Discovering
Parallelograms
by
Mary R ose Landon
Unit: Quadrilaterals
Grade 10
5 day lesson plan
Using Technology :
Geometers Sketchpad - Key Curriculum
Press
TI - 84+ Graphing Calculators
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.
2
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).
3
Unit Overview:
I.
Angles of Polygons
1. Interior Angle Sum Theorem: S = 180(n - 2)
180(n ! 2)
n
360
3. Exterior Angle (regular polygon):
n
2. Interior Angle (regular polygon):
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)
4
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.
5
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.
6
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
Sum
Triangle – 3 sides
Interior Angle
# of triangles:
sum:
Quadrilateral – 4 sides
# of triangles:
sum:
Pentagon – 5 sides
# of triangles:
sum:
7
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:
8
II. EXTERIOR ANGLES:
∠6 =
Triangle
5
∠5 =
2
∠4 =
1
3
6
Interior angle sum:
∠6 + ∠5 + ∠4 = ?
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.
9
Lessons 2 - 3: Parallelograms
(2 days)
Objectives: SWBAT
1. Construct a parallelogram using CabriJr App.
2. Use CabriJr tools to explore prope rties of a parallelogram
3.
Make conjectures about properties of a parallelogram, including
opposite sides,
angles, and diagonals
4. Test conjectures and summarize findings
Materials:
• TI - 84+ calculators (classroom set), with CabriJr App.
• Parallel ogram lab. Activity
Procedure:
I.
Day1 –
Introduce lab activity. Relate activity to previous Cabri Jr
labs. (students have already constructed segments,
parallel/perp. lines, angles, triangles, etc). Screen captures
are included as visual aids.
Instruct students to save sketches for Day 2
activities/further
exploration.
II.
Day 2 - Continue lab activity. Allow time for completion (this
could run int o 3 days)
III. Summary/Closure: Discuss lab. findings. Student s will make a
list of
Parallelogram properties in their notes.
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IV.
Assignment: pg. 415, problems #37 - 39. Use the distance and
slope formulas in a coordinate proof, to verify that . . .
a) the diagonals bisect each other
b) the opposite sides are congruent
c) t he 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: F2Ψ pointΨ point <ENTER>
2. Label the points A and B: F5Ψ alpha-num <ENTER>
3. Draw a line through points A and B:
F2Ψ line <ENTER>, select point A <ENTER>,
then drag line to point B <ENTER>
4. Plot a point above AB : F2Ψ pointΨ point <ENTER>
5. Label the point C: F5Ψ alpha-num <ENTER>
11
6. Construct a line through point C that is parallel to line AB: F3Ψ Parallel,
select point C and line AB, then <ENTER>
7. Construct line AC:
F2 Ψ Line, select A and C, <ENTER>
8. Construct a line parallel to line AC, through B: F3Ψ Parallel,
select B and AC, <ENTER>
9. Construct a point on the intersection of the two lines that contain point C
and point B: F2Ψ PointΨ Intersection, select the two lines, <ENTER>
10. Label this point of intersection point D: F5Ψ alpha-num <ENTER>
11. Construct quadrilateral ABDC: F2Ψ quad, select point A <ENTER>, drag to B,
<ENTER>,
drag to D <ENTER> and drag to C <ENTER>
12
12. Hide the original lines: F5Ψ hide/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: F5Ψ measureΨ angle, 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: F5Ψ MeasureΨ D.&Length, then
drag the measurement tool until each side is activated and hit <ENTER>.
Record your lengths below:
AB: ____________
13
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: F5Ψ Measure>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.
14
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:
F2Ψ PointΨ Intersection, <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.
15
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.
3
a) y = x ! 2
b) 3 x + 4 y = 8
Answer: neither, since equation b is
4
!3
y=
x + 2 (not = or ⊥ slopes!)
4
II. Discuss properties of a parallelogram (refer to CabriJr. lab results)
III. Introduce Sketchpad Rhombus lab. activity and outline objectives:
16
•
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.
A
B
17
#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.
D
C
B
A
#8. Construct segment DC.
Hide the circles and line DC.
D
A
C
B
18
#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 = ______.
D
C
#11. Construct diagonals AC and BD.
Plot point E at their intersection.
#12. Measure the lengths of each segment
formed by the diagonals.
E
DE = ______ EB = ______
AE = ______ EC = ______
#13. Measure the angles formed by the diagonals. Record your findings:
A
m∠DEC = ______
m∠CEB = ______
m∠BEA = ______
B
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:
19
Summary:
Including your lab. findings (and previous knowledge about parallelograms), list as
many attributes or properties of a rhombus as possible below. Be specific!
20
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.
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