Handbook m1 maths

Agastya International Foundation

Maths - 1 Handbook M1

"Do not worry about your problems with mathematics, I assure you mine are far greater."

--Albert Einstein

2

Handbook – M1 Mathematics OVERVIEW OF HANDBOOK ABL 1 2 3 4 5

CONCEPT Triangles Quadrilaterals Circles Areas and Perimeters Solids

NO. OF ACTIVITIES 6 2 4 5 4

TIME (min)

PAGE NO.

215 80 195 210 350

2 31 43 71 91

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3

ABL 1 Triangles Activity

Learning objective

1.1

What are the different types of triangles?

1.2

How can we verify the various properties of triangles?

1.3

1.4

1.5

1.6

What are congruent triangles?

What are the different criteria to test congruence of triangles?

What are similar triangles?

How can we prove the Pythagoras theorem?

Key messages



The aim of this activity is to explore the types of triangles: equilateral, isosceles, scalene, acute, right and obtuse triangles.

In this activity, we verify the following properties of triangles:  Angle sum property of triangles.  Exterior angle property of triangles.  Sum of any two sides of a triangle is always greater than the third side.  Medians of a triangle intersect at a single point.  In this activity, we learn about congruent triangles. Triangles are said to be congruent if they have same shape and size.  We study the 4 different criteria to test the congruence of triangles: SSS, SAS, ASA and AAS. We also test the RHS criteria for congruence of right angled triangles.  In this activity, we study about similar triangles. Two triangles are said to be similar if they are of same shape but different size.  In this activity, we use paper cutting to prove the Pythagoras theorem. In a right angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

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Time (min) 30

60

20

45

30

30

4 Time: 30 min

ABL 1.1

LEARNING OBJECTIVE: What are the different types of triangles?

ADVANCE PREPARATION: Material List Material 1

Number required

2

Different types of triangle models (made of 1 of each type per group either wood or cardboard) Adhesive 1 tube/box per group

3

Pair of scissors

1 per group

4

Tracing paper/coloured paper

1 large sheet per group

5

Protractor, ruler, pencil/pen

1 each per group

6

White paper sheets

A few per group

Things to do Divide the class into groups, each group containing 3 or 4 students. The number of members in a group is dependent on the amount of material available. Each group should receive the number of materials as specified in the Material List section.

Safety Precautions NA

SESSION Link to previous activity NA Procedure  Ask the students to make replicas of the triangle models. Place the model over the tracing paper; draw the triangle using a pencil or pen. Cut the shape using scissors. Use adhesive to stick the replica of the triangle on a white sheet of paper. Label them.  Measure the sides and angles of the triangles using ruler and protractor. Fill the observation table given below.

No.

1

Name of the

Measurements of

Measurements of sides

Triangle

angles (in degrees)

(in cm)

∠A = ______

AB = ______

∠B = ______

BC = ______

∠C = ______

CA = ______

∆ABC

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2

3

4

5

6

∆DEF

∆LMN

∆PQR

∆XYZ

∆GHI

∠D = _______

DE = ______

∠E = _______

EF = ______

∠F = _______

DF = ______

∠L = ______

LM = ______

∠M = ______

MN = ______

∠N = ______

LN = ______

∠P = ______

PQ = ______

∠Q = ______

QR = ______

∠R = ______

PR = ______

∠X = ______

XY = ______

∠Y = ______

YZ = ______

∠Z = ______

XZ = ______

∠G = ______

GH = ______

∠H = ______

HI = ______

∠I = ______

GI = ______

UNDERSTANDING THE ACTIVITY: Leading questions  Are there any triangles where the lengths of all sides are equal?  Are there any triangles where the lengths of any two sides are equal?  Are there any triangles where all 3 sides are different in length?  Are there any triangles where all 3 angles are less than 90°?  Are there any triangles where one of the angles is equal to 90°?  Are there any triangles where one of the angles is greater than 90°?

Fill the answers to these questions in the table given below.

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6 Points to explore

∆ABC

∆DEF

∆LMN

∆PQR

∆XYZ

∆GHI

All three sides equal (Yes / No) Any two sides equal (Yes / No) All three sides unequal (Yes / No) All three angles less than 90° (Yes / No) One angle equal to 90° (Yes / No) One angle greater than 90° (Yes / No)

Discussion and Explanation  A triangle in which all 3 sides are equal is called an equilateral triangle. Each of the angles in an equilateral triangle is equal to 60°.  A triangle in which any 2 sides are equal is called an isosceles triangle.  A triangle in which all the 3 sides are unequal is called a scalene triangle.  A triangle where all 3 angles are acute angles is called an acute triangle.  A triangle where one angle is equal to 90° is called a right triangle.  A triangle in which one of the angles is an obtuse angle is called an obtuse triangle.

KEY MESSAGES Triangles can be classified into different types based on the properties of their angles and sides. The six types of triangles that we have explored in this activity are: equilateral, isosceles, scalene, acute, right and obtuse triangles.

LEARNING CHECK:  

Draw different types of triangles on the board and ask the students to identify the type of triangle.

Ask the students if they have seen examples of triangles in real life. For example, they might recognize a nearby building/house having a triangular structure, a local sweet/candy which is triangular in shape. Ask them to guess the type of triangle in each case.

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7 Time: 60 min

ABL 1.2

LEARNING OBJECTIVE: How can we verify the various properties of triangles?

ADVANCE PREPARATION: Material List Material

Number required

1

Coloured paper sheets

1 or 2 sheets per group

2

White sheet of paper

1 or 2 sheets per group

3

A pair of scissors

1 per group

4

Adhesive

1 tube/box per group

5

Pencil/pen

1 per group

6

Thin sticks

7

Small pieces of cycle valve tube

3 per group, each of different length A few per group

Things to Do Divide the class into groups, each group containing 3 or 4 students. The number of members in a group is dependent on the amount of material available. Each group should receive the number of materials as specified in the Material List section.

Safety Precautions NA

SESSION Link to known information/previous activity: In the previous activity, we learnt about the classification of triangles based on their angles and sides.

Procedure: 1.2a Step 1: On a coloured paper, draw a triangle of any dimensions. Label the vertices A, B and C as shown in the figure below. The interior angles are ∠ABC, ∠BCA and ∠CAB.

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Step 2: Cut the triangle ABC.

Step 3: Cut the 3 angles, ∠ABC, ∠BCA and ∠CAB.

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9 Step 4: Draw a straight line on a sheet of paper. Place the 3 angles on the straight line adjacent to each other without leaving any gap, as shown in the figure.

Step 5: Ask the students to share their observation.

(We learn that ∠ABC + ∠BCA + ∠CAB = 180°, since the angles on a straight line add to 180°) 1.2b Step 1: Similar to the previous activity, draw 2 triangles of any dimensions on a coloured sheet of paper.

The 2 triangles should be of the same dimensions. Label the vertices A, B and C. The interior angles are ∠ABC, ∠BCA and ∠CAB.

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10

Step 2: Cut one of the triangles and paste it on a white sheet of paper. On the paper, extend one of the sides BC to D, as shown in the figure.

Step 3: Cut the angles ∠ABC and ∠CAB from the other triangle.

Step 4: Place these cut angles adjacent to the angle ∠BCA, on the white sheet of paper, without leaving any gap.

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Step 5: Ask the students to share their observation.

(We learn that ∠ACD = ∠ABC + ∠CAB) 1.2c Step 1: Take 3 thin sticks of different lengths. Connect them with small pieces of cycle valve tubes as shown in the figure. Place them on a white sheet of paper. Label the vertices A, B and C as shown in the figure.

Step 2: Remove the cycle valve tubes. Move the side AC to lie along the side BC. Place the side AB to lie over BC. The sticks look as shown in the figure.

Step 4: Ask the students to share their observation. (BC + CA > BA) Agastya International Foundation. For Internal Circulation only.

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12 Step 5: Repeat the experiment by comparing each side with the other 2 sides. These steps are important.

(We learn that the sum of any 2 sides of a triangle is greater than the third side)

1.2d Step 1: Draw a triangle of any dimensions on a coloured paper sheet. Cut the triangle.

Step 2: To get the midpoint of side BC, fold the triangle along BC so that B coincides with C and press the paper on the side AB to get the midpoint. Unfold the paper and mark the midpoint as M.

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Step 3: Again fold the paper from the vertex A to the midpoint M. The crease AM, which you see on the paper, is a median.

Step 4: Similarly, get the other 2 medians BN and CP by folding the triangle accordingly.

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Step 5: What do the students observe?

(We can see that the three medians meet at a point)

UNDERSTANDING THE ACTIVITY: Leading Questions:  Explain the angle sum property of a triangle.  What is an exterior angle in a triangle? Explain the exterior angle property of a triangle.  What is the relationship between sum of any two sides of triangle and its third side?  What is a median in a triangle? What do we observe when we draw all the 3 medians of a triangle?

Discussion and Explanation:  The sum of the three angles of a triangle is equal to 180°.  In a triangle, an exterior angle is formed when you extend any side of a triangle. Among the two angles formed at that vertex, one is an interior angle and the other one is called the exterior angle.

The exterior angle at the vertex of a triangle is equal to the sum of the interior angles at the other two vertices.  The sum of any two sides of a triangle is always greater than the other two sides.  In a triangle, a median is a line segment from the vertex of the triangle to the midpoint of the opposite side.

The medians of the triangle meet at a single point.

This meeting point is called a centroid.

KEY MESSAGES In this activity, we learn how to verify various properties of triangles. We discuss about relationships of angles of triangles (both interior and exterior), sides of a triangle and about medians.

LEARNING CHECK:  Draw a triangle and give values for any two angles. Ask the students to find the third angle.  Draw a triangle and give values of any two interior angles. Ask the students to find the value of the exterior angle corresponding to the third vertex.

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15

 Around the school, mark 3 points (maybe a pillar, a tree etc.). Ask the students to find the shortest distance between any two points. It is very intuitive to go directly from one point to another and not through the third point. Explain to them how this is similar to the result obtained from the 1.2c activity.  Take 3 sticks with the following lengths: 3cm, 4cm and 10cm. Try forming a triangle. Discuss.  How many medians are there in a triangle?  Draw a triangle on the board using known dimensions. Among students, invite 3 of them to draw the 3 medians, thus finding the centroid in the process.

Time: 20 min

ABL 1.3 LEARNING OBJECTIVE: What are congruent triangles?

ADVANCE PREPARATION Material List 1 2 3 4 5

Material Coloured paper sheets White paper A pair of scissors Protractor, ruler Pencil/pen

Number required 1 per group 1 per group 1 per group 1 each per group 1 per group

Things to Do Divide the class into groups of 3 or 4. Since this is a simple activity, each student can do the activity by herself. However, each student/group should receive the amount of materials as specified in the Material List section.

Safety Precautions NA

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16 Link to known information/previous activity: NA Procedure: Step 1: Cut a triangle of any dimensions on a coloured paper sheet. Place the cut triangle on a white sheet and draw the outline using a pencil/pen. Label the vertices of this outline as ABC.

Step 2: Rotate the triangle by any angle downward as shown in the figure. Draw the outline of this triangle in the new position. This transformation is called rotation. Label the vertices of this new outline as PQR.

Step 3: Flip the triangle upside down and place it somewhere above as shown in the figure. This transformation is called reflection. Draw the outline and label them as XYZ.

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17

Step 4: Move the triangle a distance away from its present position as shown in the figure. This transformation is called translation. Draw the outline and label the vertices as KLM.

Step 5: We observe that, except XYZ and KLM, the rest of the triangles look different from each other.

Now, measure the angles and sides of all the triangles and fill the table below. What do we observe now?

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18 Triangle ABC

PQR

XYZ

KLM

Sides AB = AC = BC = PR = QR = PQ = XZ = XY = YZ = KM = KL = LM =

Angles ∠BAC = ∠ABC = ∠ACB = ∠PRQ = ∠RPQ = ∠PQR = ∠ZXY = ∠XZY = ∠XYZ = ∠LKM = ∠LMK = ∠KLM =

UNDERSTANDING THE ACTIVITY: Leading Questions:  What are congruent triangles?

Discussion and Explanation:  Two triangles are said to be congruent if they are of the same size and shape. If two triangles are congruent, then their corresponding angles and sides are equal.

KEY MESSAGES: In this activity, we learn the definition of congruent triangles. Two triangles are congruent if they are of the same shape and size. If a triangle undergoes rotation, reflection or translation, or a combination of them, then the resulting triangle is congruent to it.

LEARNING CHECK: 

Draw several triangles with various dimensions, some of them congruent to each other. Ask the students to identify the congruent triangles and what transformations they have undergone?

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19 Time: 45 min

ABL 1.4

LEARNING OBJECTIVE: What are the different criteria to test congruence of triangles?

ADVANCE PREPARATION: Material List 1 2 3 4 5

Material Coloured paper sheets White paper sheet A pair of scissors Protractor, ruler, compass Cardboard models of different triangles (one per rule)

Number required 1 or 2 per group 1 per group 1 per group 1 set per group 1 set for teacher

Things to Do Divide the class into groups of 3 or 4 depending on the amount of material available. Each group should receive the amount of material as specified in the Material List section. Please note that, in order to complete this activity, the students should know how to construct triangles (using compass, ruler etc.) given simple conditions, for example, when 3 sides are known etc.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous activity, we learnt what congruent triangles are.

Procedure: In these set of activities, use cardboard models of triangles, a different one per congruency test. Without showing the triangle to students, give them information (depending on the test) using which they construct the triangle. Ask them to construct all possible triangles using the given information. Then, they cut those triangles out of the paper and give it to you. Compare all the triangles given by the students with each other as well with the cardboard model. If they all match, it means the congruency test holds true.

1.4a - Testing SSS (side-side-side) criteria Select any one triangle from your set of cardboard models. Give them the three sides. In the cardboard triangle shown below, you could give the lengths of the three sides: AB, BC and AC. Ask the students to construct as many different types of triangles possible from the given information. Cut them out and give it to you.

Compare the cut out triangles with each other and also with your cardboard model. Observe if all the triangles align. If they do, then it proves that SSS is a valid test for congruency.

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20

1.4b- Testing SAS (side-angle-side) criteria Conduct this activity similar to the previous one. However, choose a different cardboard model of the triangle. For example, in the cardboard triangle shown in the picture below, tell the students the value of sides AB and BC and the angle ∠ABC. Let the students draw as many possible triangles with the given information. Collect the triangles from them and hold them together with your cardboard triangle. If they all align together, then the SAS congruence test is valid.

1.4c- Testing ASA (angle-side-angle) criteria Select a different cardboard model of a triangle and provide the students values of any two angles and the included side. For example, in the figure shown below, you could tell the students values of ∠ACB, ∠ABC and BC. Similar to the previous two activities, let the students draw all possible triangles with the information given. Collect them all and check if they all align when held together. If they do, then the ASA congruence test is valid.

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1.4d- Testing AAS (angle-angle-side) criteria Select a different cardboard model of a triangle and provide the students values of two angles and a nonincluded side. For example, in the figure shown below, give values for ∠ACB, ∠ABC and AC. Collect all the triangles drawn by students and check if they match. If they do, then the AAS test is a valid test for congruency.

1.4e- Testing AAA (angle-angle-angle) criteria For this test, give values of the three angles of the triangle. In the figure shown below, give values for ∠ACB, ∠ABC and ∠CAB. Ask the students to draw all possible triangles. If the students tell it is not possible to draw since the length of sides is not given, ask them to assume any value for the sides. Collect all the triangles and hold them together. The triangles constructed would not match since the students would have assumed different value for the sides. Explain to the students, that AAA is not a valid test for congruency.

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22

1.4f- Testing SSA (side-side-angle) criteria In this test, provide values of two sides and a non-included angle. For example in the figure shown below, tell values for ∠ACB, AB and BC. Let the students draw all possible triangles from the information given.

Collect all the triangles drawn by students and check if they match. If they do, then the SSA test is a valid test for congruency. If they don’t then explain to the students that more than one type of triangle can be drawn from the given information, hence it cannot be a valid test for congruency.

1.4g: Testing RHS (right angle-hypotenuse-side) criteria This test is only used in case of right angled triangles. In the right angled cardboard model of the triangle, give the values for the hypotenuse and one of the sides. Let the students draw right angled triangles from the given information. Collect all the triangles and check if they match. If they do, RHS is a valid test for congruency.

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UNDERSTANDING THE ACTIVITY: Leading Questions:  Among the six criteria that we explored in this activity, which of them can be used to confirm congruency of two triangles?  What is the test for congruence in case of right angled triangles?

Discussion and Explanation:  The following criteria can be used to confirm congruency of triangles: SSS, SAS, ASA and AAS. In case of AAA and SSA, the triangles do not match and hence fail as criteria for congruency.  In case of right angled triangles, the test for congruence is called RHS. The hypotenuse and one of the corresponding sides have to be equal for the triangles to be congruent.

KEY MESSAGES: There are 4 criteria that can be used to test congruent triangles: SSS, SAS, ASA and AAS. The other two criteria: AAA and SSA are not sufficient to confirm if two triangles are congruent. For right angled triangles, the RHS criterion is used to test for congruency.

LEARNING CHECK: 



Congruent triangles are commonly used everywhere in construction (bridges, metallic structures in buildings etc.). Ask students if they have seen such structures anywhere. How can they prove that the structures are congruent?

Draw triangles (providing combinations of sides and angles) and ask students which criteria it fits in order to prove their congruence?

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24 Time: 30 min

ABL 1.5 LEARNING OBJECTIVE: What are similar triangles?

ADVANCE PREPARATION Material List 1 2 3 4

Material Graph paper (2 types with different sizes of grids) Pen/pencil Ruler/Protractor A pair of scissors

Number required 1 of each type per group 1 per group 1 set per group 1 per group

Things to Do Divide the class into groups of 2 or 3 based on the number of students and material available. Each group should receive the amount of material as specified in the Material List section. Please note that there has to be 2 types of graph paper; for example, one grid paper can be with 1 line per inch and another with 2 lines per inch.

Safety Precautions NA

SESSION: Link to known information/previous activity: NA Procedure: Step 1: Draw a triangle on one graph paper, with a known grid count for height and base. A triangle as shown in the figure can be drawn. Label this triangle ABC. Cut out the triangle from the paper.

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25 Step 2: Now, on the other graph sheet, draw another triangle with the same number of grid count for base and height. Label this triangle PQR. Cut out the triangle PQR.

Step 3: Place the 2 triangles side by side and also over each other. Ask the students to share their observation.

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26 Step 4: The two triangles, ABC and PQR are of the same shape but differ in size. Such triangles are said to be similar to each other. Fill the table below and discuss the results.

Triangle ABC PQR

AB =

Sides BC =

AC =

PQ =

QR =

PR =

PQ = AB

QR = BC

PR = AC

∠BCA =

Angles ∠CAB =

∠ABC =

∠QRP =

∠RPQ =

∠PQR

As you can see from the table, the corresponding angles are equal. The ratios of the corresponding sides are equal. This ratio is called the scale factor. This definition (of both similarity and scale factor) is applicable to other polygons and figures and not just to triangles.

UNDERSTANDING THE ACTIVITY: Leading Questions:  What are similar triangles?  With reference to similar triangles and polygons, what is a scale factor?

Discussion and Explanation:  Two triangles are similar if they have the same shape, but are different in size.  In similar triangles or polygons, the ratios of corresponding sides are equal. This ratio is called the scale factor.

KEY MESSAGES Two triangles are said to be similar to each other if they are of the same shape, but differ in size. In similar triangles, the corresponding angles are equal and the corresponding sides have same ratio.

LEARNING CHECK:    

Give examples of similar figures you have seen. Do not restrict yourself to triangles and geometrical figures. (For e.g., one can think of statues, toys that come in different sizes) Given 2 triangles ABC and PQR. Length of all the sides of ABC is known, and just the length of one side is known in PQR. How do we find the length of other 2 sides of PQR? Similar triangles and polygons are defined as figures having same shape but different size. Does it mean all squares are similar to each other? Can you use the graph sheets to verify your answer?

If you have a ruler of known length (say 30 cm), find your approximate height using your shadow.

To do this activity, the sun has to be at an angle; if it is right above your head, it might not cast a proper shadow. Refer the figure below to get a hint.

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28 Time: 30 min

ABL 1.6

LEARNING OBJECTIVE: How do we prove the Pythagoras theorem?

ADVANCE PREPARATION: Material List 1 2 3 4

Material Coloured paper/Cardboard A pair of scissors Protractor, Ruler Sketch pen/pencil/pen

Number required 1 sheet per group 1 per group 1 set per group 1 per group

Things to Do Divide the class into groups of 3 or 4 depending on the strength of the class and material available. Each group should receive the amount of material specified in the Material List section.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous few activities, we learnt about triangles and their properties.

Procedure: Step 1: Draw a right angled triangle on a coloured sheet of paper or cardboard. Either of them can be used.

Label them ABC. Let the sides AB, BC and AC be a, b and c respectively. We have to prove that a2 + b2 = c2.

Cut the triangle out of the paper/cardboard.

Step 2: Make 3 more such replicas of the triangle.

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29

Step 3: Draw a square of side equal to AC i.e. c. Cut the square out of the sheet.

Step 4: Arrange all the 5 pieces as shown in the figure below.

Step 5: Using the figure, we can prove the Pythagoras theorem in the following way.

Area of the bigger square = (Area of smaller square) + (4 x Area of the triangle) (a + b)2 = c2 + (4 x ½ x a x b) Agastya International Foundation. For Internal Circulation only.

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30 a2 + b2 + 2ab = c2 + 2ab a2 + b2 = c2

UNDERSTANDING THE ACTIVITY: Leading Questions:  What is the Pythagoras theorem?

Discussion and Explanation:  In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

KEY MESSAGES In this activity, we use paper/cardboard cutting to prove the Pythagoras theorem.

LEARNING CHECK: 



Give problems on Pythagoras theorem. Example problems are given below:  If the 2 sides of a right angled triangle are 6cm and 8cm, find the hypotenuse.  The hypotenuse of a right angled triangle is 13cm. If one of the sides is 5cm, find the third side.

Draw an isosceles right angled triangle. Draw squares using the measures of the sides of the triangle as shown in the figure. Cut the squares of the equal sides into 4 triangles each. Check if all the 8 triangles completely cover the square of the hypotenuse.

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31

ABL 2 Quadrilaterals Activity

Learning objective

2.1

What are the different types of quadrilaterals?

2.2

How can we verify the various properties of quadrilaterals?

Key messages



In this activity, we learn to identify the different types of quadrilaterals.

In this activity, we verify the following properties of quadrilaterals  Angle sum property of quadrilaterals  Properties of Parallelogram  Properties of Rhombus  Properties of Rectangle  Properties of Square

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Time (min) 20

60

32 Time: 20 min

ABL 2.1

LEARNING OBJECTIVE: What are the different types of quadrilaterals?

ADVANCE PREPARATION: Material List 1

2 3 4 5

Material Several models of quadrilaterals (either made of wood or cardboard) Ruler/Protractor 5 boxes, each labeled with a different type of quadrilateral A timer (cell phone or watch can be used) White paper sheets

Number required 1 set per group

1 each per group 1 set per group 1 for the teacher

Things to Do Divide the class into groups of 3 or 4 based on the amount of material available. Please note that there should be several models of quadrilaterals of different types (squares, rectangles, parallelograms, trapeziums and rhombi). This is a simple activity meant to teach students to identify and classify quadrilaterals. Hence, the number of models available per group is a factor in making the activity as interesting as possible.

Safety Precautions NA

SESSION: Link to known information/previous activity: NA Procedure: Step 1: Distribute randomly around 7-10 quadrilateral pieces per group. Give each group a blank white sheet of paper. Ask them to name the pieces (as they wish) and write the common attributes among the pieces in their own words. If they are not able to figure out what an attribute is, give an example such as “each piece has 4 sides�. Also, ask them to list out the differences between the pieces.

Step 2: Let each group discuss what they have written. Ask one of the group members from each group to present their discussion. Let the other groups ask questions about it.

Step 3: Draw a table on the board showing the classification of quadrilaterals and the difference between each of them. A table similar to the one shown below can be drawn. The students might not be clear about the meaning of some of the terms such as diagonal, bisect etc. Spend time to clarify their doubts. The students have to refer this table, while doing the activity.

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33

Quadrilateral Trapezium Parallelogram Rhombus

Rectangle Square

         

Features One pair of opposite sides are parallel Both the pairs of opposite sides are parallel All sides are equal Interior angles need not be 90° Diagonals bisect each other at 90° All interior angles are equal to 90° Also, a parallelogram All interior angles are equal to 90° All sides are equal Also, a parallelogram

Step 4: Distribute randomly a different set of quadrilateral pieces to each group. Looking at the table, each group should identify the type of quadrilateral they possess. Let them list the reasons (properties) they used to identify a piece.

Step 5: Let each group present their findings. A member from each group has to come forward and explain the reasons for each piece. If any other group has some doubts, let they ask questions to the presenting group.

UNDERSTANDING THE ACTIVITY: Leading Questions:  What is a quadrilateral?  What is a trapezium?  What is a parallelogram?  What is a rhombus?  What is a rectangle?  What is a square?

Discussion and Explanation:  A quadrilateral is a polygon with 4 closed sides.  A trapezium is a quadrilateral in which one pair of opposite sides is parallel.

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34    

A parallelogram is a quadrilateral in which both pairs of opposites are parallel.

A rhombus is a special kind of parallelogram in which all sides are equal.

A rectangle is a kind of parallelogram in which all the interior angles are equal to 90°.

A square is a parallelogram in which all sides are equal and all interior angles are equal to 90°.

KEY MESSAGES: In this activity, we learn to identify and classify quadrilaterals into different types.

LEARNING CHECK:  

Ask the students to think of quadrilaterals they have seen in real world. Inside the classroom, ask them to identify the different quadrilaterals they see.

For each of the questions given below, ask the students to justify their answers.  Is the square a rhombus?  Is the rectangle a parallelogram?  What is the difference between a trapezium and a rectangle?  Can the interior angles in a rhombus be all equal to 90°?  What is a rectangle called if the opposite sides are parallel but the angles are not equal to 90°.

Please ask these kinds of questions which will make the student comfortable about the relationship between different types of quadrilaterals.

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35 Time: 60 min

ABL 2.2

LEARNING OBJECTIVE: How can we verify the various properties of quadrilaterals?

ADVANCE PREPARATION: Material List Material 1 2 3 4 5

Geosticks Four 180° Protractors, One 360° Protractor Screws to hold the strips together Thread Ruler

Number required 1 set per group 1 set per group A few per group 1 pr 2 spools for the class 1 per group

Things to Do Divide the class into groups of 3 or 4. Each group should receive the amount of material as specified in the Material List section. The geosticks are made of wood and can be fixed together with the help of screws.

You can also make your own geosticks using cardboard. The protractors should have a hole at a suitable place so that it can be fixed by a screw.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous activity, we learnt about different types of quadrilateral and also to identify and classify them.

Procedure: 2.2a: Angle sum property of Quadrilateral Step 1: Fix any 4 geosticks using the screws to form a quadrilateral. You can create quadrilaterals with different shapes. In the picture below, we have shown a quadrilateral.

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36

Step 2: Fill the table below.

No.

∠1

∠2

∠ 3

∠ 4

∠ 1+∠ 2+∠ 3+∠ 4

1 2 3

Step 3: Ask the students to share their observation regarding sum of the interior angles of the quadrilateral.

2.2b: Properties of parallelogram Step 1: Similar to the previous activity, fix the sticks and protractors to form different parallelograms.

Example of a parallelogram is shown below. Label the parallelogram ABCD.

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37 Step 2: Complete the following table: Length of sides

Measure of angles

1+2

2+3

3+4

4+1

No

AB

BC

DC

AD

1

2

3

4

1 2 3 4

Step 3: Ask the students to share their observation regarding:  Opposite angles  Opposite sides  Adjacent angles Step 4: In each case, fix the diagonals of the parallelogram using 2 pieces of thread. Label the intersecting point of the diagonals as O. Complete the table that follows.

No.

Length along the diagonals AC

BD

Distance from intersection point AO

OC

BO

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OD

38 4

Step 5: Ask the students to share their observation regarding:  Length of the diagonals  Point of intersection of the diagonals 2.2c: Properties of Rhombus Step 1: Use the strips and protractors to form a rhombus as shown in the figure. Also, fix the diagonals of the rhombus using thread. Similar to the previous activity, label the rhombus ABCD. When you add the diagonals label the intersecting point as O.

Step 2: Complete the following tables: Measure of angles AOD

No 1

2

3

DOC

BOC

BOA

1+2

2+3

4

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3+4

4+1

39 2 3

No.

Length along the diagonals AC

BD

Distance from intersection point AO

OC

BO

OD

1 2 3

Step 3: Ask the students to share their observation regarding:  Opposite angles  Length of sides  Diagonals  Length of diagonals  Angle of intersection of diagonals  Point of intersection of diagonals 2.2d: Properties of Rectangle Step 1: Fix 4 geosticks to form a rectangle. Form rectangles with different length and breadth. Use thread to fix the diagonals.

Step 2: Fill the following table: No.

Length of the sides

Measure of angles

Length along the diagonals

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40 AB

BC

DC

AD

1

2

3

4

AC

AO

OC

BD

BO

OD

1 2 3

Step 3: Ask students to share their observation regarding:  Opposite sides  Interior angles  Length of diagonals  Point of intersection of diagonals 2.2e: Properties of Square: Step 1: Fix the 4 geosticks to form a square. Form squares with different side length. In each case, fix the diagonals as well.

Step 2: Complete the following table: No.

Measure of angles 1

2

3

Length along the diagonals 4

AC

AO

OC

BD

BO

OD

AOB

BOC

1 2 3

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COD

DOA

41

UNDERSTANDING THE ACTIVITY: Leading Questions:  What is the angle sum property of quadrilaterals?  What is the relationship between angles in a parallelogram? What about diagonals?  In addition to being a parallelogram, what other properties does a rhombus have?  What is special about angles in a rectangle? What about their diagonals?  What are the properties of a square?

Discussion and Explanation:  The sum of the four interior angles of a quadrilateral is equal to 360°.  In a parallelogram, the opposite angles are equal, while the adjacent angles are supplementary. The diagonals of a parallelogram bisect each other.  In a rhombus, all the sides are congruent (equal). The diagonals are equal in length and bisect each other at 90°.  In a rectangle, all the interior angles are equal to 90°. The diagonals are congruent and bisect each other but NOT at right angles.  In a square, all the sides are equal. The diagonals are congruent and bisect each other at right angles.

KEY MESSAGES: In this activity, we verify the various properties of different types of quadrilaterals. We explore properties of parallelograms, rhombi, rectangles and squares.

LEARNING CHECK: 

  

State whether the following statements are True or False:  In a rhombus, the diagonals bisect each other at right angles.  The opposite sides are parallel in a rectangle.  All the interior angles are equal to 90° in a rhombus.  Every square is also a rectangle and a rhombus.  A rectangle can also be a square.  In a parallelogram, the interior angles should always be 90°.  The 4 sides of a square are congruent.

In a parallelogram, if you are given the value of just one interior angle, can you find the other three angles? How? Do you think the square is also a rhombus? Justify your answers and have a discussion in the class.

Ask the students to identify the shape of the blackboard in the classroom. Use a measuring tape to measure the diagonals. Do they match the observations done during the activities?

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42

ABL 3 Circles Activity

Learning objective

Key messages

3.1

What are the different terminologies related to circles?

How can we verify the various properties of circles?



3.2

3.3

3.4

How do we verify properties related to circles and their tangents?

How do we verify properties of cyclic quadrilaterals?

In this activity, we learn about circles and related terminologies

In this activity, we verify the following properties of circles:  Diameter is the longest chord of a circle  The ratio of circumference to diameter is a constant  Angle subtended by an arc at the center of the circle is twice the angle subtended at any other point on the circumference of the circle  Angles in the same segment of the circle are equal  Angle in a semicircle is a right angle, angle in a major segment is an acute angle and angle in a minor segment is an obtuse angle In this activity we verify the following properties of circles and its tangents:  Radius is perpendicular to tangent of the circle at its point of contact  Lengths of tangent drawn from an external point to a circle is equal in length We verify the following properties of cyclic quadrilaterals:  The sum of interior opposite angles is 180°  The exterior angles is equal to the interior opposite angle

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Time (min) 45

90

30

30

43 Time: 45 min

ABL 3.1

LEARNING OBJECTIVE: What are the different terminologies related to circle?

ADVANCE PREPARATION: Material List 1 2 3 4 5 6

Material Models of circles (Wooden or Cardboard) Coloured paper sheets/Tracing paper Pencil/pen A pair of scissors Thin sticks Thread

Number Required A few per group 1 or 2 sheets per group 1 or 2 per group 1 per group 2 per group 1 or 2 spools for the class

Things to Do Divide the class into groups of 3 or 4 based on strength of class and amount of material available. This is a simple activity to familiarize students about various terminologies related to circles.

Safety Precautions NA

SESSION: Link to known information/previous activity: NA Procedure: 3.1a: Ask students about examples of circles they find in their surroundings (food, ornaments etc.). Using the examples, ask them what is it interesting about that shape? How is it different from other shapes such as triangles and quadrilaterals?

3.1b: Step 1: Take the model of a circle. Make the replica of the circle by using tracing or coloured sheet of paper.

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44

Step 2: Fold the replica into exactly half. Draw a line over the crease formed. Fold the replica again into half but at a different part of the circle. Draw a line over the crease formed. These 2 lines meet at the center of the circle. Explain to students the concept of diameter.

Step 3: Take a piece of thread. Hold one side of the thread on the center of the circle and cut the thread near the edge of the circle so that the length of the circle is equal to the radius of the circle. Move the thread at different positions on the edge of the circle. Explain to them the concept of radius and thus the definition of circle i.e. how it is a collection of points at the same distance from a fixed point called the center of the circle.

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Step 4: Take another piece of thread and hold it around the edge of the circle model. Cut the thread at the spot they meet so that the length of thread is the circumference. Explain to the students the concept of circumference.

3.1c: Step 1: Fold the replica at a position away from the center. The crease now formed will be a chord to the circle. Draw a line over the chord. Explain about chords.

In addition, you can explain the concept of major segment and minor segments.

Step 2: Take a piece of thread and hold it around the edge from one end of the chord to the other end.

This forms an arc. Explain to students the concept of arcs.

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Step 3: Take 2 sticks and place one near the edge of the circle. Place the other stick over the chord drawn in Step 1. Explain the concept of tangent and secants.

UNDERSTANDING THE ACTIVITY: Leading Questions:  What is a circle? What is the radius with respect to circles?  What is circumference?  What is diameter?  What is a chord?  What is an arc?  Explain major and minor segments.  What is a secant?  What is a tangent?

Discussion and Explanation:  The collection of all points at a fixed distance from a fixed point (called the center of the circle) is called a circle. The fixed distance is called the radius. Note: The line segment joining the center to any point on the circle can also be called the radius.  The length of the circle is the circumference.  A diameter is a line segment joining 2 points on the circle that passes through the center of the circle. Note: Diameter can also be defined in terms of length i.e. it is twice the radius. (Diameter is the longest chord of the circle).

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47     

Chord is a line segment joining any 2 points on the circle.

A piece of circle between the end points of a chord is an arc. The longer arc is called the major arc and the smaller arc is called the minor arc.

The region lying between the major arc and the chord is called the major segment; the region between the minor arc and the chord is called the minor segment.

A line segment that intersects a circle at 2 points is called the secant of the circle. Note that chord is a part of secant.

A line segment that touches a circle at exactly 1 point is called a tangent of the circle.

KEY MESSAGES: In this activity, we learn about terminologies related to circles.

LEARNING CHECK: 

Identify the various parts of a circle from the figure shown below.

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48 Time: 90 min

ABL 3.2

LEARNING OBJECTIVE: How can we verify the various properties of circles?

ADVANCE PREPARATION: Material List Material 1 2 3 4 5 6 7

Colored paper Tracing paper Adhesive A pair of scissors Pen/pencil/sketch pens Cardboard sheets Geometry box

Number Required A few sheets per group 2 sheets per group 1 tube/box per group 1 per group A few per group 1 or 2 per group 1 per group

Things to Do Divide the class into groups of 3 or 4. Each group should receive the amount of material as specified in the Material List section.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous activity, we learnt about terminologies related to circles.

Procedure: 3.2a Step 1: Draw a circle of any radius on a colored paper sheet. Cut the circle out.

Step 2: Fold the circle. It need not be exactly divided into two parts. A crease will be formed. Draw over the crease using a pen/pencil. The drawn line is a chord of the circle. Label it as AB.

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49

Step 3: Similarly, fold the circle at different places to get different chords. Label them as CD, EF and GH.

Step 4: To get the diameter, fold the circle exactly into half. The crease formed will divide the circle into 2 equal parts. Draw the crease. Notice that this chord passes through the center of the circle. It is called a diameter. Label it as PQ.

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Step 5: Ask the students to fill the table below and share their observation.

Chord AB CD EF GH PQ

Length of Chord

3.2b Step 1: Take a cardboard sheet and draw different sized circles on it. Cut the circle out using scissors.

Step 2: In order to find the circumference of the circle, you can use the following method. Mark a point on the edge of the circle using a pen/pencil. Take a white sheet of paper and draw a straight line on it. Label the starting point of the line as A. Place the mark on the circle at A. Rotate the circle over the line until the mark touches the line again. Mark this point as B. Measure the length of AB. This will give the circumference of the circle. Repeat this with other circles.

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Step 3: You can find the diameter by folding the circle exactly into half (similar to previous experiment).

Measure the length of the crease formed.

Step 4: Fill the table given below. Ask students to share their observation.

No.

Circumference

Diameter

Ratio Circumference/Diameter

1 2 3 4 3.2c: Step 1: Draw a circle of any radius on a coloured sheet of paper and cut the circle out. Label the center of the circle O. Step 2: Paste this circle on a white sheet of paper.

Step 3: Mark 2 points on the circumference of the circle to get the arc AB.

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Step 4: Fold the sheet of paper to get the crease OA and OB. Draw OA and OB. Step 5: Mark a point C anywhere else on the circumference of the circle (other than on the arc AB).

Step 6: Fold the paper to form the crease CA and CB. Draw the lines CA and CB.

Step 7: Take a tracing paper and place it over the circle. Trace the angle ACB, cut it out. Make 2 such replicas.

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53 Step 8: Place the 2 replicas adjacent to each other over AOB. What do you observe?

3.2d: Step 1: Draw a circle of any radius in a coloured sheet of paper. Cut the circle using scissors.

Step 2: Paste this circle on a white sheet of paper.

Step 3: Fold the circle in such a way so as to form a chord. Label the chord AB. Step 4: Mark 2 points P and Q on the circumference of the circle. Draw the lines AP, BP and AQ, BQ. Now, you can see two angles: APB and AQB in the segment of the circle.

Step 5: Make replicas of the angle APB and AQB using tracing paper.

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Step 6: Place these 2 replicas of the angles over each other. What do you observe?

3.2e: Step 1: Draw a circle of any radius on a coloured paper sheet. Cut the circle and paste it on a white paper sheet. Fold the circle into half to get the diameter. Draw the diameter and label it as AB. It should pass through the center O.

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55 Step 2: Mark a point P on the circumference. Draw the lines AP and BP. Now, we have to verify that APB is a right angle.

Step 3: Use a tracing paper to trace APB. Make 2 such replicas of APB.

Step 4: Draw a straight line on a white paper sheet. Place the 2 replicas adjacent to each other on the straight line. What do you observe? (Since the angles on a straight line add to 180°, the sum of 2 replicas has to be 180°. Hence, each angle has to be a right angle.). Keep one of these replicas carefully for use in the next experiment.

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Step 5: Now, fold the circle in such a way so as to form a chord XY. So, we form a minor segment and a major segment. Mark a point L on the major segment and a point M on the minor segment. Draw the lines XL, YL and XM, YM. The angle ďƒ?XLY lies in the major segment and the angle ďƒ?XMY lies in the minor segment.

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57 Step 6: Using tracing paper, trace and make replicas of the two angles XLY and XMY. Take the replica of the angle APB (= 90°) and place it over the replica of XLY. Similarly, place it over the replica of XMY. What do you observe? Which angle is greater than 90° and which is lesser?

UNDERSTANDING THE ACTIVITY: Leading Questions:  What is the longest chord in a circle?  What can you tell about the ratio of circumference to diameter in a circle?  What can you tell about the two angles; one subtended by an arc at the center of the circle and the other at any point on the circumference of the circle?  Are angles in the same segment always equal?  What is the nature of angles in a semicircle, major segment and minor segment?

Discussion and Explanation:  The diameter is the longest chord in a circle.  The ratio of circumference to diameter in a circle is a constant and is equal to around 3. This ratio is called Pi and denoted by the symbol π. The value of π to decimal places is given by 3.14.  The angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any point on the circumference of the circle.

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58  

Yes, angles in the same segment are always equal.

Angle in a semicircle is a right angle. Angle in a minor segment is an obtuse angle. Angle in a major segment is an acute angle.

KEY MESSAGES: In this activity, we verify various properties of circles.

LEARNING CHECK:  A school has a round playground. There is a gate to enter the playground. The distance from the



gate of the playground to the center is 20m. If a kid starts from the gate and decides to go around the edge of the playground, how long does she have to walk to come back to the gate?

(Instructions: Please let the students visualize the problem. Let them spend time thinking about the playground, drawing the circle, marking the gate, drawing the radius etc. If relevant, ask them to compare this to their playground at school.)

State whether true or false:  A circle is of radius 10cm with a center O and an arc AB. The angle AOB = 70°. If we mark a point P on the circumference of the circle in the major segment of chord AB, then the angle APB = 35°.  If the radius of the circle is reduced to 5cm. Now, the angle AOB = 70°, but the angle APB = 30°.  A circle is drawn. Then, a chord is drawn. The angle in a semicircle is 90°. The angle in the minor segment is obtuse and the angle in the major segment is acute.  Refer the figure below. ACB = ADB.

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59 Time: 30 min

ABL 3.3

LEARNING OBJECTIVE: How do we verify properties related to circles and their tangents?

ADVANCE PREPARATION: Material List 1 2 3 4 5

Material Coloured paper sheets Pen/pencil, A pair of scissors Geometry box Piece of thread Adhesive

Number required 1 or 2 per group 1 set per group 1 per group Around 1 feet per group 1 Tube or box per group

Things to Do Divide the class into groups of 3 or 4 based on the strength of class. Each group should receive the amount of material specified in the Material List.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous activity, we learnt how to verify various properties related to circles.

Procedure: 3.3a Step 1: Draw a circle of any radius on a coloured paper sheet. Cut the circle and paste it on a white sheet of paper. Label the center of the circle O. Step 2: Fold the white paper near the edge of the circle as shown in the figure. The crease formed is a tangent to the circle. Draw the tangent and label it as PQ.

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60

Similarly, fold the paper at some other part of the paper and label the new tangent as LM. Label the point of contact for PQ as A and for LM as B. Draw OA and OB.

Step 3: On another coloured paper sheet, draw a right angle, cut it out and label it as DEF. Step 4: Place the right angle DEF over PQ in such a way that EF and AQ coincide. Similarly, place it over LM such that EF and LB coincide. What do you observe?

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3.3b: Step 1: Draw a circle of any radius on a coloured paper sheet. Cut the circle and paste it on a white sheet of paper.

Step 2: Mark a point outside the circle on the white paper. Label the point P.

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62 Step 3: Fold the paper at point P, till the edge of the circle as shown in the figure. Label the point of contact as A. The crease PA is a tangent to the circle. Draw PA. Similarly draw another tangent PB.

Step 4: Take a piece of thread and hold one side at P. At point P, you could stick it to the paper too. Place the thread over PA and cut it at the point A. Now the length of the thread is same as the length of the tangent PA. Now, move the thread so that it lies over PB. Ask students to share their observation.

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UNDERSTANDING THE ACTIVITY: Leading Questions:  What type of angle does a radius form with a tangent at its point of contact?  If you draw tangents from an external point to a circle, what will be their difference in length?

Discussion and Explanation:  The radius of a circle is perpendicular to a tangent at its point. Hence, it will be a right angle.  Tangents drawn from an external point to a circle is always equal in length.

KEY MESSAGES A tangent is perpendicular to the radius of a circle at its point of contact.

Tangents drawn from an external point to a circle are equal in length.

LEARNING CHECK: 

When a vehicle is standing on the road, observe one of the wheels. Identify the tangent, radius and point of contact.

(You can ask the students to imagine this scenario and discuss it.)

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64 

In 3.3b, other than using thread, can you verify the result using some other way? (Hint: Fold the paper along OP).

Can you come up with a proof for the result?

(Try to direct the students towards identifying the congruent triangles; and see if they can come up with the proof.).

Time: 30 min

ABL 3.4

LEARNING OBJECTIVE: How do we verify properties of cyclic quadrilaterals?

ADVANCE PREPARATION: Material List 1 2 3 4 5

Material Coloured paper sheets Tracing paper Geometry box Pen/Pencil A pair of scissors

Number required 1 or 2 sheets per group 1 sheet per group 1 per group 1 or 2 per group 1 per group

Things to Do Divide the class into groups of 3 or 4 based on the strength of the class. Each group should receive the amount of material as specified in the Material List section. The students should already know about cyclic quadrilaterals. If they have not heard about it, explain to them briefly.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous activity, we learnt about properties related to circles and their tangents.

Procedure: 3.4a: Step 1: Draw a circle of any radius on a coloured paper sheet. Cut the circle out. Paste the circle on a white sheet of paper.

Step 2: Fold the paper so that you get 4 chords: AB, BC, CD and DA. Draw the lines AB, BC, CD and DA to get the cyclic quadrilateral ABCD.

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Step 3: Make a replica of the cyclic quadrilateral ABCD using tracing paper.

Step 4: Cut near the angles of the replica, so that you get the 4 angles.

Step 5: Draw a straight line on a paper. Place angles BAD and BCD adjacent to each other on the straight line. Similarly, place ABC and ADC adjacent to each other on the straight line. Ask the students to share their observation.

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66 Step 1: Use the same cyclic quadrilateral. Extend AB to E so that the exterior angle CBE is formed.

Step 2: Among the angle replicas that you had created for the previous experiment, take the replica of angle ADC. Place it on CBE. What do you observe?

UNDERSTANDING THE ACTIVITY: Leading Questions:  What is a cyclic quadrilateral?  What can you say about interior angles of a cyclic quadrilateral?  What can you say about exterior angle of a cyclic quadrilateral?

Discussion and Explanation:  A cyclic quadrilateral is a quadrilateral whose vertices lie on a single circle. This circle is called its circumcircle.  In a cyclic quadrilateral, the interior opposite angles add up to 180°.  In a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

KEY MESSAGES In a cyclic quadrilateral, the interior opposite angles are supplementary (sum is 180°).

The exterior angle of a cyclic quadrilateral is equal to its interior opposite angle.

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LEARNING CHECK:   

Given one interior angle of a cyclic quadrilateral, how can we find the other 3 angles?

Can square be a cyclic quadrilateral for some circle? How can we draw its circumcircle? Discuss.

What about rectangles and parallelograms?

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ABL 4 Areas and Perimeters Activity

Learning objective

Key messages

4.1

What is area?



4.2

How can we explore areas of different figures using a geoboard?

4.3

4.4 4.5

How to explore and find formulas to calculate areas of shapes like parallelogram, triangles and trapeziums?

How to explore areas of circles?

What is perimeter? How do we find the perimeters of squares and rectangles?

  

  

Time (min)

In this activity, we learn the concept of areas.

In this activity, we explore areas of the following figures using geoboard: Rectangle Square Right angled triangle In this activity, we explore areas of the following figures using cardboard cutouts: Parallelogram Triangle Trapezium

30

In this activity, we derive the formula to calculate the area of a circle.

In this activity we learn about the concept of perimeter. We learn to calculate the perimeter of a square and a rectangle.

30

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60

45

69 Time: 30 min

ABL 4.1 LEARNING OBJECTIVE: What is area?

ADVANCE PREPARATION: Material List 1 2 3

Material Graph paper Pen/Pencil Models of shapes of different sizes (wooden or cardboard)

Number required 1 sheet per group 1 per group A few shapes per class

Things to Do Divide the class into groups of 3 or 4 based on the strength of the class. Each group should receive the amount of material as specified in the Material List section.

Safety Precautions NA

SESSION: Link to known information/previous activity: NA Procedure: Step 1: Show models of different sizes and ask the students which one of them is bigger. If you do not have any models, you could ask them to raise their hands and show their palms. Let the students compare each other’s palms and find out which one is bigger. Ask them how they figured it out. Now, you can explain the concept of areas. (Area is the amount of space covered inside a closed flat surface such as triangle, square etc.).

Step 2: Ask the students how do they measure lines? The reply would be centimeters, feet, inches and so on. Now, ask them how they measure length and breadth of a rectangle, or sides of a square, since they are just line segments. The answer would be similar. Now, explain to them how areas are measure i.e. in terms of square units. It does not matter if it is centimeters or feet, in each case, the area would respectively be in square centimeters and square feet. You can also draw a figure of a unit square to describe the idea.

Step 3: Take a graph sheet. Each square would be 1 cm square. Draw a rectangle or square on the sheet by joining the points. Ask the students to count the number of squares inside the drawn figure. This represents the area of the figure in terms of square centimeters.

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Ask them to draw a random closed figure (a pentagon or any other polygon). If you observe, some of the squares may not be completely covered. You will get an approximate answer in this case. However you can calculate on the basis of the following 3 rules:  If the figure covers more than half of a square, then you count it as 1  If the figure covers half of a square, count it as 0.5  If the figure covers less than half of a square, then you count it as 0 Similarly, try to find the area of a circle.

UNDERSTANDING THE ACTIVITY: Leading Questions:  What is area? What is it measured in?

Discussion and Explanation:  Area is the space inside a closed flat surface such as triangle, square and so on. It is measure in square units.

KEY MESSAGES In this activity, we understand the concept of areas.

LEARNING CHECK: Agastya International Foundation. For Internal Circulation only.

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71  

If the length and breadth of a rectangular playground is measure in meters, then what will be the area measured in?

Ask the students to measure the area of their palms. Compare the result with each other.

Time: 45 min

ABL 4.2

LEARNING OBJECTIVE: How do we explore areas of various figures using geoboard?

ADVANCE PREPARATION: Material List Material 1 2 3

Geoboard Rubber bands Paper, pen/pencils

Number required 1 per group A few per group A few per group

Things to Do Divide the class into groups of 3 or 4. Each group should receive the amount of material as per the Material List section. If you have just 1 geoboard, you could use it for demonstration and the students could draw figures on dotted paper to explore areas.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous activity, we learnt the concept of areas.

Procedure: 4.2a: Step 1: Form an irregular figure using geoboard as shown below.

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Step 2: Find the area of the figure by counting the number of squares inside it. Follow the following rules:  Count the number of squares fully enclosed by the figure. Each square will count as 1 square unit.  Count the number of squares that are enclosed more than half. Add one square unit for each of such squares.  Count the number of squares that are enclosed exactly half. Add 0.5 square units for each of the counted squares.  Neglect the squares which are enclosed less than half.

Similarly, find the areas of different types of irregular figures.

4.2b: Step 1: Form different rectangles on the geoboard as show below.

Step 2: Fill the table below.

Total Number of unit squares

Length of the rectangle (units)

Breadth of the rectangle (units)

Length x Breadth (square units)

1.

2.

3.

4.

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73 Step 3: Ask the students to share their observation.

4.2c: Step 1: Form squares of different sizes on the geoboard as shown below.

Step 2: Fill the table below.

Total Number of unit squares

Side of the Square (units)

Side x Side (square units)

1.

2.

3.

4.

Step 3: Ask students to share their observation.

4.2d: Step 1: Form right angled triangles of different sizes as shown in the figure below.

Step 2: Fill the table below. Use the same rule you used in 4.2a to calculate the area.

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74 Total Number of unit squares

Height (h) in units

Base (b) in units

½ x b × h (square units)

1.

2.

3.

4.

Step 3: Ask students to share their observation.

UNDERSTANDING THE ACTIVITY: Leading Questions:  What is the formula to calculate the area of rectangle?  What is the formula to calculate the area of square?  What is the formula to calculate the area of right angled triangle?

Discussion and Explanation:  Length x Breadth  Side x Side  ½ x base x height

KEY MESSAGES: In this activity, we explore and verify the areas of formulas to calculate areas of a rectangle, square and a right angled triangle. We used a geoboard to do this activity.

LEARNING CHECK: 

 

Can you give a reason as to why the formula to calculate area of a rectangle is length x breadth.

(Arrange sticks in a rectangular way, and ask them to figure out the total number of sticks by just counting the number of rows and columns.).

Why is the formula for right angled triangle ½ x base x height? (Use another rubber band to mirror the triangle and complete the rectangle.).

Give problems on calculating areas using formulas. Example problems are given below:  Calculate the area of a rectangle if the length is 12 cm and breadth is 7 cm.  Calculate the area of a square whose side is 4cm.

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75 Time: 60 min

ABL 4.3

LEARNING OBJECTIVE: How to explore and find formulas to calculate areas of shapes like parallelogram, triangles and trapeziums?

ADVANCE PREPARATION: Material List Material 1 2 3 4

Cardboard Graph paper Pair of scissors Pen/Pencil, Rulers

Number required 1 big sheet per group 1 sheet per group 1 per group 1 set per group

Things to Do Divide the class into groups of 3 or 4. Each group should receive the amount of material as specified in the Material List section.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous activity, we used geoboards to explore areas of various figures.

Procedure: 4.3a: Step 1: Draw a parallelogram on a graph paper as shown in the figure. Cut it out of the graph sheet. Use this cutout to cut out a replica of the parallelogram on a cardboard sheet.

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76 Step 2: Use the cutout of the graph paper and the cardboard to mark the base of the height of the parallelogram on the cardboard sheet. Draw the height as shown in the figure. Cut the triangle AHD from the cardboard replica.

Step 3: Rearrange the triangle cutout and place it on the right as shown in the figure. What do you observe? You get a rectangle which will have the same area as that of the parallelogram. The length and breadth of the rectangle will be equal to the base and height of the parallelogram respectively. Can you guess the formula to calculate the area of the parallelogram?

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77 Step 1: Similar to the previous experiment, draw a triangle on a graph sheet and cut it out. Use this cutout to make 2 replicas of the triangle on a cardboard sheet.

Step 2: Arrange the 2 triangles to form a parallelogram as shown in the figure.

Step 3: What do you observe? The area of the parallelogram is twice the area of the triangle. Can you figure out the formula to calculate the area of a triangle?

4.3c: Step 1: Draw a trapezium on a graph sheet and cut it out. Use this to make 2 replicas of the trapezium on a cardboard sheet as shown in the figure.

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Step 2: Rearrange the trapeziums to form a parallelogram as shown below.

Step 3: Ask students to share their observation. What is the formula to calculate the area of a trapezium?

UNDERSTANDING THE ACTIVITY: Leading Questions:  How do we calculate the area of a parallelogram?  How do we calculate the area of a triangle?  How do we calculate the area of a trapezium?

Discussion and Explanation:  Base x Height  ½ x Base x Height  ½ x (a + b) x Height where a and b are the 2 parallel sides.

KEY MESSAGES In this activity, we learn to calculate areas of parallelograms, triangles and trapeziums. We use cardboard replicas to understand formulas used to calculate areas for the figures.

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79  

Using the cardboard technique, can you find a way to calculate the area of a rhombus? (Hint: Cut the rhombus into 4 right angled triangles.).

The students can be given problems on calculating areas to make them comfortable about using formulas. Example problems are given below.  Calculate the area of a triangle if its height is 7cm and base 10cm.  Calculate the area of a parallelogram if its base is 10cm and height 4cm.

Time: 30 min

ABL 4.4 LEARNING OBJECTIVE: How to explore areas of circles?

ADVANCE PREPARATION: Material List 1 2 3 4 5

Material Coloured paper White paper sheet A pair of scissors Glue Geometry box

Number required 1 sheet per group 1 sheet per group 1 per group 1 tube/box per group 1 per group

Things to Do Divide the class into groups of 3 or 4. Each group should receive the amount of material required as mentioned in the Material List section. The students should already be aware of properties of circles, the concept of areas, on using formulas to calculate area for parallelograms and rectangles.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous activity, we learnt to explore areas of figures such as parallelograms, triangles and trapeziums. In ABL 3, we learn about circles. Especially in 3.2, we learn about circumference and other properties related to circles.

Procedure: Step 1: Draw a circle of known radius on a coloured paper sheet. Cut the paper out of the paper.

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Step 2: Fold the paper into half repeatedly several times so that the circle is divided into 16 equal parts.

The procedure is shown in the figures below. Cut the folded circle on the creases formed so that you get 16 equal pieces.

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Step 3: Arrange the 16 pieces (you can stick them on a white paper sheet too) as shown in the figure below. The arranged formation looks like a parallelogram.

Step 4: If you cut the circle into more equal pieces (32 or 64), the formation would look more like a rectangle. Ask the students to guess the length and breadth of the rectangle formed if the circle can be cut into more such pieces. As you can see from the figure, the length would be equal to half of the circumference. The height will be equal to the radius. How?

Step 5: Ask the students if they can calculate the area of the circle provided that the area of a rectangle is given by the formula Area = length x breadth.

Area = Length x Breadth = (½ x 2 x π x r) x r = π x r2

UNDERSTANDING THE ACTIVITY: Leading Questions:  What is the formula to calculate the area of a circle?

Discussion and Explanation:  π x radius2

KEY MESSAGES Agastya International Foundation. For Internal Circulation only.

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82 In this activity, we learn how to calculate the area of a circle. The area of a circle is equal to π x radius 2.

LEARNING CHECK:  

Give problems on how to calculate the area of circles. Example of a problem is given below:  Calculate the area of a circle if its radius is 5 cm.

If the distance to walk around a circular playground is 22m. Find the area of the playground.

(Encourage the students to think about the problem and put it in mathematical terms.)

Time: 45 min

ABL 4.5

LEARNING OBJECTIVE: What is perimeter? How do we calculate the perimeters of squares and rectangles?

ADVANCE PREPARATION: Material List Material 1 2 3 4 5

Graph paper Thin sticks Small pieces of cycle valve tube Sketch pens/pencil/ruler Cutter

Number required 1 or 2 per student A few per group A few per group 1 each per group 1 per group

Things to Do Divide the class into groups of 3 or 4 depending on the strength of the class. Each group should receive the amount of material as specified in the Material List section.

The graph paper should have grids of 1 centimeter square.

Safety Precautions NA

SESSION: Link to known information/previous activity: NA Procedure: 4.5a: Agastya International Foundation. For Internal Circulation only.

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83 Step 1: Ask the students to take a graph paper each and draw random diagrams on it. The only requirement is that the diagrams should be a closed figure. An example figure is shown below. You can also ask them to colour them using sketch pens.

Step 2: Explain to the students the concept of perimeters. Ask them to calculate the perimeter of the diagrams that they have drawn.

4.5b: Step 1: Take the thin sticks. Cut them to 4 equal pieces. Take the small pieces of the cycle valve tubes and connect the 4 sticks together to form a square as shown below. You might have to use a cutter to sharpen the edges of the sticks.

Step 2: Now deconstruct the square. The perimeter of the square is the sum of the lengths of the 4 sticks.

Hold them together and explain to the students how the perimeter of a square would be equal to 4 x s where s is the length of each side of a square.

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4.5c: Step 1: Similar to the previous experiment, construct a rectangle now using the sticks. You will need 2 sticks of the same longer length and 2 of same shorter length.

Step 2: Deconstruct the rectangle. Again, the perimeter of the rectangle is the sum of the lengths of all the sticks. Place the longer sticks and shorter sticks in 2 separate groups. Explain to the students how the formula to derive perimeter of the rectangle is equal to 2 x (l + b) where l and b are length and breadth of the rectangle respectively.

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UNDERSTANDING THE ACTIVITY: Leading Questions:  What is perimeter?  How do we calculate the perimeter of a square?  How do we calculate the perimeter of a rectangle?

Discussion and Explanation:  Perimeter is the total distance around the edge of a closed figure.  4 x s where s = side  2 x (l + b) where l = length, b = breadth

KEY MESSAGES In this activity, we learn the concept of parameters. We learn how to calculate the perimeters of squares and rectangles.

LEARNING CHECK: 



Give problems on perimeter calculation. Example problems are given below:  Calculate the perimeter of a square if the side is 7 cm.  Calculate the perimeter of a rectangle if the length is 10 cm and breadth is 4 cm.

Ask the students to calculate the perimeter of their classroom. If you do not have a measuring tape, you could simply ask them to calculate in terms of steps. Ask them to guess if the classroom is square or rectangle; give hints so that they use formulas to calculate it.

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86

ABL 5 Solids Activity

Learning objective

Key messages

5.1

What are solid shapes?

What are the different examples of solid shapes?

How do we construct solid shapes using their nets?



How do we calculate the surface areas of 3D figures?



How do we calculate volumes of 3D figures?



5.2

5.3

5.4



In this activity we learn to recognize solid shapes, classify them and also learn about terminologies related to solids.

In this activity, we learn to create solid shapes using their nets. We will learn to verify the Euler’s formula using the shapes that we have made.

In this activity, by doing experiments, we demonstrate formulas to calculate surface areas of the following 3D figures: cube, cuboid, right circular cylinder, right circular cone and sphere.

In this activity, we demonstrate formulas to calculate volumes of the following 3D figures: cube, cuboid, right circular cylinder, right circular cone and sphere.

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Time (min) 50

90

120

90

87 Time: 50 min

ABL 5.1

LEARNING OBJECTIVE: What are solid shapes? What are the different examples of solid shapes?

ADVANCE PREPARATION: Material List 1

2

Material Models of different solid shapes (cube, cuboids, cylinders, prisms etc.)

Model of 2-D objects (triangle, quadrilateral etc.)

Number required One or more sets per class

One or two per class

Things to Do Divide the class into groups of 3 or 4 based on the strength of the class. This is a series of simple activities where students learn to recognize 3-D shapes and classify them based on certain properties.

Safety Precautions NA

SESSION: Link to known information/previous activity: NA Procedure: 5.1a: In this part of the activity, the students learn the difference between 2-D and 3-D shapes. Show the students, the model of a 2-D shape (e.g. a triangle). Now, show the model of a 3-D shape (e.g. a cylinder).

Ask the students the difference between the two. Let the students themselves use common terms (everyday language) to describe the difference. It is important they use their own words (such as right/left, curved/pointed, sharp/flat etc.) in doing this. Later, you can use the terms they used to teach them the concept of 2-D and 3-D shapes.

In case of 2-D shapes, the shapes are measured using just 2 parameters: length and breadth. Hence, they are called 2-D, whereas in case of 3-D, the shapes are measured using 3 parameters: length, breadth and height. So, they are called 3-D shapes. They are also commonly known as solid shapes.

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5.1b: The aim of this part of the activity is to teach the students the difference between polyhedrons and non-polyhedrons. Show the students models of different solid shapes. Now, take any 2 models (e.g. a pyramid and a sphere). Ask them the difference between the two. The students can use their own terms to tell the difference. Similarly, ask them to describe the difference between many such combinations of solids (e.g. a cube and a cylinder, a cuboid and a cone, a sphere and a pyramid etc.).

In this way, the students learn to classify solids depending on certain properties. Now, based on the terms used by students, you can teach about polyhedrons and non-polyhedrons. A polyhedron is made of flat surfaces. The surfaces are called faces; the line segments where 2 faces meet is called an edge; the point where edges meet is called a vertex. On the other hand, a non-polyhedron is any solid which has a curved surface.

5.1c: Here, the students learn to classify the solids into more categories. Ask the students to recognize faces, vertices and edges of 2 or more solids. Ask the students, if they can now differentiate the solids depending on faces, vertices and edges. Based on their responses, you can now start classifying the shapes further.

If we take polyhedrons, a prism is a polyhedron with the base and top being congruent polygons. The faces joining the base and top are parallelograms. Cubes and cuboids are special cases of prisms. A pyramid is a polyhedron with a polygon base and triangular faces that meet at a vertex. If we consider nonpolyhedrons, we can classify them into cylinders and cones.

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89 5.1d: In this part, the students play a game – “Guess the Solid”, to test their strength of identifying and classifying solid shapes.

Step 1: You can either ask the students to volunteer or choose a student to come forward. The students can take turns to come forward and play this part. The rest of the students form groups or can also play individually.

Step 2: The student who comes forward will stand in front of the class and the teacher will tell him secretly the name of a solid shape. The teacher can either give a piece of paper or show him the solid secretly.

Step 3: The rest of the students will take turns to ask questions to the one holding the secret. The questions can be of the following kind:  Does it have a curved surface?  How many faces does it have?  Are there any triangular faces on the shape?

In this way, the students narrow down options and guess the solid shape. The teacher can impose additional rules to maintain order in the classroom.

UNDERSTANDING THE ACTIVITY: Leading Questions:  What are solids? Give examples.  What are polyhedrons and non-polyhedrons?  What are prisms?  What are pyramids?  What are cylinders?  What are cones?

Discussion and Explanation:  Solids are 3-D shapes i.e. they are measured using 3 parameters: length, breadth and height.

Examples of solids are cubes, cuboids, cylinders etc.  Polyhedrons are solids that have flat surfaces called faces, line segments called edges and sharp corners called vertices. Non-polyhedrons are solids that have a curved surface.  Prisms are solids that have a base and top which are congruent polygon and lateral surfaces between the top and base that are parallelograms.  A pyramid is a solid that has a polygon base and triangular lateral surfaces that meet at a vertex.  Cylinders are solids with 2 parallel and circular bases connected by a curved surface.  A cone is a solid with a circular base and a single vertex.

KEY MESSAGES In this activity, we learn about the difference between 2-D and 3-D shapes. We learn to classify different solid shapes based on certain properties. We learn examples of solids under each classification.

LEARNING CHECK: 

Give examples of real life objects that resemble solid shapes. For example, a matchbox is a cuboid, a dice is a cube. Give examples of solids that are combinations of solid shapes. For example, a bottle

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90



might be a combination of a cylinder and a cone. To keep it simple, you could give examples of real world objects (e.g. the school building) and ask the students to recognize all the solid shapes.

Play the odd-one-out game. Make groups of 3-D shapes and ask students to choose the odd-one-out.

The students should also give reasons for their selection. For example, make a group of cube, cuboid, sphere and a triangular prism. The sphere is the odd one out since it has a curved surface (is a nonpolyhedron).

Time: 90 min

ABL 5.2

LEARNING OBJECTIVE: How do we construct solid shapes using their nets?

ADVANCE PREPARATION: Material List 1 2 3 4 5 6

Material Nets of various 3D figures Glue Pen/Pencil Chart papers/Coloured papers Pair of scissors Ruler

Number required 1 set per group 1 tube/box per group 1 per group Few sheets per group 1 per group 1 per group

Things to Do Divide the class into groups of 3 or 4 based on the amount of material available.

The diagrams of nets are given. Either you can take a printout of the diagrams or the students could themselves draw the diagram on a chart paper.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous activity, we learn to recognize 3D figures and classify them based on certain properties.

Procedure: 5.2a: Step 1: Show a demonstration on how to create 3D figures using their nets. Take the example of a cube or cuboid and show the process.

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91 Step 2: Show the students the various nets and ask them to guess what kind of figures can be made from them. The students can be made to fill a simple table with the drawing of the net and their respective guess.

5.2b: Step 1: The students have to create the 3D figure using the nets. They have to fold the nets along the edges and glue it at points where they join. The process for a prism is show in the figures below. If the students are just given the printout of the nets, they might also have to cut them out.

Step 2: Ask the students to label each of their 3D figures.

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5.2c: Step 1: Let the students keep the models aside. Now, show the students the pictures of various nets and ask them to guess the number of faces, edges and vertices. Please note that in this part of the activity, the students should guess the number using the nets and not the actual models.

Step 2: The students can again fill a simple table which will have the name of the solid along with the number of faces, edges and vertices.

5.2d: In this part of the activity we verify the Euler’s formula. The Euler’s formula for polyhedron gives a relationship between the faces, edges and vertices of a polyhedron. It is given by

F+V=E+2 F, V and E are the number of faces, vertices and edges respectively.

Step 1: Ask the students to count the number of faces, edges and vertices in the 3D figures that they have made. Fill the table shown below.

3D Figure

F

V

E

F+V

Cube Cuboid Triangular prism Pentagonal Prism Triangular pyramid Square based pyramid

Step 2: Ask students to share their observation.

UNDERSTANDING THE ACTIVITY: Leading Questions:  Among the 3D figures created, how many had curved surfaces? Name them.  What is the Euler’s formula for polyhedrons?

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E+2

93 Discussion and Explanation:  Two of them, Cones and Cylinders have curved surfaces.  F + V = E + 2, F = number of faces, V = number of vertices and E = number of edges.

KEY MESSAGES In this activity, we learn how to create 3D figures using their nets. We also verify the Euler’s formula for polyhedrons (F + V = E + 2) using the created 3D figures.

LEARNING CHECK: 

Draw nets of various 3D figures (nets have to be slightly different from the ones already shown these activities) and ask the students to guess the figure. Examples nets are shown below.

 Similar to the above, draw different nets and ask the students to guess the number of faces, edges and vertices of the solid figure.  Give simple problems on using the Euler’s formula. An example problem is given below.  If the number of faces and vertices of a polyhedron are 6 and 8. Find the number of edges using Euler’s formula.

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REFERENCES:

Cube

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Triangular Prism

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Pentagonal Prism

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Triangle based Pyramid

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Square based Pyramid

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Cylinder

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Cone

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102 Time: 120 min

ABL 5.3

LEARNING OBJECTIVE: How do we calculate the surface areas of 3D figures?

ADVANCE PREPARATION: Material List 1 2 3 4 5

Material Geometry box Nets of different 3D figures (cube, cuboid) Models of 3D figures (only sphere is mandatory) Coloured paper/Chart paper (for cylinder and cone) Glue/Adhesive tape

Number Required 1 per group 1 set per group 1 set per group 1 or 2 sheets per group 1 per group

Things to Do Divide the class into groups of 3 or 4 based on the strength of the class. In this activity, some of the models (cylinder and cones) have to be made using paper cutting/folding methods. For cube and cuboid, the nets are used to derive the formulas. In case of a sphere, we need a model (either wooden or some other material) to do the activity. Even a ball can be used instead of a sphere model. Except for the sphere, models are not mandatory to do the activities.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous two activities, we learn to recognize 3D shapes. We learn to create 3D figures using their nets.

Procedure: Before going ahead with the activity, the students should learn the meaning of surface areas. Use the 3D models to explain the concept. You can start with the 2D concept of areas and extend it to surface areas of 3D figures. The students should be able to understand what surface areas mean with respect to each 3D figure.

5.3a Step 1: As in ABL 5.2, we make a cube using the net. The nets can either be printed on a paper or the students can draw it themselves. Calculate the length, breadth and height of the cube. We can observe that the length, breadth and height of the cube is equal, let’s say a.

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Step 2: Deconstruct the cube. It will be made up of 6 similar squares, the length of its side equal to a. Can you guess the surface area of the cube?

Surface area of the cube = 6 times the Area of the square. = 6 x Area of the Square who sides equal a. = 6 x a2 5.3b Step 1: Similar to 5.3a, construct a cuboid using its net. Measure the length, breadth and height of the cuboid. Let’s label them as l, b and h respectively.

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Step 2: Deconstruct the cuboid. The surface areas of the cuboid can be measured by calculating the areas of the six rectangles shown in the figure below. Can you derive the formula?

Surface area of the cuboid = Sum of areas of the 6 rectangles = (l x h) + (l x b) + (l x h) + (l x b) + (b x h) + (b x h) = 2 x (lh + bh + lb) 5.3c In this part, we measure the surface areas of a right circular cylinder. Explain to the students the concept of curved surface area and total surface area.

Step 1: Construct a right circular cylinder using its net. You can either use a pre-printed net or create yourself by paper cutting. If you’re creating your own, construct a rectangle of length 22cms and breadth 10cms. This way, you can create the base and top circles of radius 3.5cms. The radius and height of the cylinder is shown in the figure below. Let the radius be r and height h.

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Step 2: Deconstruct the cylinder. The whole cylinder can be broken down into 1 rectangle and 2 circles.

The length of the rectangle is equal to the circumference of the circles and breadth is equal to the height of the cylinder. The curved surface area of the cylinder is the area of the rectangle whereas the total surface area is the sum of the areas of the rectangle and the two circles. Ask the students to derive the formulas.

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Curved surface area of the cylinder = Area of the rectangle = Length x Breadth = Circumference of the circle x Height of cylinder =2xπxrxh = 2πrh Total surface area of the cylinder = 2 x Area of the circle + Area of rectangle = 2 x πr2 + 2πrh = 2πr(r + h) 5.3d Step 1: In this part of the activity, we calculate the surface areas of a right circular cone. Construct a cone using its net. Instead of using the net, you can also use a given slant height (l) and base radius (r) to construct the cone. To do this, draw a circle with slant height as the base radius. Then calculate the circumference of the base of the cone (2πr) and mark an arc equal to the circumference using thread. Cut the arc out and construct the cone using adhesive tape. To add the base, draw the base circle on the paper and cut it out and stick it to the base of the cone. In the figures below, you can see the slant height and the base radius marked along with its arc/circle.

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Step 2: Deconstruct the cone. The curved surface area of the cone is equal to the area of the segment. Fold the segment into half, twice, to get 4 equal parts. Rearrange the 4 pieces to form a parallelogram.

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Step 3: As shown in the above figure, the base of the parallelogram is equal to half the circumference of the base circle. If you divide the segment into more equal pieces (even number), the height of the parallelogram will be equal to the slant height of the cone. Can you figure out the formula for the curved surface area of the cone?

Curved surface area of the cone = Area of the segment = Area of the parallelogram formed = Base x Height = Half circumference of base circle x Slant height = ½ x 2πr x l = πrl Step 4: To calculate total surface area, you have to include the area of the base circle.

Total surface area of cone = Curved surface area + Area of the base circle = πrl + πr2 = πr(l + r)

5.3e In this activity, we calculate the surface area of a sphere. To do this activity, we need a model of a sphere (wooden or any model). Use a ruler to calculate the diameter of the sphere (and then its radius from the diameter). Let the radius of the sphere be r.

Step 1: Take a sheet of coloured paper and fold it around the sphere. Now, from this paper we need to construct a right circular cylinder so that it fits around the sphere perfectly. Mark points on the paper such that the height of the cylinder is equal to diameter of the sphere. The radius of the base/top of the cylinder is equal to the radius of the sphere. Construct the cylinder.

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109

Step 2: Now, we have a sphere of radius r, and a cylinder that covers the sphere. The radius of the base/top of the cylinder is equal to the radius of the sphere. The height of the cylinder is equal to the diameter of the sphere (= 2r). If you cut the cylinder horizontally into thin pieces, with sufficient enough pieces, the cylinder will cover the sphere fully. Hence, the surface area of the sphere is equal to the curved surface area of the cylinder. This can either be explained or you could ask the students to deconstruct the cylinder, cut into thin pieces and try it.

Step 3: When you deconstruct the cylinder, you get a rectangle. Measure the length and breadth of the rectangle. The breadth of the rectangle will be equal to twice the radius of the sphere as mentioned in Step 2. The length of the rectangle will be equal to the circumference of a circle of radius r. Can you figure out the formula for the surface area of a sphere?

Surface area of the sphere = Curved surface area of the cylinder = Area of the rectangle = Length x Breadth = 2Ď€r x 2r = 4Ď€r2

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110 Leading Questions:  What is the formula to calculate the surface area of a cube?  What is the formula to calculate the surface area of a cuboid?  What is the formula to calculate the surface area of a right circular cylinder?  What is the formula to calculate the surface area of a right circular cone?  What is the formula to calculate the surface area of a sphere?

Discussion and Explanation:  6đ?‘Ž2 ,where a is the length, breadth, height of the cube.7  2(đ?‘™â„Ž + đ?‘?â„Ž + đ?‘™đ?‘?) , where l, b and h are the length, breadth and height of the cuboid respectively.  Curved surface area = 2đ?œ‹đ?‘&#x;â„Ž Total surface area = 2đ?œ‹đ?‘&#x;(đ?‘&#x; + â„Ž), where h is the height of the cylinder and r is the radius of the base and top of the cylinder.  Curved surface area = đ?œ‹đ?‘&#x;đ?‘™ Total surface area = đ?œ‹đ?‘&#x;(đ?‘™ + đ?‘&#x;), where l is the slant height of the cone and r is the radius of the base of the cone.  4đ?œ‹đ?‘&#x; 2 , where r is the radius of the sphere.

KEY MESSAGES In this activity, we learn about formulas to calculate the surface areas of different 3D figures. We demonstrate the formulas by doing experiments.

LEARNING CHECK: 





Give problems on calculating surface areas of various shapes. The students have to use the formulas learnt to calculate the areas. Example problems are given below. ďƒ˜ A box is of length 80cms, breadth 60cms and height 15cms. Calculate the surface area of the box. ďƒ˜ Find the surface area of a cone whose slant height is 10cms and base radius is 7cms. ďƒ˜ Find the surface area of a sphere of radius 14cms.

Can you figure out the curved surface area and total surface of a hemisphere? You already know the surface area of the sphere is given by 4Ď€r2. The curved surface area of a hemisphere will be half of that of a sphere. To calculate the total surface area, you need to add the area of the base circle.

A gift box has the following dimensions: 20cm length, 30cm breadth and 10cm height. If the wrapping paper has an area of 200cm2, how many such papers do you need?

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111 

Ask the students to measure the surface area of their classroom. Let them use a measuring tape and measure the length, breadth and height of the room and calculate it (Measure it in feet). If there are windows, you can subtract the area of windows. If a litre of paint can paint 150 square feet, how many litres of paint is required?

Time: 90 min

ABL 5.4 LEARNING OBJECTIVE: How do we calculate volumes of 3D figures?

ADVANCE PREPARATION: Material List 1 2 3 4 5 6 7

Material Models of 3D figures (hollow cones, cylinders, spheres) Jodo cubes (or unit cubes) A box of sand (or some grain like rice) Graph sheet Coloured paper sheet A pair of scissors Pencils/pens/Rulers

Number required 1 per group 1 set per group 1 box per group 1 per class (just for demonstration) 1 per group 1 per group 1 set per group

Things to Do Divide the class into groups of 3 or 4. Each group should receive the amount of material as specified in the Material list section. However, in this activity, there might not be sufficient models of cones, cylinders and spheres. Hence, the teacher could demo the experiment initially, and each group could take turns to conduct the experiment. In case of Jodo cubes, each group can be given sufficient amount of cubes to do the activity.

Safety Precautions NA

SESSION: Link to known information/previous activity: In the previous activity, we learnt about surface areas of 3D figures.

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112 Similar to areas, the students should be made to understand the concept of volumes. The teacher can reintroduce the concept of areas where we measure the space covered by a flat surface (2D surface).

Similarly, the space occupied by a 3D figure is called the volume. You can take a hollow model to demonstrate it. To measure areas, we use square units, whereas in case of volumes, we use cubic units.

This can be demonstrated by drawing a closed figure on a graph sheet, and counting the number of square units inside it. Each square in a graph sheet is a square unit. Now, use several Jodo cubes (unit cubes) to construct a cube (using 3x3x3 = 27 Jodo cubes). Ask the students to count the number of Jodo cubes used.

The answer, 27 cubic units, is the volume of the constructed cubic structure. Each Jodo cube is equal to 1 cubic unit.

5.4a In this part of the activity, we learn about formulas to calculate the value of cube and cuboid.

Step 1: Ask the students to construct a cube using Jodo cubes. Similar to your demonstration earlier, the students can use 27 cubes.

Step 2: The students measure the length, breadth and height of the cube, counting the number of cubes along each direction.

Step 3: The volume of the cube is got by multiplying the three: length, breadth and height. Since, in a cube all three are equal; the volume is given by the following formula: Volume of a cube = a3, where a is the side of the cube.

Step 4: Similar to the cube, ask the students to construct a cuboid.

Step 5: Ask them to measure the length, breadth and height of the cube. Count the number of Jodo cubes used.

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113 Step 6: Ask the students to share their observation. The volume is equal to the product of length, breadth and height. Thus, Volume of a cuboid = lbh l is length, b is breadth and h is height of the cuboid

At this stage, you can explain to the students that the volume of both the figures above can be seen as a product of area of the base and height. How? If you notice both the figures, the cube can be seen as a bunch of squares stacked on top of each other (or rectangles in case of cuboid). Hence, the first two variables in the formula (i.e. a 2 in case of cube and lb in case of cuboid) give the area of the base.

So, the volume of the both the figures can be generalized as: Volume = Area of Base x Height

5.4b In this part, we figure out the volume of a right circular cylinder.

Step 1: Construct a cylinder using coloured sheets. The previous activities will give you instructions on how to do this. However, you do not have to make the top and base. Let the radius of the base circle be r and the height of the cylinder h.

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114

Step 2: Take the Jodo cubes and start placing it inside the cylinder as shown in the figure. In this way, if you figure out how many Jodo cubes fill the cylinder, it is possible to calculate the volume. But, the cubes do not fill the whole cylinder completely.

Step 3: However, similar to cubes and cuboids, you can see the cylinder as a bunch of circles stacked on top of each other. As you noticed in the previous experiment, the volume can be considered as the product of area of base and height. So, can you figure out the formula for the volume of the cylinder?

Volume of Cylinder = Area of base circle x Height = πr2 h r is the radius of the base/top and h is the height of the cylinder.

5.4c In this part, we demonstrate the formula to calculate the volume of a right circular cone. To do this activity, each group should receive a set of hollow models of a right circular cone and a right circular cylinder. The height and radius of the base circle of the cylinder and cone should be equal. Let the radius of the cone be r and the height be h. If there are not enough hollow models, the students can take turns in doing the task.

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115

Step 1: Fill the cone completely with sand (or grains). Pour the contents of the cone into the cylinder.

Repeat this process until the cylinder is full.

Step 2: How many times did you do this process? The process was done 3 times. If the volume of the cylinder is given by πr2 h, can you figure out the formula for volume of the cone?

3 x Volume of the cone = Volume of the cylinder 1 Volume of the cone = 3 x Volume of the cylinder 1 = 3 đ?œ‹r2h r is the radius of the base of the cone and h is the height of the cone.

5.4d In this part, we demonstrate the formula to calculate the volume of a sphere. To do this activity, we need hollow models of a sphere and a right circular cylinder. The radius of the sphere should be equal to the radius of the base/top of the cylinder; the diameter of the sphere should be equal to the height of the cylinder. Ideally, there has to be 2 models of cylinder and 1 model of the sphere per each student group. If there are not enough models, the students can take turns to do the task.

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116 Step 1: Fill the hollow sphere completely with sand and empty it in the cylinders. Continue this process until both the cylinders are full.

Step 2: How many times did you do the process to completely fill the 2 cylinders? The process is done thrice. Can you figure out the formula for the volume of the sphere if the radius of the sphere is r?

3 x Volume of the sphere = 2 x Volume of the cylinder 2 Volume of the sphere = 3x đ?œ‹r2h (h is height of the cylinder) 2 = 3 x đ?œ‹r2h x 2r 4 = 3 đ?œ‹r3 r is the radius of the sphere.

UNDERSTANDING THE ACTIVITY: Leading Questions:  What is the formula to calculate the volume of a cube?  What is the formula to calculate the volume of a cuboid?  What is the formula to calculate the volume of a right circular cylinder?  What is the formula to calculate the volume of a right circular cone?

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117  What is the formula to calculate the volume of a sphere?

Discussion and Explanation:  đ?‘Ž3 , where a is the side of the cube.  đ?‘™đ?‘?â„Ž, where l is the length, b is the breadth and h is the height of the cube.  đ?œ‹đ?‘&#x; 2 â„Ž, where r is the radius of the base/top, h is the height of the cylinder.  

1 3 4 3

đ?œ‹đ?‘&#x; 2 â„Ž, where r is the radius of the base, h is the height of the cone. đ?œ‹đ?‘&#x; 3 , where r is the radius of the sphere.

KEY MESSAGES In this activity, we learn the concept of volumes. Volume is the space occupied by a 3D figure. We learn to demonstrate the formulas to calculate the volume of 3D figures.

LEARNING CHECK: 

  

Give problems to calculate the volumes of 3D figures using formulas.

Some example problems are given below: ďƒ˜ Calculate the volume of a cube whose side is 5cm. ďƒ˜ Calculate the volume of a sphere whose radius is 7cm. ďƒ˜ What is the volume of a right circular cylinder whose height is 12cm? The circumference of the base/top of the cylinder is 44cm.

You already know the formula to calculate the volume of a sphere. Can you figure out the formula to calculate the volume of a hemisphere?

How many cubic centimeters make a liter? Find this using a 1 liter can as shown in the figure below.

(Please note you need a 1 liter can and a measuring tape to do this problem) An ice cream shop provides ice creams in 2 sizes of spherical cups: one with 5cm radius and another 10cm radius. Can you figure out the ratio of volume of ice cream in the cups? (If you have spherical models of 2 different sizes, you can demonstrate this.)

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118

TABLES 1.1 Table-1

No.

1

2

3

4

5

6

Name of the

Measurements of

Measurements of sides

Triangle

angles (in degrees)

(in cm)

∠A = ______

AB = ______

∠B = ______

BC = ______

∠C = ______

CA = ______

∠D = _______

DE = ______

∠E = _______

EF = ______

∠F = _______

DF = ______

∠L = ______

LM = ______

∠M = ______

MN = ______

∠N = ______

LN = ______

∠P = ______

PQ = ______

∠Q = ______

QR = ______

∠R = ______

PR = ______

∠X = ______

XY = ______

∠Y = ______

YZ = ______

∠Z = ______

XZ = ______

∠G = ______

GH = ______

∠H = ______

HI = ______

∠I = ______

GI = ______

∆ABC

∆DEF

∆LMN

∆PQR

∆XYZ

∆GHI

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1.1 Table-2 Points to explore

∆ABC

∆DEF

∆LMN

∆PQR

∆XYZ

All three sides equal (Yes / No) Any two sides equal (Yes / No) All three sides unequal (Yes / No) All three angles less than 90° (Yes / No) One angle equal to 90° (Yes / No) One angle greater than 90° (Yes / No)

1.3 - Table-1 Triangle ABC

PQR

XYZ

KLM

Sides AB = AC = BC = PR = QR = PQ = XZ = XY = YZ = KM = KL = LM =

Angles ∠BAC = ∠ABC = ∠ACB = ∠PRQ = ∠RPQ = ∠PQR = ∠ZXY = ∠XZY = ∠XYZ = ∠LKM = ∠LMK = ∠KLM =

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∆GHI

120

1.5 Table-1 Triangle ABC PQR

AB =

Sides BC =

AC =

PQ =

QR =

PR =

PQ = AB

QR = BC

PR = AC

∠1

∠2

∠BCA =

Angles ∠CAB =

∠ABC =

∠QRP =

∠RPQ =

∠PQR

2.2a Table-1 No.

∠ 3

∠ 4

∠ 1+∠ 2+∠ 3+∠ 4

1 2 3

2.2b Table-1 No

Length of sides AB

BC

DC

Measure of angles AD

1

2

3

1+2

2+3

4

1 2 3 4

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3+4

4+1

121

2.2b Table-2 No.

Length along the diagonals AC

Distance from intersection point

BD

AO

OC

BO

OD

1 2 3 4

2.2c Table-1 Measure of angles AOD

No 1

2

3

DOC

BOC

BOA

1+2

2+3

3+4

4+1

4

1 2 3

2.2c Table-2 No.

Length along the diagonals AC

BD

Distance from intersection point AO

OC

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BO

OD

122

2.2d Table-1 No.

Length of the sides AB

BC

DC

Measure of angles AD

1

2

3

Length along the diagonals 4

AC

AO

OC

BD

BO

OD

1 2 3

2.2e Table -1 No.

Measure of angles 1

2

3

Length along the diagonals 4

AC

AO

OC

BD

BO

OD

AOB

BOC

1 2 3

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COD

DOA

123

3.2a Table-1 Chord AB CD EF GH PQ

Length of Chord

3.2b Table-1 No.

Circumference

Diameter

Ratio Circumference/Diameter

1 2 3 4

4.2b Table-1 Total Number of unit squares

Length of the rectangle (units)

Breadth of the rectangle (units)

Length x Breadth (square units)

1.

2.

3.

4.

4.2c Table-1 Total Number of unit squares

Side of the Square (units)

Side x Side (square units)

1.

2.

3.

4.

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124

4.2d Table-1 Total Number of unit squares

Height (h) in units

Base (b) in units

½ x b × h (square units)

1.

2.

3.

4.

5.2d Table-1 3D Figure

F

V

E

F+V

Cube Cuboid Triangular prism Pentagonal Prism Triangular pyramid Square based pyramid

Agastya International Foundation. For Internal Circulation only.

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