Experiences With Area
Teacher Materials
TC-1: area = 2 square tiles, or 2 square inches since each tile is a square inch;
perimeter = 6 edges, or 6 inches since each edge is equal to one inch. Make sure
everyone knows how to determine area and perimeter using the tiles.
TC-2: Transparent colored plastic tiles work well for this demonstration. If a tile is
outside of the rectangle, approximate the fraction of the tile that is outside, then subtract
this from the total number of tiles to get the area. For example, if the rectangle looked
like this:
and you covered it with 12 transparent tiles:
You would have approximately 6 halves that lie outside of the rectangle.
Therefore, an approximation of the area of this rectangle would be 12 tiles – 3 tiles =
about 9 tiles, or 9 square inches.
TC-3:
1. Teacher will need to provide guidance during this stage on how to approximate using
halves of tiles to measure both area and perimeter.
2. The visual diagrams the students draw of both the area and perimeter are an
essential component to building their conceptual understanding of area. Make sure
all students are careful with their diagrams. Using different colors for area and
perimeter will help the teacher to quickly assess if they understand the difference
between the two. If students are not showing evidence of understanding, the teacher
should provide feedback to students as they work to help them successfully
distinguish between area and perimeter.
3. Students may need guidance in coming up with a strategy to measure the perimeter
of the circle and their hand. Square tiles may be too difficult; provide string for those
students who want to measure out a piece of string, surround the shape, and then
either measure the length of the string to a ruler or to a line of one-inch tiles on their
desk.
Experiences with Area
Teacher Materials Page 1 of 8
4. Sample responses for each of the measurements:
Object
Diagram
Post-it
Piece of
notebook
paper
Cover of
book
Circle
(see pg.2)
Triangle
(see pg.2)
Tracing of
your hand
(see pg.2)
Experiences with Area
AREA:
Number of
square
tiles it
takes to
cover
object
9 squares
or
9 square
inches
Approx. 94
square
inches (for
an 8.5 x
11” paper)
Diagram of
area
PERIMETER:
Number of
tile edges it
takes to
surround
object
Diagram of
perimeter
12 edges
or
12 inches
Approx.
29 inches
(for an 8.5 x
11” paper)
Depends
on book
Depends on
book
Approx.
19-20
square
inches
Approx.
15-16 inches
Approx.
Approx.
10 inches
4-5 square
inches
Depends
on hand
Depends on
hand
Teacher Materials Page 2 of 8
TC-4: Possible answers to the questions from part 2: [Note: these examples all have
students completely covering their shapes. There are other possibilities, such as
covering ½ and doubling their answer.]
1. What was your strategy for measuring the area and perimeter of each shape? (In
other words, how did you find the area and perimeter of each?)
ƒ Rectangles:
Possible area strategy: Placed enough square tiles to cover the shape, and then
counted how many tiles there were.
Possible perimeter strategy: Placed enough square tiles to cover the shape and then
counted how many edges surrounded the shape.
ƒ Circle:
Possible area strategy: Placed enough square tiles to completely cover the shape.
Counted tiles. Subtracted fractions of tiles that were outside of the circle.
Possible perimeter/circumference strategy: Used string to trace the outside of the
circle. Measured the string against tile edges or a ruler to see how many inches it
was.
ƒ Triangle:
Possible area strategy: Placed enough square tiles to completely cover the shape.
Counted tiles. Subtracted fractions of tiles that were outside of the triangle.
Possible perimeter strategy: Placed tiles along the edge of the triangle. Counted how
many tile edges surrounded the triangle.
ƒ Hand:
Possible area strategy: Placed enough square tiles to completely cover the shape.
Counted tiles. Subtracted fractions of tiles that were outside of the shape.
Possible perimeter strategy: Placed tiles along the edge of the hand. Counted how
many tile edges surrounded the hand.
2. A square inch is a square that is one inch by one inch (1 in long and 1 in wide).
A linear inch is just an inch measured in a straight line.
How is the square inch shown below different from the linear inch?
The square inch takes up more space than the linear inch does.
Square
Inch
Linear Inch
3. Which unit of measurement did you use (square inches or linear inches) to measure
the area of each shape? Why?
To measure the area of each shape we used square inches, because area is a
measure of how much it takes to cover an object. Linear inches don’t cover objects.
The square inch can be used to COVER a space, but the linear inch would more
likely be used to SURROUND a space.
Experiences with Area
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4. Which unit of measurement did you use (square inches or linear inches) to measure
the perimeter of each shape? Why?
To measure the perimeter of each shape we used linear inches, because perimeter
is a measure of how much it takes to surround an object. Square inches don’t
surround objects, they cover them.
TC-5:
1. Have several students come to the front of the class to share their strategies for
measuring the area and perimeter of all the shapes. Make sure all students see a
model of appropriate strategies to measure area and perimeter. During the student
sharing, it is essential that the teacher is walking around the room checking to see if
other students are accurate in their approximations of area and perimeter. If they are
not, it is equally essential to give them time to try a new strategy and revise their
approximations.
For example, let’s say a student models an appropriate strategy for finding the area
of the circle, and comes up with an approximation of 20 square inches. However,
other students in the class have answers of 25 square units. If students have an
answer of 25, it is likely they covered their circle with a 5x5 square, and counted all
of the tiles, instead of subtracting the fractions of tiles that overlapped the circle. It is
important to stop the class and discuss why we need to subtract the number of tiles
outside of the circle to get an accurate estimate for area. After a student has
modeled how to successfully do this, give everyone time to make revisions if
necessary. Continue in this manner as strategies are shared for each shape.
2. Explicitly discuss the units. Have each table of 4 students share their answers to
questions 2-4 at their table. Make sure every student has a chance to share.
Encourage students to add or revise their own answers as they listen to others. If
you see a student sharing clear reasoning at a table, stop and ask the other students
what they think. Ask them if what the student just shared made sense to them. Then
have them take their pencils and add to their own responses to include some of what
they just heard
3. After sharing in their groups, have a few students share their responses to questions
2-4 with the entire class.
4. Emphasize that square units cover a surface, and linear units surround. Also
encourage students to visualize a square unit in their minds by picturing a literal
square, as you share answers. Visualization can be a powerful way to connect an
abstract idea (linear units vs. square units) to a concrete image that students can
then access in the future.
Experiences with Area
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TC-6:
1. The Frayer Model is a form of categorizing and defining words. The Frayer Model
allows you to analyze a word by defining its essential characteristics and nonessential characteristics, as well as provide examples and non-examples.
2. Using the student handout, discuss the Frayer Model with the students, and share the
example given. Ask students to explain why the examples (the side of a rectangle
and the side of a football field) are really examples of length. Ask students to explain
why 8 square units (the area of the rectangle in the example) is NOT an example of
length.
3. As students are working on their Frayer Models, walk around and visit groups. As you
visit each group, challenge their examples and non-examples. Ask students to
explain why they’ve chosen specific examples and non-examples – encourage them
to verbalize their reasoning as much as possible. For example, if a student has
written down ‘fence’ as a real-life example for perimeter, ask them why a fence
would be an example of a perimeter (because it surrounds a field, and perimeter
surrounds a shape). Ask them why it wouldn’t be an example of area. Ask them what
would be an example of area when thinking about fences (the field would be the
area).
4. Sample Frayer models for area and perimeter (see next page):
Experiences with Area
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Definition (own words):
Facts/Characteristics:
The number of linear units it takes
to surround an object. The distance
around an object.
A) Measured in linear units (straight
lines, not squares)
B) You can measure the perimeter
of objects using inches, centimeters,
feet, yards…
C) The perimeter of a circle is called
circumference
Math & Real Life Examples:
Non-examples:
*The perimeter of this rectangle is
14 units.
*If I count the squares in this
rectangle I get 12 square units. That
is not the perimeter.
*A fence is the perimeter around a
field.
*A field is not a perimeter – that
would be an example of area.
Experiences with Area
Teacher Materials Page 6 of 8
Definition (own words):
Facts/Characteristics:
Area is a measurement of how
many square units it takes to
cover a surface.
A) Area is always measured in
square units not linear units
B) You need to know the area
when you’re covering something,
not surrounding it
C) You can find the area of a
rectangle by multiplying the length
by the width
Math & Real Life Examples:
Non-examples:
*If I count the squares in this
rectangle I get 12 square units.
That is the area.
*If I count the number of units
surrounding this rectangle I get 14
units. That is NOT the area.
*The area of the pizza has all of
the good stuff – sauce, cheese,
toppings… it’s everything that
covers the pizza.
*The crust of the pizza above does
NOT represent the area of the
pizza… since it surrounds the pizza
the crust is the perimeter.
TC-7:
The process for sharing finished Frayer models is as follows:
Each student shares each part of their model – the definition, facts/characteristics,
examples, and non-examples. They ask the group if anyone has any questions. If
anyone doesn’t understand a part or disagrees with something, they have an
opportunity to make questions and comments at this time. Once that is done, it’s the
next person’s turn.
Experiences with Area
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TC-8:
The revised Frayer Models should combine characteristics and information from each of
the individual ones. Each group is responsible to create two models– one for area, and
one for perimeter. These can ultimately be posted around the room, reminding students
about the concepts of area and perimeter throughout their unit, or even the year.
Experiences with Area
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