Download Accelerated Math Unit 5A - Youngstown City Schools

YOUNGSTOWN CITY SCHOOLS
ACCELERATED MATH: GRADE 7
UNIT # 5A: GEOMETRY (3 WEEKS)
Synopsis: In this unit, students work with 2 –D and 3-D geometric shapes. Given certain conditions, students see what rules hold for
constructing triangles. They work with different types of angles, focusing on angles created when two parallel lines are cut by a
transversal. Students work with a variety of problems, entailing different conditions, to apply what they have learned about angles,
triangles, and side measures. Students also work with slicing 3-D shapes and analyzing the 2-D shapes that result from this slicing.
They continue work from 6th grade on area, volume and surface area of two-dimensional shapes (including composite shapes) and
three-dimensional objects comprised of polygons. Students work on understanding the relationship between circumference, diameter
and area and then working with the formulas for each.
STANDARDS
NOTE: be sure to use fractions and decimals as well as whole numbers for the activities in the unit. It is ok to start with whole numbers,
but students must move beyond this.
7.G.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing
triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or
no triangle.
7.G.3 Describe the two-dimensional figures that result from slicing three dimensional figures, as in plane sections of right rectangular
prisms and right rectangular pyramids.
7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the
relationship between the circumference and area of a circle.
7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple
equations for an unknown angle in a figure.
7.G.6 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects
composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
MATH PRACTICES
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning
LITERACY STANDARDS
L.1 Learn to read mathematical text (including textbooks, articles, problems, problem explanations)
L.2 Communicate using correct mathematical terminology
L.3 Read, discuss, and apply the mathematics found in literature, including looking at the author’s purpose
L.4 Listen to and critique peer explanations of reasoning
L.5 Justify orally and in writing mathematical reasoning
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MOTIVATION
TEACHER NOTES
1. Archimedes Puzzle (Illuminations ) do triangle puzzle on pages 6 and 7 of unit plan.
http://illuminations.nctm.org/LessonDetail.aspx?id=L720
Read Sir Cumference and the First Round Table to students to spark interest in the concepts to
be studied in the Unit. (L.1; L.3)
2. Students establish academic and personal goals for the unit.
3. Teacher previews the Authentic Assessment for the end of the unit.
TEACHING-LEARNING
Vocabulary:
Circumference
Area
Center
Lateral area
Pyramid
Radius
Surface Area
Lateral face
Prism
Complementary
Angle
TEACHER NOTES
Pi (π)
Perimeter
Slant height
Rectangular Prism
Supplementary
Angle
Diameter
Volume
Base
Triangular Prism
Adjacent Angle
Vertical Angle
Cross-section
1. Constructing Triangles from 3 Measures of Angles or Sides: Have student complete TRY
ANGLES activity attached on pages 8-9. The Activity from the Georgia Department of
Education entitled “Take the Greek Challenge” is also attached on pages 10-11. (7.G.2)
For students to classify triangles, use the orange book: Common Core Math Standards With
Hands-On Activities, pages 116-118. This activity focuses on properties of triangles. Students
work in groups of 3-4 and draw triangles according to given conditions and then classify each
triangle based on the conditions. (7.G.2)
In addition, go to http://LearnZillion.com and click on grade 7 Math, standard 7.G.2, and there will
sample problems that can be used with the concepts in 7.G.2. (e.g., draw a polygon using more
than one condition, draw triangles using given angles, draw geometric shapes given the length of
the sides, and determine if given measurements will allow you to create the appropriate shape).
Additional problems on page 12 of unit plan.
2. Work with cutting 3-D shapes and determining the 2-D shapes that emerge (equilateral triangles,
pentagons, hexagons, etc.). Go to Annenberg Learner Website listed below, and do interactive
activity for cutting 3-D shapes (7.G.3). Do the right rectangular prism and the right pyramid for
slicing. Additional problems on pages 13-14 of unit plan.
http://www.learner.org/courses/learningmath/geometry/session9/part_c/index.html
http://intermath.coe.uga.edu/dictnary/descript.asp?termID=95
3. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step
problem to write and solve simple equations for an unknown angle in a figure: (7.G.5) use the
following: additional practice included on page 15
a. Scott Foresman, Purple Book, Course 3, volume 2: Chapter 8-5, pages 416-420 on parallel
and perpendicular lines. Work through the activities on these pages to address the types of
angles formed when parallel lines are cut by a transversal: interior angles, exterior
angles, alternate angles, vertical angles, supplementary angles, adjacent angles,
complementary angles, and corresponding angles. These pages have some excellent
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TEACHING-LEARNING
TEACHER NOTES
real-world situations with these different types of angles, including street maps, etc.
b. Glencoe, Chapter 10-1: Line and Angle Relationships: pages 492-497 on Parallel lines cut
by a transversal and intersecting lines and angles. The focus is on the same terms listed
above: interior angles, exterior angles, alternate angles, vertical angles,
supplementary angles, adjacent angles, complementary angles, and corresponding
angles. Again, across these pages, are very good examples of where students would find
these angles in real-life. On page 496, the problems blend Algebra with Geometry
(something students need to be able to do later in another grade/course)
c. Teaching the Common Core Math Standards with Hands-On Activities: pages 124-126:
Solve real-life mathematical problems involving angle measure, area, surface area, and
volume: students work in pairs or three’s to identify which equations and angle measures
can be matched to a sketch of a figure. This is a hands-on activity that gets students
actively engaged.
4. Discovering π: The teacher needs to bring a tub of circular lids and cylinder containers
(Pringles, Oatmeal, empty peanut butter jar, etc.) - - the key is to have a variety of sizes. Have
students use string or measurement tapes and determine the circumference and diameter for
each item. Have students determine where the concept of π originated. Remind students what
radius is. Show YouTube video of songs about π: Pi Rap with lyrics or Pi, Pi Mathematical Pie
(which is based on the song Bye, Bye, Miss American Pie, written by Ken Ferrier and Antoni
Chan - - you may want to play the original song first so that kids can connect to the original).
Links to YouTube video “Pi Songs” - - http://www.youtube.com/watch?v=VWGGTb5pY2U
Pi Rap Video: http://www.youtube.com/watch?v=CS1WlUzjtXU
After students see videos, they should explain the relationship between circumference and
diameter (that circumference is a little more than 3 times the diameter. (7.G.4; 7.G.6) (L.2, L.3;
L.4, L.5) (MP-4; MP-5)
TEACHERS: some of
the committee suggested
that we have students
make a formula booklet
with examples to use as
a reference throughout
the unit. You can start
that here at if you choose
to do that.
Show students the formula and have students work real-life problems (you can use some OAA
released items for circumference).
The Georgia Department of Education has a good activity called Stained Glass Designs,
attached on pages 16-17
5. Illuminations Resources for Teaching Math APPLE PI: NCTM - - The Ratio of Circumference to
Diameter Lesson 1 (attached on pages 17-20) builds the circumference to diameter
relationship. This can be followed up with problems from the Glencoe Pre-Algebra Textbook,
chapter 10-7, pages 533-537. There are some really good problems on page 536! (7.G.4; 7.G.6)
6. Illuminations Resources for Teaching Math APPLE PI: NCTM - - Discovering the Area
Formula for Circles Lesson 2 (attached on pages 21-26) Next, have students solve real-world
problems for area and circumference (page 367 in text); also use How Archimedes found the
Area of a Circle attached on pages 27-29 (7.G.4; 7.G.6) (MP-1; MP-2)
7. Teacher reminds students about finding area of 2-D shapes and then goes to surface area of 3-D
objects - - prisms: rectangular, triangular and pyramid. Review area of rectangle and triangle to
lead into the 3-D shapes. Then give students a solid figure and ask them to determine the
amount of paper to cover the object (cube, rectangular prism, pyramid, triangular prism.
Students work in pairs to find a strategy to find the amount of paper. Have students explain what
surface area is (e.g., the area of the net of a 3-D object) after they complete the activity. (7.G.6)
(L.2, L.3, L.4) (MP-3; MP-7)
8. SURFACE AREA for 3-D objects:
a. rectangular prism: have students use the formula for surface area (2 (lh +lw +wh)
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TEACHING-LEARNING
TEACHER NOTES
Students compute word problems (Glencoe Pre-Algebra pages 576-577 for vinyl liners of
swimming pools where the liner covers the bottom and sides of the pool, or the amount of
glass for an aquarium where you only need bottom and sides, but no top. Have students
come to conclusion that the surface area depends on the situation. Be sure to distinguish
between surface area and volume (NOTE: don’t let students say that the surface area is the
amount outside and the volume is the amount inside; this is not a good way to phrase what
each actually is, but one we all hear ). (Attached on page 31-32: good sample problems to
use here) (MP-2; MP-8)
b. triangular prism: use Glencoe pages 575-576 (7.G.6) (MP-7; MP-8)
c. pyramids: Glencoe page 578; use Rock and Roll Hall of Fame (picture and problems are
attached on page 30) (7.G.6)(MP-2; MP-7)
9. VOLUME: of 3-D objects composed of triangles, quadrilaterals, polygons, cubes, and right
prisms. Students work with basic formula of V = (½ bh)h, which yields V = Bh, where B = the
area of the base. Website: http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/
http://shodor.org-surfaceareaandvolume/ – you can manipulate shapes and there are lots
of related problems. The site has 2 worksheets that go with the activity; attached to the
unit. on page 33-35
For the accompanying worksheets, click on the “Learner” tab at the top and scroll down to the
bottom the page.
a. rectangular prism: Teacher should use the candy Dots mini-candy box and have students
fill them with centimeter cubes (note: each box holds about 6 or 9 cubes); after students see
how many cubes fill the box, stack cubes next to box and see the number of cubes relates to
the number of cubic centimeters (cm3) the box can hold. Next have students see how many
Dots boxes fit into a bigger box (e.g., a shoe box). Have students explain that volume is the
amount of space something takes up or the number of cubic units needed to fill a container.
Students will compute problems (textbook pages 563-567; work examples 1-4 on page 563 and
564 as the teaching samples. These have unit conversions so teacher will need to remind
students about more complex conversions and work some samples together. Then students
compute real-world problems for volume of a rectangular prism. (7.G.6) (MP2; MP7)
b. triangular prism: Swimming pool problem so students see real-world example of where
triangular prisms are found; (sample attached on page 36 of unit) Additional real-world
problems on page 37 of unit plan (7.G.6)
c. pyramid: show pyramid as 1/3 of rectangular prism; use Egyptian pyramids as example;
share formula: V = 1/3(1/2bh)h - - same as V = 1/3 Bh, B = Area of the base.(7.G.6) (MP-4;
MP-1)
TRADITIONAL ASSESSMENT
TEACHER NOTES
1. Multiple-Choice Unit Test
TEACHER-MADE ASSESSMENT
TEACHER NOTES
1. Teacher Classroom Assessments
2. 2- point and 4- point Constructed Response items
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AUTHENTIC ASSESSMENT
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1. Have students evaluate their goals and give evidence
2. Students create a story or rap about circumference, area, π, etc.
3. Design a package with a given surface area and volume. For example, two boxes with same
volume but different surface areas, then two boxes with same surface area but different volumes.
4. “Saving Sir Cumference” attached on page 38
5. Teaching the Common Core with Hands-On Activities pg.127-128 “Let’s Build It” Note: When doing
the 2-D part of the activity, use grid paper.
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ARCHIMEDES’ PUZZLE
The Stomachion is a puzzle that is at least 2,200 years old. It was known to the ancient Greeks. Some people think that it
was created by the Greek scientist Archimedes, which is why it is sometimes called Archimedes’ Puzzle or the Loculus of
Archimedes.
The puzzle consists of 14 pieces of various shapes and sizes. These pieces are created by dividing a square as shown
below. The object of the puzzle is to rearrange the pieces to form other shapes.
Cut out the pieces of the Stomachion. Then, rearrange the shapes to create the triangle shown. The figure on the left
shows how the pieces must be arranged to form the triangle.
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TRY ANGLES
Math Question: Which angles form a triangle?
Materials:
Angle Mats (make copy of next page)
Counters
Protractors
Student Reminder:
all triangles have three angles. The sum of the measures of these
angles equals 180 degrees
Directions:
1. Drop the counter on the angle mat to determine which angle you will draw.
Sketch the angle or use a protractor to draw the angle in your math journal.
Label the angle in degrees.
2. The next person drops the counter, sketches or draws the angle, and labels
the angle. The new angle can be connected to one side of the first angle or
can be the start of a new triangle.
3. Players continue taking turns dropping the counter, sketching or drawing
angles, and labeling angles. Each new angle can be connected to previous
angles or can be the start of a new triangle.
4. When a player adds the third angle to form a triangle, s/he earns 100 points.
5. The game ends after 18 angles have been drawn.
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ANGLE MAT
60 degrees
15 degrees
65 degrees
10 degrees
50 degrees
90 degrees
80 degrees
35 degrees
20 degrees
75 degrees
45 degrees
30 degrees
40 degrees
70 degrees
55 degrees
60 degrees
15 degrees
65 degrees
10 degrees
50 degrees
90 degrees
80 degrees
35 degrees
20 degrees
75 degrees
45 degrees
30 degrees
40 degrees
70 degrees
55 degrees
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T-L #1
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T-L #1 7.G.2 SAMPLE PROBLEMS THAT CAN BE USED FOR STUDENT PRACTICE:
Students draw geometric shapes with given parameters. Parameters could include parallel
lines, angles, perpendicular lines, line segments, etc.
Example 1:
Draw a quadrilateral with one set of parallel sides and no right angles.
Students understand the characteristics of angles and side lengths that create a unique triangle,
more than one triangle or no triangle.
Example 2:
Can a triangle have more than one obtuse angle? Explain your reasoning.
Example 3:
Will three sides of any length create a triangle? Explain how you know which will work.
Possibilities to examine are:
a. 13 cm, 5 cm, and 6 cm
b. 3 cm, 3cm, and 3 cm
c. 2 cm, 7 cm, 6 cm
Solution for example 3: “A” above will not work; “B” and “C” will work. Students recognize
that the sum of the two smaller sides must be larger than the third side.
Example 4:
Is it possible to draw a triangle with a 90° angle and one leg that is 4 inches long and one leg
that is 3 inches long?
If so, draw one. Is there more than one such triangle?
(NOTE: Pythagorean Theorem is NOT expected – this is an exploration activity only)
Example 5:
Draw a triangle with angles that are 60 degrees. Is this a unique triangle? Why or why not?
Example 6:
Draw an isosceles triangle with only one 80°angle. Is this the only possibility or can another
triangle be drawn that will meet these conditions?
Through exploration, students recognize that the sum of the angles of any triangle will be 180°.
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7.G.3 SAMPLE PROBLEMS THAT CAN BE USED FOR STUDENT PRACTICE
Describe the two-dimensional figures that result from slicing three dimensional figures,
as in plane sections of right rectangular prisms and right rectangular pyramids.
Students need to describe the resulting face shape from cuts made parallel and
perpendicular to the bases of right rectangular prisms and pyramids. Cuts made parallel
will take the shape of the base; cuts made perpendicular will take the shape of the
lateral (side) face. Cuts made at an angle through the right rectangular prism will
produce a parallelogram; If the pyramid is cut with a plane (green) parallel to the base,
the intersection of the pyramid and the plane is a square cross section (red).
If the pyramid is cut with a plane (green) passing through the top vertex and
perpendicular to the base, the intersection of the pyramid and the plane is a triangular
cross section (red).
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If the pyramid is cut with a plane (green) perpendicular to the base, but not through the
top vertex, the intersection of the pyramid and the plane is a trapezoidal cross section
(red).
http://intermath.coe.uga.edu/dictnary/descript.asp?termID=95
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T-L #3 SAMPLE PROBLEMS THAT CAN BE USED FOR STUDENT PRACTICE
(7.G.5)
Students use understandings of angles and deductive reasoning to write and solve
equations
Example1:
Write and solve an equation to find the measure of angle x.
Solution for Example #1: Find the measure of the missing angle inside the triangle (180 – 90
– 40), or 50°. The measure of angle x is supplementary to 50°, so subtract 50 from 180 to
get a measure of 130° for x.
Example 2:
Find the measure of angle x.
Solution for Example #2: First, find the missing angle measure of the bottom triangle (180 –
30 – 30 = 120). Since the 120 is a vertical angle to x, the measure of x is also 120°.
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APPLE PI: FROM ILLUMINATIONS NCTM WEBSITE:
The Ratio of Circumference to Diameter
n
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.
y
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
y
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
n
Instructional Plan
Prior to this lesson, ask students to bring in several flat, circular objects that they can measure.
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?
[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?
[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 ¼-inch.]
3. What do we call the distance around the outside of an object?
[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.]
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.
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


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.




Description of each object
Distance around the outside of each object
Distance across the middle of each object
Distance around divided by distance across
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:
Apple Pi Activity Sheet
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.
When all groups have completed the measurements and calculations, conduct a whole-class 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.
Circle Tool
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.
y
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Questions for Students
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?
[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.]
Were any of the ratios in the last column not close to 3.14? If not, explain what might have happened.
[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.]
Describe some situations in which knowing the circumference (and how to calculate it) would be useful.
[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.]
y
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:
o According to Guinness, the world’s largest rice cake measured 5.83 feet in diameter. What is the
circumference of this rice cake?
o 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?
y
Extensions
1. In this lesson, students use a numeric approach to see the relationship between circumference and
diameter. That is, students compute the ratio of circumference to diameter and then take the average for
several objects. For a visual approach, have students plot the diameter of those objects along the
horizontal axis of a graph and plot the circumference along the vertical axis. As shown below, a line of best
fit with slope of roughly 3.14, or π, will approximate the points in the resulting scatterplot.
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2. Students can read and react to the book Sir Cumference and the First Round Table: A Math Adventure by
Cindy Neuschwander. Within their groups, students can pose questions about the book and its
mathematical accuracy, realism, and other components.
3. In their groups, students can research the history of π and its calculation, approximation, and uses. In
particular, they can research Archimedes method for estimating the area of a circle using inscribed
polygons. The students could report their findings to the class.
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APPLE PI: FROM ILLUMINATIONS NCTM WEBSITE
Discovering the Area Formula for Circles
n
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.
y
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
y
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
n
Instructional Plan
Prior to the lesson, ask students to bring in several flat, circular objects that they wish to measure with their
classmates.
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.)


Area of Circles Activity Sheet
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
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

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.)
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.
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:
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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.
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.
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Questions for Students
In your opinion, why did we use the properties of a parallelogram to discover the area formula for circles?
[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.]
When would it be necessary to know the exact area of a circle? When would an estimate be sufficient?
Explain your thinking.
[Student responses may vary.]
Why did we approximate our answers for area? Can the area of a circle ever be exact?
[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).]
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Assessment Options
1. Students can solve the following practice problem:
o 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 = pr2, so the area of the radar screen is approximately 3.14 × 122 ≈ 452.16 cm2.]
2. 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.
3. 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.
o Lessons and Worksheets on Area and Circumference – Go Math
o Perimeter, Area, and Circumference Gizmo – Explore Learning
o Circles and Pi – Learner.org
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Extensions
1. 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.
2. 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.)
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T-L #9: Worksheet to Accompany
“Surface Area and Volume”
This worksheet is intended for use with the lesson Surface Area and Volume.
For the following questions, use the rectangular prism:
1. What is the smallest volume that you can create with this prism? _______________
What is the surface area associated with this volume? _______________
What is the depth? _______________
What is the height? _______________
What is the width? _______________
2. What is the largest volume that you can create with this prism? _______________
What is the surface area associated with this volume? _______________
What is the depth? _______________
What is the height? _______________
What is the width? _______________
3. Explain why the surface area is larger than the volume in #1.
[continued on next page]
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Surface Area & Volume
Exploration Questions
Base Depth
Base Width
Prism Height
Volume
Surface Area
1. Use the applet to fill in the chart using different base depth, base width, and prism
height for a rectangular prism. Is there a pattern? Can you write a formula for volume
and surface area for a rectangular prism in terms of its base depth, base width, and
prism height? If so, write it.
a) What differences do you see in the relationship between the figure’s surface area and
volume as the figure gets larger.
b) Which dimensions give the rectangular prism the largest volume to surface area
ratio?
c) Which dimensions give the rectangular prism the smallest volume to surface area
ratio?
d) Graph the surface area of each of the cubes you can form using the applet. Then on
the same graph, graph the volume of each of the cubes you can form using the
applet. At what point on the graph does the volume grow to be greater than the
surface area? Why do you think the volume grows greater than the surface area?
Base Depth
Base Width
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2. Use the applet to fill in the chart using different base depth, base width, and prism
height for a triangular prism. Is there a pattern? Can you write a formula for volume
and surface area for a triangular prism in terms of its base depth, base width, and
prism height? If so, write it.
a) What differences do you see in the relationship between the figure’s surface area
and volume as the figure gets larger.
b) Which dimensions give the triangular prism the largest volume to surface area
ratio?
c) Which dimensions give the triangular prism the smallest volume to surface area
ratio?
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The front of the Rock and roll Hall of Fame in Cleveland, Ohio is a glass pyramid.
a. The front triangle has a base of about 230 feet and a height of about 120 feet. What
is the area?
b. How could you find the total amount of glass used in the pyramid for the Rock and
Roll Hall of Fame?
Example of a triangular prism
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AUTHENTIC ASSESSMENT: SIR CUMFERENCE
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