CBSE Class 6 Maths Understanding Elementary Shapes HOTs

HOTS

1. If the lengths of two sides of an isosceles triangle are l3cm and 6 cm, then what is the length of the third side?
Answer: 13 cm

2. If the lengths of three consecutive sides of an isosceles trapezium are 5 cm, 6 cm and 8 cm, then what is the length of the fourth side?
Answer: 6 cm

3. Can a triangle have two right angles? Explain your answer.
Answer: No, sum of three angles of a triangle should be 180°, whereas in this case two angles are making 180°.

4. Figure shows some possible fielding positions in the game of cricket. You have to describe each one's exact position to a mathematical minded first timer by telling him the angle the position is at. Take the line connecting the stumps at both ends as one of the arms of the angle. 

Answer: Point - 90°
Cover - 60°
First slip - 160°
Wicket keeper - 0°
Long on - 180°
Fine leg - 120°
Mid on - 30°

CHALLENGES

1. You have a weighing machine in a post office which weighs packets up to 300 gm. (See the figure.) 

a. Suppose a packet weighs 50 gm. What is the angle through which the pointer rotates? What type of angle is this?
b. Suppose a packet weighs 75 gm. What is the angle through which the pointer rotates? What type of angle is this?
c. If the envelope weighs 125 gm, what angle the pointer rotates? What type of angle you get?
d. If the envelope weighs 150 gm, what angle the pointer rotates? What type of angle you get?
e. If the envelope weighs 225 gm, what angle the pointer rotates? What type of angle you get?
f. If the envelope weighs 300 gm, what angle the pointer rotates? What type of angle you get?

2. In the figure, you see a speedometer of a truck. It measures speed up to 90 km/hr. The numbers shown are all multiples of 10, but there are points in between which measures even
multiples of 5. The truck starts and accelerates up to 90 km/hr. Looking at the figure find out the angle the pointer turns from the starting position when the truck attains a speed of

a. 10 km/hr; b. 15 km/hr; c. 25 km/hr; d. 50 km/hr;
e. 80 km/hr; f. 90 km/hr.

3. The figure shows a square pyramid. Suppose the faces are equilateral triangles. 

a. What are the different types of geometrical shapes you find in the figure? Name all of them.
b. How many angles are there? Name all of them.
c. How many different measures of angles are there?
d. How many different lengths are there?

4. The following figure shows a pyramid. 

a. What is its polygonal base? How many triangular faces are there?
b. Count the number of angles.

5. Suppose you have a pyramid in which the base is a 12-sided polygon. How many angles are there in the pyramid? If the base is 15-sided polygon, what is the number of angles? Can you make a guess on the total number of angles in a pyramid having n-sided polygon as its base?

6. The figure below shows a solid obtained by chopping off the top of a square pyramid by a plane parallel to the base. 

a. What are the different types of geometrical shapes you find in the figure? Name all of them.
b. How many angles are there? Name all of them.
c. If the triangular faces of the original pyramid are equilateral triangles, how many different measures of angles you get? [Hint: property of parallel lines.]

7. Look at the prism below. 

a. What is its ploygonal base? How many rectangular faces are there?
c. Count the number of angles.

8. In the figure below, a polygon is drawn and every vertex is connected to every other vertex by line segments. 

a. How many sides are there?
b. How many vertices are there?
c. How many diagonals are there?
d. How many points of intersection are there inside the polygon?
e. What are the different geometrical shapes you can find in this figure?
f. Count the number of each geometrical shape.

SUMMARY

1 Comparison of line segments means finding a relation between their lengths.

2. Line segments can be compared by - observation, using tracing paper or using a divider.

3. Length of a line segment can be measured accurately by using a ruler and a divider.

4. The size of an angle is the amount by which one of the arms needs to be rotated about the vertex so that it lies on the top of the other arm.

5. One full turn of a hand of a clock is one revolution (or rotation).

6. A right angle is 1/4 turn and a straight angle is 1/2 turn.

7. Types of angles
Acute angle — An angle whose measure lies between 0° and 90°.
Right angle — An angle whose measure is 90°.
Obtuse angle — An angle whose measure lies between 90° and 180°.
Straight angle — An angle whose measure is 180°.
Reflex angle — An angle whose measure lies between 180° and 360°.

8. Types of triangles with regard to sides
Scalene - All the three sides are unequal (in length)
Isosceles - Any two sides are equal (in length)
Equilateral - All the three sides are equal (in length)

9. Types of triangles with regard to angles
Acute angled - All the three angles are acute (less than 90°)
Right angled - One angle is a right angle ( = 90°)
Obtuse angled - One angle is obtuse (greater than 90°)

10. A triangle with one right angle and two equal sides is called an isosceles right angled triangle.

11. Names of polygons are based on the number of sides it has: 

12. A polygon is called regular if all sides have equal length and all angles have equal measure.

13. Quadrilaterals are classified with reference to their properties: 

14. Properties of a parallelogram:
• Both pairs of opposite sides are parallel.
• Both pairs of opposite sides are equal.
• Both pairs of opposite angles are equal.

15. Every rectangle, rhombus and square is a parallelogram, so each has all the properties of a parallelogram.

16. Additional properties of a rectangle:
• All angles are right angles.
• The diagonals are equal.

17. Additional properties of a rhombus:
• All sides are equal.
• The diagonals intersect at right angles.

18. Additional properties of a square:
• All sides are equal.
• All angles are right angles.
• The diagonals are equal.
• The diagonals intersect at right angles.

19. A cuboid has 6 rectangular faces, 12 edges and 8 vertices.

20. A cube has 6 square faces, 12 edges and 8 vertices.

21. A triangular prismhas 5 faces (2 triangular, 3 rectangular), 9 edges and 6 vertices.

22. Atriangular pyramid has 4 faces (triangular), 6 edges and 4 vertices.

23. A rectangular pyramid has 5 faces (1 rectangular, 4 triangular), 8 edges and 5 vertices.

24. Sphere -A football is an example of a sphere. It has only one face which is curved.

25. Cylinder -A road roller is an example of a cylinder. It has two plane circular faces and one curved face.

26. Cone -A clown's cap is an example of a cone. It has one plane circular face, one curved face and one vertex.

ERRORANALYSIS

1. Students get confused when asked to write vertices and edges especially of 3-D shapes.

2. Students get confused when asked to find the direction on moving clockwise and anti-clockwise.

3. Sometimes students are unable to understand the question when asked to find where the hands of a clock stop when moved in a particular direction through some angle.

ACTIVITY I

To make the following shapes using a pair of set squares:
a. square b. rectangle c. parallelogram d. rhombus e. trapezium Objective: Handling set squares.
Pre-requisite knowledge : Knowledge of geometrical shapes.
Material required : Two pairs of set squares, pencil and paper.

Procedure

a. Making a square
To make a square shape take two 45° -45° -90° set squares and place them on a paper such that their longest edges touch each other completely. Move the pencil around the outer edges of the set squares to make the boundary of the square.

b. Making a rectangle
Arrange two 30°-60°-90° set squares as shown in the figure below and mark the boundary to form the rectangle.

c. Making a parallelogram
Take two 45°-45° -90° or 30°-60°-90°set squares and place them such that 90° corner of one set square touches the 45° corner of the other set square.Mark the boundary to form the parallelogram.

d. Making a rhombus
Take two 30°-60°-90° set squares and arrange them to make an equilateral triangle. Make another equilateral triangle opposite the first triangle and mark the boundary to form a rhombus.

e. Making a trapezium
Make a rectangle using two identical set squares.Now place one more set square of the same shape and size in contact with one side of the rectangle obtained and draw the boundary to form a trapezium. 

Observations
Number of set squares used in making
a square = ....................
a rhombus =.......................
a trapezium = .................
a parallelogram = ...............

ACTIVITY II

Do you think it is possible to sketch
a. a right angled equilateral triangle?
b. an obtuse angled equilateral triangle?
c. a triangle with two right angles?
d. a triangle with two obtuse angles?
Think and write your conclusions.

ACTIVITYIII

To make a cube using the given net and count the number of faces, vertices and edges

Learning objective: To understand the formation of a cube and identify its faces, vertices and edges.

Pre-requisite knowledge: Knowledge of three dimensional objects, its edges, vertices and faces.

Material required: A set of given nets, tracing paper, a thick pastel sheet, a pair of scissors and gluestick.

Procedure

Step 1 : Trace the given net on a tracing paper.
Step 2 : Copy the given figure on a thick pastel sheet using the tracing paper.
Step 3 : Cut along the regular line segments and fold along the dotted segments using the pair of scissors,
Step 4 : Paste the flaps to form a cube. (Use different colours for different faces) 

Observations
1. The shape formed is......................
2. The number of faces is....................
3. The number of vertices is..................
4. The number of edges is.....................
5. The shape of each side is ..................

ACTIVITY IV

Making a right angle (RA) tester
Steps
1. Take a piece of paper (as shown in the adjoining diagram).

2. Fold the paper somewhere in the middle and press the two parts together to form a straight crease AB (as shown in the adjoining diagram). 

3. Fold the paper again in such a way that B falls on A (as shown in the adjoining diagram) and press it to form a crease. 

Your right angle (RA) tester is ready.
See a page of your notebook carefully. Use your RA tester to check that every pair of its adjacent sides contains a right angle.

4. Your geometry box has a pair of set squares. Use RA tester to check that each set square contains a right angle and the other angles are less than a right angle.

ACTIVITY V

1. Using a protractor and a ruler find the measures of the angles and the lengths of the sides of the given triangles : 

2. Fill the measures in the given table. 

In each triangle, observe the measures of the angles and the lengths of the sides. Is there anything special about them?
We find that :
• If all the angles in a triangle are equal, then its sides are also equal
• If all the sides of a triangle are equal, then its angles are also equal
• If two angles of a triangle are equal, then it has two equal sides
• If two sides of a triangle are equal then its two angles are also equal
• If none of the angles of a triangle are equal then none of the sides are equal
• If none of the sides of a triangle are equal, then none of the angles are equal