yearly plan of mathematics 2015 2016 second …yearly plan of mathematics 2015 – 2016 second...

Yearly plan of mathematics 2015 – 2016 Second semester grade 8

Teacher's Name & Signature: Senior Teacher's Signature: Supervisor's Signature: Principal's Signature:

Only minimum level of objectives are given in the yearly plan and more can be added further.

The teacher is strongly advised to give more examples and exercises (from the approved list of Math books) than the ones provided in the yearly plan.

The teacher may solve the problems in any scientific way (rather than the suggested methods in the yearly plan).

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Solving for 𝑥 in equations of the following form:

- 𝑎𝑥 = 𝑏 + 𝑐𝑥

- 𝑎(𝑥 ± 𝑏) =c

- 𝑎𝑥 ± 𝑏 = 𝑐𝑥 + 𝑑

- 𝑎(𝑏𝑥 + 𝑐) = 𝑑(𝑒𝑥 + 𝑓), where constants are rational numbers

Remember Equation Rule:

You can do anything (add/subtract/multiply/divide) to one side of an

equation, as long as you do it to the other side.

Method for Solving Equations:

1. Simplify each side:

a. clear parentheses (distributive property)

b. clear fractions (multiply both sides by the LCM)

c. collect like terms

2. Isolate the variable:

This means: add/subtract variables to get the variables on one side,

and add additive inverse to get the numbers on the other side.

3. Get 𝑥 by itself: Multiply by the multiplicative inverse.

4. Check your answer.

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e.g.(1): Solve 3 − 5𝑥 − 2 = 2𝑥 + 15 for 𝑥.

1. No parentheses, no fractions; collect like terms:

1 − 5𝑥 = 2𝑥 + 15

2. Isolate 𝑥:

1 − 5𝑥 + (−2𝑥) = 2𝑥 + 15 + (−2𝑥) add (−2𝑥) to both sides

1 − 7𝑥 + (−1) = 15 + (−1) add (−1) to both sides

−7𝑥 = 14

3. Get 𝑥 by itself:

−1

7 × −7𝑥 = 14 × −

1

7 multiply both sides by

𝑥 = −2 multiplicative inverse of −7

4. Check it: plug it back into the original equation.

3 − 5 × (−2) − 2 = 2 × (−2) + 15

3 + 10 − 2 = −4 + 15

11 = 11 Correct

e.g.(2): Solve 2𝑥 − 5(𝑥 − 2) = 4 − 8𝑥 + 1 for 𝑥 .

1. Simplify: Clear parentheses: 2𝑥 − 5𝑥 + 10 = 4 − 8𝑥 + 1

Collect like terms: −3𝑥 + 10 = 5 − 8𝑥

2. Isolate 𝑥 : 5𝑥 = −5

3. Get 𝑥 by itself, multiply both sides by 1

5 :

1

5 × 5𝑥 = −5 ×

1

5

𝑥 = −1 Check it: plug it back into the original equation

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e.g.(3): Find 𝑥 in the given figure if the perimeter = 48 𝑚 :

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Define and use the term “inequality” .

An inequality is a mathematical sentence in which two expressions

are joined by any of the following signs ( > , < , ≥ , ≤ ).

e.g.(1): What inequality does this number line show?

𝑥 ≤ −1 (𝑜𝑟 𝑥 < 0) 𝑤ℎ𝑒𝑟𝑒 𝑥 ∈ ℤ

e.g.(2): Which of the following is a solution to the inequality 𝑥 + 2 ≤ 7?

a) 𝑥 = 4 b) 𝑥 = 5 c) 𝑥 = 6

e.g.(3): Translate the following word statements into inequalities sentences:

a) A number increased by 6 is greater than or equal 2.

b) Waleed is 6 years older than Sara and their ages is less than 42

years.

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Solve simple linear inequalities (in one variable).

e.g.(1): Solve 2𝑥 − 2 < 4 𝑤ℎ𝑒𝑟𝑒 𝑥 ∈ ℤ

( Show your solution on the number line and as a roster form notation ).

2𝑥 − 2 < 4 add to both sides

2𝑥 < 6 multiply both sides by 1

2

𝑥 < 3

The answer in a roster form is ∶ {2 ,1 ,0 , −1, … }

The solution on the number line :

e.g.(2): Solve 1 − 𝑥 ≤ −5 𝑤ℎ𝑒𝑟𝑒 𝑥 ∈ ℚ

( write your solution as a set-builder notation ).

1 − 𝑥 ≤ −5 add −1 to both sides

−𝑥 ≤ −6 multiply both sides by −1

and reverse the sign

The answer is ∶ 𝑥 ≥ 6

The answer in set − builder notation is ∶ {𝑥: 𝑥 ∈ ℚ, 𝑥 ≥ 6}

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Reflections of figures on the coordinate plane.

A Reflection is a geometric transformation in which the figure is the

mirror image of the other.

The figure under reflection is called image.

If the original object was labeled with letters, such as ABCD,

the image may be labeled with the same letters followed by a

prime symbol, A'B'C'D'.

Reflection in a Point :

A reflection in a point P is a transformation of the figure such that

the image of the fixed point P is P and for all other points, the image

of A is A' where P is the midpoint of AA′̅̅ ̅̅̅ .

e.g.(1): Draw the image of

triangle ABC under a

reflection in the point O.

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Reflection in the origin (coordinate plane):

The above triangle ABC has been reflected in the point O (the origin).

Notice how the coordinates of triangle A'B'C' are the same coordinates as

triangle ABC, but the signs have been changed.

The image of point (x, y) under a reflection in the origin is (-x, -y).

e.g.(2): The vertices of a triangle ABC are given by the coordinates

A(-3,2), B(1,1), C(-3,-1). Draw the triangle then reflect it in the

origin.

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Reflection in a Line :

A reflection in a Line (L) is a transformation of the figure such that

the image of any point P on L is P and for all other points, the image

of A is A' where L is the perpendicular bisector of AA′̅̅ ̅̅̅ .

e.g.(1): Draw the image of the following figures under a reflection in

the given Line.

(a) (b)

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Reflection in the x-axis (coordinate plane):

The image of point (𝑥, 𝑦) under a reflection in the x-axis is (𝑥, −𝑦) .

e.g.(1): Find the image of reflection across the x-axis for the points :

K(−4, 3), S(5 0) , Q(2, −7)

e.g.(2): Triangle XYZ has vertices X (-1, 7), Y (6,5), and Z (-2, 2). Draw

the image of triangle XYZ after a reflection in x-axis.

Reflection in the y-axis (coordinate plane):

The image of point (𝑥, 𝑦) under a reflection in the y-axis is (−𝑥, 𝑦) .

e.g.(1): On the given coordinate plane, draw the image of triangle ABC

according to the reflection rule (𝑥, 𝑦) → (−𝑥, 𝑦) .

e.g.(2): Describe each of the following transformations:

a) F(3, 2) → F′(−3,2) b) X(1,4) → X′(−1, −4) c) A(0,7) → A′(0, −7)

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Determine the sum of the interior angles of any polygon.

Activity:

Divide up each polygon below into the fewest triangles possible and then

figure out the sum of interior angles in the polygon.

Polygon Number of

sides

Number of

triangles Sum of interior angles

3 1 = 180°

4 2

= 180° + 180°

= 2 × 180°

= 360°

5 3

= 180° + 180° + 180°

= 3 × 180°

= 540°

……. ……. …………….

……. ……. …………….

What is the relation between number of sides and number of triangles?

The sum of the interior angles of any polygon is 180° × (n − 2)

Where (n) is the number of polygon's sides

e.g.(1): Find the sum of the interior angles of a polygon with :

a) 10 sides b) 16 sides c) 21 sides

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e.g.(2): Find the number of sides in a polygon whose sum of the interior

angles is 1440?

180 × (n − 2) = 1440

180𝑛 − 360 = 1440

𝑛 = 10

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Find, given the other angles, the unknown interior angle of any polygon.

e.g.(1): Find the value of the missing angles for the following polygons:

Each interior angle of a regular polygon = 180° ( 𝑛−2 )

𝑛

Where (n) is the number of sides in the polygon

e.g.(2): A regular polygon has 15 sides.

a) What is the sum of its interior angles?

b) What is the measure of each interior angle?

e.g.(3): The size of each interior angle of a regular polygon is 135°. How

many sides does the polygon have?

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Determine the sum of the exterior angles of any polygon.

The Exterior Angle of a polygon is the angle between any side of a

shape, and a line extended from the next side.

Sum exterior angles of any polygon = 360°

The exterior and interior angles are supplementary angles.

e.g.(1): Find the sum of the exterior angles of :

a) a pentagon b) a 7 sided polygon

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Find, given the other angles, the unknown exterior angle of any polygon.

e.g.(1): Find the value of the missing angles for the following polygons:

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Each exterior angle (of a regular polygon) = 360°

𝑛

Where (n) is the number of sides in the polygon

e.g.(2): Find the measure of each exterior angle of a regular hexagon?

e.g.(3): The measure of each exterior angle of a regular polygon is

45°. How many sides does the polygon have?

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Determine the properties of congruent triangles.

Congruence of two line segments

The two line segments are congruent if they have equal length.

If the length of 𝐸𝐹̅̅ ̅̅ = the length of 𝐺𝐻̅̅ ̅̅ ,

then 𝐸𝐹̅̅ ̅̅ is congruent to 𝐺𝐻̅̅ ̅̅ and written 𝐸𝐹̅̅ ̅̅ ≡ 𝐺𝐻̅̅ ̅̅ .

Congruence of two angles

The two angles are congruent if they are equal in measure.

If the measure of ∡𝐴𝐵𝐶 = measure of ∡𝐷𝐹𝐸 ,

then ∠𝐴𝐵𝐶 is congruent to ∠𝐷𝐹𝐸 and written ∠𝐴𝐵𝐶 ≡ ∠𝐷𝐹𝐸 .

Congruence of two triangles

Two triangles are congruent if there is a correspondence between:

Their sides.

Their angles.

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Identify and state corresponding parts of congruent triangles.

If two triangles are congruent, we often mark corresponding sides and

angles as shown (the order is important):

e.g.(1): If ABC and XYZ are

congruent triangles, write

the corresponding sides

and angles.

e.g.(2): If ∆ 𝐵𝐷𝐶 ≡ ∆ 𝐶𝐴𝐵 ,

Find the measure of x?

e.g.(3): In the opposite figure,

∆ 𝐴𝐵𝑀 ≡ ∆ 𝐶𝐷𝑀 where

∡ 𝐴𝐵𝑀 = 90° , ∡ 𝐶 = 45°

𝐴𝐵̅̅ ̅̅ = 4 cm , 𝐵𝑀̅̅ ̅̅̅= 3 cm and

𝐶𝑀̅̅̅̅̅= 5cm. Find:

a) 𝐴𝑀̅̅̅̅̅, 𝐷𝑀̅̅ ̅̅ ̅ and 𝐶𝐷̅̅ ̅̅

b) ∡ 𝐴𝑀𝐵 , ∡ 𝐶𝑀𝐷 and 𝑋°

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Determine (prove) whether triangles are congruent by:

- SSS - SAS

If three sides of one triangle are congruent to three corresponding

sides of a second triangle, then the two triangles are congruent by

three sides (SSS).

e.g.(1): Given 𝑳𝑴̅̅̅̅̅ ≡ 𝑶𝑷̅̅ ̅̅ and 𝑳𝑵̅̅ ̅̅ ≡ 𝑷𝑵 and N is the midpoint of 𝑴𝑶̅̅ ̅̅ ̅ prove that ∆ 𝐿𝑀𝑁 ≡ ∆ 𝑃𝑂𝑁.

Proof: In ∆ 𝐿𝑀𝑁 and ∆ 𝑃𝑂𝑁 :

{𝑳𝑴̅̅̅̅̅ ≡ 𝑷𝑶̅̅ ̅̅ 𝑳𝑵̅̅ ̅̅ ≡ 𝑷𝑵̅̅̅̅̅ 𝑴𝑵̅̅ ̅̅ ̅ ≡ 𝑶𝑵̅̅ ̅̅̅

( 𝑠𝑖𝑑𝑒, 𝑔𝑖𝑣𝑒𝑛)( 𝑠𝑖𝑑𝑒, 𝑔𝑖𝑣𝑒𝑛)

( side , N is the midpoint of 𝑴𝑶̅̅ ̅̅ ̅)

∴ ∆ 𝐿𝑀𝑁 ≡ ∆ 𝑃𝑂𝑁 (By SSS )

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e.g.(2): In the given figure 𝑨𝑩̅̅ ̅̅ ≡ 𝑩𝑫̅̅̅̅̅ and 𝑨𝑪̅̅ ̅̅ ≡ 𝑪𝑫̅̅ ̅̅ . Prove that ∆ 𝐴𝐵𝐶 ≡ ∆ 𝐷𝐵𝐶 .

Proof: In ∆ 𝐴𝐵𝐶 and ∆ 𝐷𝐵𝐶 :

{𝑨𝑩̅̅ ̅̅ ≡ 𝑩𝑫̅̅̅̅̅

𝑨𝑪̅̅ ̅̅ ≡ 𝑪𝑫̅̅ ̅̅ 𝑩𝑪̅̅ ̅̅ ≡ 𝑩𝑪̅̅ ̅̅

(side , given)(side , given)

(common side)

∴ ∆ 𝐴𝐵𝐶 ≡ ∆ 𝐷𝐵𝐶 (By SSS)

e.g.(3): In the given figure, 𝑿𝒀̅̅ ̅̅ ≡ 𝑿𝑳̅̅ ̅̅ and 𝑳𝒁̅̅̅̅ ≡ 𝒀𝒁̅̅ ̅̅ . Prove that ∆ 𝑋𝐿𝑍 ≡ ∆ 𝑋𝑌𝑍 .

e.g.(4): In the given figure, 𝑨𝑩̅̅ ̅̅ ≡ 𝑫𝑩̅̅̅̅̅ and 𝑩𝑪̅̅ ̅̅ is

perpendicular bisector of 𝑨𝑪̅̅ ̅̅ . Prove that ∆ 𝐴𝐵𝐶 ≡ ∆ 𝐷𝐵𝐶.

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If any two corresponding sides and their included angle are the same

in both triangles, then the two triangles are congruent by (SAS).

e.g.(1): In the given figure,

𝑷𝑸̅̅ ̅̅ ≡ 𝑹𝑺̅̅ ̅̅ and ∠𝑃𝑄𝑆 ≡ ∠𝑅𝑆𝑄 .

Prove that ∆ 𝑃𝑆𝑄 ≡ ∆ 𝑅𝑄𝑆 ?

Proof:

In ∆ 𝑃𝑆𝑄 and ∆ 𝑅𝑄𝑆 ∶

{𝑷𝑸̅̅ ̅̅ ≡ 𝑹𝑺̅̅ ̅̅

∠𝑃𝑄𝑆 ≡ ∠𝑅𝑆𝑄

𝑸𝑺̅̅ ̅̅ ≡ 𝑸𝑺̅̅ ̅̅

(side , given)(angle , given)

(common side)

∴ ∆ 𝑃𝑆𝑄 ≡ ∆ 𝑅𝑄𝑆 (By SAS)

e.g.(2): In the given figure,

𝑨𝑪̅̅ ̅̅ ≡ 𝑬𝑪̅̅ ̅̅ and 𝑩𝑪̅̅ ̅̅ ≡ 𝑫𝑪̅̅ ̅̅ .

Prove that ∆ 𝐴𝐵𝐶 ≡ ∆ 𝐷𝐸𝐶 .

Proof:

In ∆ 𝐴𝐵𝐶 and ∆ 𝐷𝐸𝐶 :

{

𝑨𝑪̅̅ ̅̅ ≡ … … .̅̅ ̅̅ ̅̅ ̅ ∠ … . ≡ ∠ … .

… . .̅̅ ̅̅ ̅ ≡ 𝑫𝑪̅̅ ̅̅̅

(side , given)(𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑛𝑔𝑙𝑒)(side , given)

∴ ∆ … … . ≡ ∆ … … .. (by ……)

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e.g.(3): Determine if the two triangles are congruent. Use the result to find

x , y ,z and n?

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Define and explain the angle bisectors of any triangle and their properties.

An angle bisector is a line that divides an angle into two equal angles.

A triangle has three angle bisectors.

Properties of Angle Bisectors

The angle bisectors of a triangle intersect at a point called the

incenter.

The incenter always lies inside the triangle.

The incenter is equidistant from the sides of the triangle.

M is the incenter of

triangle ABC.

EM̅̅ ̅̅ = FM̅̅ ̅̅ = DM̅̅̅̅̅

e.g.(1): In the triangle ABC,

∡A = 80°, ∡B = 40° and M is the

incenter of the triangle. Find size

of angle ∠DMO.

e.g.(2): In the triangle DOH,

∡DHO = 30°, ∡DOM = 35° and

M is the incenter of the triangle.

Find size of angle ∠HDM.

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Define and explain the perpendicular bisectors of the sides of any triangle

and their properties.

A perpendicular bisector is a line that divides a line segment into

two equal segments at 90°.

A triangle has three perpendicular bisectors of the sides.

Properties of Perpendicular Bisectors

The perpendicular bisectors of the sides of a triangle intersect at

a point called the circumcenter.

The circumcenter is equidistant from the vertices of the triangle.

The circumcenter can lie inside the triangle or outside the

triangle or on the hypotenuse of the triangle.

e.g.(1): In the triangle ABC, BC̅̅̅̅ is

the perpendicular bisector

of AD̅̅ ̅̅ . Find the value of x.

e.g.(1): Determine the position of the circumcenter of the following

triangles:

a) Acute triangle.

b) Obtuse triangle.

c) Right angled triangle.

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Define explain medians of a triangle and their properties.

A median of a triangle is a line segment joining a vertex to

the midpoint of the opposite side.

A triangle has three medians.

Activity (Constructing the medians of a triangle):

Using a ruler, start with triangle ABC

Step 1: Choose any side to start with.

Let start with 𝐴𝐵̅̅ ̅̅ .

Measure the length of 𝐴𝐵̅̅ ̅̅ and set the

midpoint.

Step 2: Draw a line segment from vertex C to

the midpoint of 𝐴𝐵̅̅ ̅̅ .

Step 3: Repeat steps 1 and 2 for the sides 𝐵𝐶̅̅ ̅̅

and 𝐴𝐶̅̅ ̅̅ .

The three line segments drown are the medians of the triangle.

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Properties of Medians

The medians of a triangle intersect at a point called the centroid.

The centroid always lies inside the triangle.

The centroid cuts every median in the ratio 𝟐 ∶ 𝟏 (the distance

between a vertex and the centroid is twice as long as the distance

between the centroid and the midpoint of the opposite side).

The coordinates of the centroid of a triangle with

vertices(𝑥1, 𝑦1), (𝑥2, 𝑦2) and (𝑥3, 𝑦3) can be found by the

formula: (𝑥1+𝑥2+𝑥3

3,

𝑦1+𝑦2+𝑦3

3).

e.g.(1): Given that A is the

centroid of triangle XYZ and

XM̅̅ ̅̅ = 9 𝑐𝑚. Find AX̅̅̅̅ .

e.g.(2): In the given triangle ABC, If

CO̅̅̅̅ = 10𝑥 − 8 and OD̅̅ ̅̅ = 3𝑥 − 2 and

CD̅̅ ̅̅ is a median, find the value of 𝑥.

e.g.(3): In the given triangle XYZ, If CY̅̅̅̅ =1

2 𝑎 − 1 and CZ̅̅̅̅ =

2𝑎−9

2 and XC̅̅̅̅

is a median, find the value of 𝑎.

The median XC̅̅̅̅ bisects ZY̅̅̅̅ .

So, CY̅̅̅̅ = CZ̅̅̅̅

1

2𝑎 − 1 =

2𝑎−9

2

𝑎 = 7

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e.g.(4): If A(−3, −2), B(7, 1) and C(2, 7) are vertices of the triangle ABC,

find the coordinates of the centroid.

e.g.(5): If M(1,5) is the centroid of triangle ABC where A(−3, 2) and

B(5,4) , find the coordinates of C.

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Measure and calculate the surface area of a cylinder.

Revision: Find the area for the following shapes:

a) b)

c) d)

Surface area is the sum of the areas of the faces of a solid figure.

Lateral surface area of a solid is the sum of the surface areas of all

its faces excluding the bases of the solid.

□

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A cylinder is a solid that has two parallel faces which are congruent

circles. These faces form the bases of the cylinder. The cylinder has

one curved surface. The height of the cylinder is the perpendicular

distance between the two bases. The curved surface opens up to form

a rectangle.

The net of a solid cylinder consists of two circles and one rectangle.

Therefore, Lateral Surface Area = perimeter of the base × height

Surface Area = 2 × area of one base + lateral area

𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎 𝑐𝑦𝑙𝑖𝑛𝑒𝑟 = 2𝜋𝑟ℎ

𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎 𝑐𝑦𝑙𝑖𝑛𝑒𝑟 = 2𝜋𝑟2 + 2𝜋𝑟ℎ

e.g.(1): The diameter of the base of a cylinder is 12 cm and the height is 8

cm. Find :

a) The lateral surface area of the cylinder.(given that π =3.14)

b) The surface area of the cylinder.

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Measure and calculate the surface area of a prism.

Rectangular prism was taught in grade 7.

In this year, triangular prism will be considered.

A triangular prism is a solid figure whose bases are congruent

triangles and parallel to one another, and each of whose sides is a

parallelogram.

Lateral Surface Area (L.A.) = perimeter of the base × height

Surface Area (S.A.) = 2 × area of one base (triangle) + lateral area

e.g.(1): Calculate the surface area of the following prisms:

e.g.(2): Calculate the lateral surface area of the following prism:

e.g.(3): A triangular prism has a height of 5 cm .If the perimeter of its base

is 9 cm, calculate its lateral surface area.

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Calculate (using a formula) the volume of a cylinder and prism.

Volume of any solid = Area of the base × height

Volume of a cylinder = Area of circular base × height

Volume of a triangular prism = Area of triangular base × height

e.g.(1): Find the volume of the following door wedge :

e.g.(2): Calculate the volume of a cylinder with base area of 78.5 cm

2 and a

height of 8 cm. (given that π =3.14)

e.g.(3): What is the height of the prism shown in the figure:

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27 | P a g e

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Distinguish between the terms sample and population.

A population is an entire group of people or objects about which

information is desired.

Sometimes surveying everyone in an entire population is too time-

consuming or expensive. When this is the case, a sample is surveyed.

A sample is a part of a population chosen to get information about the

whole population.

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e.g.(1): Identify the population and the sample:

Residents of Residents of

a country a city in a country

5 Ants Colony of Ants

All books in a library 10 library books

e.g.(2): You want to know how many students in your school are going to

the volleyball game. You survey 50 students. 10 students are going

to the game. The rest are not going to the game. Identify the

population and the sample.

e.g.(3): Determine whether you would survey the population or a sample.

Explain.

a) You want to know the average height of students in grade seven in

Oman.

b) You want to know the favorite subject of students in your

classroom.

c) You want to know the number of students in your city who join

summer clubs.

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Recognize that the data collected (as a sample) are affected by:

a) the nature of the sample (difference in age, gender, interests, etc.)

b) the method of collection (surveys, interviews, experiments, etc.)

c) the sample size

A reasonable sample should be large enough to provide

accurate data.

A small sample size can lead to incorrect conclusions about

the population.

d) Biases :

Of either the collector of the data or the respondent to a survey can

also affect the data. Everyone has certain beliefs that can influence

the questions in a questionnaire or the observations made as well as

the responses made on a survey or at an interview. In a phone

survey or interview, the person who is conducting the activity can

influence the responses by his/her reactions to some of the

responses already given.

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Determine, from a set of data, the following measures of central tendency:

mode , mean and median.

Mode : the number which appears most often.

Ordering the numbers makes it easy to see which numbers

appear most often.

A set of data can has no mode, one mode, two or more modes.

e.g.(1): Find the mode for each of the following data :

a) 3 , 7, 5 , 13 , 20 , 23 , 39 , 23 , 40 , 23 , 14 , 12 , 56 , 23 , 29

b) 1 , 3 , 3 , 3 , 4 , 4 , 6 , 6 , 6 , 9

c) 1 , 3 , 4 , 5 , 6 , 9

Mean : is the average of the numbers.

Mean =𝑆𝑢𝑚 𝑜𝑓 𝑑𝑎𝑡𝑎

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎

e.g.(1): Find the mean for each of the following data :

a) 6 , 11 , 7

b) 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

e.g.(2): Ali received the following scores on his first 5 quizzes : 72, 86, 92,

63, and 77. What quiz score must Ali earn on his sixth quiz so that

his average (mean score) for all six quizzes will be 80?

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Median : is the "middle number" in a sorted list of numbers.

To find the Median:

1) Order the numbers in ascending or descending order.

2) If there are an odd amount of numbers , the middle

number will be the median.

If there are an even amount of numbers , add the

middle pair of numbers together and divide by two.

e.g.(1): Find the median for each of the following data:

a) 12, 3 , 5

b) 14, 23 , 23, 13, 21, 23

e.g.(2): Find the central tendency measures (mode, mean, median) for each

of the following data:

a) 5.3 , 5.7 , 5.9 , 5.4 , 4.5 , 5.7 , 5.8 , 5.7 b)

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70 80

62

90 75

0

50

100

Ali Mohd Said Ahmed Salim

Mar

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Student Name






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Determine the effect on the mean, median, or mode when :

A constant is added to or subtracted from each number.

Each number is multiplied or divided by a constant.

Data Mean Mode Median

Original

Data Set: 6, 7, 8, 10, 12, 14, 14, 15, 16, 20 12.2 14 13

Add 3 to

each data

value

9, 10, 11, 13, 15, 17, 17, 18, 19, 23 15.2 17 16

Multiply 2

times each

data value

12, 14, 16, 20, 24, 28, 28, 30, 32, 40 24.4 28 26

If you add/subtract a constant to every value, the mean, median

and mode will be increased/decreased by the same constant.

If you multiply/divide a constant to every value, the mean,

median and mode will be multiplied/divided by the same

constant.

e.g.(1): A set of scores has a mean of 16. If each score is multiplied by 3 ,

what will be the new mean?

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Determine the effect of an outlier (extreme value) on the mean, median, or

mode.

Outliers are values that "lie outside" the other values

or values that are "far away" from a set of data.

Activity:

6 students in Grade 8 got the following marks in a test :

Ali Ahmed Mohd Salim Omar Khalid

60 60 61 63 64 70

a) Find the mode , median and mean mark.

b) If the mark of Ali is 30, find the new mode , median and mean

mark.

c) If the mark of khalid is 100, find the new mode , median and

mean mark.

Comparing your results in (a), (b) and (c) how do outliers affect the

mean / median / mode?

For a set of data :

Outliers always affect the mean (affect).

Outliers never affect the median (doesn’t affect).

Outliers sometimes affect the mode.

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Determine, from frequency tables, the measures of central tendency.

e.g.(1): The marks obtained by 25 pupils on a quiz are shown below:

3 4 5 6 5

5 1 2 3 3

4 5 7 1 5

2 5 6 5 4

6 4 5 4 3

a) What is the median mark ?

b) Find the mode .

c) Find the mean of the marks.

Solution:

a) Median mark = 4 (as we need the 13th number, when in order)

b) Mode = 5 (with frequency 8)

c) We will use the following frequency table to calculate the mean:

Mark Tally Frequency 𝐌𝐚𝐫𝐤 × 𝐅𝐫𝐞𝐪𝐮𝐞𝐧𝐜𝐲

1 2 1 × 2 = 2

2 2 2 × 2 = 4

3 4 3 × 4 = 12

4 5 4 × 5 = 20

5 8 5 × 8 = 40

6 3 6 × 3 = 18

7 1 7 × 1 = 7

Totals 25 103

The average mark =𝑇𝑜𝑡𝑎𝑙 𝑜𝑓 (𝑒𝑎𝑐ℎ 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒×𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦)

𝑇𝑜𝑡𝑎𝑙 𝑜𝑓(𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦)=

103

25 = 4.12

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e.g.(2): The bar chart below shows the shoe sizes of a group of 50 children:

a) Find the mode.

b) What is the median shoe size?

c) Calculate the mean shoe size.

e.g.(3): In a season a football team scored a total of 55 goals. The table

below gives a summary of the number of goals per match :

Goals per Match Frequency

0

1

2

3

4

5

4

6

0

8

2

1

Calculate the mean number of goals per match.

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12

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Four Five Six Seven Eigth

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Create a set of numbers given a mean, median, or mode.

Introduction:

Find the mean of 5, 6 and 7.

Find the mean of 2, 4 and 12.

The means of these two data sets are 6 (it is the same mean).

e.g.(1): Construct a set of three numbers such that their mean is 6.

Solution: Suppose the three numbers are 𝑥, 𝑦 𝑎𝑛𝑑 𝑧

𝑥+𝑦+𝑧

3 = 6 ⇒ 𝑥 + 𝑦 + 𝑧 = 18

Accept any answer such as : 1, 2, 15 .

e.g.(2): Construct a set of three numbers such that their median is 4.

Solution: Accept any answer when the median is 4

such as 1, 4, 5 or 2, 4, 7 .

e.g.(3): Construct a set of four numbers such that their median is 4.

Solution: Suppose the four numbers are 𝑥, 𝑦, 𝑧 𝑎𝑛𝑑 𝑟 (𝑖𝑛 𝑜𝑟𝑑𝑒𝑟)

∴ 𝑡ℎ𝑒 𝑚𝑒𝑑𝑖𝑎𝑛 =𝑦+𝑧

2= 4 ⇒ 𝑦 + 𝑧 = 8

Accept any answer for y and z such as 3 and 5 .

Then accept any answer for the four numbers such as : 1, 3, 5, 10

e.g.(4): Construct a set of three numbers such that their mode is 3.

Solution: Accept any answer such as : 1, 3, 3 or 3,3, 9

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Calculate the theoretical probability of events.

An event is a set of possible outcomes from an experiment.

The sample space is the set of all the possible outcomes of an

experiment.

To find the total number of possible outcomes, multiply the number

of outcomes for each event.

e.g.(1): Find the total number of possible outcomes for the following

experiment: Rolling a 6 face-die, tossing a coin, then picking a

colored card Red , Blue , Green .

Total number of possible outcomes = 6 × 2 × 3 = 36 outcomes.

A tree diagram is a diagram whose branches represent possible

outcomes of a probability experiment.

Tree diagram is used to determine the sample space when the

probability experiment consists of two or more occurrences.

If A and B are events in the sample space, the possible outcomes

will be written as ordered pairs where the first element is from

event A and the second element is from event B.

e.g.(2): Draw a tree diagram to show all possible outcomes for each of the

following experiments:

a) Pick a card 𝕏 , 𝕐 , ℤ , 𝕄 . Then spin the spinner .

b) Toss a coin , then pick a card 𝔸 , 𝔹 . Then roll a 6 face-die.

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If all events in a fair game are equally likely to happen, then the

Probability of an event (A) is given by the formula :

Number of favorable outcomes (in the event A)

Total number of possible outcoms (in the sample Space)

P(A) =𝑛(A)

𝑛(S)

e.g.(1): In rolling a 6 face-die, find the probability of each of the following

events:

a) A : getting the number ' 3 '.

b) B : getting an odd number.

c) C : getting a number between 2 and 6.

e.g.(2): Jamil spun 2 spinners. He spun the first spinner which is labelled 1

and 2, then he spun the second spinner which is labelled 1, 2, 3.

a) Draw a tree diagram to list all possible outcomes (Sample Space).

b) Find the probability that:

1. Spinners stop at “1” and “3”?

2. Sum of the two numbers is 4. Solution:

The probability (Spinners stop at 1 and 3) =

1

6

The probability (Sum of the two numbers is 4) = 2

6=

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yearly plan of mathematics 2015 2016 second …yearly plan of mathematics 2015 – 2016 second...

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