7th Grade
Math 6501
Jeffery Ostrom
Geometry for the
Classroom
Executive summary
During this unit student will gain a greater understanding of the 7th grade Minnesota
Geometry & Measurement Standards, through active activities done in groups and
individual work. We will be covering standards: 7.3 Geometry & Measurement, Use
reasoning with proportions and ratios to determine measurements, justify formulas
and solve real-world and mathematical problems involving circles and related
geometric figures; 7.3 Geometry & Measurement Analyze the effect of change of
scale, translations and reflections on the attributes of two-dimensional figures We
will be using a variety of methods to learn these standards. We will be using graphs,
tables, manipulatives (concrete), verbal and formulas, to help better understand these
math standards. There will be a pre and post-test given to assess how the students have
progressed in their interaction of the unit. The assessment questions will come from
activities done in class, MCA Sample test, and other sources. Then we will work
through each activities building on prior skill and learning new ones that will help
students better understand the standards that they will need to cover in 7th grade and on.
They will be able to solve problems like:
The purpose of the first activity is to motivate students to examine relationships among
geometric properties. You will then move into translations, reflections and Rotations. In
Geometry of Circles activity student will using a MIRATM geometry tool, students
determine the relationships between radius, diameter, circumference and area of a circle.
The next lesson “Apple Pi” Students will be using estimation and measurement skills,
students will determine the ratio of circumference to diameter and explore the
meaning of π. Students will discover the circumference and area formulas based on
their investigations. Students in adding it all up students will be drawing various
polygons and investigate their interior angles. In Cubed Cans lesson, students will use
formulas they have explored for the volume of a cylinder and convert them into the same
volume for rectangular prisms while trying to minimize the surface area. In this lesson
Hay Bale Farmer, students will use dimensions of round and square hay bales to
calculate and compare volumes. They also calculate unit prices to determine which
hay bale is the better value. Finally, students explore how to fit round and square
bales into a barn to maximize volume, and decide which type of hale bale is the best
choice. Lastly in Hitting Your Mark, student will draw concentric circles, determine
angle measures given a prescribed number of equal segments, and calculate
measurements for a scale figure.
Table of Contents
(1 day) Pre-Test
(2 days) Polygon Capture
(2 days) Translations Reflection and Rotations
(1 day) Geometry of Circles
(2 days) The Ratio of Circumference to Diameter
(2 days) Discovering the Area Formula for a Circle
(1 day) Adding it All Up
(1 day) Cubed Cans
(1 day) Hay Bale Farmer
(1 day) Hitting the Mark
(1 day) Post Test
Pre-Test
Geometry
Name___________________________
What is the area of the smaller circle
A sprinkler is at the center of a lawn. The sprinkler waters the area inside the circle.
How many square units will be waters?
Using the diagram above, what would be the circumference of the circle?
Erin keeps her dog in the pen shown below. The pen is made by 2 walls of a
building and a curved fence. What is the approximate length of the fence?
What is the volume of the small hay bale below?
How much plastic would it take to wrap the hay bale above?
A 900 ft2 bag of seed cost $6. What would be the cost of seeding this yard?
Identify as many of the polygons below.
Polygon Capture
In this lesson, students classify polygons according to more than one property at a time.
In the context of a game, students move from a simple description of shapes to an
analysis of how properties are related. This lesson was adapted from an article which
appeared in the October, 1998 edition of Mathematics Teaching in the Middle School.
Learning Objectives
Students will:
 precisely describe, classify, and understand relationships among types of two- and
three-dimensional objects using their defining properties
 create and critique inductive arguments concerning geometric ideas and
relationships
 progress from description to analysis of geometric shapes and their properties
Materials
 Polygon Capture Game Rules
 Polygon Capture Game Cards, photocopied onto cardstock
 Polygon Capture Game Polygons, photocopied onto cardstock
The purpose of this game is to motivate students to examine relationships among
geometric properties. From the perspective of the Van Hiele model of geometry, the
students move from recognition or description to analysis (Fuys 1988). Often, when
asked to describe geometric figures, middle school students mention the sides ("The
opposite sides are equal") or the angles ("It has four right angles"), but they rarely use
more than one property or describe how two properties are related. For example, is it
possible to have a four-sided figure with opposite sides not equal and four right angles?
Or a triangle with three right angles? What geometric relationships make such figures
possible or impossible? By having to choose figures according to a pair of properties,
players go beyond simple recognition to an analysis of the properties and how they
interrelate.
Choosing all figures in the Polygon Capture Game Polygons sheet that have parallel
opposite sides is relatively easy. Choosing all figures with parallel opposite sides and at
least one obtuse angle requires reasoning, and a good analysis of such figures leads to the
inference that all nonrectangular parallelograms have these two properties, as does the
regular hexagon.
Another purpose of the game is to give students a format for using important geometric
vocabulary-parallel, perpendicular, quadrilateral, acute, obtuse, and right angle-in a
playful situation. The basic game is described below and is followed by warm-ups and
extensions.
Launch
To get ready for the game, distribute copies of Polygon Capture Game Rules, Polygon
Capture Game Cards, and Polygon Capture Game Polygons. You will need only one
copy of each master for every two students. Before introducing the game, have the
students cut out the polygons and the cards. They should also mark each card on the back
to designate it as an "angle" or "side" card. The eight cards from the top of Polygon
Capture Game Cards sheet should be marked with an "A" for angle property; the eight
cards from the bottom should be marked with an "S" for side property.
Before the game, assess the students' familiarity with the vocabulary used in this game,
such as parallel, perpendicular, polygon, and acute angle by engaging students in a class
discussion in which they define, illustrate, or find examples of the geometry terms.
Explore
Basic Rules of the Game
Have the students read the rules on Polygon Capture Game Rules sheet.
Teachers have found it helpful to begin by playing the game together, the teacher against
the class. You may want to do so a few times until the class is confident about the rules.
For the first game, remove the Steal Card to simplify the game.
To introduce the game as a whole-class activity, lay all twenty polygons in the center of
the overhead projector. Students may lay out their shapes and follow along. An
introductory game observed in one of the classrooms (as shown in step 4, below)
proceeded as follows.
1. The teacher draws the cards All angles have the same measure and All sides have
the same measure. She takes figures D, G, Q, and S, placing them in her pile and
out of play.
2. Students then pick the cards At least two angles are acute and It is a quadrilateral.
They choose figures I, J, K, M, N, O, and R.
3. On her second turn, the teacher picks the cards There is at least one right angle
and No sides are parallel. She chooses figures A and C and then asks students to
find a figure that she could have taken but forgot. One student points out that
figure H has a right angle and no parallel sides. Other students are not sure that
this polygon has a right angle, which leads to a discussion of how they might
check.
4. The students then proceed to take two new cards.
(a) Teacher selects cards.
Angle card: All angles have the same
measure.
Side card: All sides have the same measure.
(c) Teacher selects cards.
Angle card: There is at least one right
angle.
Side card: No sides are parallel.
(b) Students select cards.
Angle card: At least two angles are acute.
Side card: It is a quadrilateral.
(d) Students capture piece that teacher
missed.
When no polygons remain in play that match the two cards chosen, the player may turn
over one additional card-either an angle or a side card. This move calls for some planning
and analysis to determine whether an angle card or a side card is most likely to be useful
in capturing the most polygons. If the player still cannot capture any polygons, play
moves to the opponent. When all cards in a deck are used up before the end of the game,
they are reshuffled. Play continues until two or fewer polygons remain. The player with
the most polygons is the winner.
When the "Wild Card" is selected, the player may name whatever side property he or
she wishes; it need not be one of the properties listed on the cards. Again, a good strategy
to capture the largest number of polygons requires an analysis of the figures that are still
in play.
Steal Card
When the "Steal Card" comes up, a card from the deck is not drawn. Instead, the player
has the opportunity to capture some of the opponent's polygons. The person who has
chosen the Steal Card names two properties (one side and one angle) and "steals" the
polygons with those properties from the opponent. The students may select their own
properties, not necessarily those on the game cards. If the opponent has no polygons yet,
the Steal Card is put back in the deck and a new card chosen.
Share
Have the students share their keys to playing the game.
Summarize



Were the students able to precisely describe, classify, and understand
relationships among types of two- and three-dimensional objects using their
defining properties
Were the students able to create and critique inductive arguments concerning
geometric ideas and relationships
Teacher Notes
o One interesting aspect of the game is the various strategies that students
use. Some students go through the figures one at a time, using a trial-anderror method to match them to properties on the cards. Some students
perform two sorts; they find the polygons that match the first card and, of
this group, those that also match the second card. Others seem to analyze
the properties and mentally visualize the polygons that are possible. In
analyzing properties ("Is this angle acute?"), students quickly learn to use
angles and sides in other figures as benchmarks, for example, using the
right angle in a rectangle to check whether a triangle has a right angle.
Generally classes play with no time limits, although students could choose
a limit as an option.
Extensions
1. Some teachers have found that coordinating two properties is initially too difficult
for their students and have simplified the game by placing all cards into a single
pile. For this simpler version only one card is turned over, and students choose all
polygons with that property. In this adaptation, it is probably best to remove the
Wild Card and the Steal Card. The other rules are the same as described
previously. Because only one property is being analyzed at a time, this game will
go more rapidly.
2. The polygons on the Polygon Capture Game Polygons sheet can also be used for
various sorting games and activities. For example, students may work in pairs,
with one student separating the shapes into groups based on some rule or set of
rules, and the other student trying to deduce the rules. Whereas some students
may begin with simple classifications (rectangles and nonrectangles), others may
use more complex relationships (regular polygons, polygons with equal sides but
not equal angles, and other figures). With a little experience, many students will
find interesting ways to sort the polygons. You may also use the figures to review
geometry vocabulary before the game: "Find all of the figures that have a pair of
perpendicular sides." "Pick all regular polygons." These activities provide a nice
warm-up to the game and other geometry activities.
3. Several other extensions of the game are possible. More polygons can be added,
either by the teacher or by the students, including some that are more difficult to
capture, such as a kite or a nonconvex hexagon. Nonpolygons, such a figures with
curves, can be added for sorting activities. Additional property cards can also be
added to the basic deck. For example, as students learn more about polygons, you
may wish to add angle cards, such as Opposite angles have equal measure or The
number of vertices is a prime number. Similarly, questions about diagonals can be
added to the side cards, such as All diagonals have the same length. If you have a
set of geometric solids available, you can adapt this game to to three-dimensional
geometry. Instead of side and angle cards, make one set of surface and face cards
("I have one curved surface") and edge and vertex cards ("I have an even number
of vertices"). If three-dimensional solids are not available, make a third set of
picture cards. Instead of polygon cards, students choose the geometric solids.
4. The Polygon Capture game cards can also be used to generate figures. As in the
game, students turn over two cards. Instead of capturing polygons, they use a
geoboard or dot paper to make a figure that has the two properties. Rather than a
game, this is simply an activity to help students learn to coordinate the features of
a polygon.
NCTM Standards and Expectations
Geometry 6-8
1. Create and critique inductive and deductive arguments concerning geometric
ideas and relationships, such as congruence, similarity, and the Pythagorean
relationship.
2. Understand relationships among the angles, side lengths, perimeters, areas, and
volumes of similar objects
3. Precisely describe, classify, and understand relationships among types of two- and
three-dimensional objects using their defining properties.
References
Carroll, William. "Polygon Capture: A Geometry Game." Mathematics Teaching in the
Middle School, Volume 4 (Ocober 1998), pp. 90‑ 94.
Fuys, David, Dorothy Geddes, and Rosamond Tischler. The Van Hiele Model of
Thinking in Geometry Among Adolescents. Journal for Research in Mathematics
Education Monograph Series, no. 3. Reston, Va.: National Council of Teachers of
Mathematics, 1988.
This lesson prepared by William Carroll.
Polygon Capture
GAME POLYGONS
© 2008 National Council of Teachers of Mathemati cs
http://illuminations.nctm.org
Translations Reflections and Rotations
Standards:
Geometry & Measurement; 7.3.2.4; Graph and describe translations and
reflections of figures on a coordinate grid and determine the coordinates
of the vertices of the figure after the transformation.
Key terms: Segment, Midpoint, Center Point, Vertices, Coordinates, Line of
Symmetry, Translation, Reflection, and Rotation.
Supplies: Computers with internet access, worksheet NLVM Translation-RefectionRotation,
Bell Work: Write these terms in your math composition books.
Launch: Ask the student if they had good weekend, and lead them into a story about
how you went golfing and you didn’t play particularly well. That you were having
trouble with choosing the correct club/tool and taking the correct swing, and you
didn’t score very well.
Ask the students if they have ever golfed before or maybe seen it on t.v.?
What is the object or goal in golf?
I am going to introduce you to the game of golf, but not just any game of golf,
“Geometric Golfer”. Explain that is game is a little bit different than the one you
might see on t.v. In this game you don’t have clubs… you have tools like reflection,
translation, and rotations. Show them a quick demo of the game. Then explain that
before they get to play we must first learn how use these tools. We are very
concerned about safety so we need to learn how use our tools properly so no one
gets hurt by a bad shot and we will have a better chance of having a good score, so
we will not play the game until we understand the game better.
Explore: Have the students in pairs go to http://nlvm.usu.edu/ on a computer. One
student will operate the computer the other student will document their findings on
a work sheet, after each transformation you will switch roles. Have them select
Geometry : 6-8.
 You will then have to scroll down the page and select: Transformations Reflection – Dynamically interact with and see the result of a reflection transformation.
You and your partner will document your findings. What do you notice when
you select the axis box?
 You will then select the back arrow key, you still may have to scroll down the
page and select: Transformations - Translation – Dynamically interact with and see
the result of a translation transformation. Remember to switch roles and document
your findings.
 Do the same for Transformations - Rotation – Dynamically interact with and see the
result of a rotation transformation.

Share:
When you are done with those three terms go back to your desk and copy
them into your math composition books, then hand in your worksheet.
Students will share their finding with one another. Have student document anything
they had missed into their compositions books.
Summarize: “Today we defined what reflection, translation, and rotation mean.
Tomorrow we will use what we learned today to create our own shapes on a coordinate
system.
Extension: Provide students with the term dilation and see if they can define it on their
own.
National Library of Virtual Manipulative
Translation-Refection-Rotation
Name:________________________
Name:__________________________
What are some things that you and your partner are finding out about “Reflection”?
What are some things that you and your partner are finding out about “Translation”?
What are some things that you and your partner are finding out about “Rotation”?
Geometry of Circles
Using a MIRATM geometry tool, students determine the relationships between radius,
diameter, circumference and area of a circle.
Learning Objectives
Students will:
 Construct circles, and identify the diameters and centers of those circles
 Understand the relationship between diameter and circumference
 Understand the relationship between radius and the area
Materials
 MIRATM Geometry Tool
 Compass
 Geometer's Sketchpad software program on the computer (optional)
 Instructional Plan
 Finding the circumference or area of a circle depends on the diameter of the
circle. To help students develop an understanding of the characteristics of the
diameter, have them construct a circle with a compass, and examine this circle
with a a MIRA. A MIRA is a transparent geometry tool that reflects like a mirror.
It can be used to bisect angles and segments or to explore geometric
transformations. MIRAs are available from ETA Cuisenaire, Nasco, and other
educational retailers.
As an alternative, hinged mirrors can be used for this lesson. Using simple paper folding
can also work — after cutting out a circle, fold it in half, and the crease that forms is a
diameter of the circle. However, MIRATM tools are definitely better.
Launch
Have students construct a circle using a compass. (Alternatively, you may wish to
distribute a handout with circles already drawn.) Then, have them place the MIRA on the
circle and explore; when one image maps onto the other, have them draw the MIRA line.
Explain that the MIRA line is a diameter of the circle.
Then, allow students to construct several other MIRA lines for the same circle using the
same process. Explain that each of these lines is a line of symmetry because each divides
the circle exactly in half. Ask students, "How many lines of symmetry does a circle
have?" [Infinite.] Explain that any of these lines of symmetry may be called a diameter,
because each of them passes through the center of the circle.
Explore
Ask students, "How can you use the MIRA tool to find the center of the circle?" Give
students a minute to do so. [The intersection of two diameters defines the center of the
circle.] The segment from the center of the circle to the circumference is called the
radius. What is the relationship between the radius and the diameter? [The radius is half
the diameter.
Have students construct another circle using a compass. Have students mark the spot
where the point of the compass was placed. This is the center of the circle. Now, have
students draw any chord of the circle. Have students use the MIRA to determine the
perpendicular bisector of the chord. Through what special point does the perpendicular
bisector pass? [The center.]
Allow students to work in pairs to complete either of the following constructions:


Draw a circle, and then construct an inscribed square so that the vertices of the
square lie on the circumference of the circle. [One possibility is to use the MIRA
to identify a diameter. Then use the MIRA to draw the perpendicular bisector of
that diameter. The four points where the diameter and perpendicular bisector meet
the circle are the vertices of an inscribed square.]
Inscribe a regular hexagon in a circle. [One solution is to draw any diameter, and
divide it into two radii. Use the MIRA tool to draw the perpendicular bisectors of
the two radii. Connect the four points where the bisectors intersect the circle with
the two endpoints of the diameter, and a regular hexagon will be formed:
An alternative solution is to use the length of a radius to mark off segments along the
circle. Six congruent segments dictate a regular inscribed hexagon.]
Students may identify other solutions for these constructions. In addition, some students
may have difficulty finding a solution for either construction. If that happens, allow
students to struggle for a while. Eventually, however, you can have one group of students
present their solution to the class; those students who had difficulty should then be asked
to explain why the presented solution works.
Beyond these constructions, students need an understanding of diameter to examine the
circumference of a circle. Allow students to investigate the ratio of circumference to
diameter with the Circle Tool. Under the Intro tab of this tool, students are able to adjust
the diameter of a circle, and they see that a little more than three copies of the diameter
are needed to wrap entirely around the circle.
Circle Tool
Students can explore the relationship of circumference to diameter more explicitly. Under
the Investigation tab, students can view various ratios in the table. By clicking the x/y
button and selecting C as the numerator and d as the denominator, students will see
circumference (C) in the first column of the table, diameter (d) in the second column, and
the ratio of circumference to diameter (C/d) in the third column. By investigating circles
of various size, students should see that the ratio of circumference to diameter is constant
and has a value of approximately 3.14, or π.
Using this applet, lead students to see that C ÷ d = π, or C = π × d. Other ratios can be
explored in a similar manner. For instance, the ratio of d/r can be explored in the table,
and students should discern that the diameter is equal to twice the radius. This result then
leads to another formula, C = 2πr. In the applet, if the ratio C/r were investigated, the
result would be 6.28, or approximately 2π.
Finally, students can investigate the area of a circle by comparing it to the area of a
square. If a circle is inscribed in a square, as shown above, the area of the square is 4r2,
where r is the radius of the circle. Further, if a smaller square is then inscribed in the
circle, its area is half the area of the larger square, or 2r2. This will lead students to guess
that the area of the circle is approximately 3r2, which may cause some students to suspect
that π may be involved. Such a conjecture can lead to a nice discussion and
demonstration of the area formula, which can be conducted as described in the lesson
Discovering the Area Formula for Circles.
Share
Upon completion of this lesson, students should understand the relationships between
radius, diameter, circumference, and area. Have the students share their findings.
Summarize
Questions for Students
1. How do you know if a chord of a circle is also a diameter?
a. [If a chord is also a diameter, it will pass through the center of the circle.]
2. How is the diameter of a circle used to find its circumference?
a. [The value π represents the ratio of circumference to diameter of a circle.
Consequently, C = πd, so the circumference can be found by multiplying
the diameter by π.]
3. How is the radius of a circle used to find its area?
a. [The area of a circle with radius r is given by the formula A = πr2, so the
area can be found by multiplying the radius squared by π.]
Assessment Options
Provide each student with a circle of sufficient size, a ruler, and a calculator. The
radius of each circle should be 3–10", and the circle should be constructed on heavy
paper or cardboard. Do not give circles of the same size to each student. Students should
use a ruler to perform any measurements and then determine the circumference. Once
they complete their calculations, they should tell you their result. You should measure out
a piece of string in the length that they request. Students should then glue the string to the
circumference. (It is important that the teacher cut the string for this activity. If students
are allowed to cut, they tend to continually cut the string until it fits the circle, rather than
learning by doing.)
Based on the results, students should explain what they discovered. Specifically, they
should explain how well their calculations approximated the circumference and how
close their string came to making exactly one revolution. Allow students to score their
own work using the following rubric:
 Advanced. My explanation went beyond the requirements of the task, with my
reasoning communicated effectively. I was able to use logical reasoning to deduce
relationships and test conjectures. Not only did my string fit exactly around my
circle, but I used a systematic and logical process. I accurately measured the
string to the proper degree of accuracy, using the correct formula and units.
 Proficient. My string fit around the circle with no string left over and no gaps. I
had a complete and correct solution process and explanation, using correct
formula and units.
 Basic. My string was either too long or too short. I did not use the correct formula
or made computational errors, or my explanation was incomplete, unclear or
unsystematic.
 Unsatisfactory. I was unable to attempt this problem, or made an incomplete or
incorrect attempt.
Rather than revealing the area formula in class, allow students to write a conjecture
about the formula. Students should explain their reasoning. Afterwards, a class discussion
can be held; at the end of this discussion, the true formula can be revealed.
Extensions
1. To develop understanding of the area of a circle, have pairs of students cut up a
paper plate using lines of symmetry through the center, just as one slices a pizza.
Rearrange the slices as shown below. Students will realize that this configuration
almost looks like a rectangle! How would this "rectangle" help in finding the area
of a circle? [The width of the rectangle is equal to the radius of the original circle.
The length of the rectangle is half of the circumference, since the entire
circumference is both on the top and bottom. Therefore, the area is equal to the
radius times half the circumference, or A = ½Cr. Because C = =πd and d = 2r, this
formula becomes the more familiar A = πr2.]
2. Allow students to use Geometer’s Sketchpad or other geometry software to create
the constructions described in this lesson.
3. Research how hat sizes were determined! Or, check out the web site of a company
that makes and sells hats, and you might find a table like the one below. What is
the relationship between men’s head measurement (in inches) and American hat
sizes? Have students measure the circumference of their head, and divide it by π
— the result is their hat size.
Hat Size
6 3/8
6 1/2
6 5/8
6 3/4
6 7/8
7
7 1/8
7 1/4
7 3/8
7 1/2
7 5/8
7 3/4
Head Circumference
(inches)
20 1/2
20 5/8
21
21 1/2
21 7/8
22 1/4
22 5/8
23
23 3/8
23 3/4
24
24 1/2
If students were to plot these points in a scatterplot, one reasonable line of best fit is y
= 3.14x, indicating that the y‑ value (head circumference) is approximately π times the
x‑ value (hat size).
Teacher Reflection
 Which parts of the lesson had high student enthusiasm? Low? Explain why this
happened. How you could improve student enthusiasm if you teach this lesson
again?
 How do you know that students understood the material about circles? What did
students do to demonstrate understanding?
 What adjustments were needed while teaching this lesson? Were the adjustments
successful?
NCTM Standards and Expectations
Geometry 6-8
1. Precisely describe, classify, and understand relationships among types of twoand three-dimensional objects using their defining properties.
Measurement 6-8
2. Develop and use formulas to determine the circumference of circles and the
area of triangles, parallelograms, trapezoids, and circles and develop strategies
to find the area of more-complex shapes.
This lesson prepared by Rhonda Naylor.
The Ratio of Circumference to Diameter
Students measure the circumference and diameter of circular objects. They calculate the
ratio of circumference to diameter for each object in an attempt to identify the value of pi
and the circumference formula.
Learning Objectives
Students will:
 Measure the circumference and diameter of various circular objects
 Calculate the ratio of circumference to diameter
 Discover the formula for the circumference of a circle
Materials
 Pieces of string, approximately 48" long
 Circular objects to be measured
 Apple pies (or other circular food item, to be measured at the end of the lesson)
 Apple Pi activity sheet
 Calculators
 Rulers
 Instructional Plan
 Prior to this lesson, ask students to bring in several flat, circular objects that they
can measure.
Launch
As a warm-up, ask students to measure the length and width of their desktops. Ask them
to decide which type of unit should be used. Then, have students measure or calculate the
distance around the outside of their desktops.
With the class, discuss the following questions:
1. What unit did you use to measure your desks? Why?
a. [Because of the size of desks, the most appropriate units are probably
inches or centimeters.]
2. Why did some of your classmates get different measurements for the dimensions
of their desks?
a. [Measurements will obviously differ because of the units. In addition, the
level of precision may give different results. For instance, a student may
round to the nearest inch, while another may approximate to the nearest ¼inch.]
3. What do we call the distance around the outside of an object?
a. [The distance around the outside of a polygon is known as the perimeter.
The distance around the outside of a circle is known as the circumference.]
Explore
Inform the class that they will be measuring the circumference of several circular objects
during today’s lesson. Also, alert them that, just as there is a formula for finding the
perimeter of a rectangle (P = 2L + 2W), there is also a formula for finding the
circumference of a circle. They should keep their eyes open for a formula as they proceed
through the measurement activities.
Divide the class into groups of four students. Within the groups, each student will be
given a different job. (If class size is not conducive to four students per group, form
groups of three — one student can be assigned two jobs.)




Task Leader: Ensures all students are participating; lets the teacher know if the
group needs help or has a question.
Recorder: Keeps group copy of measurements and calculations from activity.
Measurer: Measures items (although all students should check measurements to
ensure accuracy).
Presenter: Presents the group’s findings and ideas to the class.
Students should measure the "distance around" and the "distance across" of the objects
that they brought to school. Students will likely have little trouble measuring the distance
across, although they may have some difficulty identifying the exact middle of an object.
To measure the distance around, students will likely need some assistance. An effective
method for measuring the circumference is to wrap a string around the object and then
measure the string. To ensure accuracy, care should be taken to keep the string taut when
measuring the outside of a circular object.
Students should be allowed to select which unit of measurement to use. However, instruct
students to use the same unit for the distance around and the distance across.
Students should record the following information in the Apple Pi activity sheet:




Description of each object
Distance around the outside of each object
Distance across the middle of each object
Distance around divided by distance across
After the measurements have been recorded, a calculator can be used to divide the
distance around by the distance across. Students should answer both questions on the
worksheet. As students are working, take note of their results. Push students to note any
numbers in the last column that seem to be irregular, and have them check their
measurements for these rows.
Share
When all groups have completed the measurements and calculations, conduct a wholeclass discussion. Rather than present each individual object, students should discuss the
average and note any interesting findings. Students should also compare their averages
with those of other groups.
You may wish to use the Circle Tool applet as a demonstration tool. This applet allows
students to see many other circles of various sizes, as well as the corresponding ratio of
circumference to diameter.
Explain that each group has found an approximation for the ratio of the distance around
to the distance across, and this ratio has a special name: π. (It may also be necessary to
explain that the "distance across" is known as the diameter and that the "distance around"
is known as the circumference. Because of this relationship, algebraic notation can be
used to write
circumference ÷ diameter = π
or, said another way,
π = C/d
which leads to the following formula for circumference:
C=π×d
Point out that groups within the class may have obtained slightly different
approximations for π. Explain that determining the exact value of π is very hard to
calculate, so approximations are often used. Discuss various approximations of π that are
acceptable in your school’s curriculum.
Summarize
Questions for Students
1. Why did we use the ratio of circumference to diameter for several objects?
Wouldn’t we have gotten the same result using just one object?
a. [If we had used just one object, an incorrect measurement would have
given an incorrect approximation for π. Using several objects ensures that
our results are correct. In addition, slight errors in measurement may give
different values of π, so using the average of several measurements will
help to eliminate rounding errors.]
2. Were any of the ratios in the last column not close to 3.14? If not, explain what
might have happened.
a. [The ratio of circumference to diameter is always the same, and the ratio is
always close to 3.14. If a value in the last column is not close to 3.14, it is
the result of a measurement or calculation error.]
3. Describe some situations in which knowing the circumference (and how to
calculate it) would be useful.
a. [Bike tires are often described by their diameter. For instance, a 26-inch
tire is a tire such that the diameter is 26". Each time the tire makes one
complete rotation, the bike moves forward a distance equal to the
circumference of the tire. Therefore, it would be helpful to know how to
calculate the circumference based on the diameter.]
Assessment Options
1. Each group can be given an apple pie (or other acceptable substitute) and will find
its circumference by measuring the diameter and using the formula.
2. Students should practice using the formula C = π × d as independent work. Their
work on such problems could be used for assessment. Two real world problems
are:
a. According to Guinness, the world’s largest rice cake measured 5.83 feet in
diameter. What is the circumference of this rice cake?
b. The tallest tree in the world is believed to be the Mendicino Tree, a
redwood near Ukiah, California, that is 112 meters tall! Near the ground,
the circumference of this tree is about 9.85 meters. The age of a redwood
can be estimated by comparing its diameter to trees with similar diameters.
What is the diameter of the Mendicino Tree?
Teacher Reflection
 What prior knowledge did students have of π (if any)? How did student’s prior
knowledge affect the delivery of the lesson? What modifications did you need to
make as a result, and how effective were these adjustments?
 How precise were student measurements? How did you assist students with their
measurements?
 How did students react to the use of 3.14 as an approximation of π? Were there
any adverse reactions due to conceptual misunderstandings?
 How did students show that they were actively learning?
 Did students understand that the ratio of circumference to diameter (i.e., π) is an
approximation? Did they understand why they had obtained different values for
this approximation during the activity?
NCTM Standards and Expectations
Measurement 6-8
 Understand both metric and customary systems of measurement.
 Select and apply techniques and tools to accurately find length, area, volume, and
angle measures to appropriate levels of precision.

Develop and use formulas to determine the circumference of circles and the area
of triangles, parallelograms, trapezoids, and circles and develop strategies to find
the area of more-complex shapes.
References
Neuschwander, Cindy, and Wayne Geehan. 1997. Sir Cumference and the First Round
Table: A Math Adventure. Watertown, MA: Charlesbridge Publishing.
This lesson prepared by Christopher Johnston.
Apple Pi Recording Chart
NAME ___________________________
Using string and rulers, measure the distance around several round objects, as well as the distance
across the middle of those objects. Record your measurements below.
OBJECT
DISTANCE AROUND THE
OUTSIDE OF THE OBJECT
DISTANCE ACROSS THE
MIDDLE OF THE OBJECT
DISTANCE AROUND DIVIDED
BY DISTANCE ACROSS
Remember: Include appropriate labels on all measurements!
1. What do you notice about the numbers in the last column?
2. What is the average of all values in the last column?
© 2008 National Council of Teachers of Mathematics
http://illuminations.nctm.org
Discovering the Area Formula for Circles
Using a circle that has been divided into congruent sectors, students will discover the area
formula by using their knowledge of parallelograms. Students will then calculate the area
of various flat circular objects that they have brought to school. Finally, students will
investigate various strategies for estimating the area of circles.
Learning Objectives
Students will:
 Measure the radius and diameter of various circular objects using appropriate
units of measurement
 Discover the formula for the area of a circle
 Estimate the area of circles using alternative methods
Materials
 Circular objects
 Calculators
 Scissors
 Compasses
 Rulers
 Area of Circles activity sheet
 Fraction Circles activity sheet
 Centimeter grid paper on overhead transparencies
 Blank copy paper
 Instructional Plan
 Prior to the lesson, ask students to bring in several flat, circular objects that they
wish to measure with their classmates.
Launch
As a warm-up, give students an opportunity to estimate the area of the circular objects
that they have brought to class. Working in groups and using the Area of Circles activity
sheet, students should individually complete the first two columns:


Description of the object
Their estimate for the area of the object
(The other two columns will be completed later in the lesson.)
Students may use any method they like to estimate the area of their objects. Some
possible methods include:

Students can trace the shape of their object on a piece of centimeter grid paper
and count how many square centimeters make up the total area of the circle.


Students can divide the circle into wedges by drawing various radii. They can
approximate the area of each wedge using the triangle formula. (This method is
similar to a method used by Archimedes, and it is the method that will be used
later in this lesson. For a connection to mathematical history, you may want to
include a brief overview of Archimedes and his method for calculating the area of
a circle.)
Students can inscribe the circle in a square, hexagon, or some other polygon.
Then, the same shape could be inscribed within the circle. Students could
determine the area of the inscribed and circumscribed shapes to get lower and
upper estimates, respectively. (You may need to provide a sample drawing of this
method, like the one shown below.)
Explore
After students have estimated the area of several objects, allow them to physically
discover the area formula of a circle. Since this is a whole-class activity, you may wish to
enlarge the manipulatives and display them on the chalkboard, or you can use them on
the overhead projector.
 Distribute the Fraction Circles activity sheet.
Have students cut the circle from the sheet and divide it into four wedges. (This can be
done if students cut only along the solid black lines.) Then, have students arrange the
shapes so that the points of the wedges alternately point up and down, as shown below:
Ask, "When arranged in this way, do the pieces look like any shape you know?" Students
will likely suggest that the shape is unfamiliar.
Then, have students divide each wedge into two thinner wedges so that there are eight
wedges total. (This can be done if students cut only along the thicker dashed lines.)
Again, have students arrange the shapes alternately up and down. Again ask if this
arrangement looks like a shape they know. This time, students will be more likely to
suggest that the arrangement looks a little like a parallelogram.
Finally, have students divide each wedge into two thinner wedges so that there are
sixteen wedges total. (This can be done if students cut along all of the dashed lines.)
Allow students to arrange the wedges so that they alternately point up and down, as
shown below:
Ask, "When the circle is divided into wedges and arrange like this, does it look like
another shape you know? What do you think would happen if we kept dividing the
wedges and arranging them like this?" Lead the discussion so students realize the shape
currently resembles a parallelogram, but as it is continually divided, it will more closely
resemble a rectangle .
You may wish to continue this activity by having students divide the wedges even
further.
Ask students, "What are the dimensions of the rectangle that is formed?" From the
Circumference lesson, students should realize that the length of the rectangle is equal to
half the circumference of the circle, or πr. Additionally, it should be obvious that the
height of this rectangle is equal to the radius of the circle, r. Consequently, the area of this
rectangle is πr × r = πr2. Because this rectangle is equal in area to the original circle, this
activity gives the area formula for a circle:
A = πr2
The figure below shows how the dimensions lead to the area formula.
Allow students to return to the objects for which they estimated the area at the beginning
of class. They should measure the radius of each object and record it in the third column
on the Area of Circles sheet. Then, students should use the formula just discovered,
calculate the actual area of each object, and record the area in the fourth column.
Share
Once all groups have completed the measurements and calculations, a whole-class
discussion and presentation should follow. On the chalkboard, the presenter for each
group should record the areas for the objects. The students should compare the results of
each group and discuss the accuracy of the areas found.
The class should also compare their original estimates with the actual measurements. On
their recording sheets, have them highlight the objects for which their estimates were
very close to their actual. Using a few sentences, have the students explain (on the
recording sheet) why some estimates were closer than others.
During the class discussion, the following are some key points to highlight:




Emphasize that 3.14 is only one approximation for π. Refer to the Circumference
lesson, and discuss the various estimates that were found for π and what caused
these variations. Also explain that there are other approximations, but typically
3.14 is used because it is accurate enough for most situations and it is easy to
remember. If students are curious, other approximations for π are given on the Pi
Approximation sheet.
The total area is almost always an approximation. Because the value of π can only
be approximated, any time the area of a circle is stated without the π symbol, it
must be an approximation. For instance, a circle with radius of 5 inches has an
exact area of 25π in.2 and an approximate area of 78.54 in.2. You might wish to
hold a "mock debate" with one student taking each position (yes, it’s always an
exact value; no, it’s not an exact value) giving examples and reasons to justify
their position.
Students should be able to calculate radius from diameter and diameter from
radius. In particular, students should realize that d = 2r.
Students should understand the area formula as described in your curriculum.
Slight variations are possible, so the version in your textbook, standards, or other
materials may be different from the formula presented in this lesson.
Questions for Students
1. In your opinion, why did we use the properties of a parallelogram to discover the
area formula for circles?
a. [Determining the area of a circle is difficult. By converting a circle to a
parallelogram, we can use the formula for the area of a parallelogram to
determine the area of the circle.]
2. When would it be necessary to know the exact area of a circle? When would an
estimate be sufficient? Explain your thinking.
a. [Student responses may vary.]
3. Why did we approximate our answers for area? Can the area of a circle ever be
exact?
a. [It is not possible to find an exact numeric value for π. Therefore, all
calculations of area must be approximations (unless the answer is left in
"exact form," which means using the symbol π to express the answer).]
Assessment Options
1. Students can solve the following practice problem:
2. The radar screens used by air traffic controllers are circular. If the radius of the
circle is 12 centimeters, what is the total area of the screen?
a. [A = pr2, so the area of the radar screen is approximately 3.14 × 122 ≈
452.16 cm2.]
3. Working in pairs or groups, have students locate manhole covers and other circles
on or near the school grounds. Have students measure the diameter of these
circles and then determine the area.
4. Have students explore the following links and answer the associated questions.
Circulate throughout the room to ensure on-task behavior and to check for
understanding.
i. Lessons and Worksheets on Area and Circumference – Go Math
ii. Perimeter, Area, and Circumference Gizmo – Explore Learning
iii. Circles and Pi – Learner.org
Extensions
 Students can use the Internet to research various methods for approximating the
area of circles throughout history. In pairs, students could try the various methods
and determine the accuracy of their results as compared to the formula that they
found. What cultures used good methods that produced accurate results? Did
anything surprise you about these methods or the results? Each pair of students
could report back to the class using a poster, overhead transparencies, or
PowerPoint presentation.

Using the Internet, students should find out the dimensions of a typical dartboard
and the sizes of each point value sector. Using their knowledge of the area of
circles, they can calculate the probability of hitting a certain point value.
(Depending on the information that they find, students may need to estimate the
area of certain sectors to find an approximate probability.)
Teacher Reflection
 When students were working in pairs to find the area of their assigned circular
objects, how precise were the students’ measurements and area calculations?
When the results were discussed as a class, did those students who were not as
precise while measuring demonstrate an understanding of how to get more precise
measurements? Or did all students get basically the same results?
 Did students use both metric and customary units of measure? With which were
they more comfortable, and would future measurement lessons make them
comfortable with the other?
 Were concepts presented too abstractly? Too concretely? How would you change
the presentation if this lesson were taught again?
 How do you know that students were actively engaged in the learning process?
 What content areas did you integrate within the lesson? Was this integration
appropriate and successful?

Did you find it necessary to make adjustments while teaching the lesson? If so,
what adjustments, and were these adjustments effective?
NCTM Standards and Expectations
Measurement 6-8
 Understand, select, and use units of appropriate size and type to measure angles,
perimeter, area, surface area, and volume.
 Develop and use formulas to determine the circumference of circles and the area
of triangles, parallelograms, trapezoids, and circles and develop strategies to find
the area of more-complex shapes.
This lesson prepared by Chris Johnston.
Area of Circles
NAME ___________________________
DESCRIPTION OF THE OBJECT
YOUR ESTIMATE OF THE
AREA (IN SQUARE
CENTIMETERS)
RADIUS OF THE
OBJECT
Remember: Include appropriate labels on all measurements!
© 2008 National Council of Teachers of Mathematics
http://illuminations.nctm.org
ACTUAL AREA
Fraction Circles
NAME ___________________________
Cut out the circle and carefully divide it into wedges, as shown.
© 2008 National Council of Teachers of Mathematics
http://illuminations.nctm.org
Adding It All Up
In this lesson, students draw various polygons and investigate their interior angles. The
investigation is done using both an interactive tool and paper and pencil to foster an
understanding of how different patterns can lead to the same solution. After comparing
results with a partner, students develop a formula showing the relationship between the
number of sides of a polygon and the sum of the interior angles.
Learning Objectives
Students will:
 Investigate the pattern between the number of sides of a polygon and the sum of
the interior angles using in two different methods
 Determine that the interior angle sum is always the same for polygons with the
same number of sides
 Create a formula to find the interior angle sum given the number of sides
 Explore interior angles in regular polygons
Materials
 Unlined paper
 Rulers
 Colored pencils or markers (optional)
 Computers with Internet access
 Angle Sum Tool
 Adding It All Up Activity Sheet
 Adding It All Up Answer Key
Launch
Begin the lesson, by reviewing key vocabulary
with students:
 Polygon: a closed plane figure made of
line segments
 Convex: the measures of all interior
angles are less than 180°
 Concave: the measure of at least one interior angle is greater than 180°
 Regular: all angles and sides are congruent
 Triangulation: the process of drawing diagonals (segments between non-adjacent
vertices) to divide a polygon into non-overlapping triangles
Tell students that they will be working with polygons. It is up to them whether they use
regular or non-regular polygons, and whether they use convex or concave polygons.
Encourage students to draw different kinds of polygons. For example, when exploring
pentagons, point out that all of the examples on the right are polygons.
Tell students that they will use the Angle Sum tool to draw polygons with different angle
measures to see what happens to the angles and the angle sum. Then, they will use a
process called triangulation to help explain their results. Angle Sum Tool
Explain to students that all polygons can be broken up into non-overlapping triangles.
Show an example such as the one below, drawing diagonals to create the triangles.
Explain that this process is called triangulation. Tell them that the triangles can help them
find the sum of the interior angles of any polygon. After they find the interior angle sum
and triangulation for several different polygons, they will find a formula for the interior
angle sum that applies to all polygons.
Launch
Pass out the Adding It All Up activity sheet to each student. You may choose to have
students work in pairs of mixed ability so they can help each other through the activity.
However, each students should complete an activity sheet.
 Adding It All Up Activity Sheet
Read the directions on the activity sheet with students and direct them to first determine
how many sides are in each polygon listed. Tell them that although they are working in
pairs, each student must draw at least one of each polygon and triangulate it on their own,
so that they have data for multiple samples of each polygon. Remind students that the
polygons do not need to be regular or convex. Encourage students to try polygons that are
not regular and concave. You may wish to offer students colored pencils or markers for
the triangulation. This may make it easier for students to make sure the diagonals are
non-overlapping.
As students work, circulate to ensure that they are drawing multiple polygons of each
type and triangulating correctly. Emphasize that the sides of the triangles must be
diagonals (segments connecting non-adjacent vertices) and the diagonals cannot intersect.
Once they finish working with the interactive tool and drawing their own polygons and
triangulations, students should work in pairs to answer the questions on the activity sheet.
Students may struggle with finding the formula 180(n – 2), but do not tell them the
pattern. Instead, encourage them with questions that lead toward the solution. You may
want to begin a discussion about triangles. For example:



How many triangles were you able to draw in that polygon? How does
that number relate to the number of sides?
o [For any polygon, there will be 2 fewer triangles than the number
of sides. For example, a hexagon has 6 sides, so a triangulated
hexagon is made of 4 triangles.]
Can you triangulate a triangle?
o [No. A triangle has no diagonals.]
What is the sum of the angles in 1 triangle? in 2 triangles?
o [The sum is 180° in a single triangle and 360° in 2 triangles
because 180 + 180 = 360. Encourage students to explore this
pattern and discover how it applies to triangulated polygons.]
Encourage students to ask themselves the same questions as they explore each polygon.
Do they see a pattern? What happens to the number of triangles as the number of sides
increases? What happens to the sum of the interior angles as the number of sides or
triangles increases?
Question 3 will be especially challenging to some students. Once students recognize the
pattern, they may still have difficulty expressing it algebraically. Help students by asking
what the variable n will represent in the formula. If they try to use a variable for number
of triangles, ask them how they can relate that back to the n-gon, which has n sides.
Remind students to ask themselves, How can I find the number of triangles if I know the
number of sides.
Share
Once students have had sufficient time to find the formula, walk through the process of
finding it as a group (the formula and the other answers can be found on the Adding It All
Up answer key). This will help those who were not able to find the formula themselves to
catch up. Ask students if the formula works every time. Check it as a class using some of
the polygons from their chart and discuss any other patterns students may have
discovered during the exploration.
 Adding It All Up Answer Key
Questions for Students
1. Are all the angle measures always the same for a single polygon?
a. [No. If all the angles are congruent, the polygon is called equiangular.]
2. As the number of sides increases, what happens to the sum of the angle measures?
a. [For each additional side in a polygon, 180° is added to the sum of the
angle measures.]
3. Does the formula work for both regular and non-regular polygons? What about
shapes like scalene triangles or trapezoids?
a. [Yes, the formula works for all polygons.]
Assessment Options
1. Have students write a journal entry describing how to find the interior angle sum
of any polygon. Students should include details such as patterns they discovered
and other questions on the topic they would like to explore.
2. Create a set of cards numbered from 50 to 100. Randomly give 1 card to each
student. Ask all students to find the sum of the interior angles of a polygon with
the number of sides shown on their card and the measure of 1 interior angle of a
regular polygon with that same number of sides.
Summarize
Develop and use formulas to determine the area of triangles, parallelograms, trapezoids,
and circles and develop strategies to find the area of more-complex shapes.
Extensions


Have students explore exterior angles. The sum of the exterior angles for
any polygon is 360°, and therefore the measure of one exterior angle in a
regular polygon is 360/n
In the bottom right-hand corner of the Angle Sum tool, there is an
animation for the triangle and square showing how the sum of the interior
angles relates to tiling. Have students watch the animations and write a
journal entry on what they demonstrate.
Teacher Reflection
Were students able to use the interactive tool without help?
 Did students have sufficient data to develop a formula? What other data could
have been provided?
 Did students work well in pairs, or would other groupings work better?
 Were students able to communicate with their classmates so that everyone had the
same understanding at the end of the activity?
NCTM Standards and Expectations
Geometry 6-8
 Understand relationships among the angles, side lengths, perimeters, areas, and
volumes of similar objects
 Draw geometric objects with specified properties, such as side lengths or angle
measures.
Measurement 6-8
 Develop and use formulas to determine the circumference of circles and the
area of triangles, parallelograms, trapezoids, and circles and develop strategies
to find the area of more-complex shapes.
This lesson was prepared by Katie Hendrickson as part of the Illuminations Summer
Institute.
Cubed Cans
In this lesson, students will use formulas they have explored for the volume of a cylinder
and convert them into the same volume for rectangular prisms while trying to minimize
the surface area. Various real world cylindrical objects will be measured and converted
into a prism to hold the same volume. As an extension, students may design and create a
rectangular prism container according to their dimensions to compare and contrast with
the cylinder.
Learning Objectives
Students will:
 use and explore volume formulas for cylinder and prisms
 create dimensions for a prism based on a fixed volume
 solve problems using the volume formulas
 explore surface areas
Materials
 Various cylindrical
cans
 Rulers
 Flexible tape
measures (optional,
if you prefer the
circumference
measured)
 Calculators
 Cubed Cans Activity Sheet
Launch
Begin your lesson by asking your students the following questions:



Why do soups and pops and other food containers come in cylindrical
containers?”
o [Students may answer that the containers are easier to open, easier to
make, cheaper, you can fit more food in a cylinder, or other various
answers]
When food companies ship these items, do they fill the shipping boxes
completely?
o [Students may answer no and point out there is gaps when cylinder
objects are stacked together]
When food companies ship these items, what type of package do they use? Is it a
cylinder or prism?
o [Students may answer or you direct them that shipping is done in boxes
because they stack and are packaged better and more securely.]
If we ship items in prism boxes for packaging, why do we not make containers that fit
this package better? What if we shipped items in prism containers?
Explore
Give students the Cubed Cans activity sheet and read through the introduction problem
together. “Why do companies choose to put cylindrical objects in containers that are
rectangular prisms? Do cylinders hold more or less volume than prisms with the same
volume? Food Containers Corporations has hired you to design a new container for
various items they currently ship in cylinders. They would like you to keep the volumes
the same, but explore various prisms. Your task is to take one of the cans provided and
convert it to a rectangular prism. You will need to prove the volumes for the containers is
the same and will want to record your notes and calculations as you report back to the
company your findings. The company would like to save money and use as little surface
area as possible.”
 Cubed Cans Activity Sheet
Once you have briefly given an overview of the task, allow students to choose which
containers they will be using. Their task will be to measure out the dimensions of their
can. Once they have the measurements, they will be calculating the volume. You will
want to point out to students that they are going to have to create prisms that involve
decimals or fractions to create a volume that is approximately the same as the cylinder.
Allow your class to direct themselves in self discovery by measurements and
calculations. If some students are struggling you can discuss with them steps need. Below
are a couple of suggestions and tips to help facilitate struggling students.
Ask struggling students, “What pieces of information will we need from this cylinder?”
[We will need to know the radius, height, and use the formula V = π·r2 to calculate the
volume.]
Once students have expressed this information, you will want to review the way volume
of a prism is calculated. Students should be refreshed that the same three dimensions are
multiplied together to create volume. We will need to know that V = l·w·h to calculate
volume of a prism.
Some students will have the volume, but find a problem working backwards to create the
dimensions of the prism. As they do, you will want to assist them in realizing it takes
three dimensions multiplied together to make that volume. Three dimensions that are the
same will create a cube, while dimensions that differ will create a rectangular prism.
Ask students, “If we know the volume of the cylinder, how can we create a prism that has
the same or similar volume?”
[We can multiply three dimensions to equal that same volume. For example, if the
volume is 45, we can make a prism that is 3 × 3 × 5 = 45.]
Once students have changed their volume to prism calculations, have them think-pairshare their results with others in class or their groups. In doing so, they will see other
people’s results.
Since we have created prisms that hold approximately the same volume as the initial
cylinder, we are going to explore if the two containers have the same surface area.
Objects with less surface area require less material and can save companies money in the
long run. Have students calculate the surface area of the cylinder and the surface area of
the cube.
Remind students:
 Surface area of a cylinder = 2·π·r2 + 2·π·r·h
 Surface area of a cube = 2lw + 2wh + 2lh
Share
Have student’s think-pair-share and compare their observations before sharing their
findings with the class. Depending on the type of prisms created by the students, their
surface areas may be in very different ranges. Allow students to discuss fellow classmates
results to see if certain prisms were close to the cylindrical surface area, or quite
different. One pattern to look for in the students dimensions are as the prisms became
closer to a cube, the surface area would continue to decrease. The least surface area for
the prism would be a perfect cubic box. Any prism that is longer on one or both sides will
increase the surface area.
Summarize
Questions for Students
1. How does the volume formula for a cylinder relate to the volume formula of a
prism?
a. [They both require calculating the area of the base first, then multiplying it
by the height to find the overall volume.]
2. How can the volume of a cylinder can be determined without filling it with
objects?
3. Do cylinders and cubes with the same volume have the same surface area?
Assessment Options
1. Have students measure and create a prism out of paper to match one they came
up with during the lesson. Have other students calculate to the volume and
surface area to check if the measurements are approximately the same as the
cylinder.
Extensions
 Have students create a cylinder based on a rectangular prism.
 Give students 36 blocks. Students will need to create a box using the dimensions
1 × 1 × 36. Next, ask students to create other boxes from the same 36 blocks. You
will want to clear up any misconceptions that 1 × 36 × 1 is a different box. It is in
fact the same as the previous box, but in a different orientation. As students create
the boxes, ask them to find the surface area as well. They can do these two ways.
They can use the length, width, and height dimension formula or students can
count the squares on each side of the box they created. This will allow for a
concrete way for students to explore finding surface area and why the formula
works. Once students have found all possible boxes, ask students if they see any
patterns in the dimensions of the boxes and the surface areas. Lead students in
discussing that as the shape of the boxes became closer to a cube, the surface area
decreased. The lowest surface area we could create would be the shape that is
closest to a cube.
Teacher Reflection
 How were students manipulating dimensions to create a prism with the same
volume as their cylinder?
 How did the students demonstrate understanding of the materials presented?
 Were concepts presented too abstractly? Too concretely? How would you change
them?
 Did you set clear expectations so that students knew what was expected of them?
If not, how can you make them clearer?
 What content areas did you integrate within the lesson? Was this integration
appropriate and successful?
NCTM Standards and Expectation
Geometry 6-8
 Create and critique inductive and deductive arguments concerning geometric
ideas and relationships, such as congruence, similarity, and the Pythagorean
relationship.
 Recognize and apply geometric ideas and relationships in areas outside the
mathematics classroom, such as art, science, and everyday life.
Measurement 6-8
1. Develop strategies to determine the surface area and volume of selected prisms,
pyramids, and cylinders.
References
Connected Mathematics, Filling & Wrapping, Prob.4.3
This lesson was prepared by Corey Heitschmidt as part of the Illuminations Summer
Institute.
Hay Bale Farmer
In this lesson, students will use dimensions of round and square hay bales to
calculate and compare volumes. They also calculate unit prices to determine which
hay bale is the better value. Finally, students explore how to fit round and square
bales into a barn to maximize volume, and decide which type of hale bale is the best
choice.
Learning Objectives
Students will:
 Calculate and compare volumes of different solid figures
 Calculate and compare unit rates
 Verbally and graphically describe arrangements of rectangular prisms and
cylinders in a given space
 Calculate percent of a given value
Materials
 Pictures of hay bales (optional)
 Cylindrical and rectangular prism blocks (optional)
 Hay Bale Farmer Activity Sheet
 Solution – Question 3 Overhead
Launch
Introduce students to the lesson through a discussion about hay bales. If students
are unfamiliar with hay bales, bring in some pictures and discuss the shapes along
with the common uses for hay. Use the following questions as a guide:
What types of hay bales exist?
 [Round and square. Be sure to mention to students that although these terms
are mathematically inaccurate, you will use them during the lesson because
they are the standard terms used in practice.]

What shape are the square bales? [rectangular prisms]

What shape are the round bales? [cylinders]
If students do not have any background knowledge of hay bales, an alternative
problem could be finding the volume of soda cans versus juice boxes, and how many
of each can fit into a large cardboard box. You could also compare round oatmeal
canisters to rectangular cereal boxes. Ask students to name similar products that
are sold in different-shaped containers and how they think the distributors choose
the shapes.
Launch
Tell students that they are going to investigate which type of hay bale is the better
deal mathematically. Students in some communities may already have some
information about their family’s preferred hay bale, and they may volunteer their
opinion and the contributing factors to the choice. If they seem eager to contribute,
allow them to make predictions, such as which takes more space and which costs
more. Remind them that a lot of factors go into deciding which type to purchase, but
today they will just be looking at two factors.
Explore
Distribute the Hay Bale Farmer activity sheet to each student. Read the introduction
and ensure that students understand the diagrams. Put students into groups of 3 or
4 of mixed ability, and have them begin working on answering the questions. You
may choose to make cylindrical and rectangular prism blocks available to help
students reason through the problems and draw their diagrams.
As students work, circulate among the groups and provide guidance as needed. If
you want to keep all students on pace with the rest of the class, you can have
students answer one question at a time and discuss their solutions as a class.
Otherwise, wait until all students are finished to discuss the answers. Have students
share their answers and draw diagrams on the board. Three-dimensional drawings
are difficult for many students, so stress the importance of communicating the
mathematics over creating an accurate diagram in the problem.
Share

Have the student work in groups to figure out their answer and share their
finding with the class.
Solutions
1. You would need to purchase approximately 12.6 square bales. If students are
stuck, you may want to suggest that they find the volume of each.


Volume of round bale 𝑉 = 𝜋𝑟 2 ℎ = π (3)2 × 4 ≈ 113 ft3
1
Volume of square bale = l × w × h = 3 × 2 × 1 2= 9 ft3

Volume of round bale Volume of square bale ≈
113 ft3
9 ft3
≈ 12.6
3. The quantities needed for 1 year are:
o 1,778 square bales
or
o 142 round bales
Note: These values are rounded up since you can only buy whole bales.
To determine which is the better value, students could find the unit rate of
dollars per ft3:
 Square bales cost $0.31/ft3.
 Round bales cost $0.18/ft3.
They could also multiply the cost per bale times the number of bales needed:
 Square bales cost $4,889.50 for a 1-year supply.
 Round bales cost $2,840 for a 1-year supply.
Either way, the round bales are far more economical.
4. The barn will fit 1,728 square bales or 108 round bales. Students should
draw 2 diagrams, as shown on the Solution – Question 3 overhead. The
square bale diagram should show 18 bales of hay fitting along the length, 12
along the width, and stacked 8 high. The round bale diagram should show 6
bales of hay along the length and 9 along the width, stacked 2 high. Square
bales fit a greater volume inside the barn.
a. Solution – Question 3 Overhead
5. 34 bales will have to be stored outside. To find this solution, take the
difference of 16,000 and the volume that can be stored in the barn, then
divide by the volume of one bale and round up. Since 34 round bales of hay
have a volume of approximately 3,845.3 ft3, the 10% loss will amount to
about 384.5 ft3, which is the equivalent of 3.4 round bales.
6. Student responses may vary. While the round bales cost less by volume, they
are harder to store. The square bales can be stored more compactly so less
hay needs to be stored outside, which results in less wasted hay due to mold.
Summarize
1. Why did we round up when answering the questions in this activity?
2. [You can only purchase whole bales, so while we may calculate that we need
13.2 bales, we cannot buy 0.2 bales. Therefore, we must buy 14 full bales of
hay.]
Assessment Options
 Have students write a journal entry, using information on their activity
sheet to provide a mathematically based argument for purchasing one type
of hay bale.
Extensions
1. Students can bring in their own data values by going outside and actually
measuring hay bales, or by using the current price for round and square
bales.
2. Students can calculate the number of round bales that would fit in a barn that
has different dimensions or a different shape than the one use in the activity
sheet. Since rectangular bales fit better in a rectangular barn, would round
bales fit better in a round barn?
Teacher Reflection
 Did students struggle with understanding the different types of hay bales?
How could you introduce farming to them?
 Was students’ level of enthusiasm and involvement high or low? Why?
 Were students able to personally connect to the information?
 How could you adapt the lesson for higher or lower achievers?
 Did students understand how different shapes can lead to different
answers? What else could you have done to emphasize this point?
NCTM Standards and Expectations
Measurement 6-8
 Develop strategies to determine the surface area and volume of selected
prisms, pyramids, and cylinders.
 Solve simple problems involving rates and derived measurements for such
attributes as velocity and density.
 Understand, select, and use units of appropriate size and type to measure
angles, perimeter, area, surface area, and volume.
Number & Operations 6-8
 Understand and use ratios and proportions to represent quantitative
relationships.
 Work flexibly with fractions, decimals, and percents to solve problems.
This lesson was prepared by Katie Hendrickson as part of the Illuminations Summer
Institute.
Hitting Your Mark
Darts is a popular game in which players throw 3 darts, one at a time, aiming for a target.
Different regions of the board give different points. In this lesson, students learn how to
change the scale of an object, and how to measure and draw angles using a protractor. By
the end of the lesson, students have created their own dartboard. The dartboard can later
serve to emphasize properties of angles and angle pairs. This activity is a good one to do
prior to a lesson in which students construct circle graphs. The practice they will get in
this lesson drawing circles and measuring angles will help them in their quest to more
accurately create circle graphs.
Learning Objectives
Students will:
 Accurately draw 7 concentric circles
 Determine angle measures given a prescribed number of equal segments
 Accurately measure angles
 Calculate measurements for a scaled figure
 Construct a scaled dartboard
Materials
 Protractor
 Compass
 Ruler
 Calculator
 12"×12" sheets of cardstock or paper
 Paper in various colors
 Markers or colored pencils
 Scissors
 Glue
 Rules of darts (optional)
 Regulation dartboard for reference (optional but very useful)
 Regulation Dartboard Overhead (if a real dartboard is unavailable)
 Hitting Your Mark Activity Sheet
 Hitting Your Mark Answer Key
This activity will give students an opportunity to work with scale, concentric circles, and
angles, and to display a little artistic creativity. Using a dartboard as the focus of this
lesson, students will practice drawing concentric circles calculated by applying a 2:1
scale to the measurements of a regulation dartboard. If possible, hang a regulation
dartboard in your classroom for students to examine before starting this activity. If you
do not have a dartboard, ask a student to bring one in if they have one at home.
Otherwise, display the Regulation Dartboard overhead so students who are unfamiliar
with the game can see what it looks like.
Launch
Distribute the paper and other craft supplies that students will need to construct their
dartboards. However, rather than distributing the activity sheet at this point, consider
beginning the lesson by asking students the following questions:






Has anyone played darts?
o [If so, have the student(s) briefly explain the rules and scoring. If not, you
may want to look up the rules in advance and familiarize yourself with the
game.]
What are some of the attributes of a regulation dartboard?
o [Attributes include concentric circles, 20 equal sized sectors, 20 congruent
central angles, numbers along the outside ring, a bull's-eye, 6 scoring
rings, and multiple colors, among others.]
How many degrees are there in a circle? If we have 20 equal segments, what is
the measure of each angle?
o [There are 360° in a circle. If there are 20 segments, each segment has an
angle measure of 18°.]
What are concentric circles?
o [Circles that share a common center]
A regulation dartboard has a diameter of 18". If we want to make a scale drawing
that will fit on a 12" by 12" piece of paper and leave at least a 1" margin on all 4
sides, what scale factor should we use?
o [To get a 1" margin, students will have to keep their scaled diameter at 10"
or less. A scale factor of 2:1, which results in a scaled diameter of 9"
would be the easiest to calculate. Other scale factors are possible,
depending on the ability level of your students. Try to discourage smaller
scales — they may make it more difficult to correctly draw the concentric
circles.]
What are some ways we could find the center of our paperto mark as the center of
our circles?
o [Some students may suggest using a ruler and making the 6-inch point
horizontally and vertically thus creating four equal quadrants. This is a
good strategy if you also have the draw the x- and y-axis since the
dartboard can be separated into 4 quadrants. Some students may suggest
folding their paper; this is fine but they need to keep in mind that their
dartboard will then have creases. If they do not want creases on their
paper, they could also fold another sheet and use it as a template.]
Explore
This activity works best with a partner but can also be done individually. Once the
students understand concentric circles, have agreed on a scale they will use, and know
how to find and mark the center they are now ready to begin working with a partner to
begin their calculations and constructing their dartboards. Depending on the class and as
an option for differentiation, all students can use the same scale or different scales. An
optional Hitting Your Mark activity sheet is provided. The activity sheet is set up for
students to use a 2:1 scale to create a dartboard with a 9 inch diameter. You may want to
make this available for students who might have difficulty keeping track of their scaled
fractions.
 Hitting Your Mark Activity Sheet
Students will use the marked center to draw the largest circle using the compass. Have
students double-check their radius and diameter for accuracy. Have them compare their
actual measurements to those calculated and documented on their measurement handout.
If you compasses cannot accommodate a circle with a radius of 4.5 inches, you may want
to create a template for this outer circle that student can trace or allow students to explore
ways to draw circles on their own.
Students should measure and check for accuracy the width of each ring on a regulation
dartboard. If an actual dartboard is not available, students can rely on the measurements
provided on the Hitting Your Mark activity sheet.
Using a protractor, students should then proceed to draw the first 18° angle, with sides
extending to the edge of the circle, representing 1 scoring sector of the dartboard. Instruct
students to verify the measure of each angle for accuracy before proceeding. Students
will get the best results if they work in quadrants dividing each quadrant into five sectors.
Once students have made, verified, and adjusted all measurements allowe them to add
design attributes and personalize their dartboards. Hanging up the finished dartboards for
viewing. Consider hanging them up in the halls with student names hidden and have
students vote for their favorites based on overall design, neatness, and accuracy. You may
want to have three or more awards so more students have an opportunity to be recognized
for their particular strength. Ask students if they would like to donate their dartboards so
that they can be displayed as examples for next year’s students.
 Hitting Your Mark Answer Key
Share
Questions for Students to share as a class.
1. If a 2:1 scale drawing of a regulation dartboard is made, will the central angle
measures change from the original to the model?
a. [Angle measures remain unchanged. To demonstrate draw a central angle
in a circle that measures 18°, and then draw a smaller concentric circle.
Ask students if the measure of the angle has changed because of the
smaller circle. They should agree that it hasn't.]
2. How do the ring widths compare between a regulation dartboard and a 2:1 scale
drawing?
a. [The rings in the scaled drawing are one half the width.]
3. How do the areas compare between a regulation dartboard and a 2:1 scale
a. [The area of a regulation dartboard is about 254.5 in2. The area of the
scale drawing would be about 63.6 in2, 1/4 the original area.]
Summarize
Where the students able to:
 Accurately draw 7 concentric circles
 Determine angle measures given a prescribed number of equal segments
 Accurately measure angles
 Calculate measurements for a scaled figure
 Construct a scaled dartboard
Assessment Options
Collect the activity sheets and finished dartboards and check for accuracy.
Change the scale and ask students to calculate the measurements for another dartboard.
Assess understanding by asking individual students questions from the activity.
Extensions
1. Consider having students research how the point values were determined. Does it
make sense that the point values differ yet the segments are all the same? Where
did the numbering scheme come from?
a. [According to Wikipedia and Ivars Peterson’s MathTrek, the numbering
scheme was devised to penalize poor shots. Both sources also claim that
there are 19! possible number schemes.]
2. Have students explore scale and how different measures vary according to scale.
In a 2:1 scale, some measurements such as width are halved, while others such as
area are quartered. Use scale models of other shapes to facilitate the exploration.
Teacher Reflection
 Did students have sufficient knowledge to draw concentric circles without
assistance? If not, what will you change in the future?
 If the class time required to complete this project is a problem, what changes
would you have to consider to make this a take-home family activity?
 If you partnered students with students of like ability and learning style would
you consider partnering students of mixed ability? Why or why not?
 Was students’ level of enthusiasm/involvement high or low? Explain why.
 Did you challenge the achievers? How?
 Did you find it necessary to make adjustments while teaching the lesson? If so,
what adjustments, and were these adjustments effective?
 What worked with classroom behavior management? What didn't work? How
would you change what didn’t work?
NCTM Standards and Expectations
Geometry 6-8
1. Draw geometric objects with specified properties, such as side lengths or angle
measures.
2. Measurement 6-8
3. Develop and use formulas to determine the circumference of circles and the area
of triangles, parallelograms, trapezoids, and circles and develop strategies to find
the area of more-complex shapes.
4. Select and apply techniques and tools to accurately find length, area, volume, and
angle measures to appropriate levels of precision.
5. Solve problems involving scale factors, using ratio and proportion.
This lesson was prepared by Julie Healy as part of the Illuminations Summer Institute.
Pre-Test
Geometry
Name___________________________
What is the area of the bigger circle
A sprinkler is at the center of a lawn. The sprinkler waters the area inside the circle.
How many square units will be waters?
Using the diagram above, what would be the circumference of the circle?
Erin keeps her dog in the pen shown below. The pen is made by 2 walls of a
building and a curved fence. What is the approximate length of the fence?
What is the volume of the small hay bale below?
How much plastic would it take to wrap the hay bale above?
A 900 ft2 bag of seed cost $7. What would be the cost of seeding this yard?
Identify as many of the polygons below.