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1
set geometric solids
Student Contract
1
Shape Up by David Adler
2-D Geometric Designs
1
pen
Polygon Names
1
Teaching Manual
Popsicle Square
3
white board markers
Popsicle Triangle
12
dice
How Many Triangles?
12
glue sticks
Prisms
12
pairs of scissors
Platonic Solids
24
pencils
Platonic Solids chart
100
square tiles
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Station Cards
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Parent Letter
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Dot to Dot
Dark Pentominoes
Light Pentominoes
Prisms
Platonic Solids
Cube
Octahedron
©
Tetrahedron
Dodecahedron
Icosahedron
1
Blokus® by Mattel
1
Pattern Block Sudoku by
Didax
1
Quizmo?™ Geometry by
Media Materials
2
IZZI® by ThinkFun
2
Blink® by Mattel
*Trademarked and/or commercially available games
are provided for educational use. Their inclusion in
this club kit does not constitute approval,
sponsorship, or endorsement of Zeno by the game
manufacturer.
2
tubs play clay
3
sheets chart paper
4
rolls transparent tape
24
journals
50
round coffee filters
100
chenille stems
150
sheets 1 inch graph paper
100
sheets 9 x 12 white
construction paper
160
sheets 60 lb paper
200
straws
275
popsicle sticks
400
toothpicks
Return envelope for evaluations
Student Club Evaluation
Teacher Curriculum Evaluation
Placing shapes into categories and using correct vocabulary to explain
categories.
Welcome students, go over the Student Contract and then ask students
what they know about shapes and geometry. Why is geometry important?
What occupations use geometry? Engineers, designers, contractors,
artists, etc.
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Remember to send home the Parent Letter at the end of class.
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Hand out the Pre-Math Club Student Evaluation sheet (copies are in the
manila envelope in the back of the Teaching Manual), and ask students to
answer the questions as best they can. Remind students that this is not a
test, but rather a way for you to understand their thinking. When students
have finished, collect the sheets and hold onto them until the last day of
club.
Divide the students into groups of 2 to 3 and pass out decks of Shapes Cards
to each group. Have each group divide their shapes into 3 categories. Ask
them to explain what their categories are and why their cards fit into each
category.
©
Pass out a die to each group. Have students roll the die and then categorize the
shapes according to the number on the die. When the students first start
grouping the cards they may not have the vocabulary to explain their categories.
Start helping to develop vocabulary by placing vocabulary words on chart
paper. Do not feel the need to use all the vocabulary listed on this page and
the next. These terms are used throughout the unit.
NOTE: If students roll a 1, a characteristic that all the shapes share is that they
are all closed figures.
Students may want to call any four-sided shape a square or rectangle. It is
helpful to describe shapes from the general to the specific. Show students an
example of what this means.
For example, in describing a student, a sequence from general to specific might
be:
Animal
Human
Child
Boy
Similarly, with a shape the sequence might be:
Closed figure
Polygon
Quadrilateral
Square
Play the game until students start to feel comfortable using the vocabulary and
understand how to categorize the shapes.
Make a copy of the
Student Contract and
Parent Letter for each
student. Make a copy
of each 2D Geometric
Shapes sheet , and
cut apart. Make
enough copies of the
Shapes cards onto
the 60 lb paper (and
then cut out) so that
there is one deck per
group.
-Student Contract
-Parent Letter
-Pre-Math Club
Student Evaluation
-Shape cards
-Dice
-Chart paper
CLOSED FIGURE—a
shape that begins and
ends at the same
point.
TRIANGLE—polygon
with three angles and
three sides.
SCALENE
TRIANGLE—triangle
with unequal sides
EQUILATERAL
TRIANGLE—triangle
with all equal sides.
ISOSCELES
TRIANGLE—triangle
with two equal sides.
QUADRILATERAL—
polygon with four
sides and four angles.
RHOMBUS—polygon
with four equal sides.
and right angles.
Hold up and flash a 2-D Geometric Design for the students to see for 3
seconds. Then ask students to draw the design in their journals from memory.
When most of the students are done drawing, flash the image again for another
3 seconds. Allow the students to make any revisions.
Show the image a final time and then discuss the different strategies used to
remember the figure. When finished, move on to the next design.
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Students may be aware of some common 2D designs in the world. Many of these
designs are symbols. A symbol is often an object, picture, or design that
represents something else. For example, the “peace” sign and the dove are
common symbols for peace, a light bulb is a common symbol for a good idea, and
a heart is often used to symbolize love.
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The following are some other common symbols:
Four leaf clover or horse shoe (luck)
Red cross on a white background (first aid)
Infinity
Male and female,
Company logos(e.g. Nike swoosh, etc.)
∞
What are some other symbols students have seen?
©
Have students create their own 2D symbols to represent something in their lives.
Give students the opportunity to create their own 2-D geometric designs for the
class to try to recreate. Use the same displaying and drawing sequence such as
seen in Activity 2.
NOTE: To avoid students drawing overly-complex designs, place some
restrictions on their creations. For instance, they may be limited to a certain
number of lines, curves, sides and/or instructed that there should be no open
figures, etc.
-Journals
SQUARE— a polygon
with four equal sides
and right angles.
TRAPEZOID—a foursided figure, with one
pair of sides parallel.
PARALLELOGRAM—a
four- sided figure with
both pairs of opposite
sides parallel.
LINE SEGMENT—part
of a line consisting of
a path between two
endpoints.
POLYGON—closed
shape with line
segments and angles.
RIGHT ANGLE—angle
measuring 90
degrees.
ACUTE ANGLE—
angle measuring less
than 90 degrees
OBTUSE ANGLE—
more than 90 degrees
VERTEX—point were
two sides of an angle
meet.
CONCAVE—“dent in”
CONVEX—“dent out”
5
6
7
8
pentagon
hexagon
septagon
octagon
9
10
11
12
nonagon
decagon
hendecagon
dodecagon
Understanding terminology such as horizontal, vertical, triangle, quadrilateral
and parallelogram. Learning what shapes are polygons and creating polygon
puzzles.
Flash a 2-D Geometric Design for 3 seconds. Have the students draw the
design in their journals. Flash the image again for 3 seconds and have them
make any necessary revisions. Show the image a final time and then discuss
the different strategies used to remember the figure.
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1. When creating triangles and angles, have them use
whole and broken toothpicks instead of slices of
American cheese and pretzels.
2. When measuring angles, have them use folded coffee
filters instead of round sheets of paper.
3. When creating quadrilaterals, have them use toothpicks
instead of pretzels.
4. When creating many sided polygons, have the students
cut them from graph paper instead of from slices of bread.
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Read the book Shape Up to the students. The students can
do the activities in the book with the following alterations:
Pass out 6 popsicle sticks to each student. Explain that that there is a way to
put these sticks together to form a square without using any glue or tape. Have
the students try to figure out how to put them together. If they are stymied, display the Popsicle Square sheet. After students have created a square, pass
out 5 more popsicle sticks and challenge them to create a triangle. If need be,
display the Popsicle Triangle sheet as a guide.
After the students have created the triangles and squares, have them test them
to see how sturdy they are. You can either have students drop them from various heights or take them somewhere to fly the puzzles like a frisbee to see how
well they hold up.
NOTE:
1. When creating the square polygon, weave the sticks together.
2. When creating the triangle, it is best to hold the three top pieces together in
the air and then slide in the other two popsicle sticks. When the sticks are
laying on the table it is harder to slot in the other popsicle sticks.
To prepare for Activity
1, read the book
Shape Up by David
Adler. For Activity 2,
practice building the
square and triangle
polygon puzzles.
-2-D Geometric
Designs sheet
-Shape Up by David A.
Adler
-Toothpicks
-Coffee filters
-Graph paper
-Popsicle Square
sheet
-Popsicle Triangle
sheet
Toothpick Polygons: Have the students use toothpicks to create the many
sided polygons in activity 1. They can use whole toothpicks to create regular
polygons or broken and whole toothpicks to create irregular polygons.
More Popsicle Puzzle Shapes: Once the students have mastered the square
and triangle popsicle puzzles, have them create other shapes. The following are
some ideas:
Adjust the angles in the square puzzle to create a rhombus.

Change the angle of an inner stick in the square puzzle to create four
internal trapezoid shapes.
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Create a rectangle by forming a puzzle with whole and broken popsicle
sticks.
Trapezoids
Rectangle
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Rhombus
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Divide the class into groups of 3 or 4. Pass out a number of toothpicks or
popsicle sticks to each group. Ask the students to create some large polygons
with the manipulatives.
Display the Polygon Names sheet and ask students to use it as a guide to name
the polygons they create.
-Popsicle sticks
-Toothpicks
-Polygon Names sheet
Playing games to learn about line segments, vertices, shape attributes and the
vocabulary to describe three-dimensional objects.
If students were unable to finish their popsicle squares and triangles in the
last lesson, have them finish up as they come into the class.
-IZZI® by ThinkFun
-Blink® by Mattel
-Dot to Dot sheet
-Geometric solids
-Station Cards
Set up the following stations in the classroom:
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STATION 1—2 IZZI® puzzles; 2 to 3 students per puzzle
Have students work together to solve each puzzle.

Make copies of the
Dot to Dot sheet and
Station Cards sheets.
Cut apart Station
cards.
What strategies did you use to solve these puzzles?
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STATION 2—2 sets of Blink®; 2 to 3 students per game
This is a very fast paced game in which students play cards that have similar
attributes.
What attribute was the most easily identified?
©
STATION 3—Dot to Dot; 2 students per dot paper
Give each student a pencil and each pair a Dot to Dot sheet. Students draw
line segments to connect one dot (vertex) to the next. On each turn they draw
one line segment either horizontally or vertically between the two points. The
object of the game is to create squares. You create a square by being the last
person to connect a line segment that will form a square. Once students form a
square they put their initials within it and take another turn. The student with
the most squares wins.

What strategies did you use to complete squares?
STATION 4—Polyhedron Structures; 2 students at a time
Give two identical sets of geometric solids to each student (Do not tell them
the names of these solids yet. Having them describe their characteristics will
help them understand the different shapes better.) The students sit back-toback and one student builds a structure with all the shapes, making sure their
partner does not see the creation. Students describe how their structure is built
as clearly as they can (using words such as “on top of”, “to the right of”,
“underneath”, “next to”, etc..) Do the two models match?

What challenges did you face with describing the shapes? Does it help
to know the exact name of the shapes?
After the students have had time at all the stations, review what they have
learned. Add any new vocabulary words to the list.
Introduce this bingo-like game to the whole class, in which students learn about
various geometric shapes and concepts.
DOT TO DOT GAME
M
Y
Understanding types of angles and angle measurements. Learning that acute,
obtuse, and right triangles are named for their angles.
Display the Triangles sheet. Have the students determine the number of
triangles in each figure. (The top triangle has 13 triangles. The bottom
triangle has 23.)
What kind of line have you made on the fold? A straight line.
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What is the special name for this line? Diameter.
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Pass out a coffee filter to each student. Have students fold them in half so the
sides match exactly.
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Have them fold the filters in half again so that the edges match exactly.
What kind of shape is formed on this fold? A right angle.
When students open up the circle, the fold makes four quarter turns (see
example below.) Have them draw lines on the folds and turn the circle so that
the lines are horizontal and vertical.
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0°
NOTE: About 6,000 years ago, the Babylonians thought
that the sun revolved around the Earth in 360 days.
Although they were wrong, because the Earth revolves
around the sun in slightly more than 365 days, 360 stayed
as the number of a full circle because it is a useful number
with many factors.
To prepare for Activity
2, cut several straws
to different lengths
(i.e. half, thirds,
fourths, etc.)
-Triangles sheet
-Coffee filters
-Scissors
DIAMETER—a line
segment that passes
through the center of
a circle, connecting
endpoints on the
circle.
DEGREE—a unit of
angular measure.
RIGHT ANGLE—an
angle equal to 90°.
ACUTE ANGLE—an
angle less than 90°.
OBTUSE ANGLE—an
angle greater than 90°.
Point out that circles are measured in degrees and that there are 360° in a full
circle. Have them mark the point where the vertical fold meets the top of the
circle (circumference) as 0°. Then have the students trace around the
circumference of the circle back to 0°. One revolution is 360°. Both 0° and
360° occupy the same point on the circumference of the circle.
Have the students trace with their fingers from the 0° to the bottom of the
vertical line.
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How much of the circle has your finger traced? One half. How many
degrees has it turned? 180° Mark that on the circle.
Now ask them to trace their fingers from the 0° clockwise to the position of the
horizontal fold.
How much of the circle has your finger traced? One fourth. How many
degrees has it turned? 90° Mark that on the circle.
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Have the students figure out the measurement of the final marking on the left
horizontal. Then have them refold the paper and then fold in half one more time
and create two more fold lines. Work with the students to figure out the degree
marks of these new folds. The final product will look like this:
315°
360° 0°
270°
90°
135°
225°
©
45°
180°
Keep this marked coffee filter compass for the next activity.
Pass out several straws previously cut to different lengths and some chenille
stems to each student. Have the students create angles by threading one
chenille stem through two straws and then bending the chenille stem where the
two straws meet (vertex.) They should create three angles; a right angle, an
acute angle, and an obtuse angle.
Now have the students estimate the measurements of their obtuse and acute
angles by using their coffee filter compass from Activity 1.
After they have formed their angles have the students add a third straw (they may
have to cut it to fit) to each angle to form a right triangle, an acute triangle and an
obtuse triangle. They can twist the ends of the pipe cleaner so that the triangle
will stay in place.
Collect the coffee filter compasses for use in the next lesson. Angles can be
sent home with students at the end of this lesson.
-Coffee filters
-Cut Straws
-Chenille stems
Looking at the properties of a triangle and learning about the sum of the angles
of a triangle.
Pass out the coffee filter compasses and ask students to stand up and all
face the same direction. Instruct students that they should hold the
compass so that the 0/360 mark faces away from their body.
Call out degree measurements and then have students rotate by that
amount. The following are some ideas for patterns to follow:
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Start at 0° then turn 90° clockwise.
Go back to 0°. Turn 180° counter-clockwise and then go back to 0°.
Rotate 360° clockwise, then 45° counter-clockwise and then back
to 0°.
Rotate 90° clockwise, 135° counter-clockwise and then 45°
clockwise.
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Divide the class into pairs; pass out two sheets of white construction paper to
each student and a pair of scissors and a glue stick to each pair.
Draw and name the six different types of triangles with the class (scalene,
isosceles, equilateral, right, obtuse and acute) Have them draw the different
types of triangles on their construction paper (make sure they draw them rather
large) and cut them out.
©
Direct students to mark each angle in all six triangles with an arrow pointing to
each vertex. Instruct students them to label the vertices (A,B,C, D, E, F), so that
all three angles in each triangle have the same letter (i.e., scalene= AAA,
obtuse= BBB, right= CCC, etc.). Then have students rip off large sections of
each corner of their triangles.
Ask students to choose any three of their triangles. Have students try
arrangements of triangles three angles with different letters so that the vertices
all point to the same spot (ABB or BBC, etc.) and arrange all the arrows
(vertices) of the angles so they point to the same spot. After they have tried
different arrangements, have students take their original sets of three angles
from each triangle (AAA, BBB, CCC, etc.), arrange them to point to the same
spot (should make a pretty straight line) and then glue these arrangements of
angles down on the other piece of construction paper. Then have them repeat
the above process with each of the six triangles they drew and label each set of
three angles with the name of the triangle they are from.
Ask some of the following questions:

What can you tell me about the angles of a triangle?

What do the measurements of the three angles in a triangle add up to? 180°

What types of triangles do you think this applies to? Every triangle.
-Coffee filter
compasses from
Lesson 4.
-White construction
paper
-Scissors
-Glue sticks
SCALENE
TRIANGLE—triangle
with unequal sides.
EQUILATERAL
TRIANGLE—triangle
with all equal sides.
ISOSCELES
TRIANGLE—triangle
with two equal sides.
RIGHT TRIANGLE—
triangle with a 90°
angle.
OBTUSE TRIANGLE—
triangle with an angle
greater than 90°
ACUTE TRIANGLE—
triangle with all angles
less than 90°
Divide the class into pairs. Pass out about 20 toothpicks to each pair. The
students will arrange the toothpicks into triangles.
Have the students start with 3 toothpicks. Ask the students what type of triangle
they made and how many pieces are on each side. Record the information on
the board while the students record the information in their journals. Then go
on to 4 pieces, then 5, then 6 and so on. Some numbers are not possible to
make into triangles and some have more than one form they can make. For
example, only one triangle is possible with 3 toothpicks, none are possible with
4 toothpicks and two different triangles are possible with 7 (a 3-2-2 isosceles
triangle or a 1-3-3 isosceles triangle.)
4
5
Equilateral
REGULAR
POLYGON—a polygon
with all sides equal
and all interior angles
equal.
None
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Types of Triangles
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Number of Pieces
PERIMETER—the sum
of the length of the
sides of a polygon.
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-Toothpicks
-Journals
-Toothpicks
-Journals
Isoceles
©
Divide the class into pairs. Pass out about 20 toothpicks to each pair. Have the
students arrange the toothpicks into other shapes beside triangles (square,
pentagon, hexagon, etc..)
Have them create and fill out a table similar to the one above for each shape.
(Point out that the number of toothpicks constitutes the perimeter of the shape.)
For example, the tables for a square and pentagon might look like this:
Pentagon
Square
Type of
Square
Number of
Pieces
Type of
Pentagon
3
None
4
None
4
1 by 1 square
5
Regular Pentagon
5, 6, 7
None
6, 7, 8, 9
Irregular Pentagon
8
2 by 2 square
10
Regular Pentagon
9, 10, 11
None
12
3 by 3 square
c
Number of
Pieces
11, 12, 13, 14 Irregular Pentagon
15
Regular Pentagon
Using spatial sense and geometrical problem solving to determine the 12
different pentomino shapes and to discover the shapes that can form an open
topped box.
Draw a large circle on the board. Explain that whatever you put in the circle
has a specific rule and whatever is outside the circle does not follow that
rule. See if students can guess the rule. The following are some figures to
draw:
Right Angles
Acute Triangles
Quadrilaterals (only 4 sides)
Polygons (closed shapes with only line segments and vertices)
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A pentomino is a shape formed by joining 5 squares together. Each square
must have at least one whole side of a square touching another whole side.
Examples:
This is not correct.
This is correct.
©
Pass out 5 square tiles and a piece of graph paper to each student. Have the
students use the square tiles to create a pentomino shape. Once they have
created a shape they may draw it out on the graph paper. Do not tell the
students how many pentomino shapes there are, but challenge them to create
and draw as many as they can.
Introduce the term “congruent” here and instruct students to test for congruence
to determine if they have found a new pentomino. If the shapes are congruent,
it is not a new pentomino.
Once students think they have found all the pentominoes, have them show you,
one on one, so others students’ searching and learning does not stop.
There are 12 different shapes; examples of them are on the right.
After the students have found all 12 pentominoes, have them try to figure out
that once folded, which ones will form open topped boxes and which will not.
Have the students cut out those pentominoes that open topped form boxes and
tape their boxes together.
-Square tiles
-Graph paper
-Scissors
-Tape
CONGRUENT—two
plane figures that
have the same size
and the same shape.
Have students look at the pentominoes and then figure out where to add a sixth
square to try to form a cube.
-Pentominoes
-Graph paper
-Blokus® by Mattel
-Pattern Block Sudoku
by Didax
In the African Congo, children of the Kuba people draw patterns of connected
squares in the sand. These patterns resemble the nets that their parents use for
fishing. Then the children trace the patterns without lifting their fingers and
without going over the same line more than once. Some of the networks they
draw are very complex.
CUBE NET—a two
dimensional shape
that can be folded to a
three dimensional
object.
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In this extension, have the students draw some of the simple networks (to the
right) on graph paper. Then see if they can trace the patterns without lifting their
pencils or going over a line more than once (they may cross the lines.)
Blokus® by Mattel
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Some solutions to the networks are shown to the right. Ask students to find
others. The numbers next to each side indicate one way that a network can be
retraced without lifting the pencil or going over the same line more than once.
1
9
10
8
4
5
7
6
Blokus® is a great strategy game that helps develop
logic, reasoning and spatial perception.
©
6
The game has some very small pieces. Let the students
know there are 21 different shapes for each color and
they should count all the pieces before they start. They
should count them again at the end the game so they
don’t lose any pieces.
5
4
3
18
2
7
8
9
17
1
14
13
12
27
28
11
10
15
16
12
11
16
Pattern Block Sudoku by Didax
14
13
10
If they lose a piece, they should figure out which shape is lost and then create
that same shape and color out of cardboard or construction paper.
This game gives students practice with problem
solving, shape recognition and spatial sense.
2
3
15
26
24
1
17
2
18
25
19
20
23 21
22
9
8
7
6
3
5
4
Exploring concepts of congruence by playing with pentomino shapes.
In this activity, students use pentominoes to explore concepts of congruence,
flips, slides, and turns.
Divide the class into pairs. Pass out a Dark and Light Pentominoes sheet and
some scissors to each pair. Have one person cut out a light shape while the
partner finds and then cuts out the matching dark shape. Have the partners
switch roles until all congruent shapes have been cut out and matched.
Practice concepts of flips, slides, and turns by displaying shapes on the
document camera. Display a shape and ask for volunteers to show a similar
shape that has been slid, flipped, or turned.
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Have each pair practice doing slides, flips and turns with their matching shapes.
This is an extension of Activity 1. In this activity, students work with a partner.
One person places a dark pentomino somewhere on a sheet of graph paper.
Then that person places a light pentomino somewhere else on the paper.
Make copies of the
Dark Pentominoes
and Light
Pentominoes sheets.
-Scissors
-Dark Pentominoes
sheet
-Light Pentominoes
sheet
-Graph paper
TRANSLATION/
SLIDE—moving a
figure without turning
or flipping it.
Next, the partner figures out the following:
1) Whether the second shape is congruent to the first (if it is not, the partners
switch roles and do the activity again.)
©
2) What has to be done to the first shape to match the second shape (e.g., a
flip and a slide to the right, a flip and a turn clockwise, etc..)
As the students get better at moving the shapes, they can get more specific
about the details of the movements (e.g., a flip and a slide two spaces to the
right, a flip, a slide 5 spaces down, and a turn 90 degrees counter-clockwise.)
REFLECTION/FLIP—
rotating an object
over a line of
symmetry, creating a
mirror image of the
original.
ROTATION/TURN—an
object is moved by
turning not flipping;
the size or shape is
not changed by the
turn.
CONGRUENT— two
plane figures that
have the same size
and the same shape.
In this game, students reflect movements that a teammate
has made.
Divide students into pairs. Each pair is a single team who
will be assigned another pair against whom they will play.
In every pair, one student is a “mover” and the other is assigned the role of “reflector.” NOTE: On the team’s second and subsequent turns, the “mover” and “reflector”
change positions.
-Dice
Teams roll a die. The team with the highest roll plays first.
The “mover” starts with a normal stance (feet together, hands at side.)
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The “reflector” rolls a die. The “mover” then moves as many body parts as are
on the rolled die. For example, if the die shows a 2, the “mover” creates a
stance with two changes from a “normal” stance (e.g., left arm up, right arm out
to the side.)
The “reflector” tries to correctly reflect each movement and gets as many points
as the number of correct reflections. If students roll a 4, 5 or 6, the mover must
move at least 3 body parts, but the team can move (and get points for) up to the
number on the die.
When the first team has finished, the second team takes a turn, repeating the
above procedure of rolling the die, then moving and reflecting.
©
The game continues until one team reaches 11 points.
Understanding what makes a 3-dimensional shape a prism. Learning to count a
prism’s faces, edges and vertices.
Pass out some play clay and toothpicks to each student. Have students
create a number of small play clay spheres by rolling them between their palms.
For best results, the spheres should be about 1/4 to 1/2 inch in diameter.
Have the class start creating prisms by first making two bases of a polygon (i.e.,
square, triangle, pentagon, hexagon etc… .) The bases should be as close to
the same size and shape as possible. After finishing two bases of the prism,
have them vertically attach toothpicks to them to complete the prism.
Write the word prism on the board and ask students if they know the definition
of the word. Show them some examples of geometric solids and take out
those that are prisms.
Can they tell you why they are prisms?
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
Make copies of the
Prisms sheet.
-Play clay
-Toothpicks
-Geometric solids
-Prisms sheet
-Prisms Answer sheet
PRISM—a 3dimensional shape
that has two
congruent bases.
Students should make at least two prisms with the play clay spheres and
toothpicks.
PLANE—a flat surface
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Prisms are 3-dimensional figures that have two identical opposite faces or
bases. A prism is named by the shape of its base. So there are triangular
prisms, hexagonal prisms, pentagonal prisms, etc…
©
Once students have created a few prisms, introduce them to the vocabulary
words face, edge, vertex. Give each student a Prisms sheet. Have them
count their prisms’ faces, edges and vertices and record the correct values in
the handout.
As a class, go over the handout and fill in any remaining information. If no one
has made a certain type of prism have the students figure out how many edges,
faces and vertices that figure will have. What patterns do they see in the
information?

How does the number of edges of each base relate to the number of total
edges?

What pattern do you see with the number of faces and edges?

What pattern do you see with the number of edges to the number of
vertices?
Discuss the famous mathematician Leonhard Euler, born in Basel, Switzerland
in 1707. Euler, one of the greatest and most prolific mathematicians of all time,
wrote nearly 900 mathematical papers in his life time (half of them after he was
blinded at the age of 60.) He also developed this formula for polyhedrons:
V + F - E = 2 (V=number of vertices, F=number of faces, E = number of edges.
Have the students use Euler’s formula to verify their answers on the prisms
table. Answers are provided to the teacher on the Prisms Answers sheet.
Have them use Euler’s formula to predict the numbers of faces, vertices, and
edges of other prisms (octagonal, nonagonal, and more.)
FACE—a plane/flat
shaped figure.
EDGE—a line
segment that joins
two faces.
VERTEX—where two
or more line segment
meet.
RECTANGLUAR
PRISM
PENTAGONAL PRISM
Identifying and creating the platonic solids.
Review prisms and Euler’s formula.
For more information on platonic solids, go to www.enchantedlearning.com/
math/geometry/solids/
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NOTE: Five regular polyhedrons were discovered by the ancient Greeks.
These solids are the cube, tetrahedron, octahedron, dodecahedron and
icosahedron. The Pythagoreans knew of the tetrahedron, the cube, and the
dodecahedron; the mathematician Theaetetus added the octahedron and the
icosahedron. These shapes are also called the Platonic solids, after the
ancient Greek philosopher Plato.
For a polyhedron to be classified as a platonic solid, all of its faces must be
congruent. Show the students the cube and tetrahedron previously made.

What polygons are on the faces of a cube and a tetrahedron? Square and
triangle

Are the faces congruent? Yes.

How many faces are there on a tetrahedron? 4. On a cube? 6.
Photocopy the Cube,
Tetrahedron,
Octahedron,
Dodecahedron and
Icosahedron sheets.
Create a cube and
tetrahedron from the
Cube and Tetrahedron
sheets to show
students.
-Scissors
-Tape
-Graph paper
-Cube, Tetrahedron,
Octahedron,
Dodecahedron,
Icosahedron sheets
-Platonic Solids sheet
-Platonic Solids
Answers sheet
©
Ask students why there is no “rectangularhedron” (because not all of a
rectangular polyhedron’s faces are congruent.)
Now explain to the students that they are going to create a cube by using the 1
inch graph paper and cutting out the shape they think makes a cube. There are
different ways to make a cube. The graphic on the right shows one example.
Pass out Tetrahedron, Octahedron, Dodecahedron and Icosahedron sheets
to each student. Have the students cut the templates out and form these
Platonic Solids.
They can work in pairs, with one student cutting out a shape and the other
preparing tape to put the shape together into a Platonic Solid. The tetrahedron
and octahedron are the easiest ones to create, so students should start with
those.
Once the shape construction has been completed, have the students fill in the
Platonic Solids sheet. Answers are provided to the teacher on the Platonic
Solids Answers sheet.
POLYHEDRON—a
solid formed by the
joining of its several
faces.
Please ask the students to fill out a Post-Math Club Student Evaluation sheet. Remind students that this is
not a test, but rather a way for you to understand their thinking.
Also, teachers please fill out the Curriculum Evaluation form so that Zeno learns how to better serve you
and the students in the future.
Please mail all the Student Evaluations (those from the first day of class, and those from the last) and your
Curriculum Evaluation back to Zeno in the self-addressed envelope provided.
Games:
Blokus® by Mattel
Adler, David A. Shape Up. Holiday House/
New York, 1998
Quizmo?™ Geometry by Learning
Advantage
Burns, Marilyn The Greedy Triangle.
Scholastic Inc./New York, 1994
Pattern Block Sudoku® by Didax
Cavon, Lucille Geometry. Enslow Publishers,
2001
Reflection Game
Activities:
©
2D Geometric Designs
Polygon Puzzles
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Dot to Dot Game
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IZZI® by ThinkFun
Burns, Marilyn About Teaching Mathematics
second Edition. Math Solutions
Publications, 2000.
Blink® by Mattel
Back to Back Polyhedrons
Long, Lynette Groovy Geometry: Games and
Activities that Make Math Easy and
Fun. J Wiley, 2003
Smoothey, Marion Let’s Investigate Angles
Marshall Cavendish Corporation,
1993
Smoothey, Marion Let’s Investigate Triangles
Marshall Cavendish Corporation,
1993
Tierney C & Russell S Ten-Minute Math Dale
Seymour Publications, 2001
VanCleave, Janice Janice VanCleave’s
Geometry for Every Kid. Wiley &
Sons/New York, 1994
www.mathisfun.com
www.enchantedlearning.com/math/geometry/
solids/
http://illuminations.nctm.org
http://nlvm.usu.edu
VanDeWalle, John A. Elementary and Middle
School Mathematics: Teaching
Developmentally, 4th Edition. Addison
Wesley Longman, Inc., 2001
Zaslavsky, Claudia Math Games and
Activities from around the World.
Chicago Review Press, 1998
Dear Parents and Caregivers,
Congratulations on enrolling your student in a Zeno math-powered club. Let the fun begin!
Research shows that students who actively engage in their learning do better in school. Enter Club
Zeno—a math adventure where students explore math skills and concepts in new and unexpected
ways. Zeno math clubs use games, movement, and investigations that pique students’ curiosity and
engagement in math.
The learning is not limited to math club! At home, you can help reinforce your child’s growth by:
positive. Never tell your child that you hated math or were not good at it—this gives
them permission to do the same.
 Encouraging growth. Participate in family math nights, robotics, chess clubs, and math
Olympiad teams.
 Making math fun. Spend time playing board games and puzzles to encourage math risktaking and persistence.
 Using real world examples. Point out ways people use math every day to budget their
money, pay bills, make change, or leave a tip.
 Preparing for a profession. Let kids know what vocations require a sound base in
mathematics.
 Encouraging problem solving. Provide assistance but let kids figure out the answers to
questions themselves. Problem solving is a lifetime skill.
©
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 Staying
As 21st century learners, technology is an increasingly present mode of learning for children. Here
are some Zeno-vetted educational web sites and phone apps to help keep your child’s screen time
fun AND educational.
Websites
Smart Phone Apps
www.mathplayground.com
Mathmateer
www.factmaster.com
Math Doodles
www.funbrain.com
Numerosity
www.funbrain.com
Glow Burst
www.coolmath4kids.com
Motion Math: Zoom or Hungry Fish
Lastly, remember to ask your children what game or activity they learned in their math club when
you pick them up. Your interest in their learning = happy, math-powered students!
Sincerely,
The Zeno Team
Student Contract
Welcome to a Zeno math club. In order for all students to have a positive experience in this club we
have developed the following guidelines. The guidelines explain the positive behaviors we expect,
as well as behaviors that are unacceptable.
Positive behavior will allow us to learn, play, grow and have fun together. Unacceptable behaviors
will be handled by the club instructor, who may choose to contact the parents, teacher, or principal
of the misbehaving student. Consequences of unacceptable behavior could include a warning or
suspension from club activities.
Positive Behaviors
Listen and cooperate with students and teachers in the program.
Wait quietly.
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Be responsible and respectful with your words and actions.
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Follow directions.
Treat the materials carefully and use them in the way that you are instructed.
Help with cleanup.
Unacceptable Behaviors
©
Not following school rules.
Put-downs, teasing, and swearing.
Roughhousing, pushing, tripping, hitting, kicking or play fighting.
Damaging materials or taking them out of the room (without teacher permission).
I agree to follow these behavior guidelines and to do my best to help everyone have a positive
experience.
___________________________________
__________
Signature
Date
Modified from
Student ID: __________________________
PRE-Exploring Geometry
Please check ONE response to each statement. Remember that there are no “right” or “wrong” answers. We
want to know how you feel about math. You do not have to write your name on this survey.
Statement
Disagree
Agree
Strongly
agree
1. I am really good at math.
☐
☐
☐
☐
2. I love math.
☐
☐
☐
☐
☐
3. I understand math.
☐
☐
☐
☐
☐
☐
☐
☐
☐
☐
5. I can solve difficult math problems.
☐
☐
☐
☐
☐
6. I enjoy doing math puzzles.
☐
☐
☐
☐
☐
7. Math is very hard for me.
☐
☐
☐
☐
☐
8. I do math problems on my own “just
for fun.”
9. Math is confusing to me.
☐
☐
☐
☐
☐
☐
☐
☐
☐
☐
10. Math is fun.
☐
☐
☐
☐
☐
11. I look forward to learning new math.
☐
☐
☐
☐
☐
12. Math comes easily to me.
☐
☐
☐
☐
☐
13. I hate math.
☐
☐
☐
☐
☐
14. I enjoy playing math games.
☐
☐
☐
☐
☐
15. I can tell if my answers in math make
sense.
16. I enjoy studying math.
☐
☐
☐
☐
☐
☐
☐
☐
☐
☐
17. Doing math is easy for me.
☐
☐
☐
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☐
18. Solving math problems is fun.
☐
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☐
☐
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4. Math is boring.
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Strongly
disagree
How I feel
Neither
agree or
disagree
☐
Please tell us a few things about you.
Are you a boy or girl? ____________________
Please circle what grade you are in:
3rd grade 4th grade
5th grade
Modified from
Student ID: ________________
PRE-Exploring Geometry
Please circle your race (you may circle more than one):
6. Middle Eastern or Arab
2. Asian American
7. Native Hawaiian or other Pacific
3. African American or Black
8. White or European American
4. Caribbean Islander
9. Other
5. Hispanic—Latino/a
10. I am not sure.
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1. American Indian or Alaska Native
1. Draw a horizontal line inside the triangle and a vertical line inside the circle.
2. What is fewest sides a polygon can have? Write the number:_______
©
Draw and name the shape with that many sides.________________
3. Circle the angle below that is a right angle.
4. How many degrees are in a circle? __________________________
5. How many faces does a rectangular prism have?_________________
Thank you for completing this survey! It will help Zeno learn and grow.
©
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Shape Cards (page 1)
Zeno: Exploring Geometry: Lesson 1 Activity 1
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Shape Cards (page 2)
Zeno: Exploring Geometry: Lesson 1 Activity 1
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Shape Cards (page 3)
Zeno: Exploring Geometry: Lesson 1 Activity 1
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Shape Cards (page 4)
Zeno: Exploring Geometry: Lesson 1 Activity 1
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Shape Cards (page 5)
Zeno: Exploring Geometry: Lesson 1 Activity 1
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2-D Geometric Designs
Zeno: Exploring Geometry: Lesson 2 Starter
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2-D Geometric Designs
Zeno: Exploring Geometry: Lesson 2 Starter
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2-D Geometric Designs
Zeno: Exploring Geometry: Lesson 2 Starter
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Popsicle Square
Zeno: Exploring Geometry: Lesson 2 Activity 2
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Popsicle Triangle
Zeno: Exploring Geometry: Lesson 2 Activity 2
POLYGON NAMES
“Gon” is derived from the Greek word “gonu” meaning “knee” which represents an “angle” in a shape
“Hedron” in “polyhedron” means “seat” which represents a “face” of shape.
A polygon is a shape with many knees or a 2-D object with many angles.
Polyhedron is a shape with many seats or a 3-D object with many faces.
To make a 2-D polygon into a 3-D polyhedron just replace the “gon” with “hedron”
Names of polygons past 12.
13–tridecagon
14–tetradecagon
15–pentadecagon
16–hexadecagon
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17–heptadecagon
18–octadecagon
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19–nonadecagon
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To construct the name of a polygon with more than 20 and less than 100 sides,
combine the prefixes as follows:
and
©
Tens
A 42 sided polygon would be:
Tens
tetraconta-
1
-hena-
20
icosi-
2
-di-
30
triaconta-
3
-tri-
40
tetraconta-
4
-tetra-
50
pentaconta-
5
-penta-
60
hexaconta-
6
-hexa-
70
heptaconta-
7
-hepta-
80
octaconta-
8
-octa-
90
enneaconta-
9
-ennea-
and
-kai-
Ones
Ones
-di-
-kai-
final prefix
-gon
final prefix
-gon
full polygon name
tetracontakaidigon
100—hectogon
1000– chiliagon
10000– myriagon
Zeno: Exploring Geometry: Lesson 2 Enrichment Extension
Dot to Dot
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Zeno: Exploring Geometry: Lesson 3 Station 3
Station Directions
For a real challenge see if you can get all the small
individual squares to form a giant square, where
only white touches white and black touches black.
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2 to 4 students per puzzle
Try to create a continuous pattern of white touching
white and black touching black.
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Station 1
Izzi
Station 2
Blink
2 students per game
Deal all the cards so that each player has an equal
amount. Place two cards face up in the middle of
the table.
Put 3 cards in your hand from your pile.
As fast as you can, place down the cards, one at a
time, in the pile that has at least one attribute that
is the same as your card.
Once you have placed a card on the pile, add more
cards to your hand from the your draw pile so you
have AT MOST 3 cards in your hand.
First one out of all their cards wins.
Zeno: Exploring Geometry: Lesson 3
Station Directions
On your turn draw a line, either horizontally or
vertically, to connect two dots. If you are able to
draw a line that creates a square, write your initials
in the square and go again.
The student with the most squares with their initials
in it wins.
©
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2 students per game
The object of the game is to create the most
squares.
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Station 3
Dot to Dot
Game
Station 4
Polyhedron
Structures
2 students at station
Players share a tub of geometric solids. They
divide the pieces so they each have an equal
number of each type.
Sit back to back. One player creates a geometric
structure out of the shapes. Then that player
describes those shapes and their locations to the
other player.
The other player tries to match the structure
without looking at it. Once done giving the
directions, the players turn around and compare
models.
Zeno: Exploring Geometry: Lesson 3
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How Many Triangles?
Zeno: Exploring Geometry: Lesson 4 Starter
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Dark Pentominoes
Zeno: Exploring Geometry: Lesson 7 Activity 1
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Light Pentominoes
Zeno: Exploring Geometry: Lesson 7 Activity 1
Prisms
Shapes
Number of Faces
Number of Vertices
Number of Edges
Triangular
Prism
©
Pentagonal
Prism
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Square Prism
(Cube)
Hexagonal
Prism
Zeno: Exploring Geometry: Lesson 8 Activity 2
Prisms Answers
Rectangular
Prism
6
9
6
8
12
6
8
7
©
Pentagonal
Prism
Hexagonal
Prism
Number of Edges
5
8
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Square Prism
(Cube)
Number of Vertices
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Triangular
Prism
Number of Faces
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Shapes
12
10
15
12
18
Zeno: Exploring Geometry: Lesson 8 Activity 2
Platonic Solids
Solid
Number of
Faces
Shape of
Faces
Number of
Faces at Each
Vertex
Number of
Vertices
Number of
Edges
Tetrahedron
Dodecahedron
©
Octahedron
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Cube
Icosahedron
Zeno: Exploring Geometry: Lesson 9 Activity 2
Platonic Solids Answers
Number of
Solid
Number of
Faces
Faces at Each
Number of
Vertex
Vertices
Shape of Faces
Number of
Edges
4
triangle
3
4
6
6
square
3
8
12
Tetrahedron
8
Octahedron
Dodecahedron
triangle
4
6
12
pentagon 3
20
30
triangle
12
30
©
12
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Cube
20
5
Icosahedron
Zeno: Exploring Geometry: Lesson 9 Activity 2
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octahedron
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Cube
Zeno: Exploring Geometry: Lesson 9 Activity 2
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tetrahedron
Zeno: Exploring Geometry: Lesson 9 Activity 2
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octahedron
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Zeno: Exploring Geometry: Lesson 9 Activity 2
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dodecahedron
Zeno: Exploring Geometry: Lesson 9 Activity 2
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n
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a
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Zeno: Exploring Geometry: Lesson 9 Activity 2
Modified from
Student ID:_________________
POST– Exploring Geometry
Please check ONE response to each statement. Remember that there are no “right” or “wrong”
answers. We want to know how you feel about math. You do not have to write your name on this survey.
How I feel
Disagree
Strongly
agree
1. I am really good at math.
☐
☐
☐
2. I love math.
☐
☐
☐
☐
☐
3. I understand math.
☐
☐
☐
☐
☐
4. Math is boring.
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Strongly
disagree
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Agree
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Neither
agree or
disagree
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Statement
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7. Math is very hard for me.
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8. I do math problems on my own
“just for fun.”
9. Math is confusing to me.
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10. Math is fun.
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11. I look forward to learning new
math.
12. Math comes easily to me.
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13. I hate math.
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14. I enjoy playing math games.
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15. I can tell if my answers in math
make sense.
16. I enjoy studying math.
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17. Doing math is easy for me.
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18. Solving math problems is fun.
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5. I can solve difficult math problems.
6. I enjoy doing math puzzles.
Please tell us a few things about you.
Are you a boy or girl? ____________________
Please circle what grade you are in:
3rd grade 4th grade
5th grade
Modified from
Student ID: _________________
POST-Exploring Geometry
Please circle your race (you may circle more than one):
6. Middle Eastern or Arab
2. Asian American
7. Native Hawaiian or other Pacific Islander
3. African American or Black
8. White or European American
4. Caribbean Islander
9. Other
5. Hispanic or Latino/a
10. I am not sure.
rv
ed
1. American Indian or Alaska Native
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1. Draw a horizontal line inside the triangle and a vertical line inside the circle.
2. What is fewest sides a polygon can have? Write the number:_______
©
Draw and name the shape with that many sides.________________
3. Circle the angle below that is a right angle.
4. How many degrees are in a circle? __________________________
5. How many faces does a rectangular prism have?_________________
Thank you for completing this survey! It will help Zeno learn and grow.
Curriculum Evaluation Form
Exploring Geometry
Dear Club Teacher,
Thank you so much for making math fun for students. In order to make these clubs most effective for students and club
teachers, we need to get some information from you on the lessons and games. Please include on the back of this form
any additional information that you would like us to know .
Thank you
Ages/Grades of Students: ______________
How many weeks was the club?______________
How many lessons were completed: ________
How many students did you teach?___________
Did you utilize Zeno’s online math club training?
YES
NO
If yes, please explain your experience.
_______________________________________________________________________________________________
_______________________________________________________________________________________________
No change
More confidence
rv
Less confidence
ed
From the beginning to the end of math club, overall, did you observe any shift in student confidence? Please
explain.
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________________________________________________________________________________________________
________________________________________________________________________________________________
What lessons and games did you find to be most helpful, and why?
________________________________________________________________________________________________
________________________________________________________________________________________________
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What lessons and games did you find to be least effective, and why?
________________________________________________________________________________________________
________________________________________________________________________________________________
Is there anything that you feel needs to be changed or restructured?
________________________________________________________________________________________________
________________________________________________________________________________________________
Do the daily lessons provide enough activities to fill an hour?
________________________________________________________________________________________________
______________________________________________________________________________________________
Were any supplies missing from the club kit?
________________________________________________________________________________________________
________________________________________________________________________________________________
Please return evaluation forms to:
Zeno
1404 East Yesler Way, Suite 204
Seattle, WA 98122
If you have any other questions or concerns please feel free to
contact: Program Director, Jennifer Gaer at 206-325-0774 or
[email protected]