Patty Paper Geometry - Wikispacestylercopd2012.wikispaces.com/.../Patty+Paper+Geometry.pdf · 2012-08-14 · original point and its corresponding image point always ... Patty Paper

8/14/2012

Region 5 ESC/CIA/Math/PattyPaper/Aug2012

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.


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.


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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…


Ready, Set…Check that you have:• Handout

• 20 sheets of Patty Paper

• Pencil

• Protractor

• Compass


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:

12

34


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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.


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.


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

______________________________________


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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.


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.



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?


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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?




– 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.

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.


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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.


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.


CSCOPE application

Gr. 6 Geometry Example:

Unit 7, Lesson 1, Day 13, Explore/Explain 5

Understanding Midpoint


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CSCOPE application

HS Geometry Example:

Unit 2, Lesson 1, Day 4, Explore 2

Understanding Midpoint


Where did I get this stuff?

Patty Paper Geometry

Michael Serra


Questions??


Region 5 ESC/CIA/Mathematics/Frayer/Aug 2012

Investigation (what you did)

Picture

Conjecture (what you think is true based on the investigation)

Official Definition

Geometric Term

VERTICAL ANGLES

The pairs of opposite angles formed by two intersecting lines are called VERTICAL ANGLES.

If two lines intersect, then each pair of vertical angles formed is ________________________.



Picture


Official Definition

Geometric Term



Picture


Official Definition

Geometric Term



Picture


Official Definition

Geometric Term

Grade 6 Mathematics

2011 Transition Unit 07 Lesson 01–Unit: 06 Lesson: 03

©2010, TESCCC 08/01/10 page 83 of 94

Half the circumference

radius

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.

• Base = 21 the circumference of the circle, or πr

• Height = the radius of the circle, or r • Area of a parallelogram: A = bh.

Geometry HS Mathematics

Unit: 02 Lesson: 01

©2012, TESCCC 05/16/12 page 1 of 2

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?

Geometry HS Mathematics

Unit: 02 Lesson: 01

©2012, TESCCC 05/16/12 page 2 of 2

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.

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Patty Paper Geometry - Wikispacestylercopd2012.wikispaces.com/.../Patty+Paper+Geometry.pdf · 2012-08-14 · original point and its corresponding image point always ... Patty Paper