Let’s Math
Under the supervision of Gabrielle Debbas Chama’a Publisher: Hachette Antoine S.A.L. All rights of reproduction and adaptation reserved in all countries. © Hachette Antoine S.A.L., 2016 Sin el Fil, Horch Tabet, Forest building, Lebanon. P.O.Box 11-0656 Riad el Solh / 1107 2050 Beirut, Lebanon. E-mail: [email protected] ISBN: 978-614-438-069-7
2 Summary
Unit 1: Decimal numbers, addition, subtraction and multiplication
1 Fractional form of a decimal p. 6
2 Rounding a decimal number p. 7
3 Comparing decimal numbers p. 8
4 Adding and subtracting decimal numbers p. 9
5 Multiplication of a decimal by 10; 100; 1,000... p. 10
6 Multiplication of a decimal by a whole number p. 11
7 Multiplication of a decimal by a multiple of 10; 100; 1,000... p. 12
8 Multiplying decimal numbers (1) p. 13
9 Multiplying decimal numbers (2) p. 14
10 Properties of multiplication (1) p. 15
11 Properties of multiplication (2) p. 16
12 Problem solving: using a diagram p. 17
13 Estimating a sum, a difference, a product p. 18
Unit 2: Division
14 Divisors of whole numbers - GCD p. 19
15 Multiples of whole numbers - LCM p. 20
16 Division of a decimal by 10; 100; 1,000... p. 21
17 Dividing a decimal number by a whole number p. 22
18 Decimal quotient of two whole numbers p. 23
19 Estimating a quotient p. 24
20 Division of two decimal numbers p. 25
21 Algebra p. 26 3 Summary
22 Problem solving p. 27
23 Order of operations p. 28
Unit 3: Fractions
24 Equivalent fractions p. 29
25 Decimal fractions p. 30
26 Rewriting two fractions with the same denominator p. 31
27 Comparing fractions p. 32
28 Adding fractions p. 33
29 Subtracting fractions p. 34
30 Multiplying fractions p. 35
31 Dividing two fractions p. 36
32 Fraction of a number: problem solving p. 37
Unit 4: Proportion, percentage and data
33 Ratio p. 38
34 Proportional sequences p. 39
35 Problem solving: proportionality p. 40
36 Percents (1) p. 41
37 Percents (2) p. 42
38 Reading a bar graph p. 43
39 Reading a line graph p. 44 4 Summary
Unit 5: Measurements
40 Measurement: length p. 45
41 Measurement: mass p. 46
42 Measurement: capacity p. 47
43 Perimeters: problem solving p. 48
44 Perimeter of a disk p. 49
45 Area of a parallelogram p. 50
46 Area of a triangle p. 51
47 Area: problem solving p. 52
48 Rectangular prism: face, area and volume p. 53
Unit 6: Geometry: angles
49 Angle: estimation and measurement p. 54
50 Angle: classification p. 55
51 Angle drawing p. 56
52 Complementary and supplementary angles p. 57
53 Adjacent angles, vertical angles p. 58
54 Angles in a triangle p. 59
55 Angles in particular triangles p. 60
56 Angles: review p. 61
57 Angles in a quadrilateral p. 62
58 Angles: finding a strategy p. 63
555 Unit 1: Decimal numbers, addition, subtraction and multiplication
1 Fractional form of Fractional form of a number / whole a decimal number / decimal / expanded form. العدد بشكل كرس \ عدد تام \ ّعرشي \ شكل ّموسع.
A decimal is a number that can be written with a decimal point. It can also have a fractional form.
Decimal form Fractional form
75 4,875 48.75 = 48 + = 100 100
can be expressed can be expressed as 48 ones and 75 as 4,875 hundredths hundredths
Example: the expanded form of 6,085.457: 1 1 1 6,≈.457 = (6 × 1,000) + (8 × 10) + 5 + (4 × ) + (5 × ) + ( 7 × ) 10 100 1,000
1 Write each decimal as the sum of a whole 2 Relate the equal numbers. number and a fraction less than one. 7 + 0.5 • 21.02 45.007 0.47 5 70 + • • 7.05 10 • 70.5 7 + 0.05 • • 75.5 755 • • 0.75 10 • 7.50 7 5 + • 10 100
3 Write in standard form.
a. Forty-four tenths: b. Twenty-five hundredths: c. Five thousand five hundred sixty-three and thirty-five thousandths: 6 Approximation / rounded value / to round down / to round up. تقريب \ قيمة ّرةمدو \ تدوير للعدد األدىن \ تدوير للعدد األعىل. 2 Rounding a decimal number
A number has many possible approximations.
Example 2,360,000 • Rounded to the nearest million: • Rounded to the nearest 100,000: 2,000,000 < 2,360,000 < 3,000,000 2,300,000 < 2,360,000 < 2,400,000
Rounded down to Rounded up to the Rounded down to Rounded up to the the nearest million nearest million the nearest 100,000 nearest 100,000 2,000,000 is the nearest million to 2,360,000. It is a rounded value (approximation) of 2,360,000 to the nearest million. 2,400,000 is the nearest hundred thousand to 2,360,000. It is another rounded value (approximation) of 2,360,000 to the nearest hundred thousand.
1 Frame each of the following decimals with two 2 For each of the following numbers, find the consecutive whole numbers. nearest decimal with one decimal place.
Frame with two consecutive Number whole numbers 13.54 12.99
75.21 … < 75.21 < … 0.987 100.099 0.65 99.45 4.111 99.5
3 Find three numbers such that each of them 4 a. Round down 325,471 to the nearest would be equal to 84,000,000 when rounded 100,000: to the nearest million. b. Round up 78,547 to the nearest 10,000:
c. Round up 325,784,235 to the nearest
million:
5 I am a decimal number less than one half and greater than one quarter. I am closer to three Maya tenths than to four tenths. Yasmina Among the decimals given below, which one matches the decimal described above? 0.136 0.28 0.37 0.345 0.39 0.45
7 To plot / ascending order / descending order / the nearest number / a digit. تحديد \ ترتيب ّتصاعدي \ ترتيب ّتنازيل \ العدد األقرب \ رقم. 3 Comparing decimal numbers
To compare decimal numbers, you can use an expanded form or plot the numbers on a number line.
1 Write all the numbers less than ten that can 2 The numbers below are arranged in ascending be formed using all the icons given below. order. Certain digits are replaced by a symbol. Find these digits. . 0 0 4 9 2.5*9 < 2.∆ £9 < 2.529 < 2.$3 < 2.63 Each icon can be used only once. Sort the numbers in ascending order.
4 Use the <, > or = signs to compare each pair of numbers. 3 Main oceanic trenches 65.7 and 657 3.45 and 3.5 Maximum depth in kilometers: 0.327 and 0.237 435.67 and 435.670 (Source: Wikipedia)
Aleutian trench: 7.679 Izu-Bonin trench: 9.78 5 Use each of the digits 0; 7; 2; 8; 9; 3; 5 only once Japan trench: 9.5 to form: Kermadec trench: 10.05 Kuril trench: 10.542 a. The greatest possible number. Peru-Chile trench: 8.065 Philippine trench: 10.54 Puerto Rico trench: 9.2 b. The least number consisting of 7 digits. Ryukyu trench: 7.460 Tonga trench: 10.882 c. The two nearest numbers to 8,000,000 such a. Which trench is the deepest? that one of them is less than 8,000,000.
b. Which trench is the shallowest? d. The nearest number to 7 million.
8 To add / to subtract / sum / difference / the terms. جمع \ طرح \ حاصل جمع \ حاصل طرح \ أطراف. 4 Adding and subtracting decimal numbers
Remark 4.3 + 4.7 is the sum of 4.3 and 4.7. 4.3 and 4.7 are the terms of the sum. While performing calculations, some steps 10.5 – 1.5 is the difference between 10.5 and 1.5. can be done mentally. Hence, it is not necessary to write them all. 10.5 and 1.5 are the terms of the difference. • If the order of the terms in a sum is changed, the sum remains the same. • If a, b and c are three numbers such that a + b = c, then c – b = a and c – a = b. • The difference between two terms does not change if we subtract or add the same number to both terms.
1 Calculate without setting up in columns. Group 2 Calculate. the numbers to make calculations easier. C = 4.6 + 27.7 – 0.6 – 7 A = 14.5 + 3.75 + 0.25 + 5.5
D = 15.7 + 3.9 – 15.7 – 1.1 B = 5.25 + 3.78 + 1.25 + 3.5
E = 150 – (85 – 15) + 35
3 Complete. G = (18 – 9.2) – (7 – 3.4)
11.25 + = 13 H = 3.75 + 0.75 – 1.25 – 2.25 9.99 + = 10
9.999 + = 10 5 Nadia had $100. She bought drawing equipment for $35.5 and books for $48.75. 4 The following rule is used to build a tower: How much money does she still have?
a + b
a b
Complete the cells with the correct numbers.
10
3.2
3 0.2 1.5 9 Product / the missing factor / multiplication. حاصل رضب \ الطرف املفقود \ عملية الرضب. 5 Multiplication of a decimal by 10; 100; 1,000...
How to multiply 3.78 by 10: To multiply a decimal number by 10, we move the decimal point one column Tens Ones Tenths Hundredths to the right. 3 7 8 × 10 How to multiply 3.78 by 100 or 1,000: 30 70 80 To multiply a decimal number by 100 or 1,000, we move the decimal point 3 7 8 two or three columns to the right.
1 Complete each product. 3 Complete.
3.5 × 10 = 11.8 × 10 = × 10 × 10
0.7 × 10 = 12.25 × 10 = 0.75
0.53 × 10 = 0.08 × 10 = 1.08
28.1 × = 2,810 0.36 × = 36 21.6
3.4 2 Complete each product. 125.3 3.5 × 100 = 2.87 × 100 = 37.8 0.7 × 100 = 2.08 × 100 =
0.53 × 100 = 0.08 × 100 =
4 Find the missing factor.
32.6 × = 3,260 0.1 × = 1 0.07 × = 7
0.2 × = 2 0.1 × = 10 0.07 × = 70
5 A remote-controlled robot makes 25.7 cm long steps. What distance does it cover after 100 steps?
10 Decimal part / integer. جزء ّعرشي \ عدد صحيح. 6 Multiplication of a decimal by a whole number
How to multiply a decimal number by an integer: Start by performing the operation disregarding the 2 decimal digits 2.25 decimal point. Then place the decimal point: the two × 14 decimal numbers have the same number of decimal digits. 900 + You can delete the unnecessary zeros from the decimal 2,250 part. 2 decimal digits 31.50
1 Calculate without setting up in columns.
207 × 0.2 = 365 × 0.4 = 743 × 0.03 =
360 × 0.05 = 7,500 × 0.02 = 5,140 × 0.7 =
2 Complete.
0.5 × 2 = 1.5 × 2 = 2.5 × 2 =
0.5 × 4 = 1.5 × 4 = 2.5 × 3 =
0.5 × 6 = 1.5 × 6 = 2.5 × 4 =
Problem solving 3 What is the length of segment AB in each of these cases?
18.5 cm 7.2 cm a. A C B
20 cm b. A 3.8 cmCB
3.2 cm c. A B
11 Decimal number / the ones' place. عدد ّعرشي \ منزلة اآلحاد. 7 Multiplication of a decimal by a multiple of 10; 100; 1,000...
• To multiply a number by 20, 30 or 40…, I have start by multiplying this number by 2, 3 or an idea! 4… then multiply the result by 10. • For 17 x 20 = • • 0, start by placing the «zero» in the ones place then multiply 17 by 2. • Do the same for 200, 300... by placing two zeros.
1 Calculate. 2 Calculate.
5.4 × 20 = 5.8 × 200 =
0.123 × 40 = 3.15 × 400 =
2.65 × 30 = 9.18 × 700 =
0.058 × 60 = 0.45 × 800 =
32.57 × 50 = 3.5 × 400 =
3 A factory produces very light balls. One ball weighs 12.5 g. What is the mass of 10 balls? of 30 balls? of 100 balls? of 130 balls?
12 Decimal point / decimal digits. فاصلة عرشيةّ \ منازل عرشيّة. 8 Multiplying decimal numbers (1)
How to multiply two decimal numbers: Multiply at first disregarding the decimal point. Then, place the decimal point so that the number of decimal digits in the product is the sum of the number of decimal digits of both numbers.
1 Calculate directly without setting up in columns.
62.14 × 0.2 = 123 × 0.03 = 1,253 × 0.03 =
0.1 × 0.04 = 25.5 × 0.2 = 2.54 × 0.2 =
2 Calculate. 32.7 × 2.4 = 27.2 × 3.21 = 305 × 4.32 =
3 The mile, a unit of length, is worth 1.852 km. A car traveled 36.5 miles. Express this distance in kilometers.
4 One liter of oil weighs 0.9 kg. a. How much does 0.5 L of oil weigh?
b. An empty bottle weighs 125 g. It is filled with 2 L of oil. How much does the full bottle weigh?
OLIVE OIL
liters c. How much would a crate containing 24 bottles weigh?
13 Product / factors. حاصل رضب \ أطراف. 9 Multiplying decimal numbers (2)
4.5 × 2 is the product of 4.5 and 2. 4.5 and 2 are the factors of the product. • If the order of the factors in a product is changed, the product remains the same. Remark Example: While performing calculations, some steps B = 25 × 36 × 4 can be done mentally. Hence, it is not B = 25 × 4 × 36 = 100 × 36 = 3,600 necessary to write them all.
1 What is the number of balls contained in 2 Calculate. this box? 5 × 4 × 7 =
9 × 50 × 4 =
35 × 6 × 2 =
50 × 8 × 2 =
8 × 9 × 25 =
75 × 12 × 4 =
3 A bookseller has 25 stacks of 30 copybooks 4 A tower is constructed according to the following each, and 180 copybooks in a crate. rule: How many copybooks does he have? a × b
a b
Complete the cells with the correct numbers.
32
3.2
1.6 1.5 14 Property / distributive / factoring / expanding. خاصيّة \ توزيعيّة \ إيجاد عامل مشرتك \ توسيع. 10 Properties of multiplication (1)
The distributive property of multiplication over addition and subtraction is used for factoring or expanding: expanding Factoring
2.3 x (15 + 7) = 2.3 x 15 + 2.3 x 7 12 x 11 + 12 x 15 = 12 x (11 + 15) 2.3 x (15 – 7) = 2.3 x 15 – 2.3 x 7 12 x 11 – 12 x 8 = 12 x (11 – 8)
1 Consider the expression: B = 5.4 × 98 + 5.4 × 2
a. What is the common factor between 5.4 × 98 and 5.4 × 2?
b. Factor B, and then perform the calculations.
2 Factor each of the following expressions and then calculate it.
Y = 4.5 × 88 + 12 × 4.5 =
Z = 67 × 14 – 67 × 4 =
R = 39 × 29 – 39 × 20 – 39 × 9 =
3 Calculate without setting up in columns (use 101 = 100 + 1 and 99 = 100 – 1).
101 × 28 = 99 × 35 =
4 29 × 15 = 435 and 29 × 16 = 464. Deduce 29 × 31.
(Hint: 15 + 16 = 31)
5 Factor each expression then calculate it.
D = 36 × 75 – 36 × 25 =
G = 3.24 × 82 + 18 × 3.24 =
X = 12 × 24 + 36 × 24 – 48 × 24 = 15 To make calculations easier / to deduce. تسهيل العمليّات الحسابيّة \ استنتاج. 11 Properties of multiplication (2)
To multiply three numbers, multiply two factors of the given product, and then multiply the result by the third number. Remark
Choose the first two factors in a way that makes calculations easier.
1 Calculate.
25 × 16 × 4 =
12 × 45 × 2 =
75 × 4 × 6 =
2 35 × 24 = 840.
Deduce 35 × 48, and then 70 × 48:
3 a. A truck carries 32 tables. Each table weighs 19 kg. What is the mass of the tables on this truck?
b. Another truck carries 64 of these tables. What is the mass of the tables on this truck?
4 In 100 g of fresh apricots, there are 86 g of water, 1.4 g of proteins, 11 g of carbohydrates, and 0.4 g of fat. The rest is made up of minerals and vitamins. What is the mass of minerals and vitamins in 200 g of apricots?
16 To use a diagram. استخدام الرسم ّالبياين. 12 Problem solving: using a diagram
Given situation Ziad, Hadi and Sami have together $120. Hadi has the double of what Ziad has, and Sami has $20 more than Ziad. How much money does each boy have? Solution Use a diagram:
Ziad
Hadi 120
Sami + 20
Calculation 120 – 20 = 100. This number is 4 times the part of Ziad. So, Ziad has $25, Hadi has $50, and Sami has $45.
1 Maya solved 42 small math problems in two days. On the first day, she solved 16 more problems than she did on the second day. Represent the given of the problem using an annotated diagram, then find the number of problems she solved each day.
2 Find two numbers whose sum is 528 and whose difference is 124. Represent the given of the problem using an annotated diagram, then find the two numbers.
3 Can you find two whole numbers whose sum is 108 and whose difference is 23? Draw an annotated diagram to justify your answer.
17 Product / estimation. حاصل رضب \ تقدير. 13 Estimating a sum, a difference, a product
1 Choose the best approximation without calculating.
• 2.6 × 2.7 is: greater than 9 less than 9
• 36 × 0.99 is: greater than 36 less than 36
• 36 × 0.5 is equal to: double of 36 half of 36
• 36.4 × 9.8 is closer to: 350 35
2 Without calculating, compare using < or >.
39 × 0.8 39 54 × 13.5 54 × 14 89 × 1.02 89
68.6 × 2.3 68 × 2 65 × 24 28 × 17 106 – 7.99 106 – 8
3 Three answers are suggested for each of the following products. Only one answer is correct. Without performing the exact calculations, circle the correct answer.
a. 42.46 × 2.9 is equal to
100.504 123.134 125.259
b. 67.5 × 4.2 is equal to
283.5 2,835 28.35
c. 198 × 0.99 is equal to
200.02 196.02 19.02 Candies 1,600
4 Circle the closest number to the given one. Candies 500 a. 687 + 0.99
687 688 68.7 787
b. 87 + 4.99
100 91 92 4.99
c. 453 – 10.99
452 53 443 442 18 Unit 2: Division
14 Divisors of whole Divisible / divisor / common divisor / greatest common divisor / GCD. يقبل القسمة \ قاسم \ قاسم مشرتك \ قاسم numbers - GCD مشرتك أكرب \ ق.م.أ.
• If a, b and q are three whole numbers such that a = b × q, then the whole number a is divisible by b and by q. • Every whole number has at least two divisors: the number itself and 1. The greatest common divisor of two numbers a and b is denoted by GCD (a, b). • To find the GCD of two numbers, we list the divisors of each number, and then we look for the greatest common divisor.
1 Find the divisors of each number. 25
60
2 Find the GCD of the numbers m and n.
a. m = 33 n = 6
b. m = 35 n = 30
3 Find the GCD of the numbers m and n. a. m = 18 n = 12
b. m = 26 n = 39
Problem solving 4 One red and one green flashing lights are at a store front. The red light flashes every 3 minutes and the green flashes every 5 minutes. Yasmina saw both of them flashing at 10:15. a. How long does she have to wait to see both of them flashing at the same time again?
b. If she waits until 12:00, at what times will she see them flashing at the same time?
19 Common multiple / LCM / to list. مضاعف مشرتك \ م.م.أ. \ إدراج ) تسجيل(. 15 Multiples of whole numbers - LCM
90 = 3 × 30. So, 90 is a multiple of 3 and of 30. 0 = 0 × 30. So, 0 is a multiple of 30. 0 is a multiple of any whole number. The least (non-zero) common multiple of two whole numbers a and b is denoted by LCM (a, b).
Example If a = 6 and b = 15, then LCM (6, 15) = 30. To determine the LCM of two numbers, we list the first non-zero multiples of each number, and then we choose the least of these common multiples appearing in each list.
1 Find the LCM of the numbers m and n.
a. m = 6 n = 9
b. m = 5 n = 20
2 Find the LCM of the numbers m and n.
a. m = 15 n = 4
b. m = 70 n = 42
Problem solving
3 A baker needs to make identical pies using all of the 27 kiwis and the 18 figs he has. He also wants to make more than three pies. How many pies can he make? How many of each fruit will he use?
20 Division / quotient / decimal. قسمة \ حاصل القسمة \ ّعرشي. 16 Division of a decimal by 10; 100; 1,000...
To divide a decimal by 10 (100; 1,000 …), we move the decimal point one (or two, or three ...) place(s) to the left. 1 1 1 Multiplying a decimal by ( ; …) is the same as dividing it by 10 (100; 10 100 1,000 …). 1,000
The quotient of two numbers does not change if the dividend and the divisor are multiplied by the same number.
Dividing a number by 0.1 ; 0.01 or 0.001 is the same as multiplying it by 10; 100; or 1,000.
1 Without performing the divisions, join up 2 Match the equal numbers in the two columns. the equal quotients. • 40 64.3 ÷ 0.5 • • 32.15 ÷ 10 4 × 0.01 • • 0.04 64.3 ÷ 50 • • 128.6 4 ÷ 0.01 • • 400 64.3 ÷ 20 • • 128.6 ÷ 100 40 × 0.01 • • 4,000 55 ÷ 110 • • (55 × 2) ÷ 100 4 ÷ 0.1 • • 4 55 ÷ 25 • • 5 ÷ 10 40 ÷ 0.01 • • 0.4 55 ÷ 50 • • (55 × 4) ÷ 100
3 Calculate directly without setting up in columns.
32 × 200 = 6.3 × 300 = 2.5 × 600 =
4 Calculate directly without setting up in columns.
2.12 × 0.1 = 34.07 × 0.1 = 12 × 0.1 =
5 Calculate directly without setting up in columns.
65.7 × 0.01 = 100 × 0.01 = 300 × 0.01 =
21 Dividend / divisor / remainder / approximately. مقسوم عليه \ قاسم \ باقي \ تقريبًا. 17 Dividing a decimal number by a whole number
To divide 680.4 by 21: 32.4 - Start by dividing the whole part; 21 680.4 - Then, when you drop 4, mark the decimal –63 point and continue the division as usual. 50 - 42 Sometimes, you will need to end the decimal part of the dividend 84 with zeros in order to get a zero remainder. - 84 00
1 Calculate the following divisions on rough paper. 211.26 ÷ 6 = 641.7 ÷31 = 215.4 ÷ 5 =
2 Calculate the following divisions. You need to get a remainder of zero. 8.25 ÷ 15 = 10.5 ÷ 14 = 3.57 ÷ 21 =
3 A merchant has 26 m of fabric. He sells 4 pieces of the same length. He is, then, left with 3.6 m.
a. What length of fabric did he sell?
b. What is the length of one piece?
4 Instruction 1: 25 identical wood cubes Instruction 2: Mia buys one kilogram of very weigh 450 g. small eggplants. She notices that she has 25 eggplants. What is the mass of one cube? What, approximately, is the mass of one eggplant?
a. These two instructions are alike. They are yet different. Read them well and answer the questions.
b. Explain the use of word “approximately” in Instruction 2.
22 Decimal quotient / decimal point / equivalent decimal numbers. حاصل القسمة ّ عرشي\ فاصلة عرشيةّ \ أعداد عرشيّة متكافئة. 18 Decimal quotient of two whole numbers
How to calculate the decimal quotient of two whole numbers:
11 11.7 11.75 12 12 141.0 12 141 141.0 - 132 - 132 - 132 90 90 9 - 84 - 84 6 60 - 60 00 The remainder is 9; it is The remainder is 6; it is not We therefore have: not zero. To continue the zero. 141 ÷ 12 = 11.75. division, use an equivalent Use an equivalent decimal: decimal 141 = 141.0 141 = 141.0 = 141.00 You know how to divide a decimal by an integer.
1 Calculate each division using equivalent decimal numbers. The quotient has 1 decimal digit (one digit after the decimal point). 448 ÷ 70 = 3,934 ÷ 35 = 198 ÷ 45 =
2 Calculate each division, using equivalent decimal numbers. The quotient has two decimal digits (two digits after the decimal point). 34 ÷ 25 = 1,071 ÷ 60 = 2,214 ÷ 75 =
Problem solving
3 Here is a recipe for 40 chocolate truffles.
Chocolate truffles • 400 g of chocolate • 100 g of butter • Orange zest
Calculate the mass of chocolate and butter needed for 30 truffles.
23 Rounded value of the quotient / the lower tenth / the lower hundredth. قيمة ّرةمدو لحاصل القسمة \ املنزلة العرشية ّاألدىن \ منزلة املئات العرشية األدىن. Estimating a quotient 19
Sometimes the division does not end. We can decide to stop the division at a certain number of decimal digits. In that case we have a rounded value of the quotient. Hence, we use the sign ≈ and write the sentence: 124 ÷ 26 ≈ 4.769230769 Calculating a quotient rounded to the lower tenth, means that the division is stopped at one decimal digit. Calculating a quotient rounded to the lower hundredth, means that the division is stopped at two decimal digits. 16 ÷ 9 ≈ 1.7 rounded to the lower tenth. 16 ÷ 9 ≈ 1,77 rounded to the lower hundredth.
1 Calculate these divisions without using the calculator. Give the quotient rounded to the lower tenth, and then write the sentence.
173 7 123 9 1.036
Sentences:
2 Estimate each quotient to the two nearest whole numbers. Then calculate the quotient.
12 ÷ 15: Estimation: 39 ÷ 60: Estimation:
Quotient: Quotient:
3 Complete the chart without setting up in columns.
× 2
11
35
47
15.5
31.5
0.5
24 Equality / a zero remainder. مساواة \ الباقي هو صفر. 20 Division of two decimal numbers
How to perform an operation of division when the divisor is a decimal number: When the divisor is a decimal number, you need to get rid of the decimal point in the divisor. Therefore: • Multiply the divisor by 10, 100, 1,000…; • Multiply the dividend by this same number; • Then perform the division. 0.65 23.4 is replaced by 65 2340
8.2 26.65 is replaced by 82 266.5
In the following exercise, complete the divisions until you obtain a zero remainder.
1 Calculate the divisions, and then write the equalities associated with each division.
2.4 29.52 8.2 25.912 2.34 8.19
2 Tick the correct answer(s). • The quotient 3.6 ÷ 1.2 is equivalent to 36 ÷ 12 360 ÷ 12 3.6 ÷ 12 • The quotient 7.2 ÷ 0.15 is equivalent to 72 ÷ 1.5 720 ÷ 15 7.3 ÷ 15
3 Calculate mentally. 4 Calculate mentally. 6 ÷ 0.2 = 15 ÷ 0.5 = 4.8 ÷ 1.2 = 7.5 ÷ 2.5 = 21 ÷ 0.3 = 0.4 ÷ 0.2 = 6 ÷ 1.5 = 7 ÷ 0.2 = 1.2 ÷ 0.3 = 2.4 ÷ 6 = 1 ÷ 0.25 = 2 ÷ 0.25 = 25 Value / pattern. قيمة \ منط. 21 Algebra
1 Calculate. (44 – 4) × (1.5 + 0.5) =
(44 – 4) × 1.5 + 0.5 =
44 – 4 × (1.5 + 0.5) =
44 – 4 × 1.5 + 0.5 =
44 – (4 × 1.5 + 0.5) =
2 Find the value of n. 25 × (n – 10) = 250
3 Given 2 × p – a = 30. Find three different values of p and a, that satisfy the given relation.
4 Mona is decorating one of her room’s walls using the pattern of flowers shown below.
She starts by drawing the first row of flowers on the top of the wall and continues downwards till she gets 10 rows of flowers. a. How many flowers will she draw in the 5th row?
b. How many flowers will she draw in the 10th row?
c. How many flowers will she draw on the wall?
26 Height / mass / cylinder / cube / liquid. ارتفاع \ كتلة \ اسطوانة \ ّمكعب \ سائل. 22 Problem solving
1 A 2 meters bamboo shoot grows by 0.45 cm per day. What will its height be after one week?
2 A snail travels 50 m in 30 min. A turtle travels 25 m in 6 min. a. Which animal is faster? b. Imagine that the snail and the turtle are in a 30 min straight race. Imagine also that in 30 min, both animals must arrive at the same point. Where should each of them start to arrive at the same point? Draw a diagram with a straight line to help you.
FINISH
3 Compare these balance situations in order to calculate the mass of one cylinder and the mass of one cube.
120 g
62 g
4 Which glass contains more liquid?
225 mL 23.1 cL 30 cL
27 An expression / to precede / to perform / order of operations. عبارة \ يسبق \ ينفّذ \ ترتيب العمليّات الحسابيّة. 23 Order of operations
If an expression has different operations, then multiplication and division are performed first. Multiplication precedes addition and subtraction. Example C = 2 + 6 x 7 = 2 + 42 = 44 Examples 16 ÷ 2 x 3 = 8 x 3 = 24 48 ÷ 2 ÷ 3 = 24 ÷ 3 = 8
1 Calculate. A = 32 – 5 × 3 × 2 + 7 =
B = 6 × ( 9 + 11) × 2 =
C = 9 × 5 + 2 × (7 + 5) =
2 Calculate. E = 100 – [5 × (3 + 7) + 50] =
F = 25 × [50 + 3 × (4 + 6)] =
3
1.5 cm 4.5 cm To calculate the perimeter of this polygon: a. First, express the solution using a single expression having many operations.
b. Then, perform the calculations and answer the question.
4 Arrange the following numbers in ascending order. A = 1 + 2.6 ÷ 0.1 B = (1 + 2.6) ÷ 0.1 E = 1 + 2.6 × 0.1 C = 1 + 2.6 ÷ 0.01 D = (1 + 2.6) ÷ 0.01 F = (1 + 2.6) × 0.1
28 Unit 3: Fractions
24 Equivalent fractions / simplest form / Equivalent fractions irreducible. كسور متكافئة \ أبسط شكل \ غري قابل لالختزال.
A fraction does not change if both of its terms are multiplied or divided by the same non-zero number n. a a × n a a ÷ n In other words, if n ≠ 0 then: = and = b b × n b b ÷ n Examples 2 2 × 3 6 22 22 ÷ 11 2 22 2 = = = = and are equivalent fractions. 7 7 × 3 21 33 33 ÷ 11 3 33 3 To simplify a fraction, we divide its two terms by one of their common divisors.
1 Which fraction does not belong? Circle it. 2 Which number does not belong? Circle it. 7 14 10 21 35 50 1 5 25 a. a. 4 8 7 12 20 100 2 10 40 12 6 3 24 10 1 25 1 10 b. b. 8 4 2 16 6 4 100 5 40 8 2 2 4 12 1 1 10 c. c. 0.1 28 7 4 14 42 10 100 100
3 a. Which of the following fractions are irreducible? Circle them. 13 25 12 56 130 40
10 9 27 21 40 120 b. Write the other fractions in their simplest form.
4 Simplify the following fractions. The simplified fraction should be an irreducible one. 35 21 48
28 6 32
29 Decimal fraction / denominator. كرس ّعرشي \ مخرج ) بسط (. 25 Decimal fractions
A fraction is called a decimal fraction whenever its denominator, or the denominator of any of its equivalent fractions, is 10, 100, 1,000... A decimal fraction is a decimal number.
1 Write each quotient in the form of a fraction.
4 ÷ 7 = 11 ÷ 9 = 12 ÷ 6 =
(2 + 4) ÷ 5 = (7 – 3) ÷ 9 = 11 ÷ (5 + 2) =
2 Write each number as an irreducible fraction.
12.5 ÷ 7 = 0.56 ÷ 1.4 = 32.7 ÷ 0.9 =
3 Replace each fraction with an equivalent fraction of denominator 100 or 1,000.
9 11 = = 20 25
27 23 = = 500 50
a 4 Find all the possible values of n, knowing 5 The fraction is irreducible, greater than n 10 that the fraction is irreducible and 1 and less than 2. 6 less than 1. What are all the possible values of a?
30 To rewrite fractions with the same denominator / LCM / to simplify. توحيد مخارج الكسور \ م.م.أ. \ اختزال ) تبسيط (. 26 Rewriting two fractions with the same denominator
1 In each case, find the LCM of the denominators, then rewrite both fractions with the same denominator. 5 7 6 2 a. and : b. and : 6 16 11 5
2 Rewrite each pair of fractions with the same denominator. 6 5 3 and : and 5 : 5 6 4
3 Simplify the fractions, then rewrite them with the same denominator.
14 6 and : 21 12 66 120 and : 55 100 24 3 and : 20 4 75 10 and : 50 40
4 Compare the fractions and justify your answer. 7 6 11 10 25 50 and and and 8 7 10 9 10 20
5 Arrange the given fractions in ascending order. Justify.
10 9 20
20 27 30
31 Term / quotient / to compare / to locate. فرد )طرف( \ حاصل القسمة \ مقارنة \ تحديد. 27 Comparing fractions
1 Compare each fraction to 1, and then deduce which fraction is greater.
3 9 12 14 a. and : b. and : 8 7 5 15
2 Compare the two fractions by calculating the quotients of their terms.
13 25 13 33 a. and : b. and : 8 16 5 15
3 Write each fraction as the sum of a whole number and a fraction less than 1, then order the fractions.
16 13
5 4
4 Complete the chart below. Fraction Fractions to be greater than Inequality compared 1 2 3 7 and 7 10 3 2 and 5 7 12 15 and 30 20 13 49 and 20 100
5 Locate the fractions on the number line, and write them in increasing order.
11 3 3 21 45 7
10 2 5 3 10 2
0 1
32 To add fractions / different denominators / a tower / a rule. جمع كسور \ مخارج مختلفة \ برج \ قاعدة. 28 Adding fractions
To add two fractions having different denominators, first rewrite the fractions so that they have the same denominator, then perform the calculations. Example
5 7 20 7 27 + = + = 6 24 24 24 24
5 4 5 4 1 Calculate: + = + = 12 3 12 9 5 5 4 + 3 = + = 12 7 3 2 Calculate. 1 1 1 1 1 1 + = + = + = 2 2 4 4 3 3 1 2 1 1 1 1 + = + = + = 3 3 6 6 2 4 3 Complete each fraction to get to 1. 7 11 10 + = 1 + = 1 + = 1 8 15 13
4 Complete the towers. Rule: a + b
a b
6 3 1 7 2 1 3 2 6 4 4
1
3 4 33 To subtract fractions / the same denominator / to share equally. طرح كسور \ مخرج مشرتك \ مشاركة بالتساوي. 29 Subtracting fractions
To subtract two fractions having different denominators, first rewrite the fractions so that they have the same denominator, then perform the calculations. Example: 11 7 33 28 5 – = – = 4 3 12 12 12
5 5 5 1 Calculate: 3 – = – = 12 2 8
2 Calculate. 4 5 3 1 – = 1 – = 1 – = 5 10 7
3 Calculate. 9 11 8 – 1 = – 1 = – 1 = 7 5 3
Problem solving
1 4 of an advertising catalog page is 6 5 dedicated for a drawing, for an 18 1 advertisement and for another 9 advertisement. The rest of the page
is covered with a text. Does the text cover more than half of the page?
5 Two brothers want to share a large field 1 A A equally. 1 12 One brother proposes to take all the 6 regions labeled A. Does this partition give the two brothers A equal shares? 1 4
34 Product of two fractions / decimal number / irreducible fraction. رضب كرسين \ عدد ّعرشي\ كرس غري قابل لالختزال. 30 Multiplying fractions
The product of two fractions is a fraction whose numerator is the product of the numerators and whose denominator is the product of the denominators. Examples 7 3 21 1 3 3 × = × = 8 5 40 4 7 28
Particular case: One of the fractions is a whole number. Example 3 11 3 33 11 × = × = 7 1 7 7
1 Calculate. 1 1 3 3 2 2 7 3 × = × = × = × = 2 2 2 2 3 5 8 2
2 Calculate. 3 7 3 8 3 × = 5 × = 6 × = 2 × = 4 2 5 7
3 Calculate. Write your answer in the form of a decimal number or an irreducible fraction.
5 7 12 × = 45 × = 4 18 9 12 × 121 = 30 × = 3 100 7 3 35 × = 50 × = 10 6
4 Calculate. Write the answer as an irreducible fraction. 5 7 × = 4 2
11 25 × = 10 33
9 2 × = 8 5
35 Quotient of two fractions / reciprocal fractions. قسمة كرسين \ مقلوب الكرس. 31 Dividing two fractions
Two fractions whose product is equal to 1 are reciprocal fractions. Examples 6 13 78 6 13 × = = 1 and are reciprocal fractions. 13 6 78 13 6 1 7 1 7 × = = 1 7 and are reciprocal fractions. 7 7 7 The quotient of two fractions is equal to the product of the first by the reciprocal of the second. Examples
3 7 3 5 15 7 7 1 7 ÷ = × = ÷ 2 = × = 4 5 4 7 28 8 8 2 16
1 3 5 14 8 ÷ = ÷ = 4 4 5 10 6 12 15 ÷ = ÷ 2 = 7 21 4
2 18 1 15 ÷ = ÷ 6 = 7 7 4 26 13 3 3 ÷ = ÷ = 5 5 4 4
3 Calculate. Write the answer as an irreducible fraction.
5 5 ÷ = 4 14 12 6 ÷ = 25 50 3 9 5 ÷ × = 2 4 7 36 Mass / length / to locate / to score. كتلة \ طول \ تحديد \ تسجيل. 32 Fraction of a number: problem solving
3 1 A recipe requires of a cup of flour. 4 Each cup contains about 225 g of flour. What is the required mass of flour? FLOUR
2 a. Draw a segment [SP] of length 8 cm. Locate the points T and F that are defined as follows: 3 • T is on [SP] and ST = SP. 4 • F is on (PS), outside [PS] and such that 1 FP = SP. 4 b. What does P represent with respect to the segment [TF]?
3 39 out of 50 students in a class passed the math test. 7 8 Which of the two fractions or 10 10 best represents this result?
4 In a shooting match, it is possible to score
a total of 152 points. A good shooter is the one who scores 3 at least of this total. 4 Here are the scores of three shooters: Ziad: 116 Yasmina: 120 Inès: 118 Who is considered as a good shooter?
37 Unit 4: Proportion, percentage and data
33 Ratio Ratio / quantities of the same kind / particular quotient / number without unit. نسبة \ كميّات من نفس النوع \ حاصل قسمة خاص \ عدد من دون وحدة.
The ratio of two quantities a and b of the same kind and expressed in the same unit, a is the number denoted by . b The ratio is thus a particular quotient: • It is a number without a unit; • It is expressed as an irreducible fraction or in the form of a decimal fraction.
1 Rami thinks that the ratio of the number of 2 Here are two dough recipes: yellow balls to the total number of balls is Recipe A Recipe B greater in the big bag. Is he right? flour: 500 g flour: 400 g sugar: 180 g sugar: 120 g water: 600 g water: 480 g
Which recipe gives the least sweet dough?
3 A shade of pink paint is obtained by mixing red paint and white paint in the ratio 3 to 5; red 3 = . white 5 a. Walid uses three measures of red paint. Each measure weighs 10 g. What is the mass of white paint that he must use to obtain the same shade of pink?
b. For his shade of pink, Ziad mixed 5 measures of red paint and 7 measures of
white paint. Who obtained a darker shade of pink, Walid or Ziad?
38 Sequence of numbers / proportional / constant of proportionality. تسلسل أعداد \ متناسب \ ثابت التناسب. Unit 4: Proportion, percentage and data 34 Proportional sequences
Two sequences of numbers are proportional when the numbers in one sequence are always multiplied by the same number, the constant of proportionality, to give the numbers of the other sequence. Example 1 Sequence a 3 5 9 10 × ÷ 3 × 3 3 Sequence b 9 15 27 30
1 Determine whether the following table 2 In the following incomplete table, the two represents a proportionality situation. If it sequences of numbers are proportional. is, calculate the constant of proportionality. 75 132 × 15 37.5 41.6 0.9 0.99 2.52 4.05 0.3 0.33 0.84 1.35 a. What is the constant of proportionality needed to move from the first row to the second row?
b. Complete the table. 3 An electric car travels 20 m in 4 s, 45 m in 9 s, and 50 m in 10 s. a. Verify that the duration is proportional to the traveled distance.
b. How much time does the car need to travel 145 m?
c. What is the distance that the car travels in 1 min 10 s?
4 a. Complete these tables with the perimeter and area of each square.
Side (in cm) 5 30 32.5 45 Perimeter (in cm) Side (in cm) 5 10 15 25 Area (in cm2)
b. Is the perimeter of the square proportional to the side?
c. Is the area of the square proportional to the side?
39 To enlarge / polygon / enlargement ratio / shade / mass. تكبري \ مضلعّ \ نسبة التكبري \ ظلّ / كتلة. 3527 Problem solving: proportionality
1 We want to enlarge the polygon PQRS (figure 1). Which of the two figures (2 or 3) corresponds to an enlargement of figure 1? What is the enlargement ratio?
G M R
S Q figure 1 figure 2 figure 3 H F N L P E K
2 45 g of blue paint are used to obtain 60 g of a certain shade of green paint. a. What is the mass of blue paint needed to get 100 g of this same green paint?
b. What is the percent of blue paint in the green paint?
3 A recipe posted online calls for 3 lb of flour to make 30 muffins. 1 lb is equal to 450 g. lb = Libra What is, in grams, the mass of flour needed to make 40 muffins?
40 Percent / the % sign / fractional form. نسبة \ إشارة نسبة مئويةّ \ شكل ّكرسي. 2836 Percents (1)
A percent is a constant of proportionality. A percent is written as a number followed by the % sign. It can also be written as a fraction or in a fractional form with denominator 100.
1 Write as a percent. a. 0.25 = b. 0.75 = c. 0.5 = d. 2 =
2 Write each ratio as a percent rounded to the nearest unit. 30 135 65 110 = = = = 75 80 120 70
3 Consider this rectangle. 4 In a bag of 70 balls, 30 % are red and the others are yellow. a. What is the percent of yellow balls?
b. How many yellow balls are there?
a. Use 4 different types of coding to represent 40 %, 30 %, 15 % and the remaining part of this rectangle. b. What percent does the remaining part represent?
5 The number of athletes subscribed at a gym
is 350. 70 % of the athletes have a yearly SPORTS CLUB subscription, whereas the others have a monthly subscription. How many athletes have a monthly subscription?
41 Dimensions / scale / rectangular parcels / to caption / areas. أبعاد \ مقياس \ أجزاء مستطيلة الشكل \ رشح \ مساحات. (Percents (2 2737
1 The dimensions of a rectangular garden are 800 m and 300 m. a. Draw this rectangle according to the following scale: 2 cm on the plan represents 100 m in reality.
b. This garden is divided into rectangular parcels: 50 % of the garden is planted with peas, 25 % with carrots, 20 % with tomatoes, and the rest is planted with lettuce. Represent these different parcels and caption the drawing. c. What percentage of the garden is reserved for lettuce?
d. Calculate the areas of the different parcels.
42 Bar graph / vertical axe / distribution table. رسم ّبياين \ محور ّعامودي \ جدول التوزيع. 2838 Reading a bar graph
1 These are the top three countries classified by number of gold medals won during the 2008 Beijing Olympic Games.
(source: fr.beijing2008.cn)
Country Gold medals Silver medals Bronze medals Total
China 28
USA
Russia 23 28
Total 282
Silver medals number of medals Medals won by USA 40 number of medals 38 39 36 34 38 32 30 37 28 26 36 24 22 35 20 gold silver bronze 18 medal 16 China USA Russia country
a. Describe the scale of the vertical axes. Is this choice justified?
b. What would the ranking of these countries be in terms of the total number of medals won? To answer this question, use the two graphs to complete the medal distribution table.
43 Line graph / to increase / the temperature recorded. الرسم ّالبياين الخطّ ّي \ زيادة \ درجة الحرارة املسجلّة. 3927 Reading a line graph
1 The graph below represents the evolution 2 Temperatures recorded in a city during the first of a population in a country between 2002 fourteen days of February. and 2007.
population in millions
140
135
130
temperature (in oC) 125 10 8 6 120 4
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2
year 0 Women 2 4 6 8 10 12 14 Men – 2 day – 4 a. What was the male population in – 6 2004? – 8 – 10
b. In what year did the female population reach 127,000,000? a. What is the temperature on the 3rd of February?
c. In what years did the female
population exceed the male population by 3,000,000 persons? b. What is the highest temperature? The lowest one?
d. What was the population in 2004?
c. On what days is the temperature less than zero? e. How much did the population increase between 2004 and 2007?
44 Unit 5: Measurements 28 40 Measurement: length Length / ascending order. طول \ ترتيب ّتصاعدي.
1 Complete. 1.03 km = m 10 mm = m
3.4 cm = m 0.005 m = cm
2 Arrange the following lengths in ascending order.
32.75 m 0.73 km 278.9 cm 9.74 km 12,547 mm
3 Calculate AC.
A BC
a. AB = 7.5 cm and BC = 3.7 cm b. BC = 8 cm and AB = 2 × BC
4 Calculate AC.
A C B
a. AB = 15.5 cm and BC = 6.9 cm b. BC = 9 cm and AB = 3 × BC
5 Tick the correct answers. 3.54 km is also equal to: 354 m 3.54 × 1,000 m 3,540 cm 45 Mass / kilogram / ton. كتلة \ كيلوغرام \ طن. 41 Measurement: mass
1 Arrange the following masses in descending order. 0.35 kg 0.003 ton 305 g 35.75 g
2 A farmer produced four blocks of white cheese. Their masses are block 1: 1,023 g block 2: 1.27 kg block 3: 987 g block 4: one kilogram and one hundred grams Arrange these blocks in descending order of mass.
3 Main producers of sunflower (Source: Campaign 2000/01)
countries
Former-USSR 6.33
Argentina 5.41
EU 3.58
millions of tons A sunflower According to the given bar graph, the mass of sunflower produced by: a. Argentina is: b. Former-USSR is: 5.41 5.41 tons 6,330,000 tons 5,410,000 tons 54 6,330 × 1,000,000 kg about 6 tons
46 Capacity / single expression / parentheses. قدرة ) سعة ( \ عبارة واحدة \ قوسان. 42 Measurement: capacity
1 Maria sees these three bottles of liquid soap in a supermarket.
0.24 L 23.5 cL 2.3 dL
Sort these bottles in ascending order of
capacity.
2 Mona pours 3 dL, and then 0.5 L, from a bottle containing 1.25 L of water. How much water is left in the bottle?
3 Rima has two jars of oil containing 18 liters each. She pours 2.5 L and then 1.8 L from each jar. She empties what is left in each jar in a barrel. To calculate the quantity of oil left in the barrel: a. First, express the solution using a single expression having many operations. Place the parentheses wherever needed.
b. Then, perform the calculations and answer the question.
47 Perimeter / equilateral triangle / rhombus / radius. محيط \ مثلث ّمتساوي األضالع \ ّمعي \ شعاع. 43 Perimeters: problem solving
1 The following figure is formed by a square whose perimeter is 14 cm, and by an equilateral triangle. The perimeter of the figure is: 24.5 cm 245 mm 17.5 cm 21 cm
2 a. What is the type of triangle ABC shown 3 The perimeter of the polygon shown below below? Justify your answer. is 43.5 cm. What is the value of x?
A 7 D 6
8.5 x 7.2 4.2 B C 11
The measurements shown on the figure are b. What is the value of AB knowing that in centimeters. AD = 3.4 cm and that the perimeter of the given figure is 17.2 cm?
4 DRUC and CDST are rhombi. UR = 12 cm
R a. What is the perimeter of the figure?
U D
C b. Which points of the figure are on the circle of S center D and radius RD?
T 48 Perimeter of a disk / length of a circle / center / diameter / number π. محيط القرص \ طول محيط الدائرة \ مركز \ قطر \ العدد π. 44 Perimeter of a disk
The perimeter p of a disk is the product of its diameter d by the number π. We have the following formulas: p = π × d and p = 2 × π × R with π ≈ 3.14 or π ≈ 3.1416 The perimeter of the disk is also the length of the circle.
1 a. Draw a circle of center A and diameter 3 cm. b. Locate a point E on this circle. What is the length of [AE]?
2 B is the midpoint of [AC]. AC = 8 cm 3 a. What is the perimeter of a disk with The shaded figure below is surrounded by a diameter of 8 cm? (Take π = 3.14). semicircles.
b. What is the perimeter of a disk of radius A B C 10 cm? Find the perimeter of this figure. Express your answer in terms of π.
4 Complete the description of the following figure.
A )C( 1.6 cm
B D O
M N a. (C) is a circle of O and of 3.2 cm. b. [OA] is a of this circle. c. [BD] is a d. [MN], [AB] and [AD] are of this circle. 49 Parallelogram / area / length / side / height. ّمتوازي األضالع \ مساحة \ طول \ ضلع \ إرتفاع. 45 Area of a parallelogram
The area of a parallelogram is the L product of the length of one side by the height corresponding height, expressed in the same unit. A B In the figure: Area (ABCD) = AH × DC height Area (ABCD) = BL × AD
HD C
1 a. What is the type of this quadrilateral? unit of length
b. Calculate its area using the area formula.
c. Verify this result on the drawing.
2 Calculate the area of this parallelogram.
9.6 cm
6.5 cm 5 cm
3 The area of the parallelogram ABCD 4 Calculate the area of the parallelogram ABCD. is 40 cm2. Calculate the distance between the straight lines (BC) and (AD). 7.2 cm 8 cm A B A 9.6 cm B
5.4 cm 6.5 cm
D C D C
50 Triangle / height / side / unit of area. مثلّث \ إرتفاع \ ضلع \ وحدة قياس. 46 Area of a triangle
The area of a triangle is equal to one-half the product of any side and the corresponding height height expressed in the same unit. h
1 Area = s × h, s being the length of a side, and 2 side c h the length of the corresponding height.
1 Calculate the area of this figure. 2 Calculate the area of this quadrilateral.
unit of area unit of area
3 a. Calculate the area of each of the green- shaded triangles.
b. Show that the area of the green-shaded region is half that of rectangle ABCD.
unit of length
A B
D C 51 Measurements / missing piece of information. مقاييس \ معلومة ناقصة. 47 Area: problem solving
1 The indicated measurements are in 2 In each case, the areas of triangles ABC, GSD centimeters. and RML can’t be calculated due to a missing Calculate the areas of the shaded triangles piece of information. whenever possible. Write a short sentence to state precisely the Remark: There is not enough information missing piece of information. to calculate the areas of certain triangles. A
F 7 R 4.7 cm G B U 3 C 6 E 8 4 B R 7 cm S 10 S 9 4 cm
A D 7 T M L
L 5 O
8 4 5 I C M 5.5
52 Rectangular prism / cube / volume / edge. ّمتوازي املستطيالت \ ّمكعب \ حجم \ طرف. 48 Rectangular prism: face, area and volume
The volume of a rectangular prism is equal vertex to the product of its three dimensions, edge expressed in the same unit: V = L x l x h. Also, V = B x h, where B is the area of a base and h is the height relative h to this base. c I The volume V of a cube of edge c is L V = c x c x c or V = c3. c base c
1 What should be the value of x so that 2 Complete this table. the rectangular parallelepiped and the Cube A Cube B Cube C cube have the same volume? Edge 0.04 m … dm … cm Face … cm2 0.36 m2 … cm2 area Volume of … cm3 … dm3 125 cm3 the cube
16 cm 8 cm
x 4 cm
Problem solving
3 a. How many identical cubes of edge b. How many identical cubes of edge 3 cm are needed to form a cube of edge 2 cm are needed to form a cube of edge 12 cm? 12 cm?
53 Unit 6: Geometry: angles
49 Angle: estimation and Angle / side / vertex / protractor / degree. زاوية \ ضلع \ رأس \ منقلة \ درجة. measurement
[OA) and [OB) are two semi-straight lines (rays) of origin O. They form an angle of vertex O. [OA) and [OB) are the sides of the angle. A The angle shown to the left is denoted by O, AOB or BOA.
O The unit of measurement for angles is the degree, B denoted by °. The protractor is an instrument used to measure angles.
1 A student measured the following angles, but he forgot to match the name of the angle with its respective measure. Match them without using the protractor. D A • • 20° B • • 130° C • • 30° D • • 90° A C B E E • • 120° G F • • 45° G • • 100° F
2 a. W ithout using a protractor, estimate the measure of each angle to the nearest ten. Write the results in the table below.
B
D A
b. Measure the angles. C c. Write the results in the table and compare the estimations to the measures.
Angles A B C D
Estimation
Measure in degrees 54 Right angle / acute / obtuse / straight. زاوية قامئة \ زاوية ّحادة \ زاوية منفرجة \ زاوية مستقيمة. 50 Angle: classification
Right angle: Acute angle: Obtuse angle: Its measure is 90°. Its measure is between Its measure is between 0° and 90°. 90° and 180°.
Straight angle: Zero angle: Its two sides form a Its two sides are confounded. straight line. Its measure is 0°. Its measure is 180°.
1 C
B
A D E
a. Without using the protractor, determine whether each angle is right, acute or obtuse. b. Measure the angles using a protractor and compare the results with the previous answers.
Angles A B C D E
Measure in degrees
2 F G E T
J M S H
L
a. Give angle TFE another name. b. A student wrote the following equalities. Are they correct? Justify.
FEM = GEH GEM = MEG
GFT = GFH MES = JEL 55 Semi-straight line / segment / compass / straight edge. شعاع \ قطعة مستقيمة \ فرجار \ طرف مستقيم. 51 Angle drawing
1 a. Draw a semi-straight line [Ax), then draw the angles yAx and xAz measuring 120° each. b. Find the measure of angle zAy using a protractor.
2 a. Draw a segment [AB] of 4 cm. b. Draw a semi-straight line of origin A that makes an angle of 40° with [AB]. Draw another one. c. Draw two semi-straight lines of origin B that make an angle of 60° with [AB].
3 Reproduce the angle xOy using only a compass and a straight edge. x
O
y
56 Complementary / supplementary / sum of two angles. ّمتممة \ ّمكملة \ مجموع زاويتي. 52 Complementary and supplementary angles
If the sum of the measures of two angles is 90°, then the angles are complementary. If the sum of the measures of two angles is 180°, then the angles are supplementary.
1 What is the complement of 35°? Of 78°? What is the supplement of 35°? Of 78°? Of 135°?
2 Match the complementary angles.
35° 45° 90° 10° 120°
80° 45° 55° 0° 60°
3 Match all the pairs of supplementary angles.
• 90° 100º • • 80° 90º • • 102° 180º • • 60° 120º • • 0° 35º • • 55° 78º • • 145°
4 Calculate the complement of each angle when possible. 47° 62° 95° 3° 45°
5 Calculate the supplement of each angle. 65° 0° 96° 125° 156°
57 Adjacent angles / vertical angles / to intersect. زوايا متجاورة \ زوايا متقابلة بالرأس \ تقاطع )تالقي(. 53 Adjacent angles, vertical angles
x Two angles are adjacent when they have the same vertex, one common side, and they do same not overlap. vertex O y
In this figure, xOy and yOz are adjacent common side angles. O z
(xy) and (zt) intersect at O. z y The angles of the same color are vertical angles. Vertical angles are equal. x O t
1 (AC) and (BD) intersect at I. A B True or False? a. DAI and AIB are adjacent angles: I b. ABD and CBD are adjacent angles: C c. BIC and AIB are adjacent angles: D d. AID and CIB are vertical angles: e. BCA and BCI are vertical angles: f. ADI and DAB are adjacent angles:
2 T is a point on (RS). E A
R T S a. Name an adjacent angle to each of the angles below. RTA TAE SET ATE
b. Given RAT= 80°, and TAE = 60°. Calculate RAE.
c. Given ATE = 90°, and ATS = 135°. Calculate ETS.
58 Sum of measures of angles / to verify. مجموع قياس الزوايا \ تحقيق. 54 Angles in a triangle
The sum of measures of angles in a triangle is 180°. A In triangle ABC: ABC + BCA + CAB = 180º C Or more simply: A + B + C = 180º B
1 Calculate angle F. D 15º 30º E
F
2 a. Calculate angle C. b. Verify the obtained result by measuring angle C. A
80º
B 50º
C
3 Draw a triangle ABC such that: 4 Draw a triangle SAT such that: BC = 6 cm, B = 50° and C = 30°. SA = 4.5 cm, ST = 3 cm and S = 110°.
59 Right triangle / isosceles / equilateral / base angles / equal angles. مثلّث قائم الزاوية \ مثلّت متساوي الضلعي \ مثلّث متساوي األضالع \ زوايا القاعدة \ زوايا متساوية. Angles in particular 55 triangles
• If a triangle is isosceles, then its base angles are equal. • If a triangle has two equal angles, then it is isosceles. These two angles are the “base angles” of the isosceles triangle. A
• The acute angles in a right triangle are complementary. equal angles
• In an equilateral triangle, each angle measures 60°. C S
Exercises 1 to 4: a. What is the type of the triangle? b. Calculate the unknown angles.
1 B
110º110º S C
2 N
M
E
3 F 48º
L A
4 C
60º E T 60 Vertex angle. زاوية رأسيّة. 56 Angles: review
1 True or False? g. The acute angles of a right triangle are supplementary. a. Two adjacent angles are supplementary. h. If an isosceles triangle has an angle of 60°, then it is equilateral. b. Two supplementary angles are adjacent. i. The triangle below is a right triangle.
c. If xOy = 60° and yOz = 40°, then 3 × a
xOz = 100°. a 2 × a d. CAB and EAB are adjacent angles. A
E C j. A = B = 45°
S B
1 e. If xOy = 135°, then yOz = xOz. A B 2
x
z k. SDF is an isosceles triangle of vertex angle D. 90º y O S
f. The two angles marked on the figure are not vertical angles.
80º F 50º D
61 A quadrilateral. ّرباعي. 57 Angles in a quadrilateral
• The sum of measures of the angles in a quadrilateral is 360°. • The square and the rectangle have four right angles.
1 ABCD is a square. A B a. What is the type of triangle ABC? Justify.
D C
b. What is the measure of angle BAC? And that of ACB?
2 Given: B DIC = 80° A ADC = 80°
ADI = 40° I I ∈ (AC) I ∈ (BD) C
D
Calculate: BIC, ICD.
I F 3 40°
50° 40° G A 4 cm a. Prove that (IA) and (IG) are perpendicular straight lines. b. What is the type of triangle FAG?
62 Collinear points / straight lines. نقاط عىل إستقامة واحدة \ خطوط مستقيمة. 58 Angles: finding a strategy
1 A, B and C are three collinear points. 2 R T S D E
A E
RTS A B C a. Calculate angle .
a. What are the measures of angles
EBC and DBA?
b. Show that the straight lines (BD) b. Are R, T and S collinear points? and (BE) are perpendicular.
63 Printed by for Hachette Antoine, January 2016