**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. |