## The Building Blocks of Geometry

### Kristy RousseauBay District Schools

#### Description

Students explore geometric building blocks in the real world in order to describe the characteristics and relationships of points, lines, line segments, rays, and planes. This is the first lesson plan in a series of lessons in geometry.

#### Objectives

The student writes notes, comments, and observations that reflect comprehension of fifth-grade or higher level content and experiences from a variety of media.

The student uses appropriate geometric vocabulary to describe properties and attributes of two- and three-dimensional figures (for example, obtuse and acute angles; radius; equilateral, scalene, and isosceles triangles.).

The student knows the characteristics of and relationships among points, lines, line segments, rays, and planes.

#### Materials

Day 1
-Chart paper labeled -What is Math?-
-Pictures of nature, architecture, and art (see Associated File)
-Chart of -Problem-Solving Steps- (see Associated File)

Day 2
-Various examples of honeycombs (see Associated File)
-Small marshmallows (six per student)
-Toothpicks (at least six per student)

Day 3
-One -Building Blocks- chart per student (see Associated File)
-Textbook
--Three-ring Venn Diagram- models (see Associated File)

Day 4
-Directions for workstations (see Teacher Preparation)
-Textbook
-Picture sources: magazines, clip art, newspapers, etc.
-4- x 6- or 5- x 8- index cards (two per student)
-Glue sticks and scissors
-Geoboards, bands, and dot paper (for recording)
-Computers and software with basic drawing capabilities
-Disks (one per student)--optional

Day 5:
-Teacher-generated or textbook quiz (see Lesson Procedures)
--Three-ring Venn Diagram- models (one per student)

#### Preparations

Have prepared for Day 1:
1. Label and post chart, -What is Math?-
2. Gather pictures of nature, architecture, and art or make overhead transparencies of the samples provided in the Associated File.
3. Post a chart of the -Problem-Solving Steps.-

Day 2:
1. Gather various pictures of a honeycomb or make an overhead transparency of the sample provided in the Associated File.
2. Gather the necessary marshmallows and toothpicks.

Day 3:
1. Copy and post a definition of geometry.
2. Make student copies and an overhead transparency of the -Building Blocks- chart (Associated File).
3. Make student copies and an overhead transparency of a -3-ring Venn Diagram- (Associated File).

Day 4:
1. Select workstations for students to complete.
2. Prepare written directions for each workstation. (Directions should be reviewed as a whole class and posted for students to reference during the small group rotations.)
3. Gather materials for each selected workstation.
4. Determine student groups for rotations.
5. If Internet computers are being used, the online student web lessons listed in the Weblinks section, may be bookmarked for easy retrieval. Preview each lesson before using it to make sure it is appropriate for the group that will be using it. Some students may need additional instruction before being able to complete the Student Web Lesson independently.

Day 5:
1. Make student copies of a teacher-generated or textbook quiz (see Assessment Part A).
2. Make student copies of a 3-ring Venn Diagram-OR-instruct students to simply draw a 3-ring Venn Diagram on a clean sheet of paper (see Assessment Part B).

************************************************************************************
OPTION: If continuing in the series of lessons, teachers may want to prepare geometry folders for students to store and organize handouts, activities, and assignments.

#### Procedures

Background: Two days of pre-assessment should occur before Day 1 if all of the lessons in the series are to be completed. For a look at the all of the lessons in this series, please see Weblinks.

Day 1
A. In order to help students understand the connection of geometry to math, teachers first need to assess students' background knowledge about -What is Math?- As an introduction, pose this question to students and record their initial ideas on the labeled chart. Post the chart and revisit it as needed throughout the week or unit (or year) to add additional ideas students uncover (i.e., problem solving, patterning, logic, spatial visualization). Note: These lessons can extend students' understanding of math if connections are consistently made between what they know and what they experience.

B. If -geometry- was not mentioned during the initial brainstorming session, explain that they will be studying another area of math known as -geometry- in order to discover how geometry is used in the building of nature, architecture, and art.

C. Show pictures of nature, architecture, and art. (An overhead transparency may be made of the sample provided in the Associated File, or other pictures may be gathered.)

D. Ask, -How is geometry used to build nature, architecture, and art?- Discuss their initial ideas to this open-ended question. (Be prepared: Students may have blank faces at first. Rest assured, ideas will begin sparking as students grapple with the connections of geometry to the world around us.)

E. Explain that mathematical problem-solving steps can be used to help answer this question. Post a classroom chart of the problem-solving steps and discuss step 1: -Understand the Problem. -

F. Let the students know that expert problem-solvers are expert question-askers in their search for answers. One question such as, -How is geometry used to build nature, architecture, and art?- may lead to many more questions before an answer is solved. Use the following questions to model the metacognition (-thinking about thinking-) used by expert problem-solvers as they take step 1 in the problem-solving plan.
1) What do we know? --List what students know about geometry on chart paper or an overhead. Possible answers might include shapes, circles, triangles, figures, angles, lines, etc.
2) What is our problem?--Help students verbalize the problem that needs to be solved. Write the problem statement generated by the students and post it for all to see. For example, -What is geometry and where can it be found in our world?-
3) What do we need to know?--Guide students to formulate questions and topics that will lead to an answer. Again, record their topics and questions; they will be used again on Day 2. Possible questions may include, -What is geometry?- and -How is geometry used in nature? Architecture? Art?-

G. Explain that as questions are answered the gathered information can be used to help solve the problem. Model how this occurs by using the geometric concepts students generated in F1 to identify where these concepts are occurring in the pictures previously provided. For example, if students mentioned specific shapes (squares, triangles), point out the shapes found in the buildings and artwork. Explain that by understanding what geometry is, it becomes easier to identify it in the world around us.

H. For practice, provide a new collection of pictures for students to review. On a clean sheet of paper, have students write 2-3 complete sentences in which they identify and explain the geometric building blocks that they think are being used in the samples.

I. Allow students to orally share their ideas and support their explanations. Encourage them as they use the geometric concepts that were brainstormed in F1.

J. For closure, review the problem statement presented in F2 and explain its relationship to one of their math goals this week: To use mathematical vocabulary to describe the geometric properties of two-and three-dimensional figures. Tell the students that the questions and topics they listed in F3 is the first step to meeting this goal and solving their problem; these questions and topics will also be one of the first things discussed tomorrow.

Day 2
A. Review the problem presented on Day 1--How is geometry used to build nature, architecture, and art?-- and the math goal for this week--To describe the geometric properties of two-and three-dimensional figures. Tell the students they will be exploring these questions today, and practicing how to organize their ideas and explanations in written paragraphs.

B. Present various examples of a honeycomb. Discuss and review from Day 1 the geometric building blocks they see being used in this nature sample. Ask, -How is geometry used to build the honeycomb?-

C. Review step 1 of the problem-solving plan: Understand the Problem. Ask, -What do we know?-

D. Record and organize students' responses on chart paper or an overhead. Possible responses might include three-dimensional shapes, repeated shapes, six-sided shapes, hexagons, rows, columns, etc.

E. Model for the students how to organize their ideas and present them in a paragraph that explains, -How is geometry used to build the honeycomb?- Begin with an introduction, provide detail sentences to support the ideas presented in the introduction, and then provide closure to the essay. Use students' ideas as you write, and think out-loud the decisions you make in each section of the paragraph. As you write, students should copy the model paragraph on a clean sheet of paper. Note: An overarching goal of the series of lessons is to teach students how to explain their ideas in expository paragraphs. The goal of the first week's lesson is to begin modeling the processes involved in expository writing.

F. Tell the students that this paragraph explains what they know about the problem at this time. Now you want them to apply what they know by building a model of the honeycomb using marshmallows and toothpicks. (Have students put their model paragraphs in a safe place during the hands-on activity. The paragraphs will be referred to again in step K.)

G. Refer back to the problem-solving plan and briefly identify and discuss step 2-- -Decide on a Plan---and step 3---Carry out the Plan.-

H. Pass out the marshmallows and toothpicks. Organize students into groups of 3-4 and have them follow steps 2 and 3 as they build a small section of the honeycomb. As groups finish their sections, use additional toothpicks to connect the sections into a larger honeycomb plane. Note: Students may construct a square with the extra toothpicks (instead of a hexagon) to connect their sections. Refer them back to the real-world examples if this occurs.

I. Use the completed honeycomb to introduce step 4---Look Back and Review.- As the students review the class' honeycomb, ask, -Have we solved our problem? Do we know how geometry is used to build the honeycomb?-

J. As the class debates and discusses the question, record any new questions that are raised. Explain that as we seek to solve one problem, new questions are often raised and the problem-solving process cycles back to step 1. Finding answers to these new questions helps to solve the larger question. Note: An insufficient geometric vocabulary may frustrate students trying to explain what they see. That's okay. Having a problem to solve motivates students to want to know more about what they are learning.

K. For homework, have students explain the steps of the problem-solving process in a paragraph. Refer students back to the problem-solving chart and the model paragraph they wrote as a class. Brainstorm introductory sentences. Discuss the details that should be included in the paragraph (problem-solving steps 1-4) and possible conclusions.

Day 3
A. Review the problem-solving steps by soliciting from students' sample introductions, detail sentences, and conclusions from their homework assignment. Provide feedback on effective writing techniques and examples that are offered by the students. (Collect paragraphs for later review. Look for strengths and weaknesses in students' attempts. Use this information to guide further instruction.)

B. Post and discuss the definition of geometry. Explain that the students will begin gathering research today in order to help them learn more about the geometric building blocks used in nature, architecture, and art.

C. Pass out copies of the -Building Blocks- chart. Discuss the concepts presented and the information to be learned. Explain that the text will be used as a resource today for gathering information. Complete the first concept--point-- with the students, modeling how to paraphrase and record the information found in the text. Explain the importance of using the mathematical symbols to correctly identify each building block. Mathematicians use these symbols as a type of -shortcut- or -shorthand- in their writing.

D. Monitor students' progress as they work on completing the chart. Identify and correct misunderstandings that may occur.

E. Review the question from day 2--How is geometry used to build the honeycomb?--in light of the information gathered on the chart. Ask, -What have we learned that is important for us to know? and -How does this information bring us closer to an answer?- Identify the building blocks that are being used in the honeycomb construction (points, line segments, planes, etc).

F. Refer back to the definition of geometry. Explain that geometry not only includes a knowledge of each building block, but also an understanding of how those building blocks are related.

G. Pass out a blank model of a -3-ring Venn Diagram.- Using an overhead copy for students to follow, label each ring with one of the following geometric concept: points, line segments, and planes.

H. Ask students to think about how they would define each concept. Using student language record the characteristics within each ring. (Students should be recording this information on their copies as well).

I. Select two overlapping rings (such as points and line segments) and ask students to explain the characteristics they share. For example, they might decide that both are made of points, or that both exist in a plane. Discuss and record their comments and observations. Continue modeling with the next two concepts. As observations are recorded, help the students to see that the overlapping rings show a relationship between the two ideas. Give students a chance to complete the last two rings on their own before discussing as a class the relationship that exists between them.

J. Finally, discuss the center ring. Ask, -How are all three of these concepts related?- Allow students to share their ideas and record a synopsis of the class' observations in the center ring.

K. Use the completed Venn Diagram to present the second goal of the week--To know the characteristics and relationships among various geometric concepts Reinforce how both characteristics (those ideas listed in single rings) and relationships (those ideas listed in overlapping rings) can be shown on the Venn Diagram.

L. For homework, have students answer -How is geometry used to build the honeycomb?- in a paragraph. Refer students back to the Venn Diagram and discuss the geometric concepts that should be included in the paragraph (points, line segments, and planes).

Day 4
A. Review the problem presented on Day 1--How is geometry used to build nature, architecture, and art?--and the problem presented yesterday--How is geometry used to build the honeycomb? Solicit answers to the second problem by having students share the introductions, detail sentences, and conclusions from their homework assignment. As students describe the geometric properties and the relationships that occur between these properties refer them back to the Venn Diagram constructed yesterday on points, line segments, and planes. Note: Do not collect students' paragraphs at this time; they will be used later on during one of the workstations.

B. Tell the students that they will be practicing and applying what they have studied over the past few days in order to learn even more about the geometric concepts that are used to build nature, architecture, and art.

C. Post written directions for the selected workstations (see Teacher Preparation). Review behavioral expectations at this time as well. 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 as needed).
1. Textbook: Select practice problems that require students to use mathematical symbols as they identify and label characteristics and relationships of the geometric concepts presented on Day 3 (points, lines--perpendicular, parallel, intersecting--line segments, rays, and planes). Provide additional problems that incorporate these concepts and are written in the standardized test format for your state (i.e., FCAT-type problems). The following problems are applicable and can be found in -The Math Advantage--Daily Practice for the FCAT- published by Harcourt Brace & Company: p.17 #3 and P.119 #8.

2. Bulletin Board Cards: Students use picture sources and index cards to locate, cut out, glue, and label real-world examples of the geometric building blocks. Specify, according to the time allotted, how many geometric building blocks should be covered (i.e., two building blocks per student).

3. Geoboards: Students use the geoboard bands to build various pictures on the geoboard. Point out that the board represents a geometric plane with specific points; the bands are the line segments which connect and close those points to build the different figures. Direct students to record and (when possible) label their constructions on dot paper.

4. Teacher: Ask students to bring their homework paragraphs on -How is geometry used to build the honeycomb?- Based on weaknesses observed in previous paragraphs, select one aspect of writing expository paragraphs (i.e, introductions, detail sentences, conclusions, transitions) and provide a mini-lesson during this time. As a group, practice rewriting selected sentences from the students' homework. When finished, collect the homework paragraphs for further review.

5. Computers:
Option 1 (for Internet-accessible computers): Students complete the online student web lessons. Please see the Weblinks section. Remember to make sure that students are ready to work on the Student Web Lessons independently. Previewing them is very important.
Option 2 (for software with basic drawing capabilities): Students draw, label (using mathematical symbols), and define the following eight geometric concepts using drawing and word-processing tools: points, lines (including intersecting, parallel, and perpendicular), line segments, rays, and planes. All work should be saved according to the teacher's directions (i.e. on the hard drive or 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.

*****The following wrap-up activities can be completed after the workstations on Day 4 or at the beginning of class on Day 5. Select the activities that will best fit the students' needs for review.*****

D. Check and discuss the textbook problems. Review any areas still causing problems for the students.

E. Select and share several bulletin board cards. (If possible, cover up the labels and ask the students what geometric building blocks are being represented in the real-world examples.) Post all cards.

F. Choose a few students who recorded especially creative figures and/or shapes to recreate these constructions on the geoboard for all to see.

G. Read the strong sentences that were written by the groups as they worked with the teacher.

H. Allow students to share with the class important things they learned while working on and with the computers.

Day 5
A. Review the problems presented this week--How is geometry used to build nature, architecture, and art? How is geometry used to build a honeycomb?--and the mathematical goals--To use mathematical vocabulary to describe the geometric properties of two- and three-dimensional figures and To know the characteristics and relationships among various geometric concepts.

B. Explain that what the students have learned this week about the geometric building blocks has helped them to develop a -foundation- for understanding geometry. In construction, the foundation must be laid first; in geometry, understanding the basic building blocks provides a foundation for learning and understanding all other geometric constructions.

C. Tell the students that in the real-world, building codes are used to ensure that all constructions are being completed according to pre-established rules. These codes ensure that strong and safe buildings are constructed.

D. Explain that they will complete a -Building Code Check-Up- today in order to ensure that the foundation they have about geometry is strong and safe. This check-up will 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.

E. Handout student copies of -Building Code Check-Up #1- and review all directions (see Assessment).

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 the series of these lessons.

#### Assessments

-Building Code Check-Up #1-
Part A: Students complete a textbook- or teacher-generated quiz with multiple choice and matching options. This quiz should require students to match descriptive properties of points, lines (parallel, perpendicular, and intersecting), line segments, rays, and planes with corresponding vocabulary and symbols.
Scoring Criteria: Score results using text- or teacher-generated answer key.
Maximum Points: 50

Part B: Students complete a -3-ring Venn Diagram- showing the characteristics of and relationships among line segments, points, and planes.
Scoring Criteria: The completed Venn Diagrams should contain three definitions (points, line segments, planes) worth 3 points each; three relationships (points and line segments, line segments and planes, planes and points) worth 4 points each; and 1 core commonality (such as geometry or their use in the construction of the honeycomb) worth 4 points.
Maximum Points: 25

Part C: Students write a paragraph explaining how geometry is used to build the honeycomb. The details of the paragraph should be drawn from and/or supported by the relationships shown on their Venn Diagrams.
Scoring Criteria: Points are earned based on the level of comprehension exhibited in the students' explanations. The use of geometric concepts (building blocks, points, line segments, planes, repeated shapes, hexagons, etc.) and the relationships drawn between these concepts should be the main focus of the paragraph. For example, a teacher may credit each concept worth 2 points and each relationship worth 3 points.
Maximum Points: 25

#### Extensions

This lesson plan represents the first week of instruction in a series of lessons on geometry. See the Weblinks to access and look at the other lessons and supplemental materials.

OPTION 1: This series of week long lessons may be taken in smaller sections (and extended over a longer part of year) if a teacher decides to provide an in-depth look at geometry during their daily math instruction.
OPTION 2: This series of lessons may be integrated with the teaching of the scientific process. Strong parallels can be drawn between the mathematical problem-solving plan and the scientific process. One teacher chooses to use her morning math time for instruction in number sense, concepts, and operations (strand A) and then uses her unit time to integrate the other math strands (one per nine weeks) with her science and social studies content. In this way, students get the year-long practice they need in skill development plus the integration of other essential math concepts.

The second lesson in the series of lessons on geometry.
Classifying and Constructing Corners

The third lesson in the series of lessons on geometry and where you are now.
The Plane! The Plane!

The fourth lesson in the series of lessons on geometry.
The Fun Polygon

This is the fifth lesson in the series of lessons on geometry.

This is an Internet research lesson (a supplemental lesson in the series of lessons on geometry.)

The online Student Web Lesson can be used as a supplement for this series of lessons on geometry.
Anglemania

This online Student Web Lesson can be used as a supplement for the series of lessons on geometry.
Triangles Side by Side

This online Student Web Lesson can be used as a supplement for this series of lessons on geometry.