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MED 308
Matthew Garozzo
Week 6
10/15/07
Lesson: Sides of an Isosceles Triangle Distance and (or Isosceles, Scalene, or
Equilateral)
Materials: white board, markers, rulers and protractors, graph paper (provided)
Lesson Overview: In this lesson we will use isosceles triangle sides to solve algebraic
expressions. By doing what?
Lesson Objective:
-The students will identify that the sides of an isosceles triangle are congruent.
This is from the Comprehension level of Bloom’s Taxonomy. Find a word from the
Analysis or Synthesis level to replace it.
-The students will learn how to solve algebraic expressions using the sides of an
isosceles triangle.
This is at the Application level of Bloom’s Taxonomy. Find a word from the
Analysis or Synthesis level to replace it.
Standards:
These need to be from the KEY IDEAS in Math A.
-G.G.29: Identify corresponding parts of a congruent triangle
Anticipatory Set: (10-15min) it’s Monday so we will play it easy in the beginning with
some recollection. Have the students identify what makes an isosceles triangle,
equilateral triangle and a scalene triangle all different, and more specifically refer to the
sides that are special about each i.e. a scalene has all three sides that are a different
length….
This should not take that long – it should be no more than 5 minutes. Remember
that class is only 40 minutes long!
Developmental Activity: (30-40min See above.) Alright we will be playing a game with
the students called: Name That Triangle!!! Give the students 1 or two sheets of graph
paper and tell them to make it into 4 separate graphs. No, you need to have the graphs
already drawn on the graph paper for the students – having the students do it
wastes time with a task that isn’t part of the objectives. You can draw the graphs on
a master sheet and then make copies of it. Each graph will have an origin and X and Y
axis going from -10 to 10. We are going to give the students three ordered pairs. The
students are going to take those three coordinates and determine if the triangle formed is
scalene, or isosceles based on the lengths of the triangle. They will have to plot the points
and measure the segments formed to get the lengths between the coordinates We could
make a quick graph on the board and use it do demonstrate the problem at hand. After a
triangle is constructed have them:
1st: take a guess
2nd: measure the angles
3rd: measure the sides
4th: compare answers and explain why they got what they got
Here are the coordinates and it might be wise to go in a different order, let the games
begin:
#1 A=(-2,0) #2 A=(1,-9) #3 A=(2,4)
#4 A=(-3,-3)
B=(2,0)
B=(0,-3)
B=(5,7)
B=(-8,-5)
C=(0,-3)
C=(-4,-5)
C=(8,4)
C=(-1,-8)
I
S
I
I
#5 A=(-8,1) #6 A=(2,-2)
B=(-2,-3)
B=(8,-3)
C=(4,-1)
C=(-6,7)
S
S
#7 A=(1,0)
B=(-1,10)
C=(-6,-3)
S
Closure: If the students are getting the correct responses according to our cheat sheet you
know they understand that an isosceles triangle has two congruent sides. And two
congruent angles!
This should be a SUMMARY activity. Ask if the students noticed anything about
all of the isosceles triangles. Allot some time for this to occur.
Assessments: students will write down what they learned about isosceles triangles. Is this
an Exit Ticket? Will it be collected? If so, allot some time for the students to write
down their responses and for the instructors to collect them.
Group: Greg Qualiana, Brittany Ponivas, Matthew Garozzo
Title: Isosceles Triangles
Date: Tuesday October 16, 2007
Grade Level: Math A, Grade 10
Materials: Handouts, writing utensil, notes, paper
Lesson Overview: Students will work with Isosceles triangles to develop an
understanding of angles. By doing what?
Lesson Objectives: By the end of the lesson students will be able to evaluate
equations to determine the measures of angles. Students will be able to
demonstrate their understanding of Isosceles triangles by filling in blank angles in
a diagram or propose an answer by evaluating equations.
Both words are from the Application level of Bloom’s Taxonomy. Find one
word from the Analysis or Synthesis level to replace one of the Application
level words.
NYS Standards:
Key Idea 4 Modeling/Multiple Representation
4A. Represent problem situations symbolically by using algebraic expressions,
sequences, tree diagrams, geometric figures and graphs
Key Idea 3 Operations
3A. Use addition, subtraction, multiplication, division, and exponentiation with
real numbers and algebraic expressions.
Anticipatory set: (5 minutes) Start off by reviewing Monday’s material. Write
three triangles on the board, one isosceles, one equilateral and the third scalene.
Have the students give some facts about each, whatever they remember, to
bring up previous knowledge.
Developmental Activity: (25 minutes) First draw a blank diagram of the triangle
and the italicized words below on the board. Have the students get their
notebooks out to take notes. Go over (Have the students help with this – they
should know some the facts – if not all – by now) every term in the box and
write the definitions next to the terms on the board (this can be difficult for the
first instance of Base Angles and the last bullet – since there aren’t any
italicized words for it). After you go over the term ask the students if they can
come up to the board and fill in the example. Have them label each angle.

An isosceles triangle has two congruent sides and two congruent angles.
The congruent sides are called legs and a third side called the base.

The vertex angle is the angle included by the congruent legs.

The other two angles are called base angles.

The base angles are congruent.

The figure below depicts an isosceles triangle with all the parts labeled.
Will the instructors work out any examples for the students about
Isosceles Triangles before handing out the worksheet?
Then go over (Review? This was studied recently. I don’t think that
you need to go into this much depth since the students spent all week on
this recently) the term exterior angle. Draw the diagram on the board along with
the definition and examples. An exterior (or external) angle is the angle between
one side of a triangle and the extension of an adjacent side. For example <ABC,
or <ECD. Have one of the students come up and give another example on the
board. Also mention the interesting fact: An exterior angle is congruent to the
sum of the non-adjacent angles in a triangle. For example, <ABC =
<BCD+<CDB.
Next, hand out the activity for the day. After handing out the activity, go
over the first example on the board. Ask the students a few questions. “Who can
tell me which angles are congruent?” What are some expected responses? or
“Tell me some characteristics of this triangle.” What are some expected
responses? This will emphasize the terms they learned earlier. Have the
students work on the worksheet individually. This can be an assessment so walk
around and see if anyone is struggling.
Closure: (5 minutes) Make sure everything is erased from the board and have
the students “Sum up the Day”, which is basically wrapping everything up by
going over the definitions and facts. Make a list on the board and tell them to
write it down.
Homework: (5 minutes) This will be the “Ticket out the door”. Hand it out and
say Good Luck!
Activity Of The Day, Yay!
Name:
Directions: Complete the worksheet and find the measurements for all angles?
for all sides? for all values of x?. See the first problem below and include
exactly what you want the students to find for all of the problems.
1.
Line Segment AB=_____
m BCA = 5x - 46
m BAC = 2x + 8
2.
Line AC =_____
m DCE = 72
m CBA = x
3.
m DBE = m DEB
m ABD = 123
m DBE = x
4.
Line DC= BD
m DCE = 3x + 16
m CBD = x
m DCB=
5.
m<BCA=_____
m<BAC=_____
Line AB=AC
m DCE = 4x + 38
m CBA = 3x + 31
6.
m CBD= m BDC
m CBD = 40
m DCE = x
Ticket Out The Door
Example 1: An isosceles triangle has one angle of 96º. What are the sizes of the
other two angles?
Example 2: A right triangle has one other angle that is 45º. Besides being right
triangle what type of triangle is this? Explain how you know.
Answers for Teachers
Activity for the day
1.
2.
3.
4.
5.
6.
Line AB = AC, x=27.25, m<BCA=90.25, m<BAC=62.5
AC=AB, m<CBA=72
<DBE=57
x=41, <DCE=139, <CBD=41, <DCB=41
x=-7, <DCE=10, <CBA=10
<DCE=100
Ticket Out The Door
1. 42
2. 45, Isosceles Triangle
MED 308
Matthew Garozzo
Week 6
10/17/07
Lesson: Isosceles Triangles: The final picture.
Lesson Overview: The students will use both the sides and angles of an isosceles triangle
to solve algebraic expressions. By doing what?
Lesson Objective:
-more practice with algebra and the use of isosceles triangles.
Standards:
These need to be from KEY IDEAS in the Math A.
-G.G.29: identify corresponding parts of a congruent triangle
-G.G.31: Investigate, justify, and apply the isosceles triangle theorem and its
converse
-G.G.32: Investigate, justify, and apply theorems about geometric inequalities,
using exterior angle theorem
Anticipatory Set: (5min) Have the students identify the properties of isosceles triangles.
This is from the Comprehension level of Bloom’s Taxonomy. Find a word from the
Analysis or Synthesis level to replace it. Make sure they understand the fact that if two
angles of a triangle are congruent than the two sides of that triangle are congruent, if the
two sides of a triangle are congruent then so are the base angles of those sides, hence the
triangle is isosceles. This is from the Comprehension level of Bloom’s Taxonomy.
Find a word from the Analysis or Synthesis level to replace it. This is basically more
application of algebra and applying it to isosceles triangles. How will this be done?
What do the instructors need to do to accomplish this? Be specific.
Developmental Activity: (30min) Hand out the practice sheets for them to work on. Give
them a few minutes to try to answer a question and than go over each question together.
Talk about why and how they came up with each answer.
Answers to practice sheets:
#1. If NL = SL (where is S?), name two congruent angles
<5 = <11
If <1 = <4, name two congruent segments
NI = IJ
If <9 = <10, name two congruent segments
MI= IK
Should be 3 isosceles triangles: IJN, IKM, NLJ Do you realize that triangle NLJ is
equilateral?
All of the symbols used in relation to angle measurement need to be congruent
symbols. Either make sure to include ~ above each = once the file is printed out
(and before making copies) or find the symbol () and use where needed.
#2
x=2 How was
#3 y=9
x found? Show your work.
<Q = 77
<C=<B=63 and <A=?
y= -9
AC=AB = 14 (x=2)
Closure: Time? If everyone can keep up with the processes involved and read what the
question is asking, it would be a great indication that everyone is on the same page. What
does this mean? What are the instructors to do for Closure? Be specific.
Assessment: Time? Homework will be given out and collected on Thursday.
Practice Sheets
K
#1
9
J
2
1 5
6
I
L
11
7
4
8
3
N
10
M
If NJ = LN, Name two congruent angles: ______________________________________
If < 1 = < 4, Name two congruent segments: __________________________________
If <9 = <10, Name two congruent segments: __________________________________
How many Isosceles triangles are on this problem (what angles are congruent? What
sides are congruent? What information will the students need in order to “find” all
the isosceles triangles?)? List them all and explain why.
#2
P
26
7-2(5+y)
O
35x+7
15
Q
Triangle OPQ is an Isosceles triangle with <P as the vertex angle.
Solve for x:
So what is the measure of <Q?
Solve for y and check your answer.
#3
A
6y
2(3+2x)
C
30-8x
7y
B
Triangle ABC is Isosceles with the vertex at <A.
Solve for y.
What are the measures of the angles?
What is the measure of the sides AC and AB?
HW
10-17-07
Name_____________________________________
B
A
C
BC is expressed as: 4a = -2(-4-a)
a=4 BC = 16
AC is expressed as: 7x = 3(2x+1)
x=3 AC = 21
AB is expressed as 5y +1 = 8y - 11
y = 4 AB = 21
What does each of the above mean? Is any of this from a Math A question? If not
use only expressions to represent each side length.
What’s (State it as “What is”) the vertex angle and (remove the word “and”) if (the)
base angles are twice the vertex (?)
what is (Use “Find” instead of “What is”) the base angle measure and the vertex angle
measure(.)
5v = 180 => v = 36
base < ‘s = 72
Do you have a Homework page without the answers?
Isosceles Triangle Review
Grade 10
Materials: -White board
-Markers
-Pencil
-Worksheet
Lesson Overview: Students will solidify their understanding of isosceles
triangles and their properties by reviewing the material covered earlier in the
week and completing an accompanying worksheet.
Lesson Objectives: Students will be able to demonstrate their understanding of
Isosceles triangles by being able to identify the uniqueness of the properties of
isosceles triangles.
NYS Standards
4A. Represent problem situations symbolically by using algebraic expressions,
sequences, tree diagrams, geometric figures and graphs
Key Idea 3 Operations
3A. Use addition, subtraction, multiplication, division, and exponentiation with
real numbers and algebraic expressions
Anticipatory Set (5 min)—Have the students list everything they have learned
about isosceles triangles. From here compose their own definition and write it on
the review worksheet
Developmental Activity (35 min)—Hand out the review worksheet. You might
want to hand this out during the Anticipatory Set since you have the
students writing their own definition on the review worksheet at that time.
Have the students work in pairs to complete the sheet. If they have any
questions throughout the problems, go over them as a group.
Assessment: (40 min) –The assessment will be ongoing throughout the lesson.
The teacher will help the students with any problems throughout the worksheet
and will be able to see the students(‘) progress. A formal assessment will be a
quiz on Friday.
Where is Closure? There needs to be a summary activity.
This totals 80 minutes and there are only 40 minutes in one class period.
Will this be continued tomorrow?
Name:
Date:
Isosceles Triangle Review
1. In your own words define what an isosceles triangle is:
2. Given that ΔABC is isosceles, find the measure of all the angles where <A=3x+2
and <C=2x+5.
3. Find the measure of angle C.
4. Find the measure of <XZY. It looks like y = 5x+1. Is that what is intended?
5. Find the measure of angles L, I, H, M in the diagram below. There are two
angles at one vertex and each angle should be referred to by three letters.
Include two more letters on the line segment that includes L, I, H, and M.
6. If <W is 2 more than 5 times <Q. What is the measure of all the angles in the
triangle?
Where are the answers? Triple-check them since I won’t be
able to!