Description

Students explore, classify, and define the various types of angles (acute, right, obtuse, and straight) that occur in the world around them. This lesson plan is the second lesson in a series on geometry.

Materials

Day 1
-Honeycomb examples (see Associated File)
-Passage from "The Honeycomb Conjecture" by Ivars Peterson (see Associated File)
-Chart paper
-Chart of Problem-Solving Steps (see Associated File)
-One Classifying Angles worksheet per student (see Associated File)
-A display method (whiteboard, chalkboard, overhead projector, etc.) will be needed for each day of instruction.

Day 2
-Completed Classifying Angles- worksheets (see Day 1)
-Constructing Angles overhead (see Associated File)
-Six toothpicks for demonstration
-One Exploring Angles Venn Diagram model per student (see Associated File)
-One Defining Angles worksheet per student (see Associated File)
-Hand-held clock with movable hands
-Clock worksheet (see Associated File)
-Textbook

Day 3
-Completed Clock and Defining Angles-worksheets (see Day 2)
-Overheads of the Constructing Angles and Note Sheet (see Associated File)
-One Constructing Angles and Note Sheet per student (see Associated File)
-One wax-paper circle per student (see Lesson Procedures and Teacher Preparation)
-Scissors (optional)

Day 4
-Completed Clock worksheets with angle measurements (see Day 3)
-Completed Defining Angles worksheet with definitions (see Day 3)
-Chart paper
-Wax paper circles from Day 3
-Geoboards, bands, and dot paper
-Picture sources: magazines, clip art, newspapers, etc.
-4" x 6" or 5" x 8" index cards (at least one per student)
-Glue sticks and scissors
-Computers and software with basic drawing capabilities
-Disks (one per student)-optional
-Practice Writing Prompt-Geometric Concepts in Architecture (see Associated File)
-One Short-Answer Question Rubric per student (see Associated File)
-One Classifying Angles worksheet per student (see Associated File)

Day 5
-Completed Classifying Angles worksheet (see Day 4)
-Short-Answer Question Rubric overhead (see Associated File)
-Teacher-generated or textbook quiz (see Lesson Procedures)
-Exploring Angles Assessment sheet (see Associated File)
-Wax-paper circles from Day 3

Procedures

Background: Students apply their knowledge of points, lines (intersecting, parallel, and perpendicular), line segments, rays, and planes in this lesson. See lessons listed in the Weblinks section for the previous lesson and the ones that follow this one.

Day 1
A. Present the honeycomb examples found in nature, architecture, and art. Use this question to review the geometric concepts learned during week 1 (points, lines, line segments, rays and planes): How is geometry used to build the honeycomb?

B. Ask: How would you describe the fundamental region (or shape) that is used in these constructions? (If necessary, highlight or outline one hexagon from each example.) Solicit students' responses and jot down their vocabulary on chart paper.

C. Present week's goal: To use appropriate geometric vocabulary to objectively define the geometric concepts that occur in the world around us (nature, architecture, art).

D. Read Ivars Peterson's passage from The Honeycomb Conjecture. (If possible, display for students to see.) Explain that the phrase, a vertical comb of six-sided, or hexagonal cells, is a model of appropriate geometric vocabulary.

E. Tell the students that an objective definition consists of two parts. First, it identifies the larger group (or general class) that the subject belongs to. Then it tells how the subject is different from all the other members that belong to the general class.

F. Dissect the phrase so students can see the two parts of the objective definition. For example: A vertical comb identifies the general class of combs that exist (part 1); while the phrase of six-sided, hexagonal cells specifies how the honeycomb is different from other vertical arrangements (part 2).

G. Redirect the students' attention to the passage and say: According to Peterson's description, how many sides are in a hexagon? (6) From what you observe, what else do the hexagons have in common? (Lead students to see that all of the shapes are closed-figures.)

H. As a class write a definition for the general class (part 1) of hexagons. The definition may look something like: A hexagon is a six-sided closed-figure shape. Then, instruct students to draw a hexagon on a sheet of paper that is different than the samples previously seen. (If necessary, provide 2-3 examples to stimulate students' thoughts.)

I. Using one of the examples, model for the students how to write a definition for the new hexagon. The definition should include the general class description (a six-sided closed-figure) and any specific details that help to set it apart from the general class.

J. After presenting the model, have students write an objective definition for their hexagon. (Remind them to include both the general class and specific details.) Choose a few students to share their definitions with the class. While they share, instruct the other students to draw the hexagons being described.

K. Discuss how closely the students' drawings matched the defined hexagons. Ask: What made it difficult to recreate the hexagons?

L. Explain that the first step to solving any problem is identifying what we know and what we need to know. Ask: What do we know about each of the hexagons? (They are six-sided, closed-figures.) and What do we need to know in order to objectively describe them? (How the sides are related or connected, the length of the sides, the opening that exists between the sides, etc.)

M. Tell the students that part of the plan this week is to study the opening that exists between the sides (i.e, the corners or angles of the shape) in order to learn more about how geometry is used in nature, architecture, and art.

N. Handout Classifying Angles and explain that you want to see what the students already know about these corners. Direct students' attention to the definition provided at the bottom of the page. For review, have students analyze this objective definition for its two parts: a general class (Angles are two-dimensional figures-) and specific details (constructed from two rays that share a common endpoint called a vertex).

O. Review the activity presented on Classifying Angles. In Part B, instruct students to objectively define their sorting method for each group. For example: In Group 1 are all the angles (general class) that (specific details). Assign as homework.


Day 2
A. Review the week's goal: To use appropriate geometric vocabulary to objectively define the geometric concepts that occur in the world around us (nature, architecture, art).

B. In partners or small groups have students share their classification schemes and explanations from the homework activity Classifying Angles. Call for a few volunteers to share their ideas with the class.

C. As students share, use an overhead copy of Classifying Angles to record their work. Model how to appropriately record angles (i.e., If a student says, I put angles b, n, and q together in Group 1, use the appropriate geometric symbol to record their responses). Explain that this symbol and the common endpoint being shared by the two rays (the vertex) can be written together to represent the angle.

D. At this time also dissect the students' explanations to review the two parts of an objective definition-general class (angles) and specific details (common properties shared by those angles).

E. Explain that by classifying angles students will be able to specifically describe the geometric concepts that occur in the world around them. (Collect students' homework papers and tell them they will revisit this activity later in the week.)

F. Use an overhead transparency of Constructing Angles to present the hexagons created from angles b, e, h, k, n and q. Redefine angles and review how points, lines, and rays interact within a plane to construct these corners.

G. Refer to the hexagons on the transparency as you review the problem from Day 1--What made it difficult to recreate the hexagons?

H. Introduce Step 2 of the problem-solving process-Decide on a Plan. Tell the students that after hearing their What do we need to know? questions from Day 1, you decided to build a lesson plan that would provide some of the information they needed. Today you will present part one of that instruction-classifying angles-as you take the third problem-solving step, Carry Out the Plan.

I. Use six toothpicks to construct a regular hexagon on the overhead. Explain that mathematicians always classify the six angles in this hexagon as obtuse. (Remove all but two of the toothpicks to focus students' attention on the obtuse angle.)

J. Ask: How do mathematicians know this angle is obtuse? Solicit students' ideas in order to access their prior knowledge about angles and measurement. (If definitions are provided, encourage students to use the objective definition format-Obtuse angles are angles that...)

K. Explain that angles are classified according to their measurement--the opening (or rotation) that exists between the two rays. Direct students' attention back to the two toothpicks and ask: How many different ways can these toothpicks move away from each other if they remained joined at one end? Solicit students' ideas and then share that ultimately the toothpicks' movements would resemble a circle-a complete rotation known as a perigon. Draw a circle on the overhead to demonstrate this point. Note: This circle is the first ring in the three-ring Venn diagram students will be creating today in order to understand the characteristics and relationships of angles.

L. Hold up a hand-held clock and display 12 o'clock. Ask the students to identify the common endpoint (vertex) of the clock's hands and the angle that currently exists between the minute hand and the hour hand. (The students will probably say that there is no angle). Explain that this angle is called a perigon and it is the starting and ending point for all other angles.

M. Pass out a copy of the Exploring Angles Venn Diagram and instruct students to label the top ring Perigon. Using the vertex provided on the diagram, model how to draw a perigon (12 o'clock) inside the ring. Instruct students to draw an example of a perigon on their diagrams as well. (FYI-1:05 and 2:11 may also be used as examples of perigons.)

N. Ask the students to find examples of this angle in the room around them. As students share, discuss how they would define or explain this type of angle to someone else.

O. Pass out the Defining Angles worksheet and review the two observations that were recorded about perigon angles in the Specific Details column. Explain that this sheet will be used over the next couple of days to record the students' observations of the various angles. Their observations will then be used to define the angles identified. Note: To ease manageability, copy the Defining Angles worksheet on the back of the Exploring Angles Venn Diagram model.

P. Tell the students that the diagram and the clock will be used today to show the different classifications of angles. Explain that when children first learn to tell time, they start with the o'clocks. Tell the students that today they will be learning different classifications for angles, but they too must start with the o'clocks.

Q. Using the clock, display 3 o'clock. Ask the students if they know the name of this angle. If not, identify it as a right angle. Model other examples of right angles using the clock hands (such as 4:05 and 5:11). In the right ring of the Venn Diagram, model how to draw a right angle. Instruct students to draw an example of a right angle on their diagrams as well. Repeat step N. Then as a class jot down two notes, comments, and/or observations for right angles in the Specific Details column.

R. Next use the clock to display 6 o'clock. Identify this angle as a straight angle. Display 7:05 and 8:11 as additional examples. In the left ring of the Venn diagram, model how to draw a straight angle. Instruct students to draw an example of a straight angle on their diagrams as well. Repeat step N and jot down two notes, comments, and/or observations for straight angles in the Specific Details column. (The Venn diagram should now contain three labels perigon, right, and straight and three examples. The Defining Angles worksheet should contain notes, comments and/or observations for each of these angle types as well.)

S. Return the hands of the clock to 12:00. Tell the students that there are angles that exist between each of these three categories. Display 1 o'clock. Have students draw it on their diagrams in the overlapping sections between perigon and right. Show 2 o'clock and explain that these angles are identified as acute. The rays have begun to spread apart, but the angle opening is not as large as a right angle. Label the right inner ring acute.

T. Move the hands to 4 o'clock. Display 4:00 and 5:00 and explain that these angles are identified as obtuse. Their angle openings are larger than a right angle, but not as large as a straight angle. Have students draw one or two examples and label the bottom inner ring (the overlapping section between right and straight angles) as obtuse.

U. For review, display the following times and ask students to identify the angle being displayed. 7:00, 8:00--Obtuse Angles; 9:00--Right Angles; 10:00, 11:00--Acute Angles; 12:00--Perigon Angles. (Note: To complete the Venn Diagram, briefly identify the angles that fall between straight and perigon. Manipulate the clock hands to show examples. Explain that these angles are known as reflex angles. Students may label the section and include one or two examples, but thorough instruction is not required at this time.)

V. Draw the students' attention back to the full circle that was made with the hour hand as you moved from 12:00 to 12:00 (as if the clock were moving from noon until midnight, for example). Review that angles are classified according to their measurement--the opening (or rotation) that exists between the two rays. For additional practice, pass out the Clock worksheet. Instruct students to use their Venn diagrams to classify each angle. If necessary, model the first few problems.

W. For homework, have students complete the Specific Details for acute and obtuse angles on the Defining Angles worksheet. Any related textbook pages may be assigned as well.


Day 3
A. Check the Clock worksheet and related textbook pages as assigned for homework. Review the notes, comments, and observations students recorded on their Defining Angles worksheet for acute and obtuse angles.

B. Show the Constructing Angles overhead and direct students to use their notes, comments, and observations to classify each angle. Call on volunteers to share their answers with the class.

C. State the week's goal: To use appropriate geometric vocabulary to objectively define the geometric concepts that occur in the world around us (nature, architecture, art).

D. Review steps 2 and 3 of the problem-solving process. Tell the students that after hearing their What do we need to know? questions on Day 1, you developed a lesson plan to provide some of the information they needed. Yesterday the class Carried Out part one of that plan as they learned to classify angles. Today, they will Carry Out part two as they learn to measure angles.

E. While the Constructing Angles overhead is still displayed, explain that mathematicians not only classify, but also measure, the opening that exists between two sides of an angle. Ask: How do you think mathematicians measure angles?

F. Review with students that just as in any measurement (i.e., length, weight, time, temperature) a standard unit is used to measure angles.
Note: The following instruction is provided to bridge a gap between knowledge of common angle measurements (45°, 90°, and 180°) and the use of a protractor (which is based on the degree-a very small unit angle). To help students conceptually understand how the unit angle is used, a larger model will be used initially. Specific instructions will not be provided for using a protractor.

G. Pass out one wax-paper circle to each student. As a review ask: What is the name of the angle whose sides complete a full rotation? (perigon) What is the name of the angle whose sides complete half of a rotation? (straight)

H. Instruct the students to fold their circle to show a straight angle. (Model as well; be sure to crease each fold tightly.) Lay the wax-paper circle on top of angle k on the Constructing Angles overhead. State, If angle k was a standard unit of measure, what other angles on this overhead could we accurately measure? (Students might say angle q, but most will recognize that the unit angle needs to be smaller to accurately measure the angles.)

I. Ask, What is the name of the angle whose sides complete a quarter of a rotation? (right) Fold the wax-paper circle to show a right angle and -measure- angle e (1 unit angle), angle k (2 unit angles), and estimate angle h (3/4 of a unit angle).

J. Tell the students that in order to measure acute angles, we'll need our unit angle to be a little bit smaller. With the students, fold the right angle in half two more times. With each fold, discuss the acute angles being created and explain that a smaller unit angle will help our measurements be more precise.

K. Open the wax-paper circle and instruct students to count the number of unit angles- that have been created (16). As a review ask, -How many unit angles are in a straight angle? (8) A right angle? (4) An acute angle? (1 up to 4 unit angles) An obtuse angle? (more than 4 but less than 8 unit angles)

L. Pass out the Constructing Angles and Note Sheet to each student. Use an overhead copy of the -Note Sheet- to review the following directions: Use your unit angles to measure each angle presented. Be sure to label the angle being measured, classify it, and write in the number of unit angles covered by its opening.

M. Model the directions with angle b. Step 1: Label. Show the various ways that angle b may be labeled using the appropriate geometric symbol for angles and the name of each point (i.e., angle b, angle abc, and angle cba). Step 2: Classify. Think outloud for the students, Is this angle acute? Right? Obtuse? Straight? Or a Perigon? (obtuse) Step 3: Measurement. Lay the circle over angle b and determine the number of unit angles being covered by the angle opening (fractional values may be included). Write 5 1/3 unit angles.

N. Provide measurement hints as necessary. For example, students need to know that one ray of the angle needs to be lined up with one fold on the circle. Also, guide the students to place the center of the circle on the vertex of the angle. (Remember that this model is being used to lay the foundation of measuring with a protractor. By understanding that an angle measurement refers to a specific number of unit angles, students have a chance to develop the concept of angle measurement without being distracted by all the tedious numbers on the protractor.)

O. Provide a chance for students to practice angle measurement with angles e, h, k, n, and q. Monitor their work and provide individual or small-group instruction as needed to correct any misunderstandings.

P. Check students' work: angle e ( right; 4 unit angles) , angle h (acute; 3 unit angles), angle k (straight; 8 unit angles), angle n (acute; 3 ½ unit angles), and angle q (obtuse; 6 1/3 unit angles). Answer any questions students may have about how to use the tool to measure angles.

Q. With the time remaining, have students pull out their Defining Angles worksheet from Day 2. Instruct students to include the number of unit angles that can be found in each type of angle. For example, under the Specific Details record that a perigon has 16 unit angles, an acute angle has up to 4 unit angles, a right angle has 4 unit angles, and so forth.

R. As a class, write an objective definition for a -perigon- angle using the General Class and Specific Details provided. Record the definition in the gray space provided on the Defining Angles worksheet. For example, A perigon angle is an angle whose opening completes a full rotation and measures 16 unit angles.

S. For an additional model, define reflex angles as well. For example, A reflex angle is an angle whose opening is larger than a straight angle and measures from 9-15 unit angles.

T. For homework, students should complete the written definitions for acute, right, obtuse and straight angles on the Defining Angles worksheet and then use their wax-paper circles to measure the angles displayed on the Clock worksheet.

Day 4:
A. Check the angle measurements recorded on the Clock worksheet and take time to discuss any discrepancies that occur between the students' answers.

B. Review week's goal: To use appropriate geometric vocabulary to objectively define the geometric concepts that occur in the world around us (nature, architecture, art).

C. Introduce step 4 of the problem-solving process Look Back and Review as students discuss the definitions they recorded for acute, right, obtuse, and straight angles on the Defining Angles worksheet. Ask, What have we learned about using appropriate geometric vocabulary to define the geometric concepts that occur in the world around us? Review with students the two parts of a definition and the labels they learned for the six classifications of angles. Record class definitions for acute, right, obtuse and straight angles on chart paper using the students' language and ideas.

D. Tell the students that they will be practicing and applying what they have studied over the past few days about classifying and measuring angles 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.
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WORKSTATIONS--Select and adapt these stations to fit the class' needs:
1. Textbook: Assign text pages that require students to classify and measure acute, right, obtuse, and straight angles. The problems should reflect the types of problems that students have encountered during the nightly homework assignments, as well as the types of problems they will be expected to answer during the Selected Response section of the Building Code Check-Up #2 (see Assessment).

2. Bulletin Board Cards: Students use picture sources, index cards and glue sticks to collect, classify, and label angles found in nature, architecture and art. Encourage the groups to try and find one example of each angle classification.

3. Geoboards: Students construct a variety of hexagons using the points and line segments (rubberbands) available in the plane (geoboard). Each student selects one hexagon to record on dot paper and then uses the wax-paper circle to measure the six angles that occur within the closed figure. As time permits, have students find a real-world use for their hexagon.

4. Teacher: Present the practice writing prompt Geometric Concepts in Architecture and allow time for students to complete. Introduce the Short-Answer Question Rubric and explain that during this geometry unit some of the written answers will be scored using this guide. Review the three sections of the rubric with the students-Answering the Problem, Showing your Work, and Explaining and Interpreting your Answer. Discuss the correct answer for the prompt (obtuse and acute angles) and help students self-evaluate their definitions of these angles using the Explaining and Interpreting your Answer section of the rubric. Refer students to the class chart of definitions and remind them about the two parts of an objective definition. As a group, write an explanation for the prompt that would score a 2.

5. Computers:
Option 1 (for Internet-accessible computers): Students complete the online Web lesson, Anglemania, from the Beacon Learning Center. See Weblinks for the URL.
Option 2 (for software with basic drawing capabilities): Students draw, label, and define the following four angle classifications-acute, right, obtuse and straight. Their work should be saved either on the hard drive or on a personal disk. Note: If students are unable to complete the computer work, a rotation schedule can be developed to provide additional computer time during the upcoming week.
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F. Reconvene as a large class to review the workstation activities. Check the textbook problems, and collect the bulletin board cards and geoboard constructions to display on a bulletin board.

G. Have students share any breakthroughs they encountered while working on the computers. (For example, by the second week during the series of lessons some students will be trying out different tools in the software program to manipulate the font, style, color, etc. of their typing and designs.)

H. Display the honeycomb examples from Day 1. Have students discuss in pairs how they would define the fundamental region that occurs in each of these constructions. Solicit students' ideas and record sample explanations on the overhead or chart paper. Explain that as problem-solvers look back and review their work, they are able to see what they have learned, how they have learned it, and what they need to learn next.

I. Pass out copies of the Short-Answer Question Rubric. Use the rubric to discuss the strengths that occur in the written explanations that were provided in step H. Explain that this rubric should be used as a guide whenever students have short-answer questions to solve.

J. Pass out clean copies of the Classifying Angles worksheet. Review the directions and assign as homework. Explain that this homework activity will be their first opportunity to use the Short-Answer Question Rubric to guide their work.

Day 5
A. Ask a student to state the week's goal: To use appropriate geometric vocabulary to objectively define the geometric concepts that occur in the world around us (nature, architecture, and art.)

B. Use peer review and the Short-Answer Question Rubric to check the Classifying Angles worksheet. (Since there is not one right answer, have the students score the paper using the criteria provided in the last two sections-Showing your Work and Explaining and Interpreting your Answer.)

C. Discuss students' answers and the peer review process. Select one of the initial concerns students have about using the rubric and provide a mini-lesson or think-aloud session to help them use this tool. (Note: Students WILL need instruction on how to use the rubric to score and complete answers. However, this rubric will be used consistently in the weeks to come, so instructional steps can be taken a little bit at a time.) Collect their work for additional review.

D. Explain that what the students have learned this week about angles builds upon the foundation they have developed about the geometric building blocks. With each new step in the building process, however, a check-up must be completed to ensure that the building is being developed according to code.

E. Pass out Building Code Check-Up #2 (see Assessment). Review the directions for each section of the assessment and remind students that this check-up will identify both the strengths and weaknesses of their foundations. Any weaknesses will be strengthened in future lessons to ensure that all students are building strong understandings of geometry.

F. Use the scoring criteria and weight values provided in the Assessment to grade 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.

G. Plan any steps and/or activities needed to address deficiencies identified by the assessment before continuing with any more lessons.

Extensions

1. This lesson plan represents the second week of instruction in a series of five lessons on geometry.
2. The writing tasks presented in these lessons can be further expanded and refined during language arts. During week 1 the students were exposed to the logical organizational pattern that exists in expository writing (i.e., an effective beginning, middle and end). Although this aspect of expository writing was not focused on during week 2, it could be addressed during the language arts time. Students could practice organizing the information they have learned about the construction and classification of angles in written paragraph(s).
3. Reference: The wax-paper circle activity was adapted from Chapter 16 of the [Elementary School Mathematics: Teaching Developmentally] by John A. Van de Walle (Longman:New York, 1994).

Attached Files

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

Answer Key     File Extension: pdf