Download Unit 1C 2013-14 - Youngstown City Schools

Youngstown City Schools
MATH: GEOMETRY
Unit 1C: THEOREMS ABOUT LINES AND ANGLES (2 WEEKS) 2013-2014
SYNOPSIS: In this unit, students learn about theorems and proofs as they relate to angles and lines. Students learn the technology
as well as the hands-on methods for solving problems that involve real-world applications of the concepts in the unit. Students apply
their learning through taking a bridge or a section of city streets and addressing the concepts from the unit and explaining each.
STANDARDS
G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel
lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line
segment are exactly those equidistant from the segment’s endpoints.
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 STANDARDS
L.1
L.2
L.6
Learn to read mathematical text - - problems and explanations
Communicate using correct mathematical terminology
Represent and interpret data with and without technology
MOTIVATION
TEACHER NOTES
1. Teacher sets up connection of this unit to the previous one on use of Theorems, moving from
congruent triangles to theorems about lines and angles.
2. Teacher gives some type of pre-assessment on vocabulary terms to be sure students have
understanding of terms needed for the unit.
3. Preview expectations for end of Unit.
4.
Have students set both personal and academic goals for this Unit or grading period.
TEACHING-LEARNING
Vocabulary Terms:
bisector
perpendicular
equidistant
supplementary angles
congruent
vertical angles
parallel
corresponding angles
TEACHER
NOTES
transversal
line segment
complementary angles
alternate interior angles
right angles
alternate exterior angles
Teachers: if needed, take one day to focus on teaching students how to use the Geometer
SketchPad or the TI-N-spire technology
6/30/2013
YCS Geometry: Unit 1C: Theorems About Lines and Angles 2013-2014
1
TEACHING-LEARNING
TEACHER
NOTES
All Teaching Learning Activities refer to G. CO.9
1. Through self-discovery or teacher explanation inform students what happens when parallel lines are
cut by a transversal; special Theorems are created because of these relationships (use Geometer
Sketchpad or Geogebra). (MP.5, MP.7, MP.8, L.2)
2. Teacher models a transversal on the board, and then provides students with handouts containing
parallel lines; students are to cut the lines with a transversal that is NOT perpendicular to the parallel
lines and label the angles formed. Students then measure and record each angle measure to
discover the relationship among the angles. Students may use a protractor or the TI-Nspire geometry
graphing program. (MP.2, MP.4, MP.5, MP.6, MP.7, MP.8, .2)
3. From previous activities, the teacher helps students label the angles as either congruent (alternate
interior, alternate exterior, vertical angles, corresponding angles) or supplementary angles
(consecutive interior, consecutive exterior, or adjacent angles). (MP.4, MP.5, MP.6, MP.7, MP.8, L.2)
4. The teacher provides students with sample problems (pp. 127-137 from text or supplementals),
stressing “real-world” applications of various shapes in the sample problems; students practice “angle”
and “relationship” skills; students supply missing information, justify answers, and use algebraic
expressions to represent missing information. Be sure to balance use of technology with hands-on
solving. (MP.2, MP.4, MP.5, MP.6, MP.7, MP.8, L.2)
5. Teacher demonstrates proving theorems about parallel lines and transversals using a two-column
proof method (e.g., p. 152 of text). Students take notes; teacher provides practice, giving students the
“given” and “what’s to be proven,” and students work with the teacher to complete the proof. (MP.1,
MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.6)
6. Students work in 2’s to complete proofs for other problems-situations using 2-column method to be
sure the students understand what they are doing. (MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.6)
7. Next the teacher models the narrative method for completing a proof, and finally moves to the flow
diagram method. This is done using demonstration problems from Activity #6, so students see how
the various methods work. Students work one problem, each, using the narrative and flow chart
methods. (MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.6)
8. Teacher provides real-world problem for the idea of a line segment with a bisector. Tampa Bay
Bridge: http://player.discoveryeducation.com/index.cfm?guidAssetId=6566F746-4FE4-4759-A15D5BA007CCE5C4&blnFromSearch=1&productcode=US. Teacher does demonstration of
perpendicular bisector of a line segment to show students that when this happens the two parts of the
line segment are equal. Students then use a protractor as a tool to prove each point on the two
halves of the line segment are equidistant from the bisector. (MP.2, MP.3, MP.4, MP.5, MP.7, MP.8,
L.2, L.6)
9. Teacher and students use SAS to prove that points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment’s endpoints: e.g., S → to show that the 2 pieces of the
line segment cut by the perpendicular bisector are congruent; A → to show that the angles formed by
the perpendicular bisector are congruent right angles; S → to show that the perpendicular bisector
creates a side for each of the triangles formed and that they are congruent. (MP.1, MP.2, MP.3,
MP.4, MP.7, MP.8, L.2, L.6)
6/30/2013
YCS Geometry: Unit 1C: Theorems About Lines and Angles 2013-2014
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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
1.
TEACHER NOTES
Students evaluate goals they set at beginning of unit or on a weekly basis
2. Sunshine Skyway Bridge in St. Petersburg, FL with United Streaming Video
3. My City of Angles – where students use concepts in the unit to create angles, lines, etc.
as per a given set of instructions below and on page 4 of Unit Plan. (G.CO.9, MP.1,
MP.4, MP.5, MP.6, MP.8. L.1, L.2)
AUTHENTIC ASSESSMENT: MY CITY OF ANGLES
Choose a city:
1.
2.
3.
4.
Find a map of your city on the internet,
Zoom in until the streets are visible,
Look for a section of town that has some parallel streets intersected by transversals, at least a total of 8 streets.
Chose an area where not all transversals are perpendicular to the parallel streets, at least one needs to be slanted
at an angle different than 900 and at least one needs to be perpendicular.
5. Print this area of town.
6. On a piece of graph paper with a straight-edge, draw this section of city.
7. Label intersection points with capital letters and number angles.
On your City Map Section - - List and identify the following:
4 pairs of vertical angles
4 pairs of corresponding angles
4 pairs of alternate interior angles
4 pairs of supplementary angles
4 pairs of complementary angles
4 pairs of alternate exterior angles
Use a protractor to find the angle measurements of all angles and show them on your drawing.
Create and answer a question to solve for a missing value (x) using alternate interior angle or same side
interior angle relationship between the angles in the city.
6/30/2013
YCS Geometry: Unit 1C: Theorems About Lines and Angles 2013-2014
3
ELEMENTS OF THE
PROJECT
Printed area of city that has
parallel streets intersected
by transversals ( at least 8
streets) that are both
slanted and perpendicular
CITY SECTION RUBRIC
2
0
1
3
4
Did not
attempt
Used section with12 streets; had
streets intersected
by either
perpendicular or
slanted transversal
Did not use straightedge, drawing; does
not resemble section
of town
Labeled only a few
of the angles and
points of intersection
Correctly listed 1
pair of vertical
angles
Used 3-5 streets; had
streets intersected by
either perpendicular or
slanted transversal
Used section with 57 streets; a
perpendicular or
slanted transversal
NA
NA
At least 8 streets, at
least one
perpendicular, at
least on intersecting
at an angle different
than 900
Used straight-edge,
drawn in likeness to
the selection of town
Labeled half of the
angles and points of
intersection
Correctly listed 2 pairs
of vertical angles
Labeled most of the
angles and points of
intersection
Correctly listed 3
pairs of vertical
angles
All points and angles
are labeled correctly
Correctly listed 1
pair of
corresponding
angles
Correctly listed 1
pair of alternate
interior angles
Correctly listed 2 pairs
of corresponding
angles
Correctly listed 3
pairs of
corresponding
angles
Correctly listed 3
pairs of alternate
interior angles
Correctly listed 4
pairs of
corresponding
angles
Correctly listed 4
pairs of alternate
interior angles
Correctly listed 1
pair of
supplementary
angles
Correctly listed 1
pairs of
complementary
angles
Correctly listed 2 pairs
of supplementary
angles
Correctly listed 3
pairs of
supplementary
angles
Correctly listed 3
pairs of
complementary
angles
Correctly listed 4
pairs of
supplementary
angles
Correctly listed 4
pairs of
complementary
angles
Listed 4 pairs of alternate
exterior angles
Did not
Correctly listed 1
attempt or pair of alternate
all were
exterior angles
incorrect
Correctly listed 2 pairs
of alternate exterior
angles
Correctly listed 3
pairs of alternate
exterior angles
Correctly listed 4
pairs of alternate
exterior angles
Find measures of all angles
on drawing
Did not
attempt or
all were
incorrect
Did not
attempt
Found measures of
1-2 angles on the
drawing
Found measures of 35 angles on the
drawing
Found measures of
6-7 angles on the
drawing
Found measures of
8 angles on the
drawing
Created problem,
did not find the
solution
Created problem, but
solution was incorrect
NA
Created problem
and found correct
solution
Prepared drawing of the
section of city with straightedge
Did not
attempt
Labeled intersection points
with letters and angles with
numbers
Listed 4 pairs of vertical
angles
Did not
attempt
Listed 4 pairs of
corresponding angles
Listed 4 pairs of alternate
interior angles
Listed 4 pairs of
supplementary angles
Listed 4 pairs of
complementary angles
Created problem and found
solution
6/30/2013
Did not
attempt or
all were
incorrect
Did not
attempt or
all were
incorrect
Did not
attempt or
all were
incorrect
Did not
attempt or
all were
incorrect
Did not
attempt or
all were
incorrect
Correctly listed 2 pairs
of alternate interior
angles
Correctly listed 2 pairs
of complementary
angles
Correctly listed 4
pairs of vertical
angles
YCS Geometry: Unit 1C: Theorems About Lines and Angles 2013-2014
4