Year III Mathematics Geometry of Size and Shape Angles of Polygons Module I — Sum of the Measures of the Angles of Polygons LEARNING GUIDE

Year III Mathematics

Geometry of Size and Shape

Angles of Polygons

Module I — Sum of the Measures of the Angles of Polygons

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.

Determine the sum of the measures of the angles of polygons the angles of a triangle

the exterior angles of a quadrilateral the interior angles of a polygon

Define, identify and name the terms related to the circle (radius, diameter and chord)

Objectives In this Learning Guide, students are expected to:

Investigate the nature and properties of interior and exterior angles of various polygons. Determine the sum of the measures of the interior and exterior angles of a polygon. Propose rules governing the behavior of those angles and test their rulesThe . Solve problems on polygons.

Reflect their learnings about the sum of angles of polygons.

It is also expected that the students should be able to come up with the following:

Sum of the measures of the interior angles of a triangle is 180 degrees.

Sum of the measures of the interior angles for any polygons is (n - 2) multiplied by 180 degrees, where n = the number of sides of the polygon.

Sum of the measures of the exterior angles for any polygon is 360 degrees.

Essential concepts, knowledge and

understandings targeted These are the esential concepts, knowledge and understanding targeted in this Learning Guide: The sum of the degree measures of the interior angles of a triangle is 180.

The sum of interior angles of a polygon = (n - 2) x 180 degrees, where n = the number of sides of the polygon.

The sum of the exterior angles of a polygon is 360 degrees.

Specific vocabulary introduced The following list of words are used in this Learning Guide:

Polygon is a closed figure made up of three or more line segments joined at their endpoints.

Triangle is a polygon formed by three noncollinear segments joined at their endpoints. An angle is a figure formed by two rays with a common endpoint, and which are not on the same line.

Interior angle is an angle formed by two consecutive sides.

Exterior angle is an angle formed by a side and an extension of an adjacent side of the polygon.

Quadrilateral is a polygon with four sides.

Suggested organizational strategies The following are suggested organizational strategies that would help the teachers in carrying out the different activities of the different stages in this Learning Guide:

Stage 1:

It is suggested that the students work in small groups to: label different types of polygons

measure and label two different angles. Stage 2:

A whole class activity relating the content of this Learning Guide to their previous work on polygons by constructing a concept map.

Stage 3:

It is suggested that the students work pairs or in groups to: find the sum of the angles of polygons

make a generalization on the sum of the angles of polygons formulate rules on the sum of angles of polygons.

Stage 4:

Students work in groups to: solve problems on polygons. Stage 5:

Students work in groups to:

determine the pattern that emerges

propose possible formula that mathematically describes the pattern Stage 6:

Individually, students will have to write a journal reflecting their learnings in the series of activities in this Learning Guide.

Activities Multiple Intelligences Skills Text Types

1 Activity 1: Polygons Revisited Logical/Mathematical Visual/Spatial

Observation and recall of information Knowledge of major ideas

Logical/Mathematical Visual/Spatial Interpersonal

Knowledge of major ideasSeeing patterns

Organization of partsGeneralize from given facts ReviewExplanationProcedure Exposition Discussion

4 Activity 4: Another side, another angle Verbal/Linguistic Logical/Mathematical Visual/Spatial Interpersonal

Observation and recall of information Understanding informationGrasp meaning Interpret facts, compare, contrast Predict consequencesUse information Seeing patternsOrganization of parts

Observation Procedure Procedural Recount Explanation Exposition Discussion

5 Activity 5: Yet another another Verbal/Linguistic Logical/Mathematical Visual/Spatial Interpersonal

Observation and recall of information Understanding informationUse information Generalize from given facts

Narrative Observation Procedural Recount Factual Recount Explanation Exposition Discussion

6 Activity 6: Lets get General Verbal/Linguistic Logical/Mathematical Visual/Spatial Interpersonal

Observation and recall of information Mastery of subject matter

Understanding informationGrasp meaning Seeing patternsOrganization of parts Generalize from given facts

Predict, draw conclusions

Observation Factual Description Explanation Exposition Discussion

7 Activity 7: Time to test the theory Logical/Mathematical Visual/Spatial Body/Kinaesthetic Interpersonal

Observation and recall of information

Understanding informationPredict consequences Use informationOrganization of parts

Generalize from given facts Predict, draw conclusions

Procedural Recount Explanation Exposition Discussion

8 Activity 8: Now for the Exterior Logical/Mathematical Visual/Spatial Body/Kinaesthetic Interpersonal

Observation and recall of information Understanding informationGrasp meaning Interpret facts, compare, contrast Predict consequences

Use methods, concepts, theories in new situations Seeing patternsPredict, draw conclusions

Narrative Observation Procedural Recount Explanation Exposition Discussion

9 Activity 9: Find the Missing Angles Logical/Mathematical Visual/Spatial Interpersonal

Mastery of subject matter Understanding information Interpret facts, compare, contrast Predict consequences

Solve problems using required skills or knowledge Seeing patternsPredict, draw conclusions

Narrative Observation Procedural Recount Factual Recount Explanation Exposition Discussion

10 Activity 10: Another way of looking at it Verbal/Linguistic Logical/Mathematical Visual/Spatial Interpersonal

Observation and recall of information Knowledge of major ideas

Understanding informationGrasp meaning Interpret facts, compare, contrast Predict consequencesUse information

Use methods, concepts, theories in new situations Seeing patternsOrganization of parts

Predict, draw conclusions

Narrative Observation Factual Description Procedural Recount Factual Recount Explanation Exposition Discussion

11 Activity 11: Let's Write About it Verbal/Linguistic Logical/Mathematical Intrapersonal

Observation and recall of information Knowledge of major ideas

Mastery of subject matter

Understanding informationGrasp meaning Translate knowledge into new context Interpret facts, compare, contrast

Organization of partsIdentification of components Relate knowledge from several areas

Narrative Observation Personal Response ReviewExplanation Exposition Discussion

Opportunities to integrate:

Other Subjects In activity 11, page 26 Reading and Writing is intgrated.

Peace Education In this Learning Guide, collaborative work is given an emphasis to promote peace and

understanding among students. They work in a small group regardless of tribe belief and status in life.

Values Education Valuing other's opinion and idea. Care for others.

Sharing ideas and helping others to accomplish tasks.

Environmental Education In activity 11, care for environment is emphasize like taking care the landscape of a land area by redesigning it.

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.

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.

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.

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

A quick revision of the competencies related to angles and types of polygons will determine whether the students are ready for the work in this Learning Guide. You may need to revise some of the earlier work if you discover misconceptions in the following activity.

Strategy: Think, Ink, Share — Cooperative Learning

What is it?

By working together as a group, students can reach consensus and check their

Activity 1: Polygons Revisited

This activity is a simple revision of some fundamental concepts related to angles and polygons.

It is suggested that the students work in small groups to:

• label different types of polygons

• measure and label two different angles.

They can then share their outputs with another group and arrive at a mutually agreed set of results.

Formative Assessment

It is important that you get a clear picture of the level of student competence in this activity. Look for misconceptions or misunderstandings while the students are involved in group discussions and during the ensuing whole-class roundup. Involve other students in the effort to correct them.

Roundup

A quick, whole class round-up of the outputs could be useful in generating comments and questions.

Note: An alternative to photocopying the Activity Sheet would be to prepare a poster on Manila paper and label the figures A, B, C etc. The students can provide their responses in their workbooks or scraps of paper.

2. Setting the Context

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

Background or Purpose

This Stage of the Learning Guide aims to extend the students' knowledge about polygons by investigating the properties associated with internal and external angles.

Strategy: Concept Mapping

It is suggested that you explore the properties of polygons as a way of:

• classifying them according to the number of sides and vertices

• determining what rules govern the angles formed at the vertices — both internal and external.

Activity

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

In this stage there are 7 activities that will allow the students to:

• investigate the nature and properties of interior and exterior angles of various polygons

• propose rules governing the behavior of those angles, and

• test their rules.

Strategy: Co-operative learning

Activities and structures are used as basic tools for group work skills. The activities have definite aims and purpose and they should not be seen in isolation but as an overall part of the learning environment. Ground rules must be established and trust between all players needs to be built.

There are five key elements of Collaborative Learning:

Positive Interdependence

occurs when all members of the group feel connected to each other in the accomplishment of common goal. All individuals must succeed for the group to succeed.

Individual Accountability

holding every member of the group responsible to demonstrate accomplishing the learning taking place.

Face to Face Interaction

Processing the Learning

when group members discuss the learning that has taken place and assess their collaborative efforts in regard to the outcomes.

Social Skills

when groups are made aware of the human interaction skills that are needed to succeed in a co-operative activity. These skills focus on developing communication, trust, leadership and conflict resolution.

These five elements can only work if a supportive environment is encouraged.

Activities

All the activities can be done as either individual task work or in small groups. It is strongly suggested that students get the opportunity to discuss their work with others and formulate rules based on that discussion. Your role as teacher is to facilitate the activities, check the students' progress through the activities and engage them in meaningful

discussion as the opportunity arises — either individually, in small groups or as a whole class.

The activities in this stage are largely self-explanatory, however the following notes are provided in support.

Activity 2: Measure Me!

Refer to page 15.

The purpose of this activity is to:

• revisit the measurement of angles using a protractor

• establish a rule relating to the sum of angles in a straight line

• provide a vehicle for beginning the cooperative learning strategies used in the next few activities, and

• provide a basis for the method used to determine the rules about the sum of interior and exterior angles in a polygon.

It is likely that students will get an answer close to, but not equal to 180 degrees.

Discuss their answers either at a group or whole-class level. Two important issues can arise that may need clarification:

1. The angle in a straight line is 180 degrees (students should make this association by themselves).

2. The accuracy of their measurements is limited by the equipment they use. By making three measurements, the errors could easily compound.

Activity 3: What's my Sum?

Refer to page 16.

At the end of this activity, students are expected to:

• find the sum of the angles of a triangle;

• make a generalization on the sum of the angles of any triangle.

The students should be able to make the connection between this activity and the previous one where they established that the angle in a straight line is 180 degrees. They are asked to form a generalization about the sum of interior angles of a triangle. Get the students to refine their statements through discussions with others. Then, as a whole class, engage the students by seeking inputs to arrive at a mutually agreeable rule. Only after you are satisfied that you have exhausted their inputs, compare their rule with the Angle Sum Theorem — “The sum of the degree measures of the interior angles of a triangle is 180.”

Activity 4: Another side, another angle

Refer to page 17.

In this activity, students get to extend their investigation to explore other polygons. Using and extending their knowledge about the sum of angles in a triangle, this activity gets them to form two triangles from a quadrilateral, as in the following diagram.

It is important to be vigilant to see if students understand the instructions properly. Students may find it difficult to make the connections between establishing two triangles and determining the sum of the interior angles of a quadrilateral.

Encourage group discussion to allow them to discover the relationship and arrive at the answer.

As possible extensions to this activity, get the students to:

• draw lines from other vertices to construct different sets of triangles

• draw other quadrilaterals and repeat the activity.

Activity 5: Yet another angle

Refer to page 18.

Now that the students have the basic idea of constructing internal triangles in a polygon as a method to calculate the sum of interior angles, this activity gives them the opportunity to use the same method on a pentagon.

This time, however, they are expected to make the leap to the formula: Sum of interior angles = number of triangles that can be formed X 180 degrees.

In this case, three triangles can be formed (ABC, ACD and ADE), so: Sum of interior angles of a pentagon = 3 X 180 degrees, or 540 degrees.

Activity 6: Lets get General

Refer to page 19.

With the background knowledge they have gained so far, it is time to consolidate and generalize.

Type of

Polygon

Number

of sides

(S)

Number of interior

triangles we can

form (T)

Difference

(S-T)

Sum of

Interior

Angles

Triangle

3

1

2

180

Quadrilateral

4

2

2

360

Pentagon

5

3

2

540

The students should arrive at the generalization:

The sum of interior angles of a polygon = (n — 2) X 180o_{, where n = the number of sides of}

the polygon.

The results they should provide in the next part of the activity are shown below.

Type of

Polygon

Number

of sides

(S)

Number of interior

triangles we can

form (T)

Difference

(S-T)

Sum of

Interior Angles

Hexagon

6

4

2

4 x 180 = 720

Heptagon

7

5

2

5 x 180 = 900

Octagon

8

6

2

6 x 180 = 1080

Activity 7: Time to test the theory

Refer to page 20.

In this activity the students will validate, as far as the accuracy of protractors will allow, The results they calculated for the sum of interior angles of a hexagon, heptagon and octagon from the previous activity.

Allowances for errors should be made. Ensure that all students are satisfied that they have discovered the rule for calculating the sum of interior angles of a polygon.

An option for extension would be to get students to calculate the sum of interior angles of further polygons. Get them to draw a graph of Number of sides vs Sum of interior angles, and ask them to interpret the graph.

Activity 8: Now for the Exterior

Refer to page 22.

The last activity in this stage provides the opportunity for the students to suggest and try a method for determining the sum of exterior angles of a quadrilateral. This involves

problem solving strategies.

• Clarify — what is the problem asking you to do or find out?

• Choose — what tools would be effective for solving the problem?

• Use — use the strategies or tools to gain an answer to the problem.

• Interpret — is the answer reasonable? Can you check it using another method?

• Communicate — depending on your audience, communicate the answer. This may be in a simple form for example a written sentence or a more complex form example writing a report.

Two likely solutions to the first part of this activity might include: 1. using a protractor to calculate the sum, or

2. cutting the polygon into pieces and assembling the exterior angles at a common vertex in the same way as in Activity 3.

It is possible that students will suggest other methods. Any ideas should be encouraged. You may wish to ask group representatives to post and report on their outputs. Get students to react, to seek clarification or offer alternatives during this time.

Also encourage students to extend this idea to more complex polygons in order to test their result, which should be that the sum of the exterior angles of a polygon = 360o_.

Formative Assessment

From your observations, you should have a clear idea of the grasp of the concepts and the level of involvement by most students.

For a more formal measure, two rubrics are provided:

1. Group Work Activity in determining and proving a rule (theorem), suitable for Activities 6 and 7 combined (refer to page 29).

2. Problem Solving, suitable in particular for Activity 8 (refer to page 30)

Roundup

Get the students to suggest extensions to the maps to cover the rules they have learned in the above series of activities.

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

Students will now have the opportunity to demonstrate that they can solve some problems based on the concepts they have encountered in this Learning Guide.

Strategy: Problem Solving

It is suggested that the students work independently to solve the problems in this stage. The results of their work will provide you with an indication of their level of mastery and of those areas you may need to revisit to clarify misconceptions.

Activity 9: Find the Missing Angles

Formative Assessment

There are many options available for you here. You can treat the activity as:

• a formal test, mark and grade the students' outputs

• an informal test, where you use the results to indicate the success of the learning in this

Learning Guide

• an opportunity for further group work, checking the results at a group rather than individual level.

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

To further enhance the learning of the students, they will be dealing with some practice and application of the concepts and skills they learned from the previous activities.

Strategy: More problem solving.

The students will be confronted with another possible way to calculate the sum of interior angles of a polygon. The activity is best done with students in small groups to facilitate discussion and cooperative learning.

Activity 10: Another way of looking at it

The students are asked to complete the following table, which has been completed for you.

Polygon

Number of Sides

Sum of Interior

Angles (degrees)

Sum of Interior

Angles in terms of

Right Angles

Triangle

3

180

2

Quadrilateral

4

360

4

Pentagon

5

540

6

Hexagon

6

720

8

Heptagon

7

900

10

Octagon

8

1080

12

Students are encouraged to:

• determine the pattern that emerges

• propose possible formula that mathematically describes the pattern.

Roundup

Engage the students at a class level to examine other groups' outputs and encourage constructive debate about the possible formula.

6. Closure

This activity encourages students to reflect on, and summarize in words or other formats what they have learned in this series of lessons.

Strategy: Journals

What is it?

Journals provide a good opportunity for students to demonstrate and reflect on their learning.

They provide a good source of assessment for teachers. Journals don't have to be written — diagrams and other drawings are authentic forms of demonstrating learning.

Activity 11: Let's write about it

Let the students work individually in producing their journal. You may want to let them share their journals with the whole class.

Formative Assessment

Check the work of the students and process their answers. To score their work you may use the suggested rubric on page 32.

Teacher Evaluation

(To be completed by the teacher using this Teacher’s Guide) The ways I will evaluate the success of my teaching this unit are: 3.

Activity 1

Polygons Revisited

Do you remember your earlier work on Angles and Polygons? Well, here's you

chance to quickly revisit it.

Instructions

Work in pairs or small groups and refer to the following diagram:

Tasks:

Label all the figures in the diagram.

Measure angle a:

degrees

What do we call this type of angle?

Measure angle b:

degrees

What do we call this type of angle?

Is a square a rectangle?

Activity 2

Measure Me!

Material:

Protractor

Instructions:

1. Work in small groups on this activity.

2. Use a protractor to measure

∠

a,

∠

b,

∠

c in the figure below, and calculate

the sum of the three angles that make up a straight line.

Angle 1 =

, Angle 2 =

, Angle 3 =

, Sum =

3. Now, write a sentence that summarizes what you have found:

4. Compare your results with other groups. Did everyone get the same answer? If

not, try and explain why.

Activity 3

What's my Sum?

Materials:

Protractor, Ruler, Pencil, Scissors, Paste, Bond/Colored Paper

Instructions:

1. Draw and cutout a triangle using bond, colored or recycled paper. Any type of

triangles will do like scalene, equilateral, isosceles, etc. Make it as big as you

like.

2. Label the three angles as a, b and c.

3. Use the protractor to measure each angle in the triangle and get the sum of

their measures.

Angle 1 =

, Angle 2 =

, Angle 3 =

, Sum =

4. Now tear the triangle into 3 pieces, making each tear from the middle of each

side to the center of the triangle, as in the diagram below.

5. Put the angles side by side such that the three angles share a common vertex

and the sides touch, as in the diagram.

Questions:

1. In what way is this activity similar to the previous activity?

2. Compare your results with other groups. Does the result depend on the type

of triangle you make:

3. Make complete the following sentence that summarizes what you have

discovered:

Another side, another angle

Instructions:

1. In the quadrilateral below. draw a line from vertex A to vertex C.

2. How many triangles are there in the figure?

Activity 5

Yet another angle

OK, so you know what the sum of the interior angles of a triangle and

quadrilateral are. How about a pentagon?

Instructions

1. Draw lines from vertex A to other vertices to form triangles in the following

figure.

2. How many triangles have you formed?

3. The sum of interior angles of each triangle is

4. So, the sum of interior angles of a pentagon is:

Activity 6

Lets get General

We have examined three polygons — a triangle, a quadrilateral and a pentagon.

Let's see if we can make a rule about the sum of interior angles of a polygon.

1. Summarize what you have found so far in the following table:

Type of

Polygon

Number of

sides (S)

Number of interior

triangles we can

form (T)

Difference

(S-T)

Sum of Interior

Angles

Triangle

Quadrilateral

Pentagon

2. Can you see a pattern? Complete the following sentences:

3. The number of triangles we can form inside a polygon:

= the number of sides —

4. Therefore, the sum of interior angles of a polygon:

= (the number of sides —

) X 180

If we represent the number of sides of a polygon with the letter n, then the rule

is:

5. Use your rule to complete the table below for other polygons.

Type of

Polygon

Number of

sides (S)

Number of interior

triangles we can

form (T)

Difference

(S-T)

Sum of Interior

Angles

Hexagon

Heptagon

Activity 7

Time to test the theory

You have discovered the rule for calculating the sum of the interior angles of a

polygon. Now let's do a rough test of the rule.

Materials:

Protractor

Instructions:

1. For each figure below, measure and mark the interior angles.

2. Then find the sum of the interior angles.

3. Complete the table:

Polygon

Sum of interior angles using the

Activity 8

Now for the Exterior

Now that we know about interior angles, it is time to discover something about

exterior angles.

The exterior angles of a polygon are the angles you get if you extend the sides.

The exterior angles a, b, c and d of a quadrilateral ABCD are shown:

Instructions:

1. Your task is to use this shape to find the sum of the exterior angles of a

quadrilateral.

2. Work with your fellow group members to decide on a method and arrive at a

solution.

3. Describe what you did and show your results below:

4. Now, make a prediction about the sum of the external angles of all polygons

by completing the following sentence:

5. Use any piece of paper to draw another polygon. It can be any polygon —

hexagon, nonagon or whatever. Show its external angles. Use and describe a

method to test your prediction by finding the sum of the exterior angles.

6. Compare your results with those from other groups.

Activity 9

Find the Missing Angles

Activity Proper:

You are given the situations below.

Show your solutions and give your justifications.

Situation 1

Situation 2

A group of architecture

students are planning to

redesign the landscape as

shown in the illustration.

For them to make a good

design they need to determine

the measure of the angle

indicated by 4x.

Help the students solve their problem.

Situation 3

I went to check a lot that

was for sale and made the

following rough map.

Can you calculate the

Activity 10

Another way of looking at it

Problem 1

You already know that the sum of the interior angles of a triangle is 180

_{or two}

right angles.

Examine and complete the following table:

Polygon

Number of Sides

Sum of Interior

Angles (degrees)

Sum of Interior

Angles in terms of

Right Angles

Triangle

3

180

2

Quadrilateral

4

360

4

Pentagon

5

Hexagon

Questions

1. Is there a pattern in the results? What do you think it is?

Activity 11

Let's write about it

Name:________________________ Year and Section:_________ Date:_________

Activity Proper:

Individually, demonstrate and reflect on your learning about the sum of the angles of a

polygon. Indicate your answers on the space after each unfinished sentence. You can

use diagrams and other drawings to demonstrate your learning.

I have learned that...

I find the lesson...

CRITERIA

Very Satisfactory

4-5

Satisfactory

2-3

Not Satisfactory

0-1

Group Participation

All students enthusiastically

participate.

At least half of the students

confer or present ideas

Only one or two persons

actively participate

All members of the group

contributed their ideas during

the activity

Some members of the group

did not share their ideas with

the team during the activity

No member of the group

shared their ideas

Theorem proved logically

Completely logically approach

_taken

Majority of the steps show

_logic

Illogical approach used

Quality Interaction

All students reflect awareness

of other views and opinions in

their discussion. All students

are very participative and

active in sharing their

opinions.

Some students reflect

awareness of other views and

opinions in their discussion.

Some students are very

participative and active in

sharing their opinions.

Very few students reflect

awareness of other views and

opinions in their discussion.

Very few students are

participative and active in

sharing their opinions.

Quality of Work

It appears that a strong effort

was made and great pride was

taken in meeting the highest

standards

It appears that a good effort

was made and the work

appears close to meeting the

highest standards

Rubric: Problem Solving

CRITERIA

Excellent

4

Very Satisfactory

3

Satisfactory

2

Needs

Improvement

1

Mathematical Concepts

Explanation shows

complete understanding

of the mathematical

concepts used to solve

the problem/s.

Explanation shows

substantial

understanding of the

mathematical concepts

used to solve the

problem/s.

Explanation shows some

understanding of the

mathematical concepts

used to solve the

problem/s.

Explanation shows very

limited understanding

of the underlying

concepts needed to

solve the problem/s OR

is not written.

Mathematical Reasoning

Uses complex and

refined mathematical

reasoning.

Uses effective

mathematical reasoning

Some evidence of

mathematical

reasoning.

Little evidence of

mathematical

reasoning.

Mathematical errors

90-100% of the steps

and solutions have no

mathematical errors.

Almost all (85-89%) of

the steps and solutions

have no mathematical

errors.

Most (75-84%) of the

steps and solutions have

no mathematical errors.

More than 75% of the

steps and solutions have

no mathematical errors.

Working with others

Student was an engaged

partner, listening to

suggestions of others

and working

cooperatively.

Student was an engaged

partner, but had

trouble listening to

others and/or working

cooperatively.

Student cooperated

with others, but needed

prompting to stay

on-task.

Point / Score

Description

5

With correct answer and complete solution and justifications

4

With correct answer but with 1 or 2 missing solutions and/or justifications

3

With correct answer only (NO SOLUTION)

With incorrect answer but with right solution and justifications

2

Had made an attempt to solve the problem (incorrect answer and unreasonable solutions and

_{justifications)}

Rubric for Journal Writing

Point / Score

Description

10

With complete demonstration and reflection of learning with diagrams and other drawings

8

With complete demonstration and reflection of learning but with 1 or 2 missing informations

6

With complete reflection only (NO DEMONSTRATION like diagrams or drawings)

With complete demonstration (like diagrams or drawings) only (NO WRITTEN REFLECTION)

4

Had made an attempt to demonstrate and reflect on the learning (incorrect and unreasonable

_{demonstration and reflection of learning)}

1.

2.

Setting the

Context

3.

Learning

Activity Sequence

4.

Check for

Understanding

5.

Practice and

Application

6.

Closure

Strategies

Activities from the Learning Guide

Extra activities you may wish to include

Materials and planning needed

Estimated time for this Stage