Rebecca
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Fosdick—Geometry—exterior/interior angles
Good to give prerequisite information
Warm-up problems on the board
Very good problems/questions in the warmup!
Nice handling of calling on Jennifer to respond
o Is True…but why? More than just the definition of
alternate exterior angles. In essence, this is the chance
for the students to explain or clarify WHAT exterior
angles are, WHAT alternate angles are, WHAT
consecutive angles are, etc.
o Good that you immediately ask “why” and that you give
time for them to think about their response
o Good that you used colors to draw in the “Z” principle—
this is the chance with students to review their
understanding of the Z principle.
o You are having them reason through the questions, but
you don’t seem to have your understanding of the
warm-up responses clarified (maybe just because
you’re fatigued)
NOTES you gave:
Triangle:
 Def: a triangle is a figure formed by three line segments
connected by three non-collinear points. (this is redundant).
 A triangle is a plane figure formed by the three line segments
connecting three non-collinear points.
 “Two ways of classifying triangles:”
Classification by angle:
 ACUTE triangle is a triangle whose interior angles are all acute
 OBTUSE triangle is a triangle that has 1 obtuse interior angle.
 RIGHT triangle is a triangle that has 1 right interior angle.
Careful—you know the content—be prepared with your wording of how
you present definitions
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Very good—you had a good way of helping Jennifer think
through her question about more than one obtuse angle in a
triangle. Very visual—can you draw the connection to a
numerical argument as well?
Here’s where the presentation medium will help. Rather than
drawing one triangle at a time, which takes whole group time,
have a host of triangles ready-drawn on a sheet and let them
identify which ones are which…then, they can be reused for
the other classification. Also, you are more inclined to think
about and include more subtle examples when preparing
them ahead of time rather than just drawing them on the
overhead.
 Question: Are isosceles triangles acute? Response: “Well,
actually that is our next one” What is ONE?  “part of our
next classification scheme”
 Actually, the question about “isosceles” brings up an
interesting issue: Even though these students are in
geometry in HS, they will have seen and heard about these
terms before. They are covered in the elem and middle
school curriculum just to recognize shapes and properties
about those shapes. So, they are not starting from scratch.
In other words, this lesson may go quickly.
 Classification by Sides:
 Isosceles triangle is a triangle that has at least 2 congruent
sides.
o Be careful about the “at least” you might want to
discuss the importance of this nuance phrase!
o Some texts will say “exactly 2 congruent sides” rather
than “at least”. I prefer “at least” myself.
 Question: What if the triangle were upside down? Good
diagram.
o This is a time that you can emphasize that “bottom”
and “top” are not mathematical terms, but relative to
where it is positioned.
o Question about base. Actually, it is the same as the
base of a jar…if the jar is lying on its side, it is still a
base.
Language: “The other two…classification by sides…” The other
two types of triangles in the classification by sides are: “…
Rebecca,
Overall you have a good lesson and idea of how to introduce the topic. Because
you have been “fatigued”, I am not sure that it went exactly the way that you
wanted it to go. Many things are very good and they should be highlighted:
First of all, your warmup problems are good selections for the content that they
cover. They are not just straightforward computational problems, but ask them to
think about and verify in their own minds the validity of certain statements about
exterior, interior, alternate and corresponding angles. My question is how does
this connect to the lesson that is forthcoming on triangles. Not every warm-up
will connect so easily and it may merely be a review of what was done previously
before embarking on a new topic. However, this one can connect. The
understanding of angles formed when transversals cut parallel lines leads to
understanding the parts of a triangle. Your two classification schemes are
classifying triangles by lengths of sides and by types of angles. Use the natural
connections there.
The body of your lesson focuses on clear definitions of various types of triangles
and being able to distinguish between them. You showed careful attention to the
terminology and precision needed in forming definitions. With students, you
might need to spend a little more time emphasizing the nuances of the definitions
and why the precision is needed. E.g., the difference between “at least” and
“exactly” is not one that students immediately see the need for in mathematics.
The concept is not hard to attain, but the importance is not immediate for them.
You had a good interchange with Jennifer responding to “can a triangle have two
obtuse angles?” You described it quite well and gave a visual explanation. Now,
take the time to emphasize and make the connection to a numerical explanation.
If Angle A > 90 and if Angle B > 90, then …
You handled this interchange very well, and it gives an opportunity for us to think
about how to adapt your lesson. Knowing that students will have some
experience with this terminology prior to the geometry class, the intro lesson
should not take long. You can deepen the lesson by asking THEM questions
such as the ones that Jennifer asked you. Get them to think outside the box, and
ask them questions to clarify their understanding. E.g., Can you have a right
triangle that is equilateral?
Overall, good job on the lesson and the presentation, even though you were very
tired. I am most worried about this tiredness in terms of your consistency next
semester in student-teaching. I hope that you are making plans for how to handle
this on a regular basis. On the positive side, you showed that you have good
content knowledge, are able to handle questions effectively, and that you ask
thoughtful problems/questions of the students.
Teaching Feedback Form
MAT 764, Fall 2005, Hendrix
Teacher:
Rebecca Fosdick
Activity:
Classification of Triangles
Category
Comments
Organization—
Warm-up prepared, planned
ideas of what to do with lesson,
but had not thought all the way
through the lesson. Just had
basic definitions and outline of
what to do, it seemed.
Warm-up review questions were
very good ones. The content of
the lesson is very worthwhile
mathematically, classifying the
types of triangles from two
different perspectives. The
objectives weren’t completely
clear. What big ideas do you
want them to take from the
lesson?
Very calm and clear leadership
as a teacher. Able to explain
ideas and to respond to student
questions well. Used terminology
precisely. Could emphasize the
nuances of the terminology
more.
Pace was a little slow for this
lesson. Most of that comes
from more planning.
Evidence of planning, thought,
preparedness, materials ready, etc.
Mathematically Worthwhile
Task—Does the activity help
students address an important
mathematics concept in a meaningful
way? Are the objectives for the
activity clear? Is the mathematics
treated accurately in the activity?
Delivery of the lesson —
Classroom leadership, presence in
teaching a lesson and leading
students in instruction, clear
instructions/explanations, precise use
of terminology, etc.
Time Management —
Pace of the lesson; were the goals of
the intended lesson/activity met?
Overall Comments
Provider of feedback:
Overall, good lesson and good
ideas. You are starting to
develop your teacher
personality. More planning and
preparation is needed.
Dr. Hendrix
Presentation Grade:
B