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Transcript
Youngstown City Schools
MATH: GEOMETRY
Unit 1D: THEOREMS ABOUT TRIANGLES / PARALLELOGRAMS / CONSTRUCTIONS (2 WEEKS) 2013-2014
SYNOPSIS: In this unit, students continue their work with theorems and work specifically with triangles and parallelograms.
Students make constructions of triangles and parallelograms and relate these to real-life experiences. Finally, students apply the
concepts in the unit by plotting three con-collinear points and finding the point that is the shortest distance from the three points; then
find a fourth point that will form a parallelogram with the three non-collinear points.
STANDARDS
G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles
triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; the
medians of a triangle meet at a point
G.CO. 11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the
diagonals of parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
G.CO. 12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices,
paper folding, dynamic geometric software, etc.) Copying a segment; copying an angle; bisecting a segment; bisecting an angle;
constructing perpendicular lines; including the perpendicular bisector of a line segment; and constructing a line parallel to a given line
through a point not on a line
G.CO. 13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
MATH PRACTICES:
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning
LITERACY
L.2
Communicate using correct mathematical terminology
L.4
Listen to and critique peer explanations of reasoning
L.5
Justify orally and in writing mathematical reasoning
L.6
Represent and interpret data with an without technology
L.8
Read appropriate text, providing explanation for mathematical concepts, reason or procedures
L.9
Apply details of math readings/use information found in texts to support reasoning and develop a “works cited document” for research
done to solve a problem
6/30/2013
YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014
1
MOTIVATION
TEACHER NOTES
1. Teacher sets up connection of this unit to the previous ones on use of Theorems.
2. Teacher gives some type of pre-assessment on vocabulary terms to be sure students have
understanding of terms needed for the unit
3. Tom Reardon- TI Inspire activity. Can be done with TI Inspire Player or TI Inspire Calculator
http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5024&t=9123&id=13138
4. Preview expectations for end of Unit.
5. Have students set both personal and academic goals for this Unit or grading period.
TEACHING-LEARNING
Vocabulary Terms:
Midpoint
Median
Circumscribe
Centroid
TEACHER NOTES
Parallelogram
Isosceles Triangle
Quadrilateral
Equilateral Triangle
Inscribe
Bisector Angle
1.
Teacher asks students to draw different size triangles (on TI-Nspire, Geometer SketchPad, by
hand, or cut out triangles); find the measure of all 3 angles to discover sum will always be 180°.
(G.CO.10; G.CO.12) (MP-4; MP-6; MP-8)
2.
Teacher reminds students that if 2 sides of an isosceles triangle are congruent, its opposite
angles are also congruent; Teacher gives students a ruler and paper to construct an isosceles
triangle; students fold it to “prove” the theorem (p. 216 of Geometry text); students discuss
reasoning and proof of the theorem. (G.CO.10) (MP-1; MP-7; MP-8) (L.2; L.4)
3.
Teacher gives examples for students to work out in their notebooks/grid paper/grid board, for the
following: (G.CO.10)(MP-2; MP-4; MP-6) (L.5)
 An example for students to find the distance between two points: Distance Formula (round
to the nearest tenth)
( x2  x1 )2  ( y2  y1 )2
 An example for students to find the midpoint: Midpoint Formula
 x1  x2 y1  y2 
,


2 
 2
 An example for students to find the slope: Slope Formula
m
y2  y1 rise

x2  x1 run
4. Teacher provides three points to plot on a coordinate plane, connecting to form a triangle; Students
will follow along in notebooks or on grid paper/grid boards to draw a segment joining two sides of
the triangle, at their midpoint. (G.CO.10) (MP-5; MP-7) (L.5)
Students should use the midpoint formula, to find the midpoint of the two sides of the triangle.
Students will be instructed to connect the joining segment
Using the slope formula, students should recognize the slopes are the same and thus proving
6/30/2013
YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014
2
TEACHING-LEARNING
TEACHER NOTES
that they are parallel; Question to the students: How do we prove parallel?
5. Use OGT Released Test 2006, question 40 as a practice problem. Teacher gives more examples,
for students to work on. (G.CO.10; G.CO.12) (L.5; L.6)
6. Have students use Geometer SketchPad or TI-Nspire to create a triangle, its midpoints,
and then connect the vertex with the midpoint to show all 3 medians meet at a point. Have
the students choose a point and make new triangle. (G.CO.10; G.CO.12) (MP-5; MP-7)
7. Students will use coordinate grid paper to draw their own individual triangle. Instruct the students to
find the midpoints, using midpoint formula; connect the vertex to the midpoint. The three medians
will meet at a point. (G.CO. 10, G.CO. 12) (MP-4; MP-8)
8. Teacher introduces students to two different terms: parallelogram and quadrilateral. Give
examples, students take notes and offer practical examples. Have students look for figures in the
classroom, and discuss the attributes. (G.CO.11) (MP-4; L.2, L.4)
9. Give students coordinates for two different parallelograms; have students draw on grid paper. Have
students prove that the opposite sides are congruent. Ask the students: How do you prove two
lengths are congruent? Students construct and prove the two opposite sides are congruent.
(Students should use the distance formula to solve these problems). Discuss results with students.
(G.CO.11; G.CO.12) (MP-6; MP-8) (L.2; L.4)
10. Ask the students: What do we know about the angles A and B in parallelogram ABCD? Students
should recall consecutive interior, parallel, transversal, etc. leading them to the sum of the interior
angles of a parallelogram and the idea that angles of the line segment equal 180°…proving
opposite sides are congruent, opposite angles are congruent. Demonstrate for students that if
angle A = 30°, then angle B = 150°. Since angle B is 150°, then angle C must = 30°. Hence,
Angles A and C are congruent, each being 30°. If angle A + angle B = 180°, and angle B + angle
C = 180°, then angle A + angle B = angle B + angle C (subtraction); and angle A = angle C.
(G.CO.11) (MP-3; MP-4) (L.5; L.6)
B
C
A
D
11. Students will use Geometer SketchPad or TI-Nspire to construct a visual of proving opposite sides
and opposite angles of a parallelogram are congruent. Also spend time working on the converses
of these theorems. (G.CO.11; G.CO.12) (MP-5; MP-7) (L.5; L.9)
12. Use coordinate grid paper/grid board to create parallelogram. Draw the diagonals with E being the
intersection of the diagonals. Use distance formula to prove that line segment AE is congruent to
line segment EC and line segment DE is congruent to line segment EB. …proving diagonals
bisect each other. (G.CO.11; G.CO.12) (MP-3; MP-8) (L.5)
13. Extend Example to rectangle and prove diagonals are congruent by using the distance formula.
Use Geometer SketchPad or TI-Nspire to prove. (G.CO.11; G.CO.12) (MP-2; MP-8) (L.2)
14. Review the vocabulary terms: inscribe and circumscribed. Give examples, students take notes,
include practical examples. (G.CO.12; G.CO.13) (MP-1)
15. Using Geometer SketchPad or TI-Nspire, have students construct an equilateral triangle inscribed
in a circle. Then have students inscribe a square and regular hexagon in a circle. (G.CO.12;
G.CO.13) (MP-2; MP-3)
6/30/2013
YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014
3
TEACHING-LEARNING
TEACHER NOTES
16. Students create construction book or are familiar with all basic constructions listed below:
(G.CO.12, G.CO.13, MP.4, MP.5, MP.6, MP.7, L.2, L.5)
 Copy a segment
 Copy an angle
 Angle bisector
 Perpendicular bisector
 Line perpendicular to a given line through a point on the given line
 Line perpendicular to a given line through a point not on the given line
 Line parallel to a given line through a point not on the given line
 Divide a segment into a given number of congruent segments
 Equilateral triangle
 Regular hexagon
 Square
 Equilateral triangle, square, and regular hexagon inscribed in a circle
TRADITIONAL ASSESSMENT
TEACHER NOTES
1. Paper-pencil test with M-C questions and 2-and 4-point questions
TEACHER CLASSROOM ASSESSMENT
TEACHER NOTES
1. Quizzes
2. In-class participation and practice problems for each concept
AUTHENTIC ASSESSMENT
TEACHER NOTES
1. Students evaluate goals they set at beginning of unit or on a weekly basis.
2. Students solve a series of real-life geometry problems where they apply the distance formula,
work with perpendicular bisectors, mid-point and properties of parallelograms (attached on
page 5 and rubric on page 6) (G.CO.10; G.CO.11; G.CO.12; G.CO.13) (L.8; L.9)
6/30/2013
YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014
4
AUTHENTIC ASSESSMENT GEOMETRY UNIT 1D
THEOREMS ABOUT TRIANGLES, PARALLELOGRAMS, CONSTRUCTIONS
STANDARDS: G.CO.10, G.CO.11
You have been hired by a landscaper to create a drawing of a garden on graph paper using coordinate geometry.
The garden must have the following:
1. The garden is in the shape of a parallelogram with a possible area of 120 to 156 square boxes. Draw
the parallelogram and assign capital letters to the vertices and then assign coordinates to the vertices.
2. Use the distance formula to find the length of the sides of the parallelogram.
3. Determine if the drawing is truly a parallelogram. Support your answer with a theorem.
4. The garden is to have four walkways that are formed by connecting the midpoints of the adjacent sides
of the parallelogram. Find the midpoints and label with capital letters and coordinates. Then connect
them to form the walkways.
5. Find the lengths of the walkways using the distance formula.
6. Determine if the quadrilateral formed by connecting the four midpoints is a special type of
quadrilateral? If so, what type?
7. Next to the garden will be a triangular area for roses. Draw an isosceles triangle using one side of the
parallelogram as a side of the triangle. Label any points that have not been previously labeled on the
triangle with capital letters and coordinates.
8. Use the distance formula to prove that your triangle is isosceles.
9. The triangular area for roses is to be divided in half to accommodate two different types of roses.
Draw the perpendicular bisector to the base of the isosceles triangle to divide the triangle into two
equal parts. Label the point where the perpendicular bisector intersects the base of the triangle with a
capital letter and coordinates.
10. Find the length of the perpendicular bisector using the distance formula.
This is great. You now have helped your employer design a garden that he can build for his client.
6/30/2013
YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014
5
RUBRIC
ELEMENTS OF THE
PROJECT
0
1
3
Labeled 2 of the 4
vertices with capital
letters and
coordinates
Used the distance
formula, but had errors
in two of the
calculations
Determined if drawing
is parallelogram; did
not support with
correct theorem
Labeled 3 of the 4
vertices with capital
letters and
coordinates
Used the distance
formula, but had
errors in one of the
calculations
Determined if
drawing is
parallelogram; had
correct theorem but
did not explain it
correctly
Found and labeled 3
midpoints with
capital letters and
coordinates; then
connected midpoints
incorrectly
Drew parallelogram and
labeled the vertices with capital
letters and coordinates
Did not
attempt
Used the Distance Formula to
find the lengths of sides of
parallelogram
Did not
attempt
Determined if the drawing is
truly a parallelogram and
supported answer with a
theorem
Did not
attempt
Determined if drawing
is parallelogram; did
not support with
theorem
Found four walkways formed
by connecting midpoints of
adjacent sides of
parallelogram; labeled
midpoints with capital letters
and coordinates
Did not
attempt
Found and labeled 2
midpoints with capital
letters and
coordinates; then
connected midpoints
to form a walkway
Used distance formula to find
the lengths of walkways
Did not
attempt
Found and labeled
midpoints with capital
letters or coordinates;
did not connect
midpoints of adjacent
sides to form
walkways
Did not use distance
formula to find length
of walkways
Determined if quadrilateral
formed was a special type of
quadrilateral and supported
with theorem
Did not
attempt
Attempted to
determine if special
quadrilateral was
formed; no theorem
Determined if special
quadrilateral was
formed, but could not
cite a theorem
Drew isosceles triangle and
label points with letters and
coordinates
Did not
attempt
Drew isosceles
triangle, and labeled
points or coordinates
Use Distance Formula to prove
triangle is isosceles
Did not
attempt
Drew isosceles
triangle, but did not
label points or
coordinates
Used wrong formula.
Draw perpendicular bisector
and label with capital letter and
coordinates
Did not
attempt
Use the distance formula to
determine the length of the
perpendicular bisector
Did not
attempt
Drew perpendicular
bisector, but did not
label point or give
coordinates
Did not know distance
formula, and
attempted to use
another strategy
Drew perpendicular
bisector, labeled either
the point or give
coordinates
Used distance formula
but with multiple
calculation errors
6/30/2013
Labeled 1 of the 4
vertices with capital
letters and
coordinates
Did not use the
distance formula to
find lengths of sides
2
Used distance formula
but had multiple errors
in calculating length of
walkways
Used distance formula
but had error in
calculations
Used distance
formula but had
minor errors in
calculating length of
walkways
Determined if
quadrilateral was
formed, and cited a
theorem but there
was an error
NA
Used distance
formula, but did not
connect result to
isosceles triangle
NA
Used distance
formula but had one
calculation error
YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014
4
Labeled vertices with
capital letters and
coordinates
Used the distance
formula and found the
length of sides
correctly
Determined if drawing
is parallelogram;
supported with correct
theorem and
explanation
Found and labeled the
4 midpoints with
capital letters and
coordinates; then
connected midpoints
of adjacent sides to
form walkways
Used distance formula
to find the lengths
Determined if
quadrilateral was
formed and gave a
theorem for support
Drew isosceles
triangle, and labeled
points and coordinates
Used distance formula
correctly and proved
triangle was isosceles
Drew perpendicular
bisector and labeled
point correctly and
listed coordinates
Used distance formula
correctly
6