Chapter 1; Geometry of shape and size
CHAPTER 1
LESSON 1.1 UNDEFINED TERMS
I. Mathematical concepts and skills
A. The undefined terms in geometry are point, line and plane.
B. A point has no length, width, or thickness.
C. A line has length, but no width and no thickness
D. A plane has infinite width and length but no thickness.
E. A good definition has the following characteristics:
1. It should contain ordinary words and geometric terms that have been previously defined or accepted as undefined.
2. A good definition should list only the essential properties of the term being defined.
3. A good term is reversible.
F. Segment AB
G. Point P is between A and B if and only if A, P, and B are distinct points of the same line AP + PB = AB
H. Collinear points are points on the same line.
I. A ray starts at one point of a line and goes on infinitely in one direction.
J. A postulate is a statement, which is accepted as true without proof.
K. A theorem is a statement that needs to be proved.
L. Two points exactly one line.
M. Two distinct lines intersect in only one point.
N. Three points are contained in at least one plane and three non-collinear points are contained in exactly one plane.
O. If two distinct planes intersect, then their intersection is a line.
P. If two points of a line are in a plane, then the line is in the plane.
Q. If a line contained in a plane intersects the plane, then the intersection contains only one point.
R. Exactly one plane contains a given line and a point not on the line.
S. Exactly one plane contains two intersecting lines.
II. Objectives
A. To name and describe the ideas of a point, line, and plane
B. To state the characteristics of a good definition
C. To define, identify, and name the subsets of a line
D. To states the postulates related to points, lines, and planes
E. To prove in paragraph from the theorems related to points, line, and planes
III. Values Integration
The importance of cooperation in doing a given task
IV. Materials
Manila paper
Markers
Rulers
V. Instructional Strategies
A. Whole class discussion
B. Small group discussion
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the attendance.
2. Discuss briefly the history of geometry.
B. Lesson Paper
1. Ask the students to give examples of symbols and the ideas of symbols and the ideas they represent.
Examples: numeral-number
heart-love
2. Introduce the three undefined terms of geometry. Write the three terms on a manila paper, and above each term draw the corresponding symbols.
3. Tell the students to form small groups. Tell each group to give as many real objects as possible that suggest points, lines, and planes. Tell them the importance of cooperation in doing a given task.
4. Tell the students to describe a point, a line, and a plane.
5. Answer the Mental Mathematics of Exercise 1.1.
6. Illustrate how to name a point, a line, and a plane.
7. Discuss the three characteristics of a good definition
8. Guide the students in defining the terms between, segment, collinear points, and ray.
C. Assignment
A. Define the following terms:
1. Postulate
2. Theorem
B. Write on your notebooks the postulates and theorems related to points, lines, and planes. Refer to the textbook pages 6-8
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Ask the students to enumerate the characteristics of a good definition.
B. Follow-up Lesson
1. Tell the students that geometry, just like any other mathematical system, is based not only on undefined terms but also postulates and theorems.
2. Distinguish a postulate from a theorem.
3. Discuss the postulates related to points, lines, and planes.
4. Discuss the proof of Theorem 1-1, Theorem 1-2, Theorem 1-3, and Theorem 1-4.
5. Ask the students to summarize what they have learned on their mathematical journals.
6. Practice exercises
a. Answer the Mental Mathematics of Exercise 1.1 (numbers 1-10) on pages 10-11 of the textbook.
b. Solve the Written Mathematics of Exercise 1.1 (numbers 1-10) on page 11 of the textbook.
C. Checking for Understanding
Solve the Written Mathematics of Exercise 1.1 (numbers 11-12) on page 12 of the textbook.
D. Assignment
1. Solve the Challenge Problems 1 and 2 on page 12 of the textbook.
2. Bring wires, ruler, and protractor.
Lesson 1.2 ANGLES
I. Mathematical concepts and skills
A. An angle is the union of two non-collinear rays with a common endpoint.
B. A. angle in a plane separates the plane into three sets of points: the points in the interior of the angle, the points in the exterior of the angle, and the points on the angle itself.
C. To every angle there corresponds a unique real number r where 0<r<180
D. In a half-plane H, through the endpoint of a ray BC lying in the edge of the half-plane, there is exactly one other ray BA, with A in H, such that an angle ABC formed by two rays has a given measure between 0° and 180°.
E. If D is in the interior of
∠ABC, then m∠ABC = m∠ABD + m∠CBD .
F. Two angles are congruent if and only if their measures are equal.
G. Angles can be classified according to their measures.
1. Acute angle - greater than 0° but less than 90°
2. Right angle - equal to 90°
3. Obtuse angle - - greater than 90° but less than 180°
II. Objectives
A. To define, illustrate, and name an angle
B. To identify an angle and its parts
C. To determine the measure of an angle using a protractor
D. To define and identify the different kinds of angles: acute, right, and obtuse
III. Values Integration
The importance of a joint effort in doing a task
IV. Materials
Wires
Marker
Protractor
Ruler
V. Instructional Strategies
A. Practical work
B. Small group discussion
VI. Procedure
A. Preliminary activities
1. Check the Assignment
2. Recall the definition of ray.
Session 1
B. Lesson Paper
1. Arrange the students into small groups. Have them sit around a table or joined chairs. Make them realize the importance of joint effort in doing a task. Have the wires they bought be cut to 35 cm. tell them to bend the endpoints of each wire to indicate that the wire has no endpoint.
2. Tell the students to mark a point between the two arrows using a marker. Ask: “ into how many rays does the point divide the line?”
3. Tell the students to form angles by bending their wires.
4. Ask the following questions
a. How many rays are there in each angle?
b. Do the two rays have a common endpoint?
c. Are the two rays collinear
5. Draw the following figures on the board:
6. Ask the students which of the following figures are angles. Then ask them to define an angle on their own words.
7. Tell the students to draw three angles on their notebooks and ask them to label the angles’ parts
8. Illustrative Examples
Give illustrative examples on how to name angles
9. Ask the students to use their protractors in measuring the angles they drew on their notebook. Move around the room and check their work.
10. Practice Exercises
a. Answer the Mental Mathematics of Exercise 1.2 (numbers 1-10) on pages 19-20 of the textbook.
b. Solve the Written Mathematics of Exercise 1.2 (numbers 1-3) on page 20 of the textbook.
C. Assignment
Define the congruent angles, right angles, acute angles, and obtuse angles. Explain why your answers contain the characteristics of a good definition on page 18 of your textbook.
Session 2
A. Preliminary Activities
3. Check the assignment.
4. Recall the definition of angle.
B. Follow-up Lesson
1. Have the students copy the two coplanar angles with a common vertex and a common side but no interior points in common as shown below .
2. Ask the students to measure ∠ABD, ∠CBD , and ∠ABC. Ask them what they notice on the measures of the angles.
3. Illustrate by means of sticks that if BD is not coplanar with BA and BC, then m∠ABC≠m∠ABD + m∠CBD . Illu8strate congruent angles by using a protractor.
4. Practice exercises
Solve the Written Mathematics of Exercise 1.2 (4-8) on page 21 of the textbook.
C. Checking for Understanding
Solve the Written Mathematics of Exercise 1.2 (numbers 10, 12, 14, and 16) on page 22 of the textbook.
D. Assignment
Solve the Written Mathematics of Exercise 1.2 (numbers 11, 13, 15, and 17-19) on pages 22-23 of the textbook.
If there is enough time, solve the Challenge problems on page 23 of the textbook
LESSON 1.3 POLYGONS
I. Mathematical concepts and skills
A. A polygon is the union of three or pore coplanar segments, which intersect only a endpoints, with each endpoint shared by only two non-collinear segments.
B. Polygons are classified in the number of their sides.
C. A polygon is a convex if and only if the lines containing the sides of the polygon do not contain points in its interior.
D. A polygon is non convex if and only if at least one of its sides is contained in a line, which contains also points of the interior of the polygon.
The number of diagonals that can be drawn of n sides is given by the formula: Nd= n(n-3).
2
E. A regular polygon is a polygon that is both equilateral and equiangular.
F. A triangle is a figure formed by three segments whose joining three non-collinear points.
G. An angle bisector of a triangle is a segment contained in a ray that bisects the angle of the triangle, and whose endpoints are the vertex if this angle and a point on the opposite side.
H. A segment is an altitude of a triangle if and only if it is perpendicular from the vertex to the line that contains the opposite side.
I. A segment is a median of a triangle if and only if its endpoints are the vertex and the midpoint of the opposite side.
J. Triangle can be classified according to the measure of their angles and according to the lengths of their sides.
II. Objectives
A. To define a polygon
B. To define and identify the different kinds of polygon
C. To illustrate and identify convex and non convex polygons
D. To identify the parts of the regular polygon
E. To determine the number of diagonals that can be drawn in a polygon of n side
F. To identify, define, and name a triangle
G. To define the basic and secondary parts of a triangle (e.g. vertices, sides, medians, angle bisector, and altitude)
H. To classify triangles according to the measure of their angles
I. To classify triangles according to the lengths of their sides
III. Values Integration
The importance of carefulness
IV. Materials
Wires
Picture of structural designs
V. Instructional Strategies
A. Group work
B. Discussion
VI. Procedure
Session 1
D. Preliminary activities
1. Check the Assignment
2. Recall the definition of a segment.
E. Lesson Paper
1. Arrange the students into small groups. Ask them to make three different closed figures using wires with a length of 35 cm. each figure should not contain a curve. In performing the activity, it is most important that they must exercise utmost care.
2. Tell the students that the figures they formed are called polygons. Explain the root word of polygon.
3. Present some figures on the board. Ask them which of the figures are polygons and which are not.
4. Direct the students’ attention to their polygons that are made of wires. Have them count the number of sides of their polygons.
5. Introduce the terms triangle, quadrilateral, pentagon, etc.
6. Ask the students to define their own words the different kinds of polygons
7. Draw a polygon on the board. Ask the students to draw a point in its interior, and also on the polygon itself.
8. Move around and inspect the students’ polygons made of wires. Chose one convex and one non convex.
9. Draw the two chosen polygons on the board. Ask one student to draw the lines containing the sides of the convex polygon.
10. Ask another student to draw the lines containing the sides of the non convex polygon.
11. Based on the drawing on the board, ask the students to define a convex and a non convex polygon.
12. Direct the students to read the subtopic on the Parts of a Regular Polygon on pages 28-30. Ask them to define a regular polygon and guide them in identifying and defining its parts.
13. Illustrative Examples
Guide the students in determining the number of diagonals of a polygon. Use the example on page 25 of the textbook.
14. Practice Exercises
Answer the Mental Mathematics of Exercise 1.3 (numbers 1-4) on page 36 of the textbook.
Session 2
A. Preliminary Activities
5. Check the assignment.
6. Recall the definition of a triangle.
B. Follow-up Lesson
1. Tell the students that polygons are used in structural designs. Show them pictures of structures and then ask them to name all the types of polygons in the pictures.
2. Draw three triangles on the board, one acute, one right, and one obtuse. Then draw bisectors of their angles using a protractor.
3. Tell the students to be careful in bisecting the angles.
4. Guide the students in defining angle bisectors.
5. Draw another set of triangles. This time ask the students to draw the medians or each triangle.
6. Ask how many medians are there in every triangle. Tell the students to define median based on the illustrations on the board.
7. Draw another set of triangles. This time draw the altitudes of each triangle. Ask the students to identify the altitudes of each triangle. Then ask them to define the altitudes in their own words.
8. Guide the students in defining a scalene triangle, an isosceles, and an equilateral triangle.
9. Tell the students to make a summary of the lesson.
C. Checking for Understanding
Identify the geometric figures described by each statement.
1. A triangle with no congruent side.
2. A triangle in which all angles are acute.
3. A triangle with at least two congruent sides.
4. A triangle in which one of the angles are acute.
5. A triangle with all sides congruent.
6. A triangle in which all angles are congruent
7. A segment perpendicular from a vertex of a triangle to the line containing the opposite side.
8. A polygon that is both equilateral and equiangular.
9. A polygon in which at least one of the sides of a polygon is contained in a line which contains also in the interior of the polygon
10. A polygon with ten sides.
D. Assignment
Solve the Written Mathematics of Exercise 1.3 (numbers 5-8) on page 36 of the textbook.
LESSON 1.4 QUADRELATERALS
I. Mathematical concepts and skills
A. A quadrilateral is a four sided polygon.
B. A parallelogram is a quadrilateral with two pairs of parallel sides.
C. A rectangle is a parallelogram with four right angles.
D. A square is a rectangle with four congruent sides.
E. A rhombus is a parallelogram with four congruent sides.
F. A trapezoid is a quadrilateral with exactly one pair of parallel sides.
G. The sum of the measures of the angles of a triangle is 180°.
H. The sum of the measure of the exterior angles of a convex quadrilateral, one at each vertex, is 360°.
I. The sum of the measures of the interior angles of a convex polygon with n sides is (n-2)180.
The sum of the measures of the interior angles of a regular polygon with n sides is equal to (n-2)180.
The measure of each interior angle of a regular polygon with n sides is (n-2)180.
N
The measure of each exterior angle of a regular polygon with n sides is 360.
n
II. Objectives
A. To illustrate, name, and define a quadrilateral and its parts
B. To illustrate, name, and define, the different kinds of quadrilaterals
C. To find the sum of the angles of a triangle
D. To find the sum of the measures of the interior angles of a convex polygon with n sides.
E. To find the sum of the measures of the exterior angles of a convex quadrilateral, one at each vertex
F. To find the measure of each interior angle of a regular polygon with n sides
G. To find the measure of each exterior angle of a regular polygon of n sides
VII. Values Integration
The importance of being diligent
VIII. Materials
Boxes
Sticks
IX. Instructional Strategies
C. Practical work in small groups
D. Discussion
X. Procedure
Session 1
A. Preliminary activities
3. Check the Assignment
4. Recall the definition of quadrilateral.
B. Lesson Paper
1. Play the game “Meeting the Deadline.”
i. Divide the students into two groups. Give each group a box containing 20 sticks: four 10 cm, thirteen 6 cm, one 8cm, one 7 cm, and one 12 cm in length.
ii. Using the 20 sticks, have each group construct quadrilaterals with the following properties:
· Quadrilateral with two pairs of parallel sides
· Quadrilateral with four right angles
· Quadrilateral with four congruent sides
· Quadrilateral with exactly one pair of parallel sides
The group that could be able to construct the quadrilaterals ahead of the other groups wins the game.
2. Have the students copy or draw the quadrilaterals on their notebooks. Tell the students to exercise diligence.
3. Introduce the terms parallelogram, rectangle, rhombus, and trapezoid
4. Using the same sticks, ask the students to construct a rhombus with congruent angles. Tell the students that the resulting figure is a special kind of rhombus called square.
5. Ask the students to define square in their own words.
sections to form a figure that resembles a parallelogram. Refer to the textbook.
2. Illustrative Examples
Use the examples on page 66 of the textbook.
3. Practice Exercises
Solve Written Mathematics of Exercise 1.7 (numbers 21 and 22) on page 71 of the facebook.
C. Checking for Understanding
1. Find the area of the parallelogram
a. base = 10 cm, altitude= 6 cm
b. base= 14cm, altitude= 10 cm
2. Find the value of x
Area = 60cm2
3. Find the area of the shaded region.
ABCD is a square
AB = 8 cm
Use pi = 3.14
D. Assignment
Solve Written Mathematics of Exercise 1.7 (numbers 23 and 14) on page 71 of the textboo.
If there is enough time, solve the Challenge Problems on page 71 of the textbook.
Lesson 18
Surface Areas
I. Mathematical Concepts and Skills
A. The surface area of a cube is the sum of the areas of its bases and the faces.
B. The surface area of a rectangular prism is the sum of its lateral area and the area of its bases.
C. The surface area of square pyramid is the sum of the lateral area and the areas of the base.
D. The surface area of a cylinder is SA= 2pi2 + 2piRH
E. The total surface area of a cone is equal to pi times the radius squared plus pi times the radius times the slant height
F. The surface area of a sphere is four times the area of a circle with the same radius.
II. Objective
To fine the surface area of a cube, rectangular prism, square pyramid, cylinder cone and sphere
III. Values Integration
Being prudent to avoid errors
IV. Material
Compass
Models of different prism
Tape measure
V. Instructional Strategies
A. Discussion
B. Practical work
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment.
2. Recall the formula for finding the area of a square, rectangle, and triangle.
B. Lesson Proper
1. Bring models of a cube and a rectangular prism. Divide the class into four groups. Give the first two groups a models of a cube and the other two groups a model of a rectangular prism.
a. Cube(make one using cardboard)
b. Rectangular Prism (make a box using cardboar)
2. To avoid errors in this activity, tell the students to be prudent. Tell them that the area that covers all the surfaces of a solid is called surface area.
3. Tell the groups to find the surface area of their models. They may open up their models if they want to.
4. When “opened up”, the cube looks like this. Tell the group to apply the Area Addition Postulate.
5. When “opened up”, the rectangular prism looks like this. Tell the group to apply the Area Addition Postulate
6. Illustrative Examples
Use the examples on pages 72-77 of the textbook.
7. Practice Exercises
a. Answer Mental Mathematics of Exercise 1.8(numbers 1-10) on page 77 of the textbook.
b. Solve Written Mathematics of Exercise 1.8 (numbers 1 and 2) on page 78 of your textbook.
C. Assignment
Solve for Written Mathematics of Exercise 1.8 (numbers 3 and 4) on page 78 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment
2. Divide the class into two groups. Tell the first group to formulate about areas of triangles. The second group will be asked to solve problems. Then tell the second group to formulate problems related to areas of squares. The first group will be asked to solve the problems.
B. Follow-up Lesson
1. Bring models of a square pyramid made from cardboard
2. Tell the students to find the surface areas of their models by opening them up.
3. When “opened up”, the square pyramid looks like this. Tell the group to apply the Area Addition Postulate.
4. Practice Exercises
Solve Written Mathematics of Exercise 1.8(numbers 5 and 6) on page 78 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 1.8 (numbers 7 and 8) on page 78 of the textbook.
Session 3
A. Preliminary Activities
1. Check the assignment
2. Recall the area formula of a circle.
B. Follow-up Lesson
1. Bring models of a cylinder made from cardboard.
2. Tell the students to find the surface area of their models by opening them up
3. When “opened up”, the cylinder looks like this. Tell the group to apply the Area Addition Postulate. Tell them that they need a compass to determine the center and the radius of each circular base.
4. Illustrative Examples
Use the examples on pages 75-76 of the textbook.
5. Practice Exercises
Solve Written Mathematics of Exercise 1.8 (number 9 and 10) on page 79 of the Textbook
C. Assignment
Solve Written Mathematics of Exercise 18 (numbers 11 and 12 ) on page79 of the textbook.
Session 4
A. Preliminary Activities
1. Check the assignment
2. Review the lesson about areas of circles
B. Follow up Lesson
1. Bring models of a cone made of cardboard.
2. Ask the students to open up their model of a cone.
3. When “opened up”, the cone looks like a piece of pie.
s
2 pi r
4. The pie-like cardboard can be divided into smaller pie-shaped pieces. Tell the group to cut these pieces and rearranged to look like a parallelogram. Then, ask the group what area of a curved surface is. Tell the students to add this to the area of the Circular base.
5. Practice Exercises
Solve Written Mathematics of Exercise 1.8 (numbers 13 and 14) on page 79 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 1.8 (numbers 15 and 16) on page 79 of the textbook.
Session 5
A. Preliminary Activities.
1. Check the Assignment.
2. Tell the students who Archimedes was and tell them what he said about the area of a sphere.
B. Follow-up Lesson
1. Tell the students to use a ball in the next activity.
2. Tell the group to find the circumference of the (ball) sphere’s greatest circle by using a tape measure.
3. The group can then calculate the radius of the ball by using the formula for finding the circumference of a circle.
C= 2 pi r
Therefore, r = C / 2 pi use= 3.14. The surface area of a sphere = 4 times the area of the greatest side SA= 4(pi r2 )
C. Practice Exercises
Solve Written Mathematics of Exercise 1.8 (numbers 17 and 18) on page 80 of the textbook.
D. Checking for Understanding
E. Assignment
Solve Written Mathematics of Exercise 1.8 (numbers 19 and 20) on page 80 of the textbook.
If there is enough time, solve the Challenge Problems on page 80 of the textbook.
Lesson 19
Volumes
I. Mathematical Concepts and skills
A. The volume (V) of a rectangular prism is the area of the base (B) times the height(h) (V= Bh, where B = lw)
B. The volume (V) of triangular prism is the area of the base (B) times the height (h) (V= Bh, where B= area of the triangular base).
C. The volume of any pyramid is equal to 1/3 the area of the base times the height (v = 1/3Bh, where B = area of the base)
D. The volume (V) of a cylinder is the area of the base (B) times the height (h)
(V= Bh, where B= pi r 2)
E. The volume (V) of a cone equals one-third the area of the base (B) times the height (h) (V= 1/3Bh, where B= pi r 2)
F. The volume of sphere is equal to of 2/ 3 of the volume of a circumscribed cylinder.
II. Objectives
To find the volume of a rectangular prism, a triangular prism, a pyramid, a cylinder, a cone, and a sphere.
III. Values Integration
Realize the importance of having a good reputation
IV. Materials
Manila paper
Ruler
V. Instructional Strategy
Whole Class Discussion
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment
2. Recall the definition of surface area
B. Lesson Proper
1. Explain the meaning of volume. Tell the students that the volume of a solid is the number of cubic units contained in the solid. Draw this figure on the board.
2. Ask the following questions:
A. How many layers of unit cubes are in the figure?
B. How many cubes are in the first layer?
C. How many cubes are in the second layer?
D. How many cubes are in two layers?
E. What is the volume of the solid figure?
3. Change the length of the base to 3cm. Ask the same questions then tell the students to write a formula finding the volume of a rectangular prism.
4. Show the following illustrations:
Ask the students how they can find the area of the base of the rectangular Prism
5. Practice Exercises
a. Answer Mental Mathematics of Exercise 1.9 (numbers 1- 14) on pages 87-88 of the textbook.
b. Solve Written Mathematics of Exercise 1.9 (numbers 1-14) on page 88 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 1.9 (numbers 5-10) on page 89 of the textbook.
Session 2
A. Preliminary Activity
1. Check the assignment.
2. Recall the formula for finding the area of a circle.
B. Follow-up Lesson
1. Direct the students top the subtopic Volume of a Cylinder and Volume of a Cone on page 84 of the textbook.
2. Illustrative Examples
Use the sample on pages 84-85 of the textbook.
3. Practice Exercises
Solve Written Mathematics of Exercise 1.9 (numbers 11 and 12) on page 89 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 1.9(numbers 13-16) on page 89 of the textbook.
Session 3
A. Preliminary Activity
1. Check the assignment
2. Recall the formula for finding the volume of a cylinder.
B. Follow-up Lesson
1. Direct the students to the subtopic Volume of a Sphere. Ask what Archimedes said about the relationship between the volume of a sphere and the volume of a circumscribed cylinder.
2. Illustrative Examples
Use the example on page 86 of the textbook.
3. Practice Exercises
Solve Written Mathematics of Exercise 1.9(numbers 15 and 16) on page 89 of the textbook.
C. Checking for Understanding
Solve Written Mathematics of Exercise 1.9(numbers 19 and 20) on page 89 of the textbook.
D. Assignment
Solve Written mathematics of Exercise 1.9(numbers 19 and 20) on page 89 of the textbook.
If there is enough time, solve the Challenge Problems on page 90 of the textbook.
6. Guide the students in identifying a quadrilateral and in determining its parts.
7. Direct the students to work on the Explore on page 38 of the textbook. This time tell them to work in pairs.
8. Illustrative Examples
Use the examples on page 41-45 of the textbook.
9. Practice Exercises
a. Answer Mental Mathematics of Exercise 1.4 (numbers 1-10) on page 45 of the textbook.
b. Solve Written Mathematics of Exercise 1.4 (numbers 1-4) on page 45 pf the text book.
C. Assignment
Solve Written Mathematics of Exercise 1.4 (numbers 5-10) on page 46 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Recall the definition of a regular polygon.
B. Follow-up Lesson
1. Investigate Corollary 1-7.1
a. Draw an equilateral triangle.
b. Find the sum of the measures of its angles using the formula Sa = (n-2)180.
c. Find the measure of each angle using a protractor.
d. What did you notice?
e. Draw a square.
f. Find the sum of the measures of its angles using the formula Sa = (n-2)180.
g. Find the measure of each angle using a protractor.
h. What did you notice?
2. Investigate Corollary 1-7.2
a. Draw an equilateral triangle and extend one of its sides.
Using a protractor, find m<CBD.
Divide 360 by 3.
What do you notice?
b. Draw a square and extend one of its sides.
Using a protractor, find <DAE.
Divide 360 by 4.
What do you notice?
c. Draw a regular pentagon and extend one of its sides.
Using a protractor, find <CDF.
Divide 360 by 5.
What do you notice?
3. Illustrate how to apply Corollary 1-7.1, Corollary 1-7.2,and Corollary 1-7.3 b explaining the examples on pages 41-45 of the textbook.
4. Tell the students to write a summary of what they have learned about polygons on their math journals.
5. Practice Exercises
Solve Written Mathematics of Exercises 1.4 (numbers 11-16) on page 47 of the textbook.
C. Checking for Understanding
Solve Written Mathematics of Exercise 1.4 (numbers 17-20) on page 47 of the textbook.
D. Assignment
Solve Written Mathematics of Exercise 1.4 (numbers 21-24) on page 47 of the textbook.
If there is enough time, solve the Challenge Problems on page 47 of the textbook.
LESSON 1.5
MEASUREMENTS
(Textbook pages 48-56, 2 sessions)
I. Mathematical Concept and Skills
A. A circle is the set of all points in a plane with a given distance from a given point in the plane.
B. A radius of a circle is a segment from the center of the circle to a point on the circle.
C. A chord is a segment whose endpoints lie on the circle.
D. A diameter is a chord that contains the center of the circle.
E. A cylinder is a space figure with tow circular bases that are congruent and parallel.
F. A cone is a space figure with one circular base and a vertex.
G. A sphere is a set of points in space that are of the same distance from a given point called the center.
H. A rectangular prism is a polyhedron with two rectangular bases that are congruent and parallel.
I. A triangular prism is a polyhedron with two triangular bases that are congruent and parallel.
J. A rectangular pyramid is a polyhedron with one rectangular base and four triangular faces
K. A triangular pyramid is a polyhedron with one triangular base and three triangular faces.
L. The perimeter of polygon is the distance around it.
M. The circumference of a circle is 2πr, where the r = radius.
II. Objectives
A. To define a circle and to be able to identify and define the terms related to it.
B. To identify the common solids and their parts.
C. To find the perimeter of a triangle, square, rectangle, and parallelogram.
D. To find the circumference of a circle.
III. Values Integration
The need to be independent
IV. Materials
Manila paper
Tape measure
Compass
Ruler
V. Instructional Strategies
A. Discussion
B. Individualized Instruction
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment.
2. Recall the definition of triangle, square, rectangle, and parallelogram.
B. Lesson Proper
1. Show the drawings of the following figures.
· Cylinder
· Cone
· Rectangular prism
· Triangular prism
· Rectangular pyramid
· Triangular pyramid
· Sphere
2. Ask the students to compare the figures with polygons. Tell them to do this individually. Tell them that it is not always good to be always dependent to other people.
3. Label the figures and guide the students in defining them.
4. Draw he following figures on manila paper.
Post the Manila paper on the board.
5. Guide the students in determining the formula for finding the perimeter of triangle, square, rectangle, and parallelogram.
6. Practice Exercises
a. Answer Mental Mathematics of Exercise 1.5 (numbers 1-4) on page 54 of the textbook.
b. Solve Written Mathematics of Exercise 1.5 (number 1-6_ on page 55 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 1.5 (numbers 7-10) of the textbook on page 56.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Recall the irrational numbers.
B. 1. In the follow-up activity, divide the students into group of five. Give each group a tape measure, a compass, and a ruler. Then, tell the students to do the following activities:
· Draw several circles.
· Measure the circumference of each circle.
· Measure the diameter of each circle.
· Divide the circumference by the length of its diameter.
2. Tell them briefly about π.
3. Illustrative Examples
Use the examples on page 53 of the textbook.
4. Practice Exercises
Solve Written Mathematics of Exercise 1.5 (numbers 11-14) on page 56 of the textbook.
C. Checking for Understanding
1. Find the perimeter of each of the following:
a. a square 10.5 cm on one side
b. a rhombus 3.5 cm on one side
c. a parallelogram with sides of 10.5 cm and 8.5 cm
2. A rectangular garden is 5 meters long and 26 meters wide. Find the perimeter of the garden.
3. Find the circumference of a circle with a radius of 12 cm long.
D. Assignment
Solve Written Mathematics of Exercise 1.5 (numbers 7-10) on page 56 of the textbook.
If there is enough time, solve the Challenge Problems on page 56 of the textbook.
LESSON 1.6
Areas of Squares and Rectangles
(Textbook pages 57-62, 1 session)
I. Mathematical Concept and Skills
A. Every polygonal region has an area, which is a unique positive real number.
B. The area (A) of a square is the square of the length of its sides.
C. The area (A) of a rectangle is the product of its base (b) by its height (h).
D. If two polygonal regions do not overlap, then the area of their union is equal to the sum of their individual areas.
II. Objectives
A. To find the area of a square and a rectangle
B. To apply the Area Addition Postulate in computations
III. Values Integration
Being friendly
IV. Materials
graphing paper
manila paper
V. Instructional Strategy
Discussion
VI. Procedure
Session 1
1. Check the assignment.
2. Recall the definition of square and rectangle.
3. Provide exercises on factoring quadratic trinomials and quadratic equations.
B. Lesson Proper
1. Illustrate the basic concept of area. Use the following figures.
From the illustrations, the students should see that every polygon encloses a surface. And to every surface bounded by a polygon there corresponds a unique positive number called area.
2. Explain further the Area Postulate. Use the following illustrations.
3. Help the students discover the Area Addition Postulate.
22 + 22 =2(4) |
4. Illustrative Examples
Use the examples on pages 58-59 of the textbook.
5. Practice Exercises
a. Answer Mental Mathematics of Exercise 1.6 (numbers 1-4) on page 60 of the textbook.
b. Solve Written Mathematics of Exercise 1.6 (numbers 1-6) on page 61 of the textbook.
C. Checking for Understanding
Solve Written Mathematics of Exercise 1.6 (number 7-10) on page 62 of the textbook.
D. Assignment
Solve Written Mathematics of Exercise 1.6 (numbers (11-12) on page 62 of the textbook.
If there is enough time, solve the Challenge Problems on page 62 of the textbook.
LESSON 1.7
Areas of Parallelograms, Triangles, Trapezoids, and Circles
(Textbook pages 63-71, 4 sessions)
I. Mathematical Concept and Skills
A. The area (A) of parallelogram is equal to the product of the length of its base (b) and the length of the corresponding altitude (h).
B. The area (A) of a triangle is equal to one-half the product of the length of any base (b) and altitude (h).
C. The area (A) of a trapezoid is equal to one-half the product of the length of its altitude and the sum of the lengths of its bases.
D. The area of a circle = πr2 , where r = radius.
E. A circular region is the union of a circle and its interior.
II. Objectives
A. To discover and apply the formula for the area of a parallelogram,, a right triangle, a triangle, a trapezoid, and a circle.
B. To differentiate a circle from a circular region.
III. Values Integration
Carefulness in doing one’s work
IV. Materials
Activity sheets
Pair of scissors
V. Instructional Strategies
A. Practical Work
B. Discussion
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment
2. Recall the definitions of altitude, parallelogram, triangle, trapezoid, and
circle.
B. Lesson Proper
1. Give each student a photocopy of this activity.
a. On a separate piece of paper, draw a rectangle similar to the rectangle drawn below.
b. Label rectangle ABCD.
c. Draw a line through point E between A and B.
d. Cut the drawn rectangle on your paper.
e. Cut along DE.
f. Move the left piece to the right until point A coincides with point B and point D coincides with point C.
g. What geometric figure was formed?
h. What can you say about the area of this figure?
i. Write a general statement using symbols.
2. Practice Exercises
a. Answer Mental Mathematics of Exercise 1.7 (numbers 1-8) on page 67 of the textbook.
b. Solve Written Mathematics of Exercise 1.7 (numbers 1-4) on page 68 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 1.7 (numbers 5-8) on page 68 of the text book.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Recall the Area Formula for a Rectangle.
B. Follow-up Lesson
1. Give each student a photocopy of this activity.
a. Label your rectangle ABCD. How do you find the area of the rectangle ABCD?
b. Draw DB.
c. Cut along DB.
d. What happened to your rectangle?
e. What do you call this geometric figure? Why?
f. Are they congruent? Why?
g. What can you say about their areas?
h. How do you find the area of this geometric figure?
2. Give each student a photocopy of this activity.
a. Draw two parallelograms similar to figures 1 and 2 below.
b. Label figure 1, ABCD and figure 3, EFGH.
c. In figure 1, plot any point E between A and B. Draw De and CE.
d. How many triangles are formed? Name the first triangle, ADE, the second, DEC, and the third, EBC.
e. What kind of triangles are formed? Why?
f. Cut out rectangle ABCD. Cut along DE and CE.
g. Show the sum of the areas of ∆ ADE and ∆ EBC is equal to the area of ∆DEC.
h. Make a conjecture about the area of a triangle.
i. In figure 2, draw diagonal EG. Into how many triangles is the parallelogram divided?
j. What kind of triangles are formed?
k. Do you think the areas of the two triangles are equal? Why?
l. Make a conjecture about the area of the triangle.
3. Practice Exercises
Solve Written Mathematics of Exercise 1.7 on page 69 of the text book.
C. Assignment
Solve Written Mathematics of Exercise 1.7 (numbers 15-16) on page 69 of the textbook.
Session 3
A. Preliminary Activities
1. Check the assignment.
2. Review factoring polynomials.
B. Follow-up Lesson
1. Give each student a copy of this activity.
d. What postulate did you use?
e. Is the right side of the equation factorable?
f. What is the common factor?
g. Factor the right side of the equation.
Therefore, the Area of a Trapezoid=
2. Illustrative Examples
Refer to the textbook.
3. Practice Exercises
Solve Written Mathematics of Exercise 1.7 (numbers 17 and 18) on page 70 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 1.7 (numbers 17and 18) on page 70 of the textbook.
Session 4
A. Preliminary Activities
1. Check the assignment.
2. Recall the area formula for a rectangle.
B. Follow-up Lesson
1. Guide the students in deriving the formula for the area of a circle by separating a circle into equal sections and then arranging these
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