Download LESSON OVERVIEW

Geometry Concept Lesson Unit 1: Amazing Amanda
2008 - 2009
LESSON OVERVIEW
Overview: Students explore the sum of the interior angles of polygons with an increasing number of sides (beginning with three) and, using
inductive reasoning, discover the formula for the sum of the interior angles of an n-gon.
After an initial discussion involving the interior angle sum of triangles, the students work in groups of four to collaboratively develop a
formula based upon drawing polygons, splitting them into non-overlapping triangles, writing the information on a group graphic organizer
and looking for patterns in the resulting data. This hands-on investigation is followed by three questions requiring full explanations to check
the students’ understanding.
Note: Students may possibly use one of two correct methods to split a polygon into non-overlapping triangles. The intent is that the students
are NOT shown either of these methods but come to one of these conclusions while investigating in their groups.
1. An n-gon is split into n triangles by
connecting all vertices to a point inside
the polygon. If students use this method
they should develop a formula based on
n • 180 − 360 , i.e., the sum of the
angles in all n of the triangles less the
360 degrees of the sum of the angles
around the interior point.
2. An n-gon is split into (n-2)
triangles by choosing a single vertex
and drawing lines to the other
vertices. If students use this method
the should develop a formula based
on (n − 2) • 180 , i.e., the sum of all of
the angles in all (n-2) of the triangles.
CA Standards Addressed: 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify
figures and solve problems.
Mathematical Goals of the Lesson:
• Students develop their own theorem involving the interior angle sum of polygons
• Students develop their mathematical reasoning skills:
o Pattern recognition and inductive reasoning
o Making connections between different representations of ideas (algebraic formula to geometric shape)
Academic Language Goals of the Lesson:
• Develop academic vocabulary to be used in the verbal and written explanations.
• Describe the conditions in a situation algebraically, verbally or in writing.
• Explain the process used in solving the task, verbally or in writing.
Template developed by the LEARNING RESEARCH AND DEVELOPMENT CENTER
© LAUSD Secondary Mathematics 2008
page 1 of 7
Access Strategies: Throughout the document you will see icons calling out use of the access strategies for English Learners, Standard English Learners,
and Students With Disabilities.
Access Strategy
Cooperative and Communal Learning
Environments
Instructional Conversations
Academic Language Development
Advanced Graphic Organizers
Template developed by the LEARNING RESEARCH AND DEVELOPMENT CENTER
Icon
Description
Supportive learning environments that motivate students to engage
more with learning and that promote language acquisition through
meaningful interactions and positive learning experiences to achieve
an instructional goal. Working collaboratively in small groups, students
learn faster and more efficiently, have greater retention of concepts,
and feel positive about their learning.
Discussion-based lessons carried out with the assistance of more
competent others who help students arrive at a deeper understanding
of academic content. ICs provide opportunities for students to use
language in interactions that promote analysis, reflection, and critical
thinking. These classroom interactions create opportunities for
students’ conceptual and linguistic development by making
connections between academic content, students’ prior knowledge,
and cultural experiences
The teaching of specialized language, vocabulary, grammar,
structures, patterns, and features that occur with high frequency in
academic texts and discourse. ALD builds on the conceptual
knowledge and vocabulary students bring from their home and
community environments. Academic language proficiency is a
prerequisite skill that aids comprehension and prepares students to
effectively communicate in different academic areas.
Visual tools and representations of information that show the structure
of concepts and the relationships between ideas to support critical
thinking processes. Their effective use promotes active learning that
helps students construct knowledge, organize thinking, visualize
abstract concepts, and gain a clearer understanding of instructional
material.
© LAUSD Secondary Mathematics 2008
page 2 of 7
Assumption of Prior Knowledge:
• Definition of a polygon (students must also
draw polygons)
• Knowledge of types of polygons and their
characteristics, including regular and irregular
polygons.
• Sum of interior angles of a triangle = 180 º
• Sum of angles in a circle = 360 º
Academic Language:
polygon
names of polygons
• triangle
• quadrilateral
• pentagon
• hexagon
• heptagon
• n-gon
• interior angle
• exterior angle
• regular figure
• irregular figure
Materials:
•
•
•
•
•
Task
Recording Sheet (provided in task)
Straight-edge
Pencil
Paper and scissors for triangle introduction
activity (if necessary)
Connections to the LAUSD Algebra 1, Unit 1, Instructional Guide
Understand
polygons and
angles
12.0, 13.0, 16.0
• Construct the bisector of an
angle
• Construct an angle congruent to
a given angle
• Solve problems using angle and
side measures for triangles and
polygons
• Prove relationships between
angles in polygons
Template developed by the LEARNING RESEARCH AND DEVELOPMENT CENTER
© LAUSD Secondary Mathematics 2008
page 3 of 7
Students use reading strategies to decode
key vocabulary
Key: Suggested teacher questions are shown in bold print.
*Note: Questions and strategies that support ELs are underlined identified by an asterisk.
Possible student responses are shown in italics
Phase
S
E
T
U
P
SET UP PHASE: Setting Up the Mathematical Task—Part 1
INTRODUCING THE TASK
• Distribute page one of Amazing Amanda to the class
• Have a student read the first two paragraphs of Amazing Amanda
• Ask for any words or terms that the students do not understand. Clarify the meanings of the words or terms for the students, using
visuals where possible, or having other students provide answers (for example have a student come to the board and draw a polygon
of their choosing and have another student label on the same diagram their understanding of an interior angle). *A Frayer model
could be used to assist students in developing a clear understanding of the term “Interior Angles”.
• Have a student read “Getting Started: Triangles”
• Give the students individual time to think about this question and begin a class discussion
a) If the students are very familiar with the answer to this question (i.e., the sum of the angles in any triangle is 180 degrees)
then have them individually fill out the table in part 2) and move into the main task itself distributing page 2 “Next Step:
Investigating polygons”
Angle A
Angle C
b) *If the students are NOT able to easily answer the getting started
Angle B
question then it may be appropriate (especially for EL, SEL and SWD)
to spend time doing an additional hands on activity to reinforce
concept. Pass a piece of blank paper to each student and have them
draw a point anywhere on the top edge of the paper. Connect the point
to both the bottom corners of the paper and label the angles a, b and c
(see diagram). Now have the students either cut down the two lines or
fold and tear the paper along the lines. Have the students align both
angle A’s and angle C’s together and ask them, “What do you
notice?” “Why do they think this works?” Draw their attention to the
idea that the piece of paper has parallel lines along the top and bottom
Angle C
Angle A
and thus develop the idea that the angles in the triangle are congruent to
the three that lie along the line which is the top of the paper. Hence the
sum of the angles in any triangle is 180 degrees.
To assist ELLs’ participation in the class discussion*:
• Allow time for students to first talk in small groups (pairs) and then have the groups report to the whole class.*
• Reinforce appropriate language as students communicate their ideas (e.g. re-voice a student’s contribution in complete, grammatically
correct language). Ask students if you have captured what they said*.
• Create work groups that are heterogeneous according to language proficiency*
• Model appropriate mathematical language, emphasizing vocabulary used in appropriate context.*
Template developed by the LEARNING RESEARCH AND DEVELOPMENT CENTER
© LAUSD Secondary Mathematics 2008
page 4 of 7
The Group Recording sheet helps students’
organize their work to see the inductive pattern
Students work in small groups to explore the sum of the interior angles of various polygons
Phase
E
X
P
L
O
R
E
E
X
P
L
O
R
E
E
X
P
L
O
R
E
E
X
P
L
O
R
E
Small group discussion guided by teacher questions
EXPLORE PHASE: Supporting Students’ Exploration of the Task
STRUCTURE
PRIVATE THINK TIME
• Distribute page two of Amazing Amanda. Students should be organized into collaborative groups of four. Have a student read the
task directions
• Ask for any words or terms that the students don’t understand. Clarify the meanings of the words or terms for the students.
• Assign a student to be individually responsible for completing the table for one of the polygons with 4, 5, 6 or 7 sides.
• Ask students to work individually for about 15 minutes so that they can make sense of the problem for themselves.
• Circulate around the classroom and clarify confusions. Be careful to NOT give away too much information or suggest conjectures.
SMALL GROUP (4 students) WORK
• After allowing 15 minutes for individuals to finish their tables distribute one copy of page three of the task to each group, the
“Group Recording Sheet”
• Using a “round robin/roundtable” collaborative strategy have the students transfer their work from their individual tables to the
group recording sheet, explaining their thinking to the group as they do so.
• Distribute page four of Amazing Amanda and allow students time to discuss the questions in their groups
• As students are working, circulate around the room.
o Be persistent in asking questions related to the mathematical ideas (see question suggestions in the following section),
exploration strategies, and connections between representations.
o Be persistent in asking students to explain their thinking and reasoning.
o Be persistent in asking students to explain, in their own words, what other students have said.
o Be persistent in asking students to use appropriate mathematical language.
*What do I do if students have difficulty getting started?
Ask them to draw their assigned polygon. Ask, “How many sides does your polygon have?” “How can we use the sum of the
angles in a triangle to figure out the sum of the interior angles of this polygon?” “How many non-overlapping triangles are
contained in your polygon?”
What do I do if students finish early?
• Challenge students to investigate whether their formula works for both concave and convex polygons.
• Ask a “quiet” group member to explain the work of the group. If s/he can’t explain, challenge the group to make sure that that each
group member understands, leave, and return later to ask again.
MONITORING STUDENTS’ RESPONSES
• As you circulate, attend to students’ mathematical thinking and to their conjectures, in order to identify those responses that will be
shared during the Share, Discuss, and Analyze Phases.
• Identify students who can correctly explain and demonstrate a formula for the sum of the measurements of the interior
angles of any polygon. Find students who used different methods, including different triangulation methods, and find
students with different formulas.
Identify a group member of several different groups that can explain the answers to each question.
Template developed by the LEARNING RESEARCH AND DEVELOPMENT CENTER
© LAUSD Secondary Mathematics 2008
page 5 of 7
Phase
Possible Solutions
E
X
P
L
O
R
E
• Correctly drawn polygons partitioned into
non-overlapping triangles
E
X
P
L
O
R
E
• Discussion of the patterns in the “Group
recording Sheet” table (i.e., as you circulate
you should hear students noticing that the
pattern of triangles is increasing by one as
they move down each row)
Look for indictors of students’ effective
exploration:
Look for indicators of students’
understanding:
Possible Questions
Ask questions such as:
• How many sides does your assigned
polygon have?
• Can you draw this polygon with
your straight-edge?
• Did you draw a regular or irregular
polygon?
• How can you partition it into nonoverlapping triangles?
• How many triangles are there?
• How does this relate to the sides?
• How can you use the triangles to
compute the total number of degrees
in the interior angles of your
polygon?
• What is the total number of degrees
in the interior angles of your
polygon?
• Look at the group table-are there
any patterns?
• Does your formula work for a
triangle and other polygons in your
table?
• Does everyone in the group have an
equivalent formula?
Misconceptions/
Errors
Questions to Address
Misconceptions/Errors
• Not using
inductive reasoning
• Not understanding
the definition of
interior angles
• Not partitioning
the polygons
correctly.
• Overlapping
triangles
• If students
partition the
polygon using a
central point (see
overview) they may
not subtract the
angles around the
central point in their
formula
• Not writing a
correct formula
relating number
of sides to the
sum of the interior
angles
• Can you see any patterns in
your “Group Recording
Sheet”?
• Can you show me an
interior angle in your
diagram?
• How did you decide to split
the polygon into triangles?
• Are any of your triangles
overlapping?
•
• Does your formula work
for triangles, and all the
other polygons in your table?
• Are all of the interior
angles included in your
partitioning?
*Explain in your own words what
_______(another student) said.
Template developed by the LEARNING RESEARCH AND DEVELOPMENT CENTER
© LAUSD Secondary Mathematics 2008
page 6 of 7
Sharing, Discussing, and Analyzing
Orchestrating the mathematical discussion: a possible Sequence for sharing student work, Key Questions to achieve the goals
of the lesson, and possible Student Responses that demonstrate understanding.
Revisiting the Mathematical Goals of the Lesson:
The purpose of this sharing/discussion is to make explicit the conclusions of the exploration.
Phase
S
H
A
R
E
D
I
S
C
U
S
S
A
N
A
L
Y
Z
E
Sequencing of Student
Work
Rationale and
Mathematical Ideas
The work of at least two
groups should be chosen.
If possible choose work
from one group that
partitioned their polygon
by using an interior point
and another group that
partitioned their polygons
from a single vertex.
All students should develop a
rich understanding of the
polygon interior angle sum
formula.
If the work can be
displayed, then have the
class look at the students’
work, without any
explanation, and ask
students other than those
in the contributing groups
how they think the
problem has been solved.
Students should be able to
make connections between
different solution paths.
Students should be able to
explain where inductive
reasoning is occurring in the
solution process.
If it is not possible to
display the work to the
whole class, have groups
present and explain their
own solutions.
Template developed by the LEARNING RESEARCH AND DEVELOPMENT CENTER
Possible Questions and Student Responses
How did you partition your polygon to determine the sum of the
interior angles?
• I used diagonals from one vertex to partition the polygon into
triangles and used the fact that triangles have 180º to determine the
sum.
• I used triangles from a central point with each side of my polygon
as the base of each triangle but I had to subtract 360º from my sum.
What patterns did you notice?
• I always had 2 less triangles than the number of sides
• I had the same number of triangles as sides but I had to subtract
360 which is like subtracting two triangles
How did you generate a formula?
• I used a variable for the number of sides and subtracted 2 then
multiplied by 180.
• I used a variable for the number of sides then multiplied by 180
and subtracted 360.
Why did you subtract 360°?
• Because the angles around the central point are not interior
angles of the polygon. And, since these angles form a circle, I knew
that the sum of their measures was 360°.
Are these two formulas equivalent? Why or why not?
• S=180n-360 and S=180(n-2) are equivalent if you distribute the
180 across the parenthesis then you have 180n – 360.
• Subtracting 360 is like subtracting two triangle totals from the
sum.
What are the advantages/disadvantages of each formula?
• S= 180(n-2) is simpler to use.
© LAUSD Secondary Mathematics 2008
page 7 of 7