1

Contents Unit 1: Numbers and

Lesson 1: Approximating to the nearest hundredth

Lesson 2: Approximating to the nearest thousandth

Lesson 3: Multiplying decimals

Lesson 4: Multiplying decimals by10, 100 and 1000

Lesson 5: Dividing by - 3 digit number

Lesson 6: Lesson 6 Infinite division

Lesson 7: Dividing decimals by 10, 100 and 1000

Lesson 8: Dividing by a decimal

Lesson 9: Comparing and ordering fractions

Lesson 10: Multiplying fractions

Lesson 11: Dividing fractions

Unit 2: Sets

Lesson 1: Introduction to sets

Lesson 2: Set notation

Lesson 3: Types of sets

Lesson 4: Representing sets by Venn diagram

Lesson 5: Subsets

Lesson 6: Operations on sets

Unit 3: Geometry

Lesson 1: Geometric patterns

Lesson 2: Constructing a circle

Lesson 3: Constructing a triangle

Lesson 4: Constructing the heights of the triangle

Unit 4: Probability

Lesson 1: Investigating experiments and outcomes

Lesson 2: Certain and impossible events

2Revision(1)

1 meter = 100 centimeters

1 decimeter = 10 centimeters

1 centimeter = 10 millimeters

1 kilometer = 1000 meters.

1 kilogram = 1000 grams.

1 ton = 1000 kilograms.

1 litre = 1000 millilitres

1 day = 24 hours

1 hour = 60 minutes

1 minute = 60 seconds

(1Perimeter of a square = side length ×4 (2Perimeter of a rectangle = (length + width) × 2 (3Area of a square = side length × itself (4 Area of a rectangle = length × width (5Prime numbers =( 2,3,5,7,11,13,17,…..) 6)Even numbers =(0,2,4,6,8,……..) 7)Odd numbers =(1,3,5,7,9,……..(

3Revision (2)

First: Choose the correct answer:

1)The value of 3 in the number 347 is ......... [700 , 400 , 300]

2)The place value of 9 in 3972 ......... [units, tens, hundreds]

3)667 – ......... < 498 + 152 [7 , 17 , 27]

4)The height of the classroom door in Meters is......... [2 , 4 , 6]

5)The height of the greatest pyramid in Meters is... [50 ,180 ,400]

6)50 > 5 × ......... [11 , 10 , 9]

7)When it is seven o'clock, the angle between the hands of the clock is.........

[Acute, right, obtuse]

8)The probability of appearance of 2 on the upper face of a die when it is

thrown once is......... [½, ⅓, 1/6]

9)The H.C.F. of 12 and 18 is ...... [3, 6, 9, 72]

10) The L.C.M. of 12 and 20 is ......... [4, 30, 36, 60]

11) The perimeter of the square whose side length 6 cm equals..... cm.

[12 , 18 , 24 , 36]

12)7 m² =......... [70 dm² , 700 cm² , 700 dm², 49 dm²]

13)12.585 ≈…...... to the nearest tenth [13, 12.6, 12.59, 12.5]

14) 7 251 309 + 748 691=….. [8 milliard, 8 million, 8 thousand]

15) XYZ is a triangle in which, m (< X) = 40°, m (<Y) = 30°, then it is

…......triangle. [a right angled, an obtuse angled, an acute angled]

16) The value of 7 in 123.579 is ......... [7, 70, 0.07, 700]

17) 256.104 = 256 + 0.1 + …...... [0.04, 0.4, 0.004]

18) Number of axes of symmetry of a square is [0, 2, 3, 4]

4

Second: Complete:

1)47.85 + ......... = 100

2)33.3 – ......... = 12.008

3)93608.2 + 18905 =.............. ≈.............. To the nearest hundred.

4)453.64 – 72.317 =.............. ≈.............. To the nearest tenth.

5)1, 5, 9, 13,......,……,…….

6)The smallest number from the following numbers 1.3 , 3.2 , 10.04 ,

7)3.12, 3.215, and 1.12 is.........

8)2.8 >.........

9)The name of the figure is .........

10) ¼ of a day = ......... hours = ......... minutes.

11) The prime factors of 350 are ........,........,…….,……

5

1)Complete:

1) The place value of the digit 2 in the number 3.256 is……..

2) A mixed number is more than ………………..

3) 73 thousandths = …………… ( in decimal form)

4) 758 hundredths = …………….. ( in decimal form)

5) 3 = ………………… ( in decimal form)

6) 5 units and 4 tenths = ……………..

7) The 4.25 = (in the simplest form)

8) 0.225 = (in the simplest form)

9) 49.87 app. to the nearest unit is …………….

10) 7.358 app. to nearest tenths is ………………

11) 299.9 ≈ …………… to nearest unit.

12) 75.2 – (19 x 3) = …………… ≈ to nearest ten.

13) 58.6 ≈ 60 is app. to nearest ………………

14) 6.081 ≈ ……………… (To the nearest tenth)

15) ≈ ……………… (To the nearest unit)

16) 4927 ≈ ……………… (To the nearest ten)

2) Write the place value of the digit 5 in each:

a. 6.25 b. 18.54 c. 1.835 d. 528.087

6

1 2 4 8 . 5 6 7

7

.

thousands

hundreds

tens

units

Tenth( 0.1 ,one decimal place , )

Hundredth ( 2 decimal places , 0.01 , )

Thousandth ( 3 decimal places , 0.001 , )

Decimal point

The place value

Unit one

Lesson ( 1)and (2)

Approximate to the nearest hundredths and thousandths

a) 5.614 + 9.25 = ………………… (To the nearest hundredths)

b) 45.678 ≈…………….. ( )

c) 34.1834 ≈ ……………. ( to the nearest thousandths )

d) 8.459 ≈ …………. (two decimal places)

e) The decimal 0.84699 ≈ ……… ( to the nearest thousandths)

f) 7.358 app. To nearest hundredth is ………………

g) 0.9998 ≈ …………… . to the nearest thousandths

2)Complete:

1. Number 42.652 ≈42.7 to nearest------

2. Number 152.23 ≈150 to nearest-------

3. Number 1.4256 ≈1.426 to nearest-------

4. Number52.113 ≈52.1 to nearest-------

8

3)Complete:

1. 0.7351 ≈---------- (to nearest hundredths)

2. 152.3017 ≈-------- (to nearest thousandths)

3. 2 ⅛ ≈ --------- (to nearest hundredths )

4. 39⅙ ≈ --------- (to nearest thousandths )

5. ≈--------- (to nearest hundredths )

4)Choose

a) 165.128 app.to the nearest ≈ (165.13 , 165.12 , 165.1 , 165.2)

b) 8.016 ≈ …………… (to the nearest 0.01)

c) 8.454 app. to the nearest hundredths is …………… (8.54 , 8 , 8.45)

d) 0.7 + 0.07 + 0.0076 ≈ … (to nearest ) [0.776 , 0.777 , 0.77

e) 32.497≈ 32.50 to nearest --------(unit,tenth,hundredths,thousandths)

5)Carry out the following operations:

a) 0.398+0.2232 = ---------(to nearest thousandths)

b) 567.8+78.53= --------(to nearest tenths)

9

c) 3405 – 9.543 = -------(to nearest 0.01)

d) 5367 ÷ 1000 = -------(to nearest )

6- Approximate:

a) 0.637 m ≈ ……….. cm b) 0.58 m ≈ ………. dm

c) 3.7995 km ≈ ……… m d) 4.38 dm ≈ ……….. cm

e) 5.92 cm ≈ ………. dm f) 85.6 m ≈ ……….. dm

g) 74 cm ≈ ………… m h) 15.68 km ≈ ……… m

i) 7.584 m ≈ ……….. cm j) 8.7 dm ≈ ………… m

k) 5.63 cm ≈………. mm l) 39 days ≈ …….week.

m) 478.5 p.t. ≈ ……..L.E n) 6.895 L.E ≈ ………p.t.

o) 674 gm ≈ ……. Kg p) 3.5641 kg ≈ ………gm

10

Lesson 3

Multiplying decimals

1)Estimate to place the decimal point in the Prodfuct.

( a ) 7.5 × 23 = 1725 ( d ) 4.25 × 33 = 14025

( b ) 10.2 × 24 = 2448 ( e ) 15.6 × 204 = 31824

( c ) 88 × 6.3 = 5544 ( f ) 1.22 × 34 = 4148

2)If 345 x23=7935 then…

1) 34.5x23=… 4) 345 x2.3=…

2) 345 x0.23=…. 5) 0.345 x23=….

3) 345 x0.023=

3)Find the product

a) 52.3x6 =

b) 0.891x8 =

c) 6.78x31 =

d) 0.4 x0.3 =

e) 8.6x 24.7 =

f) 0.963x0.9 =

11

g) 4.3x4.8 =

4)Compare using (< , > , = ) :

a) 14.25x3.5 ……… 1.425 x3.5

b) 2.6 x2.8 ………. 26 x0.28

c) 12.23 x1.3 ……….. 12.32 x1.3

d) 12x0.8 ……….. 12x0.80

e) 7.5 x 2.7 ………. 0.75 x2.7

5)Multiply:

a) 0.15 b) 0.67 c) 1.374

X 2 x 3 x 6

_______ _______ ________

d) 57 e) 4.27 f) 17.6

X 0.003 x 0.7 x 0.05

_________ _________ _________12

Lesson 4

Multiplying BY 10,100,1000,…….

1-Find the result:

a) 8.72 x 100 = ……… b) 0.095 x 10 = ………

c) 298 x 1000 = ……… d) 3.009 x 10000 = ……..

e) 85.93 x 1000 = …….. f) 9.84 x 10 = ……….

2- Complete:

a) 5.2 x ………. = 520 b) 2.73 x ………. = 2730

c) 0.869 x ……. = 8.69 d) 1.403 x ……… = 14030

e) 0.083 x …….. = 83 f) 0.8025 x …….. = 802.5

g) ........ x 10 = 29.4 h) ...... x 0.254 = 2.54

3- Put the suitable sign:

a) 0.084 x 10 ………… 0.084 x 1000

b) 56.03 x 100 ……… 56.03 x 100

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c) 0.87 x 1000 ……… 0.78 x 1000

d) 7.52 x 10 ……….. 0.752 x 100

4- Find:

a) 0.63 m = ……….. cm b) 0.58 m = ………. dm

c) 3.79 km = ……… m d) 4.38 dm = ……….. cm

e) 5.92 dm = ………. Cm f) 85.6 m = ……….. dm

g) 74 cm = ………… m h) 15.68 km = ……… m

i) 7.584 m = ……….. cm j) 3.7 dm = ………… m

k) 5.63 cm = ………. mm

5-solve

1)An insect moves 13.8 meters a minute. How many meters does it moves

during 10 minutes?

………………………………………………………………………………………………………………………

2)A merchant bought 100 bicycles for his store. Each bicycle costs L.E 115.5.

How much money did he pay?

………………………………………………………………………………………………………………………

3)Engy saves L.E 1.5 daily. After 10 days she wanted to buy a toy for L.E 12.5

How much money did Engy have left?

………………………………………………………………………………………………………………………

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Lesson (5)

Dividing by 3 - digit number

Divide

a) 71604 ÷ 221 =

b) 34506÷ 142 =

c) 78084÷ 241 =

1) A tourist group travelled from Cairo to Aswan to visit its ancient monuments. There were 325 tourists, and the total cost of the trip for the whole group was 48750 pounds. Find the cost of the trip for each tourist.

-----------------------------------------------------------------------------------------------

2) A truck can carry 225 watermelons. Find the number of trip needed to transport 97200 watermelons.

------------------------------------------------------------------------------------------------- 3)A factory produces 205 pieces of cloth monthly.

In how many months does it produce 70110 pieces of cloth?

----------------------------------------------------------------------------

4)A shopkeeper saves LE 332 each month which he deposits in his bank account. After how many years will he have saved LE 122508?

-------------------------------------------------------------------------------------

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Note : the division operation is carried out without a

remainder is called finite division

Lesson 6

Infinite division

Example :

Divide 30 by 4 , then the result is 7 0.7

and the remainder is 2 4 3.0

- - 2.8

0.2

Divide each of the following, approximating the quotient to

one decimal place.

1)3 ÷ 8 = ≈

2)11 ÷ 125 = ≈

3)921 ÷ 12 = ≈

4)5 ÷ 11 = ≈

5)3 ÷ 123 = ≈

16

Note : the division operation which carried out with remainder is called infinite division

Lesson( 7)

Dividing decimals by 10,100, and 1000

1- Find the quotient:

a) 3.78 ÷ 10 = e) 52.9 ÷ 100 =

b) 0.98 ÷ 10 = f) 35.14 ÷ 1000 =

c) 68.38 ÷ 1000 = g) 756.8 ÷ 100 =

d) 94.65 ÷ 100 = h) 9.4 ÷ 10 =

2- Complete:

a) 82.58 ÷ ……… = 8.258 e) 32.9 ÷ …… = 0.0329

b) 0.395 ÷ ……. = 0.00395 f) 68.9 ÷ …….. = 0.689

c) ……… ÷ 10 = 78.42 g) …….. ÷ 1000 =5.384

d) 8145 ÷ ………. = 8.145 h) 1859 ÷ …….. = 18.59

3- Put the suitable sign:

a) 84.57 ÷ 10 …… 8.457 ÷ 100

b) 2843 ÷ 1000 .…. 284.3 ÷ 10

c) 712.8 ÷ 100 ….. 712.8 ÷ 10

d) 5.08 x 100 …… 50800 ÷ 100

e) 61.59 x 10 …… 615.9 ÷ 100

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4- Complete:

a) 34.7 cm = ………………. meters.

b) 458 meters = ………………. Km

c) 5278 cm = …………………. m

d) 6451780 cm = ………….. km

e) 3000 m = ……………. Km

f) 2.485 m = …………… cm

g) 786 cm = ………………. meters.

Word Problems

1-Hoda has 13.6 meters of cloth. She cuts it into 10 equal pieces. How long is

each piece?

……………………………………………………………………………………………………………………

2-7.65kg of juice were distributed over 10 equal bottles. Find the weight of

the juice in each bottle.

………………………………………………………………………………………………………………………

3-Ahmed is on holiday. He paid L.E 60.5 for 10 days to the hotel. How much

money did he pay per day?

……………………………………………………………………………………

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Lesson 8

1) Dividing by a decimal

a) 4 . 4 ÷ 0 . 4 =

b) 175 ÷ 0 . 25 =

c) 30.24 ÷ 3.6 =

d) 76.5 ÷ 7.65 =

e) 2.16 ÷ 7.2 =

2) Find the quotient of each of the following.

( a ) 98.4 ÷ 8.2 =

( b ) 4.794 ÷ 1.7 =

( c ) 18.45 ÷ 4.5 =

( d ) 114.45 ÷ 1.09 =

3) Find the result of each of the following.

( a ) (42.566 – 25.36) ÷ 0.7 =

( b ) 5.78 + (228.92 ÷ 9.7) =

( c ) (67.495 + 23.45) ÷ 0.05 =

4) A cyclist covered 38.7 km in 4.5 hours. How many kilometers can he cover in one hour?

5) The length of an orbit on one flight of the space shuttle was 25905.24 miles. The shuttle traveled at a speed of 285.3 miles per minute. How long did it take the space shuttle to make one orbit?

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Lesson ( 9)

Comparing and ordering fractions

1- Put the suitable sign < , > or = :

a) 1 ………… b) 1 …………

c) 1 ………… 2 d) ………… 1 e) 9 ………… 6

f) 6 ………… g) ………… h) 2 ……… 2

i) …………… 1 j) 1 ………… k) 3 ………… 3

l) …………… 1 m) ………… n) 6 ……… 4

o) 3 ……… p) ………… 6 q) 9 ………

2) Arrange in an ascending order :

a) , , c) , ,

b) , , d) , , 1 ,

3) arrange in a descending order :

a) 52 , , , 2 b) 1 , 1 ,1 , 2

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Lesson 10

Multiplying Fractions1) multiply then write the result in simplest form:

a) x =……… b) x 5 =………

c) x = ……… c)1 x = ……..

d) x =………….. e) x =………

f) x = …………. g) x =………

h) x = ………… i) x = ………

k) x = ………..

2)find the result of each of the following as in the example:

Example:

. of = x = . of 8 = x =

= 6

a) of b) of c) of 12

d) of 16 e) of 25 d) of 36

3 ) find the missing fraction : 21

a) x = b) x ……. = 1

c) x …… = d) ……. X =

Dividing Fractions

Example : ÷ = x =

Exercise:-

5 ÷ = …………… ÷ 6 = ……………

36 ÷ = …………… ÷ 4 = ……………

30 ÷ = …………… ÷ 7 = ……………

16 ÷ =…………… ÷2 = ……………

18 ÷ = …………… ÷ 4 = ……………

3 ÷ = …………… ÷15 = ……………

27 ÷ = …………… ÷5 = ………

9)complete each of the following:

1) x ………… = 1 2) ……….. x = 1

22

The reciprocal of a fraction is the fraction you get when you exchange the numerator and the denominator of the fraction

3) ………x 1 = 1 4) ÷ ……….= 1

5) ÷ ………..= 6) ……. = 5

7) is the reciprocal of ……………………………

8) is the reciprocal of …………………………..Word Problems

1) Shady bought 3 books having the same price and he paid 3 pounds,

What is the price of each book?

………………………………………………………………………………………………………………………

2) Samir covers km in 5 minutes, how many km does he cover in one

minute?

………………………………………………………………………………………………………………………

3) Ahmed bought 3 km of apple for L.E 25. How much was the cost of one kg?

…………………………………………………………………………………

4) Noura bought 5 m of cloth for L.E 19 . Find out the price of one meter?

…………………………………………………………………………………

5) A worker earns L.E 2 per hour, when he finished his work he got L.E 14 .

How many hours did he work?

7) A big cane has 113 kg of oil, and we want to distribute the oil in bottles

the capacity of each one is kg of oil. How many bottles are needed?

23

8) A traffic officer collect L.E 1 from each person who doesn’t follow the

traffic rules. At the end of the day he got L.E 20. Find the number of

persons who did not follow the traffic rules?

24

Unit test on unit 1 ) (

1 – ) complete :

1 - 0.9571 ) …………… ≈To nearest hundredths(

2 -3 ) .…………… ≈ To nearest (

3 - 99.997 ) .........……… ≈ To nearest two decimal places(

4 - 15.7801) …………… ≈ To nearest 0.01 (

5 -17 +71.0075 ) ..……… ≈ To nearest thousands(

6 -The reciprocal of is………………

2-( choose :

a- 97.5x100 = …………………. ( 9750 - 975 - 975.1)

b-62.5 2.5 =………………….. (25 - 35 - 70)

c- x=..…… ) , , (

d-( = , < , >)

e- x………= 1 (2 , 3 , )

f- =1 (8 , , )

25

3)Arrange the following in Ascending order

, , 1 , and

4 – (Arrange in Descending order

, 1, and

5 ( Find the result with Steps:

a-1.……………=

b-………………=

6 (If of a 24 pupils class play football .How many pupils play football?

7 (A factory produces 265 pieces of cloth monthly .In how many months does it produce 9275 pieces of cloth?

26

Lesson (1 )Introduction to sets

Definition of sets :

Remarks :

1) Behavior ,manners ,tall, short, clever ,intelligent ,nice ,bad….. (not set ).

2) We take only one from repeated digits or letters .

3) We must write elements of set in capital letters .

4) Equal sets same number of elements ,same elements

EX: {2,3,1} = {1,3,2}

27

Denote is not element of set.

Subsets with braces. } {

Not subset.

Empty set ( null ) no elements.

The set is the collection of well-defined objects which have common elements.

5) Equivalent sets same number of elements

Ex: {1,3,2} ,{5,6,7} .

State weather the each of following represents a set or not :

1- Good schools in Cairo.

2- Planets of the solar system.

3- Months of the Arabic calendar.

4- Spectrum colors.

5- Lovely colors.

6- Creative teachers.

7- Digits of no. 12358

8- Letters of the word Shereen

9- Egyptian league teams.

10- Lovely days of the week.

11- Months which have good weather.

12- Names of the students in my class.

13- Clever students in my class.

14- Good Egyptian league teams.

15- Lovely songs in the radio.

16- Delicious meals.

28

17- Days of the week.

18- Months of the year.

Lesson 2

Set notationEXERCISE :

1)List the elements of each of the following sets:-

1. letters of the word Nada.

2. Counting numbers less than 7.

3. Days of the week.

4. Letters of the word mathematics.

5. The African countries which have a capital named Cairo.

6. The digits of no. 1231157.

7. The numbers divisible by 3.

2) List the elements of sets :

1) Numbers divisible by 2

................................................................

2) Counting numbers less than 5

.....................................................................

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3) Digits of number 566788

.......................................................................

2- complete using or

1) 3……………… 4,3,2}

2) 15…………….{2,4,6,………..}

3) 36…………….{1,2,3,…………}

4) 69…………….{96,6,9}

5) M……………..{x,m,l}

6) A……………….{AB,BA}

7) 12……………..{21,12}

8) 12………………{1,2}

9) 0…………………{102,201}

10) 3…………………{1,2,………….}

_______________________3)say equal or equivalent sets

1) {6,8,3} and {8,6,1}

2) {4,8,20,1} , {2,0,8}

3) {1,3,5} and {5,3,1}

4) {2,3,4} and {7,8,9}

5) {2,5,8} and {8,5,2}

30

4)complete

1) if x {1,2,3} then x=……..or……….or…….2) if 3 {2,x,5} then x=………3) if 7 {a,4,5} then a=……..4) x {5,8,9} then x=…..or……or…….5) if 9 {4,8,z} then z=………6) if y {1,2} then y=………or ……………….

31

1)Complete each of the following using Or :

1. 3 ………. { 2 , 4 , 3 }

2. 2 …………{ digits of the no. 314 }

3. 5 ………. The set of even numbers

4. 70 ………. { 10 , 20 , 30 , 40 , 50 ………….. }

5. 12 …….. g, is a whole number : 3 < g < 16

6. 15 …….. { 2 , 4 , 6 , 8 , 10 ,………..}

7. 178 ………{ 1 , 7 , 8 }

8. 10 …….. { 100 }

9. 7 ……..

10) 34……… { 1 , 2 , 3 , 3 , 4 , 5 ,6 , 7,…………}

11)33 ….. { 3 , 6 , 9 , 12 , 15 ,18 , 21 ,………………… }

2)complete to get correct statement :

1. If x { 1 , 2 , 3 } then x = ….. or …… or …….2. If 4 { 2 , x , 5 } then x = ……

3) List the elements of each of the following sets:

1) 5 > a A = {......., ...,...., ....,...}2) 4 + X < 8 X = {......., ......., ......., .......}3) b < 2 B = {.......}4) Y< 1 Y = {…..}

32

( Do it by yourself )

Put the suitable symbol: ( Or )

Equal sets & Equivalent sets

( a ) Are {2, 3, 9} and {9, 3, 2} equal sets?

Are they are equivalent sets?

( b ) Are {6, 8, 3} and {8, 6, 1, } equal sets?

Are they are equivalent sets?

( c ) Are {4, 8, 20} and {2,0, 8, 4} equal sets?

Are they are equivalent sets

33

Lesson( 3 ) Types of sets

Finite Set Infinite Set Null Set It has fixed no. of elements

it has infinite number of elements

it is empty { } with no elements

with no element

1)Find the type of each of the following sets :

1. Days of the week.

2. Numbers divisible by 2.

3. Numbers divisible by 2 less than 20.

4. Sheep in your class.

5. Whole numbers between 2 and 3.

6. Numbers more than 1500.

7. Countries in Asia.

8. Odd numbers.

9. Living people over 200 years old.

2)Give the first three elements of each of the following sets.

( a ) { Whole numbers greater than 5}

.

( b ) {odd numbers greater than 200}

.

( c ) {numbers that can be divided by ten without a remainder}

.

34

( d ) {Prime numbers greater than 7 }

Lesson (4) Representing sets by Venn diagrams

1)Represent the following sets on a Venn diagram

a) M = { 4 , 5 , 6 } M N

N = {3 , 4 , 6 }

_____________________________________________________________

b) X = { 7 , 9 ,15 , 3 , 5 }

Y = { 3 , 5 , 11 , 13 , 19 }

X Y

_______________________________________________________________

c) A = {7 ,5 ,9 } B={ 4 , 5 , 6 }

A B

35

Lesson( 5 ) Subsets :1)Complete The Following Using , :

a){ 5 } ……… { 4 , 5 , 6 }

b){ 3 , 4 , 7 , 9 } ………. { 9 }

c){ b } …….. { letters of the word “day” }

d){ 1 , 0 } …….. { digits of no. 1157 }

e){ 5 , 7 , 1 } …….. { digits of no. 1571 }

f)If A B , B A then B ……. A

2) Represent the following sets on a Venn diagram

K = { 1 , 3 , 5 , 7 , 9 } N = { 3 , 5 , 7 }

( a ) Is N K? ( b ) Is K N?

_______________________________________________________________

3 )Represent the following on a Venn diagram

A = { 5 ,4 , 3 , 7 , 8 , 9 } , B = { 3 , 4 , 5 , 8 } , C = { 5 , 4 }

36

A B

C

Complete : C …….. B …….. A

4)Represent the following sets on a Venn diagram :

1 ) X = { 1 , 2 } ,, Y = { 1 , 2 , 3 , 4 } ,

Put the suitable sign ( or ). X ....... Y , Y ....... X_______________________________________________________________

5)If A = { 1, 4 , a , b } , B = { a , b , 5 , 6 }State whether the following are true or false. Give reasons(1) A B (3) {a, b, k} A(2) B A (4) {a} B

6)List

( a ) The elements of U( b ) The elements of A( c ) The elements of B( d ) The elements of A that are in B

37

7)Complete the following using , , ,

8)If X = {Letters in the word "cover" } , Y = { Letters in the word"Recover"}.(a) Is X Y?(b) Does X = Y?Give reasons for your answers

a)3 …..…… { 2 , 3 , 4 }

b){ 2 } …..…… { digits of no. 222 }

c){ 1 , 0 }…..…… { digits of no. 1070 }

d)7……..{7, 8 , 10 ,77 }

e){ 23 } ………. { is prime no, 19}

f)105 …… { 1 , 0 , 5 }

g){ 1 , 5 } …… { digits of 105 }

h){ 7 , 6 , 3 } …… {6 , 3 }

i) ……… {2,4,7}

j)3 … { whole numbers }

k){} ……… { , , , }

l){ 19 } ……… { 1 , 2 , 9 }

38

Lesson ( 6 ) Operations on sets

Cases of intersection ;

1- The 2 sets have common elements

A B =

2- The 2 sets are subsets A B = ,then B A

3- The 2 sets are disjoint A B =

4- The 2 sets are equal X Y = , then X = Y

1)Represent the following sets on a Venn diagram :

A = { 1 , 4 , 5 , 6 , 7 }

B = { is a digit of number 55789 }

Then find

A B

B A=

Is A B= B A

39

B

32

3)Complete the following to get a correct statement :

If A B = then A , B are ……………………….. If A B = { 1 ,2 } then A , B are …………………..

4) If D = {Letters of "cat"}, E = {Letters of "act"} Then D E = {... , ... , ...}.

Then E …..D

_______________________________________________________________

5) C = { , , , , }

,F = { , , , }

Then C F = .........................

Then C and F are……………

______________________________________________________________

6)If A = { e, f, g ,h } B = { e , m , a , n }

Find A U B

------------------------------------------------------------------

7)A = { 1 , 2 , 3 , 4 , 5 , 6 }

B = { 2 , 4 , 6 }

Find A U B

_______________________________________________________________

8)A = { 1 , 4 , 5 } B = { 5 , 1 , 4 }

Find A U B

11) Complement40

IF U = { 1 , 2 , 3 , 4 , 5 , 6 } , A = { 2 , 4 , 5 }A′ find = {…. , ….. , …… }

________________________________________________________

12) If U = { 1 ,2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } ,A = { 1 , 2 , 7 , 3 }

B = { 1 , 2 , 5 , 8 } , C= { 1 , 2 , 8 , 9 }Draw Venn diagram representing sets then find the following

A′ =

B′ =

C′=

-----------------------------------------------------------------------------

13)For the following sets U , X , Y , Z ,

U = { 1 , 2 , 4 , 6 , 7 , 8 , 9 }

X = { 1 , 2 , 4 , 7 }

Y = { 7 , 9 }

Z= { 2 }

Represent them in one Venn diagram

find each of the following shading corresponding areas .

a. X′ = {…………. e) ( X Z )/ =

b. Y′ ={………………. F ) / =

c. Z′={…………………… g) / =

d. (XY) ′ ={………………..

( General Exercise On Sets )41

1- From the opposite Venn diagram complete the following :1. X = { }

2. Y = { } z

3. Z = { }

4. X …….

5. Y ……

6. Z-Y = { }

7. X-Y = { }

8. Z- (XY) ={ }

2- If :

X = { 1 , 3 , 5 } Y = { 3 , 4 , 7 } U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 }

Draw Venn diagram to represent X , Y , U

Then find :

1. X ′ = 5. (XY) ′

2. Y′ = 6. X′ Y′ =

3. XY =

4. (XY) ′ =

3)Find the value of x in each of the following :

1. 5 { 2 , 3 , x } x = …………42

X 5 3 Y

1 2 4

2. 6 { 7 , x , 5 } x = …………3. { 2 , 5 , 7 } { x , 5 , 7 } x = …………4. x { 1 , 3 , 5 } { 4 , 3 , 7 } x = …………5. x { 1 , 4 , 7 } { 2 , 4 . , 6 } x = …………6. x { 4 ,3 , 5 } - { 3 , 5 } x = …………

4) complete the following using , , ,

1 39 ……………. {3 , 9 } 6 …….. {0}

2 4 ……………. { 44 } 7 0 ……..

3 {0} ……….. {10} 8 { 2 , 4 , 6 , 8 } …….. { even numbers }

4 17 { 1 , 3 , 5 , 7 , 9 , ..} 9 { 1 , 2 , 5 } ……. { 1 , 2 , 5 }

5 ……. { 1 , 2 ,3 } 10 Y …….. { hany , sky , yellow }

5)complete:

43

1. 5 {2,3,x} then x = -----

2. 6 { 7,x ,5} then x= -----

3. X {1,3,5} {4,1,7} = ----

4. X {4,3,5} – {3,5} = ------

6)Compete :1. X Y = Y then X-----Y

2. X Y = X then X -----Y

3. X Y then X Y = ----

4. Y X then X Y =-----

5. A B =A, A B=B then A---B

6. 7 A , 7 B then 7----A B

7. A B = then A, B are ----

8. A B = {1,2} then A,B are ----

9. (A`)` =

10))U =`)`

44

45

( Test on unit 2 ) 1-PUT , , ,

1)3 ----- {2,3,5}

1) ----- {4 }

2) ----- The sets of prime numbers .

3) ----- { 66 } .

4) ----- { 2,3,4,5,8} .

5) ----- { 1,2,4}

6) 5,2,0} ------ .

---------------------------------------------------------------------------------

2 ) State the type of following sets :

a) The set of whole numbers lying between 3 and 4 .

{ =

b) The set of prime numbers between 1 and 3 .

{=

c) The set of stars in the sky .

= {

3 )If U = { 1,2,3,4,5,6 }

X = {1,2,3 }

Y = {2,4,6 }46

*Represent these sets by venn diagram.

4 )Let X = { 3,4,5 } , Y = {2,3,4 }

1. X Y =

2. X Y =

3. X Y =

4. Y` =

5. X` =

6. (X Y)` =

------------------------------------------------------------------------------

5) Complete using , , :

1. { 3,4 } ---- X Y

2. 5 ---- X Y .

3. 33 ---- X

4. {20,3,5 } ---- X Y

5. { } ---- X Y

6. {3,4,5 } ---- X Y

7. 2 ---- X Y

8. X Y ---- X Y

47

Unit 3

Lesson 1 Geometric patterns Circle the shape that continues the pattern

48

Discover the sequence

ACTIVITY

Draw the next figure in each pattern.

49

________

Lesson 2 ( Constructing a circle ) Definitions

1)A line segment that joins any point on the circle to its centre is called a

radius. 2)All the Radii of a circle are equal in length.

3) Infinite number of radius can be drawn in a circle

4)A line segment that joins any two points on the circle and passes through its

centre is called a diameter.

5)A line segment that joins any two points on the circle is called a chord.

50

6) The diameter is the longest chord.

7) The length of the diameter of a circle is twice the length of its radius.( D =2 r )

1)Mark (√) for the correct statement and (X) for the incorrect one, using the

opposite figure:-

1) XY is a diameter of circle (M). ( )

2) YZ is a chord of circle (M). ( )

3) MYZ is a central angle. ( )

4) XY = 2ML. ( )

5)Any chord that passes through the centre of a circle is called a diameter.( )

6)The length of the radius of a circle = the length of the greatest chord in the

circle. ( )

7)If the vertex of an angle lies at the centre of circle then it is a central

angle. ( )

2)Use the opposite figure to answer:-

1)The length of CM is greater than the length of AB.( )

2)AB is a diameter of the circle of centre M. ( )

3)The radius = the length of the longest segment drawn inside

the circle. ( )

4)The length of AM = the length of MC. ( )

5)(MAC) is a central angle. ( )

3)Complete each of the following:-

a)If the length of the radius = 15cm, then the diameter = ………cm

51

M

LY

ZX

MA

C

B

b)Any line segment whose end points are on the circle is called ……………

c)The diameter of the circle = ………… x the length of the radius

4)Draw:

1)Draw the circle M whose radius=5cm. Draw diameter AB.

2)Draw the circle M whose radius=4cm. Draw diameter AB

3)Draw the circle M whose radius=3.5cm. Draw diameter AB

4)Draw the circle M whose radius=3cm. Draw diameter AB

5)Draw the circle M whose diameter =8cm.

6)Draw the circle M whose diameter =10cm.

7)Draw the circle M whose diameter =6cm.

8)Draw the circle M whose diameter =9cm.

General Exercise: (Circle ) C

1)AB is called -------

2)MB is called ------ A B

3)BC is called ------

4)This circle is called ------ with center ------52

5)-----,-----,----- are radii of circle M .

6)Radius = ----- Diameter .

7)Diameter = -----Radius .

8)If length of radius = 5 cm then longest chord is ------

9)All diameters are ------ in same circle .

10) We can draw ------of diameters in same circle .

11) Any chord passes through center is called ------.

12) ------- angle with vertex lies at the center of circle .

13) Any line segment whose end points on circle is -----

14) If diameter =12 cm then radius =------cm .

15) If radius =6 cm then diameter =------cm .

16) The longest chord in circle is ------.

17) Line segment joins center to any point on circle is called -----.

18) Diameter of circle is chord that -----.

53

Kinds of triangles

According to lengths of its sides According to the measure of its Angles

1- Equilateral triangle 1- Right – angled triangle

a) It has 3 equal sides. a)It has only one right angle

b) It has 3 equal acute angles. b) 2 acute angles

each equal to 60º B = 90 A + B + C = 180º

AB = BC = CA

B = A = C = 60º 2) Obtuse angled triangle

2) Isosceles triangle a) It has one obtuse angle

a) It has 2 equal sides. , and 2 acute angles.

b) It has 2 equal angles b) A + B + C = 180º

AB = CA B = C

3) Scalene triangle 3) Acute angled triangle

a) It has 3 different sides. a) It has 3 acute angles

b) It has 3 different angles b) It has neither a right angle nor

obtuse angle.54

ABC

60º

60º60º

C

BA

A

B

C

A

B C

Notes:

1. A right angle = 90º

2. An acute angle is < 90º

3. An obtuse angle is > 90º and less than 180º

4. The sum of angles in any triangle is = 180º

5. Any triangle has at least 2 acute angles.

1)Draw ΔXYZ in which = XY = 7cm, m( <X) = 100°,

and m(<Y) = 50°.

Measure (< Z), then answer:

a What is the sum of the measures of angles of ΔXYZ? ………°

b What is the type of the triangle XYZ according to the measures

of its angles? ………

2) Draw the triangle ABC in which <B is a right angle, m(<C) = 60°

and BC = 4 cm. Measure <A, then check that the sum of

measures of angles of a triangle is 180°.

3) Draw ΔABC in which AC = 7 cm, m< (A) = 45°, and m< (C) = 75°.

a Calculate, mentally, m(<B), then check your answer using the protractor.

b What is the type of the triangle according to the measures of its angles?

c What is the type of the triangle according to its side lengths?

55

4) Complete:

1)If ABC is a triangle in which m(<B) = 98°, then it is……………………. triangle.

2) If XYZ is a triangle in which m(<X) = 100° and m(<Y) = 58°, then m(<Z) = …………………………

5)Draw :

1. The triangle ABC in which AB = 7cm, BC = 6cm, and AC = 6cm.

2. The triangle XYZ in which XZ = XY = YZ = 6cm. then find measure of angles of the triangle, what do you notice.

3. The triangle ABC with AB = 10 cm, BC = 8 cm and CA = 6 cm find its perimeter and its type according to its sides.

4. Try to draw a triangle of sides 3, 4, and 6cm. what can you deduce.

5. The triangle XYZ in which XY = ZY = 7cm and XZ = 6cm find. Measure of angles ( X and Z ), what do you notice.

6)Draw the equilateral triangle ABC of perimeter 24cm. Find the measure of each angle. What do you notice? Determine its type.

56

Lesson( 4 ) Constructing the altitudes of the triangle

Note that: Any triangle has (3) three heights

1)Put ( ) or ( ):-

a) The acute angled triangle has three heights. ( )

b) The right angled triangle has only one height. ( )

c) The obtuse angled triangle has 2 heights. ( )

2)Complete:-

a) The obtuse angled triangle has ……… heights. Which intersect at a point

that lies ……… the triangle.

b) The right angled triangle has ……… heights which intersect at ……… of

the right angle.

c) A triangle can be drawn if …………… are given

57

Heights of triangle:

1) Draw MON such that MN = 7cm, MO = 5cm and ON = 3cm. Draw its heights.

2) Draw ABC in which AB = 4cm, BC = 3cm, AC = 5cm. Draw the heights of this triangle?

3) Draw the equilateral triangle ABC in which its side = 4cm, draw its heights?

4) Draw triangle ABC in which AB = 6cm, BC = 5cm, AC = 4cm. draw the three heights of this triangle

5) Draw XYZ in which, XY = YZ = 5cm, XZ = 8cm. Draw the perpendicular segment XE , YF and XH on its sides.

6) Draw the triangle XYZ in which XY = 6cm, m(X) = 45o and m(Y) = 25o

.Draw the heights of the triangle .

7) Draw the triangle ABC where AB = BC = CA = 5cm.

8)Draw AD BC to cut it at D. Find the length of AD Draw ABC where BC = 6cm, AB = 5cm, m(B) = 120o

Draw its heights .

Unit test on unit 3

1 ) Draw the next two figures

a-

b-

c -

58

2 -) complete

1 )AB is a ……………….in the circle. A B

2 )AB is a………………..in the circle D

3(The point …………is the centre of the circle

4(The line segments …………….,…………..and ……..are radii in circle

5 (The angle ……………is the central angle of the circle

3 – Draw the Triangle ABC in which AC = 10 cm ,BC=8 cm , and AB= 6 cm

Draw BD altitude on AC.

Find length of BD ?

Probability

Investigating Experimentsand Outcomes Lesson( 1) The probability is a number from 0 to 1 that measure

the likelihood or The chance of an event occurring

The probability of the impossible event = 059

The probability of the certain event = 1

The probability of the possible event is between

Zero and 1 )fraction)

Generally:

Probability of an event A

P)a) =

1)Complete

a) The sum of probabilities of all possible events = ………………

b) Probability of appearance of head or tail when tossing a coin = …

c) Probability of getting any number on the upper face when rolling a

dice =………

d)The probability of an event is always 0, or ........, or a number between ........ and ....

e)The figure opposite shows a disk divided into equal sectors from 1 to 4 probability of the pointer pointing at one of the sectors is ……

f)The probability of appearance of the number 5 when rolling a die once =…………

_________________________________________________________

2)A letter of the word "Ahmed" is selected randomly. what is theprobability of selecting the letter "d"?_________________________________________________________

3)A box contains 4 white balls, 3 blue balls and 5 red balls, all ofwhich are of equal size. When one ball is drawn randomly fromthe box find the probability of:

60

1 2

3 4

a) The drawn ball being white

b) The drawn ball being red

_______________________________________________________

4)In a class of 45 students, there are 5 more girls than boys. If oneof the students is chosen randomly, Find the probability of thisstudent being a girl.

_________________________________________________________

5) 1) The probability of getting an odd number is……2) The probability of getting a number less than 6 is…3) The probability of getting a number greater than 5……………4) The probability of getting an even number is…..

5) probability The of this card being a prime number is….

6) A box contains 5 black balls, 3 white balls, and 2 stripped balls if a ball is drawn blindly, complete .

a) Probability of drawing a white ball is ……………

b) Probability of drawing a black ball is ……………..

c) Probability of drawing a stripped ball is ………………61

d) Probability of drawing a non-black ball is 1- ……….

e) Probability of drawing a non-white ball is …………….

7) A box contains 4 blue balls, 2 red balls, and 3 green balls of the same size.

a - The probability of getting a blue ball = …..

b- The probability of getting non green ball = ….

8) A box contains ten balls , 4 are red and the rest are white. If you blindly draw one ball

what is the probability that it will be white ?

9)Flip a coin 50 times and tally how many HEADS and TAILS are turned up.

10) If you have a cards with numbers 3,2,4,5,8.if one card is drawn blindly,

What is the probability of having a number between 2 and 5 ?

Some of the language of probability:

perhaps, might, likely, unlikely, equally likely, chance, certain, uncertain, probable, possible, , good chance, poor chance, no chance, equal chance, even chance, evens, fifty- fifty, likelihood, probability, possibility,fair,unfair,.

Doubtful, expected

13)Choose

62

1) The probability of an impossible event = …… )1 – 0 – 2)

2) The probability of certain event =……….. )0 – 1 – 2)

3) The probability of visiting the sun is ) Zero – 1 – between 0

and 1 )

14)Tell whether each of the following means that the probability

Is 0, close to 0, close to 1 or 1?

1) It will get dark tonight

2) Ali will turn into a fish

3) If I roll a dice I will get an even number

4) If I toss a coin it will land heads up

5) You will drink something today.

6) You will see the Queen on your way home

7) The sun will set in the evening.8) Summer vacation will last 4 months.

9) If you toss a coin you have an equal chance of getting a head or a tail.

10) If a baby is born it has an equal chance of being a boy or a girl .

) Final exam1 ( 1- Complete :

1. The probability of an impossible event = …….

2. 82.795 ……… ( to the nearest tenths )

3. The longest chord in the circle is called ………..

4. If U = then A’=…..63

5. All radii of a circle are ……………

2- Choose the correct answer :

1( ( )

2) …… ) (

3(13.75 10 = 1.375 x …….. )10 , 100 , 1000 , 0.01 (4)The set of prime numbers that are even

(finite , infinite , empty)

3-a) Arrange in a descending order

c)Mona went to the market . she bought 4.5 kg of fish for L.E 12 each and 6kg of apples for L.E 5.5 each . how many pounds did she pay.

4-a( A box contains 4 white balls , 3 blue balls ,and 5 red balls . find the probability of getting :1)blue ball 2) red ball

4-b) Draw a triangle ABC in which AB = BC =5 cm ,and AC = 6 cm, then draw its altitudes.

( Final EXAM2 ) 1)

a) The ------ is the longest chord in the circle .

64

( radius – chord –diameter )

b) 7 ---- 2

c) ---- { 4,5,9 } (

d) ----Thousandths(64.59 ,64.6,64.597)

e) The set of even numbers between 11 and 12 is -------- ( finite ,infinite ,empty ) set .

2) Complete :

a) If u = { 1,2,3,4} and A = { 1,2,3 } then A` = ----

b)215.68 ------- Tenths .

c) 2 ---- { 12, 22} put ( .

d)The 3 altitudes of the acute angled triangle intersect at one point ---- the triangle .

e) 46.583 x ----- = 4658.3

3 ) Arrange in ascending

,

B)- Draw the triangle ABC where AB=8 cm . AC= 6 cm and BC = 6 cm . Draw the altitude from C AB then find the type of the triangle

according to their sides.

65

4 -) A- Using the opposite Venn diagram complete :

U

a)- = ………………

b)- x = ………………

c)- Y = ………………

d)- XY = ………………

e)- XY = ………………

f)- X- Y = ………………

B)- Karim bought 3 Kg of meat . If the price of each Kg is L.E 40 How much did Karim pay?

*************************************************************

5 – )A)- A box contains 4 white balls , 3 blue balls and 5 red balls , all of which are of equal size . When one ball is drawn randomly from the box find the

probability of:

a)- The probability that the drawn ball is blue ball=

b)- The probability that the drawn ball is white ball=

c)- The probability that the drawn ball is yellow ball=

B)- Draw the circle (M) with radius 4 cm , then draw the diameter

66

7

42

6

1

8

5

) final Revision 1 (

1 ( Complete :

is the reciprocal of ….

b) The probability of certain event is …… c) 4 2, x, 5 } then x =…..

d) ÷ = ….. e) The right –angled triangle has ….altitude and intersects at …. f) , ,……..

g)If radius = 5 cm ,then longest chord ……cm .

h) In a box , there are 4 blue balls , 2 red , 3 green .1) Outcomes = …..

2) Probability of red balls = …..

3) Probability of blue balls = ….

4) Probability of green balls =….

5) Probability of non – red balls .

--------------------------------------------------------------2)Choose :

1) Set of even numbers ( finite – in finite - )

2) Days of the week ( set – not set )

3) Probability of impossible event ( 0 – 1 – 2 )

4) 6 { 2 , x , 5 } then x = …..

5) 8 ≈ …… ( nearest hundredths ) is ( 8.13 – 8.12 -8.125 )

67

6) The sum of angles of triangle ( 108 – 180 – 100 )

68