Discovering Discovering ParallelogramsParallelograms

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Mary Rose LandonMary Rose Landon

Unit: QuadrilateralsUnit: Quadrilaterals Grade 10Grade 10

5 day lesson 5 day lesson planplan

Using TechnologyUsing Technology::

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Geometers SketchpadGeometers Sketchpad-- Key Curriculum Key Curriculum PressPress

TITI--84+ Graphing Calculators84+ Graphing Calculators CabriJr Applications CabriJr Applications

Unit Objectives: Upon the completion of this unit, students should be able to . . .

• find the sum of the measures of the interior and exterior angles of a

polygon

• recognize the properties of regular polygons

• recognize and apply the properties of sides, angles, and diagonals of

parallelograms

• prove that a quadrilateral is a parallelogram

• recognize and apply properties of rectangles

• recognize and apply properties of rhombi and squares

• recognize and apply the properties of trapezoids

• use quadrilaterals in coordinate proofs

• prove theorems using coordinate proofs

• apply properties of parallelograms to problem-solving

NY State Standards: G.R.P.2 Recognize and verify geometric relationships of perpendicularity, parallelism, and congruence, using algebraic strategies. G.R.P.3 Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.

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G.R.P.9 Apply inductive reasoning in making and supporting mathematical conjectures. G.CM.11 Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams. G.CM.12 Draw conclusions about mathematical ideas through decoding and interpretation of mathematical visuals (Mathematics Core Curriculum, MST Standard 3, Geometry, March 2005).

NCTM Standards:

• Analyze properties and determine relationships of two-and-three

dimensional objects. • Explore relationships (including congruence and similarity) among classes

of two-and-three dimensional geometric object, and make and test conjectures about them.

• Establish the validity of geometric conjectures using deduction, prove

theorems, and critique arguments made by others (Principles and Standards for School Mathematics, NCTM, 2000, Reston, VA).

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Unit Overview:

I. Angles of Polygons 1. Interior Angle Sum Theorem: S = 180(n - 2)

2. Interior Angle (regular polygon): n

n )2(180 !

3. Exterior Angle (regular polygon): n

360

II. Parallelograms

1. Construct a parallelogram 2. Properties a. opposite sides

b. opposite and consecutive angles c. diagonals as bisectors d. congruent triangles formed by diagonals

3. Tests for Parallelograms a. opposite sides test b. opposite angles test c. distance formula (coordinate graphs) d. slope formula

4. Rectangles a. constructions b. properties

i. 4 Right angles ii. congruent diagonals (using slope and distance)

5. Rhombi

a. constructions b. properties

perpendicular diagonals opposite angles bisected by diagonals

c. squares (properties)

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III. Trapezoids 1. Properties

a. bases - parallel sides (one pair) b. legs (non-parallel) c. median of a trapezoid

2. Special trapezoids

a. isosceles b. right trapezoid

IV. Coordinate Proof With Quadrilaterals 1. Slope (formula) 2. Midpoint 3. Distance

Materials Used:

Glencoe Mathematics Geometry, Boyd, Cummins, Malloy, the McGraw-Hill Co, Columbus, OH, Copyright 2005, Chapter 8, pp 402-437.

Mathematics Core Curriculum, MST standard 3, Geometry, The University of the State of NY, revised March, 2005.

Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, Copyright 2000, Reston, VA.

Geometer’s Sketchpad V4, Key Curriculum Press, 2005. TI-84+ Graphing Calculators (class set), Texas Instruments.

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Lesson 1: Angles of Polygons Objectives: SWBAT 1. Find the sum of the measures of the interior angles of a polygon 2. Find the sum of the exterior angles

Materials: • Glencoe Mathematics text-Geometry Ch 8, pp 404-410 • Polygon angle worksheet discovery activity

Procedure:

I. Warm-Up activity: Find each angle in a triangle, if the angles have a ratio of 4:5:6. Show your solution process.

II. After discussing the warm-up activity, connect to today’s lesson:

Q: How can you determine how many degrees are inside a polygon? Hint: Try to draw as many triangles as possible by connecting a vertex to each other vertex (see polygon activity).

III. Students will work in groups to draw polygons and connect vertices. They

should “discover” the Angle Sum Theorem: S = (n - 2)180

IV. Compare interior to exterior angles (connect to prior knowledge of “linear pair” angles). Then students should complete activity and compare their findings.

V. Summary/Closure: How can your method be used without drawing the

polygon? Examples: 1. Find the angle sum measure in a 20-gon. 2. Find an interior and exterior angle measure of a “regular” decagon.

VI. Assignment: textbook applications. Pg 407, #1-11.

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POLYGON ACTIVITY I. INTERIOR ANGLES To find the interior angle sum, draw triangles inside each polygon:

#1. Connect one vertex to each other vertex in the polygon. #2. Count the number of triangles formed. #3. Interior angle sum: Sum = #triangles · 180º

Polygons Interior Angle Sum Triangle – 3 sides # of triangles: sum: Quadrilateral – 4 sides # of triangles: sum: Pentagon – 5 sides # of triangles: sum:

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Hexagon – 6 sides # of triangles: sum: Heptagon – 7 sides # of triangles: sum: Octagon – 8 sides # of triangles: sum: “n-gon”: If n = # of sides, then how many triangles are formed? ___________ An expression (using “n”) for the interior angle sum is:

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II. EXTERIOR ANGLES: ∠6 = Triangle 5 ∠5 = 2

∠4 = 1 3 6 4 Interior angle sum: ∠6 + ∠5 + ∠4 = ? Experiment ~ Find the exterior angle sum of a polygon: 1. Find the measure of an interior angle in a regular pentagon. Then find the measure of each exterior angle (use linear pairs!). What is the sum of the exterior angles? 2. Try a rectangle; what is the angle sum of the four exterior angles? 3. Make a conjecture about the sum of the exterior angles in any convex polygon.

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LessonsLessons 2 2--3: Parallelograms 3: Parallelograms (2 days) (2 days) Objectives: SWBATObjectives: SWBAT 1. Construct a parallelogram using CabriJr App.1. Construct a parallelogram using CabriJr App. 2. Use CabriJr tools to explore prope2. Use CabriJr tools to explore properties of a parallelogramrties of a parallelogram 3. Make conjectures about properties of a parallelogram, including 3. Make conjectures about properties of a parallelogram, including opposite sides, opposite sides, angles, angles, and and diagonalsdiagonals 4. Test conjectures and summarize findings4. Test conjectures and summarize findings Materials:Materials:

•• TITI--84+ calculators (classroom set), with CabriJr App.84+ calculators (classroom set), with CabriJr App. •• ParallelParallelogram lab. Activityogram lab. Activity

Procedure:Procedure:

I.I. Day1 Day1 –– Introduce lab activity. Relate activity to previous CabriJr Introduce lab activity. Relate activity to previous CabriJr labs. (students have already constructed segments, labs. (students have already constructed segments, parallel/perp. lines, angles, triangles, etc). Screen captures parallel/perp. lines, angles, triangles, etc). Screen captures are included as visual aids.are included as visual aids.

Instruct students to save sketches for Day 2 Instruct students to save sketches for Day 2 activities/further activities/further exploration. exploration.

II.II. Day 2Day 2-- Continue lab activity. Allow time for completion (this Continue lab activity. Allow time for completion (this could run into 3 days)could run into 3 days)

III.III. Summary/Closure: Discuss lab. findings. StudentSummary/Closure: Discuss lab. findings. Students will make a s will make a

list of list of Parallelogram properties in their notes.Parallelogram properties in their notes.

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IV.IV. Assignment: pg. 415, problems #37Assignment: pg. 415, problems #37--39. Use the distance and 39. Use the distance and slope formulas in a coordinate proof, to verify that . . . slope formulas in a coordinate proof, to verify that . . . a)a) the diagonals bisect each other the diagonals bisect each other b)b) the opposite sides are congruentthe opposite sides are congruent c)c) tthe opposite sides are parallelhe opposite sides are parallel

Cabri Lab Activity: Properties of a Parallelogram

Objectives:

In this lab, you will use TI-84 CabriJr construct tools to • construct parallel lines and transversals • investigate properties of angles formed by parallel lines • construct and explore a parallelogram • Use your sketch to investigate other properties of a parallelogram

Day #1 Construct a Parallelogram

1. Plot two points: F2ΨpointΨpoint <ENTER>

2. Label the points A and B: F5Ψalpha-num <ENTER>

3. Draw a line through points A and B:

F2Ψline <ENTER>, select point A <ENTER>,

then drag line to point B <ENTER>

4. Plot a point above AB : F2ΨpointΨpoint <ENTER>

5. Label the point C: F5Ψalpha-num <ENTER>

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6. Construct a line through point C that is parallel to line AB: F3ΨParallel, select point C and line AB, then <ENTER>

7. Construct line AC: F2 ΨLine, select A and C, <ENTER>

8. Construct a line parallel to line AC, through B: F3ΨParallel, select B and AC, <ENTER>

9. Construct a point on the intersection of the two lines that contain point C and point B: F2ΨPointΨIntersection, select the two lines, <ENTER>

10. Label this point of intersection point D: F5Ψalpha-num <ENTER>

11. Construct quadrilateral ABDC: F2Ψquad, select point A <ENTER>, drag to B,

<ENTER>, drag to D <ENTER> and drag to C <ENTER>

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12. Hide the original lines: F5Ψhide/show, select each line and <ENTER>

13. Now look at quadrilateral ABDC. Is it a parallelogram?

How can you prove that ABDC is (or that it is not) a parallelogram?

List as many properties of a parallelogram as possible.

__________________________________________________

__________________________________________________

Day #2: Investigating Properties of Parallelograms Investigate the properties of the angles of parallelogram ABDC:

1. Measure each angle of ABDC: F5ΨmeasureΨangle, select three consecutive vertices and each time hit <ENTER>

2. Record your measurements:

γ A: ________ γ B:________ γ C:________ γ D:________

3. Make conjectures about the relationships between angles: Adjacent angles are__________________________.

Opposite angles are __________________________.

4. Test your conjectures by selecting a vertex and dragging it, using the hand

key (green ALPHA key). Do your conjectures hold true? ________ Will they be true for any parallelogram?______ How do you know this, or how could you prove that this is always true?

5. Measure each side of your parallelogram: F5ΨMeasureΨD.&Length, then drag the measurement tool until each side is activated and hit <ENTER>. Record your lengths below:

AB: ____________

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BD: ____________

DC: ____________

CA: ____________

Now drag a vertex (Select and use the hand tool – Green ALPHA key) and make a conjecture about opposite sides of a parallelogram:

_________________________________________________________

6. Verify that opposite sides of ABCD are parallel.

Measure the slope of each side: F5ΨMeasure>Slope, then select side.

mAB = ________ mAC = ________

mCD = ________ mBD = ________

7. Draw diagonal AD: F2Ψ Segment, then select point A, hit <ENTER>, drag to point D and hit <ENTER>.

Are the two triangles congruent? _______ If yes, name the pair of congruent triangles:

∆______ ≅ ∆_______

How do you know (or how can you prove) that they are congruent triangles?

Can you use SAS, ASA, or SSS theorems? Explain how you would prove that the triangles are congruent.

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8. Construct diagonal BC (follow the same steps as above).

9. Measure the length of BC and the length of AD: MeasureΨ D.&Length, then select each segment and hit <ENTER>. Record measurements below:

AD: ________ BC: ________

10. Now plot a point on the intersection of diagonals AD and BC: F2ΨPointΨIntersection, <ENTER>, then activate both diagonals and <ENTER>. Label the point of intersection E: F5Ψ Alpha-Num, then <ENTER>, and type E.

11. Measure the following lengths: F5Ψ D.&Length, <ENTER>, then select a

segment (select one point at a time, and hit <ENTER>, ). Record measurements:

AE:_______ CE: _______ BE: _______ DE: _______ What do you notice about the measurements of the diagonals and the segments formed by the intersection? Make a conjecture about the diagonals of any parallelogram, based on your measurements.

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Drag a vertex and investigate the measures of these segments and diagonals. Does your conjecture hold true? How could you prove that this is always true? (Hint: think about triangles formed by intersecting diagonals) Summary: In conclusion, write a short paragraph about the properties of a parallelogram, including properties of opposite and consecutive angles, opposite sides, and diagonals.

Lessons 4-5 (1-2 days) Objectives: SWBAT 1. Construct a rhombus using Sketchpad tools. 2. Investigate properties of a rhombus. 3. Make conjectures based on constructions, then test conjectures. Materials:

• Geometer’s Sketchpad, V4 • TI-84+ graphing calculators • Textbook and notes (Glencoe Geometry—Ch 8)

Procedure: I. Warm-Up activity: determine whether the lines given by equations a and b are parallel, perpendicular, or neither.

a) 24

3!= xy b) 843 =+ yx Answer: neither, since equation b is

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3+

!= xy (not = or ⊥ slopes!)

II. Discuss properties of a parallelogram (refer to CabriJr. lab results) III. Introduce Sketchpad Rhombus lab. activity and outline objectives:

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A B

• Each student will construct and examine a rhombus, following explicit directions and using Sketchpad tools. Students have prior experience with Sketchpad constructions (angles, lines & triangles).

IV. Lab. Activity:

Day#1: Complete sketches and record all data (measurements of angles, sides, diagonals). Day #2: Summarize and share findings, investigate properties of a square.

V. Closure/Summary: Q: What properties are true for any parallelogram? Q: What makes a rhombus “special”? VII. Assignment: pp 434-5. Problems #1-10 and #26-31. Rhombus Discovery Activity Introduction:

By definition, a rhombus is a parallelogram with four congruent sides. In this CabriJr lab. activity, you will construct a rhombus and discover its properties, make conjectures based on your findings, and then test your knowledge. Follow each direction carefully and record all findings. Be sure to save your final sketch in order to demonstrate your findings! #1. Draw and label segment AB.

A B

#2. Use the compass tool to draw a circle with center at point A and with radius AB.

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CD

A B

CD

A B

#3. Repeat step #2 process to draw another circle, with center at B and radius BA. #4. Plot a point at the intersection of the two circles. Label this point D. #5. Construct a line through •D that is parallel to AB. #6. Draw segment AD. #7. Construct a line through B, parallel to AD. Then plot a point at the intersection of •B and this line. #8. Construct segment DC. Hide the circles and line DC.

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E

CD

A B

#9. Measure and record the length of each side to verify that it is a rhombus. AB = ________, BC = ________, CD = ________, DA = ________.

#10. Measure each angle: m∠ABC = ______, m∠BCD = ______,

m∠CDA = ______, m∠DAB = ______.

#11. Construct diagonals AC and BD. Plot point E at their intersection. #12. Measure the lengths of each segment formed by the diagonals.

DE = ______ EB = ______

AE = ______ EC = ______

#13. Measure the angles formed by the diagonals. Record your findings: m∠DEC = ______ m∠CEB = ______ m∠BEA = ______ m∠AED = ______ Make a conjecture about the diagonals of any rhombus:

______________________________________________________________

______________________________________________________________

Test your conjecture by dragging an angle. As the sides of the rhombus become larger (or smaller), does your conjecture about the angles hold true? How could you use your segment lengths (in steps #9 and #12) to verify your conjecture algebraically (ie. using a theorem or formula)? To show that your conjecture is true, demonstrate the algebraic steps below:

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Summary: Including your lab. findings (and previous knowledge about parallelograms), list as many attributes or properties of a rhombus as possible below. Be specific!

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Day #2: Now that you have constructed a rhombus, try to answer the following questions:

• What is the definition of a rhombus?

• How are the diagonals of a rhombus related?

• What special triangles do you see on your constructed rhombus?

• What kind of triangle congruencies do you see? (Be specific!)

• How is a square related to a rhombus? • How do you think the diagonals of a square are related?

Sketchpad directions: #1. Construct square ABCD. Does it have the properties of a rhombus? Measure the diagonals and angles to verify. What other special properties does it have? Make as many conjectures as possible about your square (include angles, diagonals, and triangles formed). _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ #2. Check your conjectures by measuring the angles, diagonal segments, etc.. Then summarize your findings.