PRIMARY 6 THIRD TERM LESSON NOTES MATHEMATICS

WEEK 2

OPEN SENTENCE

(a). If 40 note are to be shared Among 5 pupils, how many books Will be given to a pupil?

5 pupils 40 notes

1 pupils (40 + 5) notes

= 8 notes

(b). Find the letters e.g.

2y + 6 = 30

2y = 30 –6

2y = 24 divide both sidesBy 2

2Y/2=24/2

= y = 12

Closed and open sentences

Study the following mathematical statements:

13 + 6 = 19 23 + 12 = 35

42 − 20 = 22 63 − 49 = 14

7 × 5 = 35 11 × 12 = 132

40 ÷ 5 = 8 120 ÷ 10 = 12

The mathematical statements above are called closed number sentences.

Closed number sentences can either be true or false.

Examples

15 + 7 = 22 (True mathematical statement) 18 + 3 = 19 (False mathematical statement)

3 × 6 = 12 (False mathematical statement) 42 ÷ 6 = 7 (True mathematical statement)

Study each of the following mathematical statements:

[]+ 9 = 13 11 +[] = 25 [] − 4 = 11 20 − = 7

[]× 5 = 15 4 ×[] = 24 [] ÷ 6 = 5 48 ÷ = 12

In each of the statement above, there is a missing number called unknown represented by

.[] They are called open sentences.

An open sentence is a mathematical statement that involves equality signs and a missing

quantity represented by[] that the four arithmetic operations of addition, subtraction,

multiplication and division can be applied to solve.

Open sentences can either be true or false depending on the value [].

Exercise

A. Write True (T) or False (F) for each of the following closed number sentences.

1. 15 + 16 = 31 2. 54 + 4 = 68 3. 18 + 10 = 38 4. 51 + 47 = 98

5. 29 + 60 = 82 6. 42 + 54 = 84 7. 55 − 23 = 33 8. 54 − 11 = 43

9. 64 − 43 = 21 10. 98 − 45 = 53

B. Write True (T) or False (F) for each of the following open sentences if is replaced by 4.

1.[] + 2 = 9 2[]. + 3 = 7 3.[] + 7 = 12 4.[] − 3 = 1

5. 12 –[] = 7 6. 8 –[] = 4 7. 4 × []= 16 8.[] × 2 = 10

9.[] ÷ 2 = 2

Operation of addition and subtraction involving open sentences (Revision)

Examples

Here the number represented by in each of the following has been found.

1.[] + 14 = 36 2. 12 +[] = 8 3[]. − 4 = 30 4. 15 –[] = 9

Solution

1.[] + 14 = 36 can be interpreted as “what can be added to 14 to get 36?”

[]+ 14 = 20 + 16

[]+ 14 = 20 + 2 + 14

[]+ 14 = 22 + 14

[]= 22

Check:

22 + 14 = 36

Short method

If[] + 14 = 36

then []= 36 − 14

= 22

[]∃ = 22

Check:

22 + 14 = 36

2. 12 +[] = 20 + 10

12 +[] = 12 + 8 + 10

12 + []= 12 + 18

= 18

Check:

12 + 18 = 30

Short method

If 12 +[] = 30

Then[] = 30 – 12

= 18

∃ = 18

Check:

12 + 18 = 30

Note: Since the problem is addition, the number is subtracted from each other to find .

3 [] . − 4 = 8 can be interpreted as “what number minus 4 gives 8?”

[]− 4 = 12 – 4

[]= 12

Check:

12 − 4 = 8

Short method

If []− 4 = 8

Then[] = 8 + 4

= 12

Check:

12 − 4 = 8

Note: The numbers 8 and 4 are added to get the number represented by[] .

4. 15 –[] = 9 can be interpreted as ‘when a number is subtracted from 15, the answer is 9’

15 –[] = 9

15 –[] = 15 – 6 [15 = 9 + 6]

[]= 6

Check:

15 − 6 = 9

Short method

If 15 –[] = 9

Then[] = 15 – 9

= 6

Check:

15 − 6 = 9

Note: 9 is subtracted from 15 to get the number represented by[] .

. ”

Exercise

A. Find the number represented by in each of the following.

1. 9 +[] = 16 2[]. + 25 = 34 3[]. + 3 = 14

4. 8 = 5 +[] 5[] + 17 = 25 6. 7 +[] = 13

B. Find the number represented by in each of the following.

1.[] − 16 = 13 2[]. − 7 = 23 3. 19 –[] = 11

4. 77 =[] – 39 5. 17 =[] – 59 6[]. − 17 = 39

Operation of multiplication and division involving opensentences (Revision)

Examples

Find the number represented by in each of the following:

1. 7 ×[] = 56 2.[] × 4 = 48 3. 60 ÷[] = 12 4[]. ÷ 8 = 9

Solution

1. 7 × = 56 can be interpreted as “7 multiplied by a certain number equals 56”

7 ×[] = 7 × 8

= 8

Check:

7 × 8 = 56

Short method

If 7 × = 56

then =

56/7

=8 × 7/7 = 8

Check:

7 × 8 = 56

2. × 4 = 12 × 4

= 12

Check:

12 × 4 = 48

Short method

If × 4 = 48

then =

48/4

= 12 × 4/4 = 12

Check:

12 × 4 = 48

3. 60 ÷ = 12 can be interpreted as

‘what number divides 60 gives 12?’

60 ÷ = 12

60 = 5 × 12

60 ÷ 5 = 12, 60 ÷ 12 = 5

60 ÷ = 60 ÷ 5

∃ = 5

Check:

60 ÷ 5 = 12

4. ÷ 8 = 9 can be interpreted as ‘when

a number is divided by 8, the answer is 9’

÷ 8 = 9

÷ 8 = 72 ÷ 8

= 72

9 × 8 = 72

72 ÷ 8 = 9

72 ÷ 9 = 8 Check: 72 ÷ 8 = 9

Exercise

Find the number represented by in each of the following.

1. 6 × = 48 2. × 8 = 96 3. × 5 = 45 4. 6 × = 60

5. 4 × = 36 6. × 4 = 28 7. × 11 = 33 8. 12 × = 84

9. ÷ 5 = 7 10. 14

of = 16 11. 12

of = 18 12. 3 × = 18

13. ÷ 8 = 32 14. 1

10 of = 9 15. 680 ÷ = 34 16. 13

of = 12

17. 448 ÷ = 56 18. 1

10 of = 42

Use of letters to find the unknown

Activity

Study the following mathematical statements.

+ 5 = 11, a + 5 = 11 6 + = 15, 6 + y = 11 − 3 = 2, x − 3 = 2

× 2 = 12, 2 × m = 12 32 ÷ = 8, 32 ÷ n = 8

By comparing each statement, you will discover that the box is replaced with a letter of

the alphabet. That is;

+ 5 = 11 is the same as a + 5 − 3 = 2 is the same as x − 3 = 2 and so on.

Mathematical statements containing simple letters and numbers are called simple equations.

When the value of the letter is solved, the equation is solved.

Examples

1. x + 5 = 12 2. y − 12 = 3 3. 2m = 14 4. a5

= 6

Hint: Write a sentence to show the meaning of each equation.

Solution

1. x + 5 = 12 can be interpreted as “If a number is added to 5 we get 12”

3. 2m = 14 (2 m means 2 × m) can be interpreted as ‘what number multiplied by 2 gives 14?’

2 × m = 2 × 7

m = 7

Check:

2m = 2 × m = 2 × 7 = 14

Short method

If 2m = 14

then m = 14

= 7

Check: 2m = 2 × m = 2 × 7 = 14

= 6 5 × 6 = 30

= 30

5 30 ÷ 5 = 6, 30 ÷ 6 = 5 a = 30

Check: a5

= 30

5 = 6 5 × 6 = 30

Short method

If a5

= 6

then a = 5 × 6 = 30

2. y − 12 = 3 can be interpreted as “If 12 is subtracted from a number, the answer is 3”

x + 5 = 7 + 5

x = 7

Check:

7 + 5 = 12

Short method

If x + 5 = 12

then x = 12 − 5

= 7

Check:

x + 5 = 7 + 5 = 12

y − 12 = 3

y − 12 = 15 − 12

y = 15

Check:

15 − 12 = 3

Short method

If y − 12 = 3

then y = 3 + 12

= 15

Check:

y – 12 = 15 − 12 = 3

Check: a5

= 30

5 = 6

4. a5

= 6 can be interpreted as ‘when a number is divided by 5 we get 6’

Exercise

Solve the following equations.

1. m + 5 = 8 2. p + 6 = 13 3. d + 8 = 17 4. c + 2 = 12

5. e + 8 = 18 6. 5 + x = 9 7. 1 + q = 25 8. 12 + t = 30

9. m − 6 = 13 10. p − 5 = 15 11. q − 7 = 21 12. k − 12 = 35

Examples

1. Think of a number, add 7 to it, and the result is 21. Study how the number is found.

Word problems

Solution

The number I think of + 7 = 21

Let m stand for the unknown number then,

m + 7 = 21

m + 7 = 10 + 10 + 1

m + 7 = 11 + 3 + 7

m + 7 = 14 + 7 m = 14

Short method

m + 7 = 21

m = 21 − 7

= 14

Check:

m + 7 = 14 + 7

= 21

2. If 43 is subtracted from a number, we get 38. Study how the number is found.

Solution

Unknown number − 43 = 38

Let x stand for the unknown number, then

x − 43 = 38

x − 43 = 81 − 43

x = 81

Short method

x – 43 = 38

x = 38 + 43 = 81

Check:

x – 43 = 8 1

− 4 3

3 8

3. I think of a number, multiply it by 3 and the result is 36. Study how the number is found.

Solution

Unknown number × 3 = 36

Let y be the unknown number, then

y × 3 = 36

y × 3 = 12 × 3

y = 12

4. When a number is divided by 7 we get 9. Study how the number is found.

Solution

Unknown number ÷ 7 = 9

Let q be the unknown number

q ÷ 7 = 9

q ÷ 7 = 63 ÷ 7 q = 63

7 × 9 = 63

63 ÷ 7 = 9

63 ÷ 9 = 7

Check: q ÷ 7 = 63 ÷ 7 = 9

Exercise

1. When 79 is added to a number, we get 124. Find the number.

2. When 71 is added to a number, we get 214. Find the number.

3. When I subtract 19 12

from a certain number, the result is 9 12

. What is the number?

4. When 31 kg of meat is removed from the part of the cow, there is 25 kg left. What is the

weight of the cow?

5. A poultry farmer took four crates of eggs to the market. He had 45 eggs left after

market hour. How many eggs were sold?

6. When 564 is added to a certain number, the result is 801. Find the number.

7. 6 times an unknown number gives 72. Find the number.

8. When a number is multiplied by 12, we get 108. Find the number.

9. I think of a number, divide it by 8 and get 32. Find the number.

10. A certain number of oranges was shared equally among 6 children. Each child

received 14 oranges. How many oranges were shared?

[mediator_tech]

1. 15 + [] = 22
a) 6
b) 7
c) 8

2. 11 + [] = 25
a) 10
b) 12
c) 14

3. [] – 4 = 11
a) 12
b) 14
c) 15

4. 20 – [] = 7
a) 10
b) 11
c) 13

5. [] × 5 = 15
a) 2
b) 3
c) 4

6. 4 × [] = 24
a) 4
b) 6
c) 8

7. [] ÷ 6 = 5
a) 25
b) 30
c) 36

8. 48 ÷ [] = 12
a) 3
b) 4
c) 6

9. [] + 9 = 13
a) 1
b) 2
c) 4

10. 7 + [] = 14
a) 6
b) 7
c) 8

11. [] – 5 = 9
a) 9
b) 10
c) 11

12. 16 – [] = 8
a) 6
b) 7
c) 9

13. [] × 3 = 18
a) 4
b) 5
c) 6

14. 6 × [] = 30
a) 4
b) 5
c) 6

15. [] ÷ 2 = 7
a) 12
b) 14
c) 16

16. 56 ÷ [] = 8
a) 5
b) 6
c) 7

17. [] + 4 = 12
a) 6
b) 7
c) 8

18. 16 + [] = 28
a) 10
b) 12
c) 14

19. [] – 6 = 9
a) 12
b) 13
c) 15

20. 24 – [] = 16
a) 6
b) 8
c) 10

Feel free to ask for explanations or further assistance with any of the questions!

WEEK 3

ANGLES

Complementary Angle

Two Angles are Complementary when they add up to 90 degrees (a Right Angle).

They don’t have to be next to each other, just so long as the total is 90 degrees.

Examples:

• 60° and 30° are complementary angles.

• 5° and 85° are complementary angles.

Supplementary Angles

Two Angles are Supplementary when they add up to 180 degrees.supplementary angles 40 and 140

These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°:Notice that together they make a straight angle. supplementary angles 60 and 120But the angles don’t have to be together.

These two are supplementary because60° + 120° = 180°

Acute angle:

An angle whose measure is less than 90 degrees.

Right angle

angle whose measure is 90 degrees.

Obtuse angle:

An angle whose measure is bigger than 90 degrees but less than 180 degrees. Thus, it is between 90 degrees and 180 degrees.

Straight angle

An angle whose measure is 180 degrees.Thus, a straight angle look like a straight line.

Reflex angle:

An angle whose measure is bigger than 180 degrees but less than 360 degrees.

In worksheet on angles you will solve 10 different types of questions on angles.

1. Classify the following angles into acute, obtuse, right and reflex angle:

(i) 35°

(ii) 185°

(iii) 90°

(iv) 92°

(v) 260°

2. Measure these angles:

3. Use your protractor to draw these angles:

(i) 40°

(ii) 125°

(iii) 25°

4. Identify which of the following pairs of angles are complementary or supplementary?

(i) 70°, 20°

(ii) 20°, 170°

(iii) 50°, 145°

(iv) 125°, 55°

(v) 105°, 75°

(vi) 55°, 35°

5. Find the complement of each of the following angles:

(i) 40°

(ii) 27°

(iii) 35°

6. Find the supplement of each of the following angles?

(i) 100°

(ii) 90°

(iii) 110°

(iv) 107°

7. Draw a pair of supplementary angles such that one of them measures:

(i) 120°

(ii) 90°

8. Construct the angles of the following measures with the help of a compass:

(i) 150°

(ii) 90°

(iii) 120°

9. An angle whose measure is less than 90° is called an ……………… .

10. An angle measure 0° is called a …………….. .

[mediator_tech]

1. An angle of 70 degrees is classified as a(n) ______________ angle.
a) Acute
b) Right
c) Obtuse

2. Two angles measuring 45 degrees each are classified as ______________ angles.
a) Complementary
b) Supplementary
c) Reflex

3. A 90-degree angle is classified as a ______________ angle.
a) Acute
b) Right
c) Obtuse

4. An angle measuring 110 degrees is classified as a(n) ______________ angle.
a) Acute
b) Right
c) Obtuse

5. If two angles add up to 180 degrees, they are classified as ______________ angles.
a) Complementary
b) Supplementary
c) Reflex

6. An angle measuring 135 degrees is classified as a(n) ______________ angle.
a) Acute
b) Right
c) Obtuse

7. Two angles measuring 60 degrees and 120 degrees are classified as ______________ angles.
a) Complementary
b) Supplementary
c) Reflex

8. An angle of 180 degrees is classified as a ______________ angle.
a) Acute
b) Right
c) Straight

9. An angle measuring 75 degrees is classified as a(n) ______________ angle.
a) Acute
b) Right
c) Obtuse

10. Two angles measuring 150 degrees and 30 degrees are classified as ______________ angles.
a) Complementary
b) Supplementary
c) Reflex

11. An angle of 270 degrees is classified as a(n) ______________ angle.
a) Acute
b) Right
c) Obtuse

12. An angle measuring 100 degrees is classified as a(n) ______________ angle.
a) Acute
b) Right
c) Obtuse

13. If two angles add up to 90 degrees, they are classified as ______________ angles.
a) Complementary
b) Supplementary
c) Reflex

14. An angle measuring 45 degrees is classified as a(n) ______________ angle.
a) Acute
b) Right
c) Obtuse

15. Two angles measuring 80 degrees and 100 degrees are classified as ______________ angles.
a) Complementary
b) Supplementary
c) Reflex

16. An angle of 360 degrees is classified as a ______________ angle.
a) Acute
b) Right
c) Reflex

17. An angle measuring 120 degrees is classified as a(n) ______________ angle.
a) Acute
b) Right
c) Obtuse

18. If two angles add up to 270 degrees, they are classified as ______________ angles.
a) Complementary
b) Supplementary
c) Reflex

19. An angle measuring 60 degrees is classified as a(n) ______________ angle.
a) Acute
b) Right
c) Obtuse

20. Two angles measuring 200 degrees and 160 degrees are classified as ______________ angles.
a) Complementary
b) Supplementary
c) Reflex

WEEK 4

POLYGON AND PLANE FIGURES

A polygon is a shape with straight sides. Triangles and quadrilaterals are polygons. Some

other polygons have special names.

Number of straight sides Name of polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

Regular polygon

A polygon is said to be regular if all of its sides are equal. Also all its interior angles are

equal. Consequently all exterior angles are also equal.

Irregular polygon

If all the interior angles are not equal, then the polygon is not regular. It is then known as an

irregular polygoSn.

Study the table drawn below:

Name of Polygon No of sides No of angles. No of triangles

Triangle. 3 3 1

Quadrilateral. 4. 4. 2

Pentagon 5. 5. 3

Hexagon. 6. 6 4

Heptagon. 7 7 5

Octagon. 8. 8. 6

Triangles

Triangles can be classified by sides or angles.

Draw three triangles on your not book. Name them as ∆PQR, ∆ABC and ∆LMN. With the help of protector measure all the angles the angles and find them:

In ∆ABC

∠ABC + ∠BCA + ∠CAB = 180°.

In ∆PQR

∠PQR + ∠QRP + ∠RPQ = 180°

In ∆LMN

∠LMN + ∠MNL + ∠NLM = 180°

Angle Properties of Triangles

Here, we observe that in each case, the sum of the measures of three angles of a triangle is 180°.

Hence, the sum of the three angles of a triangle is equals to 180°.

For Example:

1. In a right triangle, if one angle is 50°, find its third angle.

Solution:

∆ PQR is a right triangle, that is, one angle is right angle.

Given, ∠PQR = 90°

∠QPR = 50°

Therefore, ∠QRP = 180° – (∠Q + ∠ P)

= 180° – (90° + 50°)

= 180° – 140°

∠R = 40°

2. Draw a ∆ABC. Measure the length of its three sides. Let the lengths of the three sides be AB = 5 cm, BC = 7 cm, AC = 8 cm. Now add the lengths of any two sides compare this sum with the lengths of the third side.

(i) AB + BC = 5 cm + 7 cm = 12 cm

Since 12 cm > 8 cm

Therefore, (AB + BC) > AC

(ii) BC + CA = 7 cm + 8 cm = 15 cm

Since 15 cm > 5 cm

Therefore, (BC + CA) > AB

(iii) CA + AB = 8 cm + 5 cm = 13 cm

Since 13 cm > 7 cm

Therefore, (CA + AB) > BC

two sides of a triangle is greater than the length of the third side.

For Example:

1. Is it possible to have a triangle whose sides are 5 cm, 6 cm and 4 cm?

Solution:

The lengths of the sides are 5 cm, 6 cm, 4 cm,

(a) 5 cm + 6 cm > 4 cm.

(b) 6 cm + 4 cm > 5 cm.

Hence, a triangle with these sides is possible.

We will solve some of the examples of properties of triangle.

1. In the triangle, given write the names of its three sides, three angles and three vertices.

Solution:

Three sides of ∆PQR are: PQ, QR and RP

Three angles of ∆PQR are: ∠PQR, ∠QRP and ∠RPQ

Three vertices of ∆PQR are: P, Q and R

2. Measures of two angles of a triangle are 65° and 40°. Find the measure of its third angle.

Solution:

Measures of two angles of a ∆ are 65° and 40°

Sum of the measures of two angles = 65° + 40° = 105°

Sum of all three angles of ∆ = 180°

Therefore, measure of the third angle = 180° – 105° = 75°

3. Is the construction of a triangle possible in which the lengths of sides are 5 cm, 4 cm and 9 cm?

Solution:

The lengths of the sides are 5 cm, 4 cm, 9 cm.

5 cm + 4 cm = 9 cm.

Then the sum of two smaller sides is equal to the third side. But in a triangle, the sum of any two sides should be greater than the third side.

Hence, no triangle possible with sides 5 cm, 4 cm and 9 cm.

1. Scalene Triangle:

A triangle in which all the three sides are unequal in length is called a scalene triangle.

Scalene Triangle

AC > BC >AB

6 cm > 5 cm > 4.5 cm.

2. Isosceles Triangle:

A triangle in which two of its sides are equal is called isosceles triangle.

Isosceles Triangle

PQ = PR = 6 cm.

3. Equilateral Triangle:

A triangle in which have all its three sides equal in length is called an equilateral triangle.

Equilateral Triangle

LM = MN = NL = 5.5 cm

. Classify the triangle into acute triangle, obtuse triangle and right triangle with the following angles:

(a) 90°, 45°, 45°

(b) 60°, 60°, 60°

(d) 130°, 40°, 10°

(e) 90°, 35°, 55°

(f) 92°, 38°, 50°

3. Classify the triangle according to sides, that is, equilateral, isosceles and scalene triangles

(a) 6 cm, 3 cm, 5cm.

(b) 6 cm, 6 cm, 6 cm.

(d) 8 cm, 12 cm, 10 cm.

(e) 3 cm, 4 cm, 5 cm.

(f) 3.5 cm, 3.5 cm, 4.5 cm.

4. Is it possible to have a triangle with the following angles and sides? Give reason in support of your answer.

(a) 110°, 60°, 30°

(b) 70°, 70°, 70°

(d) 7 cm, 3 cm, 4 cm.

(e) 50°, 50°, 90°

(f) 10 cm, 12 cm, 2 cm.

5. Find the perimeter of a triangle when its sides are:

(a) AB = 7.6 cm, BC = 4.5 cm, CA = 6.3 cm.

(b) PQ = 4.00, QR = 3 cm, RP = 5 cm.

6. If in a triangle LMN:

(a) ∠L = 80°, ∠M = 50°, find ∠N,

(b) ∠M = 60°, ∠N = 60°, find ∠L,

(d) ∠N = 90°, ∠L = 45°, find ∠M.

Mention the kind of triangle also.

7. The angle of a triangle is in the ratio of 2: 3: 4. Find the measure of each angle of the triangle.

8. One of the two equal angles of an isosceles triangle measures 55°. Determine the other angles.

9. The perimeter of a triangle is 24 cm. Two of its sides are 8 cm and 9 cm. Find the length of its third side.

10. Each side of a ∆ is one third of its perimeter. What kind of triangle is this?

11. One of the acute angles of a right triangle is 48°. Find the other acute angle.

12. Say whether the following statements are true or false:

(a) All the angles of an isosceles triangle are equal.

(b) If one angle of a triangle is obtuse, the other two angles must be acute.

(d) Equiangular triangle has its three sides also equal.

(e) A triangle can have two obtuse angles.

Drawing polygons not exceeding octagons

1. Acute Angle:

An angle whose measure is less than 90° is called an acute angle.

Acute Angle

∠MON shown in adjoining figure is equal to 60°. So, ∠MON is an acute angle.

2. Right Angle:

An angle whose measure is 90° is called right angle.

Right Angle

In the above figure, ∠AOB is a right angle. In this case, we say that the arms OA and OB are perpendicular to each.

Therefore, ∠AOB shown in adjoining figure is 90°.

So, ∠AOB is a right angle.

3. Obtuse Angle:

An angle whose measure is greater than 90° but less than 180° is called an obtuse angle.

Obtuse Angle

6Save

∠DOQ shown in the above figure is an obtuse angle.

Quadrilaterals

These are 4-sided plane shapes. They are rectangles, parallelograms, squares, rhombus,

kite and trapeziam.

Rectangles

Opposite sides are equal.

Each of the interior angle is 90°.

Opposite sides are parallel

AB // CD

AD // BC

The two diagonals are AC and BD. Is AC = BD?

The two lines of symmetry are MN and ST. Is MN = ST?

Parallelograms

Opposite sides are equal and parallel.

A B

D C

None of the angles is equal to 90°

Is diagonal AC = BD?

How many lines of symmetry has parallelogram ABCD?

Squares

Opposite sides are equal parallel.

A B

D C

AB = CD = AD = BC

All the sides are equal.

Each of the interior angle is 90°.

Draw the lines of symmetry. How many are they?

Is AC = BD?

Rhombus

Opposite sides are equal and parallel. A B

C D

AB = CD and AD = BC

All the sides are equal.

AB = CD = AD = BC

None of the angles is equal to 90°.

Draw the diagonals AC and BD.

Measure AC and BD.

Is AC = BD?

How many lines of symmetry does a rhombus have?

Kite

Opposite sides are not equal.

How many lines of symmetry does a kite have?

Name them.

Draw the diagonals. How many are they?

Name the diagonals.

None of the interior angles is equal to 90°.

Trapezium

A pair of the opposite sides are parallel.

Two of the interior angles may be 90°.

How many lines of symmetry does a trapezium have?

These are two different types; see fig 1 and fig 2.

Draw the diagonals. How many are they?

Pentagon

A regular pentagon has 5 equal sides

and angles.

Here AB = BC = CD = DE = EA.

How many diagonals does a pentagon

have?

How many lines of symmetry does a

pentagon have?

Hexagon

A regular hexagon has 6 equal sides and

angles.

How many lines of symmetry does pentagon ABCDEF have?

Heptagon

A regular heptagon has 7 equal sides and angles.

Octagon

A regular octagon has 8 equal sides and

angles.

Properties of different prisms

Properties of a cube

It has 6 equal faces. Each face is a square

It has 12 straight edges

All the edges are equal

It has 8 vertices (corners).

Properties of a cuboid

A cuboid has 6 faces

Some faces are either a

square or a rectangle

It has 12 straight edges

It has 8 vertices (corners).

Properties of a cylinder

A cylinder has 2 plane faces

The 2 plane faces are circular

It has 1 curved face

It has 2 curved edges.

Properties of a triangular prism

A triangular prism has 5 faces

It has 9 edges

The 2 opposite faces are

parallel and equal. Theyare triangularIt has 6 vertices (corner).

Properties of pyramids

A pyramid is a 3-dimensional shape with a square, rectangular or triangular base. It has

some slanting edges which meet at a point. Examples could be seen on modern day rood,

cocoa and groundnut pyramids. The Egyptian pyramids are also good examples. There are

different types of pyramids. Each derives its name from its type of base.

Properties of pyramids

Square based pyramids

A square based pyramid has:8 edges

5 faces: 1 square face + 4 triangular faces5 vertices.

Rectangular based pyramids

A rectangular based pyramid has:8 edges

5 faces: 1 rectangular face + 4 triangular faces5 vertices.

Triangular based pyramids (tetrahedron)

A triangular based pyramid has:6 edges

4 triangular faces 4 vertices.

Pentagonal pyramid

A pentagonal pyramid has:10 edges 6 faces 6 vertices.

[mediator_tech]

1. A polygon is a shape with ______________ sides.
a) Curved
b) Straight
c) Mixed

2. A shape with 4 straight sides is called a ______________.
a) Triangle
b) Quadrilateral
c) Pentagon

3. A polygon with 5 sides is called a ______________.
a) Triangle
b) Quadrilateral
c) Pentagon

4. A regular polygon has ______________ equal sides.
a) No
b) Some
c) All

5. An irregular polygon has ______________ interior angles.
a) Equal
b) Unequal
c) No

6. A polygon with 6 sides is called a ______________.
a) Hexagon
b) Heptagon
c) Octagon

7. A polygon with 7 sides is called a ______________.
a) Hexagon
b) Heptagon
c) Octagon

8. A polygon with 8 sides is called a ______________.
a) Hexagon
b) Heptagon
c) Octagon

9. In a triangle, the number of angles is ______________.
a) 2
b) 3
c) 4

10. In a quadrilateral, the number of triangles formed is ______________.
a) 1
b) 2
c) 3

11. A triangle has ______________ sides.
a) 2
b) 3
c) 4

12. A quadrilateral has ______________ angles.
a) 3
b) 4
c) 5

13. A pentagon has ______________ triangles formed.
a) 2
b) 3
c) 4

14. A hexagon has ______________ sides.
a) 5
b) 6
c) 7

15. A heptagon has ______________ angles.
a) 6
b) 7
c) 8

16. An octagon has ______________ triangles formed.
a) 7
b) 8
c) 9

17. A regular polygon has ______________ equal interior angles.
a) Some
b) All
c) None

18. An irregular polygon has ______________ equal sides.
a) Some
b) All
c) None

19. A triangle can be classified by its ______________.
a) Sides
b) Angles
c) Color

20. A triangle has ______________ angles.
a) 1
b) 2
c) 3

WEEK 7

EVERYDAY STATISTICS

Exercise 2

Write these figures in tally marks.

1. 13 2. 22 3. 19 4. 25 5. 31 6. 4

7. 29 8. 24 9. 36 10. 43

Mode

Mode is the item or number that occurs most frequently in a given data. To find the mode of

a given data, arrange the numbers in order of magnitude, then count the number of times

each number occurs.

Example

The following are the scores of fifteen pupils in a mathematics test: 100%, 84%, 92%, 84%,

67%, 88%, 84%, 45%, 35%, 84%, 24%, 20%, 54%, 64% and 10%. Study the solution to find

the mode of the score.

Solution

Arrange the scores in order of magnitude to enable you to note the mode clearly.

100%, 92%, 88%, 84%, 84%, 84%, 84%, 67%, 64%, 54%, 45%, 35%, 24%, 20%, 10%

The mode of the scores = 84%.

We can also prepare a table as follows.

Scores % Number of times Scores % Number of times

100 1 45 1

92 1 35 1

84 6 24 1

67 1 20 1

58 1 0 1

Unit 1

255

Exercise 1

Find the mode of each set of data.

1. 1, 6, 7, 5, 8, 9, 6, 3, 2 2. 3, 7, 8, 4, 0, 0, 0, 6, 5, 9, 10

3. 5, 2, 5, 6, 8, 6, 2, 2 4. 4, 5, 6, 7, 7, 0, 2, 1, 7, 6, 8

5. 4, 7, 6, 7, 5, 9, 5, 7, 8 5 6. 2, 5, 7, 6, 8, 3, 8, 9, 7, 8

7. 9, 3, 4, 5, 9, 6, 7, 8, 0, 2 8. 10, 2, 4, 6, 8, 7, 9, 10, 10, 0

9. 7, 5, 6, 4, 5, 6, 3, 8, 7 10. 4, 6, 9, 7, 9,

MEAN

Mean is the average of a given set of numbers. To find the mean, add up all the numbers,

then divide by how many numbers that are in the set.

Mean = Sum of the numbers/Numbers in the set

Example

The mean of the following set of numbers has been calculated:

36, 59, 54, 20, 82, 61, 48, 42, 58 and 40

Solution

Mean = 36 + 59 + 54 + 20 + 82 + 61 + 48 + 42 + 58 + 40/10

= 500/10

= 50

256

Exercise

Find the mean of each set of the following scores.

1. 50%, 85%, 70%, 45%, 90%, 68%, 98%, 25%, 63%, 39%

2. 14%, 91%, 40%, 73%, 88%, 60%, 55%, 59%

3. 32%, 24%, 53%, 68%, 92%, 81%, 47%, 76%, 40%, 87%, 61%, 23%

Answer these questions.

4. The pulse rates of six patients are as follows: 59, 36, 48, 51, 62, 68. Find the mean of

the pulse rate.

5. The following are the scores of fifteen pupils in a mathematics test out of 10

5, 8, 6, 9, 7, 4, 10, 2, 5, 8, 9, 0, 1, 6, 9.

Find the mean score.

6. The ages of eight pupils are shown below. Find their mean age:

13 years, 14 years, 12 years, 10 years, 11 years, 9 years, 15 years, 8 years

7. The allowances of ten students are as follows: 􀎏800.00, 􀎏1000.00, 􀎏2000.00, 􀎏1500.00,

􀎏700.00, 􀎏1800.00, 􀎏2200.00, 􀎏1700.00, 􀎏2500.00, 􀎏600.00. What is the mean

allowance?

8. The distance covered by eight cars are 120 km, 200 km, 350 km, 150 km, 250 km, 90 km,

500 km, 420 km. Find the mean of the distance covered.

Median

The median of a set of numbers is that number that is exactly at the middle of the set of

numbers when they are arranged in order of size.

To find the median, arrange the numbers in order of size, from the largest to the smallest.

The median is then the middle number.

Examples

Here the median of each of the following sets of numbers has been found.

1. 20, 30, 10, 90, 80, 100, 70

Solution

Arrange the numbers in order of size.

100 90 80 |70| 30 20 10

The median = 70

We have three numbers to the left and three to the right and 70 is at the middle.

2. 18, 20, 14, 15, 12, 16, 24, 28, 30, 25

Solution

Arrange the numbers in order of size

30 28 25 24 |20 18| 16 15 14 12

Since there is no middle number the median is the mean of the two middle numbers.

Median = 20 + 18/2

= 38/2

= 19

Exercise

Find the median of each of the following sets of numbers:

1. 9, 12, 14, 20, 15, 32, 41, 35, 40

2. 218, 314, 420, 117, 510, 250, 340, 480, 550, 180, 390

3. 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180

4. 8.2, 3.5, 9.7, 10.1, 13.4, 5.5, 17.6, 12.3, 11.4

5. 40, 20, 80, 60, 100, 140, 120, 180, 160, 200, 240, 220

6. 170, 230, 190, 150, 130, 250, 240, 300, 330

7. 111, 222, 333, 414, 515, 616, 720, 820, 920, 1120

8. 8 cm, 10 cm, 14 cm, 18 cm, 22 cm, 24 cm, 12 cm, 26 cm, 11 cm, 17 cm, 25 cm

9. 27 kg, 30 kg, 14 kg, 32 kg, 45 kg, 54 kg, 23 kg

10. 40 litres, 30 litres, 53 litres, 26 litres, 64 litres, 70 litres, 82 litres, 90 litres, 100 litres

Frequency tables can be drawn from your experiments of:

􀁏 rolling dice

􀁏 tossing coins several times.

Examples

1. Bisola tossed a coin 20 times and recorded his results as follows. Study the solutions.

H T T H H

T T H T H

T H T T H

H T H T T

a) Draw a frequency table to show the experiment.

b) How many times did heads appear?

c) How many times did tails appear?

d) Express b) and c) as fraction of total outcomes.

Solution

a) Face Tally Frequency

H //// ///// 9

T //// //// / 11

Total 20

b) Heads appears 9 times.

c) Tails appears 11 times.

d) i) The fraction of heads appearing is 9/20

ii) The fraction of tails appearing is 11/20

Bar Graph Examples

To understand the above types of bar graphs, consider the following examples:

Example 1: In a firm of 400 employees, the percentage of monthly salary saved by each employee is given in the following table. Represent it through a bar graph.

Savings (in percentage) Number of Employees(Frequency)

20 105

30 199

40 29

50 73

Total 400

Solution: The given data can be represented as

Bar graph – Vertical

This can be also represented using a horizontal bar graph as follows:

Bar Graph – Horizontal

Example 2: A cosmetic company manufactures 4 different shades of lipstick. The sale for 6 months is shown in the table. Represent it using bar charts.

Month Sales (in units)

Shade 1 Shade 2 Shade 3 Shade 4

January 4500 1600 4400 3245

February 2870 5645 5675 6754

March 3985 8900 9768 7786

April 6855 8976 9008 8965

May 3200 5678 5643 7865

June 3456 4555 2233 6547Swipe left

Solution: The graph given below depicts the following data

Bar Graph

Example 3: The variation of temperature in a region during a year is given as follows. Depict it through graph (bar).

Month Temperature

January -6°C

February -3.5°C

March -2.7°C

April 4°C

May 6°C

June 12°C

July 15°C

August 8°C

September 7.9°C

October 6.4°C

November 3.1°C

December -2.5°C<

Solution: As the temperature in the given table has negative values, it is more convenient to represent such data through a horizontal bar graph.

Bar Graph Example

Uses of Bar Graphs

Bar graphs are used to match things between different groups or to trace changes over time. Yet, when trying to estimate change over time, bar graphs are most suitable when the changes are bigger.

Bar charts possess a discrete domain of divisions and are normally scaled so that all the data can fit on the graph. When there is no regular order of the divisions being matched, bars on the chart may be organised in any order. Bar charts organised from the highest to the lowest number are called Pareto charts.

Bar Graph Question

Question: A school conducted a survey to know the favourite sports of the students. The table below shows the results of this survey.

Name of the Sport Total Number of Students

Cricket. 45

Football. 53

Basketball 99

Volleyball 44

Chess 66

Table Tennis 22

Badminton 37

From this data,.

1. Draw a graph representing the sports and the total number of students.

2. Calculate the range of the graph.

3. Which sport is the most preferred one?

4. Which two sports are almost equally preferred?

5. List the sports in ascending order.

PIE CHART

Constructing Circle Graphs or Pie Charts

A pie chart (also called a Pie Graph or Circle Graph) makes use of sectors in a circle. The angle of a sector is proportional to the frequency of the data.

The formula to determine the angle of a sector in a circle graph is:

circle graph pie chart formula

Study the following steps of constructing a circle graph or pie chart:

Step 1: Calculate the angle of each sector, using the formula

Step 2: Draw a circle using a pair of compasses

Step 3: Use a protractor to draw the angle for each sector.

Step 4: Label the circle graph and all its sectors.

Example:

In a school, there are 750 students in Year1, 420 students in Year 2 and 630 students in Year 3. Draw a circle graph to represent the numbers of students in these groups.

Solution:

Total number of students = 750 + 420 + 630 = 1,800.

Draw the circle, measure in each sector. Label each sector and the pie chart.

pie chart

Example:

The following pie chart shows a survey of the numbers of cars, buses and motorcycles that passes a particular junction. There were 150 buses in the survey.

a) What fraction of the vehicles were motorcycles?

b) What percentage of vehicles passing by the junction were cars?

c) Calculate the total number of vehicles in the survey.

d) How many cars were in the survey?

Solution:

a) Fraction of motorcycles

[mediator_tech]

1. A rectangle has ______________ sides.
a) 2
b) 3
c) 4

2. Opposite sides of a rectangle are ______________.
a) Parallel
b) Perpendicular
c) Diagonal

3. In a rectangle, each interior angle measures ______________ degrees.
a) 45
b) 90
c) 180

4. Parallelograms have ______________ sides that are equal and parallel.
a) Opposite
b) Adjacent
c) Diagonal

5. In a parallelogram, the diagonals ______________.
a) Are equal
b) Are parallel
c) Are perpendicular

6. A square has ______________ lines of symmetry.
a) 1
b) 2
c) 4

7. In a rhombus, all the sides are ______________.
a) Equal
b) Parallel
c) Perpendicular

8. A kite has ______________ sides that are equal.
a) Opposite
b) Adjacent
c) Diagonal

9. A trapezium has ______________ lines of symmetry.
a) 1
b) 2
c) 0

10. A pentagon has ______________ equal sides and angles in a regular pentagon.
a) 3
b) 5
c) 7

11. A hexagon has ______________ lines of symmetry.
a) 1
b) 3
c) 6

12. A heptagon has ______________ equal sides and angles in a regular heptagon.
a) 5
b) 7
c) 9

13. A regular octagon has ______________ equal sides and angles.
a) 6
b) 8
c) 10

14. A cube has ______________ equal faces.
a) 4
b) 6
c) 8

15. A cuboid has ______________ straight edges.
a) 8
b) 10
c) 12

16. A cylinder has ______________ plane faces.
a) 1
b) 2
c) 3

17. The faces of a cuboid can be either a square or a ______________.
a) Circle
b) Rectangle
c) Triangle

18. A cylinder has ______________ curved edges.
a) 1
b) 2
c) 3

19. A cube has ______________ vertices (corners).
a) 4
b) 6
c) 8

20. A cylinder has ______________ curved faces.
a) 1
b) 2
c) 3

Third Term Examinations Primary 4 Mathematics

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