Description

Students use hands-on manipulatives to explore and describe the properties and attributes of the “fundamental” polygon: triangles. This is the fourth lesson in a series of five on geometry.

Materials

An overhead, chart paper, and/or white board will be needed all week.

Day 1:
-“Polygons A-F” (See Associated File.)
-Scissors (one pair per student)
-“Classifying Angles Review” sheet (See Associated File.)
-“Classifying Angles: Using Degrees” instructional model (See Associated File.)
-Protractors (one semi-circle and one circle protractor— if possible)
-“The ‘Fun’ Polygon” (See Associated File.)
-“The ‘Fun’ Polygon Classification and Measurement Chart” (See Associated File.)
-Chart paper
-“Trying Triangles” (See Associated File.)
-“Triangle Trivia” (See Associated File.)--optional

Day 2:
-Completed homework “Trying Triangles” (See Day 1.)
-Long-Answer Question Rubric (See Associated File.)
-“The ‘Fun’ Polygon” (from Day 1)
-“The ‘Fun’ Polygon Classification and Measurement Chart” (from Day 1)
-Rulers (one per student)
-Textbook

Day 3:
-Long-Answer Question Rubric (from Day 2)
-“The ‘Fun’ Polygon Manipulatives” handout (See Associated File.)
-Scissors
-“The ‘Fun’ Polygon” (See Day 1.)
-Textbook

Day 4:
-Long-Answer Question Rubric (See Associated File.)
-“The ‘Fun’ Polygon” (from Day 1)
-“The ‘Fun’ Polygon Classification and Measurement Chart” (from Day 1)
-Textbook
-“Tricky Triangles” and “True Triangles” (See Associated File.)
-“Traveling Triangles” (See Associated File.)
-Crayons and/or colored pencils
-“Equilateral Triangles” manipulatives (See Associated File.)
-Computers and software with basic drawing capabilities
-Disks (one per student)—optional
-Student web lessons “Let’s Learn Symmetry” and “Congruent or Not” from Beacon Learning Center—optional (See Weblinks)

Day 5:
-Building Code Check-Up #4 (See Associated File.)
-“Equilateral Triangle” manipulatives (See Associated File.)
-Rulers
-Scoring Criteria (See Associated File.)
-Tessellation Coloring Page (See Extensions.)
-Crayons and/or colored pencils

Procedures

Day 1:
A. Pass out student copies of “Polygons A-F” and scissors. Pose the following statement for students to think about as they cut out each polygon: “The ‘point’ is the fundamental building block from which lines, line segments, rays, planes and angles are all constructed. There is also a fundamental polygon from which all other polygons can be constructed.”

B. As students are cutting and thinking, take time to review the names of the polygons shown on the handout (hexagon, rectangle, regular pentagon, quadrilateral, regular hexagon, and dodecagon).

C. Once all polygons have been cut out, have students fold the polygons to find the various lines of symmetry that exist. (There will not be a line of symmetry for the quadrilateral.) (Symmetry was introduced during lesson 3.)

D. Have students unfold the polygons and lay them on their desks. Pose the following question, “What is the fundamental polygon from which all other polygons can be constructed?”

E. Solicit students’ hypotheses and explanations. After sufficient discussion (and if consensus has not been reached), use a ruler and line segments to dissect each polygon to reveal the fundamental building block of all polygons— the triangle! Explain that all other polygons can be divided into smaller triangles. (Encourage the students to test this “truth” with the polygons they encounter during the upcoming week.)

F. Present week’s goal: To construct, classify* and describe the properties and attributes of the “fun” polygon. (*Paragraphs will be written to explain classifications.)

G. Draw a triangle on the overhead or whiteboard. Review that the prefix “tri-” means “three,” and write the following definition: “A triangle is a polygon with three sides and three angles.”

H. Ask students to classify the angles shown in the triangle drawn. If needed, use the “Classifying Angles Review” sheet to review the various types of angles studied so far: straight, right, acute, and obtuse. If the Review sheet is used, explain that the area of “Acute Angles” represents the angles that are less than a “Right Angle,” and that the area of “Obtuse Angles” represents those angles greater than a “Right Angle.”

I. Note: If wax-paper circles were used in previous lessons to measure angles, use steps J, K and L to transition into protractors. If students are already familiar with using degrees to refer to common angles—0, 45, 90, 180, and 360—feel free to skip these steps.

J. Explain that there is a “very small unit angle,” known as a “degree,” that can also be used to measure angles. Tell the students that there are 360 of these small unit angles in a complete circle. Show students a circular protractor to reinforce this statement, if possible.

K. Use the “Classifying Angles: Using Degrees” sheets to provide visual models as you share the following information: A protractor is a tool that is used to measure the number of degrees in an angle. Many protractors are in the form of a semi-circle. Ask, “What angle is formed by a semicircle?” (straight angle) Tell the students that there are 180 degrees in a straight angle; it takes 180 very small unit angles to make a straight angle. Ask, “Have you ever heard of a 180 degree turn?” Allow students to share where they have heard these words used before. (For example, many skateboarders use this term to indicate a specific turn/movement.)

L. Say, “If there are 360 degrees (360º) in a complete circle, and 180 degrees (180º) in a straight angle, how many degrees are in a right angle?” Allow students time to determine the answer. Then use the measure of right angles (90º) as a reference point for helping students understand that acute angles measure less than 90º and obtuse angles measure more than 90º (but less than 180º). Note: The goal of this instruction is to help students build reference points for understanding common angle measurements. For this lesson, exact measurements will not be needed to accurately classify angles.

M. Pass out copies of “The ‘Fun’ Polygon” and the accompanying “Classification and Measurement Chart.” Read the directions for Step 1 and model how to classify and record the angles shown in Triangle 1. Have students complete the chart for Triangle 1 as you model. For Triangle 1, their charts should show a similar version of this “regular” triangle, and the labels, “acute,” “acute,” and “acute” for angles a, b, c. (For fun, tell the students that these are “cute little angles;” that is why they are classified as “acute.”)

N. Have students try Triangle 2 on their own. As a class, check their answers (acute, right, acute) and allow students to correct their papers as needed.

O. Allow time for students to complete Step 1 for Triangles 3-7. (Take this time to monitor students’ progress and provide assistance as needed.)

P. When most have finished, have the students share their classifications for Triangles 3-7. (An Answer Key is available in the Associated File.) If discrepancies arise about the answers, allow students to defend their work. Also be ready to direct them back to the “Classifying Angles: Using Degrees” sheet for reference. Be sure the class reaches a consensus before moving on to the next triangle.

Q. Ask the class, “How would you classify these triangles based on what you now know about their angles?” Solicit students’ ideas and if necessary, direct them to search their charts for any patterns that occur among the answers. Students should notice that Triangles 1, 3, 6 have three acute angles, Triangles 2 and 4 have one right angle and two acute angles, and Triangles 5 and 7 have one obtuse angle and two acute angles.

R. Explain that one way to classify triangles is according to their unique angle measurements. Triangles 1, 3, 6 are known as “acute triangles” because of their three acute angles; Triangles 2 and 4 are known as “right angles” because of their one right angle; and Triangles 5 and 7 are known as “obtuse triangles” because of their one obtuse angle. Have students write these labels in the gray sections on their “Classification and Measurement Chart.”

S. Refer the students back to the weekly goal: To construct, classify, and describe the properties and attributes of the -fun- polygon. Explain that as we describe the properties and attributes of these polygons, patterns emerge and classification becomes easier.

T. As a class, write and post definitions for “acute triangles,” “right triangles,” and “obtuse triangles” using the definition format followed in the previous weeks (general class + specific details) and the students’ language. A definition for “acute triangles” may be, “An acute triangle is a triangle (general class) that contains three acute angles (specific details).”

U. Homework: Pass out student copies of “Trying Triangles.” Tell the students that tonight they will have an opportunity to try their hand at constructing different triangles. Review the directions for both pages, and provide the following model for the first criteria: Use toothpicks, pencils, pipe cleaners, etc. and try to construct a triangle with only one acute angle (the other two angles have to be a combination of right and/or obtuse angles). After several failed attempts, students should be able to see that it is not possible, and they may check the “NO” column. Record one or two drawings of the failed attempts in the “Evidence” column.

V. Remind students that after completing the experiment, they should write a paragraph (page 2) to explain the number (one, two, three) and kinds (right, acute, obtuse) of angles that can be used to construct a triangle. Review with students that specific facts and information from their experiment should be used to help support the ideas presented in the paragraph. (Note: In the previous weeks, students have practiced using facts and information to support their ideas. If writing in math is a new area for your students—and if you have not been working through this entire unit—adjust this writing exercise to meet your students’ needs.)

W. Option: The handout “Triangle Trivia” may be introduced as a mini-research project. See “Extensions” for further details.

Day 2:
A. Review the evidence gathered on “Trying Triangles.” Students’ results should show that triangles can be constructed with two acute angles, three acute angles, one right angle, and one obtuse angle. If discrepancies arise, afford the students an opportunity to defend their ideas by providing them with manipulatives to reconstruct the triangle. Help the students to see why the criterion in question does or does not work.

B. Ask for two or three volunteers to share their written paragraphs. As they read, record on the board or overhead the specific facts and information that were presented to support their ideas. Commend students who remembered to include an introduction and/or conclusion to their paragraphs as well.

C. Tell the students that paragraphs will be written this week to explain their classifications of the “fun” polygon. Two specific areas of paragraph writing will be practiced in order to help them clearly and completely explain their ideas.

D. Pass out copies of the “Long-Answer Question Rubric” and if possible, use an overhead transparency as you instruct on the first area of paragraph writing.

E. Direct students’ attention to the “Explaining and Interpreting your Answer” and have them silently read the “4” point response. Explain that the Long-Answer Question Rubric is very similar to the Short-Answer Question Rubric they have used in the previous weeks; it simply helps to assess the writing that occurs in longer passages (one or more paragraphs).

F. Have students reread their written homework paragraphs and identify the score that reflects the type of explanation they provided. (Tell the students not to change what they have written; this writing sample won’t be officially scored.)

G. Explain that one goal of writing is to explain an idea so “clearly and completely” that someone else can use the facts and information in the paragraph to understand a new situation. This is one area of paragraph writing that they will be practicing this week.

H. Share that another goal of writing is to present ideas in an organized fashion so that they make sense to the reader. Including an effective beginning (introduction), middle (supportive facts and information), and ending (conclusion) does this. Explain that this is the second area of paragraph writing that they will be practicing this week. Note: Students have practiced using supportive facts and information, and they have been introduced to correct paragraph form in previous lessons. The instruction provided this week will help them strengthen both areas of writing.

I. Collect students’ homework papers for further review (see Assessment) and instruct them to take out “The ‘Fun’ Polygon” and “Classification and Measurement” chart from Day 1.

J. Review the three classifications identified by Step 1 (acute, right, and obtuse triangles) and explain that triangles can also be classified according to their side measurements.

K. Have students brainstorm some possible classifications based on what they see. Jot down their hypotheses before having them measure the sides of each triangle.

L. Pass out rulers and model how to record the measurements for Triangle 1. (Note: It is assumed that students have knowledge of and background experiences with using rulers to measure length. Be prepared to provide additional mini-lessons as needed.)

M. Have students try Triangle 2 on their own. As a class, check their answers (5.3 cm, 2.5 cm, and 5.9 cm) and allow students to correct their papers as needed. Note: Inches may also be used, but the centimeter measurements are a little easier to read.

N. Allow time for students to complete Step 2 for Triangles 3-7. (Take this time to monitor students’ progress and provide assistance as needed.)

O. When most have finished, have the students share their measurements for Triangles 3-7. (An Answer Key is available in the Associated File.)

P. Ask the class, “How would you classify these triangles based on what you now know about their sides?” Solicit students’ ideas and, if necessary, direct them to search their charts for any patterns that occur among the answers. Students should notice that Triangle 1 has three sides that are the same length; Triangles 4, 5, 6 have two sides that are the same length; and Triangles 2, 3, 7 don’t have any sides with the same length. Compare these results with the students’ initial hypotheses.

Q. Explain that another way to classify triangles is according to their unique side measurements. Triangle 1 is known as an “equilateral triangle” because all of its sides are equal in length; Triangles 4, 5, 6 are called “isosceles triangles” because two of their sides have the same length; and Triangles 2, 3, 7 are known as “scalene triangles” because none of their sides have the same length. Have students write these labels in the gray sections on their “Classification and Measurement Chart.”

R. Review the week’s goal: To construct, classify, and describe the properties and attributes of the -fun- polygon. Reiterate that as we describe the properties and attributes of these polygons, patterns emerge and classification becomes easier.

S. As a class, write and post definitions for “equilateral triangles,” “isosceles triangles,” and “scalene triangles.” Use the definition format and the students’ language as you record the definitions. A definition for “equilateral triangles” may be, “An equilateral triangle is a triangle (general class) with three equal sides (specific details).”

T. Homework: Part A: Assign applicable textbook pages or worksheets that give students an opportunity to classify triangles according to their side and angle measurements. If necessary, review with students the degree of an acute angle (less than 90), right angle (90), and obtuse angle (greater than 90 but less than 180).

U. Part B: Instruct students to write a paragraph to explain how angle and side measurements can be used to classify triangles. Be sure to review the two areas that they should focus on as they write-- a) including an effective beginning, middle, and ending, and b) using supportive facts and information.

Day 3:
A. Check students’ homework and discuss the classifications identified. Use this initial review time to clarify any misunderstandings that are represented in the students’ work and writings. (See Assessment-Day 2.)

B. Instruct students to use their “Long-Answer Question Rubric” to self-assess their paragraph and then peer assess a neighbor’s paragraph. (Each score should be written on the paper and initialed.)

C. While students still have their neighbor’s paper, ask for volunteers to share how the paragraphs were started (introductions). Discuss a strength of each introduction and record on the board or overhead novel approaches or techniques that were used.

D. Ask for volunteers to share how the paragraphs were concluded. Again, discuss strengths and record novel approaches and techniques.

E. Collect students’ paragraphs for further review (see Assessment) and pass out student copies of “The ‘Fun’ Polygon Manipulative” sheet and scissors.

F. Instruct students to carefully cut out each polygon. As they are cutting, use questions to review the classifications assigned to each triangle. For example, “Which of these triangles are acute?” “How do you know?” “Which are isosceles?” “How do you know?” etc.

G. Review the week’s goal: To construct, classify and describe the properties and attributes of the “fun” polygon. Tell the students that today they will classify these “fun” polygons according to symmetry.

H. Review what “symmetry” means. As a class, write and post a definition for symmetry using the students’ language and ideas.

I. Hold up Triangle 1 for the students and ask, “Do you think this triangle has a line of symmetry?” Allow students to respond with a “thumbs-up” sign for “yes” and a “thumbs-down” sign for “no.”

J. With the students, test the hypotheses by folding Triangle 1. Open the triangle back up to reveal the two congruent triangles formed along the fold. If necessary, review with the students the definition of “congruent” (same size and same shape). Explain that the presence of congruent figures gives evidence to the fact that this triangle has a line of symmetry. Label each half of the triangle with a “C” to show that congruent figures have been formed.

K. Ask, “Do you think this triangle has another line of symmetry?” Allow students to respond (via thumbs) and then test their hypotheses.

L. With the triangle folded, show that the parts match exactly, and with it open, help the students to identify the two congruent triangles formed along this second line of symmetry.

M. By this time, students may realize that there is even a third line of symmetry. Allow them to make the third fold and test their hypotheses.

N. Instruct students to take out their copy of “The ‘Fun’ Polygon” from Day 1 and record (with dashed lines) the three lines of symmetry that exist within this equilateral triangle. (Model this step for students as well on the board or overhead. Remember to label all congruent triangles with a “C.”)

O. Review the class definition for symmetry and the fact that congruent figures are always present in symmetrical polygons.

P. Have the students find the line(s) of symmetry for Triangle 2 by folding the paper manipulative. Discuss that sometimes there will not be any lines of symmetry. In cases like Triangle 2, instruct the students to simply record “none” on their copy of “The ‘Fun’ Polygon” sheet from Day 1.

Q. Allow time for students to complete Triangles 3-7. (Take this time to monitor students’ progress and provide assistance as needed.)

R. When most have finished, have the students share their results for Triangles 3-7. (Triangle 1: three lines; Triangle 2: “none;” Triangle 3: “none;” Triangle 4: one line; Triangle 5: one line; Triangle 6: one line; and Triangle 7: “none.”)

S. Ask the class, “How would you classify these triangles based on what you now know about their lines of symmetry?”

T. Most students will recognize that Triangles 2, 3, and 7 should be grouped together because they have no lines of symmetry, and that Triangles 4, 5, and 6 also go together because they have one line of symmetry.

U. However, some students may also see a correlation between the lines of symmetry and the side measurements of the triangles. For example, Triangles 2, 3, 7 are scalene (and have zero lines of symmetry), while Triangles 4, 5,6 are isosceles (and have one line of symmetry). Encourage the students to find and discuss as many patterns and relationships that they can see between the triangles and their classifications.

V. Homework: Part A: Assign applicable textbook pages or worksheets that give students an opportunity to practice finding lines of symmetry. Part B: Instruct students to write a paragraph to explain how to find lines of symmetry. Review the two areas that they should focus on as they write-- a) including an effective beginning, middle, and ending, and b) using supportive facts and information. As time permits, brainstorm various beginnings for the paragraph.


Day 4:
A. Check students’ homework and use the Long-Answer Question Rubric to score the written explanations from Part B. Use this initial review time to clarify any misunderstandings that are represented in students’ work and writings. (See Assessment—Day 2 and 3.)

B. Review the week’s goal: To construct, classify, and describe the properties and attributes of the “fun” polygon. Reiterate that the triangle is the “fundamental” polygon because it is the smallest polygon from which all other polygons can be constructed.

C. Explain that triangles are “named” according to their angle and side classifications. Have students refer to “The ‘Fun’ Polygon” sheet from Day 1 and as a class, name each triangle by combining its angle and side classifications. For example, Triangle 1 is an “acute equilateral triangle,” Triangle 2 is a “right scalene triangle,” etc.

D. Tell the students that they will be practicing and applying what they have learned over the past few days about triangles in today’s workstations.

E. Post written directions for the workstations (see Teacher Preparation) and review behavioral expectations. Note: If this is the first time stations have been used, allow time for both the students and yourself to become acclimated to the process. Take small steps and clearly model the outcomes you expect.

WORKSTATIONS—Select and adapt these stations to fit the class’ needs:
1. Textbook: Assign practice pages and activities that will help students hone their skills at classifying triangles and identifying lines of symmetry.

2. Problem-Solving Investigation #1--“Tricky Triangles:” Students work as a group and follow specific problem-solving steps to tackle an assigned task. Students are individually responsible for writing a paragraph to explain their solution (Note: Their solution does not have to be the same as the other member’s solutions.)

3. Problem-Solving Investigation #2—“True Triangles:” Students work as a group and follow problem-solving steps to answer a specific question. Students are individually responsible for writing a paragraph to explain their solution (Note: Their solution does not have to be the same as the other member’s solutions.)

4. Teacher: Provide the necessary instruction that will meet the specific needs of the small group (See Assessment.) For enrichment, pass out an equilateral triangle to each student and introduce two transformations, or movements, that triangles can take within the plane—flips and slides. Use “Traveling Triangles” as an instructional model and for individual student practice.

5. Computers:
Option 1 (for Internet-accessible computers): Students complete the student Web lessons “Let’s Learn Symmetry” and “Congruent or Not” available from the Beacon Learning Center. (“Carol’s Congruent Concentration” is also applicable if it was not used during lesson 3.)
Option 2 (for software with basic drawing capabilities): Students use the drawing tools (and congruent copies of a triangle) to construct, name, print (optional) and save an interesting or creative polygon. Note: Computer time may also be used for research if students are working on the mini-research project “Triangle Trivia.”

F. Reconvene to review the workstation activities. (If more time is needed to complete the textbook problems or write the solution paragraphs, assign these tasks as homework.) If not, check the textbook problems and discuss the students’ solutions to the problem-solving investigations. Collect all completed work for further review. (See Assessment.)


Day 5:
A. Review: Check assigned homework and review the students’ solution paragraphs from Day 4’s workstations. Use this initial review time to clarify any misunderstandings that are represented in students’ work and writings.

B. Review the week’s goal: To construct, classify, and describe the properties and attributes of the “fun” polygon.

C. Explain that what the students have learned this week about this “fundamental” polygon builds upon the foundation they developed during the previous weeks about the geometric building blocks (points, lines, line segments, rays, and planes), angles, and polygons.

D. Reiterate that with each new step in the building process, a check-up is completed to ensure that the building is being developed according to “code.”

E. Pass out “Building Code Check-Up #4,” a ruler, and one equilateral triangle per student. Review the directions for each section of the assessment and remind students that this check-up will help to identify both the strengths and the weaknesses of their “foundations.” (Any weaknesses will be strengthened in future lessons to ensure that all students are building strong understandings of geometry.)

F. Optional: As students finish, provide them with a triangular tessellation coloring page to reinforce the concepts covered during the week (see Extensions).

G. Optional: When all students have completed the assessment, discuss the answers to the mini-research project, “Triangle Trivia.” (See Extensions.)

H. Use the Scoring Criteria provided in the Associated File to grade the students’ work. Based on the extent of mastery shown on the assessment, provide feedback to the students that will help them to reflect on where they are in the learning process.

I. Plan any steps and/or activities needed to address deficiencies identified by the assessment before continuing with the next lesson in the series.

Extensions

1. This lesson plan represents the fourth lesson in the series on geometry instruction.

2. Triangle Trivia: Have students search the Internet, electronic encyclopedias, and other references for real-world information (not pictures) about triangles in nature, architecture, and art. Use the questions on the “Triangle Trivia” handout as a starting point for this mini-research project.

Answers: 1. The Bermuda Triangle* covers an area from the southern Virginia coast to Bermuda to the Bahama Islands that is known as the “graveyard of the Atlantic.” *Additional information on the Bermuda Triangle can be viewed at the following web site: http://tqjunior.advanced.org/3717/index.html. 2. Sao Paulo, the largest city in Brazil, has a business district called the Triangle. “This name dates back to the 1500s, when three mission buildings that stood in the area were connected by paths that formed the shape of a triangle.” (“Sao Paulo.” World Book Multimedia Encyclopedia. 1999 edition.) 3. Cuneiform is an ancient form of writing that uses triangular or wedge-shaped characters. This form of writing was practiced by ancient Middle Eastern civilizations.)

3. Sample coloring pages of triangular tessellations may be used to reinforce the concepts covered during the week (congruency, similarity, flips, slides, turns). -Tessellations: The Geometry of Patterns- by Stanley Bezuska (published by Creative Publications, 1977) and -Introduction to Tessellations- by Dale Seymour and Jill Britton (published by Dale Seymour, 1986) contain several appropriate designs.

Attached Files

The worksheets identified in the Materials Required section.     File Extension: pdf

Answer Key     File Extension: pdf