# Mathematics Guides for SS 1 Geometry – Construction, Mensuration, Prove of Some Basic Theories and Trigonometry Ratio

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Mathematics Guides for SS 1 Geometry – Construction, Mensuration, Prove of Some Basic Theories and Trigonometry Ratio

**MATHEMATICS **

**THEME – GEOMETRY **

**TOPIC 1 – CONSTRUCTION **

**INSTRUCTIONAL MATERIALS**

1. Chalkboard

2. Mathematical set

3. Student’s mathematical set

**LEARNING OBJECTIVES **

By the end of the lesson, students should be able to:

1. Construct a triangle with given sides.

2. Bisect a given line segment.

3. Bisect a given angle.

4. Bisect special angles (30°, 45°, 60°, and 90°).

5. Construct equivalent angles.

6. Construct 4 sided plane figures given certain conditions.

7. Construct the locus of moving points equidistant from 2 points, 2 lines, and a fixed distance from a point, etc.

**CONTENTS OF THE LESSON**

**FOCUS LESSONS **

1. Revision of construction of triangles with given sides and bisection of special angles (30°, 45°, 60°, and 90°).

2. Construction,

- an angle equal to a given angle

- 4 sided plane figures given certain conditions

- locus of moving points equidistant from 2 points, 2 lines, and a fixed distance from a point, etc.

**LESSON PRESENTATION**

TEACHER’S ACTIVITIESTeacher,

1. Guides students to review the steps involved in constructing a triangle, bisecting a line segment, an angle and special angles.

2. Leads students to construct a given triangle and bisect a given line segment, an angle and some special angles.

3. Inspects the students’ construction activity and makes necessary corrections.

4. Guides students to list and explain the steps involved in constructing equivalent angles, 4 sided plane figures, and locus of moving points equidistant from 2 points, 2 lines, etc.

5. Demonstrates the construction of equivalent angles, 4 sided plane figures given certain conditions, and the locus of points equidistant from 2 points, 2 lines, etc on the chalk board using the chalkboard mathematical set.

6. Inspects the students construction activities, and makes necessary corrections.

4. Demonstrates again and explains the steps most students found difficult on the chalkboard and lays emphasis on essential steps.

STUDENT’S ACTIVITIESStudents,

1. Recall and list the steps involved in constructing a triangle with given sides, bisecting a line segment, an angle and special angles.

2. Construct the given triangle and bisect a given line segment, an angle and some special angles.

3. Label the constructions carried out correctly.

4. Attempt to list and explain the steps involved in the constructions.

Write the steps listed and explained by the teacher, and ask questions for clarification were necessary.

5. Follow the teacher’s demonstration on the chalkboard by carrying out similar activities in your exercise book using your mathematical set.

6. Participate in the teacher’s second demonstration and take note of salient steps.

**LESSON EVALUATION**

Students to,

1. List the steps involved in constructing of a triangle with given sides, bisecting a line segment, an angle and special angles.

2. Construct a given triangle and bisect a given line segment, an angle and some special angles.

3. Label the triangle, line segment, angle, and special angles.

4. Recall and explain the steps involved in constructing:

- equivalent angles

- 4 sided plane figures

- locus of moving points equidistant from 2 points, 2 lines, etc.
5. Construct,

- equivalent angles

- 4 sided plane figures

- locus of moving points equidistant from 2 points, 2 lines, etc.
6. Do assignments on the construction of,

- equivalent angles

- 4 sided plane figures

- locus of moving points equidistant from 2 points, 2 lines, etc.
7. Do assignments on the construction of,

- equivalent angles

- 4-sided plane figures and the locus of moving points.

**MATHEMATICS **

**THEME – GEOMETRY **

**TOPIC 2 – MEASURATION **

**INSTRUCTIONAL MATERIALS**

1. Pie demonstration board

2. Strings

3. Ruler

4. Cut out papers (sectors and segments)

5. Cardboard papers

6. Cut out shape of a circle containing a sector and a segment

7. Cut out shapes of the triangle and the segment

8. Cardboard

9. Rulers

10. Strings

11. Shapes of cube, cuboid, cylinder, prism and pyramid.

12. Models of fractions of cones and pyramids

13. Models of relevant compound shapes

**LEARNING OBJECTIVES **

By the end of the lesson, students should be able to:

1. Revise the components of a circle.

2. Find the lengths of arcs practically.

3. Find the length of arcs using formula.

4. Determine the perimeters of sectors of circles.

5. Determine the perimeters of segments of circles.

6. Find the area of a sector.

7. Determine the area of a segment.

8. Determine the relationship between the sector of a circle and the surface area of a cone.

9. Find the surface areas and volumes of cubes, cuboids, cylinders, cones, prisms, pyramids.

10. Find the surface area and volume of a frustum of a cone and pyramid.

11. Find the surface area and volume of compound shapes.

**CONTENTS OF THE LESSON**

**FOCUS LESSONS **

1. Lengths of arcs of circles.

2. Perimeters of sectors and segments.

3. Areas of sectors of circles.

4. Areas of segments of a circle.

5. Relationship between the sector of a circle and the surface area of a cone.

6. Surface areas and volumes of solids:

- cube

- cuboids

- cylinder

- cone

- prism

- pyramid
7. Surface area and volume of a frustum of a cone and pyramid.

8. Surface area and volume of compound shapes.

**LESSON PRESENTATION**

TEACHER’S ACTIVITIESTeacher,

1. Guides students to find the lengths of arcs of circles using a pie demonstration board.

2. Leads students to deduce the formula for finding the lengths of arcs.

3. Solve problems using formula.

4. Guides students to determine the perimeters of sectors of circles.

5. Asks students to cut out segments of circles.

6. Guides students to measure and determine their perimeters.

7. Gives exercises on perimeters of sectors and segments.

8. Guides students to cut a circle into a number of sectors of equal angles at the center, say 60°, 72° and 90°.

9. Leads students to measure the angles and compare the ratios.

10. Leads students to deduce the formula.

11. Guides students to find the area of the sector.

12. Leads students to find the area of triangle and subtract it from the area of the sector.

13. Gives exercises on areas of segments.

14. Guides students to cut out a sector and fold it into a cone.

15. Leads students to determine the relationship between the sector of a circle and the surface area of a cone.

16. Revises the areas and volumes of the cube, cuboids, cylinder and cone.

17. Leads students to find the surface areas and volumes of cube, cuboids, cylinder and cone.

18. Gives problems on the surface areas and volumes of cubes, cuboids, cylinders and cones.

19. Leads students to solve problems on the surface areas and volumes of fractions of a cone and pyramid.

20. Leads students to solve problems on surface areas and volumes of compound shapes.

STUDENT’S ACTIVITIESStudents,

1. Find the lengths of arcs of circles using a pie demonstration board.

2. Participate in deducing the formula.

3. Solve problem on lengths of arcs of circles.

4. Determine the perimeters of sectors of circles.

5. Cut out various segments.

6. Measure the cut out segments and determine their perimeters.

7. Solve exercises.

8. Cut a given circle into a number of sectors of equal angles.

9. Measure the angles and compare their ratios.

10. Deduce and note the formula.

11. Find the area of the sector.

12. Determine the area of triangle and subtract it from the area of the sector.

13. Cut out a sector and fold it into a cone.

14. Determine the relationship between the sector of a circle and the surface area of a cone.

15. Participate in the revision of areas and volumes of the cube, cuboids, cylinder and cone.

16. Solve problems on surface areas and volumes of fractions of a cone and pyramid.

17. Solve problem on surface areas and volumes of compound shapes.

**LESSON EVALUATION**

Students to,

1. Find lengths of given arcs of circles using the pie demonstration board.

2. Find the lengths of given arcs using the formula.

3. Determine the perimeters of given sectors.

4. Find the perimeters of given segments of circles by measurements.

5. Find the areas of given sectors.

6. State the relationship between the sector of a circle and the surface area of a cone.

7. Find the surface area of the cube, cuboids, cylinder and cone.

8. Solve problems on surface areas and volumes of fractions of a cone and pyramid.

9. Solve problem on surface areas and volumes of compound shapes.

**MATHEMATICS **

**THEME – GEOMETRY **

**TOPIC 3 – PROOFS OF SOME BASIC THEOREMS**

**INSTRUCTIONAL MATERIALS**

1. Cardborad paper.

2. Cut out triangles.

3. Protractor.

4. Models of parallelogram, parallel lines, congruent triangles, polygons, cut out papers, protractors

**LEARNING OBJECTIVES **

By the end of the lesson, students should be able to:

1. Write out formal proofs of some basic theorems in Euclidean geometry.

2. Apply the proofs in solving practical problems involving Euclidean geometry.

3. Apply the skill of deduction in proving riders on,

- angles of parallel lines

- angles in a polygon

- congruent triangles

- properties of parallelograms

- intercept theorem.
4. Solve problems on riders in Euclidean geometry.

**CONTENTS OF THE LESSON**

**FOCUS LESSONS **

1. Proofs of:

– the angle sum of a triangle is 180°

2. The exterior angle of a triangle is equal to the sum of two interior opposite angles.

3. Riders including –

- angles of parallel lines

- angles in a polygon

- congruent triangles

- properties of parallelograms

- intercept theorem.

**LESSON PRESENTATION**

TEACHER’S ACTIVITIES1. Leads students to explain the format for carrying out proofs in Euclidean geometry, i.e. explain the concepts:

- given

- required to prove

- construction

- proof

- Q.E.D.
2. Guides students to prove the two theorems one after the other on the chalkboard with necessary diagrams.

3. Assists students to carry out practical demonstrations of the proofs by using cut-out paper.

4. Leads students to solve examples and gives them some tasks to solve; and inspects them.

5. Demonstrates on the chalkboard how to prove the following:

- angles of parallel lines

- angles in a polygon

- congruent triangles

- properties of parallelograms

- intercept theorem through deductive reasoning and axioms using relevant models of plane shapes.
6. Leads students to demonstrate the properties of the riders using paper cut outs, protractors, models of the parallelogram, polygon, congruent triangles, etc.

7. Guides students to solve problems on the riders helping them reproduce arguments based on reasons (theorems and axioms).

STUDENT’S ACTIVITIESStudents,

1. Participate in the discussion of the format for proving Euclidean geometry; take notes and ask questions for clarity.

2. Solve the tasks given.

3. Participate in the teacher’s demonstrations by contributing in making deductions, and write down essential points agreed upon, i.e. essential points of:

- angles of parallel lines

- angles in a polygon

- congruent triangles

- properties of parallelograms

- intercept theorem.
4. Carry out practical demonstration of the properties of the riders along with the teacher using paper cut outs, models, and protractors.

5. Construct the models of plane shapes.

6. Apply deductive reasoning to solve the given practical problems on the riders, and call the teacher’s attention where necessary.

**LESSON EVALUATION**

Students to,

1. List the format for proving basic theorems in Euclidean geometry.

2. Prove that the angle sum of a triangle is 180°, and the exterior angle of a triangle is equal to the sum of the two opposite interior angles.,br>3. Demonstrate with cut out cardboard papers the proofs practically.

4. Do assignments on application of problems leading to the basic theorems in Euclidean geometry.

5. List and explain the properties of:

- angles of parallel lines

- angles in a polygon

- congruent triangles

- properties of parallelograms

- intercept theorem.
6. Demonstrate the properties of the riders practically using paper cut outs, models, and protractors.

7. Solve tasks on riders in Euclidean geometry.

**MATHEMATICS **

**THEME – GEOMETRY **

**TOPIC 4 – TRIGONOMETRIC RATIOS**

**INSTRUCTIONAL MATERIALS**

Charts showing trigonometric ratios of a right angled triangle.

2. Pencil and ruler

3. Protractor

4. Cut-out shapes of right-angled triangles showing angles 30°, 45° and 60° respectively

5. Chart showing unit circle.

6. Graph board

7. Broom stick/twine

9. Graph book

**LEARNING OBJECTIVES **

By the end of the lesson, students should be able to:

1. Solve problems involving the use of sine, cosine, and tangents in right angled triangles.

2. Derive trigonometric ratios of special angles 30°, 45° and 60°.

3. Apply the use of trigonmetric ratios of 30°, 45° and 60° to solve problems without the use of calculating aids.

4. Relate sine and cosine ratios to the unit circle.

5. Draw graphs of sine and cosine of angles.

**CONTENTS OF THE LESSON**

**FOCUS LESSONS **

Basic Trigonometric ratios:

- Sine

- Cosine

- Tangent with respect to right angled triangles.
2. Trigonometric ratios of,

- angle 30°

- angle 45°

- ang60°
3. Application of trigonometric ratios of special angles to simple problems.

4. Trigonometric ratios related to the unit circle.

5. Graphs of sine and cosine of angles.

**LESSON PRESENTATION**

TEACHER’S ACTIVITIES1. Shows students a chart of a right angled triangle with a clearly marked angle.

2. Guides students to identify ratios forming sine, cosine, and tangent of the marked angle.

3. Guides students to use sine, cosine, and tangents to solve problems involving calculations of lengths, angles of elevation, angles of depression, etc.

4. Leads students to construct right angled triangles of 30°, 45° and 60°.

5. Guides students to use the above shapes to derive trigonometric ratios of 30°, 45° and 60°.

6. Leads students to draw right-angled isosceles triangle of side I unit on equal. sides.

7. Guides students to derive trigonometric ratios of 45° from the triangle.

8. Leads students to draw an equilateral triangle of side 2 units.

9. Guides students on how to derive trigonometric ratios of 30°, 45°, and 60° from the equilateral triangle.

10. Guides students to draw diagrams related to trigonometric ratios involving angles of elevation, depression, and problems relating to school subjects like physics, geography, etc.

11. Leads students to solve problems on the above.1. Displays unit circle charts, also containing right-angled triangles.

12. Asks students to draw the unit circles.

13. Leads students to draw right-angled triangles inside the unit circles.

14. Guides students to measure the opposite, adjacent and hypotenuse of the triangle.

15. Leads students to measure the two other angles in the right-angled triangle.

16. Leads them to obtain sines and cosines of various angles using the measured lengths, e.g. sine = opposite/hypotenuse , etc.

17. Guides them to see the relationship between the calculated sine and cosine of trigonometric ratios and the angles measured with a protractor in the unit circle.

18. Guides students to construct tables of values for sines and cosines using intervals of 30°.

19. Guides learners to draw graphs of sines and cosines.

20. Leads them to observe that values of sine and cosine lie between +1 and -1.

STUDENT’S ACTIVITIES1. Study the chart.

2. Identify ratios forming sine, cosine, and tangents of marked angles on the chart.

3. Draw right angled triangles and use it to solve problems involving calculations of lengths, angles of elevation, angles of depression, etc.

4. Construction of right-angled triangles of 30°, 45° and 60°.

5. Derive trigonometric ratios of 30°, 45° and 60° under the teachers supervision.

6. Draw right-angled triangles.

7. Derive the trigonometric ratio of 45° using the drawn diagram under the teacher’s supervision.

8. Draw the equilateral triangle of side 2 units.

9. Derive the trigonometric ratios of 30°, 45° and 60° under the teacher’s supervision.

10. Draw diagrams involving elevation and depression and other related areas using a ruler and pencil.

11. Solve problems on the practical application of trigonmetric ratios under the guidiance of the teacher.

12. Study the unit circle chart.

13. Draw the unit circle.

14. Draw a right-angled triangle inside the unit circle.

15. Measure the two other angles in the right angled-triangle.

16. Obtain sine and cosine of various angles.

17. Identify the relationship between the trigonometric ratios and the measured values.

18. Construct tables of values with intervals of 30°.

19. Draw graphs of sines and cosines using broomstick or twine.

20. Identify the values of sine and cosine using a broomstick or twine, and identify that lie between +1 and -1.

**LESSON EVALUATION**

Students to,

1. Construct right angled triangles.

2. Identify trigonometric ratios in such triangles.

3. Use trigonometric ratios to calculate lengths, angles of elevation, angles of depression, etc.

4. Construct shapes of right-angled triangle of 30°, 45° and 60°.

5. Use shapes to derive trigonometric ratios of 30°, 45° and 60°.

6. Draw diagrams involving angles of elevation depression and other topics relevant to trigonometric ratios.

7. Solve practical problems on the use of trigonometric ratios.

8. Draw unit circle.

9. Draw right angled triangle.

10. Measure angles and sides.

11. Calculate ratios.

12. Identify relationship between trigonometric ratios and unit circle.

13. Construct tables of value for sine and cosine using 15°, 20°, 60°, etc.

14. Draw graphs of the constructed tables.

15. Identifies that the values of sine and cosine lie between +1 and -1.