The "Real Math" behind Geometric Solids

by Alice Hom and Barbara Rosenblum, P.S. 6

Warm-Up:  Number Strings

 A number string is a set of related equations designed to help students discover "big ideas" in number operations.  Problems in a string are written horizontally rather than vertically because vertically written problems usually suggest a splitting strategy based on place value.  Number strings give students an opportunity to notice patterns that might suggest certain strategies.

Goal:         To elicit quick and efficient strategies for computation.

Materials: Chart Tablet Paper/Dry erase board /Blackboard

                   Marker/Chalk

Process:    Number strings should be done in a small group or whole class situation.  This will enable all participants to articulate and share their strategies with each other.

1.   Begin with a single problem written horizontally on the paper or board.  "Think time" is given in order for students to mentally solve the problem and prepare to explain how they solved it.

2.  Call on a few students (no more than three or four) to share his/her ideas and represent visually what is said (i.e., number line, array model, numerical equations).  Thus, students can hear and see representations of each other's strategies, which will foster a discussion of the various approaches.

3.  Introduce the next problem in the string.  The same procedure of sharing and recording is followed for the subsequent problems.

Examples:

          150 - 75                        This string is likely to elicit

                   151 - 76                        a strategy of using constant

                   149 - 74                        difference.

                   294 - 100

                   291 - 97

                   301 - 107

                   6 x 3                              Doubling and Halving

                   12 x 3                            If you double one factor,

                   12 x 6                            product doubles.  If you

                   24 x 3                            double one factor and half

                   24 x 6                            the other factor, the product

                   12 x 12                          remains the same.

Number strings done regularly help students develop flexibility and fluency in computation.  They engage students to focus their thinking, to discuss computational strategies, and to develop strong number sense that will enable them to determine the most efficient strategy for solving a problem.

 Lesson 1 - Sorting and Describing Solids

Goal

To explore, sort, and discuss different attributes of various geometric solids.

Objectives

1.  Students will discover characteristics of various geometric solids.

2.  Students will observe and verbalize similarities and differences between common geometric solids.

3.  Students will develop a common vocabulary in order to describe and discuss the properties of geometric solids.

4.  Students will distinguish between 2D and 3D shapes.

5.  Students will distinguish between polyhedron and non-polyhedron.

6.  Students will distinguish between prisms and pyramids.

Vocabulary

Sphere                  Hemisphere                   Triangular Prism

Cube                    Cone                              Hexagonal Prism

Cylinder               Square Prism                Rectangular Prism

Pyramid               Vertex                           Octagonal Prism

Face                     Edge                              Polyhedron

Base

*Note: Teachers should be familiar with vocabulary and know basic definitions to facilitate discussions.  Pictures of the geometric solids and the words should be displayed in the classroom for children to refer back to.  A mathematical dictionary would be useful.  

Websites: 
http://mathforum.com/dr.math/faq/formulas/index.html - Pictures and formulas of geometric solids

 - Geometry Glossary

Materials

Sets of geometric solids corresponding to those on Worksheet 1

(1 set per group of 4-6 students)

Paper for recording

Marker

Preparation

Use small adhesive labels to number the solids 1 - 12.  All sets of models must be numbered the same (i.e., the square pyramid in each set might be numbered 4). 

Procedure

1.   Students work in cooperative learning groups of 4-6.  Each person has a role determined by group members (for example, Timekeeper, reporter, recorder, facilitator).

2.   From a baggy or basket containing the pre-labeled solids, students will first sort the shapes into two groups so that all the shapes in each group are similar in some way. 

3.   Record how solids are sorted (using the numbers) and label each group by its particular attribute.

                    Points    

       
                     No points

4.   Repeat process using other attributes and record results.

5.   After 20 minutes of group exploration, students then regroup as a whole class.  A reporter from each group will share one way his/her group sorted the solids and describe their common attributes.

Evaluation

1.   Using a worksheet depicting the various geometric solids, each student will sort them into 2 groups with attributes different from the ones used during the class exploration session.  (Challenge:  Sort using 3 different attribute groups)

2.  Look at school, at home and in magazines for examples of the various geometric solids and make a list. (Worksheet 2)

3. Students will describe the differences between polyhedra and non-polyhedra.

 Lesson 2 - Exploring Characteristics of Geometric Solids

Goal

To identify specific characteristics of different geometric solids.

Objectives

1.  Students will develop a common vocabulary of the names of various geometric solids.

2.  Students will identify properties of the geometric solids and look for relationships/patterns among them.

Vocabulary

Names of geometric solids (see Lesson 1)

Edge                    Face                     Vertex (Vertices)

 Materials

Set of 12 geometric solids

Worksheet to fill in

Mathematical dictionary (optional)

Procedure

1.    Students will work in cooperative groups of 4 - 6.

2.    After looking at each geometric solid, they will fill in chart.  (Worksheet 3)

This will be used as a reference for future activities in the unit.  Some discussion may arise regarding properties and general definitions of certain solids.  For example, is the base of the cone considered a face?  Do the circular bases (faces?) of a cylinder have circular edges?  Use of a mathematical dictionary or web site would be helpful - try Math Forum's Defining Geometric Figures at: http://mathforum.com/dr.math/faq/formulas/faq.figuredef.html 

Evaluation

1.    When shown a geometric solid, students will verbally identify it by name and describe its features (number of faces, edges, vertices).  

Further activities:

Manipulatives for the Upper Elementary Grades 4-6:  http://mathcentral.uregina.ca:  type "geometric solids" in the Resource Room search box to get to this page

 Lesson 3 - Guess My Solid

Goal

To identify a mystery solid through questions regarding characteristics of that solid.

Objectives

1.   Students will verbalize descriptive characteristics of geometric solids.

2.   Students will logically determine the identity of the mystery solid by using the clues gathered from the questions asked and through a process of elimination.

Materials

Pictorial chart of 12 geometric solids (poster size for whole class to view or xeroxed copies, 1 per student)

Chalkboard/Dry erase board/Chart Tablet Paper

Chalk/Marker

Procedure

1.  Explain the rules for Guess My Solid and play with the whole class for two to three rounds.  Then students will have an opportunity to play game in small groups.

2.  Teacher will choose a geometric solid and write its number on a piece of paper, out of sight of the students. 

Rules for Asking Questions

·  Students must take turns asking questions.

·   Only questions that have yes or no as an answer may be asked.

·   Student cannot ask directly what the shape is (ex.  Is the shape a sphere?  Is it number 4?)

·   Student cannot point to a shape.

·  Student cannot name an object that looks like the shape (ex.  Does the shape look like an ice cream cone?)

3.  As students ask their questions, the teacher can record responses on a Yes/No T-chart.

  Ex.                                 YES                    NO                      

                        Any flat surfaces?                  Square faces?

                        Rectangular faces?

4.  The second time the game is played, have a student be the "chooser" and the teacher can model effective types of questions by asking one or two him/herself.  The teacher can, by example, help students learn what is worth noticing about the solids.

5.  At the end of the class period, discuss the game, vocabulary used, good questions that were asked and helpful, as well as problems with the game.

Evaluation

1.  Students will play game in small groups and record questions asked.  Listen to how students communicate and discuss ways to reach consensus on terminology and descriptive attributes.

2. Students will write in their journals descriptions of various geometric solids, share them in small groups, and determine specific characteristics of each of the shapes.  These can be made into posters to be displayed in classroom and used as references for other activities in the geometric unit.

3.  Homework - Students can practice communicating by playing Guess My Solid at home with relatives or friends.  A list of questions asked should be recorded.

 Lesson 4 - Building Polyhedra

Goal

To understand structures of shapes, recognize relationships between parts of geometric shapes, and build polyhedra.

Objectives: 

1.  Students will recognize parts of polygons.

2.  Students will recognize components of polyhedra--vertices, faces, edges.

3.  Students will recognize how parts of polyhedra form whole shapes.

4.  Students will determine spatial relationships among faces, edges, and corners in order to build polyhedra. 

Materials:

Worksheet with pictures of geometric solids OR sets of geometric solids;  

Straws cut into the following lengths - 10-20 of each length:  two inch, three inch, four inch, six inch, OR

Toothpicks
Connectors - with hollow straws, you can use paper clips or thick pipe cleaners, bent and stuck inside the straw ends to connect two or more together.  With toothpicks you can use clay, play dough, mini-marshmallows, or small jelly beans as connectors

Teacher Resources

http://coolmath.com/polyhedra.htm - Polyhedra Gallery

http://mathforum.com/alejandre/workshops/unit14.html - Polyhedra in the Classroom

Procedure:

1.  Students work in pairs with a set of building materials (straws or toothpicks and connectors)

2.  Distribute worksheet of 12 geometric solids and/or a set of the geometric solids

3.  Allow students to work approximately 20-30 minutes to construct the 3 dimensional figures

4.  On another sheet of paper, after a figure is successfully built, students will write how many edges and corners it has as well as name the polyhedron.

5.  During whole class discussion, have students share their strategies for counting the faces of different polyhedra and explain how they constructed the figures.  Talk about the shapes of the different faces and the geometric solids which could not be constructed.

Extension:

1.  Provide verbal descriptions of a polyhedron and have students (in pairs) build it.  When finished, compare figures with another pair of students.  Are they the same?  How are they different?  What strategies (i.e. visualization, building parts and then trying to it them together) were used to build the polyhedron?

2.  Choose a "mystery" polyhedron and write a description.  Have another student/adult read and try to build the polyhedron or guess which polyhedron matches the description.