Download What are Constructions?

Lesson plan: Review Common Core Geometry
Lesson question: What are constructions? How do I create different constructions?
Common Core Standards Addressed:
 Make geometric constructions:
o
CCSS.MATH.CONTENT.HSG.CO.D.12
Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
Lesson objective:
Assessment:
Students will correctly construct
angles/lines/ angle and line bisectors.
Students will have correct constructions on
scrap paper.
Teacher will be collecting student’s
construction Multiple Choice.
Teacher will observe students and ask for
the level of difficulty for each construction.
Opening:
Students will discuss what types of constructions they remember preforming
throughout the class year. This warm up will be a review of what they have been
learning over the year. Students will use background knowledge to have a class
discussion on the topic of constructions.
Procedure:
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Students will complete warm up
Teacher will discuss and go over warm up
Teacher will ask if anyone has any questions about the warm up.
Teacher will instruct students to open up the review packet handed out to them.
Teacher will discuss topic of the day
Topic that will be discussed will be how to construct a line and an angle
Teacher will show a video on how to construct a line segment and an angle
Teacher will discuss with class the steps that are necessary for them to create a line
segment and an angle
9. Students will create their own line segments and angles.
10. Teacher will go through example constructions with students
11. Students will independently work on additional examples.
12. Students will discuss example constructions
13. Students will be asked about how they got their answers and why
14. Teacher and students will discuss example constructions
15. Teacher will introduce homework.
16. Teacher will discuss directions.
Tiered by complexity: Each student will have separate homework assignments that will
challenge each student and also allow for them to see success.
Gabby- Angle Bisector
Kenneth- Line Bisector
Lenny- Line Bisector
Closure:
Teacher will discuss with students the overall lesson. Teacher will ask for student input on
whether or not they liked how the lesson played out or if I could teach it better to meet all
of their needs. We also discuss the different homework assignments. I tell the students that
their homework is not all the same and they will need to attempt it on their own while
providing me with two example constructions. They are to read the steps and create 2
constructions while using the steps to assist them.
Materials:
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Notes
Pen/pencil
Calculator
Promethean Board
Internet
Regnetsprep.org
Scrap sheet for constructions
Scrap sheets for homework
Ruler
compass
Construction directions
Review Geometry
Copy a line segment
Given: (Line segment)
Task: To construct a line segment congruent to (line segment)
.
Directions:
1. If a reference line does not already exist, draw a reference line
with your straightedge upon which you will make your construction.
Place a starting point on the reference line.
2. Place the point of the compass on point A.
3. Stretch the compass so that the pencil is exactly on B.
4. Without changing the span of the compass, place the compass point on the starting
point on the reference line and swing the pencil so that it crosses the reference line.
Label your copy.
Your copy and (line segment)
are congruent. Congruent means equal in length.
Explanation of construction: The two line segments are the same length, therefore
they are congruent.
Copy an angle
Given:
Task: To construct an angle congruent
to
.
Directions:
1. If a reference line does not already exist, draw a reference line with
your straightedge upon which you will make your construction. Place
a starting point on the reference line.
2. Place the point of the compass on the vertex of
(point A).
3. Stretch the compass to any length so long as it stays ON the angle.
4. Swing an arc with the pencil that crosses both sides of
.
5. Without changing the span of the compass, place the compass point
on the starting point of the reference line and swing an arc that will
intersect the reference line and go above the reference line.
6. Go back to
and measure the width (span) of the arc from where it crosses
one side of the angle to where it crosses the other side of the angle.
7. With this width, place the compass point on the reference line where your new arc
crosses the reference line and mark off this width on your new arc.
8. Connect this new intersection point to the starting point on the reference line.
Your new angle is congruent to
.
Explanation of construction: When this construction is finished, draw a line segment
connecting where the arcs cross the sides of the angles. You now have two triangles
that have 3 sets of congruent (equal) sides. SSS is sufficient to prove triangles
congruent. Since the triangles are congruent, any leftover corresponding parts are also
congruent - thus, the angle on the reference line and
are congruent.
Bisect a line segment
(Also know as Construct a Perpendicular
Bisector of a segment)
Given: (Line segment)
Task: Bisect
Directions:
.
1. Place your compass point on A and stretch the compass MORE THAN
half way to point B, but not beyond B.
2. With this length, swing a large arc that will go BOTH above and
below
.
(If you do not wish to make one large continuous arc, you may simply place one small arc
above
and one small arc below
.)
3. Without changing the span on the compass, place the compass point
on B and swing the arc again. The two arcs you have created should
intersect.
4. With your straightedge, connect the two points of intersection.
5. This new straight line bisects
. Label the point where the new line and
cross as C.
has now been bisected and AC = CB. (It could also be said that the segments are
congruent,
.)
(It may be advantageous to instruct students in the use of the "large arc method"
because it creates a "crayfish" looking creature which students easily remember and
which reinforces the circle concept needed in the explanation of the construction.)
Explanation of construction: To understand the explanation you will need to label the
point of intersection of the arcs above segment
as D and below segment
as E.
Draw segments
,
,
and
. All four of these segments are of the same
length since they are radii of two congruent circles. More specifically, DA =
DB and EA = EB. Now, remember a locus theorem: The locus of points equidistant
from two points, is the perpendicular bisector of the line segment determined by the
two points. Hence,
is the perpendicular bisector of
.
The fact that the bisector is also perpendicular to the segment is actually MORE than
we needed for a simple "bisect" construction. Isn't this great! Free stuff!!!
Bisect an angle
Given:
Task: Bisect
.
Directions:
1. Place the point of the compass on the vertex of
(point A).
2. Stretch the compass to any length so long as it stays ON the angle.
3. Swing an arc so the pencil crosses both sides of
. This will create
two intersection points with the sides (rays) of the angle.
4. Place the compass point on one of these new intersection points on the
sides of
.
If needed, stretch your compass to a sufficient length to place your pencil well into the
interior of the angle. Stay between the sides (rays) of the angle. Place an arc in this
interior - you do not need to cross the sides of the angle.
5. Without changing the width of the compass, place the point of the compass on
the other intersection point on the side of the angle and make the same arc. Your two
small arcs in the interior of the angle should be crossing.
6. Connect the point where the two small arcs cross to the vertex A of the angle.
You have now created two new angles that are of equal measure (and are each 1/2 the
measure of
.)
Explanation of construction: To understand the explanation, some additional labeling
will be needed. Label the point where the arc crosses side
as D. Label the point
where the arc crosses side
as E. And label the intersection of the two small arcs
in the interior as F. Draw segments
and
. By the construction, AD = AE (radii
of same circle) and DF = EF (arcs of equal length). Of course AF = AF. All of these
sets of equal length segments are also congruent. We have congruent triangles by
SSS. Since the triangles are congruent, any of their leftover corresponding parts are
congruent which makes
equal (or congruent) to
.
Parallel
-through a point
Given: Point P is off a given line
Task: Construct a line through P parallel to the given line.
Directions:
1. With your straightedge, draw a transversal through point P.
This is simply a straight line which runs through P and intersects
the given line.
2. Using your knowledge of the construction COPY AN ANGLE,
construct a copy of the angle formed by the transversal and the
given line such that the copy is located UP at point P. The vertex
of your copied angle will be point P.
3. When the copy of the angle is complete, you will have two parallel lines.
This new line is parallel to the given line.
Explanation of construction: Since we used the construction to copy an angle, we now
have two angles of equal measure in our diagram. In relation to parallel lines, these
two equal angles are positioned in such a manner that they are called corresponding
angles. A theorem relating to parallel lines tells us that if two lines are cut by a
transversal and the corresponding angles are congruent (equal), then the lines are
parallel.
Perpendicular - lines (or segments) which meet to form right
angles.
Perpendicular
from a point ON a line
Given: Point P is on a given line
Task: Construct a line through P perpendicular to the
given line.
Directions:
1. Place your compass point on P and sweep an arc of any size that crosses
the line twice (below the line). You will be creating (at least) a semicircle.
(Actually, you may draw this arc above OR below the line.)
2. STRETCH THE COMPASS LARGER!!
3. Place the compass point where the arc crossed the line on one side and
make a small arc below the line. (The small arc could be above the line if
you prefer.)
4. Without changing the span on the compass, place the compass point where the arc
crossed the line on the OTHER side and make another arc. Your two small arcs
should be crossing.
5. With your straightedge, connect the intersection of the two small arcs to point P.
This new line is perpendicular to the given line.
Explanation of construction: Remember the construction for bisect an angle? In this
construction, you have bisected the straight angle P. Since a straight angle contains
180 degrees, you have just created two angles of 90 degrees each. Since two right
angles have been formed, a perpendicular exists.
Given: Point P is off a given line
Task: Construct a line through P perpendicular to the given
line.
Directions:
1. Place your compass point on P and sweep an arc of any size that
crosses the line twice.
2. Place the compass point where the arc crossed the line on one side
and make an arc ON THE OPPOSITE SIDE OF THE LINE.
3. Without changing the span on the compass, place the compass point
where the arc crossed the line on the OTHER side and make another
arc. Your two new arcs should be crossing on the opposite side of the
line.
4. With your straightedge, connect the intersection of the two new arcs to point P.
This new line is perpendicular to the given line.
Explanation of construction: To understand the explanation, some additional labeling
will be needed. Label the point where the arc crosses the line as points C and D.
Label the intersection of the new arcs on the opposite side as point E. Draw
segments
,
,
, and
. By the construction, PC = PD and EC =
ED. Now, remember a locus theorem: The locus of points equidistant from two
points (C and D), is the perpendicular bisector of the line segment determined by the
two points. Hence,
is the perpendicular bisector of
.
The fact that we created a bisector, as well as a perpendicular, is actually MORE than
we needed - we only needed to create a perpendicular. Yea, free stuff!!!
Construct an isosceles
triangle whose legs and
Construct an Isosceles Triangle
base are of the predetermined lengths
Using Given Segment Lengths:
given. Construct the
new triangle on the
reference line.
When constructing an isosceles triangle, you may
Using your compass,
measure the length of
the given "base".
be given pre-determined segment lengths to use
for the triangle (such as in this example), or you
may be allowed to determine your own segment
lengths. Either way, the construction process will
be the same.
Do not change the size Using your compass, measure Without changing the
You now have three
of the compass. Place the length of the given "leg".
size of the compass, points which will define
your compass point on Place the compass point where move the compass point the isosceles triangle.
the reference line point the previous arc crosses the
to the point on the
and scribe a small arc
reference line and scribe
reference line. Scribe
which will cross the another arc above the reference an arc above the line
line.
line
such that it intersects
with the previous arc.
Construct an Equilateral Triangle Using a
Given Segment Length:
Construct an equilateral
triangle whose sides are of
given length "a".
Construct the new triangle
on the reference line.
When constructing an equilateral triangle, you
may be given a pre-determined segment
length to use for the triangle (such as in this
example), or you may be allowed to determine
your own segment length. Either way, the
construction process will be the same.
Using your
Do not change the size of Do not change the size
You now have three
compass,
the compass. Place your of the compass. Place points which will define
measure the
compass point on the
the compass point
the equilateral triangle.
length of the
reference line point and
where the arc crosses
given segment, scribe an arc which will the reference line and
"a".
cross the line and will rise scribe another arc which
above the line.
crosses the previous arc.
Alternate Method for Constructing an
Equilateral Triangle:
Draw a circle and place a point
on the circle. Do not change
the size of the
An equilateral triangle can be easily constructed from a circle.
The secret to this method is to remember to keep the compass set
at the same length as the radius of the original circle.
compass.
With the
compass
still set at
the same
size as the
radius of
the circle,
place the
compass
point on
the point
on the
circle and
mark off a
small arc
on the
circle.
You now have a circle with six equally Connect every other point on
divided sections on its
the circle to form the equilateral
circumference.
triangle.
Now,
move the
compass
point to
this new
arc and
mark off
another
arc.
Continue
around the
circle.