7/8 Grade Geometry Project
Summer 2009
Angela Tollefson, Lynnea Salscheider ,
Marilyn Delaney
Pre Test
Day 1 Mini Metric Olympics
Day 2 Mini Metric Olympics II
Day 3 Mini Metric Scavenger Hunt
Day 4 Angle Relationships
Day 5 Create your own Angle Relationships
Day 6 Pythagoras –patty paper
Day 7 Pythagoras
Day 8 Pythagoras
Day 9 Pythagorean Theorem Proof -Geometer’s Sketch Pad
Day 10 Geometer’s Sketch Pad Continued
Day 11 and 12 “Let’s Have Some pi”
Day 13 and 14 “What’s is the area of a circle?”
Day 15 Constructing a Labyrinth
Mini Metric Olympics
(Copyright 1987 AIMS Education Foundation)
6.3.3.2
Estimate weights, capacities and geometric measurements using benchmarks in measurement
systems with appropriate units.
For example: Estimate the height of a house by comparing to a 6-foot man standing nearby.
I. Objectives: Students will be able to estimate weights, capacities, and geometric
measurements of everyday objects.
II. Launch: Students will have the opportunity to show their athletic and estimation skills!
III. Explore: The link posted below provides a good outline for the activity. Modifications
may be made to fit your class room…here are a few that I made.
Under Procedures, use groups of two, and have a designated recorder instead of a team
captain. Students may move from station to station, but remind students to be conscious
of the time they spend waiting in line. If there is a free station they should go there
instead of waiting in line.
http://cmase.uark.edu/teacher/workshops/AIMS-lessons/mini-metrics.pdf
IV. Summary: Have students calculate their scores. How close were the estimates to the
actual measurements? (Discuss) Were the guesses mostly too high or too low? (Discuss)
Does this help give a better feel for the metric system? Why or why not? (Discuss)
Special Angle Practice
8.3.2.1
9.3.3.1
Understand and apply the relationships between the slopes of parallel lines and between the
slopes of perpendicular lines.
Know and apply properties of parallel and perpendicular lines, including properties of angles
formed by a transversal, to solve problems and logically justify results.
For example: Prove that the perpendicular bisector of a line segment is the set of all points equidistant from the two
endpoints, and use this fact to solve problems and justify other results.
Know and apply properties of angles, including corresponding, exterior, interior, vertical,
complementary and supplementary angles, to solve problems and logically justify results.
9.3.3.2
For example: Prove that two triangles formed by a pair of intersecting lines and a pair of
parallel lines (an "X" trapped between two parallel lines) are similar.
I. Objectives: Students will know and apply properties of angles formed by a transversal
through two parallel lines.
II. Launch: Brainstorm with students a list of special angles and their definitions.
 Parallel Lines → lines that never intersect, or lines that have the same slope
 Transversal → a line that cuts across two or more (usually parallel) lines
 Corresponding Angles → angles that are at the same position at a each intersection
 Vertical Angles → angles opposite of each other when two lines cross
 Supplementary Angles → two angles that add to 180 degrees
 Adjacent Angles → angles that have a common side and common vertex
 Alternate Exterior Angles → the pairs of angles on opposite sides of the
transversal but outside the two lines
 Alternate Interior Angles → two congruent interior angles that lie on different
parallel lines and on opposite sides of a transversal
 Congruent Angles → angles that have exactly the same measure
(these definitions can be manipulated at http://www.mathopenref.com a very cool site)
III. Explore: After generating list of angles with definitions, complete the first Angle
Practice problem together as a whole group. Clarify any questions. Students will work in
pairs to complete questions 2, 3, and 4.
IV.
Summary: Check answers by having students check off the clue and write the correct
letter/number in the correct angle on the smart board. Discuss any angles with multiple
answers.
V. Extension: Students will create their own puzzle. The puzzle must contain a set of
parallel lines, a transversal, and eight clues.
Pythagorean Theorem Using Geometer’s Sketch Pad
8.3.1.3
Informally justify the Pythagorean Theorem by using measurements, diagrams and
computer software.
I. Objectives: Using Geometer’s Sketch Pad, students will animate a right triangle in order
to informally justify the Pythagorean Theorem.
II. Launch: “The next 2 days we get to practice what we have been learning about the
Pythagorean Theorem on the computer. When drawing a triangle to justify the
Pythagorean Theorem in Sketch Pad, what are some important aspects to keep in mind?”
 must draw a right triangle
 label all points
 only 2 points move
 etc.
III. Explore: Previous sketchpad experienced is assumed. Allow students to work in pairs.
Students must include names and a title on their project. Let students make discoveries on
how to complete the task. Use scaffolding when helping and giving clues. Directions for
an example informal justification are included below.
1. Place a point on the bottom left side of the screen, and draw a horizontal line through the
point.
2. Select the point and line, and then construct a perpendicular line.
3. Label points.
4. Construct line segments over the lines already drawn, so that when original lines are
hidden the edges of the triangle will not disappear.
5. Hide original lines (leaving the hypotenuse).
6. Measure all line segments and ensure the right angle is indeed 90 degrees.
7. Calculate “c^2”.
8. Calculate “a^2 + b^2”.
9. Calculate “a^2 + b^2 – c^2”.
10. Animate points b and c.
11. Should be left with a measurement of 0.
12. To see example click HERE
IV.
Conclusion: Ask students “How does this activity show the Pythagorean Theorem?”
Sample answer… even though two points of the right triangle may be moving and the
measurements are changing, a^2 + b^2 – c^2 always = 0.
The Theorem of Pythagoras 12.1
Objective
Use the Pythagorean Theorem to solve problems involving right triangles.
8.3.1.1
For example: Determine the perimeter of a right triangle, given the lengths of two of its sides.
Another example: Show that a triangle with side lengths 4, 5 and 6 is not a right triangle.
Materials
 Each student will need a piece of 2 pieces of patty paper, a pair of scissors, a pencil, and a
small amount of tape.
 Handout Open Investigation 12.1 – The Pythagorean Theorem
Launch
 Pythagorean Theorem Video
Explore
Step 1: Fold or draw a line segment from one corner of a patty paper to one of the nonadjacent
sides, forming a right triangle. Cut out this triangle.
Step 2: Cut a square out of the patty paper so that each side of the square is equal in length to the
shorter leg of the right triangle.
Step 3: Locate the center of a second patty paper by bringing each pair of opposite vertices
together and pinching.
Step 4: Temporarily tape this patty paper to the longer leg of the right triangle.
Step 5: fold the perpendicular from this center to the hypotenuse of the right triangle.
Step 6: Fold the line through this center parallel to the hypotenuse of the right triangle. These
two folds should divide the square patty paper into four parts.
Step 7: Remove the tape, and cut out the four parts of the square patty paper. You should now
have five pieces: one small square (shoes side has the same length as the short leg of the right
triangle) and the four regions from the square patty paper. Arrange these five parts into a square.
How does the length of each side of the square formed by these five parts compare to the length
of the hypotenuse of the right triangle? Compare your results with the results of others near you.
Summarize
Write a conjecture which shows the relationship between the lengths of the legs and the
hypotenuse of the right angle.
The Theorem of Pythagoras 12.2
Objective
Use the Pythagorean Theorem to solve problems involving right triangles.
8.3.1.1
For example: Determine the perimeter of a right triangle, given the lengths of two of its sides.
Another example: Show that a triangle with side lengths 4, 5 and 6 is not a right triangle.
Part 1
Materials
 Each student will need a piece of patty paper, a pencil and a ruler.
 Handout Open Investigation 12. 2 – The Pythagorean Theorem Revisited
Launch
 Why do carpenters measure from corner to corner when installing a door?
Explore
Step 1: Place a mark on the one edge of a patty paper. Label the two parts a and b. Use this patty
paper to mark all four edges of another patty paper as shown. Mark the four corners as right
angles.
Step 2: Fold or draw the four segments between pairs of points on adjacent sides. The four right
triangles formed are congruent, so the four segments are congruent. Label each segment c. The
quadrilateral has four congruent sides.
Explain why this quadrilateral must be a square.
Summarize
Write the area of the patty paper square algebraically in two different ways. Explain why this
implies that a2 + b2 = c2.
Part 2
Materials
 Each student will need a piece of patty paper, a pencil, and a pair of scissors.
 Handout Open Investigation 12. 3 – The Return of the Pythagorean Theorem Revisited
Launch
Step 1: Fold a corner of a patty paper so that the crease passes through an adjacent corner as
shown on the right.
Step 2: Fold a perpendicular to the first fold passing through a second corner of the square.
Step 3: Repeat this two more times. This gives you a patty paper with four right triangles and a
square in the center.
Step 4: Mark the right angles and label the sides. Cut out the four right triangles and the square
as shown.
Use algebra to again show that a2 +b2 = c2
Summarize
Replace a, b, and c with the following number combinations (3, 4, 5 and 6, 8, 12). Will either of
these sets help prove if it is a right triangle.
The Theorem of Pythagoras 12.4
Objective
Use the Pythagorean Theorem to solve problems involving right triangles.
8.3.1.1
For example: Determine the perimeter of a right triangle, given the lengths of two of its sides.
Another example: Show that a triangle with side lengths 4, 5 and 6 is not a right triangle.
Materials
 Each student will need at least 2 pieces of patty paper, a ruler with measurements in
millimeters, a pair of scissors, and a pencil.
 Handout Open Investigation 12.4 – The Converse of the Pythagorean Theorem
Launch
 Do all right triangles prove this formula? Are all answers rational numbers?
Explore
Step 1: Select of one of the Pythagorean triples from the list above. Check that the three numbers
satisfy the Pythagorean formula. Draw three segments on your patty paper, measuring their
lengths in millimeters.
Step 2: Cut the patty paper into three strips with one of the segments on each strip. Arrange the
three segments into a triangle.
Step 3: Place a second patty paper over the three strips, and mark a dot over each vertex of the
triangle. Draw the triangle.
What kind of triangle do the three segments form? How can you prove this?
If given two of the three integers, can you find the third?
__, 12, 13
21, __, 75
24, __, 145
24, 45, __
Summarize
Write a conjecture about what kind of triangle is formed by sides whose lengths are a
Pythagorean triple.
Extension
 Complete Exercise Set 12A
 Find the length of the missing leg and the perimeter and area of each triangle.
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
7.3.1.1
Demonstrate an understanding of the proportional relationship between the diameter
and circumference of a circle and that the unit rate (constant of proportionality) is  .
Calculate the circumference and area of circles and sectors of circles to solve problems
in various contexts.
Day eleven and twelve – “Let’s have some pi”
Launch – Teacher has brought in approximately 10 – 15 items with varying sizes of diameter. Note: we are
learning about circles, but it is not easy to measure the circumference of a flat piece of paper – thus, your circles
will actually be cylinders. Things that work well are pop cans, coffee mugs, soup/tuna cans, lids from ice cream
buckets. It is also fun and interesting for the students to do large objects such as tires or hula-hoops. Have the
items numbered or labeled so that students have the correct measurements for each object. Students will also
need a piece of rope, yarn or string to measure the circumference of the items. A rope will work well on a tire,
but may be too clumsy to use on a pop can. Yarn works well on a pop can, but because it has some “give”, it
will not allow for a good measurement on a tire. Each team will also need a meter stick. Ask the students what
they know about finding area and perimeter of rectangular objects. Students are familiar with what to do. Now,
show them the lid from the ice cream bucket and ask them how to find the perimeter and area of this shape. At
this point you will have to explain that we are dealing in two-dimensional, not three-dimensional. Discussion
will turn to “How do we measure something that isn’t a straight line?” Allow students time to brain storm, and
guide them to the string and meter stick. Model for the students how they will be measuring the “perimeter”
and diameter of each circular object. They are to record their results on the attached student data sheet.
Explore – Students will spend the rest of the class period collecting their data. They will hang on to their data
sheets for discussion on the following day.
Day twelve –
Launch – Have students look at their data from the previous day. Have the students look for any interesting
patterns.
Explore – Allow students ample time to see the “about 3 times” ratio between the diameter and the
“perimeter”.
Summary – After students have discovered the “3 times” ratio, they are ready to be introduced to the concept
of “pi”. I’ve also attached a little history to go along with my explanation to my students.
(WikiAnswers, 2009)
The ancient Egyptians and Babylonians knew that there was a constant ratio between the diameter of a circle
and its circumference. The Egyptians used the approximate value 3 and 1/8 (=3.125) around 2000 BC.
The ancient Greek mathematician Archimedes (287-212 BC) proved that pi is between 3 and 1/7 and 3 and
10/71, in around 240 BC. In decimal notation this puts pi between 3.1408 and 3.1429, which is good enough for
most practical purposes. In fact 3 and 1/7, or 22/7, is good enough for most purposes.
There is a lot more information at http://en.wikipedia.org/wiki/Pi.
The actual symbol, the Greek letter, was not used until 1706, but the concept is 4000 years old.
Note that pi is not exactly equal to 22/7, or to any ratio of whole numbers. See the answer below for quite a few
digits, and note the three dots at the end; pi goes on for ever without repeating.
[Note: In my country, BC is common usage. In yours, it may be BCE.]
This number has been known since ancient times, although we have arrived at more accurate values lately.
The ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more
than 3. This fact was known to ancient Egyptian, Babylonian, Indian and Greek mathematicians.. The earliest
known approximations of Pi are from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both
within 1% of the true value
The first person that used pi would be the Greeks and the Archimedes from Syracuse. The mathematician
figured out that pi = 3.14 or 3 1/7.
Pi is the ratio between of circle's circumference to its diameter.
It is a constant irrational value. It's most accurate value is 3.14159.....
(WikiAnswers, 2009)
You can also show students the website with the non-terminating, non-repeating decimal.
Day thirteen and fourteen – “What is the area of a circle?”
Launch – Discuss with students what they know about finding area of rectangular shapes. Hold up the icecream bucket lid and ask them how to measure the area. Allow sufficient time for the students to think of using
the graph paper (you may want to hold off on the hula-hoop or tire)
Explore - Students spend the remainder of class tracing the objects onto graph paper. They will then find the
area by counting the squares on the grid paper to their best approximation. This data is added to their data sheet
from day one.
Day 4 –
Explore – Have students display their grid paper and data sheets. During a silent “gallery walk”, using “I
notice”, “I wonder”. Have students look for patterns. There should be a few students who have used the
formula for the area of rectangles to count all of the full boxes on the grid. Lead a group discussion on the
“length and width” of a circle. Using the attached worksheet, guide students to discover the area of a circle
using the square as your focus.
Summary – students should be to tell you how the formulas for area and circumference of a circle work. By
this age, they are very familiar with perimeter and area of rectangles, and using that information as background
knowledge, it should now be clear how the formulas for circles work and why they work!!
Day fifteen…..
Launch – During our social studies unit on Ancient Civilizations, students have learned about labyrinths,
their purpose and history. This is our culminating project. It usually takes about three class periods to
complete. The labyrinth that I use is from a cathedral in France. I have attached a copy that I use in the
classroom.
Explore – Hand out the worksheet and have students work in pairs to complete accurately. The supplies are
expensive and you do not want to over-spend. You will be using
NAME:____________________________
Calculate approximately how many rolls of plastic tape we will need. If tape comes in 200ft.
rolls, how many should I buy? Rolls of tape are $1.69 each. How much will this cost? Each
path is 1 meter wide. Round numbers to the nearest whole meter. (You may use a calculator,
but you must SHOW ALL WORK, that is, I need to know what numbers you put into the
calculator and why)
Amount Needed
# of Rolls Needed
Total Price
If I wanted to paint our Labyrinth on canvas, how much canvas would I need?
http://www.seton.com/seton/catalog/refineResults.do/79179/~0/0/10/~0/false/0/P_MasterNu
mberDescription%7C0/400749.html#79179
I the future, I will purchase my flags from this website. The ones that purchased at a local
big box lumber company were MUCH more expensive.
This is an example of the type of flags and tape that I use. You may be able to get a local
telephone or electric company to donate some to you! You will need a flag approximately
every 1 – 1 ½ meters.
Pre Test and Post Test- Geometry
3. Estimate the mass of a pencil.
4. Pythagorean Theorem
a. Construct a right triangle with patty paper.
b. Prove using one of the three methods that a2 +b2 = c2.
c. Staple patty paper to your test.
5. Pythagorean Theorem
a. Describe a real life situation where this theorem would be relevant.
6. Pythagorean Theorem
a. Find c (hypotenuse) If a= 3 and b= 4.
7. Pythagorean Theorem
a. Find a (leg) If c= 10 and b= 8
8. Pythagorean Theorem
a. Find the diagonal of the rectangle. Assume the lengths of the legs are 5 by 12.
9.
10.
Light
*
sprinkler
A sprinkler is at the center of a lawn.
The sprinkler waters the area inside
the circle. How many square units will
be watered? (Use 3.14 for pi )
A. 25.12
B. 50.24
C. 100.48
D. 200.96
Kelly installed a light fixture on her
house to light part of her yard. The
light shines onto the yard in a
semicircle with a radius of 20 feet, as
shown above. What is the area of
Kelly’s yard that is lit by the new
light? (Use 3.14 for pi)
A. 62.8 feet squared
B. 125.6 feet squared
C. 628 feet squared
D. 1256 feet squared
Erin’s Dog Kennel
16 ft.
Fence
11.
The circumference of a basketball is
30 inches. What is the approximate
diameter of a basketball?
(Use 3.14 for )
A. 3.1 inches
B. 4.8 inches
C. 6.2 inches
D. 9.6 inches
12.
Erin keeps her dog in the pen shown
above. The pen is made by 2 walls of
a
building and a curved fence. What is
the approximate length of the fence?
(Use 3.14 for )
A. 25 feet
B. 50 feet
C. 100 feet
D. 201 feet