Note :
A = the Area /
b = the base 58
/
h = the height
To get the area we have this rule :
Area of parallelogram =
base x height
A =
h
b x h
h
base
base
From this rule we can deduce these rules to find the base or the height.
A b=
h
b
A
h=
h
A b
Exercise ( 6 ) A. Find the area of each of the following figures :
6cm
3cm
2cm
10 cm
3cm 4.5 cm
B. Complete : 1. The base of a parallelogram is 10 cm and its height is 4 cm, then its area = ……….. cm². 59
2. The area of the parallelogram if its height is half of its base length, where its base is 12 cm long = …………………… cm². 3. If the area of a parallelogram with base 5 cm is 35 cm², then its height is ……………. cm long. 4. The area of a parallelogram is 28 cm² and its height is 4 cm, then its base = ………. cm. 5. The base of a parallelogram is ……… when its area is 24 cm² and it height is 6 cm. 6. If the base of a parallelogram is 7 cm and its area is 49 cm², then its height is ………. cm. 3. Which is bigger in area, a triangle of base 4 cm, and height 6cm, or a parallelogram with base 9 cm and 3.5cm? 4. The area of a triangle equals the area of a parallelogram with base 7 cm and 4 cm height, find the length of the triangles base if its height is 3cm.
Note : It is a parallelogram with sides equal in length.
It has 2 perpendicular diagonals.
Its 2 diagonals are not equal in length. Note : A = the Area / L = side length / h = the height / d = diagonal To get the area we have these 2 rules : Area of rhombus =
side length x height
A= L X h Area of rhombus = half the product of its two diagonals 60
A =
X
x
A
1st rule :
h
L
4cm 6cm Area of rhombus = L x h = 6x4 = 54 cm²
L=
h=
4 cm nd
2 rule :
A 5 cm
Area of rhombus =
x d1 x d2
d2 = =
A
d1 =
x5x4
0.5 x d2
A
d1 d2 1 d1 = 2A 1 d2 1 d2 = 2A
0.5 x d1
d1
= 10 cm²
Exercise ( 7 )
A. Complete : Rhombu s
Side length
Height
Diagonal 1
Diagonal 2
Area
1
9 cm
… cm
12 cm
6 cm
… cm²
2
… cm
4 cm
8 cm
… cm
16 cm²
61
3
4.2 cm
5 cm
6 cm
… cm
… cm²
1. The perimeter of rhombus 1 = ………………………………….. 2. The perimeter of rhombus 2 = ………………………………….. 3. The perimeter of rhombus 3 = …………………………………..
B. Find the area of the rhombus its two diagonals are 20, 12 cm.
C. If the height of a rhombus is 9 cm long and its side length is 14cm, find its area.
D. The length of the diagonals of a rhombus are 10, 8 cm, If its height is 5 cm, calculate the side length of this rhombus.
E. The side length of a rhombus is 6.5 cm and its height is 3.2 cm; If the length of one of its diagonal is 6 cm, calculate the length of the other diagonal.
Note :
The square is a quadrilateral.
It is a rectangle in which the length and the width are equal. 62
It has 2 equal and perpendicular diagonals.
Note : A = the Area / S = the side length / h = the height / d = diagonal To get the area we have these rule :
Area of square =
side length x side length
A= s x s Area of square = half the product of its two diagonals
A =
x d xd
3.5 3.5
5 cm Area = side x side
Area =
A = 5 x 5 = 25 cm²
A=
xdxd x7x7
= 24.5 cm ≃ 25 cm
S=
d= Exercise ( 8 )
A. Answer the following : 1. The side length of a square is 12 cm. Find its area. 63
2. The diagonal length of a square is 8 cm long, find its area. 3. The perimeter of a square is 40 cm , calculate : a. Its side length.
b. Its area.
c. The length of its diagonal.
4. A square and a rectangle are equal in area; the length and width of the rectangle are 7cm and 2 cm. Find the perimeter of the square. 5. The area of a square is 144 cm², Find its side length. 6. The area of a square is 84 cm²; Find the length of its diagonal. 7. Which is larger in area, a square of side length 13 cm or a rhombus of diagonals 4.5 cm and 11 cm? 8. If the side length of a square equals the radius of a circle with circumference 28 cm, find : a. The radius of the circle.
(π =
)
b. The area of the square. 9. The perimeter of a square is 40 cm, if its side length equals the diameter of a circle, find : a. The side length of the square. b. The circumference of the circle. ( π = 3.14 )
64
Study well :
The Rules The Circle C=π xd
or
C= 2 x π x r
d=
&
r=
The Triangle A= xbxh
b=
A 0.5 ×h
or
b=
h=
A 0.5 ×b
or
h=
Per. = C
Per. = s1 + s2 + s3
The Rectangle
The Parallelogram
A=lxw
A=bxh
l =
b =
w=
h=
Per. = ( l + w ) x 2
Per. = ( l + w ) x 2
The Square
The Rhombus
A=sxs
&
s=
A= lxh
65
&
l =
h=
A= xdxd &
d=
A= xd1 xd2
d1=
A 0.5 × d 2
d2=
A 0.5 ×d1
or
or
d1=
d2=
Per. = l x 4
Per. = s x 4
66
Unit test A. Complete : 1. The area of the parallelogram if its height is 5 cm, where its base is 12 cm long = …………………. 2. The area of the rhombus its two diagonals are 14cm, 12 cm = ……. 3. The area of a triangle its base is 6 cm. long and its height = 3.5 cm is ………… 4. The base of a parallelogram is 10 cm and its height is 4 cm, then its area = ……….. cm². 5.
The circumference of a circle with diameter 28 cm is …….. ( π =
)
6. The diagonal length of a square is 8 cm long, then its area is ………. 7. The circumference = ………….. = ………..... 8. The area of the square = …………… or …………… 9. The area of the rhombus = …………… or ………… 10. The area of the parallelogram = ……………………. 11. The area of the triangle = ……………………. 12. The base of the parallelogram = ……………………
B. Find the area of the rhombus its two diagonals are 25, 15 cm. C. If the height of a rhombus is 10 cm long and its side length is 16 cm, find its area.
D. Which is bigger in area, a triangle of base 5 cm, and height 7 cm, or a parallelogram with base 8 cm and 4.5cm?
67
E. Find the difference between the circumference of a circle of radius 21cm, and another circle of diameter 14 cm.
68
(Ď&#x20AC; =
)
Worksheet ( 4 ) A. Find the area of the figure : ABCDEFGH
B
A
Where : AH = 6 cm AB = DC = 10 cm
H
BC = 7cm
C
G
D F
E
ED = FG = GH = 1cm
B. Calculate the perimeter of the of the opposite figures : ( Ď&#x20AC; = 3.14 ) 1.
4cm Radius =4cm
2.
5cm Diameter = the side of the square = 5cm
3.
4.
20cm
10cm
7 cm
69
C. Choose the correct answer :
( π=
)
1. The circumference of a circle with diameter length 42 cm is …… cm. ( 48 , 96 , 168 , 132 ) 2. The circumference of a circle with radius length 3.5 cm is …… cm. ( 44 , 22 , 14 , 11 ) 3. The side length of a rhombus is 5 cm, its height is 4.2 cm, and the length of one of its diagonals is 6cm, then the length of the other diagonal = ….cm. ( 7 , 21 , 14 , 6 ) D. Complete : 1. The area of the shaded triangle shown = …………cm² 4cm
8 cm 12cm 2. The area of the shaded triangle shown = …………cm². 10 cm 6 cm
4. In the figure opposite, the base of the triangle is XC and its height is …………..
C base
X B
A 70
E. Choose the correct answer : 1. Twenty four is three times a number is expressed by …………. ( 24 = n + 3 , 24 + 3 = n , 24 = 3 x n , 24 ÷ 3 = n ) 2. 1 + 1 , 1 + 1 + 3 , 1 + 1 + 3 + 5 , ………. ( 1+3+5+7 , 1+1+3+5+7 , 1+1+3+5+6 ) 3. 2y ÷ 3 = …… ; y = 6 ( 3 , 4 , 5 ) 4.
The area of the triangle = ………
(bxh , 2xbxh ,
5. 6m + 12 = 30 , then m = …… 6. The set of natural numbers between 3 & 4 is ……
xbxh )
( 3 , 4 , 5 ) ( Ø , 1 , 0 )
B. Complete : 1. 23 x ( 32 + 32 ) = ( …… x …… ) + ( …… x …… ) 2. If n = 3 and h = 45 ÷ n, then h = …… 3. ( 35 x 27 ) + ( 27 x 53 ) = …… x ( …… + …… ) = …… + …… = …… 4. The ……… and ……… properties are impossible in division and subtraction operations. 5. The ………. property is only possible in multiplication. 6. If we turn an image then it is ………… transformation. 7. Flipping an image is called ………… 8. The two numbers which give the location of a point are called ……… 9. The circumference of a circle = …... x …… x …… 10. If we turned the letter d it will be …… but if we flip it, it will be ……… 11. The coordinates ( 2 , 4 ) by adding 4 will be ( ….. , ….. ). 12. The rotation of the number 696 is …………
C. Use the number line to graph the following : 1. 2 + 4 2. 7 – 5 3. 3 – 3 4. The set of prime numbers between 1 and 15. 5. The set of natural numbers equal to or less than 6. 71
6. The set of even numbers greater than zero.
Lesson (1) : Collecting and organizing data. Lesson (2) : Displaying data. Lesson (3) : Representing data by pie charts. Lesson (4) : 72 Reading and interpreting data.
A. The ages of 50 pupils are given in the opposite table : Age
12
13
14
15
16
No. of pupils
11
14
17
6
2
Represent these data by : 1. A histogram.
2. A frequency polygon.
B. How many meals did you eat out last month? No. of meals
0-
4-
8-
12 -
16 -
Frequency
10
20
45
30
25
Show these data by : 1. A histogram.
2. A frequency polygon.
C. How many hours do you sleep each night? No. of hours
5-
7-
9-
11 -
Frequency
6
12
9
3
Show this data by : 1. A histogram.
2. A frequency polygon.
73
A. The following table shows the favourite TV programs for some persons : Sports
News
Art
1 2
1 4
1 4
Represent these data by pie chart graph then answer the following question : In which program were the greatest viewers?
B. The following table shows the production of sugar in four factories : First
Second
Third
Fourth
500
750
250
500
Represent these data by pie chart graph then answer the following questions : 1.
Which factory produces most ?
2.
Which factory produces lowest ?
C. A group of 360 Children was asked to Choose their favoriite color, the table below shows their choises : Black White
Color No. of children
45
90
1. How many children choose blue? 2. What fraction of the Children Choose pink? 3. Represent these data by a pie chart.
Unit test 74
Pink
Blue
135
……
A. The ages of 60 pupils are given in the opposite table : Age
11 -
12 -
13 -
14 -
15 -
No. of pupils
12
15
20
8
5
Represent these data by :
1. A histogram. 2. A frequency polygon.
B. How many hours did you study out last year? No. of hours
0-
2-
4-
6-
8-
Frequency
20
50
30
40
10
Show these data by : 1. A histogram.
2. A frequency polygon.
C. The following table shows the production of Tea in four factories : First
Second
Third
Fourth
150
200
100
150
Represent these data by pie chart graph then answer the following questions : 1. Which factory produces most? 2. Which factory produces lowest?
D. The following table shows the production of sugar in four factories : First
Second
Third
Fourth
1 2
1 4
1 8
……..
Complete the table then represent these data by a pie chart graph.
Second term revision tests Test ( 1 ) 75
A. Choose the correct answer : 1. { 1 , 2 }
( ∈ , ∉ , ⊂ , ⊄ )
…… C
2. The circumference of a circle = ……
( 2πr , πr , 4πr , 2πd )
3. Set of prime numbers ⊂ N.
( true , false )
4. 37 + 19 + 63 = 37 + (19 + 63 )
( …………. Property )
5. ……….. is the additive neutral ( identity ) element.
( 0,1,Ø )
B. Use the property to find the result of the following : 1.
54 + 72 + 46
2.
33 + 67 + 75
3.
29 + 74 + 71
4.
5 × 37 × 2
5.
64 × 99
C. Solve the equation for the given replacement set : 1.
89 – a = 80
where a = { 8 , 9 , 10 }
2.
X + 2 = ……
where X = 34
D. Find : 1. X + 4 = 24 2. The value of Z where four times the number Z is 16. 3. Twice the sum of a number and twelve. 4. 5 , 10 , 20 , 35 , …. , …. , ….
E. 1. Graph the figure ABCD by giving the following ordered pairs to each point : Point A = ( 3 , 1 )
,
Point B = ( 7 , 1 ) 76
Point C = ( 7 , 5 )
,
Point D = ( 3 , 5 )
a. What is the name of the figure ABCD ? b. What is the length AB ? c. Find the perimeter of the figure.
2. Tell whether each transformation is the result of : ( Turn or Flip or Slide ) a.
b.
c.
F. 1. The area of a square is 64 cm² , Find : a. Its side length.
b. Its perimeter.
c. The length of its diagonal. A
2. Calculate the perimeter of the opposite figure : AB = 14 cm, and AC = BC = 10 cm. ( π = 3.14 )
C B
4. Ahmed bought 500 stalks of flowers on a certain day, the pie chart shows the number of each type of flower sold . a. How many tulips were sold? b. What fraction of the flowers sold were carnations? Types of flowers
Roses
Tulips
No. of flowers
……
……
77
Roses Tulips orchids carnations
Carnation s 100
Orchids
Total
125
500
Test ( 2 ) A. Choose the correct answer : 1. Which symbol makes the sentence true? 3x5–2=7+(3
2)
( + , – , x , ÷ )
The circumference of a circle with diameter 21 cm long is …..cm.
2.
( π=
)
( 44 , 22 , 11 , 14 )
3. 1 , 4 , 16 , …….
( 32 , 64 , 72 )
4. 5m + 4 = 14, then m = ……
( 1 , 4 , 2 , 6 )
5. The area of a triangle its base 12cm and its height 5 cm,= …. cm². ( 60 , 34 , 30 )
B. Complete : 1. The set of natural numbers smaller than 5. 2. 250 + 75 + 25 ( use the addition properties ) 3.
2 × 16 × 50
( use the multiplication properties )
4. The set of natural numbers ≤ 3. 5. The lengths of the diagonals of a rhombus are 10 cm & 8 cm, then its area = ……
C.1. Write the following sets : a. b.
0
1
2
3
4
5
6
0
1
2
3
4
5
6
2. Place parentheses to make the statement true : 78
a.
5+5÷5–5=1
b.
5+5÷5–2=0
3. Find the area of the shaded part. 6 cm
4 cm
D. 1. Which has larger area a parallelogram of base 7.5 cm and its height is 5 cm or a rhombus its two diagonals are 6 cm and 8.2 cm long.
2. In the figure opposite, calculate the perimeter. L M
( π = 3.14 )
Z
7 cm
M
5 cm
X Y E. 1. If the length of a rectangle is 8 cm, and its width is 4 cm, find : c. Its perimeter. b. Its perimeter if its length is increased by 2 cm.
2. Which of the following represents the number zero, and which represents meaningless? a. 0 ÷ 10 c.
b. 90 ÷ 0 d.
3. How many hours do you study daily ?
79
No. of hours
5-
7-
9-
11 -
Frequency
10
18
6
3
Show these data by : a. Histogram.
b. Frequency polygon.
Test ( 3 ) A. Answer the following questions : 1.
Underline the natural numbers from
23 , 4.98 , 4 , 0.5 ,
2. { 0 , 1 , 2 , 3 , 4 , ………. } ⊂ N
( true or false )
3. What is the natural numbers greater than 6 but less than 8?
4.
The natural number between
and
is ………………..
5. Make a number-line graph for { 3 , 4 , 5 , 6 }.
B. Complete : 1. The set of natural numbers is closed under the ……… and ………. operations. 2. 34 + 48 + 66 = …………… ( use the properties of addition ) 3. ……….. is the multiplicative identity element in N. 4. ( 12 x 4 ) x …… = 12 x ( 4 x 7 ) 80
5. 4 x 1005 = ……………………… ( use the distributive property )
C. Find the result of the following : 1. Use the inverse operation x + 9 = 24 2. Mona is “ f ” years old, use f to write an expression for : a. Her age 3 years ago.
b. Her age 10 years from now.
3. Write an equation for : Ten is 8 more than twice the number z . 4. A transformation of a figure produced by ………, ……… or …………. the figure. 5. Graph the figure for the coordinates given, then graph a congruent figure by adding 3 to each of the coordinates : (1,2) , (3,4) , (7,4) , (1,2)
D. Find : 1. The circumference of a circle with a radius 0f 5 cm.
( π = 3.14 )
2. The distance covered when the tyre of bicycle makes 50 complete rotations if the radius is 40 cm.
( π = 3.14 )
3. The diameter of a circle if half of the circumference of the circle is equals to 157 cm.
( π = 3.14 )
4. The difference between the area of a squared shaped land with side length 6m and the area of triangle shaped garden with base 7m and height 8m .
F. 1. If ABCD is a parallelogram of area 750 cm², E is a point on find the area of the triangle AEB.
D
E C
81 A
B
,
2. A survey team asked 100 persons about how many hours a week they watched TV. No. of hours
0-
2-
4-
6-
8-
Persons
8
22
50
16
4
Show these data by : a. Histogram.
b. Frequency polygon.
Test ( 4 ) A. Complete : 1. The diagonal length of a square is 5 cm long, then its area =………… 2. The area of a triangle its base 10 cm long & its height is 3 cm.= ……. 3. The set of natural numbers ≤ 7 is ……………... 4.
0
1
2
3
4
5
6
Represents the set { ………………… }
5. The circumference of a circle = ……………x …………………
B. Choose : 1. X + 4 = 24
X = ………
( 2 , 20 , 28 ) ( ∈ , ∉ , ⊂ , ⊄ )
2. 4.5 …….N 3.Five times the number B.
( ( 5 x B ) , ( 5 – B ) , ( B+ 5 ) )
4.The height of a parallelogram if its area = 64cm² and the base =16cm ( 82
8 ,
6 , 16 , 4 )
5.The additive neutral element is ……………
( 0 , 1 , 2 , 10 )
C. Find: 1. The area of the shaded part shown = …………cm² 8 cm
4 cm
2. Use the associative and the commutative property of multiplication to simplify a.
25 x 62 x 4
b.
16 x 43 x125
D. 1. Find the no. that will make the following statements true : 1.50 x ( 12 + 18 ) = …….x …….. + …….. x ……. 2.( …. X 10 ) x 5 = 20 x ( 10 x 5 )
2. Name the property of addition and multiplication illustrated by each statements : 1. 3 + ( 2 + 3 ) = ( 3 + 2 ) + 5 2. 32 + 0 = 0 + 32 = 32 3. 7 x 9 = 9 x 7
E .1. Write the equation : 1. The difference of four times a number and two. 2. The product of five and three more than a number. 3. Twice the sum of a number and three. 83
4. Half the number y.
2. Find : 1. The area of a parallelogram its base is 17 cm and the height is 12 cm. 2. The solution of the equation 38 x G = 912 3. Six times the difference of a number and five. 4. The value of 43 x 12 using the distributive property. 5. The transformation of this figure
2. Use the following table of data to make a histogram : Number of hours
0-
4-
8-
12 -
Frequency
6
10
4
8
Test ( 5 ) A. Complete using ∈ , ∉ , ⊂ , ⊄ : 1.
1 5
……
N
2.
{1,2}
……
C
3.
Ø
……
C
4.
2.2
……
C
5.
{ 0, 1, 2 ….. } ……
C 84
B. 1. Find by sing the distributive property: 1. 52 × 101 = 2. 15 × 412 = 3. 74 × 99 =
2. State whether " true" or " false" : 1. 63 ÷ 9 = 9 ÷ 63 2. 135 ÷ 3 ∈ N 3. ( 23 – 3 ) – 5 = 23 – ( 3 – 5 )
C. 1. Use the number line to find the following products : 1. The set of even numbers. 2. The set of natural numbers bigger than 0 and less than6. 3.
The set of natural numbers equal to or bigger than 2.
4. The set of natural numbers between 2 and 4.
2. Calculate the area of a rhombus if its diagonals are 4 cm, 7 cm long. D. 1. After graphing the following points and connecting them in order : (2,3) , (7,3) , (9,6) , (4,6) Find the following : 1. The name of the drawn figure. 2. The length of each side. 3. The perimeter of the drawn figure.
2. Write the type of the transformation of each figure: Figure 1
Figure 2 85
E. 1. Calculate the perimeter of the figure opposite: if AB = AC = 7 cm. and the diameter of the circle = 3.5 cm . B
(π =
C
) A
2. After graphing the following points and connecting them in order ( 4, 1 ) , ( 4 , 3) , ( 8, 3 ) a. Switch the numbers then draw the coordinate. b. What did your new figure look like? c. Tell the transformation of the figure produced.
3. A group of 250 children was asked to choose their favourite colour , the table below shows their choices. Represent these data by pie chart. Colour
Blue
Red
Green
Yellow
Frequency 100 50 25 a. How many children chose yellow? b. What fraction of the children chose green?
……..
Test ( 6 )
A. Name the property : 1. ( 6 x 4 ) x 2 = 6 x ( 4 x 2 ) 2. 7 x 5 = 5 x 7 3. 3 x 99 = 3 x ( 100 – 1 ) = ( 3 x 100 ) - ( 3 x 1 ) = 297
B. Express each of the following in the form of equations : 86
1.
Five less than half of a number .
2.
The product of 6 and three more than a number.
C. If X = { 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 } then put true or false : 1.
25
∈
N
3.
35
∉
N
2.
64
∉
N
4.
100 ∈
N
D. Complete using all the numbers 1 , 3 , 5 , 7 and 9 : 1.
x
÷
+
-
= 1
2.
÷
x
+
-
= 5
E. Show the possible operation in N : 1.
40 ÷ 5
2. 140 - 14
F. 1. Noha is M years old .How old will she be after 18 years? 2. Sara and her sister have 20 books. If Sara has X books, How many books does her sister have?
G. In the figure opposite ABCD is a
D
A
rectangle of area 36 cm² and EC = 5cm. Calculate the area of figure the AECD.
4cm B
C E
H. In the figure opposite , give the ordered pair for each letter in the graph : A
87
3 B C 2 1 0 1
2
3
I. Choose the correct answer : 1. The area of a rhombus 24 cm² and the length of one of its diagonal is 8 cm then the length of the second diagonal =…………..cm ( 3 , 6 , 8 , 12 ) 2. The circumference of a circle with radius 21 cm is ………..cm. ( 48 , 96 , 168 , 132 ) 1. The length of the diagonal of the square with area 24.5 cm² is ……. (
9 , 7 ,6 )
2. The circumference of a circle with diameter 21cm is ………. cm. (
22 , 44 , 66 )
3. The circumference of a circle= ……………. ( 2π r , π r , 4 π r , 2π d ) 4. a x b = b x a is …………. Property 7. D + 9 = 35 ; D = ………..
( associative , commutative ) ( 26 , 18 , 44 )
J. The following table shows the sales of some books in one month. Represent these data by pie chart graph.
88
Types of books
Story
Monthly sales
500
Text books Magazines 250
250
Others 200
Test ( 7 )
A. Choose the correct answer: a.
= ......
b. zero .... N
(10 or zero or 1 or meaningless) (∈ , ∉ , ⊂ , ⊄ )
C. The additive identity element in N is .... (0 or 1 or 2 or 3 ) d. Area of a square whose diagonal length 8 cm is ........ cm². (16 or e. If 3x = 27 then x = ..... ( 3 or 8 or 9 81 )
64 or 32 or 128 )
B. Complete: a. A rhombus of diagonals 8 and 10 cm , its area = ...... cm² b. 15 x 37 + 15 x 63 = 15 x ............ 89
c. The circumference of a circle with diameter length 49 cm is ..... cm (π =
)
d. 2 , 8 , 32 , ...... , ........ e. if A ( 2 , 1 ) and B ( 4 , 3 ) ,then the coordinates of the midpoint between
them is (...,....)
C. (1) Find the perimeter of opposite figure Where
14 cm
(π = )
7cm
(2) solve the equation: 3 X + 4 = 19
D. (1) In the coordinates plane : Draw the triangle XYZ where X ( 3 , 1 ) , Y ( 5 , 1 ) , Z ( 6 , 6 ) , then draw the image of this triangle in YZ. (2)
In the opposite figure:
ABCD is a rectangle of area 42 cm²
A
B
, DE = 5 cm. And BC = 4cm Calculate the area of ABED D E. (1) Use the commutative and associative properties to simplify : 74 + 43 + 26 + 57 90
E
C
(B) The following table shows the profits of a certain shop within 4 months:
FIRST
SECOND
THIRD
FOURTH
L.E. 500
L.E. 750
L.E. 250
L.E. 500
Represent these data by a pie chart.
Test ( 8 )
1) Choose the correct answer: a) The sum of any two natural numbers ...... N (∈ , ∉ , ⊂ , ⊄ ) b) The opposite transformation is ....... (flip, slide , turn)
c) Which equation means: fourteen is five less than a number ? ( 14 = x + 5 , 14 = 5 – x , 14 – x = 5 ) d) The circumference of a circle is : ..... 91
(2πd, πr 2πr)
e) The multiplicative identity element in N is ..... ( 0 , 1 , 2 , 3 )
2) Complete: a) The height of a parallelogram with area 40 cm² and base 5 cm is ..... cm b) 7 - 34 is .... in N C) 10000 , 1000 , 100 , ....... , ........ d) The area of triangle = 0.5 x base x ...... e) 3 + 4 = 4 + 3
(.......... property )
3)(a) Use the properties of multiplication in N to find the value of : 25 x 465 x 4
(b) Which is greater in area : A square of side length 9 cm or a rhombus of diagonals 10 cm and 7 cm.
4)(a) Calculate the perimeter of this figure :
6cm
Where (Ď&#x20AC; =3.14)
(b) On the coordinates plane : Draw the figure ABCD in which A ( 1 , 1) , B ( 1 , 5 ) , C ( 4 , 5 ) , D ( 4 , 1 ) , what the name of this figure ? 16 cm (5) (a) In the opposite figure : 92
Find the area of the shaded p 5c m
12cm
4cm
(b) Use the following table of data to draw the frequency polygon:
Sets
0-
4-
8-
12-
Frequency
6
12
9
8
93