HOW MANY
SQUARES?
Getting Ready
What You’ll Need
Geoboards, 1 per child
GEOMETRY • PATTERNS/FUNCTIONS
• Using patterns
• Spatial visualization
Overview
Children search for all the squares that can be made on the Geoboard that
have sides parallel to the Geoboard’s sides. In this activity, children have the
opportunity to:
Rubber bands
make and compare squares of different sizes
Geodot paper, page 92 (optional)
use spatial visualization
Overhead Geoboard and/or geodot
paper transparency (optional)
collect and analyze data
discover patterns and use them to make predictions
The Activity
Introducing
Ask children to use one rubber band to make a square on their
Geoboard. Specify that the sides of their square must be parallel
to the sides of the Geoboard.
Explain that Geoboard squares can be identified by their dimensions. Then display a one-by-one square on your Geoboard.
Confirm that the length of each side is one unit, and that the square
is a one-by-one square.
Invite children to identify the dimensions of their squares.
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On Their Own
How many squares can be made on a Geoboard?
• Work with your group to make as many squares as you can on the Geoboard.
Make only squares that have sides parallel to the sides of the Geoboard.
• Keep track of the different-sized squares you make, and the number of squares of
each size. Record this information and the total number of squares you made.
• Look for patterns in your work.
The Bigger Picture
Thinking and Sharing
Have children help you create a class chart showing the dimensions of the squares and the
number of each size that can be made. If there is disagreement, let children work together
to resolve their differences.
Use prompts such as these to promote class discussion:
◆
How many squares did you find in all?
◆
What size squares did you find? How many of each size?
◆
How did you go about finding your squares?
◆
Were some squares easier to find than others? If so, which ones, and why?
◆
How did you keep track of your squares?
◆
◆
Do you think that you have found all the possible squares? What makes you
think so?
Did you need to make all the squares to know how many there would be? Explain.
Writing
Ask children to figure out how many squares they could make on a six-by-six Geoboard and
to explain their reasoning.
Extending the Activity
Have children use a rubber band to mark off a three-by-three square region. Ask them to
find the number of squares that can be made in this region.
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Teacher Talk
Where’s the Mathematics?
To find all 30 squares, children need to consider both size and location of
the squares. It is possible to make squares of four different sizes: one-byone, two-by-two, three-by-three, and four-by-four. As children experiment,
they come to realize that squares can overlap as well as be adjacent. Most
children can find the adjacent two-by-two squares (see A) with little difficulty, but usually need to work harder to find the five other two-by-two squares
(see B and C) that overlap with these squares.
Visualizing overlapping squares is not
an easy task. Children may find it
helpful to use different-colored
crayons to record each square on
geodot paper as they work.
1
4
2
5
3
A
B
C
One way children may deal with finding all the two-by-two squares on the
Geoboard is to start with a two-by-two square in the upper left-hand corner,
then move it one unit at a time until all the places where a two-by-two
square can be made have been found.
© ETA/Cuisenaire®
The same procedure can be used to find the four three-by-three squares that
can be made on the Geoboard.
As they work, some children may notice that the number of squares of any
one size that can be made in one row working across the Geoboard is the
same as the number of rows of that size square that can be made going
down the board. Specifically, there are four one-by-one squares in each of
four rows, three two-by-two squares in each of three rows, and two threeby-three squares in each of two rows.
Making a table, like the one at the right, provides an opportunity to introduce children to the term square numbers. (These are the numbers in the
second column on the table.) Greek mathematicians named the sequence
of numbers 1, 4, 9, 16, 25, ... square numbers because they, unlike other
numbers, could be represented as square arrays of dots.
1
4
9
16
Size of
Square
Number of
Squares
1x1
2x2
3x3
4x4
16
9
4
1
25
The table also gives children the opportunity to talk about the numerical
patterns that come from squares produced on the Geoboard. Children may
notice that the differences between 1 and 4, 4 and 9, and 9 and 16 are odd
numbers, 3, 5, and 7, respectively. Investigating further, children can conclude that consecutive square numbers always differ by an odd number.
Another pattern children may observe is that the product of the dimensions
of the four-by-four square is the number of one-by-one squares that can be
made (16). Similarly, the product of the dimensions of the three-by-three
square is the number of two-by-two squares that can be made (9), the product of the dimensions of the two-by-two square is the number of the threeby-three squares that can be made (4), and the product of the dimensions of
the one-by-one square is the number of four-by-four squares that can be
made (1).
Being able to find patterns is an extremely useful skill for children to have.
It enables them to solve problems, make predictions, formulate conjectures,
and draw generalizations. Being able to distinguish overlapping shapes is
another useful skill. Children will encounter overlapping shapes many times
in geometry problems.
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