Discovering Discovering Parallelograms Parallelograms
Discovering Discovering Parallelograms Parallelograms
Discovering Discovering Parallelograms Parallelograms
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<strong>Discovering</strong><br />
<strong>Parallelograms</strong><br />
by<br />
Mary Rose Landon<br />
Unit: Quadrilaterals<br />
Grade 10<br />
5 day lesson plan<br />
Using Technology:
Geometers Sketchpad- Key Curriculum<br />
Press<br />
TI-84+ Graphing Calculators<br />
CabriJr Applications<br />
Unit Objectives:<br />
Upon the completion of this unit, students should be able to . . .<br />
• find the sum of the measures of the interior and exterior angles of a<br />
polygon<br />
• recognize the properties of regular polygons<br />
• recognize and apply the properties of sides, angles, and diagonals of<br />
parallelograms<br />
• prove that a quadrilateral is a parallelogram<br />
• recognize and apply properties of rectangles<br />
• recognize and apply properties of rhombi and squares<br />
• recognize and apply the properties of trapezoids<br />
• use quadrilaterals in coordinate proofs<br />
• prove theorems using coordinate proofs<br />
• apply properties of parallelograms to problem-solving<br />
NY State Standards:<br />
G.R.P.2 Recognize and verify geometric relationships of perpendicularity,<br />
parallelism, and congruence, using algebraic strategies.<br />
G.R.P.3 Investigate and evaluate conjectures in mathematical terms, using<br />
mathematical strategies to reach a conclusion.<br />
2
G.R.P.9 Apply inductive reasoning in making and supporting mathematical<br />
conjectures.<br />
G.CM.11 Understand and use appropriate language, representations, and<br />
terminology when describing objects, relationships, mathematical<br />
solutions, and geometric diagrams.<br />
G.CM.12 Draw conclusions about mathematical ideas through decoding and<br />
interpretation of mathematical visuals (Mathematics Core Curriculum,<br />
MST Standard 3, Geometry, March 2005).<br />
NCTM Standards:<br />
• Analyze properties and determine relationships of two-and-three<br />
dimensional objects.<br />
• Explore relationships (including congruence and similarity) among classes<br />
of two-and-three dimensional geometric object, and make and test<br />
conjectures about them.<br />
• Establish the validity of geometric conjectures using deduction, prove<br />
theorems, and critique arguments made by others (Principles and<br />
Standards for School Mathematics, NCTM, 2000, Reston, VA).<br />
3
Unit Overview:<br />
I. Angles of Polygons<br />
1. Interior Angle Sum Theorem: S = 180(n - 2)<br />
180 ( n ! 2)<br />
2. Interior Angle (regular polygon):<br />
n<br />
360<br />
3. Exterior Angle (regular polygon):<br />
n<br />
II.<br />
<strong>Parallelograms</strong><br />
1. Construct a parallelogram<br />
2. Properties<br />
a. opposite sides<br />
b. opposite and consecutive angles<br />
c. diagonals as bisectors<br />
d. congruent triangles formed by diagonals<br />
3. Tests for <strong>Parallelograms</strong><br />
a. opposite sides test<br />
b. opposite angles test<br />
c. distance formula (coordinate graphs)<br />
d. slope formula<br />
4. Rectangles<br />
a. constructions<br />
b. properties<br />
i. 4 Right angles<br />
ii. congruent diagonals (using slope and distance)<br />
5. Rhombi<br />
a. constructions<br />
b. properties<br />
perpendicular diagonals<br />
opposite angles bisected by diagonals<br />
c. squares (properties)<br />
4
III.<br />
Trapezoids<br />
1. Properties<br />
a. bases - parallel sides (one pair)<br />
b. legs (non-parallel)<br />
c. median of a trapezoid<br />
2. Special trapezoids<br />
a. isosceles<br />
b. right trapezoid<br />
IV.<br />
Coordinate Proof With Quadrilaterals<br />
1. Slope (formula)<br />
2. Midpoint<br />
3. Distance<br />
Materials Used:<br />
Glencoe Mathematics Geometry, Boyd, Cummins, Malloy, the McGraw-Hill Co,<br />
Columbus, OH, Copyright 2005, Chapter 8, pp 402-437.<br />
Mathematics Core Curriculum, MST standard 3, Geometry, The University of<br />
the State of NY, revised March, 2005.<br />
Principles and Standards for School Mathematics, National Council of<br />
Teachers of Mathematics, Copyright 2000, Reston, VA.<br />
Geometer’s Sketchpad V4, Key Curriculum Press, 2005.<br />
TI-84+ Graphing Calculators (class set), Texas Instruments.<br />
5
Lesson 1: Angles of Polygons<br />
Objectives: SWBAT<br />
1. Find the sum of the measures of the interior angles of a polygon<br />
2. Find the sum of the exterior angles<br />
Materials:<br />
• Glencoe Mathematics text-Geometry Ch 8, pp 404-410<br />
• Polygon angle worksheet discovery activity<br />
Procedure:<br />
I. Warm-Up activity: Find each angle in a triangle, if the angles have a ratio<br />
of 4:5:6. Show your solution process.<br />
II.<br />
III.<br />
IV.<br />
After discussing the warm-up activity, connect to today’s lesson:<br />
Q: How can you determine how many degrees are inside a polygon?<br />
Hint: Try to draw as many triangles as possible by connecting a vertex to<br />
each other vertex (see polygon activity).<br />
Students will work in groups to draw polygons and connect vertices. They<br />
should “discover” the Angle Sum Theorem: S = (n - 2)180<br />
Compare interior to exterior angles (connect to prior knowledge of<br />
“linear pair” angles). Then students should complete activity and compare<br />
their findings.<br />
V. Summary/Closure: How can your method be used without drawing the<br />
polygon?<br />
Examples:<br />
1. Find the angle sum measure in a 20-gon.<br />
2. Find an interior and exterior angle measure of a “regular” decagon.<br />
VI. Assignment: textbook applications. Pg 407, #1-11.<br />
6
POLYGON ACTIVITY<br />
I. INTERIOR ANGLES<br />
To find the interior angle sum, draw triangles inside each polygon:<br />
#1. Connect one vertex to each other vertex in the polygon.<br />
#2. Count the number of triangles formed.<br />
#3. Interior angle sum: Sum = #triangles · 180º<br />
Polygons<br />
Sum<br />
Triangle – 3 sides<br />
Interior Angle<br />
# of triangles:<br />
sum:<br />
Quadrilateral – 4 sides<br />
# of triangles:<br />
sum:<br />
Pentagon – 5 sides<br />
# of triangles:<br />
sum:<br />
7
Hexagon – 6 sides<br />
# of triangles:<br />
sum:<br />
Heptagon – 7 sides<br />
# of triangles:<br />
sum:<br />
Octagon – 8 sides<br />
# of triangles:<br />
sum:<br />
“n-gon”:<br />
If n = # of sides, then how many triangles are formed? ___________<br />
An expression (using “n”) for the interior angle sum is:<br />
8
II. EXTERIOR ANGLES:<br />
Triangle<br />
∠6 =<br />
5 ∠5 =<br />
2<br />
∠4 =<br />
1 3 6<br />
4<br />
Interior angle sum: ∠6 + ∠5 + ∠4 = ?<br />
Experiment ~ Find the exterior angle sum of a polygon:<br />
1. Find the measure of an interior angle in a regular pentagon.<br />
Then find the measure of each exterior angle (use linear pairs!).<br />
What is the sum of the exterior angles?<br />
2. Try a rectangle; what is the angle sum of the four exterior angles?<br />
3. Make a conjecture about the sum of the exterior angles in any<br />
convex polygon.<br />
9
Lessons 2-3: <strong>Parallelograms</strong> (2 days)<br />
Objectives: SWBAT<br />
1. Construct a parallelogram using CabriJr App.<br />
2. Use CabriJr tools to explore prope rties of a parallelogram<br />
3. Make conjectures about properties of a parallelogram, including<br />
opposite sides,<br />
angles, and diagonals<br />
4. Test conjectures and summarize findings<br />
Materials:<br />
• TI-84+ calculators (classroom set), with CabriJr App.<br />
• Parallel ogram lab. Activity<br />
Procedure:<br />
I. Day1 –<br />
Introduce lab activity. Relate activity to previous CabriJr<br />
labs. (students have already constructed segments,<br />
parallel/perp. lines, angles, triangles, etc). Screen captures<br />
are included as visual aids.<br />
activities/further<br />
Instruct students to save sketches for Day 2<br />
exploration.<br />
II. Day 2- 2 Continue lab activity. Allow time for completion (this<br />
could run into 3 days)<br />
III. Summary/Closure: Discuss lab. findings. Student s will make a<br />
list of<br />
Parallelogram properties in their notes.<br />
10
IV. Assignment: pg. 415, problems #37-39. 39. Use the distance and<br />
slope formulas in a coordinate proof, to verify that . . .<br />
a) the diagonals bisect each other<br />
b) the opposite sides are congruent<br />
c) the opposite sides are parallel<br />
Cabri Lab Activity: Properties of a Parallelogram<br />
Objectives:<br />
In this lab, you will use TI-84 CabriJr construct tools to<br />
• construct parallel lines and transversals<br />
• investigate properties of angles formed by parallel lines<br />
• construct and explore a parallelogram<br />
• Use your sketch to investigate other properties of a parallelogram<br />
Day #1 Construct a Parallelogram<br />
1. Plot two points: F2ΨpointΨpoint <br />
2. Label the points A and B: F5Ψalpha-num <br />
3. Draw a line through points A and B:<br />
F2Ψline , select point A ,<br />
then drag line to point B <br />
4. Plot a point above AB : F2ΨpointΨpoint <br />
5. Label the point C: F5Ψalpha-num <br />
11
6. Construct a line through point C that is parallel to line AB: F3ΨParallel,<br />
select point C and line AB, then <br />
7. Construct line AC:<br />
F2 ΨLine, select A and C, <br />
8. Construct a line parallel to line AC, through B: F3ΨParallel,<br />
select B and AC, <br />
9. Construct a point on the intersection of the two lines that contain point C<br />
and point B: F2ΨPointΨIntersection, select the two lines, <br />
10. Label this point of intersection point D: F5Ψalpha-num <br />
11. Construct quadrilateral ABDC: F2Ψquad, select point A , drag to B,<br />
, drag to D and drag to C <br />
12
12. Hide the original lines: F5Ψhide/show, select each line and <br />
13. Now look at quadrilateral ABDC. Is it a parallelogram?<br />
How can you prove that ABDC is (or that it is not) a parallelogram?<br />
List as many properties of a parallelogram as possible.<br />
__________________________________________________<br />
__________________________________________________<br />
Day #2: Investigating Properties of <strong>Parallelograms</strong><br />
Investigate the properties of the angles of parallelogram ABDC:<br />
1. Measure each angle of ABDC: F5ΨmeasureΨangle, select three consecutive<br />
vertices and each time hit <br />
2. Record your measurements:<br />
γ A: ________ γ B:________ γ C:________ γ D:________<br />
3. Make conjectures about the relationships between angles:<br />
Adjacent angles are__________________________.<br />
Opposite angles are __________________________.<br />
4. Test your conjectures by selecting a vertex and dragging it, using the hand<br />
key (green ALPHA key). Do your conjectures hold true? ________<br />
Will they be true for any parallelogram?______<br />
How do you know this, or how could you prove that this is always true?<br />
5. Measure each side of your parallelogram: F5ΨMeasureΨD.&Length, then<br />
drag the measurement tool until each side is activated and hit .<br />
Record your lengths below:<br />
AB: ____________<br />
13
BD: ____________<br />
DC: ____________<br />
CA: ____________<br />
Now drag a vertex (Select and use the hand tool – Green ALPHA key) and<br />
make a conjecture about opposite sides of a parallelogram:<br />
_________________________________________________________<br />
6. Verify that opposite sides of ABCD are parallel.<br />
Measure the slope of each side: F5ΨMeasure>Slope, then select side.<br />
m AB = ________<br />
m CD = ________<br />
m AC = ________<br />
m BD = ________<br />
7. Draw diagonal AD: F2Ψ Segment, then select point A, hit , drag to<br />
point D and hit .<br />
Are the two triangles congruent? _______<br />
If yes, name the pair of congruent<br />
triangles:<br />
∆______ ≅ ∆_______<br />
How do you know (or how can you prove) that they are congruent triangles?<br />
Can you use SAS, ASA, or SSS theorems?<br />
Explain how you would prove that the triangles are congruent.<br />
14
8. Construct diagonal BC (follow the same steps as above).<br />
9. Measure the length of BC and the length of AD: MeasureΨ D.&Length,<br />
then<br />
select each segment and hit . Record measurements below:<br />
AD: ________<br />
BC: ________<br />
10. Now plot a point on the intersection of diagonals AD and BC:<br />
F2ΨPointΨIntersection, , then activate both diagonals and .<br />
Label the point of intersection E: F5Ψ Alpha-Num, then , and type E.<br />
11. Measure the following lengths: F5Ψ D.&Length, , then select a<br />
segment (select one point at a time, and hit , ). Record measurements:<br />
AE:_______ CE: _______ BE: _______ DE: _______<br />
What do you notice about the measurements of the diagonals and the segments<br />
formed by the intersection?<br />
Make a conjecture about the diagonals of any parallelogram, based on your<br />
measurements.<br />
15
Drag a vertex and investigate the measures of these segments and diagonals.<br />
Does your conjecture hold true?<br />
How could you prove that this is always true? (Hint: think about triangles<br />
formed by intersecting diagonals)<br />
Summary: In conclusion, write a short paragraph about the properties of a<br />
parallelogram, including properties of opposite and consecutive angles,<br />
opposite sides, and diagonals.<br />
Lessons 4-5 (1-2 days)<br />
Objectives: SWBAT<br />
1. Construct a rhombus using Sketchpad tools.<br />
2. Investigate properties of a rhombus.<br />
3. Make conjectures based on constructions, then test conjectures.<br />
Materials:<br />
• Geometer’s Sketchpad, V4<br />
• TI-84+ graphing calculators<br />
• Textbook and notes (Glencoe Geometry—Ch 8)<br />
Procedure:<br />
I. Warm-Up activity: determine whether the lines given by equations a and b are<br />
parallel, perpendicular, or neither.<br />
a)<br />
3 y = x ! 2<br />
4 b) 3 x + 4y<br />
= 8 Answer: neither, since equation b is<br />
! 3<br />
y = x + 2 (not = or ⊥ slopes!)<br />
4<br />
II. Discuss properties of a parallelogram (refer to CabriJr. lab results)<br />
III. Introduce Sketchpad Rhombus lab. activity and outline objectives:<br />
16
• Each student will construct and examine a rhombus, following<br />
explicit directions and using Sketchpad tools. Students have prior<br />
experience with Sketchpad constructions (angles, lines & triangles).<br />
IV. Lab. Activity:<br />
Day#1: Complete sketches and record all data (measurements of angles,<br />
sides, diagonals).<br />
Day #2: Summarize and share findings, investigate properties of a square.<br />
V. Closure/Summary:<br />
Q: What properties are true for any parallelogram?<br />
Q: What makes a rhombus “special”?<br />
VII. Assignment: pp 434-5. Problems #1-10 and #26-31.<br />
Rhombus Discovery Activity<br />
Introduction:<br />
By definition, a rhombus is a parallelogram with four congruent sides. In<br />
this CabriJr lab. activity, you will construct a rhombus and discover its properties,<br />
make conjectures based on your findings, and then test your knowledge.<br />
Follow each direction carefully and record all findings. Be sure to save your final<br />
sketch in order to demonstrate your findings!<br />
#1. Draw and label segment AB.<br />
A<br />
B<br />
#2. Use the compass tool to draw a<br />
circle with center at point A<br />
and with radius AB.<br />
A<br />
B<br />
17
#3. Repeat step #2 process to draw<br />
another circle, with center at B and radius BA.<br />
#4. Plot a point at the intersection of the two circles. Label this point D.<br />
#5. Construct a line through •D that is parallel to AB.<br />
#6. Draw segment AD.<br />
#7. Construct a line through B, parallel to AD. Then plot a point at the<br />
intersection of •B and this line.<br />
D<br />
C<br />
A<br />
B<br />
#8. Construct segment DC.<br />
Hide the circles and line DC.<br />
D<br />
C<br />
A<br />
B<br />
18
#9. Measure and record the length of each side to verify that it is a rhombus.<br />
AB = ________, BC = ________, CD = ________, DA = ________.<br />
#10. Measure each angle: m∠ABC = ______, m∠BCD = ______,<br />
m∠CDA = ______, m∠DAB = ______.<br />
#11. Construct diagonals AC and BD.<br />
Plot point E at their intersection.<br />
D<br />
C<br />
#12. Measure the lengths of each segment<br />
formed by the diagonals.<br />
E<br />
DE = ______ EB = ______<br />
AE = ______ EC = ______<br />
#13. Measure the angles formed by the diagonals. Record your findings:<br />
A<br />
B<br />
m∠DEC = ______ m∠CEB = ______ m∠BEA = ______ m∠AED = ______<br />
Make a conjecture about the diagonals of any rhombus:<br />
______________________________________________________________<br />
______________________________________________________________<br />
Test your conjecture by dragging an angle. As the sides of the rhombus become<br />
larger (or smaller), does your conjecture about the angles hold true?<br />
How could you use your segment lengths (in steps #9 and #12) to verify your<br />
conjecture algebraically (ie. using a theorem or formula)?<br />
To show that your conjecture is true, demonstrate the algebraic steps below:<br />
19
Summary:<br />
Including your lab. findings (and previous knowledge about parallelograms), list as<br />
many attributes or properties of a rhombus as possible below. Be specific!<br />
20
Day #2:<br />
Now that you have constructed a rhombus, try to answer the following questions:<br />
• What is the definition of a rhombus?<br />
• How are the diagonals of a rhombus related?<br />
• What special triangles do you see on your constructed rhombus?<br />
• What kind of triangle congruencies do you see? (Be specific!)<br />
• How is a square related to a rhombus?<br />
• How do you think the diagonals of a square are related?<br />
Sketchpad directions:<br />
#1. Construct square ABCD.<br />
Does it have the properties of a rhombus? Measure the diagonals and angles<br />
to verify.<br />
What other special properties does it have? Make as many conjectures as<br />
possible about your square (include angles, diagonals, and triangles formed).<br />
______________________________________________________________<br />
______________________________________________________________<br />
______________________________________________________________<br />
_____________________________________________________________<br />
#2. Check your conjectures by measuring the angles, diagonal segments, etc..<br />
Then summarize your findings.<br />
21