unit 3 organizer: - compton coaches€¦ · web viewcone: a hollow or solid object that has 1...

Grade 2 Mathematics Frameworks

Unit 4Plane and Solid Figures

MA

TH

EM

AT

IC

S

Georgia Performance Standards FrameworkGrade 2 Mathematics Unit 4 1st Edition

Unit 4: Plane and Solid Figures (4 Weeks)

TABLE OF CONTENTS

Overview..............................................................................................................................3

Key Standards and Related Standards.................................................................................3

Enduring Understandings.....................................................................................................5

Essential Questions..............................................................................................................5

Concepts and Skills to Maintain..........................................................................................6

Selected Terms and Symbols...............................................................................................7

Classroom Routines.............................................................................................................8

Strategies for Teaching and Learning..................................................................................8

Evidence of Learning...........................................................................................................9

TasksGreedy Shapes...............................................................................................................12Pattern Block Creations................................................................................................16 Shapes From Shapes......................................................................................................20Human Angles...............................................................................................................24Angle Shape Sort...........................................................................................................29Is This the Right Angle?...............................................................................................35 3-D All Around Me........................................................................................................39Cubes or Spheres............................................................................................................433-D Detective.................................................................................................................47Building 3-D Figures.....................................................................................................51Culminating Task: Our Dream Home/Castle..............................................................54

Georgia Department of EducationKathy Cox, State Superintendent of Schools

MATHEMATICS GRADE 2 UNIT 4: PLANE AND SOLID FIGURESJanuary 2010 of 56

Copyright 2010 © All Rights Reserved


OVERVIEW

In this unit students will cultivate spatial awareness by: further developing understandings of basic geometric figures identifying plane figures and solid figures based on geometric properties describing plane figures and solid figures according to geometric properties expanding the ability to see geometry in the real world beginning to explore area and volume by recognizing, covering, and filling

objects investigating the outcomes when geometric figures are covered investigating the outcomes when geometric figures are filled identify right, acute and obtuse angles within geometric figures

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as counting, time, money, positional words, patterns, and tallying should be addressed on an ongoing basis through the use of calendar, centers, and games. This first unit should establish these routines, allowing students to gradually understand the concept of number and time.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources.

Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.

KEY STANDARDS

M2G1. Students will describe and classify plane figures (triangles, square, rectangle, trapezoid, quadrilateral, pentagon, hexagon, and irregular

polygonal shapes) according to the number of sides and vertices and the sizes of angles (right angle, obtuse, acute).

M2G2. Students will describe and classify solid geometric figures (prisms, pyramids, cylinders, cones, and spheres) according to such things as the number of edges and vertices and the number and shape of faces and angles.

a. Recognize the (plane) shapes of the faces of a geometric solid and count the number of faces of each type.

b. Recognize the shape of an angle as a right angle, an obtuse, or acute angle.





M2G3. Students will describe the change in attributes as two and three-dimensional shapes are cut and rearranged.

RELATED STANDARDS

M2N1. Students will use multiple representations of numbers to connect symbols to quantities.c. Use money as a medium of exchange. Make change and use decimal notation and the

dollar and cent symbols to represent the collection of coins and currency.

M2N4. Students will understand and compare fractions.a. Model, identify, label, and compare fractions (thirds, sixths, eighths, tenths) as a

representation of equal parts of a whole or of a set.

M2P1. Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving.

M2P2. Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.

M2P3. Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.b. Communicate their mathematical thinking coherently and clearly to peers, teachers,

and others.c. Analyze and evaluate the mathematical thinking and strategies of others.d. Use the language of mathematics to express mathematical ideas precisely.

M2P4. Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to

produce a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics.

M2P5. Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical

ideas.b. Select, apply, and translate among mathematical representations to solve problems.





c. Use representations to model and interpret physical, social, and mathematical phenomena.

ENDURING UNDERSTANDINGS

In this unit students will: further develop understandings of basic geometric figures identify plane figures and solid or hollow figures according to geometric

properties describe plane figures and solid or hollow figures according to geometric

properties develop an understanding of the relationship between solid or hollow figures and

plane figures understand that the faces of solid or hollow figures are plane figures further develop spatial awareness of geometric solids and figures investigate what happens when geometric figures are combined investigate what happens when geometric figures are cut apart understand that a flat surface can be covered by a collection of squares understand that rectangular prisms can be filled with a collection of cubes expand the ability to see geometry in the real world recognize plane and solid figures in the real world

ESSENTIAL QUESTIONS

How are plane figures and solid figures related? How are they different? Where can we find geometric figures in the world around us? How do we use the terms: angle, vertices, faces, sides, and edges to describe geometric

figures? How do we describe geometric figures? How do we describe an angle? How do we describe a vertex? How do we describe a face? How do we describe a side? How do we describe an edge? What is an angle? How can I tell if an angle is right, obtuse, or acute? What makes an angle a right angle? How can you use only a right angle to classify all angles? How can we sort plane figures by their angles? How can plane shapes be cut, rearranged, and combined to create new shapes? What happens when three-dimensional figures are cut and rearranged? How can we cover a flat surface? Why would we cover a flat surface with squares rather than circles? How can we fill a rectangular prism? Why would we fill a rectangular prism with cubes rather than spheres?





CONCEPTS/SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

Fluency with single digit addition/subtraction facts to 18 Fair trades with coins or bills Duration and sequence of events Number patterns: skip count, odd/even Fact families Fractions: halves, fourths Tally marks Picture graphs Estimation to nearest ten Telling time Measurement : estimating, comparing, and ordering Basic geometric figures and spatial relationships

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The definitions below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

Angle: Formed when two lines meet at a common point, the vertex, or by two planes meeting along an edge.

Concave: All possible segments can be drawn so that part of the segment is outside the figure.

Convex: All possible segments can be drawn so that they are entirely inside the figure.

Cone: A hollow or solid object that has 1 flat, round face and narrows to a surface that has a point at the top called the vertex.

Cube: A special kind of rectangular prism with all six faces being squares.





Cylinder: A hollow or solid object shaped like a round pole or tube that has 2 round, flat face.

Edges: Where two surfaces meet on a 3-D figure.

Face: The plane figure(s) or flat surface (s) that makes up the surface of a solid or hollow figure.

Hexagon: A six-sided polygon.

Irregular Polygon: A polygon that has sides and angles of differing sizes.

Pentagon: A five-sided polygon.

Plane figure: A figure whose points all lie in the same plane. Plane figures included in the Grade 2 GPS are triangles, square, rectangle, trapezoid, quadrilateral, pentagon, hexagon, and irregular polygons.

Polygon: A closed plane figure (no gaps or openings) made of 3 or more sides and angles.

Quadrilateral: A four-sided polygon.

Rectangular prism: A hollow or solid object with six rectangular faces.

Regular Polygon: A polygon that is equiangular (all of the angles are congruent) and equilateral (all of the sides are congruent).

Solid figure: A three-dimensional figure that can be either hollow or solid. Solid figures in Grade 2 GPS include prisms, cylinders, cones, and spheres.

Sphere: A hollow or solid circular object; a three-dimensional figure with all points on the surface equidistant from the center (examples include a basketball or a globe).

Trapezoid: A quadrilateral with exactly one pair of parallel sides.

Vertices: The corners of a geometric figure; the point where edges or sides meet

CLASSROOM ROUTINES

The importance of continuing the established classroom routines cannot be overstated. Daily routines must include obvious activities such as taking attendance, doing a lunch count, determining how many items are needed for snack, lining up in a variety of ways (by height, age, type of shoe, hair color, eye color, etc.), daily questions, and calendar activities. They should also





include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, how to open and close a door, how to do just about everything! An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of the routines are important to the development of students’ number sense, flexibility, and fluency, which will support students’ performances on the tasks in this unit. See Unit 1 for suggestions concerning specific ideas for classroom routines.

STRATEGIES FOR TEACHING AND LEARNING

Students should be actively engaged by developing their own understanding. Mathematics should be represented in as many ways as possible by using graphs, tables,

pictures, symbols, and words. Appropriate manipulatives and technology should be used to enhance student learning. Students should be given opportunities to revise their work based on teacher feedback,

peer feedback, and metacognition which includes self-assessment and reflection. Math journals are an excellent way for students to show what they are learning about a

concept. These could be spiral bound notebooks that students could draw or write in to describe the day’s math lesson. Second graders love to go back and look at things they have done p so journals could also serve as a tool for a nine week review.

EVIDENCE OF LEARNING

By the conclusion of this unit, students should be able to demonstrate the following competencies:

Describe plane figures according to their characteristics (edges, corners, angles). Describe solid figures according to their characteristics (faces, edges, vertices). Describe and understand the relationships (similarities and differences) between solid

figures and plane figures. Recognize the relationship between geometry and the environment. Compare geometric figures to similar objects in everyday life. Identify right, acute, and obtuse angles within geometric figures. Understand that irregular polygons are classified according to the number of sides and

angles. Prior to teaching the unit, you can use the plane shapes graphic organizer as a whole class

assessment or give each child a copy and have them list everything they know about the given shapes.





hexagon

TASKS




Triangle

Triangle

Plane Shapes

Parallelogram SquareTrapezoid

Pentagon

Hexagon Circle Rectangle


TASKS

The following tasks represent the level of depth, rigor, and complexity expected of all second grade students. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance task, they also may be used for teaching and learning (learning task).

Task Name Task Type/Grouping Strategy Content Addressed

Greedy Shapes Learning TaskLarge Group/Partners Describing geometric figures

Pattern Block Creations Performance TaskLarge Group/Individual

Rearranging and combining plane shapesFractions and money

Shapes from Shapes Learning TaskLarge Group/Partners Cutting and Rearranging 2-D shapes

Human Angles Learning TaskLarge Group/Individual Describing types of angles

Angle Shape Sort Performance TaskPartners Sorting shapes by angles

Is this the Right Angle? Learning TaskLarge Group/Individual Comparing angles

3-D All Around Me Learning TaskPartners Classifying solid figures

Cubes or Spheres Learning TaskSmall Group Filling shapes

3-D Detective Learning TaskSmall Group Describing geometric solids

Building 3-D Figures Performance TaskPartners Building and describing geometric solids

Shapes from Shapes Revisited

Learning TaskLarge Group Cutting and rearranging 3-D figures

Culminating Task:Our Dream Home/Castle

Performance TaskSmall Group Building and describing geometric solids





Learning Task: Greedy Shapes

STANDARDS ADDRESSED

M2G1. Students will describe and classify plane figures (triangle, square, rectangle, trapezoid, quadrilateral, pentagon, hexagon, and irregular polygonal shapes) according to the number of sides and vertices and the sizes of angles (right angle, obtuse, acute).




ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical

phenomena.

ESSENTIAL QUESTIONS

How do we use the terms: angle, vertices, faces, sides, and edges to describe geometric figures?

How do we describe geometric figures?

MATERIALS

The Greedy Triangle by Marilyn Burns or similar book Geoboards Rubber bands (optional) Virtual Manipulative Geoboard:

http://nlvm.usu.edu/en/nav/vlibrary.html Describing Plane Shapes Class Chart Describing Plane Shapes Student Chart

GROUPING

Large Group, Partners TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION




http://nlvm.usu.edu/en/nav/vlibrary.html


Background KnowledgeStudents should have had prior experiences and/or instruction with plane figures from first grade and kindergarten. Students should be familiar with circles, triangles, quadrilaterals (squares, rectangles,) pentagons, and hexagons.

Teachers may want to spend some time watching this video to assist in teaching the necessary vocabulary. http://gadoe.georgiastandards.org/mathframework.aspx?PageReq=MathName

Part IRead students the book, The Greedy Triangle by Marilyn Burns. Before reading, discuss the terms side and vertex. Draw a triangle on the board and have students determine the number of sides and vertices. As you read the book, have the students predict the shape which the greedy triangle will become next.

Ask questions as you read such as: How many sides did the triangle have to begin with? How did you figure that out? How many vertices did he have to begin with? How are these different from sides? How many sides did the shape have when he became a (quadrilateral, pentagon,

etc.)? How many vertices did the shape have when he became a (quadrilateral, pentagon,

etc.)? What do you notice about the number of sides and vertices for each shape?

Part IIIntroduce the term polygon to the students. This word is not contained in the book, but is an important term for the students to use and understand. Use a word web to deconstruct the meaning of the word polygon. Break apart the word using prior knowledge of shapes.

Tell students that they will now use geo-boards to recreate the story of The Greedy Triangle. Each student should have his or her own geoboard. However, students can sit in partner groups as they create shapes. This will encourage dialogue about the geometry and allow students to comment on each other’s work.

VariationToothpicks or pretzels sticks could be used to create the shape instead of using the geoboard.

Begin reading the book again, however, this time stop at each shape and allow the students to create that shape on their geo-boards using rubber bands. (Students who can create the shapes quickly may explore creating various sizes and irregular examples of the shapes).

While students are working, ask questions like: How many sides does your shape have now? What shape have you made? How do you know it is that shape? How did your shape change?




http://gadoe.georgiastandards.org/mathframework.aspx?PageReq=MathName

http://gadoe.georgiastandards.org/mathframework.aspx?PageReq=MathName


What are differences between a (triangle) and a (quadrilateral)? Can you make that shape smaller? Larger? What would happen if you made that edge longer? Would it still be a (triangle?)

Once all students have created the shape, allow a partner group to come to the board. Have one student demonstrate how to make the shape using the virtual geoboard: http://nlvm.usu.edu/en/nav/vlibrary.html. Discuss with students the meaning of the word “regular polygon”; that the shape is a regular polygon if all the sides are equal. However, show the students that you can make the figure an irregular polygon by grasping one or more of the vertices and extending the sides. Ask the students, “What do we call this three sided shape that does not have all sides the same length? (a triangle) Why do we still call it a triangle? (because it still has three sides).” Make sure that students understand and can explain that making sides longer or shorter does not change the name of the shape (triangle, quadrilateral, pentagon, etc.) because the number of sides and vertices is still the same.

Allow the other student in the partner group to record the number of sides and vertices by the shape name on the class chart. Allow all the students in the class to record the number of edges, vertices, and several of their favorite examples of each shape on their student chart. Continue with the book, stopping at each shape and repeating the process as above.

NoteWhen students are creating quadrilaterals, encourage them to create various kinds of quadrilaterals, including trapezoids and parallelograms.

Part IIIOrganize students into small groups. Distribute the sets of cut-out figures, one set per group. “What’s My Rule?” task sheet cards should be distributed. Then review the rules of the game. One participant in each group is the sorter. The sorter writes down a "secret rule" to classify the set of figures into two groups and uses that rule to slowly sort the pieces as the other players observe. At any point in the game, the players can call "stop" and guess the rule. After the correct rule identification, the player who figured out the rule becomes the sorter. The correct identification is worth five points. A correct answer, but not the written one, is worth one point. As a variation, each incorrect guess results in a two-point penalty. The winner is the first one to accumulate ten points.

As students are sorting the cards and making decisions about the sort, the teacher should be listening for student’s descriptions of the shapes.

When students have had accumulated 10 points or allotted time has been used, gather students together as a large group and play “What’s My Rule?” again. This time the teacher should lead students to a rule about convex and concave shapes. These terms should be simply discussed. The teacher may find it helpful to use red string as pictured below to show convex and concave figures. Dot paper could also be used.




Concave: All possible segments can be drawn so that a part of the segment is outside the figure.

Convex: All possible segments can be drawn so that they are entirely inside the figure.



Questions for Teacher Reflection What strategies were students using as they changed from one shape to the next? Were students able to explain their strategies? Could students describe the shapes they were creating by their edges and vertices? Were students able to differentiate between the shapes?

DIFFERENTIATION

Extension Students who demonstrate an understanding of the shapes presented in this lesson may be

introduced to heptagons (7 sides), octagons (8 sides), nonagons (9 sides), decagons (10 sides), and dodecagons (12 sides). Students can create these shapes using their geo-boards and add information regarding these shapes to their charts.

Choose a polygon and create a picture using the shape. Describe it with mathematical words and then create a story about your picture.

Intervention Some students may have difficulty using the geobards with rubber bands. These students

can use dot paper instead. They can draw lines between the dots to create the various shapes.

Use dot paper to model various shapes such as triangles, quadrilaterals, pentagons, hexagons, etc.





Name ___________________________________ Date ______________

Describing Plane Shapes

Shape Name Number of Sides

Number of Vertices Examples

Triangle

Quadrilateral

Pentagon

Hexagon





“What’s My Rule?” Cards for Greedy Shapes





Performance Task: Pattern Block Creations

STANDARDS ADDRESSED



M2N1. Students will use multiple representations of numbers to connect symbols to quantities.c. Use money as a medium of exchange. Make change and use decimal notation and the

dollar and cent symbols to represent the collection of coins and currency.

M2N4. Students will understand and compare fractions.a. Model, identify, label, and compare fractions (thirds, sixths, eighths, tenths) as a

representation of equal parts of a whole or of a set.





a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce

a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics.


ideas.b. Select, apply, and translate among mathematical representations to solve problems.





c. Use representations to model and interpret physical, social, and mathematical phenomena.

ESSENTIAL QUESTIONS

How can plane shapes be rearranged and combined to create new shapes?

MATERIALS

Pattern Blocks (optional) Virtual Manipulative Pattern Blocks: http://nlvm.usu.edu/en/nav/vlibrary.html Papers

GROUPING

Large Group, Individual

TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION

Background KnowledgeStudents should have had prior experiences and/or instruction with plane figures from first grade and kindergarten. Students should be familiar with circles, quadrilaterals (squares, rectangles), triangles, pentagons, and hexagons.

Part I Give students a set of pattern blocks. Review each shape in the set. Discuss the name of the shape, the number of sides, and the number of vertices with the students. Give students the following task: Using the pattern blocks, show me as many ways as you can think of to create or cover a hexagon?

Students should manipulate the various pattern blocks on their desks to create the hexagon. After they have created a hexagon, they should set it aside and work with different pattern blocks to find additional ways to create a hexagon. Make sure to have a handout with numerous hexagons on it available for students to use to actually cover up the hexagon if they need to. While students are working, ask questions like:

What shape are you trying to create or cover? How many vertices will your shape need to have? What happens to many of the vertices and sides of your shapes when you combined

them with other shapes? What other shapes might you be able to put together to create or cover the hexagon? How do you know this is a hexagon? Are there any other ways you could create or cover a hexagon?






Why didn’t you use the orange square or the tan parallelogram? Can you create or cover a hexagon using two different shapes? Three different

shapes? Extension question: Can you make a hexagon with the tan parallelograms? How many trapezoids does it take to cover a hexagon? How much of the hexagon is

covered? What fraction name could we use to describe the trapezoid in relation to the hexagon? In other words, how much of the hexagon does one trapezoid cover?

Could we give fraction names to the other pattern blocks (blue parallelogram and green triangle) in relation to how much of the hexagon one piece would cover? Why is it harder to do this with the orange square and the tan parallelogram?

If the hexagon is considered one whole, then what fractional amount name would we give to the trapezoid? Rhombus? Triangle?

After students have had time to explore the pattern blocks and create hexagons, allow several students to share their hexagons using the Virtual Manipulatives Pattern Blocks: http://nlvm.usu.edu/en/nav/vlibrary.html. Allow the students in the audience to ask the student how he or she knew this was a hexagon and how he or she created the hexagon.

Repeat the process with the students creating quadrilaterals making sure to include regular rectangles (squares), rectangles, and trapezoids.

Part II

NoteUse die-cut shapes if available.

Tell students that they will now be able to create a picture using the pattern blocks. Give students a piece of paper and a pencil. Have students create a picture first with the pattern blocks. Then, encourage students to draw their picture on the paper, outlining each shape. Have students label each of the shapes. The names of the shapes should be posted on the math word wall.

Part IIIAssign a value (monetary) to each shape and have students total the picture. For example, triangles would be worth 1¢, squares 5¢, rhombi 2¢, or trapezoids 3¢.

Allow several students to share their pictures with the class. Encourage students to describe the shapes they combined to create the larger pictures. Student pictures could then be sorted into pictures that are worth more than 25¢, less than 25¢ or more than $1.00 etc.






Questions for Teacher Reflection What strategies were students using to determine which shapes could be combined to

make a larger shape? Were students able to explain their strategies? Could students describe the shapes they were creating by their edges and vertices? Could students explain what happened to the edges and vertices of the shapes when they

joined the shapes together? Were students able to differentiate between the shapes? Were students able to determine fraction names for the pattern block pieces?

DIFFERENTIATION

Extension Graph their own picture based on the number of each shape used. Create a class graph that depicts the amount of the different shape pictures the students

made. Then find the total amount of all the shape pictures.

Intervention Some students may need to work with a partner in order to “see” another hexagon being

“created” or covered up in order to understand the task.





Learning Task: Shapes from Shapes

STANDARDS ADDRESSED







phenomena.

ESSENTIAL QUESTIONS

How do we use the terms: vertices and sides to describe geometric figures? How do we describe geometric figures?

MATERIALS

Shapes from Shapes Task Chart Shapes from Shapes Student Task Sheet Scissors Glue Large teacher-sized paper, foam, or other 2-D shapes

GROUPING

Large Group, Partners TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION





Background KnowledgeStudents should have had prior experiences and/or instruction with geometry, including the discussion of vertices and sides. These concepts will be important for the students as they make connections with cutting and rearranging shapes.

Part IGather students together on the class meeting area. Use student task or draw these shapes on large pieces of construction paper. Ask two students to each choose a shape from task sheet and describe its individual attributes. Make a chart with this information. As students create new shapes, the number of sides or corners etc. may change; therefore emphasize this in your class discussion.

Part IIThe teacher should create another chart with the shape name and picture and then a column for a picture of cut shapes. The teacher can model the cuts or students can demonstrate. It is important to describe the ways to cut shapes. Horizontal, vertical, and diagonal vocabulary could be used. Emphasize the lines that students cut have to be straight horizontal, vertical, or diagonal and then demonstrate these to the students. Example cuts should include ones that are not just straight through the middle, instead the teacher should snip off one corner demonstrating a small cut. This will show students their cuts can be of various lengths. Take turns having one student demonstrate a cut and then other students model the same cut. As one piece is cut off, the teacher will lead students in a discussion of vocabulary terms of possible shapes such as: pentagon, triangle, rectangles, and quadrilaterals. Be sure to save pieces that are snipped off in the individual zippered plastic bags for students, so that they can use them later in a center to compose and decompose shapes. As students create various shapes, they can label for future investigations.

Examples of cuts that can be made:





Part IIIGather students together on meeting area to share and discuss differences in two parts of the task. Students should be developing a more sophisticated understanding of the shapes and their attributes.

Questions for Teacher ReflectionIn this activity, the goal is for students to see the relationship between different plane shapes, as well as for students to identify how the characteristics change as new shapes are cut and/or rearranged. Becoming familiar with the characteristics of these shapes will help them to make the connection to solid figures and their similar characteristics.

Are students able to describe what makes shapes different from each other? Can students describe how do shapes fit together and come apart?





Shapes from Shapes Student Sheet





Learning Task: Human Angles

STANDARDS ADDRESSED



b. Recognize the shape of an angle as a right angle, an obtuse angle, or acute angle.


M2P2. Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.





phenomena.

ESSENTIAL QUESTIONS

What is an angle? How can I tell if an angle is right, obtuse, or acute?





MATERIALS

3 bendable straws/Wikki Sticks/Pipe Cleaners per student Construction paper folded into three sections (1 per student)

GROUPING



Background KnowledgeStudents should have had prior experiences and/or instruction with geometry, including the discussion of vertices and sides. These concepts will be important for the students as they make connections with angles.

Part ITell students that today you will learn about something called angles. Explain to students that an angle is formed when two lines or sides share a vertex. Show students several angles on the board. Ask students to look for angles throughout the room. After students have found several angles, tell students that there are three types of angles that we will discuss this year: acute, obtuse, and right.

Show students how sometimes you can create angles through different parts of your body, like your arms or your ankles. Show students a 90o angle with your ankle. Tell students that this is called a right angle. Next, show them an acute angle by pulling your toes up toward your shin. Say “acute” with a very high pitched voice. Last, show them an obtuse angle by pointing your toes and stretching them away from your shin. Say “obtuse” with a big, deep voice. Allow the students to try showing the angles with their ankles as you say the words “right angle”, acute angle”, or “obtuse angle”. You can also do this with your arms. Have them make a strong bicep “muscle” to demonstrate a right angle. Then draw your fist closer to your shoulder to create an acute angle and extend your forearm moving the fist away from the shoulder to create an obtuse angle. Using the regular, high pitched, and deep voices help them to make a verbal connection to the size. Ask the students if the length of their foot or leg changes the size of the angle. How about the length of the arm? Why or why not? Talk with the students about the fact that an angle represents the size of the opening between your foot and leg or your upper and lower arm.

Discuss how an angle forms when two planes intersect. Guide students in a discussion about this intersection. Have students engage in finding these angles in their classrooms. Tell students that you are going to play a game called “Angles All Around.” Have students look for angles in the classroom and then go and stand by them. The teacher should do a visual sweep of the room to see if students are identifying angles correctly. Have students come back together on meeting area. Ask “Who thinks they were standing beside an acute angle?” Ask those students to go and





stand by the acute angle they located. Discuss each one and have students describe how they determined the angle type moving on to obtuse and right.

Part II Say to the students, “Besides acute angles, what other types of angles are there?” Continue discussion to include obtuse and right angles in a similar way. You are leading them to discuss right angles and obtuse angles as well. Model for students how to use their fingers to show differences in angles. Demonstrate angle types using your body (or you can use a student to model.)Then have the students spread out so that they can put their arms out straight on both sides and not touch anyone. Show students how they can use their entire bodies to make a right angle. Ask students to demonstrate a 90o angle (right angle) as well. Continue with the acute and obtuse angles.

Explain that you are going to play a variation on Simon Says, “Simon Sees.” They will need to demonstrate the angle that Simon sees. Play this game a few different times and allow the winner of each game to be the next Simon.

Part III After the game has been played, give students a sheet of construction paper that has been folded into three sections and three bendable straws, Wikki Sticks or pipe cleaners. Have students glue one straw in each section in the shape of each of the three angles discussed today (acute, obtuse, right) and label each angle. Students can then fill the rest of the section with drawings of objects that they see around the room (or that they cut from magazines) with the corresponding angle.

While students are working, ask questions like: What angle have you created? How do you know? How does this angle differ from the others? What is an angle? How would you explain this to someone who does not know what

an angle is? What can you find around the room that has this angle?

Allow several students to share their angle charts with the class. Encourage students to describe each of their angles and objects in the room that were comprised of those angles.

CommentTo continue doing angle “hunts” throughout the year you can set up a station where you change out the picture or item to be assessed each week. You can use die cut-outs of the upper case letters of the alphabet, pictures from magazines, various objects found at home or in the classroom, etc.

Questions for Teacher Reflection Were students able to form each of the three types of angles? Could students distinguish between the three types of angles? Did students correctly locate acute, obtuse, and right angles in the room?





DIFFERENTIATION

Intervention Students may need to create the angles with a partner using a variety of materials,

including bendable straws, play dough, marker boards, and/or their bodies.





Performance Task : Angle Shape Sort

STANDARDS ADDRESSED





b. Recognize the shape of an angle as a right angle, an obtuse angle, or acute angle.





phenomena.

ESSENTIAL QUESTIONS

How can we sort plane figures by their angles?

MATERIALS 3 bendable straws/Wikki Sticks/Pipe

Cleaners per student Paper shape cutouts Angle sorting student task sheet





GROUPING

Partners


Background KnowledgeStudents should have had prior experiences and/or instruction with plane figures and angles.

Part I Review the three types of angles with students. Give each student three bendable straws, Wikki Sticks, or pipe cleaners. Have students use the material to form each type of angle (acute, obtuse, or right). Have them show their angles to a partner to check. Then give each set of partners a set of sticks (coffee stirrers etc.) and ask them to play “pick up” sticks. Students will gather a fist full of straws and then carefully drop them from a kneeling position. Once all sticks have dropped, they should locate angles. The teacher should circulate and ask students to identify angles they found. This game time should only be a few minutes.

Part II Give each student a sorting sheet and shape handout. Have students cut out each of the shapes. Then, give each student two straws/Wikki sticks/pipe cleaners. Students can measure one straw using the corner of their paper and tape it at a 90 degree angle. Students can then manipulate the other straw to match the angles of each shape. Another option is to use an index card to locate a right angle. Next, they can compare the manipulated straw to the right angle straw to determine if the angle is right, obtuse, or acute. After measuring, encourage students to draw the shape in the correct section of the chart.

While students are working, ask questions like: What shape are you working with? How did you know its name? How many angles does your shape have? What types of angles does your shape have? How did you figure that out? Where will you place your shape on the chart? Did you have to use the straws each time, if not how did you determine what the

angle was?

Part III Have students come together to share the placement of each of the shapes. The teacher should prepare larger versions of each shape and the sorting sheet. Allow partner groups to place the shapes in the correct sections. Students should justify the placement of each shape by explaining their strategies for determining the types of angles. Encourage the audience to ask questions and make comments about the placement of the shapes.

Questions for Teacher Reflection Could students distinguish between the three types of angles?





Were students able to determine the types of angles in each shape? Could students explain and justify their thinking as they sorted the shapes by types of

angles?

DIFFERENTIATION

Extension Ask the students write descriptors for bingo style game using large student task sheet

from this task. Students can take angle hunt task sheet around school for a scavenger hunt. Challenge

them to find various angles.

Intervention Play a bingo style game with different variations of task sheet. Partner students together for an Angle Hunt scavenger hunt around the school.





Name ________________________________ Date ____________________

Only Right

Angles

Only Acute Angles

Only Obtuse Angles

Acute and Right

Angles

Acute and Obtuse Angles

Right and Obtuse Angles

Angle Shape Sort





Cut the shapes out to place on the Sorting Angles Task Sheet.

Use these shapes to determine the types of angles that make up each shape.





Learning Task: Is This the Right Angle?

STANDARDS ADDRESSED








phenomena.

ESSENTIAL QUESTIONS

What makes an angle a right angle? How can you use only a right angle to classify all angles?

MATERIALS

One piece of irregularly shaped paper per student Is This the Right Angle? Task Sheet

GROUPING






TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION

CommentsIn this task, students will explore one way to make a right angle and to use that angle to classify other angles around them. This task gives students a chance to use previous knowledge. Square corners are easily found in the classroom and in the school. An important element of this task is for students to use a square corner to measure the angles in their world.

Background KnowledgeStudents should know what a right angle is and have learned the terms right, acute, and obtuse angles and be able to locate some examples of each.

Task DirectionsGive each student a piece of irregularly shaped paper. Have them work to determine how to fold it to create a square corner. The students can create a square corner by making any two perpendicular folds. The figures show one way of folding the square corner:

Once students have folded their square corners, they can use this to find right, acute, and obtuse angles in the classroom (or take a right angle field trip throughout the school with cameras and record them on the chart.) If a student is having difficulty, encourage group members to help. When the students compare their angles to their group members’ angles, they should notice all the right angles are the same size. The groups can present the angles they found to their classmates to make sure they agree on the comparative sizes of the angles. Let students discuss the angle that was easiest to find. Ask them to tell why they think this angle is so common. Generally, students will have the easiest time finding right angles.

Questions/Prompts for Student Assessment Can you make a right angle using anything? How? Which angle is the easiest to find? Why?





Why is a right angle an important angle to know? How can you use the right angle to help you determine whether other angles are acute or

obtuse?

Questions for Teacher Reflection Were students able to construct the right angle from the paper? Can students accurately determine whether an angle is right, acute or obtuse?

DIFFERENTIATION

Extension Using a digital camera, have students go on a scavenger hunt and take pictures of

different angles. Use the pictures to create a slide show of angles.

Intervention Pair students to work together and compare answers. Give students a hand-made angle

(two strips of paper and a brad) to use when searching for angles.





Name_________________________________ Date ________________

Is This the Right Angle?Directions: Find right, acute, and obtuse angles in the

classroom (or take a right angle field trip throughout the school with cameras and record them on the chart.)

Angles that are right angles

Angles that are smaller than right

anglesAngles that larger than right angles





Learning Task: 3-D All Around Me

STANDARDS ADDRESSED











phenomena.

ESSENTIAL QUESTIONS

Where can we find geometric figures in the world around us? How do we use the terms; angle, vertices, faces, sides, and edges to describe geometric

figures?





MATERIALS

Captain Invincible and the Space Shapes by Stuart J. Murphy or similar book

3-D Shape Characteristic Chart Various 3-D figures for students to explore ( spheres, prisms,

cylinders, cones, and pyramids) NCTM Geometric Solids Manipulative:

http://illuminations.nctm.org/ActivityDetail.aspx?ID=70 Paper for riddle Pencils

GROUPING

Partners


Background KnowledgeStudents should have had prior experiences and/or instruction with plane figures and angles. They should also be familiar with the beginning concept of three-dimensional shapes from first grade.

Part IRead aloud Captain Invincible and the Space Shapes by Stuart J. Murphy or similar book with students. As you are reading, review each of the shapes with the students. Have students volunteer to share the definition of a face, vertex, and edge. Also demonstrate for students how to find each of these on the 3-D shapes.

After reading, pass out a variety of 3-D shapes to students. Allow students to explore the shapes and locate the faces, vertices, and edges on each solid figure. Encourage students to use their sense of touch as they identify these characteristics.

Part IIGive each student the 3-D Shape Characteristic Chart. Have the students work with a partner to explore the shapes and complete the chart. Students may benefit from using the NCTM web resource: http://illuminations.nctm.org/ActivityDetail.aspx?ID=70 to help them count the faces of their shapes. When using this website, students can color faces which they have already counted. As students are completing the chart, ask questions like:

What shape are you holding? How many (faces) does it have? What strategies are you using to count the number of (faces)? Are there any shapes with similarities? What are some of the differences with (rectangular prisms) and (cubes)?




http://illuminations.nctm.org/ActivityDetail.aspx?ID=70



Where might you find this shape in the real world?

Part III After students have filled in their charts together, allow them to come up with 3-D riddles about their shapes. The riddles may look something like this, but does not have to rhyme:

My number of faces, vertices, and edges are all the same. Zero is that number, can you guess my name? Answer: Sphere

Some of my faces are squares. I can be stacked, but not rolled. None of my faces are circles. I have lots of right angles. Answer: Rectangular Prism

Allow students to read the riddles they made with their partners in front of the class so the class can guess their 3-D shape.

Questions for Teacher Reflection Could students use the terms: angle, vertices, faces, sides, and edges to describe

geometric figures? Were students able to compare and contrast various 3-D shapes? Could students explain their thinking as they counted the faces, vertices, and edges of

each 3-D shape?

DIFFERENTIATION

Extension Have students cut out pictures from magazines that are examples of spheres, cones,

cylinders, prisms, and pyramids.

Intervention The teacher can type up the riddles students created on cards and have students match the

cards to the correct shape.





Name __________________________________ Date________________

3-D Shape picture

3-D Shape name

# of FacesA face is a flat

surface.

# of VerticesA vertex is a

corner.

# of EdgesAn edge is where two surfaces

meet.

1.

2.

3.

4.

5.

6.





Learning Task: Cubes or Spheres?

STANDARDS ADDRESSED








phenomena.

ESSENTIAL QUESTIONS

How can we fill a rectangular prism? Why would we fill a rectangular prism with cubes rather than spheres?

MATERIALS

Very small rectangular boxes; need some that are exactly the same for each group to work with in Part II of the lesson and then some that are different for Part III of the lesson.

marbles base ten blocks (ones) Cubes or Squares? Student Task Sheet

GROUPING

Small GroupsTASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION





Background KnowledgeStudents should have had prior experiences and/or instruction with rectangular prisms, cubes, and spheres.

Part I Give students the following task:Leila owns a company that makes bubble gum. Her brother, Jorge’, owns a candy store in a different state. Leila wants to ship some of her bubble gum to her brother’s candy store. She already has the box she wants to use to send the bubble gum, and she wants to send the most pieces of gum possible within the box. Leila wants to know if she should form the bubble gum into small spheres or cubes in order to fit the most pieces into the box.

Part II Give each small group of students a small rectangular box, marbles, and base ten blocks (ones) or small cubes. Have the students predict which solid will be able to fill more space within the rectangular prism, and why they believe this to be true.

Once they have made their prediction have students fill the box with one material at a time, counting the marbles or cubes as they place them into the rectangular prism. Have students record the totals on the recording sheet and explain how Leila should form her bubble gum to be shipped to her brother’s candy store.

Part III Have students select another box from the set of “other” boxes you brought in to again determine which 3-D shape is able to fit more of inside the rectangular prism. Encourage students to explain why this is the case. Allow several groups to share their ideas and thinking strategies and solutions with the class. Ask: Do you think it will always be best to form the gum in cubes rather than spheres? Most likely the students will say yes based on this experiment, so then you can pose this question for further thought… “Then why do we see so many gum ball machines in restaurants and candy stores?!” Perhaps one of the students will suggest that there are other ways to form the “box” so that it is not in the shape of a rectangular prism (see extension idea below). If not no worries, it’s always good to leave them with something to think about!

Questions for Teacher Reflection Were students able to identify various 3-D shapes? Could students explain their strategies as they were completing the task? Were students able to explain why a rectangular prism should be filled with cubes rather

than spheres?





DIFFERENTIATION

Extension Students may be challenged to design a box that Leila could use to ship circular

pieces of gum more efficiently.

Intervention Some students may need assistance keeping track of the counts of marbles and

rectangular prisms. These groups may need to make tally marks on a sheet of paper each time they put in a marble or cube.





Name _______________________________________ Date______________

Cubes or Spheres?




Before: (Prediction)

When filling the box, will the sphere or cube be able to fit more 3-D figures inside the rectangular prism?

Why did you choose this shape?

During:

About how many cubes fit inside the rectangular prism? _________________

About how many spheres fit inside the rectangular prism? _________________

After:

When filling the box, were you able to fit more spheres or cubes (3-D figures) inside the rectangular prism?

Why do you think this is true?


Learning Task: 3-D Detective

STANDARDS ADDRESSED










phenomena.

ESSENTIAL QUESTIONS


Where can we find geometric figures in the world around us?

MATERIALS

Various 3-D figures Brown Bag for each Student





Paper for Riddle Pencils 3-D Shape Scavenger Hunt Recording Sheet

GROUPING

Partners


Background KnowledgeStudents should have had prior experiences and/or instruction with three-dimensional figures and how to describe them using faces, edges, and vertices.

Part IAsk students what it means to be a detective. Allow students to share stories about detectives and what they do. Tell students that they will get to be a detective in two different activities today. First, tell students that they will go on a scavenger hunt to find various 3-D shapes in their school environment. Tell students that detectives must be very observant as they explore their surroundings searching for objects. Review the 3-D shapes (prisms, cylinders, cones, and spheres) with students, and discuss the characteristics of each shape.

Give students the 3-D Shape Scavenger Hunt recording sheet. Lead students through the school building, allowing them to work in pairs to identify solid figures in their everyday environment. Encourage students to write the names or draw pictures of the various shapes they see. Allow students to share the shapes they detected upon completion of the activity.

Part II

Special NoteThis section of the task is similar to what students did with partners in “3-D All Around Me.” The difference is this time they will make up the clues on their own.

Tell students that they will now get to be a detective in a different way. In this activity, students will take turns giving each other clues and guessing various 3-D shapes. Give each student pair a brown bag and several 3-D figures. One student will discretely place a solid figure in the bag. The student will write down clues about the figure related to its attributes including the number of vertices, faces, edges, and types of angles. The other student must then try to guess the figure. If the student cannot figure out the type of figure based on the clues of his or her partner, the student can use his or her sense of touch to determine the solid figure by feeling within the bag without looking at the figure.





The teacher can close the lesson by having several students present their clues to the class to see if the class can guess the hidden solid figure. The audience can guess the figure, ask questions, and make comments.

Questions for Teacher Reflection Could students use the terms: angle, vertices, faces, sides, and edges to describe

geometric figures? Were students able to identify various 3-D shapes in the environment? Could students identify solid figures based on the types of angles and number of vertices,

faces, and edges described by their partner?

DIFFERENTIATION

Extension Some students may be ready to attempt drawing the three-dimensional figures.

Intervention Students may need visual access to examples of the three dimensional shapes when

playing the brown bag game. Or they may need to work with a partner to develop the clues and then together with their partner they can present the clues for the class to guess the 3-D shape in their bag.





Name _____________________________________ Date ___________

3-D Shape picture

3-D Shape name

Examples of these shapes found in the real world

1.

2.

3.

4.

5.

6.

3-D Shape Detective





Performance Task: Building 3-D Figures

STANDARDS ADDRESSED












phenomena.

ESSENTIAL QUESTIONS


How are plane figures and solid figures related? How are they different?





MATERIALS

Clay Toothpicks Math Journal NCTM 3-D Figure Learning Tool: http://illuminations.nctm.org/ActivityDetail.aspx?ID=70.

GROUPING

Large Group


Background KnowledgeStudents should have had prior experiences and/or instruction with identifying three-dimensional figures and describing the figures using faces, edges, and vertices.

Part ICreate a chart by brainstorming with students “What do we know about 3-D Shapes?”It is important to include the concepts related to three-dimensional figures, including the shapes of their faces and the number of faces, vertices, and edges.

Part IITell students that they will now have the opportunity to explore three-dimensional shapes by creating their own figures using clay and toothpicks. Another option is to use marshmallows and toothpicks instead. Call student’s attention to the NCTM Illuminations tool at the following website: http://illuminations.nctm.org/ActivityDetail.aspx?ID=70. Click the small box on the screen to make the shapes transparent so the students can see the edges and vertices of the shapes more clearly. Explore a few of the solids with the students before releasing them to explore with the materials.

Directions to use website: To rotate the shape, click on the shape and drag the mouse while holding down the mouse button. To color the shape, hold the Shift key and click on a face, an edge, or a vertex. (The color palette allows you to choose the color for the faces; however, the edges will always be colored white, and the vertices will always be colored black.) The Reset Shape button removes all coloring. Drag the slider up or down to adjust the size, and highlight Transparent to view an outline of the shape. The New Shape button allows you to choose one of the six polyhedron that can be investigated with this tool.

It is important that you do not “teach” the students how to make the 3-D shapes. Allow the students time to explore the materials and discover how to create the shapes themselves. If students begin to experience frustration, lead them into beginning by making the plane figures or faces of the 3-D shape. Then, students can brainstorm on how to join these faces to create the







solid figure. It is imperative that students are allowed to use “accountable” or “math” talk during this time. They should discuss their shapes with their classmates, using geometric terms such as faces, vertices, angles, and edges. Struggling students will benefit from verbal and visual clues as they watch and speak with each other. While the students are working, the teacher should circulate the room, questioning students regarding their three dimensional shapes. This will help solidify the student’s understanding of the geometry concepts. Suggested questions include:

Which solid figure are you working to create? How do you plan to create this shape? What plane figure do you need to create for the _____’s faces? How are plane and solid figures related? How are they different? How could you rearrange the edges and vertices of this 3-D shape to create a new shape? What 3-D shape would you have if you cut your shape here? How many (faces, edges, vertices) does your 3-D shape have? What new 3-D figure would you have if you joined these two figures? What smaller 3-Dshapes could you fill this with? Which would fit better? How do you

know?

Part IIIAfter students have created their shape using toothpicks and clay, they should draw the shape in their math journal to the best of their ability. Students should then describe the process they used as they created the figure and characteristics of the figure. After the work session, allow the students to take a gallery walk around the classroom. During this time, students are allowed to walk around the classroom and observe their classmates geometric creations without touching.

Questions for Teacher Reflection Could students use the terms: angle, vertices, faces, sides, and edges to describe the

geometric figures they were creating? Were students able to describe the change in attributes of their figures as they were being

created? Could students explain their processes as they were creating the 3-D figures? Can students explain how plane and solid figures are related and different?

DIFFERENTIATION

Extension Within this lesson, students should be able to take responsibility for providing themselves

a challenge, by creating more difficult 3-D figures including dodecahedrons, icosahedrons, and irregular polyhedrons as shown through the Illuminations website.

Intervention For students who are having difficulty allow them to work with a partner who can

articulate their mathematical thinking clearly.





CULMINATING TASK: Dream Castle

This culminating task represents the level of depth, rigor, and complexity expected of all second grade students to demonstrate evidence of learning.

STANDARDS ADDRESSED



















phenomena.

ESSENTIAL QUESTIONS

How are plane figures and solid figures related? How are they different? Where can we find geometric figures in the world around us? How do we use the terms; angle, vertices, faces, sides, and edges to describe geometric

figures? What is an angle, vertex, face, side, edge? What happens when three-dimensional figures are cut and rearranged?

MATERIALS

Boxes in a variety of sizes (cubes and rectangular prisms) Cans (cylinders) Various Balls (spheres) tape glue paper

GROUPING

Small Groups


Background KnowledgeStudents should have had prior experiences and/or instruction with identifying three-dimensional figures and describing the figures using faces, edges, and vertices.

Part IBegin by discussing prior knowledge about what an architect does. Ask if architects use shapes to build things. Are those shapes 3-D or 2-D? Tell students that they are going to work with a team of other architects to design a model of a dream home or castle. Allow students to view the materials that they will be able to use. Tell students that every 3-D figure used in the castle/dream home must be accounted for by students being able to name the figure; determine the number of faces, edges, and vertices of each figure; and name the plane figures and types of angles on each 3-D figure.





Have students draw a plan (or blue print) for their castle or dream home and begin gathering materials for the construction.

Part IIAllow students to build their creations. Have students label the 3-D shapes in their castle/dream home using the method of choice. This may include creating little labels for each figure or students may choose to create a key for their creation.

Part IIIHave an open house, where students can share their creations. Invite parents, and have each group share their dream home/castle and the characteristics of the shapes from which it is made.

VariationInstead of making Dream Castle have the class create a city. Students work with a partner to make one of the buildings for the city. For example: library, hospital, school, bank, grocery, store, church, movie theater, etc.

Questions for Teacher Reflection Could students use the terms; angle, vertices, faces, sides, and edges to describe the

geometric figures they were creating? Could students describe and classify plane figures? Could students describe and classify 3-D figures? Could students describe how the 3-D figures changed as they were combined them to

form the castle?




unit 3 organizer: - compton coaches€¦ · web viewcone: a hollow or solid object that has 1...

Documents