FAMILY OF QUADRILATERALS

TEACHER EDITION

List of Activities for this Unit:

ACTIVITY / STRAND / DESCRIPTION
1 - Squares / GS / Properties of 2-D squares
2 - Rectangles / GS / Properties of 2-D rectangles
3 - Parallelograms / GS / Properties of 2-D parallelograms
4 - Trapezoids / GS / Properties of 2-D trapezoids
5 – Rhombi / GS / Properties of 2-D rhombi
6 – Kites / GS / Properties of 2-D kites
7 – Comparing Quadrilaterals / GS / Comparing properties of quadrilaterals
8 – Venn are We Going to
Organize these
Quadrilaterals / GS/CU / Comparing properties of quadrilaterals/organizing information
9 – Whitebeard’s Treasure / ME / Using midpoints & mid-segments to investigate properties of special quadrilaterals
10 – Midpoints and Mid-
segments Applied to
Quadrilaterals / ME/GS / Determine the midpoint of a line segment, Investigate properties of mid-segments
11 – Practice Problems / ME/GS / Multiple Choice Practice
COE Connections / Alex’s Front Window
Alex’s Rear Window
MATERIALS / Grid Paper
Patty Paper
Protractors
Graphing Calculators
Warm-Ups
(in Segmented Extras Folder)

Vocabulary: Mathematics and ELL

accessible / cube / perimeter
adjacent / diagonals / quadrilateral
alphabetical order / intersection / rhombus
archeologists / kite / sketch
axis / location / square
characteristics / midpoint / transformation
compare / midpoint / trapezoid
conjecture / mid-segment / Venn diagram
connects / overlapping / vertices
coordinate plane / parallel
coordinates / parallelogram

Essential Questions:

  • What is a quadrilateral?
  • What are examples of quadrilaterals?
  • What are the properties of a rectangle? A square? A parallelogram? A trapezoid? A rhombus? A kite?
  • What is a diagonal?
  • How do you use a protractor?
  • What is a set? A subset?
  • How is the perimeter of a polygon determined?
  • What does it mean to do a translation?
  • What is a Venn diagram?
  • How important is precision in measurement?

Based on the optional activities:

  • How does a person locate the midpoint of a segment? What is meant by midpoint?
  • What is a midsegment?
  • What is a conjecture?
  • How can a conjecture be validated?

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? In 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.

Performance Expectations:

5.3.AClassify quadrilaterals.

5.3.GDraw quadrilaterals and triangles from given information about sides and angles.

8.2.GApply the Pythagorean Theorem to determine the distance between two points on the coordinate plane.

8.3.GSolve single- and multi-step problems using counting techniques and Venn diagrams and verify the solutions.

G.3.GKnow, prove, and apply theorems about properties of quadrilaterals and other polygons.

Performance Expectations and Aligned Problems

Chapter 15 “Dependent and Independent” Subsections: / 1-
Squares / 2-
Rec-tangles / 3-
Parallel-ograms / 4-
Trap-ezoids / 5-
Rhombi / 6-
Kites / 7-
Com-paring Quadri
-laterals / 8-
Venn are We Going… / 9-
White-beard’s Trea-sure / 10-
Mid-points & Mid-Segments / 11-
Practice Prob-lems
Problems Supporting:
PE 5.3.A / 1 – 9 / 1 – 9 / 1 – 9 / 1 – 9 / 1 – 9 / 1 – 9 / 1, 2 / 4
Problems Supporting:
PE 5.3.G / 1 – 9 / 1 – 9 / 1 – 9 / 1 – 9 / 1 – 9 / 1 – 9 / 1 – 3 / 1 – 3 / 1 – 3, Sit-uation 2
Problems Supporting:
PE 8.2.G / 4
Problems Supporting:
PE 8.3.G / 1 - 4
Problems Supporting:
PE G.3.G / 1 – 9 / 1 – 9 / 1 – 9 / 1 – 9 / 1 – 9 / 1 – 9 / 4 / 1 – 3 / 1 – 3, Sit-uation 2

Assessment: Use the multiple choice and short answer items from Measurement and Geometric Sense that are included in the CD. They can be used as formative and/or summative assessments attached to this lesson or later when the students are being given an overall summative assessment.

Squares

The set of quadrilaterals has the following subsets: rectangles, squares, parallelograms, kites, trapezoids, rhombi and of course quadrilaterals themselves.

1. Measure the sides and angles of each square.

2. List two or more examples of where you can find squares in the world around you.

Answers will vary.

______

3. Describe a square in your own words. ______

A square is a quadrilateral with four right angles and four congruentsides that have congruent diagonals that bisect one another.

4. Draw and measure both diagonals(line segment connecting opposite corners) for each square in problem #1. What did you notice about the measurements of the diagonals of a square?

See the figures above; the diagonals of a square are congruent (the same length).

5. Measure the angles of intersection(point in common)of the diagonalsfor each square in problem #1. What did you notice about the angles of intersection of the diagonals of a square?

The diagonals are perpendicular to one another and form four right (90°) angles.

6. Name as many characteristics(things that have to be true) of a square as you can.

A square is a regular polygon, quadrilateral and a parallelogram with 4 right angles, congruent sides, diagonals are congruent, and diagonals are perpendicular.

7. Draw a picture of a square on the coordinate plane so that one pair of sides is parallel to an axis. List the coordinates(x-value, y-value)of the vertices(the point where two sides of the square come together) of your square and label the vertices: A, B, C, and D.

.An example is given: (answers will vary).

8. What are the coordinates of the midpoint(the point at the middle of the segment)of each side of your square? ANSWERS WILL VARY. (an example is given.)

Midpointis (0, 5) Midpoint (5,0) Midpointis (0,-5) Midpointis (-5,0)

9. Draw line segments that connect the midpoint of one side to the midpoint of each adjacent(sides that have a common endpoint) side.

What new figure have you created? The new figure is a square of side lengths 5 (and has half the area of the original square).

Rectangles

1. Measure the sides and angles of the rectangles.

2. List two or more examples of where you can find rectangles in the world around you.

ANSWERS WILL VARY

3. Describe a rectangle in your own words. A rectangle is a quadrilateral with four right angles, opposite sides are parallel and congruent which has congruent diagonals that bisect each other.

4. Draw and measure both diagonals for each rectangle in problem #1. What did you notice about

the measurements of the diagonals of a rectangle? The diagonals are congruent (same length) and bisect each other.

5. Measure the angles of intersection of the diagonals for each rectangle in problem #1. What did

you notice about the angles of intersection of the diagonals of a rectangle? The angles of intersection come in pairs one angle measure from each pair add to 180°. If the rectangle is also a square the diagonals intersect forming four right angles; i.e. all the angles are congruent and measure 90°.

6. Name as many characteristics of a rectangle as you can.

Rectangles are a quadrilateral and a parallelogram with four right angles where opposite sides are congruent and the diagonals are also congruent.

7. Draw a picture of a rectangle on the coordinate plane so that one pair of sides is parallel to an axis. List the coordinates of the vertices of your rectangle and label the vertices: H, I, J, and K.

An example is given: (answers will vary).

8. What are the coordinates of the midpoint of each side of your rectangle?

ANSWERS WILL VARY. (an example is given.)

Midpointis (-3,5) Midpoint (0,2) Midpointis (-3,-3) Midpointis (-6,2)

9. Draw line segments that connect the midpoint of one side to the midpoint of each adjacent side.

What new figure have you created? The new figure is a parallelogram; specifically a rhombus.

Parallelograms

1. Measure the sides and angles of the parallelograms.


2. List two or more examples of where you can find parallelograms in the world around you.

Answers will vary.

______

3. Describe a parallelogram in your own words. A parallelogram is a four sided plane figure that has opposite sides parallel and the diagonals bisect each other.

4. Draw and measure both diagonals for each parallelogram in problem #1. What did you notice

about the measurements of the diagonals of a parallelogram? All the diagonals bisect each other and the diagonals are congruent if the parallelogram is a rectangle.

5. Measure the angles of intersection of the diagonals for each parallelogram in problem #1. What did you notice about the angles of intersection of the diagonals of a parallelogram? The angles formed by the diagonals of a parallelogram come in pairs and one angle measure from each pair adds to 180°. If the parallelogram is a square then the diagonals form four 90° angles where they intersect.

6. Name as many characteristics of a parallelogram as you can. A parallelogram has opposite sides congruent and parallel, opposite angles are congruent and adjacent angles add to 180°. The diagonals bisect each other with supplementary angles and are congruent if the parallelogram is a rectangle.

7. Draw a picture of a parallelogram on the coordinate plane so that one pair of sides is parallel to an axis. List the coordinates of the vertices of your parallelogram and label the vertices: L, M, O, and P.

An example is given: (answers will vary).

8. What are the coordinates of the midpoint of each side of your parallelogram?

ANSWERS WILL VARY. (an example is given.)

Midpointis (6, 4) Midpoint (5.5, 1) Midpointis (-1, -2) Midpointis (-0.5, 1)

9. Draw a line segment that connects the midpoint of one side to the midpoint of an adjacent side.

What new figure have you created? The new figure is another parallelogram.

Trapezoids

1. Measure the sides and angles of the trapezoids.

2. List two or more examples of where you can find trapezoids in the world around you.

Answers will vary.

3. Describe a trapezoid in your own words. A trapezoid is a four sided plane figure (quadrilateral) with one pair of opposite sides parallel.

4. Draw and measure both diagonals for each trapezoid in problem #1. What did you notice about

the measurements of the diagonals of a trapezoid? The diagonals form supplementary angles and only the isosceles trapezoid has congruent diagonals.

5. Measure the angles of intersection of the diagonals for each trapezoid in problem #1. What did you notice about the angles of intersection of the diagonals of a trapezoid?

The diagonals for a supplementary set of angles.

6. Name as many characteristics of a trapezoid as you can. The trapezoid is a four sided plane figure (quadrilateral) that has one pair of opposite sides that are parallel. Angles formed by the parallel lines and a side of the trapezoid are supplementary.

7. Draw a picture of a trapezoid on the coordinate plane so that one pair of sides is parallel to an axis. List the coordinates of the vertices of your trapezoid and label the vertices: Q, R, S, & T.

An example is given: (answers will vary).

8. What are the coordinates of the midpoint of each side of your trapezoid?

ANSWERS WILL VARY. (an example is given.)

Midpointis (0.5, 6) Midpoint (8.5, 2.5) Midpointis (5, -1) Midpointis (-3, 2.5)

9. Draw a line segment that connects the midpoint of one side to the midpoint of an adjacent side.

What new figure have you created? The new figure is a parallelogram.

Rhombus

1. Measure the sides and angles of a rhombus.

THIS IS A RHOMBUSTHIS IS A RHOMBUS

THIS IS NOT A RHOMBUSTHIS IS NOT A RHOMBUS

2. List two or more examples of where you can find rhombi in the world around you.

Answers will vary.

______

3. Describe a rhombus in your own words. A rhombus is a four sided plane figure (quadrilateral) that has opposite sides parallel and all sides are congruent, adjacent angles are supplementary and the diagonals are perpendicular and bisect one another.

4. Draw and measure both diagonals for each rhombus in problem #1. What did you notice about

the measurements of the diagonals of a rhombus? The diagonals of a rhombus bisect one another and they are perpendicular.

5. Measure the angles of intersection of the diagonals for each rhombus in problem #1. What did you notice about the angles of intersection of the diagonals of a rhombus? The diagonals intersect at 90° which implies perpendicular lines.

6. Name as many characteristics of a rhombus as you can. A rhombus is a four sided plane figure (quadrilateral) with four congruent sides; diagonalsare perpendicular and congruent only if the rhombus is a square.

7. Draw a picture of a rhombus on the coordinate plane so that one pair of sides is parallel to an axis. List the coordinates of the vertices of your rhombus and label the vertices: W, X, Y, and Z.

An example is given: (answers will vary).

8. What are the coordinates of the midpoint of each side of your rhombus?

ANSWERS WILL VARY: (an example is given) Midpointis (1.5, 3.5)

Midpoint (1.5, -3.5) Midpointis (-1.5, -3.5) Midpointis (-1.5, 3.5)

9. Draw a line segment that connects the midpoint of one side to the midpoint of an adjacent side.

What new figure have you created? The new figure is a rectangle. If the original figure was the special rhombus “a square” then the new figure would be a square; as we noticed in the lesson about squares.

Kites

1. Measure the sides and angles of the kite.

THESE ARE KITES:

THESE ARE NOT KITES:

2. List two or more examples of where you can find kites in the world around you.

Answers will vary.

3. Describe a kite in your own words. A kite is a four sided plane figure (quadrilateral) that has perpendicular diagonals where only one diagonal is bisected and has two pair of adjacent sides that are congruent; though not necessarily all sides are congruent.

4. Draw and measure both diagonals for each kite in problem #1. What did you notice about the

measurements of the diagonals of a kite? The diagonals do not bisect each other unless the kite is arhombus and the diagonals are not congruent unless the kite is a square.

5. Measure the angles of intersection of the diagonals for each kite in problem #1. What did you notice about the angles of intersection of the diagonals of a kite? The diagonals of the kite are perpendicular.

6. Name as many characteristics of a kite as you can. A kite is a four sided plane figure (quadrilateral) that has two pairs of adjacent sides congruent and the diagonals are perpendicular with at least one diagonal bisected.

7. Draw a picture of a kite on the coordinate plane. List the coordinates of the vertices of your kite and label the vertices: G, E, F, and N. An example is given: (answers will vary).

8. What are the coordinates of the midpoint of each side of your kite?

ANSWERS WILL VARY: (an example is given) Midpointis (-1.5, 5.5)

Midpoint (-1.5, -2.5) Midpointis (-4.5, -2.5) Midpointis (-4.5, 5.5)

9. Draw a line segment that connects the midpoint of one side to the midpoint of an adjacent side.

What new figure have you created? The new figure is a rectangle.

Comparing Quadrilaterals

1. Write descriptions of relationships between the pairs of figures given below.

Example: All squares are rectangles but some rectangles are not squares

a. rectangles and parallelograms All rectangles are parallelograms, but not all parallelograms are rectangles.

b. rhombi and kites All rhombi are kites, but not all kites are rhombi

c. squares and rhombi All squares are rhombi, but not all rhombi are squares.

2. Based on your ideas about paired relationships between the special quadrilateral sets, squares are

the intersection of which two sets. Explain.

Squares are the intersection of rectangles and rhombi. Because a square must have four right

angles and four congruent sides. Squares are a special case of both rectangles and rhombi.

Venn are We Going to Organize these Quadrilaterals

A Venn diagram is a drawing that shows relationships among sets of data.

1. Write sisters in the circle on the left and brothers in the circle on the right. If you have a sister, put your name in the circle labeled sisters. If you have a brother, put your name in the circled labeled brothers.

a. What should a student do who has both brothers and sisters? We could overlap the circles and have a place where we could write something like “both”. Those students who have both brothers and sisters put your name in the correct place—inside the both area of the overlapped circles.

b. What does a student do who has neither? We could draw a rectangle around the circles and put the names of those with neither in the rectangle. Those students who have neither brothers nor sisters put your name in the correct place—inside the rectangle and not in either circle.

Siblings for Some Students

Answers will vary.

c. Now check with 10 other students in the class and fill in the Venn diagram below.