Chapter 11: Basic Geometry
Children should be introduced to the basic shapes as early as possible, in order for them
to practice discernment, classification, and vocabulary. Do not overwhelm them, however,
because it will be a long time before this knowledge is critical for their education. Introduce a
few of the concepts at first, and add more when the kids are comfortable and ready for more.
The basic shapes are circles, squares, rectangles, and triangles:
Even these can be a bit tricky for very young children, for the difference between a square and a
rectangle is rather subtle. Ask the children to point out all the examples of these they can find, in
the room, outside, and elsewhere.
When a child has an easy time with these shapes, you can begin to introduce the concept
of a polygon. A polygon is a closed loop made of straight lines that does not intersect itself.
These aspects are generally easiest to understand by looking at counter-examples:
Again, this technical definition is something that a high school student ought to know. In
elementary school, it is enough for a child to know the names of basic polygons.
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Most of geometry concerns itself with lines and polygons. There are a few shapes with
curved sides that children ought to know:
Polygons are most generally classified by the number of straight parts (called either edges
or sides) which make up the loop. A 3-sided polygon is called a triangle. A 4-sided polygon is
called a quadrilateral. A 5-sided polygon is called a pentagon. A 6-sided polygon is called a
hexagon. A polygon with 7 sides is a heptagon, with 8 sides an octagon, with 9 sides a nonagon,
with 10 sides a decagon, and with 12 sides a dodecagon. For young children, it is sufficient to
count the sides and not necessary to learn the Greek and Latin prefixes for numbers. In fact,
most polygons have names which come straight from the number. For example, a polygon with
25 sides is a 25-gon, and one with 43 sides is a 43-gon.
A polygon that looks like it has a dent in it (a bit like a cave) is called concave.
Otherwise, it is called convex:
Nearly all of the shapes used in mathematics are convex, but it is useful to know these words.
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Polygons are also classified depending upon how many of their sides have the same
length. A triangle with two sides the same length is called isosceles, with all three sides the same
length equilateral, and with three different lengths scalene. We use little tick-marks to indicate
when two sides have the same length:
A quadrilateral with all sides the same length is called a rhombus:
Nearly all of the polygons with 5 or more sides that appear in mathematics are regular polygons,
where all the sides have the same length and all the angles are the same:
A square, for example, is a regular quadrilateral because it has four sides all the same length and
the angles at the four corners are all the same.
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Another means of classifying a polygon depends on the concept of parallel lines.
Mathematically, this is a very tricky concept, but it can be introduced rather simply to children.
Basically, two lines are parallel if they run in the same direction. All horizontal lines are
parallel, for example, as are all vertical lines:
Slanted lines can be parallel, too, but only if they slant in the same way:
Actually, only quadrilaterals are classified by parallel lines. A quadrilateral with two
pairs of parallel lines is called a parallelogram. A quadrilateral with only one set of parallel
lines is called a trapezoid. We put little arrows on lines to indicate that they are parallel:
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The last way to classify polygons depends on right angles. The easiest way to explain a
right angle is that it is the right way to place a block on the ground if you don't want it to fall
over. There are technically no "wrong" angles in math, but the idea is useful in introducing right
angles:
A triangle with a right angle is called a right triangle. A quadrilateral with all angles right is
called a rectangle. Because the angles of a square are all right angles, we put a little square in
the corner of an angle to indicate that it is right:
For the most part, the shapes confronting elementary school children are all drawn to
scale, so they should recognize when an angle is right or when two sides are the same length,
even if the squares and tick-marks are absent. It is only in later grades when figures are drawn
contrary to their information, just to test analytical skills.
Once again, the vocabulary listed in this chapter should not be given to a child all at once,
but only gradually as the child shows a readiness and capacity for learning more. Start with
circles, squares, rectangles, and triangles. Move on to regular polygons – saying "eight-sided
shape" instead of "octagon" at first. Later introduce the idea of a right angle as something found
in squares and rectangles, then define right triangles. When the child is comfortable with all of
this, you can introduce the concept of parallel lines, parallelograms, and trapezoids. Concavity
and convexity are not too difficult to differentiate. The different kinds of triangles, rhombi, and
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other properties based on length can be introduced as children work through the edge-measuring
exercises discussed later in this chapter.
As an early exercise in discernment, make a set of index cards with something like the
following drawn on them, one symbol per card:
Shuffle these up and deal out a dozen or so to each kid and have them separate them into
groups. There are a variety of ways in which this can be done, so accept a lot of different
answers. Some might separate them by color, some by shape. Some might put the ones with a
similar pattern together. Some might put the four-sided shapes all together. Have the students
discuss the way they did this, to share all the ways it could be done. If everyone misses an
obvious and important category, you can mention it, but try to let them figure them all out on
their own.
As a later exercise, deal out 2 cards to each kid. The new challenge is to name all the
ways in which the two objects are similar (shape, color, pattern, number of sides, etc.) and all the
ways in which they are different. For a more advanced challenge, deal out 3 cards to each kid
and see if they can find something that all the cards have in common.
As another exercise, have a child draw out a card from one of these decks. The child
must name one of the aspects of the figure and then go through the deck and pull out all the cards
that share that attribute. To make this more challenging, have the child name two aspects (green
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triangle, for example, or red and striped) and then find only those cards that share both
properties.
For children who have played extensively with this deck, you can give them a much
bigger challenge: take one of the cards out of the deck. See if the child can deduce the card you
took, by sorting out the cards and deciding which is missing. If a kid wants a bigger challenge,
take two or three cards out and see if he or she can figure out which they are. A disciplined child
could even play this independently, by dealing the card face-down and only flipping it back over
to verify if the guess is correct.
If the children enjoy the challenge of these games, you (or they) can add more shapes,
colors, and patterns to the deck. You could also introduce size with smaller versions of the
figures.
Games like these are excellent for introducing deductive reasoning, the process of
elimination, the ability to recognize attributes, and the ability to classify, among other things.
All of these are essential skills for success in mathematics, logic, science, and more.
This game also teaches children that things can have a large number of different
attributes. A square, for example, is a rectangle, a quadrilateral, a parallelogram, a rhombus, and
a regular polygon. To force a child to recite this as a math fact would be a pointless and cruel
exercise in rote memorization. With experience playing the pattern game (and gradual exposure
to the definitions), however, a child might recognize these attributes as easily as noticing that an
index card had a small, blue, striped circle drawn on it.
As soon as children have learned how to count on the number line and add multi-digit
numbers, they are ready for the concepts of measurement and perimeter.
To do this, make a number of different shapes (each fairly large) out of a durable material
like oak-tag. For a first assignment, have the children label each one with as many attributes as
they know (polygon, triangle, etc.).
Next, give them each a ruler and teach them how to measure the length of each side.
Initially, have them measure only to the nearest inch. Only later on should you worry about halfinches and other fractions of inches, unless, of course, your kids ask about them. It is always a
good idea to answer questions honestly, but make it clear that the knowledge is advanced and not
to be expected of them at the time. To avoid confusion, though, try to provide your students with
shapes whose lengths are whole-inch lengths.
One important trick is to teach your students how to find the "start" mark on a ruler. For
some rulers, this is at the very end, but on others it is a small distance in:
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Have the children write the length along each edge of the figure. They can also calculate
the perimeter of the shape – the sum of all the sides. For example:
Many children confuse the concepts of perimeter and area. To help avoid this, use the
word "perimeter" as much as possible.
One fun exercise is called "walk/scout/check the perimeter." A fidgety child could be
asked to walk the perimeter of a rug, of the room, or even of the whole building, if you trust the
child out of your sight for that long. This, of course, means to walk entirely around the outside
edge of the area, following the corners as closely as possible. For example, the following might
be an overhead-view of your classroom:
To "mark off a perimeter," have children lay out objects all the way along the perimeter
walk, either yarn, clothesline, sticks, or something of the sort. You could also have them mark
off the perimeter with cones or blocks, but it is better to use a long one-dimensional object like
string or rope because the perimeter is a one-dimensional object. If this gets out of control or
tangles up the room with yarn, you can have kids mark off the perimeter of a smaller things: a
throw rug, a beach blanket, or a shape drawn with masking tape on the ground. Ideally, try to get
your hands on the yellow tape used by construction workers to mark off the perimeter of a
worksite, and have the students use this.
As a group project, you should have the class measure the perimeter of the room. Have
them draw a rough map of the shape of the classroom (don't worry if it is not to scale – that
comes much later) and then copy down the lengths of each side as they measure it. Have them
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measure several times, to catch any mistakes (and practice). It is generally best to measure only
to the nearest foot. This way, when a child walks the perimeter of the classroom, they can tell
you the distance walked! As an extra assignment, have the children make rough maps of their
bedrooms and measure out the dimensions. If they do not have their own rulers, you can have
them make rulers out of oak-tag – a very useful exercise in itself – by copying the measurements
off a classroom ruler.
As a warm-up for multiplication, you can have children walk the perimeter of the room
several times (or a rug, or a shape taped to the floor, if that is more manageable) and then
calculate the total distance walked.
Another excellent thing to measure is the kids themselves. Measure their heights with
several units – to the nearest foot, to the nearest inch, and to the nearest centimeter. This helps to
illustrate how a single measurement can have several different numbers, depending upon the unit
used. Small units lead to big numbers, and big units lead to small numbers. Of course, see if
you can get the class to discover this pattern on their own, perhaps with a little prompting, after
measuring something with several units.
You can also measure all sorts of different parts of the kids – the length of a foot, the
distance from thumb to pinky-finger on an outstretched hand – by tracing these first onto paper.
Practice in measurement is not only a good geometry exercise, but it also reinforces the number
line. By making the objects very personal, you can increase the children's interest significantly.
For a particularly fun exercise, you can have them lie down on large sheets of paper (or
on a sidewalk) and trace their own perimeter. You could invite the students to try to measure
their personal perimeter, but be kind – this is nearly impossible to do accurately.
In general, geometry offers all sorts of possibilities for entertaining, hands-on activities
for students.
Questions:
(1) Write out all the definitions for all vocabulary words in this chapter.
(2) Drawn a concave decagon.
(3) Draw a hexagon that has exactly one pair of parallel lines.
(4) Draw two figures that have 2 attributes in common and 2 dissimilar attributes.
(5) Draw (to scale!) an isosceles (non-equilateral) triangle with a perimeter of 12 inches.
(6) Name all of the different geometry vocabulary terms that apply to squares.
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