Download What is patty paper?

8/14/2012
Region 5 ESC/CIA/Mathematics/Aug 2012
What is patty paper?
• Waxed squares of paper used by fast
food restaurants to separate
hamburger patties.
• You can write on them with pencils
and felt-tip pens.
• Perfect size for students to use to
discover geometric properties by
folding and tracing.
Region 5 ESC/CIA/Mathematics/Aug 2012
Group Structure
•
It is recommended to use pair-share cooperative learning for
students practicing patty paper geometry.
– Students who ordinarily work in groups of 4 break into 2 groups
– One student from each pair reads the instructions to the
partner while the partner does the folding.
– The pair compares its results with the results from the other
group.
– The group then makes the conjecture.
– For the next investigation, both the pairs and the roles switch.
– In pair-share, everyone has a role and shares in the pride of
discovery.
– Pair-share also helps reduce students’ math anxiety.
Region 5 ESC/CIA/Mathematics/Aug 2012
Region 5 ESC/CIA/Math/PattyPaper/Aug2012
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Investigations
• Students should keep a geometry notebook to organize
all the investigations they do, the vocabulary they use,
the conjectures they make, and the exercises they
complete.
• See Frayer Model…
Region 5 ESC/CIA/Mathematics/Aug 2012
Ready, Set…
Check that you have:
• Handout
• 20 sheets of Patty Paper
• Pencil
• Protractor
• Compass
Region 5 ESC/CIA/Mathematics/Aug 2012
Let’s roll!
Guided investigation 1: Vertical Angles
– Fold a line on a piece of patty paper.
– Unfold.
– Fold a second line intersecting the first
line.
– Label the angles as in the diagram:
2
1
4
3
Region 5 ESC/CIA/Mathematics/Aug 2012
Region 5 ESC/CIA/Math/PattyPaper/Aug2012
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Investigation 1: Vertical Angles (cont.)
– Place a second piece of patty paper over the
first and copy one angle of a pair of vertical
angles.
– Rotate the copy to see how well it fits over
the second angle of the vertical angle pair.
– Make a conjecture about the measure of
vertical angles on your Frayer Model.
– Definition: The pairs of opposite angles
formed by two intersecting lines are called
vertical angles.
Region 5 ESC/CIA/Mathematics/Aug 2012
Investigation 2: Translations
– Construct a simple shape on your patty paper.
– Place a dot in its interior.
– Draw a ray from that dot to the edge of your patty paper.
This will be the direction of the translation.
– Place a second dot on the ray. The distance from the first dot
to the second dot will be the translation distance.
– Place a second piece of patty paper over the first, and trace
the figure, the interior dot, and the ray.
– Slide the second patty paper along the path of the ray until
the dot on the tracing paper is over the second dot on the
original.
– Use another piece of patty paper (or a ruler) to measure the
distance between a point in the original shape and it’s
corresponding point in the translated image.
– Try this with another set of corresponding points.
Region 5 ESC/CIA/Mathematics/Aug 2012
Investigation 2: Translations (cont.)
– What do you observe? Is the distance between the
original point and its corresponding image point always
the same?
– Compare your results with the results of others near you.
– Finish this sentence on your Frayer Model to make a
conjecture:
In a translation transformation, the distance between any
point and its image is
______________________________________
Region 5 ESC/CIA/Mathematics/Aug 2012
Region 5 ESC/CIA/Math/PattyPaper/Aug2012
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Investigation 3: Circumcenter
– Draw a large acute scalene triangle on your
patty paper.
– Fold to construct the perpendicular bisector
of each side of your triangle.
– On another piece of patty paper, repeat
steps 1 and 2 with an obtuse scalene
triangle.
• If you have difficultly getting the
perpendicular bisectors to intersect on the
patty paper, try relocating your triangle.
Region 5 ESC/CIA/Mathematics/Aug 2012
Investigation 3: Circumcenter (cont.)
– What seems to be true about the
perpendicular bisectors of the sides of a
triangle?
– Compare your results with the results of
others near you, and make a conjecture
about the perpendicular bisectors of a
triangle on your Frayer Model.
– Definition: The point of intersection of the
three perpendicular bisectors of the sides of
a triangle is called the circumcenter of the
triangle.
Region 5 ESC/CIA/Mathematics/Aug 2012
Investigation 3: Circumcenter (cont.)
Which distances from the circumcenter of a
triangle to the edge of the triangle are all the
same?
– Place a second piece of patty paper over the
acute triangle.
– Mark the distance from the circumcenter to
one of the 3 vertices.
– Compare this distance with the distances of
the other 2 vertices.
– How do they compare?
Region 5 ESC/CIA/Mathematics/Aug 2012
Region 5 ESC/CIA/Math/PattyPaper/Aug2012
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Investigation 3: Circumcenter (cont.)
– Make a conjecture about the circumcenter of
the triangle in relationship to the vertices of
the triangle.
Where is the circumcenter of your acute triangle
located?
• Of the obtuse triangle?
– Compare with others around you and see if
they had the same results.
– Where do you think it is located in a right
triangle?
Region 5 ESC/CIA/Mathematics/Aug 2012
Investigation 3: Circumcenter (cont.)
– Fold a new piece of patty
paper to get a right triangle.
– Draw the triangle on the
folds.
– Fold the perpendicular
bisectors of the sides of the
triangle.
Region 5 ESC/CIA/Mathematics/Aug 2012
Investigation 3: Circumcenter (cont.)
– Where is the circumcenter of your right
triangle located?
– Did your results match your original
conjecture?
– Complete the conjecture below on your
Frayer Model:
The circumcenter is ___________ an acute
triangle, _____________ an obtuse triangle, and
_____________ a right triangle.
Region 5 ESC/CIA/Mathematics/Aug 2012
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Investigation 4:
Measure of Inscribed Angles
– Definition: An inscribed angle is an angle whose vertex lies
on the circle and whose sides are the chords of the circle.
– Use your compass to draw a circle on a piece of patty paper.
– Fold or draw an inscribed angle.
– Fold or draw a central angle intercepting the same arc as the
inscribed angle.
– With another piece of patty paper, make a copy of the
inscribed angle.
– Slide the copy of the inscribed angle on the second patty
paper over the central angle on the original patty paper.
Region 5 ESC/CIA/Mathematics/Aug 2012
Investigation 4: Inscribed Angles (cont.)
– How many times will the inscribed angle fit in the
central angle?
– Compare your results with the results of others
around you.
Recall that the measure of an arc is equal to
the measure of the central angle that
intercepts it.
– Make a conjecture about the measure of an
inscribed angle on your Frayer Model.
Region 5 ESC/CIA/Mathematics/Aug 2012
CSCOPE application
Gr. 6 Geometry Example:
Unit 7, Lesson 1, Day 13, Explore/Explain 5
Understanding Midpoint
Region 5 ESC/CIA/Mathematics/Aug 2012
Region 5 ESC/CIA/Math/PattyPaper/Aug2012
8/14/2012
CSCOPE application
HS Geometry Example:
Unit 2, Lesson 1, Day 4, Explore 2
Understanding Midpoint
Region 5 ESC/CIA/Mathematics/Aug 2012
Where did I get this stuff?
Patty Paper Geometry
Michael Serra
Region 5 ESC/CIA/Mathematics/Aug 2012
Questions??
Region 5 ESC/CIA/Mathematics/Aug 2012
Region 5 ESC/CIA/Math/PattyPaper/Aug2012
Investigation
Picture
(what you did)
Geometric Term
Conjecture
VERTICAL ANGLES
Official Definition
(what you think is true based on the investigation)
If two lines intersect, then each pair of vertical angles formed is ________________________. Region 5 ESC/CIA/Mathematics/Frayer/Aug 2012
The pairs of opposite angles formed by two intersecting lines are called VERTICAL ANGLES. Investigation
Picture
(what you did)
Geometric Term
Conjecture
(what you think is true based on the investigation)
Region 5 ESC/CIA/Mathematics/Frayer/Aug 2012
Official Definition
Investigation
Picture
(what you did)
Geometric Term
Conjecture
(what you think is true based on the investigation)
Region 5 ESC/CIA/Mathematics/Frayer/Aug 2012
Official Definition
Investigation
Picture
(what you did)
Geometric Term
Conjecture
(what you think is true based on the investigation)
Region 5 ESC/CIA/Mathematics/Frayer/Aug 2012
Official Definition
Grade 6
Mathematics
2011 Transition Unit 07 Lesson 01–Unit: 06 Lesson: 03
Area of a Circle – Challenge KEY
You can use what you know about circles and pi to learn about the area of a circle.
Hint 1: The radius of a circle is half its diameter. Use a coffee filter as a model for your circle. Be sure
to flatten it out as much as possible.
Fold the circle in half three times as shown below. Be sure to fold carefully.
Hint 2: Cut along the folds, and fit the pieces together to make a figure that looks like a parallelogram.
(See figure below.)
Hint 3: Think of this figure as a parallelogram. The base and height of the parallelogram relate to the
parts of the circle.
radius
Half the circumference
•
1
the circumference of the circle, or πr
2
Height = the radius of the circle, or r
•
Area of a parallelogram: A = bh.
•
Base =
©2010, TESCCC
08/01/10
page 83 of 94
Geometry
HS Mathematics
Unit: 02 Lesson: 01
Understanding Midpoints
1. On the attached coordinate grid, plot and label the following points: P(-7, -6) and Q(5, 8).
2. Using a straight edge, draw segment PQ.
3. Using a straight edge, draw a dotted vertical line passing through point P and another
through point Q.
4. Fold and crease the paper so that the two vertical lines are perfectly aligned on top of
each other. Use patty paper to compare the lengths of the two segments formed. What do
you observe?
5. Mark the intersection of the vertical crease and segment PQ.
6. In terms of the distance between the vertical lines, what does the crease in the paper
represent?
7. Using a straight edge, draw a dotted horizontal line passing through point P and another
through point Q.
8. Fold and crease the paper so that the two horizontal lines are perfectly aligned on top of
each other.
9. Mark the intersection of the horizontal crease and segment PQ. (If you were careful, you
will notice that the vertical crease and the horizontal crease intersect on segment PQ in
the same spot!)
10. In terms of the distance between the horizontal lines, what does the horizontal crease in
the paper represent?
11. Mark the point of intersection of the crease lines on segment PQ and label it M.
12. What are the coordinates of M?
13. Fold and crease the paper so that point P and point Q are perfectly aligned on top of each
other. What does this lead you to believe about point M?
14. Find the average of the x-coordinates of point P and Q and record below. How does this
value relate to the coordinates of M?
15. Find the average of the y-coordinates of point P and Q and record below. How does this
value relate to the coordinates of M?
©2012, TESCCC
05/16/12
page 1 of 2
Geometry
HS Mathematics
Unit: 02 Lesson: 01
Understanding Midpoints
16. Based on your previous answers, write a conjecture about how to find the midpoint of any
line segment given the coordinates of its endpoints. Record your conjecture below.
17. Write a formula for finding the midpoint of segment AB given A(x1, y1) and B(x2, y2).
Record your answer below.
©2012, TESCCC
05/16/12
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