Class 07 Project : Visualizing solid shapes

 Project : Visualizing solid shapes

Objective: 

Drawing cubes and cuboids.

Materials Required: 

Some sheets of isometric dot paper, sketch pen, pen, pencil, etc.

Procedure:

 I. To draw a cube of given dimensions (say a cube of edge 3 cm)

(In an isometric drawing, the measurements also agree with those of the solid.)

Take an isometric dot paper and draw two line segments AB and BC as shown below. 

Since the edge of the cube is 3 cm, we join four dots along each line to get the length of each line as 3 cm.

2. Now, draw three vertical lines AD, BE and CF as shown below. Here again, we join 4 dots along each line to get the length of each line as 3 cm.

3. Finally, join DE, EF. Also, draw DG || EF and FG || ED. The solid so obtained is a cube of edge 4 cm. Using a ruler measure each edge of the cube and verify.

You can also draw the cube by interchanging the steps 1 and 3 as shown below.

II. To draw a cuboid of given dimensions (say a cuboid of 4 cm x 2 cm x 3 cm)

1. Take an isometric dot paper and draw two line segments AB and BC as shown below. Here AB = 4 cm and BC = 2 cm. 

2. Now, draw three vertical line segments AD = BE = CF = 3 cm as shown below.

3. Finally, join DE and EF. Also, draw DG || EF and 

FG || ED. The solid so obtained is a cuboid of dimensions 4 cm x 2 cm x 3 cm. Measure each edge of the cuboid and verify.

Class 07 ACTIVITY 2 – TRIANGLES- Pythagoras Theorem

 ACTIVITY 2 – TRIANGLES- Pythagoras Theorem

Objective: 

To verify the Pythagoras theorem by paper cutting and pasting method.

Materials required: 

Squared paper, Colour pencils, a pair of scissors, glue stick, geometry box etc.,

Procedure:

1. On a squared paper, draw a right triangle ABC, right angled at B.

2. Draw squares on each side of the triangle as shown below.

3. Locate the centre of the square drawn on the longer leg of the ∆ABC. Mark it as 0. Draw DE ∥ AC, which passes through O. Draw FG such that ∠FOD = 90 °.

4. Cut out the square on side BC and the four pieces of the square on side AB.

5. Paste these five pieces over the square on the side AC as shown below. 

Observations:

In figure 5, we see that the square on side BC and the square on side AB (four pieces)completely cover the square on side AC. Or square on side AC = square on side AB + square on side BC. Or 

AC2 = AB2 + BC2

Conclusion: 

From the above activity, we can say that in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. 

Do Yourself: 

On a squared paper, draw two different right triangles. In each case verify the Pythagoras theorem by using the paper cutting and pasting method.

Class 07 ACTIVITY – TRIANGLES- Pythagoras Theorem

 ACTIVITY – TRIANGLES- Pythagoras Theorem

Objective: 

To verify the Pythagoras theorem by using a squared paper and shading the squares.

Materials Required: 

Squared papers, colour pencils, geometry box, etc.

Procedure:

On a squared paper, draw a right triangle ABC whose legs 

(sides forming the right angle) are 3 cm and 4 cm,

 i.e.,(AB = 3 cm and BC = 4 cm. 

Measure the side AC of ∆ABC.

It is 5 cm. 

Shade the triangular region ABC. 

2. On another squared paper, draw three squares

 having sides 3 cm, 4 cm and 5 cm. 

Shade each square using different colours and cut them out.

3. Paste these squares along the sides of triangle ABC such that one side of square (a) (green coloured) falls along AB, one side of square (b) (blue coloured) falls along BC and one side of square (c) (red coloured) falls along AC.

Observations : 

In figure,

area of the square on side AB = number of small squares inside the square on AB = 9 cm2

Area of the square on side BC = number of small squares inside the Square on BC = 16 cm2

Area of the square on side AC = number of small squares inside the Square on AC = 25 cm2

We find that 9 + 16 = 25 or area of the square on side AB + area of the square on side BC = area of the Square on side AC or AB2 + BC2 = AC2

Conclusion

From the above activity, we can that in a right triangle, the square on the hypotenuse is equal to the sum of the squares on other two sides.

Do Yourself: 

on a squared paper, draw the following right triangles (a) AB = 8 cm, BC = 6 cm, ∠ B = 90 °

(b) PQ = 5 cm, QR = 12 cm, ∠ Q = 90 °

In each case, verify the Pythagoras theorem by shading the squares.

Class 07 ACTIVITY 4 - TRIANGLES

 ACTIVITY 4 - TRIANGLES

Objective: 

To medians get the pass medians through of a a common triangle by point paper. folding. Also, to verify that in a triangle medians pass through a common point.

Materials Required: 

Tracing paper, colour pencils, geometry box, etc.

Procedure:

1. On a. tracing paper, trace the following triangles. Cut out each triangle from the tracing paper. Note that triangle (a) is an equilateral triangle, triangle (b) is isosceles as well as right angled triangle and  triangle(c) is as a scalene triangle. We can also say that triangle (a) is an acute angled triangle, (b) is right angled triangle, (c) is an obtuse angled triangle.

2. Fold each triangular cut out such that the vertex Q coincides with the vertex R.

3. Unfold each tracing paper. In each case mark the point of intersection of QR and the crease as X. Draw PX as dotted line.

4. Now, fold each triangular cut out such that the vertex P and R coincide.

5. Unfold each tracing paper. In each case mark the point of intersection of PR and the crease P as Y. Draw QY as a dotted line.

6. Finally fold each triangular cutout such that the vertex P coincides with the vertex Q.

7. Unfold each tracing paper. In each case mark the point of intersection of PQ and the crease as Z. Draw RZ as a dotted line. 

Observations :

In figure 3, X is the mid point of QR. So PX is a median of  each triangle.

2. Similarly in figures 5 and 7,  Y and Z are mid points PR and PQ respectively. So QY and RZ are medians of ∆PQR in each case.

3. Also  in figure 7, we see that in each case all the three medians pass through a common point O.

Conclusion

A triangle has three medians.

2. All the three medians of a triangle pass through a common point. This point is called the centroid of the angle.

Do Yourself: 

Draw an acute angled, a right angled and an obtuse angled triangle. By paper folding, verify that in each case the three medians are  concurrent.

Class 07 ACTIVITY 3 - TRIANGLES

 ACTIVITY 3 - TRIANGLES

Objective: 

To verify that an exterior angle of a triangle is equal to the sum of the two interior opposite angles by paper cutting and pasting.

Materials Required: 

White sheets of paper, colour pencils, geometry box, a pair of scissors, glue stick etc.

Procedure:

On a white sheet of paper, draw a triangle ABC. Produce its side BC to D as shown in the figure. Using Colour pencils, mark its angles as shown.

2. Using a pair of scissors, cut out the two marked angular regions as shown below.

3. Paste the angular cutouts A and B over exterior angle C such that vertices A, B and C coincide as shown.

Observations:

In figure, ∠ACD is an exterior angle of ∆ABC and  ∠ A and ∠ B are its two interior opposite angles

2. In figure, we see that the angular cutouts neither overlap nor leave any gap between them. In other words, the angular cut outs A and B completely cover exterior angle C.

or ∠ A + ∠ B = exterior angle C. or ∠ A + ∠ B = ∠ ACD

Conclusion: 

From the above activity, we can say that an exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Do Yourself: 

Copy each of the following triangles. In each case verify that an exterior angle of a triangle is equal to the sum of two interior opposite angles.

Class 07 ACTIVITY 2 - TRIANGLES

 ACTIVITY 2 - TRIANGLES

Objective: 

To verify the angle sum property of a triangle by paper folding.

Materials Required: 

White sheets of paper, tracing paper, geometry box, a pair of scissors, etc.

Procedure:

On a white sheet of paper, draw a fairly large triangle ABC. Using scissors, cut it out. Mark A, B and C on both sides of the cutout.

2. Fold the side BC of ∆ABC such that the folding line passes through A.

3. Unfold it and draw a line along the crease. This line cuts BC at D.

4. Fold the vertices (three corners) of ∆ABC such that A, B and C meet at D.

Observations: 

In figure 4, we see that the angles A, B and C form a straight angle 

i.e., ∠A + ∠ B + ∠ C = 180°

Conclusion: 

From the above activity, it is verified that the sum of the angles of a triangle is 180 °

Do Yourself:  

Draw an acute angled triangle, a right angled triangle. By paper folding, verify the angle sum property in each case.

Class 07 ACTIVITY - Triangles

 ACTIVITY - Triangles

Objective: 

To verify that the sum of all interior angles of a triangle is 180 °, by papercutting and pasting.

Materials Required: 

White sheets of paper, colour pencils, a pair of scissors, geometry box, etc.

Procedure:

On a white sheet of paper, draw a triangle ABC. Using colour pencils mark its angles as shown.

2. Using a pair of scissors, cut out the three angular regions.

3. Draw a line segment XY and mark a point on it.

4. Paste the three angular cut outs so that the vertex of each falls at O as shown in the figure.

Observations:

In figure 4, we see that the angular cutouts neither overlap nor leave any gap between them. But, XOY is a straight angle So, ∠A + ∠ B + ∠C = 180 ° Or sum of the angles of a triangle is 180 °.

Conclusion: 

From the above activity, it is verified that the sum of all interior angles of a triangle is 180 °

Do Yourself: 

Draw an acute angled triangle, a right triangle and an obtuse angled triangle. By paper cutting and pasting, verify the above property for each triangle.

Class 07 ACTIVITY3 - LINES AND ANGLES

 ACTIVITY3 - LINES AND ANGLES

Objective: 

To verify that if two parallel lines are cut by a transversal, then each pair of interior angles on the same side of the transversal are supplementary by paper cutting and pasting

Materials Required: 

White sheets of paper, colour pencils, geometry box, a pair of scissors, gluestick, etc. 

Procedure:

1. cutting On a white sheet of paper, draw two parallel lines AB and CD and a transversal EF cutting them at P and Q resp., Mark a point O on PQ. 

2. Cut the angular regions ∠OQC, ∠ OQD ∠ OPA and ∠ OPB.

3. Draw a line segment and take a point X on it. Paste the angular cut out OQD such that Q coincides with X and QD falls along the Straight line.

Now, paste the angular cut out OPB such that P coincides with X and PO falls along QO.

4. Draw another line segment and take a point Y on it. Paste the angular cut out OQC such that Q coincides with Y  and QC falls along the straight line. Now, paste the angular cut out OPA such that P coincides with Y and PO falls along QO.

Observations:

In figure 1, AB || CD and EF is a transversal.

So, (∠ APO, ∠ CQO) and (∠ BPO, ∠ DRO) are two pairs of interior angles on the same side of the transversal.

2. In figure 3, BD is a straight line. So, ∠ BXD = 180 °or

 ∠ BPO + ∠ DRO = 180 °

3. In figure 4, AC is a straight line.

So, ∠ AYC = 180 °or ∠ APO + ∠ CQO = 180 °

Conclusion: 

From the above activity, we can say that if two parallel lines are cut by a transversal, then each pair of interior angles on the same side of the transversal are supplementary.

Class 07 ACTIVITY2 - LINES AND ANGLES

 ACTIVITY2 - LINES AND ANGLES

Objective:

 To verify that if two parallel lines are cut by a transversal, then each pair of alternate interior angles are equal by paper cutting and pasting

Materials Required : 

glue stick, White sheets of paper, colour, pencils, geometry box, a pair of scissors, etc., 

Procedure:

On a white sheet of paper, draw a pair of parallel lines AB and CD. Also, draw a transversal EF cutting them at P and Q resp., Mark a point O somewhere in the middle of PQ. Mark the angles as shown.

2 . Cut the angles ∠ OQD and ∠ OQC.

3. Paste the angular cutouts ∠ CQO and ∠ DQO over ∠BPO and ∠ APO respectively such that in each case the vertex Q coincides with vertex P and one arm of each angle falls along one arm of the corresponding angles.

Observations: 

In figure 1, AB || CD and EF is a transversal. So, (∠APO, ∠ DRO) and (∠ BPO, ∠ CQO) are two pairs of alternate interior angles.

2. In figure 3, we see that if vertex Q of ∠ CQO coincides with vertex P of ∠ BPO and arm QC falls along PB, then QO  falls along PO, i.e., ∠ CQO completely overlaps ∠ BPO.

So, ∠ CQO = ∠ BPO 

Similarly, ∠ DQO completely overlaps ∠ APO.

So, ∠ DQO = ∠ APO transversal 

Conclusion : 

From the above activity, interior, we can say that if two parallel lines are cut by a transversal, then each pair of alternate interior angles are equal.

Class 07 ACTIVITY1 - LINES AND ANGLES

 ACTIVITY1 - LINES AND ANGLES

Objective: 

To verify that if two parallel lines are cut by a transversal, then each pair of corresponding angles are equal, by paper cutting and pasting.

Materials Required: 

White sheets of paper, colour pencils, geometry box, a pair of scissors, glue stick etc.

Procedure:

1. On a white sheet of paper, draw a pair of parallel lines AB and CD. Also, draw a transversal EF, cutting AB and CD at P and Q respectively. Mark the angles as shown in the figure. Mark a point O somewhere in the middle of PQ.

2. Cut the figure along the dotted lines to get four angular cut outs as shown below.

3. Paste the angular cutout ∠ DQO over ∠ BPE such that Q coincides with P and QD falls along PB.

4. Similarly, paste the angular cut outs ∠DQF,   ∠ CQF and ∠ CQO over ∠ BPO, ∠ APO and ∠ APE resp., such that in each case Q coincides with P and one arm of each angle falls along one arm of the corresponding angle.

Observations:

In figure1, AB ∥ CD and EF is Transversal. So, (∠ EPB, ∠ DQO), (∠ BPO, ∠ DQF), (∠ APE, ∠CQP) and (∠ APQ, ∠ CQF ) are four pairs of corresponding angles.

2. In figure, we see that if vertex Q of ∠ DQO coincides with the vertex P of ∠ BPE and QD falls along PB, then QO falls along PE, i.e., DQO completely overlaps BPE. So, ∠ DQO = ∠ BPE.

3. Similarly, in fig, we see that ∠ DQF, ∠ CQF and ∠ CQO completely overlap ∠ BPO, ∠ APO and ∠ APE resp.,

So, ∠ DQF = ∠ BPO, ∠ CQF = ∠ APO and ∠ CQO = ∠ APE

Conclusion : 

From the above activity, we can say that if two parallel lines are cut by a transversal, then each pair of corresponding angles are equal.

Do yourself: 

Verify the above property by drawing a pair of parallel lines which are 5 cm apart.

Class 07 Fun Activity – Algebraic Expression CROSS NUMBER PUZZLE

 Fun Activity – Algebraic Expression
CROSS NUMBER PUZZLE

Find the root of the given equations and complete the following cross number puzzle

Fun Activity – Algebraic Expression -  solution

CROSS NUMBER PUZZLE

Find the root of the given equations and complete the following cross number puzzle

CROSS NUMBER PUZZLE

Find the root of the given equations and complete the following cross number puzzle