LESSON PLAN IN GEOMETRY

S.Y. 2011-2012

Lorena M. Masbaño September 23, 2011

Student Teacher Date

Mr. Sonny Chiu 9:45- 11:05 AM

Student Teaching Mentor Time

I. Objectives:

Through thorough discussions, proof finding, and varied illustrative examples, the third year

high school students are expected to do the following with at least 85 % accuracy:

a. Define Geometry according to mathematician, structures, and examples;

b. Differentiate the following terms: a definition, a postulate, a theorem, an axiom, and

corollary;

c. Write the conditional statements into a converse, inverse, contrapositive and

biconditional;

d. Use inductive and deductive reasoning to prove mathematical statements.

II. Subject Matter:

a. Topic: Introduction to Geometry and Logical Reasoning

b. Pre-requisite: Theorems, postulates, definitions and corollary

c. Skills to develop: Logical Reasoning

d. Subject Integration: Social Studies

e. Values Integration: Cooperation, Speed and Accuracy to form statements

f. Materials: Picture of Euclid, three-dimensional pictures, Strips of Cartolinas and Manila

papers

g. References:

i. Local Author:

Oronce, Orlando A. and Mendoza, Marilyn O. E-Math, Revised Edition 2010, Rex

Bookstore, Inc., Sampaloc, Manila, Philippines, Pages 2-7.

Oronce, Orlando A. and Mendoza, Marilyn O. E-Math, 2007, Rex Bookstore, Inc.,

Sampaloc, Manila, Philippines, Pages 18-28.

ii. Foreign Author:

Bass, Laurie E., et.al. Geometry: Tools for a Changing World, 2005, Pearson

Education, Inc., Philippines, pages 104-108.

Swift, Ben. Logical Reasoning. Autumn 2003. Word Press, New York.

iii. Internet Sources:

GRE Sample Logical Reasoning Question. Best Sample Question, 2004.

www.bestsamplequestion.com/gre-questions/logical-reasoning.html

h. Method: Expository Method

III. Procedure:

a. Approach:

Geometry comes from the two Greek words, geo, meaning “earth”, and metri meaning

“Measurement”.

Euclid- an ancient Greek philosopher who first developed Geometry around 300 B.C.

Elements- this is the book of Euclid which contains the fundamentals and concepts in

Geometry.

Geometry deals with shapes that we see in the world each day.

What do you think will happen if Geometry was not discovered or introduced to the

World?

What would its effects to the infrastructure? to houses? to businesses?

FOR LOGICAL REASONING:

The students will be asked regarding the current issues on RH Bill.

1. Who among you have heard or read anything about the issue on RH Bill?

2. What do you know about the said issue?

3. What is your stand about the Bill? Are you pro-RH Bill? Or are you

against it?

4. Why do you say so?

Every time we expressed an argument, we used statements that would really hit the idea that we

want to express. That is why we need to think carefully and logically so that the statement would

be acceptable and true.

b. Presentation:

EXAMPLE 1:

1. Given the statement: Glass objects are fragile.

Conditional Statement- has two parts: a hypothesis denoted by p, and a conclusion,

denoted by q.

2. Conditional: If the objects are made of glass, then they are fragile. (TRUE)

3. Converse Statement:

-“If q, then p” is written as 𝑞 → 𝑝

Converse: If the objects are fragile, then they are made of glass. (FALSE)

4. Inverse Statement:

- “If not p, then not q” is written as ~𝑝 → ~𝑞

Inverse: If the objects are not made of glass, then they are not fragile. (FALSE)

5. Contrapositive Statement:

-“If not q, then not p” is written as ~𝑞 → ~𝑝

Contrapositive: If the objects are not fragile, then they are not made of glass.

(FALSE)

6. Biconditional: is form when a conditional and its converse are both true.

In symbols: “p if and only if q” is written as 𝑝 ↔ 𝑞

Biconditional: No biconditional statements can be drawn since the converse

statement is false.

For BICONDITIONAL:

ORIGINAL: Mammals have mammary glands

CONDITIONAL: If an animal is a mammal, then it has a mammary gland. (TRUE)

CONVERSE: If an animal has mammary gland, then it is a mammal. (TRUE)

BICONDITIONAL: An animal is a mammal if and only if it has a mammary gland.

(TRUE)

Conditional statement may be true or false. To show that a conditional statement is

TRUE, you must construct a logical argument using reasons.

Definition- a statement of a word, or term, or phrase which made use of previously

defined terms

Postulate- is a statement which is accepted as true without proof.

Theorem- is any statement that can be proved true.

Corollary- to a theorem is a theorem that follows easily from a previously proved

theorem.

EXAMPLE 2:

1. ORIGINAL: Complementary angles are any two angles whose sum of

their measure is 90°. TRUE

2. CONDITIONAL: If two angles are complementary, then the sum of their

measure is 90°. TRUE

3. CONVERSE: If the sum of the measures of two angles is 90°, then they

are complementary. TRUE

4. BICONDITIONAL: Two angles are complementary if and only if the sum

of their measure is 90°. TRUE

5. INVERSE: If two angles are not complementary, then the sum is not 90°.

TRUE

6. CONTRAPOSITIVE: If the sum of the measures of two angles is not 90°,

then they are not complementary.

EXAMPLE 3:

1. ORIGINAL: The sum of two odd numbers is even.

2. CONDITIONAL: If two numbers are odd, then their sum is even. TRUE

3. CONVERSE: If the sum of two numbers is even, then they are odd

numbers. TRUE

4. BICONDITIONAL: Two numbers are odd if and only if their sum is even.

TRUE

5. INVERSE: If two numbers are even, then their sum is odd. FALSE

6. CONTRAPOSITIVE: If the sum of the numbers is odd, then they are odd

numbers. FALSE

DEDUCTIVE REASONING

-from deduce means to reason form known facts;

-use in proving theorem;

-using existing structures to deduce new parts of the structure.

-“if a, then b”

SYLLOGISM- an argument made up of three statements: a major premise, a minor

premise (both of which are accepted as true), and a conclusion.

EXAMPLES OF SYLLOGISM:

1. Major Premise: If the numbers are odd, then their sum is even.

Minor Premise: The numbers 3 and 5 are odd numbers.

Conclusion: the sum of 3 and 5 is even.

2. Major Premise: If you want good health, then you should get 8 hours of sleep a day.

Minor Premise: Aaron wants good health.

Conclusion: Aaron should get 8 hours of sleep a day.

3. Major Premise: Right angles are congruent.

Minor Premise: < 1 𝑎𝑛𝑑 < 2 are right angles.

Conclusion: < 1 𝑎𝑛𝑑 < 2 are congruent.

4. Major Premise: Diligent students do their homeworks.

Minor Premise: Amy and Andy are diligent students.

Conclusion: Amy and Andy do their homeworks.

INDUCTIVE REASONING:

- It is a process of observing data, recognizing patterns, and making generalizations

from observations.

- Geometry is rooted in inductive reasoning. The geometry of ancient times was a

collection of procedures and measurements that gave answers to practical

problems.

- Used to calculate land areas, build canals, and build pyramids.

- Using inductive reasoning to make a generalization called conjecture.

Use inductive reasoning to find the next term/figure of each sequence.

c. Application:

Each student will work with her/his seatmate for this activity.

I. Underline the hypothesis once and the conclusion twice. Identify if it is true or

false.

1. If you see lightning, then you hear thunder.

2. If three points lie on a line, then they are collinear.

3. If you are 13 years old, then you are a teenager.

4. If 5𝑥 + 2 = 12, then 𝑥 = 2.

5. If two angles are supplementary, then the sum of their measure is 180.

II. Write the following statement to its (a) conditional, (b) converse, (c) inverse and

(d) contrapositive. Identify its truth value.

1. Obtuse angles have measures more than 90.

2. A basketball player is at least 5’9” tall.

3. An angle measuring 90 is a right angle.

4. All rectangles are quadrilaterals.

5. Good citizens pay taxes.

d. Generalization:

1. What should we consider in order to obtain a conditional statement?

2. Is reasoning skills important to our life? Why?

3. How important is logical reasoning in proving arguments? Explain.

e. Evaluation:

I. Rewrite each statement in “if p, then q” form. Underline the hypothesis once and

the conclusion twice.

1. An angle measuring 180° is a straight angle.

2. Numbers that have 2 as a factor are even.

3. 3𝑥 − 7 = 14, then 3𝑥 = 21.

4. An isosceles triangle has two congruent sides.

5. A polygon with four sides is a quadrilateral.

II. Write each conditional statement to its (a) converse, (b) inverse, and (c)

contrapositive.

1. If a triangle is a right triangle, then it has a 90 degrees angle.

2. If you can’t work, you will not get paid.

3. If a conditional statement is false, then its contrapositive is false.

III. Write the converse of each statement. Determine the truth value of the statement

and its converse. If both statements are true, write a biconditional.

1. If a point is in the first quadrant, then its coordinates are positive.

2. If a substance is water, then its chemical formula is H2O.

IV. Assignment:

i. On your assignment notebook, write the conditional statement, its converse,

inverse, contrapositive and biconditional, if possible, of the given statement.

1. A quadrilateral that has exactly two congruent sides is not a rhombus.