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FEATURE LESSON PLAN
Project Sirius: Curriculum Development
Angles (Grade 7/8)
By Daniel Zhu
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Par t I: Teach er M ater ial s
Objectives
This lesson aims to teach students:
-
Angle relationships for parallel lines and transversals
Angle sum in a triangle is always 180°
How to solve problems involving interior, complementary, supplementary, opposite, alternate, and/ or
corresponding angles
Preparation
-
Materials: paper, scissors, rulers, protractors; cardboard and markers are also helpful.
Prerequisites: basic understanding of what an angle is. If students have never seen an angle before, you can
begin this lesson by introducing what an angle is and then proceed with the activities.
Overview
One of the best ways to teach students geometric concepts is for them to
visualize and explore the concepts themselves---students will be much more
engaged if you give them the opportunity to draw, measure, and manipulate
things, creating a more effective learning experience. These activities aim to
go beyond that---to see why the angles are congruent (have the same size),
students will be asked to cut out the angles and spatially rearrange them.
W henever you instruct students to draw a diagram or explore, do it with
them on the board, and at the same time, you can ask them engaging
questions or to share their examples.
To make it easy, we have provided a student handout to guide them through
the lesson. They will need paper/ cardboard and scissors to cut up the angles.
Depending on your students' knowledge and preferences, you may find it
helpful to adjust or supplement the activities.
Angles and Lines
"One of the best
ways to teach
students geometric
concepts is for them
to visualize and
explore the
concepts
themselves"
Divide the students into small groups and, according to the student handout,
explore with them what happens when you cut and rearrange the
angles---refer to Table 1 at the end of Part I for detailed teaching instructions
and/ or if students seem stuck on a concept.
W hen the students begin to grasp the idea that opposite angles are
congruent, ask students to share their ideas with the rest of the class and see
if they can piece together some reasons as to why that is true. Remind them how they moved one angle on top of
another if they are stuck.
W hen most students understand why this is true, wrap it up with an example similar to the following diagram:
label one of the angles, and ask them if they can figure out the rest:
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"Too often we
give childr en
a nswer s to
r emember
r ather tha n
pr oblems to
solve."
The same goes for transversals that intersect parallel lines: the goal is
for students to visualize that corresponding angles are congruent (have
the same size). Draw diagrams and cut them with the students, making
sure that they see the top half can stack perfectly on top of the bottom
half.
-Roger Lewin
Once students come to that realization, ask them, "what angles in the
above diagram are equal?" Notice that we have in the above diagram:
? a = ? d = ? e= ? h
? b= ? c= ? f = ? g
Angles and Triangles
For triangles, we have prepared an interactive activity for the students, where you will play an important role of
guidance.
Ask the students to draw their favorite triangle and then measure its angles. You can play the following game with
eager students a few times until they discover/ want to learn the pattern:
1. Claim that you are a magician and can guess their favorite triangle
2. Ask the students for the measure of two of the angles, then, after some acting, guess the third angle using
the fact that the angles of a triangle sum to 180°
3. Repeat & vary as necessary to engage your students
W hen some students appear to know what you are doing, ask them to play the game with their neighbors. For
more advanced students, draw the following diagram on the board and ask them to use it to show why your
"magic" works:
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Table 1: Summary of Concepts & Ideas
Concept
Supplementary angles
Example Diagram
Potential Guiding
Questions
-
Complementary angles
-
Opposite angles
-
Corresponding angles
-
Alternate angles
-
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Possible Student
Conjectures
W hat is ? a+? b?
W hat if a, bare under the horizontal line?
Do the orientations of the angles matter?
W hat if ? a is less than
? b?
? a+? b=180°
Or: ? a+? b?180° due
to measurement error
The orientations,
locations, and relative
size of the angles don't
matter
W hat is one necessary
feature of this
diagram?
W hat is ? a+? b?
Do the orientations of
the angles matter?
W hat if ? a is less than
? b?
-
There must be a right
angle
? a+? b=90°
Or: ? a+? b?90° due to
measurement error
The orientations,
locations, and relative
size of the angles don't
matter
W hich is bigger, ? a
or ? c?W hat about
? band ? d?
Do the orientations,
sizes, and/ or locations
of the angles matter?
W hy is this true?
Think of supplementary
angles!
-
W hich is bigger, ? a
or ? e?W hat about
? band ? f?
W hich is bigger, ? cor
? g?W hat about ? d
and ? h?
Do the orientations,
sizes, and/ or locations
of the angles matter?
-
? a=? e
? b=? f
? c=? g
? d=? h
The orientations,
locations, and sizes of
the angles don't matter
W hich is bigger, ? cor
? f?W hat about ? d
and ? e?
Do the orientations,
sizes, and/ or locations
of the angles matter?
W hy is this true?
Think of oppositeand
corresponding angles!
-
? c=? f, ? d=? e
The orientations,
locations, and sizes of
the angles don't matter
? b=? c(opposite) and
? b=? f (corresponding),
so ? c=? f
Similar reasoning for
? d=? e
-
-
-
? a=? c
? b=? d
The orientations,
locations, and sizes of
the angles don't matter
? a+? b=180°=? a+? d,
so ? b=? d.
? a+? b=180°=? b+? c,
so ? a=? c.
Table 1: Summary of Concepts & Ideas
Concept
Example Diagram
(Same-side) Interior
angles
Potential Guiding
Questions
-
-
Sum of angles in a triangle
-
W hat is ? d+? f?
W hat is ? c+? e?
Do the orientations,
sizes, and/ or locations
of the angles matter?
W hy is this true?
Think of supplementary
and corresponding (or
alternate) angles!
W hy do you think
these angles are named
same-side interior
angles?
W hat is ? a+? b+? c? W hat if the triangle
has a right angle?
W hat if the triangle
has an obtuse angle?
W hat if all three angles are acute?
Do the orientations of
the angles matter?
Possible Student
Conjectures
? d+? f=180°
? c+? e=180°
The orientations,
locations, and sizes of
the angles don't matter
? d+? b=180° and
? b=? f (corresponding),
so ? d+? f=180°
Or: ? d+? c=180° and
? c=? f (alternate), so
? d+? f=180°
Similar reasoning for
? c+? e=180°
? a+? b+? c=180°?
If ? a=90°, ? b+? c=90°
The orientations and
sizes of the angles don't
matter
Or: ? a+? b+? c?180°
due to measurement
error
Supplementary Materials: For More Advanced Students
For more advanced students, draw the following
diagram on the board to introduce them to the concept
of exterior angles:
triangle is 180°-100°-50°=30°, and that x+30°=180°, so
x=150°. More algebraically inclined students can
directly reason that x=50°+100°=150° instead.
If students seem stuck, prompt them to find the third
angle of the triangle. If they still seem stuck, prompt
them to look for a diagram in their table that looks like
the region around x in the diagram on the left. This
should be enough to guide them to the correct answer.
By now, these students should have a good
understanding of the reasoning behind why the angle
sum of a triangle is always 180°. For this concept, it's
best for students to reason out the correct answer
(x=150°) algebraically or by inspecting the angles in the
diagram, rather than cutting-and-placing which has
been the teaching tool for all of the previous concepts.
Student that have gotten this far should be able to
reason that the third, unknown angle of the above
Finally, if there is still time, you can add a layer of
abstraction to the situation by replacing the 50° and
100° with variables a and b, and then asking your
students to express x in terms of a and b. Students who
can grasp this level of abstraction would thus have
(implicitly) proven the property that they have just
observed.
Try your best not to bore your more advanced
students. If a student has mastered all the concepts up
until here, give him/ her the Student Problem Set
(homework) and instruct him/ her to get a head start on
it. The rest of your students can receive Student
Problem Set as homework at the end of the class.
Students are welcome to collaborate with their parents
on the problem set, but otherwise should work alone.
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Par t II: Stu den t M ater ial s
Supplementary & Opposite Angles
Angles are cool---you see them everywhere: they form whenever you cross two or more lines, and 3 angles make a
triangle. But what interesting facts can we discover about them?
Draw two intersecting lines. Then, measure the angles formed by these lines using your protractor. Do you notice
anything interesting? Draw another picture and do the same. Do you have a hypothesis of what's going on? Are
there any angles that add up to 180°?Are there any angles that are equal to each other?W rite your thoughts in your
table.
Now, take one of your drawings and cut out the four angles. Can you
rearrange some of them and stack them on top of each other? W hat do you
notice?Does this support your hypothesis?
Try doing the same for other sets of intersecting lines, then share your ideas
with the class. Move on to the next section when you are comfortable with
stating your hypothesis and explaining why you think you are right to others.
Complementary Angles
Draw two lines that intersect at a right angle (90°) and a third line that cuts
this right angle into two parts (not necessarily the same), as follows:
"Sorry, we are not
seeing any angels
today."
Do you notice anything interesting about these two parts (? a and ? babove)
that you just created?W rite your thoughts in your table.
Transversals & Parallel Lines
You may have discovered that you get equal angles when two lines intersect, but what happens when the lines are
parallel? Well, parallel lines don't form angles, so we need to draw a transversal that cuts (or intersects) both of
them!
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Does this remind you of what you just saw? W hat does the above diagram look like if we only look at the angles a,
b, c, and d?
Measure the angles of the above diagram (a, b, c, d, e, f, g, and h) and write them down next to their labels. W hat do
you notice? Now, draw a bigger picture on cardboard or paper and cut out the angles (you should have 8 pieces).
Can you somehow rearrange the angles and stack some them on top of each other again?
Do you see something similar between the top part (? a, ? b, ? c, ? d) and the bottom part (? e, ? f, ? g, ? h)?
These angles are called corresponding angles, since they are in the same region with respect to the two parallel lines.
To see that, draw another diagram similar to the one on the previous page (with two parallel lines and a
transversal) and cut your diagram between the two parallel lines. Can you translate the top four angles to lie on top
of the bottom ones?
There are other sets of angles that are related to each other. These are called alternate angles. Think about what this
word might mean geometrically and use the guiding questions in your table to guide your search for these angles.
Finally, there's a third set of angles, known as (same-size) interior angles, that have a special property associated
with them. Think about what this word might mean geometrically and use the guiding questions in your table to
guide your search for these angles. W hat other type of angles that you've played around with before does
(same-size) interior angles remind you of?
W hich angles are equal? W hich add up to 180°? W rite your thoughts in your table and share your ideas with your
classmates.
Triangles
Let's take a closer look at triangles. Triangles come in all shapes and sizes, so draw your favorite triangle!
Let's play a game! Measure the 3 internal angles of your triangle and, without letting your teacher know which
triangle you drew, tell him/ her two of its internal angles. Do you believe that he/ she can guess the third?
Try it out a few times. Do you notice any patterns? If you think you figured it out, take turns playing it with your
neighbor: can you figure out the measure of their 3rd internal angle?W rite your thoughts in your table.
Can you draw a triangle whose 3 internal angles sum up to 200°? W hat happens when you try to draw such a
triangle?W hat about a triangle whose 3 internal angles sum to 150°?
It turns out, much like how angle ? a and ? b are supplementary angles in the above diagram and add up to
________°, the angles of a triangle always sum up to ________° (fill in the blanks!). Otherwise, the three sides simply
don't come together the right way. Can you convince yourself why?Hint: usethefollowing diagram!
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Table: Observations & Hypotheses
Concept
44
Example
W hat do you notice?
Supplementary
angles
-
W hat is ? a+? b?
W hy do you think this is true?
Complementary
angles
-
W hat is ? a+? b?
W hy do you think this is true?
Opposite
angles
-
W hich is bigger, ? a or ? c?W hat about ? band ? d?
W hy do you think is this true?Think of supplementary
angles!
Corresponding
angles
-
W hich is bigger, ? a or ? e?W hat about ? band ? f?
W hich is bigger, ? cor ? g?W hat about ? d and ? h?
W hy do you think this is true?
Alternate
angles
-
W hich is bigger, ? cor ? f?W hat about ? d and ? e?
W hy do you think this is true?Think of oppositeand
corresponding angles!
(Same-side)
Interior angles
-
W hat is ? d+? f?W hat about ? c+? e?
W hy do you think this is true?Think of supplementary and
corresponding (or alternate) angles!
Sum of angles
in a triangle
-
W hat is ? a+? b+? c?
Does the type of the triangle matter?
Challenge: W hy do you think this is true?
Par t III: Stu den t Pr obl em Set
This is your homework. It's due ______________________________
Name: ______________________________
1. Using the table that you filled out in class, fill in the following blanks: Date: ______________________________
a. Opposite angles have the __________ measure.
b. Supplementary angles add up to __________.
c. ____________________ angles add up to 90°.
d. W hen two parallel lines are cut by a transversal, ____________________ angles and
____________________ angles are formed, which have the same measure.
e. Same-side interior angles add up to __________.
f. The angles of a triangle sum to __________.
2. Figure 1 on the right shows a diagram with 9 unknown angles labeled. Determine each of their measures
(in degrees). Hint: dothem in theorder given & usetheother hintsgiven below!
a: __________
d: __________
g: __________
b: __________
e: __________
h: __________
c: __________
f: __________
i: __________
Hints:
a: corresponding
b: triangle
c: opposite
d: alternate
e: complementary
f: supplementary
g: triangle
h: alternate
i: alternate
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Figure 1
3. Using your answers to the previous problem and the given hints, briefly explain how you got each of
your answers. Make sure to use the right words to identify the angles!
a: ____________________________________________________________________________________
b: ____________________________________________________________________________________
c: ____________________________________________________________________________________
d: ____________________________________________________________________________________
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e: ____________________________________________________________________________________
f: ____________________________________________________________________________________
g: ____________________________________________________________________________________
h: ____________________________________________________________________________________
i: ____________________________________________________________________________________
4. Figure 2 on the right shows an isosceles right-angled triangle on top of a parallelogram with 4 unknown
angles labeled. Determine each of their measures (in degrees). Hint: do them in the order given and use the
fact that thetriangleisisoscelesand right-angled!
a: __________
c: __________
b: __________
d: __________
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Figure 2
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Par t III: Stu den t Pr obl em Set
This is your homework. It's due ______________________________
Name: ______________________________
Date: ______________________________
5. In the space below, draw your favorite diagram that contains each of the following features:
? a, ? b: A pair of supplementary angles
? g, ? h: A pair of corresponding angles
? c, ? d: A pair of complementary angles
? i, ? j: A pair of alternate angles
? e, ? f: A pair of opposite angles
? k, ? l: A pair of same-side interior angles
Label one example of each of the features above in your diagram using the angle labels given (you can
give more than one label to the same angle). Make sure to only draw one diagram and add numerical
angle measures where appropriate!
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6. Briefly explain how you know ? cand ? d in your diagram from problem 5 are complementary:
__________________________________________________________________________________________
__________________________________________________________________________________________
7. Briefly explain how you know ? eand ? f in your diagram from problem 5 are corresponding:
__________________________________________________________________________________________
__________________________________________________________________________________________
8. Briefly explain how you know ? k and ? l in your diagram from problem 5 are same-side interior:
__________________________________________________________________________________________
__________________________________________________________________________________________
9. Can a triangle have two parallel sides? If so, draw a triangle that has two parallel sides and label all of its
angles and side lengths. If not, use the properties of parallel lines and triangles to explain why not.
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Par t IV: Qu iz
No calculators, rulers, or protractors.
Time limit: 20 minutes.
Name: ______________________________
Date: ______________________________
1. Figure 1 on the right shows a diagram with 5 unknown angles labeled. Determine each of their measures
(in degrees). Hint: noticetheequilateral triangleand theright anglein thegiven diagram!
a: __________
b: __________
c: __________
d: __________
e: __________
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Figure 1
2. Using your answers to the previous problem, briefly explain how you got each of your answers. Make
sure to use the right words to identify the angles!
a: ____________________________________________________________________________________
____________________________________________________________________________________
b: ____________________________________________________________________________________
____________________________________________________________________________________
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c: ____________________________________________________________________________________
____________________________________________________________________________________
d: ____________________________________________________________________________________
____________________________________________________________________________________
e: ____________________________________________________________________________________
____________________________________________________________________________________
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