Lesson Plans on Math 14

COMPILATION OF

MANIPULATIVE-INTEGRATED

LESSON PLANS IN MATHEMATICS

KHARELLE MAE M. NADUMA

BSED-MATH 3

MATH 14

AUGUST 2014

Table of Contents

First Exam

Squares and Square Roots ……………………………………………………………….….. 2

Attachment 1 …………………………………………………………………………….….. 5

Attachment 2 …………………………………………………………………………….….. 7

Multiplication and Division of Integers ………………………………………………….…. 8

Attachment 3 …………………………………………………….………………….……… 11

Attachment 4 …………………………………………………….…………………..……... 13

Second Exam

The Sum and Difference of Two Terms ……………………….………………….………... 15

Attachment 5 …………………………………………………….………………….……… 19

Attachment 6 ………………………………………………….……………………….…… 21

Third Exam

Perpendicularity ……………………………………………………………………….….... 24

Attachment 7 …………………………………………………………………….…………. 30

Attachment 8 …………………………………………………………………….…………. 32

Fourth Exam

Special Right Triangles ………………………………………………………….………… 35

Attachment 9 …………………………………………………………………….………… 41

Attachment 10 ………………………………………………………………….………….. 43

Attachment 11 ………………………………………………………………….………….. 45

Experimental and Theoretical Probability ……………………………………….………... 46

Attachment 12 …………………………………………………………………….……….. 50

Attachment 13 …………………………………………………………………….……….. 51

FIRST EXAM

2

GRADE 7

Lesson Plan in Numbers: Squares and Square Roots

Content Standards

Performance Standards

Learning Competencies

The learner demonstrates

understanding of…

estimation/approximation

of a square of a number

and the measures of

quantities

The learner is able to…

apply various

procedures and

manipulations on the

different subsets of the

set of real numbers,

particularly on squares

and square roots

The learner…

determines what two

integers the square

root of a number is

*K to 12 Curriculum Guide – version as of January 31, 2012

I. Objective

At the end of the 60-minute lesson, the students will be able to determine the

squares of numbers and the square root of perfect squares.

II. Learning Task

A. Subject Matter: Squares and Square Roots

B. References:

Math is Fun: http://www.mathsisfun.com/square-root.html

Math.com: http://www.math.com/school/subject3/lessons/S3U3L3GL.html

C. Materials:

a. LCD Projector

b. Laptop

c. Virtual Manipulative – Geoboard

d. Whiteboard and marker

e. Worksheets for Activity and Quiz (see attachment)

III. Procedure

A. Preliminary Activities

a.Opening Prayer (2 min.)

The teacher will ask a student to lead the prayer.

b. Greetings (3 min.)

The teacher will greet the students a good morning/afternoon and check the

attendance.

http://www.mathsisfun.com/square-root.html

http://www.math.com/school/subject3/lessons/S3U3L3GL.html

3

B. Lesson Proper

a. Class Activity (15 min.)

In this activity, the students will determine the squares of numbers and the

square root of perfect squares with the help of Geoboard.

Guidelines:

1. Group the students by pairs. Each pair should have a laptop/tablet/iphone

with the Geoboard App. Note that a day before the lesson the students

have already downloaded the app on their laptop/tablet/iphone.

2. The teacher will then introduce the virtual Geoboard to the students. The

teacher will also review the following terms:

o Square (shape), its area and side length

o Exponents, radicals

3. After the introductionof the Geoboard and the short review, the students

will be left working with their pairs as they answer the activity sheet (see

attachment 1) given by the teacher. The teacher will roam around the class

to monitor the performance of each student.

b. Discussion (25 min.)

After the allotted time for the activity, the teacher will collect the activity

sheets and will start to ask about the students’ answers and observations in the

activity. He/she will then probe questions until the students arrive to the

concept of squares and square roots.

Given the side length, how did you get the area of the square?

Given the area, how did you get the side length of the square?

What can you say about the square’s area in relation to its side length?

What can you say about the side length in relation to its area?

What rules can you come up with these observations? (squares and

square roots)

What if we square a negative number, say -3, what do we get? (9)

Can we consider -3 as a square root of 9?

From this, how many square roots do you think a number has?

How will you define squares (not the shape) and square roots now?

C. Generalization (5 min.)

The teacher will give a brief generalization or summary of the lesson.

IV. Rationale

It is important that students know and understand squares and square roots because

they are used in practical application, especially when we talk about areas. Through

square roots also, it becomes easier for us to represent some of the irrational numbers.

4

IV. Evaluation (8 min.)

The teacher will give the students a quiz as their individual assessment (see

attachment 2).

V. Homework: Non-Perfect Squares

The teacher will project the homework on the board.

Copy and answer the following on a ¼ sheet of paper.

1. √12 = 3. √45 = 5. √2 =

2. √80 = 4. √150 =

5

Attachment 1

ACTIVITY SHEET

Names: ____________________________ Date: ________________

_____________________________ Grade & Section: ________________

Direction: Read and do the following instructions.

1. Create a square on your Geoboard. What is its side length? Its area?

2. Create another square, this time, the smallest square you can on your Geoboard. What is

its side length? Its area?

3. Now, create the largest square you can. What is its side length? Its area?

4. Create the squares with the following side lengths and areas and fill in the table.

SIDE LENGTH AREA

1 ______

2 ______

_____ 9

_____ 16

5 ______

6 ______

_____ 49

_____ 64

9 ______

Answer the following:

Given the side length, how did you get the area of the square?

_____________________________________________________________________

_____________________________________________________________________

__________________________________________________________________.

Given the area, how did you get the side length of the square?

_____________________________________________________________________

_____________________________________________________________________

__________________________________________________________________.

What can you say about the square’s area in relation to its side length?

_____________________________________________________________________

_____________________________________________________________________

__________________________________________________________________.

6

What can you say about the side length in relation to its area?

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________________.

With your observations, what pattern/s can you create? Write it below or draw a table.

7

Attachment 2

QUIZ

Name: ________________________________ Date:________________

Grade & Section: _______________________

A. Square the following numbers.

1. 11 =

2. 13 =

3. 15 =

4. 17 =

5. 19 =

To square a number means to

__________________________________________________________________________.

B. Find the square root of the following perfect squares.

6. 196 =

7. 256 =

8. 324 =

9. 484 =

10. 625 =

To find the square root of a number means to

__________________________________________________________________________.

8

GRADE 7

Lesson Plan in Numbers: Multiplication and Division of Integers

Content Standards




understanding of…

estimation/approximati

on of a square of a

number and the

measures of quantities


apply various

procedures and

manipulations on the

different subsets of the

set of real numbers,

particularly on squares

and square roots

The learner…

performs fundamental

operations on integers


I. Objective

At the end of the lesson, the students will be able to multiply and divide integers.

II. Learning Task

A. Subject Matter: Multiplication and Division of Integers

B. Reference/s:

Benigno, G. D. (2006). Basic Mathematics. Quezon City: Rex Printing Company,

Inc.

C. Materials:

a. Traditional Manipulative – Integer Tiles

b. Whiteboard and marker

c. Worksheets for Activity and Quiz (see attachment)

d. LCD Projector

III. Procedure

A. Preliminaries

a.Opening Prayer (2 min.)



The teacher will greet the students a good morning/afternoon and check

the attendance.

9

B. Lesson Proper

a.Class Activity (15 min.)

In this activity, the students will be able to multiply and divide integers wit the

help of Integer Tiles.

Guidelines:

1. Before the activity starts, the teacher will review the following terms:

Multiplication

Division

Integers

2. The teacher will then group the students through counting. Each group has

3 members.

3. After that, the teacher will give each group a bag of “Integer Tiles” and

introduce it to them.

- represents one positive

- represents one negative

-

How will you represent (+5)? (-4)?

+5 =

-4 =

What about zero? How will you represent it using these tiles?

- represents zero; also referred to as a “zero-pair”

4. Once the students know how to represent integers using the tiles, the

teacher will give each group an activity sheet (see attachment 3) and leave

them work on it. The teacher will roam around the classroom to monitor

each group, making sure that each member cooperates and each has the

chance to manipulate the tiles.

b. Discussion (20 min.)

After the time allotted for the activity, the teacher will start to ask each

group about their answers. He/she will project on the board the charts that

the group have answered in the activity as an aid, collect the answers of

the students and write it on the charts, and will ask the class to check if the

answers are right. He/she will then probe questions until the students arrive

to the concept of multiplication and division of integers.

10

Have you observed patterns?

On multiplication, what is the result if we multiply both positive

integers? Both negative integers? A positive and a negative integer?

What rule can you make on the multiplication of integers based on your

observations?

On division, what do we get if we divide a positive integer by another

positive integer? A negative integer by another negative integer? A

positive integer by a negative integer? A negative integer by a positive

integer?

What rule can you make on the division of integers based on your

observations?


The teacher will give a brief generalization or summary of the lesson.

IV. Rationale (3 min.)

It is important that students know and understand operations like multiplication and

division of integers because this lesson does not just allow students to use their

knowledge about addition and subtraction but it also prepares them for the next years

where they will have to solve problems involving money, distance, movement, and

measurement, among others.

V. Evaluation (10 min.)

The teacher will give the students a quiz (see attachment 4) which they will answer

for 5 minutes. After that, they will exchange papers with their seatmates and check it.

VI. Homework

Answer the following on a ¼ sheet of paper.

Direction: Determine whether the product/quotient of the following is positive or

negative. No need to perform actual multiplication/division.

1. (-x)(y)(-z) __________________

2. (1)(-2)(-3)(-4) __________________

3. 48 ÷ (-4) __________________

4. (-120) ÷ (-60) __________________

5. (6)(3)(a)(-b)(9) __________________

11

Attachment 3

ACTIVITY SHEET

Group No. _____

Names: ________________________________ Date: ________________

________________________________ Year & Section: ________________

________________________________

Direction: Using the Integer Tiles, fill in the table:

A. Multiplication

Note:

The sign of the multiplicand tells us if we are adding (+) or taking away (-) groups.

The sign of the multiplier tell us what tiles the group consists of.

If you “take away” groups of tiles you don’t have, you could add as many zero-pairs

as needed until you have enough tiles.

Student 1 Student 2 Student 3

(+3) • (+2) = ____ (+4) • (+5) = ____ (+6) • (+3) = ____

(+3) • (-2) = ____ (+4) • (-5) = ____ (+6) • (-3) = ____

(-3) • (+2) = ____ (-4) • (+5) = ____ (-6) • (+3) = ____

(-3) • (-2) = ____ (-4) • (-5) = ____ (-6) • (-3) = ____

B. Division

Note:

Rewrite into a multiplication problem, then identify the number of groups you have to

add/take away.

Student 1 Student 2 Student 3

(+8) ÷ (+2) = ____ (+16) ÷ (+4) = ____ (+12) ÷ (+6) = ____

(+8) ÷ (-2) = ____ (+16) ÷ (-4) = ____ (+12) ÷ (-6) = ____

(-8) ÷ (+2) = ____ (-16) ÷ (+4) = ____ (-12) ÷ (+6) = ____

(-8) ÷ (-2) = ____ (-16) ÷ (-4) = ____ (-12) ÷ (-6) = ____

12

C. Observation

What patterns have you noticed?

With the result

_____________________________________________________________________

____________________________________________________________________.

With the sign

_____________________________________________________________________

____________________________________________________________________.

On Multiplication

When does the product become positive?

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________________.

When does the product become negative?

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________________.

On Division

When does the quotient become positive?

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________________.

When does the quotient become negative?

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________________.

13

Attachment 4

QUIZ

Name: ____________________________________ Date: _______________

Year & Section:_____________________________

Answer the following:

1. -3 • 5 =

2. -2 • -7 =

3. 18 • -3 =

4. -13 • 4 =

5. 7 • 12 =

6. 108 ÷ -27 =

7. -98 ÷ 14 =

8. -135 ÷ -9 =

9. 136 ÷ -17 =

10. 341 ÷ 11 =

Fill in the blanks.

11. Multiplying both positive integers yields ____________________.

12. Dividing both negative integers yields ____________________.

13. Multiplying a positive integer and a ____________________ yields a negative integer.

14. Dividing a ___________________ and a positive integer yields a negative integer.

15. Multiplying both negative integers yields _____________________.

SECOND EXAM

15

GRADE 8

Lesson Plan in Algebra: Special Products – The Product of the Sum and

Difference of Two Terms

Content Standards




understanding of…

special products and

factors of polynomials

and operations on

rational expressions


recognize special

products and factor

polynomials

The learner…

finds special products

of certain

polynomials (e.g.

square of a binomial,

product of sum and

difference of two

terms)


I. Objective

At the end of the 60-minute lesson, the students will be able to find the special

product of the sum and difference of two terms.

II. Learning Task

A. Subject Matter: Special Products – The Product of the Sum and Difference of

Two Terms

B. Reference/s:

Abuzo, E. P., et. al. (2013). Mathematics – Grade 8 Learner’s Module. Quezon

City: Book Media Press, Inc.

http://www.mathsisfun.com/algebra/polynomials.html

http://mathworld.wolfram.com/Polynomial.html

http://www.intmath.com/factoring-fractions/1-special-products.php

C. Materials:

a. Paper and scissors

b. Activity sheet

c. Whiteboard and marker/Chalkboard

d. LCD Projector

e. Flash Quiz (PPT)

III. Procedure

A. Preliminaries

a. Opening Prayer (2 min.)


http://www.mathsisfun.com/algebra/polynomials.html

http://mathworld.wolfram.com/Polynomial.html

http://www.intmath.com/factoring-fractions/1-special-products.php

16



attendance.

B. Lesson Proper

a. Review (5 min.)

Teacher: What is a polynomial?

Student: A polynomial is a mathematical expression involving a sum of

powers in one or more variables multiplied by coefficients. It means “many

terms”. It can have constants, variables and exponents, but never division by a

variable.

Teacher: What are special products?

Student: Special products are products of certain polynomials that can be

easily found through patterns than by multiplying all of the terms of the

polynomials. They are "special" because they are very common, and they're

worth knowing.

Teacher: What special products have we already discussed?

Student: Square of a binomial and square of a trinomial

Teacher: Okay. Today, we will have an activity to discover another type of

special products.

b. Class Activity: Transformers (10 min.)

In this activity, the students will model the product of the sum and difference

of two terms. The teacher will group the students by pairs base on who their

seatmates are. Note that a day before the activity the students were asked to bring

scissors and a piece of paper, with an ideal size of 8”x11” in any color. The

teacher will ask the students to bring out the said materials and will give each pair

an activity sheet (see attachment 5) having the instructions and guide questions for

their discovery. Each student must perform the instructions in the activity sheet

individually but must answer the guide questions collaboratively with their pair.

The teacher will roam around the class to monitor the performance of each

student.

c. Discussion (25 min.)

After the time allotted for the activity, the class will discuss what they did and

what they have observed. The teacher will ask the following questions and each

pair will be given a chance to answer.

TEACHER STUDENT/S

How’s the activity? What did you

observed? Who among you here saw a pattern? For those who raised their hands, I

want you to come here in front and write

the pattern/s you’ve observed on the board.

Expected Pattern: (x + y)(x – y) = x2 – y

2

http://mathworld.wolfram.com/Power.html

http://mathworld.wolfram.com/Variable.html

http://mathworld.wolfram.com/Coefficient.html

17

Okay, thank you. Good observations, but

we’ll get back to that later and see if those are related patterns. For now let’s go back

to the activity that you did. What was the

first thing that you did?

We made a square, named it as square

ABCD, and labeled its side as x.

And…? We cut a small square from it, named it as QRCS, and labeled its side as y.

Okay. So now we have two squares, square

ABCD and QRCS. What are the areas of

those squares?

The area of square ABCD is x2 while the

area of square QRCS is y2.

x2 and y

2, correct. Now you formed a new

figure, right? What is the area of that new figure?

x2 – y

2.

How did you get that answer? Since the new figure was formed when we

cut square QRCS from square ABCD, we

just subtracted the areas of the two squares to get the area of the new figure.

Very good. Now what did you do next? We cut the new figure as instructed and

rearranged the pieces to form a rectangle.

Okay. So what is now the length and width of that rectangle? How did you represent

the dimensions in terms of x and y?

Length = (x + y), width = (x – y)

How about its area? Can you show on the

board how you got the area of the rectangle?

We used the formula on getting the area of

a rectangle. Area = length x width

A = (x + y)(x – y)

A = x2 – y

2

Okay, very good. Now let’s go back to the

patterns you’ve written on the board earlier.

The last question on your activity sheet is

“Do you see any pattern in the product of the sum and difference of two terms? What

is it?” The sum of the two terms is the

length of the rectangle while the difference is the width. When we multiplied the length

and width we got x2 – y

2.Most of you seem

to observe the same pattern, only it was expressed in different ways. Now how will

synthesize the patterns you’ve written into

one?

(x + y)(x – y) = x2 – y

2

Do you think this pattern applies to any polynomial that is in this form? Try to

solve the following on your notebook and

observe their product.

1. (2m + n)(2m – n) 2. (xy + 8)(xy – 8)

3. (x + 3y)(x – 3y)

Yes, it applies to any polynomial in this form

Does the order matter? What if it’s (x – y)(x + y), will we still get the same answer?

Yes, we’ll still get the same answer. The order does not matter. It’s commutative.

What do you notice with its product? How

many terms does it have?

Two.

What do we call a polynomial that has two terms?

A binomial.

18

Is the product always a binomial? Yes.

Do you have to do the FOIL method to get

the product of polynomials in this form?

No.

Alright. And because this pattern makes

our solving easier, this means that this is

another type of a special product - the product of the sum and difference of two

terms.


The teacher will ask the students to summarize what they have learned.

Expected Generalization: One type of special products is the product of the

sum and difference of two terms, which is the difference of the squares of

the terms. In symbols,

(x + y)(x – y) = (x – y)(x + y) = x2 – y

2.

Notice that the product is always a binomial and the order does not matter.

IV. Rationale

It is important that students know how to identify special products because their

patterns can be used to easily find the product of algebraic expressions and is very

essential in factoring. More than that, special products can be applied in real-life

situations. A specific example is the Punnet square.


The class will have a “Flash Quiz”. Using the pattern that the students have learned,

they must find the product of the sum and difference of two terms that the teacher will

flash on the projector. They will write their answers to the 10-item quiz on a ¼ sheet

of paper. They will only be given 15 seconds to answer each item. After 15 seconds,

the slide will automatically flash another problem and it will not flash the previous

problems again. After the 10th

item, they will exchange papers with their seatmates

and check it. (See attachment 6 for the print screen of the Flash Quiz in ppt.)

VI. Assignment

The teacher will give the students a research assignment:

Research at least two real-life applications of the special products we have discussed

so far – square of a binomial, square of a trinomial, and the product of the sum and

difference of two terms. Write it on a one whole sheet of paper.

19

Attachment 5

TRANSFORMERS

ACTIVITY SHEET

Names: ________________________________ Date: ________________

________________________________ Year & Section: ________________

Direction: Do the following instructions individually and answer the guide questions

with your pair.

Materials: Paper and scissors

1. Make a square out of the paper you have. Name it as square ABCD and label its side as x.

2. Cut a small square from any of the corners of square ABCD. Name it as QRCS and label

its side as y, like the figure below.

In terms of x and y…

What is the area of square ABCD?

What is the area of square QRCS?

How will you represent the area of the new figure (ABRQSD)?

3. Cut the new figure (ABRQSD) on the broken line shown below.

S

R Q

D

B A

C

Q R

S

y

x

20

4. Rearrange the pieces to form a rectangle.

In terms of x and y…

How will you represent the length and width of the rectangle?

What is its area? Show how you get it.

Do you see any pattern in the product of the sum and difference of two terms? What is

it?

21

Attachment 6

Print Screen of Flash Quiz

THIRD EXAM

24

GRADE 8

Lesson Plan in Geometry: Perpendicularity

Content Standards




understanding of…

parallel and

perpendicular lines


prove results involving

parallel and

perpendicular lines

The learner…

defines perpendicular

lines

determines and

proves the conditions

under which lines and

segments are

perpendicular


I. Objective

At the end of the 60-minute lesson, the students will be able to:

illustrate perpendicular lines

determine the conditions under which lines and segments are perpendicular

II. Learning Task

A. Subject Matter: Perpendicularity

B. Reference/s:



C. Materials:

a. The Geometer’s Sketchpad

b. Worksheets for Activity and Quiz

c. Whiteboard and marker/Chalkboard

d. LCD Projector

e. Protractor

III. Procedure

A. Preliminaries





attendance.

25

B. Lesson Proper

a. Review: Am I Perpendicular? (7 min.)

The teacher will test if the students can still recall what perpendicular lines are and

how they look like. The teacher will project four different figures on the

whiteboard and ask the following questions:

Figure 1 Figure 2

Figure 3 Figure 4

1. What is common in the four given figures?

The lines in each figure intersect.

2. What makes figure 3 and figure 4 different from the first two figures?

They form a right angle.

3. Let’s measure the angles formed by these intersecting lines. (The teacher

will call four students to come in front and measure the angles using

their protractor and write the measurements on the board.) Which among

the four figures show perpendicularity?

Figures 3 and 4.

4. When are lines said to be perpendicular?

When the lines intersect each other at right angles.

5. Does perpendicularity apply only to lines?

No, segments and rays can also be perpendicular.

b. Class Activity (20 min.)

In the class activity, the students will determine the conditions under which lines

and segments are perpendicular with the help of a virtual manipulative – The

Geometer’s Sketchpad. It is assumed that the students already know how to use

the said manipulative before the activity.

Guidelines:

1. The teacher will group the students into 10 groups through counting. Each

group will have 3 members (assuming that the class size is 30).

26

2. There are 5 tasks and two groups will work on the same task, as shown

below:

Groups 1 & 2 – Task 1





3. On the first part of the activity, each group will do the task assigned to them

using The Geometer’s Sketchpad (see attachment 7 for the Tasks). The teacher

must remind the students that each should have the chance to manipulate and

everyone should cooperate in doing the task.

4. After 15 minutes, the groups that have the same assigned task will merge and

will be given 5 minutes to compare their observations. They must write their

observations on a ½ sheet of paper with their names and task number.

c. Discussion/Sharing (15 min.)

After the allotted time for the class activity, a member from each group (the new

formed group) will share to the class what their task is and what they have

observed. The member that will share will be picked randomly so each student

must prepare.

TEACHER STUDENT

Let’s have groups 1 and 2 to share about their task, Task #1. (random calling of name)

(The chosen student will share to the class what their task is. Please refer to Task 1 in attachment

1.)

What can you say about 𝐴𝐵 and 𝐷𝐸 in terms of the angles formed?

When we measured the angles, it formed right

angles so we can say that 𝐴𝐵 and 𝐷𝐸 are perpendicular to each other.

Very good. 𝐷𝐸 is the bisector, right? And it is

perpendicular to 𝐴𝐵 , the given line segment. For

this reason, we call 𝐷𝐸 the “perpendicular

bisector” of 𝐴𝐵 . A perpendicular bisector of a line segment is a line or a ray or another line

segment that is perpendicular to the line segment

and intersects it at its midpoint.

Now, have you identified other lines, segments,

or rays that bisect 𝐴𝐵 the same way that 𝐷𝐸

bisects it? Does 𝐴𝐵 have other perpendicular bisector?

No, ma’am. We haven’t identified other

perpendicular bisector of 𝐴𝐵 other than 𝐷𝐸 .

Oh, only 𝐷𝐸 is the perpendicular bisector of 𝐴𝐵 . What does this mean?

This means that a line segment has only one

perpendicular bisector.

Very good! Remember this class: A line segment

has a unique perpendicular bisector.

Let’s go to the next, groups 3 and 4, with Task #2. (random calling of name)

(sharing)

How many lines that passes through A are

perpendicular to 𝐵𝐶 ?

One.

27

What does this mean?

Given a line and a point not on the line, there is, … what?

This means that there is only one line/segment

perpendicular to the given line that passes through the given point.

Okay, very good! Take note of this class: Given

a line and a point not on the line, there is exactly

one line perpendicular to the given line that passes through the given point.

(continuation of sharing)

So, what can you say about the distance in terms

of the angles between the distance and the given

line?

The angles formed between the distance and the

given line are right angles so we can say that the

distance is perpendicular to the given line.

Okay, nice observation. And for that reason, we

call that distance as the “perpendicular distance”

between the given point and line.

The next groups are groups 5 and 6, with Task #3. (random calling of name)

(sharing)

What can you say about the angles formed by the

perpendicular lines? Compare the measures of the angles formed by

the perpendicular lines to the measures of the

angles formed by the non-perpendicular lines.

The angles formed by the perpendicular lines are

all congruent, 90°. However, on the non-perpendicular lines, not all

are equal. Only the two opposite or vertical

angles are congruent.

What angles did the perpendicular lines formed again?

Right angles.

How many? Four.

Therefore,…? What conclusion can we make out

of these observations?

Perpendicular lines form four right angles.

Very good! If two lines are perpendicular, then

they form four right angles. Take note of that.

Now let’s proceed to Task #4 by groups 7 and 8. (sharing)

Tell us about your observations in the linear pair whose angles are congruent.

We have observed that 𝑂𝐵 , the adjacent side, is

perpendicular to 𝑂𝐴 , 𝑂𝐶 , and 𝐴𝐶 .

Are your observations with the linear pair whose

angles are congruent (Figure 1) the same with the linear pair whose angles are not congruent

(Figure 2)?

No, ma’am. It’s different. The adjacent side in

Figure 2 is not perpendicular to any of the other sides in the linear pair.

Does this mean that, among the two figures, only Figure 1 shows perpendicularity?

Yes, ma’am.

What can you say about the sides of a linear pair

whose angles are congruent?

The adjacent side of a linear pair whose angles

are congruent is perpendicular to the other sides

(sides that are not adjacent).

Very good! Remember this: If the angles in a

linear pair are congruent, then the lines

containing their sides are perpendicular.

Let’s have the last, Task #5 by groups 9 and 10. (sharing)

Tell us about your observations in the 3 figures

in terms of the non-common sides.

In Figure 1, the non-common sides formed a

right angle.

In Figure 2, the non-common sides formed an

angle of (any angle not equal to 90°). Among the two figures, of course, Figure 1

shows perpendicularity because the non-common

sides intersect at right angle. Figure 3 does not show perpendicularity because

there is no right angle in the figure.

Okay, only Figure 1 shows perpendicularity.

What is in Figure 1 or what characteristics do the

The angles are complementary and adjacent.

28

angles have that showed perpendicularity?

Does this mean that if two angles are adjacent and complementary, the non-common sides are

perpendicular?

Yes, ma’am.

Yes, that’s right. Remember that, okay? If two angles are adjacent and complementary, the non-

common sides are perpendicular.

Alright, let’s give a big hand for everybody. You

all did a great job. Actually, what you have discovered in Task 4 and 5 are the conditions

under which lines and segments are

perpendicular.

C. Closure/Generalization (5 min.)


Expected Generalization:

Perpendicular lines are lines that intersect each other at right angles.

A perpendicular bisector of a line segment is a line or a ray or another line

segment that is perpendicular to the line segment and intersects it at its

midpoint. This follows that, in a plane, a segment has a unique perpendicular

bisector.

The distance between a line and a point not on the line is the length of the

segment perpendicular to the line from the point, called the perpendicular

distance.

Given a line and a point not on the line, there is exactly one line perpendicular

to the given line that passes through the given point.

If two lines are perpendicular, then they form four right angles.

If the angles in a linear pair are congruent, then the lines containing their sides

are perpendicular.

If two angles are adjacent and complementary, the non-common sides are

perpendicular.

The teacher will then ask the students to cite a specific example of the use of

perpendicularity in real life.

IV. Rationale

It is important that students know what perpendicularity is and determine the

conditions under which lines and segments are perpendicular because these are

needed for the next lessons: quadrilaterals and proving of geometric relationships.

Moreover, perpendicular lines, together with parallel lines, are very useful in real life.

We can see them almost everywhere. These two are crucial elements in construction,

such as the windows and walls of a room, roads and rails, building of a flagpole, and

many more.

29


The students will be assessed individually through the assessment task (see

attachment 8).

30

Attachment 7

Task 1:

1. On the Geometer’s Sketchpad, make a line segment using the Straightedge Tool and label its endpoints as A and B

using the Text Tool.

2. Bisect the line segment using the Compass Tool.

3. Using the Point Tool, place a point on the intersection of the line segment and the bisector, and label it as C.

4. Label the endpoints of the bisector as D and E.

5. Measure the angles formed - ∠𝐴𝐶𝐷, ∠𝐵𝐶𝐷, ∠𝐴𝐶𝐸, ∠𝐵𝐶𝐸.

What can you say about 𝐴𝐵 and 𝐷𝐸 in terms of the angles formed?

_______________________________________________________________________________________________

_______________________________________________________________________________________________

Can you identify other lines, segments, or rays that bisects 𝐴𝐵 the same way 𝐷𝐸 bisects it?

_______________________________________________________________________________________________

---------------------------------------------------------------------------------------------------------------------------------------------

Task 2:

1. On the Geometer’s Sketchpad, construct a line and a point not on the line. Label the point as A and the line as 𝐵𝐶 .

2. Make a line perpendicular to 𝐵𝐶 that passes through A using the Compass Tool.

3. Identify how many line/s that passes through A is/are perpendicular to 𝐵𝐶.

___________________________________________________________

4. On a new sketch, construct another line and a point not on the line.

5. Name the line as 𝑃𝑅 and the point as S.

6. Find the distance between 𝑃𝑅 and S. (Hint: To find the distance, do the same procedure you did in Step 2.)

7. Measure the angles formed between the distance and the given line (𝑃𝑅 )

What can you say about the distance in terms of the angles formed?

_______________________________________________________________________________________________

_______________________________________________________________________________________________

----------------------------------------------------------------------------------------------------------------------------------------------

Task 3:

1. On the Geometer’s Sketchpad, construct two lines that are perpendicular.

2. Name the lines as 𝐴𝐵 and 𝐷𝐸 , and the point of intersection as C.

3. Measure the angles formed - ∠𝐴𝐶𝐷, ∠𝐵𝐶𝐷, ∠𝐴𝐶𝐸, ∠𝐵𝐶𝐸.

4. Construct another pair of lines that intersect but are not perpendicular.

5. Name the lines as 𝐹𝐺 and 𝐼𝐽 , and the point of intersection as H.

6. Measure the angles formed - ∠𝐹𝐻𝐼, ∠𝐹𝐻𝐽, ∠𝐺𝐻𝐼, ∠𝐺𝐻𝐽.

What can you say about the angles formed by the perpendicular lines?

Compare the measures of the angles formed by the perpendicular lines to the measures of the angles formed by the

non-perpendicular lines.

_______________________________________________________________________________________________

_______________________________________________________________________________________________

31

Task 4:

1. In the Geometer’s Sketchpad, construct a linear pair* whose angles are congruent. Consider this as your Figure 1.

2. Name the angles as ∠𝐴𝑂𝐵 and ∠𝐶𝑂𝐵, with 𝑂𝐵 as the adjacent side of the two angles.

3. Measure and compare ∠𝐴𝑂𝐵 and ∠𝐶𝑂𝐵.

What do you observe with 𝑂𝐵 in relation to 𝑂𝐴 ?

___________________________________________________________________________________________

What do you observe with 𝑂𝐵 in relation to 𝑂𝐶 ?

___________________________________________________________________________________________

What do you observe with 𝑂𝐵 in relation to 𝐴𝐶 ?

___________________________________________________________________________________________

4. Construct a linear pair whose angles are not congruent. Consider this as your Figure 2.

5. Do steps 2 and 3 but change the name of the angles, and answer the questions in bullet.

Are your observations with the linear pair whose angles are congruent (Figure 1) the same with the linear pair

whose angles are not congruent (Figure 2)?

___________________________________________________________________________________________

* two angles that are adjacent (share a leg) and supplementary (add up to 180°)

----------------------------------------------------------------------------------------------------------------------------------------------

Task 5:

1. Construct any two angles that are complementary and adjacent. Consider this as your Figure 1.

What do you observe with the non-common sides of the angles? Measure the angle it forms.

___________________________________________________________________________________________

___________________________________________________________________________________________

2. Construct any two angles that are adjacent but are not complementary. Consider this as your Figure 2.

What do you observe with the non-common sides of the angles? Measure the angle it forms.

___________________________________________________________________________________________

___________________________________________________________________________________________

Which among Figure 1 and Figure 2 shows perpendicularity?

________________________________________________

3. Construct any two angles that are complementary but are not adjacent. Consider this as your Figure 3.

Does this figure show perpendicularity? Measure the angle it forms.

________________________________

http://www.mathopenref.com/anglesadjacent.html

http://www.mathopenref.com/anglesupplementary.html

32

Attachment 8

Assessment Task

Name: ________________________________ Date: ________________

Year & Section: ________________________

Directions:

For numbers 1-5, draw a segment that is perpendicular to the given side. The desired segment

should have an endpoint given below. You can use either a protractor or a compass.

1. Endpoint: B 2. Endpoint: R 3. Endpoint: L

Side: 𝐴𝐶 Side: 𝑇𝑆 Side: 𝑁𝐺

4. Endpoint: A 5. Endpoint: E

Side: 𝑀𝐻 Side: 𝑅𝐴

A

R

O

G

N

E H

T A

M

L

E

G

N

A T S

R Q

C

B

A

33

For numbers 6-10:

On Column B, write the condition of perpendicularity that makes the statement in Column A

true based on the given.

Given Column A Column B

6. x⊥y.

∠1, ∠2, ∠3, and ∠4 are

right angles.

7.

𝐼𝑆 ⊥ 𝐼𝑁 .

8. ∠1 ≅ ∠3.

q ⊥ p.

9. B – midpoint

Line m is the only

perpendicular bisector of

𝐴𝐶.

10.

Line k is the only line

perpendicular to line a

through B.

E

S

k

N I

4 3

2 1

y

x

1 2

3

A

4

q

p

m

C B A

FOURTH EXAM

35

GRADE 9

Lesson Plan in Trigonometry: Special Right Triangles

Content Standards




understanding of…

basic concepts of

trigonometry


explore the concept of

trigonometric ratios and

use these to solve

problems on angles of

elevation and depression

and navigation

The learner…

finds the

trigonometric ratios

of special angles


I. Objectives


determine the characteristics of the two special right triangles, namely 45°-

45°-90° and 30°-60°-90°

find the trigonometric ratios of special angles

II. Learning Task

A. Subject Matter: Special Right Triangles

B. Reference/s:

http://www.regentsprep.org/Regents/math/algtrig/ATT2/Ltri30.htm

http://www.regentsprep.org/Regents/math/algtrig/ATT2/Ltri45.htm\

http://www.wyzant.com/resources/lessons/math/geometry/triangles

C. Materials:

a. Laptop

b. GeoGebra

c. Worksheets for Activity and Quiz

d. Whiteboard and marker

e. LCD Projector

III. Procedure

A. Preliminaries





attendance.

B. Lesson Proper

http://www.regentsprep.org/Regents/math/algtrig/ATT2/Ltri30.htm

http://www.regentsprep.org/Regents/math/algtrig/ATT2/Ltri45.htm/

http://www.wyzant.com/resources/lessons/math/geometry/triangles

36

a. Review (7 min.)

The teacher will ask the students to recall what trigonometric ratios are from

their discussion yesterday.

Student: Trigonometric ratios are the ratios of the sides in a right triangle with

respect to its acute angles.

Teacher: Yes, very good. And the three basic trigonometric ratios are…?

(Note that in this lesson we limit the discussion of trigonometric ratios to sine,

cosine, and tangent since the other trigonometric ratios are just the reciprocals

of the three.)

Student: Sine, cosine, and tangent, which are given by

sinθ =length of leg opposite the angle

length of hypotenuse

cosθ =length of leg adjacent to the angle

length of hypotenuse

tanθ =length of leg opposite the angle

length of leg adjacent to the angle

Exercise. Given the lengths of two sides of a right angled triangle, find the

length of the third side using the Pythagorean Theorem. Then find the values of

the three basic trigonometric ratios coreesponding to the angle θ.

b. Class Activity (20 min.)

In this activity, the students will discover the characteristics of special

triangles with the help of GeoGebra, a virtual manipulative, and find the

trigonometric ratios of special angles. Note that before this activity the

students were already oriented on how to use the said manipulative. The

students will be grouped into groups of four. The teacher will assign each

group a group number. After that, the teacher will ask each group to get their

laptops and prepare the GeoGebra. He/she will then give each group the

activity sheet. The “odd groups” (1, 3, 5, etc.) will be do the Task 1 (see

attachment 9), and the “even groups” (2, 4, 6, etc.) will do the Task 2 (see

attachment 10). Each student in the group must cooperate in the activity. The

teacher will roam around the class to facilitate and monitor the performance of

each student.

x

9

5

B A

G

θ

37


After the allotted time for the activity, the teacher will now ask the students

questions which will lead to the concept of special triangles and angles.

TEACHER STUDENT

On Task 1

What polygon is formed?

Triangle.

Okay, ∆ABC. What type of triangle is ∆ABC in terms of its sides and angles?

∆ABC is an equilateral and equiangular triangle.

Very good. When you drew a segment connecting

C and D, what happened?

∆ABC was divided into two.

Okay. You measured the sides and angles of the two triangles formed, right? What have you

observed with the corresponding sides and angles

of the two triangles?

The corresponding sides and angles of the two triangles are congruent.

Yes. This means that they are congruent triangles. And since they are congruent, let’s focus on one

triangle. Let’s take ⊿BDC. What angles are formed

in ⊿BDC?

30°, 60°, and 90°.

So what type of triangle is ⊿BDC in terms of its angles?

Right triangle.

That’s right. And because it’s a right triangle this

means that it has trigonometric ratios, let’s first talk about its sides. What are the measurements of its

sides?

The hypotenuse is 2, the shorter leg is 1,

and the longer leg is √3.

Okay, let’s consider that as Triangle 1. How about

the other triangles that you’ve made based on the given table, the Triangle 2 whose vertices are on

(0,0) and (4,0), and the Triangle 3 whose vertices

are on (0,0) and (5,0), what are the measurements of their sides?

Triangle 1 Triangle 2

Shorter leg 2 5

2

Longer leg 2√3 5√3

2

Hypotenuse 4 5

What have you observed with the values of the

shorter leg and the hypotenuse?

Shorter leg Hypotenuse

Triangle 1 1 2

Triangle 2 2 4

Triangle 3 5

2

5

What pattern do you see?

The hypotenuse is twice the shorter leg.

Yes, very good. The hypotenuse is twice the shorter

leg or c = 2a, where c is the hypotenuse and a is the

shorter leg. This follows that the shorter leg is half the

hypotenuse or 𝑎 =𝑐

2.

How about in the values of the shorter leg and the

longer leg? What pattern do you see?

Shorter leg Longer leg

Triangle 1 1 √3

Triangle 2 2 2√3

The longer leg is √3 times the shorter leg.

38

Triangle 3 5

2

5√3

2

Yes, that’s right. The longer leg is √3 times the

shorter leg or b = a√3, where b is the longer leg

and a is the shorter leg. This follows that 𝑎 = 𝑏

√3=

𝑏√3

3.

Alright, with these patterns we can now easily find the other sides of this kind of triangle given only

one side. And because of this, this 30°-60°-

90°triangle is considered as a “special triangle”. But

that is not the only reason why it is considered as special. Let’s look at its angles. What are the

trigonometric ratios of angles 30° and 60°?

sin30° =1

2

cos30° =√3

2

tan30° =√3

3

sin60° =1

2

cos60° =1

2

tan60° = √3

Observe their values. Are they approximate values? No, they’re exact.

Yes. Usually, when we look for the trigonometric ratios of other angles like 10°, 65°, 83°, etc. the

values are expressed in approximate values but in

these two angles the trigonometric values are exact and easy to remember, right?

For this reason, angles 30° and 60° are referred to

as “special angles”, and this is another reason why

30°-60°-90° triangle is considered special.

On Task 2


Square.

When you drew a segment connecting A and C,

what happened?

It formed to triangles.

What type of triangles are they in terms of their

sides and in terms of their angles?

They are both isosceles and right triangles.

Are they congruent? Yes.

Okay, so now let’s focus on one triangle like we did

earlier. Let’s take ⊿ABC. What angles are formed

in ⊿ABC aside from 90°

45°

Okay. Now, just like in Task 1, you also made three

⊿ABC that have different side lengths, right? What are the measurements of the sides of the three

triangles.

Leg 1 Leg 2 Hypotenuse

Triangle 1

2 2 2√2

Triangle

2

3 3 3√2

Triangle 3

4 4 4√2

Alright. Since it’s an isosceles triangle, the two legs

are just equal. Now, what have you observed with

the leg (leg 1 or leg 2) and the hypotenuse? What pattern do you see?

The hypotenuse is √2 times the leg.

Correct! The hypotenuse is √2 times the leg or

c = a√2, where c is the hypotenuse and a is the leg. From this pattern, how will you create a pattern for

finding a in terms of c?

c = a√2 𝑐

√2=

𝑎√2

√2

𝑎 =𝑐

√2=

𝑐√2

2

Okay, very good. Because of these patterns, 45°-

45°-90° triangle is also considered a “special

triangle” together with 30°-60°-90° triangle.

Now let’s look at its angles and see if those are also

sin45°√2

2

cos45°√2

2

tan45° = 1

39

special. What are the trigonometric ratios of 45°?

Can we consider 45° as a “special angle”? Yes, because just like 30° and 60°, it’s

trigonometric ratios are exact and easy to remember.

Very good. Now, let’s remember that not all

triangles have these kinds of characteristics where a side can be easily found given the other sides

without using the Pythagorean Theorem, and not all

angles have an exact and easy to remember

trigonometric values. For this reason, these right triangles and angles are called “special”.

C. Closure/Generalization (5 min.)

The teacher will ask the students to summarize what they have learned and cite

situations when and where they can use what they’ve learned.


The 30°-60°-90° and 45°-45°-90° triangles are special right triangles. The special

nature of these triangles is that they consist of sides that can be easily found without

using the Pythagorean Theorem. The patterns are as follows.

30°-60°-90° Triangle 45°-45°-90°

Shorter Leg/Leg 1 𝑎 =𝑐

2 or 𝑎 =

𝑏 √3

3 𝑎 =

𝑐 √2

2

Longer Leg/Leg 2 b = a√3 𝑏 =

𝑐 √2

2

Hypotenuse c = 2a c = a√2

These special right triangles also consist of special angles, 30°, 45°, and 60°that yield

exact answers instead of decimal approximations when dealing with trigonometric

functions. Their numerical values are easy to remember.

sin30° =1

2 sin60° =

1

2 sin45°

√2

2

cos30° =√3

2 cos60° =

1

2 cos45°

√2

2

tan30° =√3

3 tan60° = √3 tan45° = 1

IV. Rationale

The importance of the triangle to mathematics and the world around us is absolutely

tremendous. And the trigonometry behind triangles has advanced some of the latest

developments in construction and in technology including GPS systems, satellite

imagery, and cell phone frequencies so it is really important that students know the

trigonometric ratios of angles in triangles, especially those of the special triangles

which are commonly used not just in trigonometry but also in mechanics and other

branches of mathematics.

40



attachment 11).

41

Attachment 9

ACTIVITY SHEET

Names: ________________________________ Date: ___________________________

________________________________ Year & Section: __________________

________________________________

________________________________

TASK 1

Directions:

1. On GeoGebra, click “Regular Polygon” then click the coordinates (0,0) and (2,0), respectively. A

box will appear. On that box change the vertices into “3”and click “OK”.


The polygon formed is automatically labeled as ABC. Measure its sides and angles. What type

of triangle is ABC in terms of its sides and angles?

2. Click the “Midpoint or Center” then click the vertices A and B. A point will appear between A and

B and is automatically labeled as D. Click the “Segment” then click C and D.

How many triangles are there now?

3. Measure the following angles and sides.

⊿ADC ⊿BDC

𝐴𝐷 ______ ∠𝐴𝐶𝐷 ______

𝐶𝐷 ______ ∠𝐷𝐴𝐶 ______

𝐴𝐶 ______ ∠𝐶𝐷𝐴 ______

𝐵𝐷 ______ ∠𝐷𝐶𝐵 ______

𝐶𝐷 ______ ∠𝐶𝐵𝐷 ______

𝐵𝐶 ______ ∠𝐵𝐷𝐶 ______

How to measure a side: Click “Distance or Length” then click the endpoints of the side.

How to measure an angle: Click “Angle” then, click the points that form the angle. Example,

∠𝐴𝐵𝐶 , click “Angle”, “A”, “B”, “C”.

What have you observed with the corresponding angles and sides of the two triangles,

⊿ADC and ⊿BDC?

4. Complete the table on the next page using the same steps above.

42

_ _ _ _ _ _ _

Given

Vertices

Find:

EQUILATERAL TRIANGLES

A (0,0)

B (2,0)

A (0,0)

B (4,0)

A (0,0)

B (5,0)

𝐵𝐷

𝐶𝐷

𝐵𝐶

∠𝐷𝐶𝐵

sinC

cosC

tanC

∠𝐶𝐵𝐷

sinB

cosB

tanB

Let a = 𝐵𝐷 (shorter leg), b = 𝐶𝐷 (longer leg), and c = 𝐵𝐶 (hypotenuse).

What is a in terms of c?

What is c in terms of a?

What is b in terms of a?

43

Attachment 10

ACTIVITY SHEET

Names: ________________________________ Date: ___________________________

________________________________ Year & Section: __________________

________________________________

________________________________

TASK 2

Directions:

1. On GeoGebra, click “Regular Polygon” then click the coordinates (0,0) and (2,0), respectively. A

box will appear, click “OK”. The polygon formed is automatically labeled as ABCD. Measure its

sides and angles.


2. Click the “Segment” then click the vertices A and C.

How many triangles are there now?

3. Measure the following angles and sides.

⊿ADC ⊿ABC

𝐴𝐷 ______ ∠𝐷𝐶𝐴 ______

𝐶𝐷 ______ ∠𝐶𝐴𝐷 ______

𝐴𝐶 ______ ∠𝐴𝐷𝐶 ______

𝐴𝐵 ______ ∠𝐵𝐴𝐶 ______

𝐶𝐵 ______ ∠𝐴𝐶𝐵 ______

𝐴𝐶 ______ ∠𝐶𝐵𝐴 ______

How to measure a side: Click “Distance or Length” then click the endpoints of the side.

How to measure an angle: Click “Angle” then click the points that form the angle. Example,

∠𝐴𝐵𝐶 , click “Angle”, “A”, “B”, “C”.

What have you observed with the corresponding angles and sides of the two triangles,

∆ADC and ∆ABC?

4. Complete the table on the next page using the same steps above.

44

_ _ _ _ _ _ _

Given

Vertices

Find:

ISOSCELES TRIANGLES

A (0,0)

B (2,0)

A (0,0)

B (3,0)

A (0,0)

B (4,0)

𝐴𝐵

𝐶𝐵

𝐴𝐶

∠𝐵𝐴𝐶

∠𝐴𝐶𝐵

sinA

cosA

tanA

Let a = 𝐴𝐵 (leg 1), b =𝐶𝐵 (leg 2), and c = 𝐴𝐶 (hypotenuse)

What is a in terms of b?

What is a in terms of c?

What is c in terms of a?

45

Attachment 11

ASSESSMENT TASK

Name: ________________________________ Date: ________________

Year & Section: ________________________

Answer the following and show your solution.

1. Given: In ⊿BAD, ∠B = 60°, 𝐵𝐷 = 26.

Find: ∠𝐷, 𝐵𝐴 , 𝐴𝐷 , and the three trigonometric ratios of ∠B and ∠𝐷.

2. Given: In ⊿ABC,∠𝐴= ∠𝐶 , and 𝐴𝐵 = 5.

Find 𝐵𝐶 , 𝐴𝐶 , and the three trigonometric ratios of ∠𝐴 .

46

GRADE 8 MATHEMATICS

Lesson Plan in Probability: Experimental and Theoretical Probability

Content Standards




understanding of…

measures of variability

of data and basic

concepts of probability


discuss the basic

concepts of probability

and solve simple

problems involving

probabilities of events

The learner…

explain and interpret

the probability of an

event

differentiate between

an experimental

probability and a

theoretical probability


I. Objectives


determine the experimental and theoretical probability of an event/s

differentiate experimental and theoretical probability

II. Learning Task

A. Subject Matter: Experimental and Theoretical Probability

B. Reference/s:



C. Materials:

a. Worksheets for Activity and Quiz

b. Whiteboard and marker/Chalkboard

c. LCD Projector

d. Number cube or dice

III. Procedure

A. Preliminaries





attendance.

47

B. Lesson Proper

a. Review (5 min.)

TEACHER STUDENT

Yesterday, we talked about probability.

What is probability again?

Probability is a measure of the

likeliness that an event will occur.

Yes, that’s right. If I tossed a coin, what is

the probability that it will land on head?

½.

Why? Because probability is the ratio of the

number of favorable outcomes to the

total number of outcomes. The number

of favorable outcome is 1, the head,

and the total number of outcome is 2,

the head and tail, that’s why its ½.

Very good!

b. Class Activity: It’s Probably Probability (20 min.)

In this activity, the students will discover the difference between experimental

and theoretical probability through an experiment that uses a number cube. The

students will be grouped into groups of three and the teacher will give each

group a number cube (dice) and an activity sheet (see attachment 12). Each

student in the group must cooperate in the activity/experiment. The teacher will

roam around the class to facilitate and monitor the performance of each student.


After the allotted time for the activity, the teacher will now ask the students

questions which will lead to the concept of experimental and theoretical

probability.

TEACHER STUDENT

In the activity, you used a number cube or

dice and in (a) you were asked to

calculate the number of times you expect

to obtain each possible number when it is

to be rolled 60 times. What is your

answer here?

10 times or 6%.

How did you get that? Please show it on

the board.

p(1)= p(2) =p(3)= p(4)= p(5)= p(6) =

1

6

Expected number of

times each number

will be obtained =

1

6

× 60 = 10

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Okay, very good. You first took the

probability that the number cube will land

on each number which is 1/6 and

multiplied it to the times you’ll roll the

dice, that is 60. Now what you just found

out is the theoretical probability of a

number cube landing on each possible

number (1, 2, 3, 4, 5, and 6).

Okay, now let’s go to what you did on

(b). What did you found out? Is the

probability of getting each number really

10 times?

No ma’am. And that is what’s

confusing us. Each number has

different probabilities unlike the

expected number of times each number

will be obtained. Why do our results

from rolling the number cube aren’t the

same as the theoretical probability of

rolling the number cube?

Theoretical probability is a way of

estimating what could happen based on

the information that you have, it is a

calculation. It cannot predict what the

actual results will be but it does give you

an idea of what is likely to happen in a

situation.

So the results of rolling the number

cube should be somewhere around 6%

1, 6% 2, 6% 3, 6% 4, 6% 5, and 6% 6,

since that is the theoretical probability.

Now what were the results you got on

your experiment?

When you rolled the number cube 60

times, what is the probability that the

number cube landed on…

o 1 =

o 2 =

o 3 =

o 4 =

o 5 =

o 6 =

(Each group has different answers

because they have different results

though they did the same experiments.)

Sample result:

o 1 = 1/5 or 20%

o 2 = 2/15 13.33%

o 3 = ¼ or 25%

o 4 = 13/60 or 21.67%

o 5 = 7/60 or 11.67%

o 6 = 1/12 or 8.33%

Alright, what you just found is another

form of probability, the experimental

probability. Experimental probability is

the ratio of the number of trials with a

certain outcome to the total number of

trials. Let’s take for example the case of

1. The number cube landed on 1 twelve

times. The number of outcomes is 12 and

the number of trials is 60, so the

probability is 12

60=

1

5 or 20%.

Each number has different probabilities.

Two or three numbers could have the

same probabilities but not all have the

same probabilities. And these

probabilities do not precisely match with

49

the theoretical probability you’ve

calculated earlier. However, they are not

too far off.

Now you rolled the 180 times more right?

What are now the experimental

probabilities of each number?

(This depends on the results of the

experiment of each group.)

Did the probabilities increased or

decreased?

The probabilities of each number

increased.

What educated guess can you make out of

this observation?

With more number of rolls, the

resulting experimental probabilities

increase and are closer to the

theoretical probabilities.

That’s right! As the amount of trials

increases, the experimental probability

gets closer to the theoretical probability.

C. Closure/ Generalization (5 min.)



Theoretical probability is the probability that is calculated using math formulas. This

is the probability based on math theory. It is the ratio of the number of favorable

outcomes to the total number of outcomes. On the other hand, experimental

probability is calculated when the actual situation or problem is performed as an

experiment. In this case, you would perform the experiment, and use the actual results

to determine the probability. It is the ratio of the number of times an event occurred to

the number of times tested (trials). And as the number of trials increases, the

experimental probability gets closer to the theoretical probability.

IV. Rationale

Probability is one of the important mathematics concepts to understand as it is

applicable and useful to our day to day living like in weather forecasts, lotteries, and

even human behavior. It is important that students know and understand the

difference between experimental and theoretical probability as likely events

sometimes do not occur and unlikely ones do.



attachment 13).

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Attachment 12

Activity

It’s Probably Probability!

Names: ________________________________ Date: ____________________________

________________________________ Year & Section: ___________________

________________________________

Material: A number cube

Directions:

1.A number cube is to be rolled 60 times.

(a) Calculate the number of times you expect to obtain each possible number. Show your solution.

(b) Roll the number cube 60 times and record how many times (frequency) each number came up on

the second column of the table below. Then, try to roll the dice 60 times more and record the results

on the third column of the same table. After that, add the two frequencies.

Numbers on

the cube

Frequency Frequency Total

Frequencies

1

2

3

4

5

6

Total 60 180 240




1 =

2 =

3 =

4 =

5 =

6 =




1 =

2 =

3 =

4 =

5 =

6 =

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Attachment 13

Assessment Task

Name: ________________________________ Date: ________________

Year & Section: ________________________

Answer the following and SHOW YOUR SOLUTION.

1. Mark tossed a coin. What is the theoretical probability of getting a head?

2. Mark tossed the coin 20 times.

Outcomes Frequency

Heads 12

Tails 8

Total 20

What is the experimental probability of landing on tails?

Use the table below to determine the probability of each number on a number cube. Then, on

the space provided before each item, write whether it is an experimental or theoretical

probability.

__________________ 3. Rolling an odd numberOutcome Frequency

1 4

2 5

3 9

4 18

5 8

6 16

Total 50

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__________________ 4.Rolling a 6

__________________ 5.Rolling a 2 or 4

How does experimental differ from theoretical probability? (5 pts)

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

Lesson Plans on Math 14

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