QUADRILATERAL MIDSEGMENT CONJECTURE

Enduring Understanding: Develop a better understanding of how to make and test conjectures made regarding quadrilaterals and a better understanding of the relationship of the midsegments of quadrilaterals to other aspects of quadrilaterals.

Essential Questions:

 How is the perimeter of a polygon determined?  How does a person locate the midpoint of a segment?  What is a midsegment?  What is a segment vs. a line or ray?  What is meant by the term “conjecture”?  How can a conjecture be validated?  How important is precision in measurement?

Lesson Overview:

 Before allowing the students the opportunity to start the activity: access their prior knowledge with regards to properties of quadrilaterals. Discuss what is meant by a diagonal. Warm-up exercises, discussion in collaborative groups, problems on the wall around the room can be used.  What tools can be used to aid in the precision of the measurements? Is this case is it better to measure the lengths in centimeters or inches? Why choose one over the other?  What is being asked by the questions in the problem? How do you decode what the problem is asking you to do?  The teacher could have the students do their drawings initially on a larger sheet of paper, then transfer a proportional drawing to the paper with the questions. That would allow for a larger and easier drawing to evaluate.  The teacher could have different students (in groups or individually) do the last 4 shapes so that each student only has to draw one shape. Then compare with others who have the same shape. Then could do a jigsaw activity to share their thinking.  How can the students make their thinking visible?  How can you support a conjecture that you make?  Use resources from your building. Technology could be used to support this investigation— especially sketchpad or cabri geometry.

EALRs/GLEs:

1.3.1 1.3.2 1.3.3 2.2.2 3.2.1 5.1.1

Item Specifications: ME04; GS01; GS02; SR04; SR05; MC01 Assessment:

 Use WASL format items that link to what is being covered by the classroom activity  Include multiple choice questions Quadrilateral Midsegment Conjecture

Adapted from the Charles A. Dana Center at the University of Texas at Austin, © 2002

Situation 1

Use paper, pencil, construction and measuring tools

1. Sketch and label Quadrilateral ABCD. a. Draw all possible diagonals for Quadrilateral ABCD. b. Create a polygon (in this case a quadrilateral) by connecting the midpoints of the sides of quadrilateral ABCD. Label the midpoints of the segments as follows:

AB has a midpoint W; BC has a midpoint of X; CD has a midpoint of Y; has a midpoint of Z

2. Take the appropriate measurements, and write a conjecture that relates the perimeter of the midpoint quadrilateral to the diagonals of the original quadrilateral. Show all calculations.

AC = ______BD = ______WX = ______XY = ______

YZ = ______ZW = ______3. Repeat the steps in #1 (including the measurements) with two more quadrilaterals that are different from your original quadrilateral.

Situation 2

Earlier you proved the Triangle Midsegment Conjecture:

The midsegment of a triangle is parallel to one side of the triangle and measure half the length of that side.

Since you have proved this conjecture, you can use it to help prove and explain why other conjectures may or may not be true.

Use the Triangle Midsegment Conjecture to write an explanation of why your quadrilateral conjecture is true. Provide a labeled diagram with your explanation.

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______Do you think your conjecture will be true for other polygons besides quadrilaterals? Why or why not?

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Sketch, measure, and investigate the relationship between a polygon’s diagonals and the polygon formed by connecting the midpoints of its sides. Do this with a pentagon, hexagon, heptagon and octagon. Diagram of pentagon:

Relationship between the diagonals and the polygon formed by connecting the midpoints of its sides:

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______Diagram of hexagon: Relationship between the diagonals and the polygon formed by connecting the midpoints of its sides: ______

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Diagram of heptagon:

Relationship between the diagonals and the polygon formed by connecting the midpoints of its sides: ______

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Diagram of octagon: Relationship between the diagonals and the polygon formed by connecting the midpoints of its sides:

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Situation 3

Modify your original conjecture (or write a new conjecture) to take into account other polygons besides quadrilaterals.

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______Whitebeard’s Treasure

Whitebeard, the notorious pirate of the West Bay, buried treasure on Tiki Island over 200 years ago. Archeologists recently discovered a map showing the location of the treasure. The location has generated quite a bit of media attention, much to the dismay of the archeologists. In order to allow both the media and archeologists to work together, officials have decided to erect two fences around the location, allowing the media access to the site, yet allowing the archeologists room to work. One fence encloses the actual area where the archeologists will work. Another fence surrounds the enclosed dig area.

Descriptions of the fencing locations have been provided to the media so they may indicate accessible areas for their employees. Use the given information to draw and label a quadrilateral on graph paper indicating the location of the two fences.

1. Corners of the first fence are located at points A(11,3), B(3,-11), C(-13,-9) and D(-5,9). The media must stay within this fence’s area. Connect the points in alphabetical order, and then join point D to Point A. 2. Find and label the midpoints of each segment of quadrilateral ABCD, showing all work. Label the midpoints of the segments as follows:

AB has midpoint Q, BC has midpoint R, CD has midpoint S, DA has midpoint T.

3. Connect the four midpoints in alphabetical order to create a new quadrilateral QRST. This quadrilateral represents the fence surrounding the archeological dig site.

4. Quadrilateral ABCD was an ordinary quadrilateral, but QRST is a special one. Determine the special name for quadrilateral QRST, and justify your answer using coordinate geometry in two different ways.

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______5. A vertical cut is made through the cube along the diagonal of the top face as shown below.

Which two identical solids will be produced?

O A. rectangular pyramids O B. rectangular prisms O C. triangular pyramids O D. triangular prisms

6. Which transformation always preserves the dimensions of a figure?

I. translation II. rotation III. reflection IV. dilation

O A. I, II, and III O B. I, II, and IV O C. I, III, and IV O D. II, III, and IV

7.

In the figure above, the radius of the inscribed circle is 6 inches (in).

Which is the perimeter of square ABCD?

O A. 12p in O B. 36p in O C. 24 in O D. 48 in