SHAPES FROM
FOUR SQUARES
Getting Ready
What You’ll Need
Geoboards, 1 per child
GEOMETRY • NUMBER
• Comparing
• Counting
• Transformational geometry
• Congruence
• Spatial visualization
Overview
Children try to make all the different shapes that can be made by putting
together four unit squares on their Geoboards. In this activity, children have
the opportunity to:
Small rubber bands, 4 per child
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use spatial visualization to build shapes
Geodot paper, page 90
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develop strategies to find new shapes
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test to find shapes that are flips or turns of other shapes
Geodot writing paper, page 96
Overhead Geoboard and/or geodot
paper transparency (optional)
The Activity
Introducing
Make and display this Geoboard shape.
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Ask children to tell something about the
shape. Determine that it is made up of three
of the smallest Geoboard squares, and that
every square shares at least one side with another square.
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Now have children use rubber bands to make a different Geoboard
shape with three squares. Tell children to be sure that at least one
whole side of each square touches that of another square.
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Have children hold up their Geoboards and discuss the differences
and similarities among the shapes.
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Model how to record your shape on geodot paper, then cut it out.
Then have children record and cut out their shapes.
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Collect the cut-out shapes, making two piles, one for each of the
two possible arrangements. Show how, by flipping or turning the
pieces in each pile, you can fit them over each other exactly to
make a neat stack.
You will probably
note many positions
for each of the two
arrangements of the
three squares.
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© ETA/Cuisenaire®
On Their Own
What different shapes can you make with 4 of the smallest squares on
your Geoboard ?
• Work with a partner. Each of you put 4 of the smallest
Geoboard squares together to make a shape. At least 1 side
of each square must touch 1 whole side of another square.
Okay
Smallest square
Not okay
• Compare your shapes.
• If you and your partner made the same shape, copy it only once onto geodot
paper. If you made 2 different shapes, copy both shapes.
• Make more shapes with 4 squares. Each time, compare your shape with your
partner’s. If the 2 are the same, record only 1.
• Record and cut out each different shape.
• Continue until neither of you can think of any new shapes to make.
• Count all your different shapes. Record that number.
• Compare all your shapes. Be ready to talk about them.
The Bigger Picture
Thinking and Sharing
Ask pairs to tell how many different shapes they found. List these numbers on the chalkboard. If necessary, help children to reach agreement on the number of different shapes that
are possible. Then have volunteers post their shapes across the board, one at a time. As they
bring up additional shapes, ask children to test each new shape against those already posted
to make sure theirs is unique.
Use prompts such as these to promote class discussion:
◆
Did some of your shapes turn out to be the same as others? What did you do to find
out that these were the same?
◆
Did you use any of your old shapes to find new ones? How did you do this?
◆
How can you be sure there are no more different shapes?
Drawing and Writing
Suggest that children look back at all the shapes and choose one. Have them sketch the
shape three times so that it is facing a different way each time. Then have them tell things
they could do to prove that the shapes are all really the same.
© ETA/Cuisenaire®
Extending the Activity
Have pairs play a game describing the shapes from four squares. The first
player chooses one of the five shapes, secretly draws it on geodot paper,
then describes it using words like above, below, and next to. The other
Teacher Talk
Where’s the Mathematics?
In this activity, children are given a hands-on introduction to transformational geometry. As children create shapes with four squares, then move the
basic squares around to create new shapes, the idea of transformation
becomes concrete.
There are five possible shapes that can be made by arranging four squares
so that at least one side of each square touches one side of another square
completely.
In the process of creating unique shapes and eliminating duplicates, some
children may simply try to visualize two Geoboard shapes in different positions. Other children may compare two shapes by actually turning their
Geoboards before they record a new shape. Still others may cut out their
paper shapes, then turn and flip them to see whether or not they match
other shapes.
As children discuss their strategies for determining whether or not two
shapes are the same, you may want to informally introduce the terms
turning, flipping, and sliding —terms used in transformational geometry.
Turning the Geoboard on the left a quarter turn clockwise until the two
shapes have the same orientation makes it easy to see that the two shapes
are identical.
© ETA/Cuisenaire®
player tries to make that shape on their Geoboard based on the descriptive
words. Players then compare their work to see if they have a match.
Flipping is a term used to describe picking up a figure and turning it over.
Many children in grades K-2 are likely to have difficulty imagining what a
Geoboard shape would look like if it was flipped over. Thus, it is probably
best to use the paper shapes to demonstrate flipping.
Sliding describes the movement of a shape from one place in a plane to
another.
Asking children to describe how they can turn, flip, and/or slide a shape on
their Geoboard or on a geodot-paper drawing so that it will exactly match
another child’s shape will contribute to children’s spatial reasoning ability.
During On Their Own, children may ask you if they have found all the possible shapes that can be made with four squares. Encourage children to
work to find all the possible shapes themselves (although it is not necessary
that each group do so). Later, during the class discussion, some children
may find it helpful to hear others describe their process of searching for
unique shapes. A child might describe, for example, starting with the shape
that has all four squares in a row, then leaving three of the squares and
moving just the fourth square, to find all the positions in which it could be
placed to create new shapes. Another child might then tell of leaving only
two of the squares in a row and moving the other two squares around until
all those possibilities have been exhausted.
© ETA/Cuisenaire®