BASIC EDUCATION ASSISTANCE FOR MINDANAO THIRD YEAR MATHEMATICS PYTHAGORUS' THEOREM ON PYTHAGOREAN THEOREM Information about this Learning Guide

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Third Year Mathematics

Similarities in Right Triangle

Module: On Pythagorean Theorem

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Basic Education Assistance for Mindanao (BEAM) project. Prior approval must be

given by the author(s) or the BEAM Project Management Unit and the source must

be clearly acknowledged.

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Information about this Learning Guide

Recommended number of lessons for this Learning Guide: 4

Basic Education Curriculum Competencies

Year 3 Mathematics: Pythagorus

• Apply the definition of similar triangles to derive the Pythagorean Theorem.

• If a triangle is a right triangle, then the square of the hypotenuse is equal to the sum of the squares of the legs

• Derive the relationships between the sides of particular triangles using the Pythagorean Theorem.

• isosceles right triangle

• 30-60-90 triangle

• Solve problem involving similar triangles.

Objectives

At the end of this module, students should be able to:

1. Use AAA similarity theorem to derive the relationship among the sides of a right triangle.

2. Derive the relationship among the sides of any right triangle.

3. Find the length of any side of a right triangle using the Pythagorean Theorem.

4. Apply the concept of Pythagorean Theorem to find the relationship among the sides of a right triangle applied to other objects or situations.

Essential concepts, knowledge and understandings targeted

The following concepts were used, discussed and included in this module:

1. AAA Similarity Theorem - If the three angles of two (or more) triangles are congruent correspondingly, then the triangles are similar.

2. Pythagorean Theorem - the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs.

3. For isosceles right triangle, the two legs are congruent and the length of the hypotenuse is equal to the length of the legs times the square root of 2.

4. In a 30º-60º-90º triangle, the length of the side opposite the 30º angle is one-half the length of the hypotenuse; while the length of the side opposite to 60º is equal to the length of the side opposite to 30º times the square root of 3.

Specific vocabulary introduced

The following vocabularies were used in this module:

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2. similar triangles - two or more triangles whose corresponding angles are congruent and corresponding sides are proportional.

3. Pythagorean Theorem - a theorem that states that the square of the hypotenuse is equal to the sum of the squares of the legs of the right triangle.

4. isosceles right triangle - a special right triangle whose legs are congruent.

5. 30º-60º-90º right triangle - a special right triangle whose angle measures are 30º, 60º and 90º.

Suggested organizational strategies

This module incorporates organizational strategies to be used in the actual classroom student teaching. These include:

1. Teacher as facilitator of the whole learning experience of the students in implementing this module.

2. Facilitator will prepare ahead the necessary materials needed in each activity in this module.

3. Most activities make use of small grouping, hence students must be grouped ahead for easier facilitation and smooth sailing session.

4. Students with mathematical advantage should be distributed in each group to have maximum participation from all the groups and to help facilitate the learning activities in their respective group.

Activities in this Learning Guide

Activity 1: What's in these numbers?

Multiple Intelligences • Logical/Mathematical

Skills

• Organization of parts

• Order, group, infer causes

• Generalize from given facts

• Seeing patterns

• Predict, draw conclusions

Text Types

• Observation

Activity 2: Forming Triangle

Multiple Intelligences

• Visual/Spatial

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Skills

• Seeing patterns

• Verify the value of evidence

Text Types

• Procedure

• Observation

Activity 3: Growing Right Triangles

• Visual/Spatial

• Logical/Mathematical

Skills

• Knowledge of major ideas

• Knowledge of dates events, places

• Seeing patterns

Text Types

• Exposition

• Factual Description

• Procedure

• Observation

Activity 4: AAA Similarity Theorem in Right Triangles

• Visual/Spatial

Skills

• Use old ideas to create new ones

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• Interpret facts, compare, contrast

Text Types

• Procedure

• Observation

Activity 5: Pythagorean Theorem

• Visual/Spatial

Skills

• Seeing patterns

Text Types

• Procedure

• Observation

Activity 6: The Isosceles Right Triangle

• Verbal/Linguistic

Skills

• Seeing patterns

Text Types

• Procedure

• Observation

Activity 7: The 30º-60º-90º Right Triangle

Multiple Intelligences • Visual/Spatial

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Skills

• Seeing patterns

• Understanding information

• Interpret facts, compare, contrast

Text Types

• Procedure

• Observation

Activity 8: What's the Word? That's the Word

• Visual/Spatial

Skills

• Use information

• Assess value of theories, presentations

• Use methods, concepts, theories in new situations

• Understanding information

Text Types

• Personal Response • Observation

Activity 9: Let's Try New Things Out

Multiple Intelligences • Visual/Spatial

Skills

• Solve problems using required skills or knowledge

• Use information

• Assess value of theories, presentations

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Text Types

• Factual Recount

• Personal Response

• Procedure

Activity 10: Pythagorgrams

• Visual/Spatial

Skills

• Seeing patterns

• Assess value of theories, presentations • Translate knowledge into new context

Text Types

• Personal Response

• Procedure

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Mind Map

The Mind Map displays the organization and relationship between the concepts and activities in this Learning Guide in a visual form. It is included to provide visual clues on the structure of the guide and to provide an opportunity for you, the teacher, to reorganize the guide to suit your particular context.

Assessment

All six Stages of Learning in this Learning Guide may include some advice on possible formative assessment ideas to assist you in determining the effectiveness of that stage on student learning. It can also provide information about whether the learning goals set for that stage have been achieved. Where possible, and if needed, teachers can use the formative assessment tasks for summative assessment purposes i.e as measures of student performance. It is important that your students know what they will be assessed on.

Stages of Learning

The following stages have been identified as optimal in this unit. It should be noted that the stages do not represent individual lessons. Rather, they are a series of stages over one or more lessons and indicate the suggested steps in the development of the targeted competencies and in the achievement of the stated objectives.

Literacy

Before we discuss anything about finding perimeters and circumference, we need to unlock the following words which are used in this module:

altitude of a

triangle a line segment drawn from the intersection of two the legs of a right triangle perpendicular to the hypotenuse similar triangles two or more triangles whose corresponding angles are congruent and

the corresponding sides are proportional Pythagorean

Theorem a theorem that states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs of a right triangle

isosceles right

triangle a right triangle with two congruent legs

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1. Activating Prior Learning

This stage aims to engage or focus the learners by asking them to call to mind what they know about the topic and connect it with their past learning. Activities could involve making personal connections.

Background or purpose

Students learned a lot when they are engaged and involved in deriving the concepts of a particular topic presented in the class. The use of mind boggling games like number games, mental mathematics and “mathe-magics” are just some of the few mathematical activities that the students found enjoyable, interesting and meaningful. Their learning is more meaningful and relevant if they are actively involved in the class activities.

Strategy

Investigation and Exploration

Investigationis a process of obtaining information by observation or making a study by close examination and systematic inquiry.

Explorationis trying to investigate some properties or phenomenon and trying to make an attempt to develop an initial, rough understanding of some phenomenon.

Materials

paper and pencil

Activities

1. “What's in these numbers?”

Students (individually) will be performing a task on mathematical computations by carrying out a given set of mathematical procedures.

They will then be grouped by four or five to investigate the resulting significant numbers. Procedures:

1. Choose and write any two whole numbers. These will be your first and second numbers respectively.

2. Find the sum of these two numbers. This will be your third number.

3. Repeat step 2 using your second and third number. Their sum will be your fourth number.

4. Multiply the second and third numbers. Double their product. Call this resulting number as your first significant number.

5. Multiply the first and fourth number. This product is your second significant number.

6. Multiply the second number by itself. Do the same with the third number. Find the sum of these two products. This number (sum) is your third significant number. Questions:

1. How are these three significant numbers related to each other? 2. State their relationship mathematically.

Formative Assessment

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Roundup

Students will be asked to state the mathematical relationship among the three numbers obtained from the activity in this stage.

2. Setting the Context

This stage introduces the students to what will happen in the lessons. The teacher sets the objectives/expectations for the learning experience and an overview on how the learning experience will fit into the larger scheme.

Background or purpose

The relationships that students established in the previous stage will be used to let them construct or form the desired triangle using a straight broom stick or a soft drink straw or any other indigenous materials available in the locality. The purpose of this stage is to challenge students to construct a right triangle given the corresponding length of the sticks or straws obtained in the previous stage.

Strategy

Geometric Construction

This is a strategy where basic concepts and skills of students about a certain topic or lesson delivered in the class are put or placed in actual hands-on testing or practicum.

Materials

ruler, meter stick, paper, pencil, broom stick or soft drink straw, glue or paste or scotch/masking tape, pair of scissors

Activity

2. “Forming Triangle” (refer to page 18 for this activity)

Using the same grouping used in the first activity, students will perform an activity on forming triangles resulting from the significant numbers generated in the first activity. They will choose two least sets of significant numbers and use them as the corresponding lengths of the the broom stick or straw. They will then form the desired triangle using the three lengths which are represented by the significant numbers. (see attached activity sheet 1 on page 9.) Describe the geometric figure that you formed.

Did you find any difficulty in forming the desired figures? How did you resolve such difficulties?

3. “Growing Right Triangles” (refer to page 19 for this activity)

With the triangles formed in activity 2, the students will explore further these right triangles by constructing a line segment from the point of tangency between the legs to the

hypotenuse. The said line segment must be perpendicular to the hypotenuse. Students will then find the length of this line segment. (see attached activity sheet 2 on page 11.)

Formative Assessment

How do we know that the given lengths will form the sides of a right triangle?

Roundup

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3. Learning Activity Sequence

This stage provides the information about the topic and the activities for the students. Students should be encouraged to discover their own information.

Background or purpose

At this point, students will establish similarities in right triangles based from the figure they constructed in stage 2. The AAA similarity theorem will be used to derive the basic similarity theorems in right triangles that eventually lead to the concept of Pythagorean theorem, The isosceles right triangle, and the 30º-60º-90º right triangle.

Strategy

Geometric Construction and Exploration This strategy enables students to construct geometric figures and explore their properties.

Materials

Pentel pen, manila paper (table of values for lengths or dimensions)

Activity

4. “AAA Similarity Theorem in Right Triangles” (refer to page 20 for this activity)

For the students to establish the basic relationships among the sides of any right triangle and other special right triangles, they must be able to look at the similarities between two right triangles.

The AAA similarity theorem will help the students establish the similarity concept for any two right triangles.

Once similarities were established, students will identify the proportions that exist among the corresponding parts of similar triangles.

Please refer to activity sheet number __ attached on page __.

5. “Pythagorean Theorem” (refer to page 21 for this activity)

Here the students will use the concept obtained from the preceding activity (similarities in right triangles) to find out the relationships among the sides of any right triangle.

6. “The Isosceles Right Triangle” (refer to page 22 for this activity)

In this activity, students will identify the relationships among the sides of the isosceles right triangle using the concept obtained from the preceding activity (similarities in right

triangles.)

6. “The 30°-60°-90° Right Triangle” (refer to page 23 for this activity)

In this activity, students will identify the relationships among the sides of the 30°-60°-90° right triangle using the concept obtained from the preceding activity (similarities in right triangles.).

Formative Assessment

Students will answer the following questions.

1. How are the three sides of a right triangle related?

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Roundup

Students will be asked to write down mathematically the relationships among the sides of any right triangle and other special right triangles.

4. Check for Understanding of the topic or skill

This stage is for teachers to find out how much students have understood before they apply it to other learning experiences.

Background or purpose

This stage will enhance students' understanding of the concepts considered in the preceding activities. They will be working in some tasks where they will find the length of the sides of any given right triangle and special right triangle – isosceles right triangle and the 30°-60°-90° right triangle.

Strategy

Collaborative and Cooperative Learning.

Students will collaboratively help one another to come up with a more feasible and reliable answer by presenting their respective idea(s) and each one will agree or disagree and present a more appropriate and sensible answer to a problem.

Materials

Activity sheets, task cards that contain set of problems, set of letters representing the possible answers to the problems

Activity

7. “What's the Word? That's the Word.”

Let the students discover the secret message by giving them a group task where they will solve a given set of problems – similarities in right triangle; Pythagorean theorem; isosceles right triangle; and the 30º – 60º - 90º right triangle in which answers to the problems have corresponding letters that they can use to decode the secret message.

Please refer to the attached activity sheet number 7 on page 25.

Formative Assessment

Students' outputs will be checked based on the possible answers below. Further discussion may be drawn for other possible answers that will be given by the students.

To the teacher facilitator:

The following suggested problems with the corresponding possible sets of answers can be given:

Given a = 9 cm and b = 3 cm, find u.

a b

u t

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In the figure below, if b = 30 cm and c = 18 cm, then find a.

Given the isosceles triangle below, if RS = 15 cm, find PS and PR.

In the figure below, m = 26 cm. Find a and s.

The set of possible answer are as follows:

A = 15√3 cm G = 13√3 cm M = 24√2 cm

B = 26 cm H = 12 cm N = 52 cm

D = 26√3 cm I = 15√2 cm O = 3 cm

E = 15 cm K = 48 cm P = 3√3 cm

F = 6 cm L = 34 cm R = 24 cm

Roundup

It is important to determine the difficulties of students in the given tasks for clarification. Further discussion may be done to clear misconceptions.

a b

c

A B

C

P

R S

30°

60°

M

S A m

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5. Practice and Application

In this stage, students consolidate their learning through independent or guided practice and transfer their learning to new or different situations.

Background or purpose

At this stage, allow the students to work in groups by considering some problems involving geometrical figures where they can apply the concepts they learned on similarities in right triangle, Pythagorean theorem, isosceles right triangle, and 30º – 60º – 90º right triangle.

Strategy

Collaborative and Cooperative Learning

Students at various performance levels work together in small groups towards a common goal. The students are responsible for one another's learning as well as their own. Thus, the success of one student helps other students to be successful.

Each member of a team is responsible not only for learning what is taught but also for helping teammates learn, thus creating an atmosphere of achievement. Students work through the assignment until all the group members successfully understand and complete it.

Materials

circular paper cut-out; square cut-out; rectangular chalk box; and 3-piece set of semi-circles

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Activity

8. “Let's Try New Things Out”

Students will be working by group where they will be provided with some tasks and a set of materials.

Please refer to the attached activity sheet number 8 on page 27.

Formative Assessment

Encourage group presentation to check their outputs and peer assessment to justify answers.

Roundup

Students were able to connect and apply the concepts learned in other situations. Interactive discussion should be facilitated here to clear confusions and prevent misconceptions.

6. Closure

This stage brings the series of lessons to a formal conclusion. Teachers may refocus the objectives and summarize the learning gained. Teachers can also foreshadow the next set of learning experiences and make the relevant links.

Background or purpose

To bring this session to its close, students in group will be performing some puzzles involving Pythagorean concepts.

Strategy

Working a Puzzle

Working a puzzle is a strategy where pupils and students will manipulate geometrical shapes, plane and figures to derive a specified and prescribed form.

Materials

four pieces identical right triangles, a piece of square board

Activity

9. “Pythagorgrams”

Students in group will solve a puzzle involving four identical right triangles that they can join to form the desired figure instructed in the activity sheet.

Please refer to the attached student activity sheet number 9 on page 30.

Formative Assessment

Roundup

Students were able to solve puzzles involving Pythagorean concepts. Teacher needs to clear up some misconceptions and difficulties if there are any.

Teacher Evaluation

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The ways I will evaluate the success of my teaching this unit are: 1.

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Activity 1

“Forming Triangles”

Objective:

At the end of the activity, the participants should be able to form the desired

geometric figures.

Materials:

–

broom stick (or soft drink straw), ruler/tape measure/ meter stick,

protractor

Activity Proper:

1. Choose two sets of significant numbers (preferably the least) obtained in

activity 1.

2. Use these chosen sets of significant numbers as the corresponding lengths

of the pieces of broom stick or straw. Cut them into pieces. Each set is

comprised of three lengths (representing the three numbers in a set.).

3. Connect these three lengths end to end to form any geometrical figure.

4. Identify the measure of each angle of the resulting figure in 3.

Questions:

1. Describe the figure that you formed.

2. What are the measures of each angle in the figure?

3. Find the sum of the measures of these angles.

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Activity 2

“Growing Right Triangles”

Objectives:

At the end of the activity, the students should be able to construct a

perpendicular line from the hypotenuse to the opposite vertex of a triangle and

identify its length.

Materials:

triangle (formed in the previous activity), ruler

Instructions:

1. Trace the triangle you formed in the previous activity in a given sheet of

paper.

2. Label your triangle as triangle ABC, where B is the right angle, segment AB as

the longer leg, segment BC as the shorter leg and segment AC as the

hypotenuse. Let DB be the segment drawn from the hypotenuse AC to vertex

B. 3. Identify the lengths of each segment and summarize your findings by

completing the table below:

AB

AC

BC

DB

AD

CD

Questions:

1. Describe the resulting figure after a line was drawn from the hypotenuse to

the opposite vertex, B.

2. Identify and draw the resulting right triangles formed after constructing a

perpendicular line from the hypotenuse to the opposite vertex, B.

3. Identify the different parts of the three resulting right triangles by

completing the table below. Some parts were provided as examples.

∆

ABC

∆

BDC

∆

ADB

right angle

∠

B

hypotenuse

Segment AB

longer leg

Segment BD

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Activity 3

“AAA Similarity Theorem in Right Triangles”

Objective:

At the end of the activity, the students should be able to establish the basic

similarity theorem of right triangles using the AAA Similarity Theorem.

Materials:

paper, ball pen, triangle (formed in the previous activity), ruler

Activity Proper:

1. Using the data obtained from the previous activity, identify the length of each

segment.

2. Summarize your data by completing the table below.

∆

ABC

∆

BDC

∆

ADB

right angle

hypotenuse

longer leg

shorter leg

Questions:

1. Compare

− ∠

B in

∆

ABC with

∠

D in

∆

BDC.

-

∠

C in

∆

ABC with

∠

C in

∆

BDC.

-

∠

A in

∆

ABC with

∠

B in

∆

BDC

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Activity 4

“Pythagorean Theorem”

Objective:

At the end of the activity, the participants should be able to establish the

relationship among the sides of any right triangle.

Materials:

paper, ball pen, triangle (formed in the previous activity), ruler

Activity Proper:

1. Using the data obtained in the previous activity, identify the length of each

segment.

2. Summarize your data by completing the table below.

∆

ABC

∆

BDC

∆

ADB

shorter leg

longer leg

hypotenuse

(shorter leg)

2

(longer leg)

2

(hypotenuse)

2

Questions:

1. Get the sum of the squares of the shorter and longer legs of each triangle.

2. Compare the sum of these legs with the square of the hypotenuse.

3. Is your observation true to the other two triangles?

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Activity 5

“Isosceles Right Triangle”

Objective:

At the end of the activity, the participants should be able to establish the

relationship among the sides of any isosceles right triangle.

Materials:

paper, ball pen, isosceles triangle cut-out (as shown below), ruler, protractor

Instruction:

1. Using a ruler, measure the lengths of the sides of the given right triangle cut

out. With the protractor, measure its angles . Record your findings in the

table below.

Parts

Measurement

Segment AB

Segment BC

Segment AC

∠Α

∠

B

∠

C

Questions:

1. Describe the length of the legs of the given triangle.

2. What about the angles?

3. Now, how would you classify this triangle? Support your answer.

A

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Activity 6

“The 30

°

-60

°

-90

°

Right Triangle”

Objective:

At the end of the activity, the participants should be able to establish the

relationship among the sides of any 30

°

-60

°

-90

°

right triangle.

Materials:

paper, ball pen, 30

°

-60

°

-90

°

right triangle cut-out (as shown below), ruler,

protractor

Activity Proper:

1. Using the ruler, measure the lengths of the sides of the given right triangle.

Measure also the angles of the given right triangle. Record your findings in the

table below.

Parts

Measurement

Segment AB

Segment BC

Segment AC

∠Α

∠

B

∠

C

B C

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Questions:

1. Compare the measures of the angles in the given triangle.

2. Compare the length of the side opposite the 30

°−

angle to the length of the

hypotenuse.

3. Compare also the length of the side opposite to the 60

°−

angle to the length

of the side opposite the angle whose measure is 30

°.

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Activity 7

“What's the Word? That's the Word!”

Objective:

At the end of the activity, the students should be able to solve the given

problems and be able to decode the hidden word.

Materials:

activity sheets, task cards (set of problems), set of letters representing the

possible answers to the problems

Instructions:

1. Each group will be given a set of task cards and a set of letters with the

corresponding possible answers to the given problem.

2. Solve each problem by finding the indicated missing length.

3. Then, look for the letter that corresponds to your answer and paste this in the

appropriate blank below. Find the Magic Word that will be formed out of

these letters after solving all the given problems.

The Magic Word is ___ _ _ _ _ ___

u a

2

PS PR a

4

s

Problems:

1. Given a = 9 cm and b = 3 cm. Find u.

a b

u t

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2. In the figure below, b = 30 cm and c = 18 cm. Find a2.

3. Given below is an isosceles triangle. If RS = 15 cm, find PS and PR.

4. In the figure below, m = 26 cm. Find a4 and s.

The set of possible answers are as follows:

A = 15√3 cm G = 13√3 cm M = 24√2 cm

B = 26 cm H = 12 cm N = 52 cm

D = 26√3 cm I = 15√2 cm O = 3 cm

E = 15 cm K = 48 cm P = 3√3 cm

F = 6 cm L = 34 cm R = 24 cm

a

2

b c

A B

C

P

R S

30°

60°

M

S A m

a4

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