Download 8th Math Community and Communication

Communities and Communication
Three Weeks
Math
Lesson Plan
Grade:
8th Grade
Teacher:
th
8 Grade Math Teacher
Lesson Title:
Design and Reasoning Through Similarity and Mathematical Proofs
STRANDS
Congruence
Similarity, Right Triangles, and Trigonometry
Expressing Geometric Properties with Equations
LESSON OVERVIEW
Summary of the task, challenge, investigation, career-related scenario, problem, or community link.
The unit lessons will first begin with similarities in figures along with different ways in which to construct those figures. After students are able to identify and construct
similar figures, they will scaffold their learning by using this knowledge to write both algebraic and geometric proofs for lines, parallelograms and triangles. Students will also
learn the significance of these mathematical proofs and the STEM professions in which they are vital.
After four lessons centered on writing proofs for different situations, the students will begin their first project day by designing a food wagon. The food wagon will carry
items from their assigned colony and students will classify the food contents on the food wagon as part of their science standards, but they will have historical context in line
with the social studies standards. The English standards will encompass a script of the commercial containing the food wagon. The design on the food wagon body must be
rectangular in shape and must also contain a triangle somewhere within the wheel design, whether it is structural or aesthetic in nature. Students will use this design for
their final mathematical project assessment to complete a proof concerning the rectangular body and one concerning the triangle within the wheel.
MOTIVATOR
Hook for the week unit or supplemental resources used throughout the week. (PBL scenarios, video clips, websites, literature)
Day 1 -"Congruence and Similarity":
This motivator will utilize the following video clip – “Congruence and Similarity” (Appendix A). The students will begin by watching this clip and then discussing the
similarity of all American Flags. The students will discuss why maintaining this similarity is necessary. We will also talk about the main properties the flag must have to
maintain this similarity (e.g., seven stripes next to the rectangle with stars, 13 stripes total, the relationship between length of width of the flag, etc.). This will lead into
the lesson on defining similarity for Day one while showing students how we use similarity as a part of designing and constructing objects for our everyday lives.
Day 3 -"How to Prove a Mathematical Theory":
This motivator will utilize the following video clip – “ How to Prove a Mathematical Theory ” (Appendix C). The students will then watch the following clip and have a
discussion about why proofs are important to our scientific and mathematical advancements. Students will then begin an activity that they will not produce a proof, but
will investigate how to analyze a given figure in the appropriate way to lead to writing mathematical proofs.
DAY Objectives
(I can….)
1
I can represent
transformations
in the plan and
describe them
as functions
that take points
in the plane as
inputs and give
other points as
outputs.
I can compare
transformations
that preserve
distance and
angle to those
that do not.
Materials &
Resources
“Congruence
and Similarity”
(Appendix A)
Ruler (or
straight edge)
Calculator
Graph paper
Laptop
Triangular and
rectangular cut
pieces of patty
paper (one of
each per group)
“Station 1”
Activity
(Appendix A)
“Station 3”
Activity
(Appendix A)
Instructional Procedures
Essential Question(s):
1. How can I represent transformations in the plan and describe them as functions
that take points in the plane as inputs and give other points as outputs?
2. How can I compare transformations that preserve distance and angle to those that
do not?
Set:
Teacher will begin by showing the following video clip: “Congruence and Similarity.” The
students will begin by watching this clip and then discussing the similarity of all American
Flags. The students will discuss why maintaining this similarity is necessary. We will also
talk about the main properties the flag must have to maintain this similarity (e.g., seven
stripes next to the rectangle with stars, 13 stripes total, the relationship between length of
width of the flag, etc.). This will lead into the lesson on defining similarity for Day one while
showing students how we use similarity as a part of designing and constructing objects for
our everyday lives.
Teaching Strategy:
1. Have students construct a triangle or quadrilateral and switch drawings with another
student.
2. Have students recreate the figure they were given from another students, but with the
dimensions being half as large.
3. Have students discuss what is necessary for figures to be similar. Use this as an
opportunity to clarify any misconceptions.
4. Place the students in heterogeneous group of 3-4.
Differentiated Assessment
Instruction
Remediation:
Peer Tutoring
Heterogeneous
Grouping
Use of laptops
and websites to
provide support
Prompting
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Use of laptops
and websites to
provide a
enrichment for
deeper
investigation at
the student’s
pace
Formative
Assessment:
Teacher
observations of
opening activity
Teacher
observations of
students
creating similar
figures
Teacher
observations of
students
completing
stations
Performance
Assessment:
Discussions
about similarity
in the American
Flag
Discussion
about creating
Similar Figures
Homework
(Appendix A)
5. Have students rotate through the stations and complete each of the activities. Station 4
should be a computer with the site
http://www.mathopenref.com/similartriangles.html. They should explore the site and
make a list of properties they notice make triangles similar.
6. Come back together as a group and discuss your findings.
similar figures
Ending
discussion of
angles and
sides when
transformations
are applies
Summarizing Strategy:
As an exit ticket have students answer the following questions:
Station work
1. What transformations preserve distance and angles? Which do not?
2. What has to remain true for figures to be similar? Congruent?
Homework:
Summative
Assessment:
Exit Ticket
Students will be assigned homework for further investigation.
Homework
Adapted from: Howard County Public Schools (HCPSS) Secondary Mathematics Office Curricular Projects (v2.1)
2
I can make
formal
geometric
constructions
with a variety
of tools and
methods.
“Geometric
Construction”
Stations
(Appendix B)
Compass
Straight edge
Graph paper
Patty paper
Marker
String
Chart Paper
Laptops/iPad
Geogebra (or
Essential Question(s):
Remediation:
Peer Tutoring
How can I make formal geometric constructions with a variety of tools and methods?
Set:
Have students write, draw or discuss the definitions of bisect and perpendicular.
As a class, come to a consensus. Allow students to share their drawings and definitions.
Teaching Strategy:
1. Place the students in heterogeneous group of 3-4. Set up the 4 stations described
below. Have groups visit each station and complete the task. Each station should take
about 10 minutes depending on ability level
2. Have students complete the “Geometric Construction” Stations. For an example of how
to complete Station 1, see the following link:
http://www.mathopenref.com/constbisectline.html.
3. When students are finished, discuss the methods used at each station, and how they
are different. Which did students think were easiest to use and why?
4. If the students were unable to finish all of the stations, have them available online to
complete for homework.
Heterogeneous
Grouping
Use of
laptops/iPads
and websites to
provide support
Formative
Assessment:
Teacher
observations of
opening activity
Teacher
observations of
students
completing
stations
Prompting
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Use of
Performance
Assessment:
Discussions
about
definitions of
bisector and
perpendicular
lines
geometric
software)
Summarizing Strategy:
As an exit ticket, have students answer the following questions.
1. Name 2 ways to create a perpendicular bisector
2. What is created when each side of a triangle is bisected?
Adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement.
Portsmouth, NH: Heinemann.
3
Graph Paper
I can prove
theorems about
Patty Paper
lines and
angles.
Geogebra (or
geometric
software)
“Congruent
Sailing”
(Appendix C)
laptops/iPads
and websites to
provide a
enrichment for
deeper
investigation at
the student’s
pace
Remediation:
Peer Tutoring
Set:
Teacher will begin by showing the following video clip-“How to Prove a Mathematical
Theory ” (Appendix C). The students will then watch the following clip and have a
discussion about why proofs are important to our scientific and mathematical
advancements. Teacher will then lead into why proofs may be necessary in other STEM
professions.
Heterogeneous
Grouping
Summarizing Strategy:
As an exit ticket, ask students:
Station work
Summative
Assessment:
Exit Ticket
Essential Question(s):
How can I prove theorems about lines and angles?
Teaching Strategy:
1. Place students in heterogeneous groups of 3.
2. Present “Congruent Sailing.” Have students work through all parts of the task.
Provide adequate time for students to complete the task.
3. Jigsaw students into different groups. Each member must show the convincing
argument for the guy-wire having equal lengths. Have the other members of the
new group rate the argument (scale 1-5) for its ability to convince them of the
correctness.
4. Have students return to their original group. Ask groups to tally their scores.
Recognize the high scoring group. Allow groups with various scores to explain their
method. Ask for alternate approaches. Allow students to defend their group’s
argument.
5. Discuss with students which approach they think was best and why?
Discussions
about
constructing
bisectors
Prompting
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Formative
Assessment:
Teacher
observations of
students
completing
proofs
Performance
Assessment:
Discussions
about the
importance of
proofs in the
opening activity
Discussions
about different
methods of
proving
Final Proofs
Summative
Assessment:
Exit Ticket
1. Why do we need to be able to write mathematical proofs?
2. Name at least 3 STEM professions that would require proofs.
4
I can prove
theorems about
lines and
angles.
“Pythagorean
Theorem Proof”
Opening Activity
(Appendix D)
Patty paper
Essential Question(s):
Remediation:
Peer Tutoring
1. How can I prove theorems about lines and angles?
2. How can I prove theorems about parallelograms?
3. How can I prove theorems about triangles?
I can prove
theorems about Ruler
parallelograms.
Protractor
I can prove
theorems about
triangles
Heterogeneous
Grouping
Prompting
½ Project Day – See Unit Plan
Colonial Commercial Project – Planning
Set:
Explain to students that Pythagorean’s Theorem was observed and utilized long before
Pythagoras’s time. However, he has credit for the theorem because he was the first to be
able to prove it! His proof was geometric, but the class will prove it algebraically today.
Have students complete the “Pythagorean Theorem Proof” Opening Activity. Circulate and
offer guidance if students seem to feel challenged, but do not give them the answer. Give
the chance for self-discovery. Compare proofs. Are they all the same? If not how are they
different? Which method was best?
Teaching Strategy:
1. Place students in heterogeneous groups of 3-4 where they will do an investigation
using patty paper.
2. Using paper, have students create a right triangle and draw an altitude from the
right angle. There should now be three total triangles, all of which are similar.
3. Have students determine whether or not the triangles are similar. Ask students: “If
all three triangles are similar, how can we verify this?”
4. Give the students adequate time to measure each angle and side and determine if
they have the same ratios, making them similar.
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Formative
Assessment:
Teacher
observations of
students
completing
proofs
Performance
Assessment:
Discussions
about methods
used for
proving
Pythagorean’s
Theorem
Final Proofs
Summative
Assessment:
Exit Ticket
Summarizing Strategy:
As an exit ticket, ask students:
1. What are two ways to prove Pythagorean’s theorem?
2. Can you think of a third way to prove Pythagorean’s theorem?
5
I can prove
Protractor
theorems about
lines and
Graph Paper (or
angles.
patty paper)
Essential Question(s):
I can prove
GeoGebra
theorems about
triangles.
Rulers
Set:
Have each student draw a triangle with a ruler. Have the students measure the three
angles of the triangle, and discuss the findings. What were their sums? What can we
conclude about the sum of a triangle?
1. How can I prove theorems about lines and angles?
2. How can I prove theorems about triangles?
Paper
Scissors
Proof Blocks
Laptops/iPads
Teaching Strategy:
1. Give each of the students a triangle. Using the triangle, have them tear off all three of
the angles. Line each of the angles up back to back. (Angle to angle)
2. Discuss what happened when the students do this. (The three different angles a
straight line. Students should be able to conclude that the angles of a triangle are
supplementary since a line is 180 degrees.
3. Assign students to small heterogeneous groups. Have groups work through a more
formalized activity for the proof using geometry software such as GeoGebra. Students
should start with a triangle and parallel lines. Then, have the students manipulate the
vertices of the triangles and parallel line to see what happens so they may document
and discuss what occurs. A tutorial that demonstrates this is found at the following
link: https://docs.google.com/file/d/0B_2_-NMZ5KYqRzV4MHhOOUdYeHM/edit
For additional resources for students needing support with the triangle proof, please see:
this tutorial
Summarizing Strategy:
As an exit ticket, ask students:
1. What are two ways to prove Pythagorean’s theorem?
2. Can you think of a third way to prove Pythagorean’s theorem?
Adapted from tasks available through the Common Core Website (2013).
Remediation:
Peer Tutoring
Heterogeneous
Grouping
Use of
laptops/iPads
and websites to
provide support
Use of
GeoGebra can
assist students
in taking
precise
measurements
easily
Prompting
Enrichment:
Peer Tutoring
Formative
Assessment:
Teacher
observations of
opening
triangle design
and
calculations
Teacher
observations of
lining up angles
Teacher
observations of
students using
proof blocks
Performance
Assessment:
Discussions
about findings
Final proofs
Heterogeneous
Grouping
Use of
laptops/iPads
and websites to
provide a
enrichment for
Summative
Assessment:
Exit Ticket
deeper
investigation at
the student’s
pace
6
I can prove
“Practical
theorems about Pennants, Inc.”
triangles.
Activity
(Appendix E)
Graph Paper
Isometric dot
paper
Rulers
Essential Question(s):
The use of
GeoGebra
software allows
students in
need of a
challenge a tool
to further
explore and
deepen their
understanding
Remediation:
Peer Tutoring
How can I prove theorems about triangles?
Set:
Begin by asking students to list ways to classify triangles and how to prove triangle
congruence. In addition, have them sketch an isosceles triangle and label the base angle,
vertex angle, legs, and base. This will serve as a review for the upcoming lesson.
Discuss as a class the different classifications of triangles, proof methods and have them
share the different sketches with labels. Discuss the differences and clear up any
misconceptions before the lesson begins.
Protractors
Teaching Strategy:
1. Place students in heterogeneous groups of 3-4.
2. Allow students approximately 10 minutes to create a rough sketch of their pennant
designs and determine the scale they will use for their blueprints. Encourage
students to keep their designs simple, placing more focus on the dimensions and
angles of their designs.
3. Provide students with graph paper, isometric dot paper, rulers, and protractors to
begin their formal drawings, reminding them to include all side and angle
measurements.
4. Once designs are complete, encourage students to begin proof writing, reviewing
the acceptable forms of proof for the project.
5. Come back together as a class. Have a gallery walk so that all the students can see
Heterogeneous
Grouping
Prompting
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Formative
Assessment:
Teacher
observations of
opening
triangle design
and label
Teacher
observations of
pennant design
Teacher
observations of
students
completed
proofs
Performance
Assessment:
Discussions
about isosceles
triangles and
their properties
the designs.
6. Discuss the best designs and proofs. Which type of proof was most clear? Which
one would be of most importance when trying to make a client happy?
Summarizing Strategy:
As an exit ticket, ask the students to answer:
1. Why is it important to have a proof with your design?
2. Name at least 2 types of companies that would require a proof with a proposed
design. Be specific, and tell why they would require a proof.
Homework:
Instruct students to create two pennants of different size, but with identical angle
measurements. Explore similarity and the proportional relationships that exist between
side measurements. What is the minimum number of angles between the two pennants
that must be identical? Discuss AA Similarity and why it does not need to be AAA Similarity?
7
Project Day – See Unit Plan
Colonial Commercial Project – Script Writing and Building
8
Project Day – See Unit Plan
Colonial Commercial Project – Filming
Discussion
about designs
and proofs
Final designs
Final Proofs
Summative
Assessment:
Exit Ticket
Homework
9
I can prove
GeoGebra
theorems about
triangles.
Patty Paper
Rulers
I can prove
theorems about
Index Cards
lines and
angles.
Scissors
“Medians of a
Triangle”
Geogebra
Example
(Appendix F)
Essential Question(s):
Remediation:
Peer Tutoring
1. How can I prove theorems about triangles?
2. How can I prove theorems about lines and angles.
½ Project Day – See Unit Plan
Colonial News Network – Planning
Set:
Present students with the following statement:
“A median is defined as a segment with an endpoint on the vertex and the another
endpoint on the midpoint of the opposite side. Draw a triangle on your patty paper and
construct all three medians. Then, record what you notice about the relationship of the
medians in a triangle.”
Possible student responses: The medians intersect at one point. (You can remind students
that this is called a point of concurrency, specifically, the centroid of the triangle.) Some
students may also realize that the length of the median from the point of concurrency to
the vertex is twice the distance of the length of the segment from the point of concurrency
to the midpoint of the opposite side.
Teaching Strategy:
1. Have students take turns discussing the conjectures they developed about the medians
of a triangle. Let them know that today they will have the chance to test these
conjectures for all types of triangle.
2. Have students work in pairs to create a triangle in GeoGebra, construct the medians of
all three sides, and label.
3. Give students ample time (10-15 minutes) to complete the assignments of measuring
the side lengths and testing their conjectures.
4. Next, have students share their findings “In It to Win It!” Provide students with an
index card, straightedge, and scissors. Tell students they will have two minutes to draw
and cut out a triangle and then figure out how to balance the triangle on a pencil tip. If
Heterogeneous
Grouping
Use of
GeoGebra can
assist students
in taking
precise
measurements
easily
Prompting
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
The use of
GeoGebra
software allows
students in
need of a
challenge a tool
to further
explore and
deepen their
understanding
Formative
Assessment:
Teacher
observations of
opening activity
Teacher
observations of
students
conjectures
Performance
Assessment:
Discussions
about medians
and their
properties
Discussions
about
GeoGebra
findings
Discussions
about centroids
GeoGebra
figure
Index card
triangle
Summative
Assessment:
Exit Ticket
they were able to do this, they must mark the approximate “balancing point” on the
triangle where they were able to balance it.
5. After the two minutes are up, give students two minutes to construct the centroid of
their triangle, using only their triangle and a straightedge. Then, have them determine
the relationship between their “balancing point” and the centroid. It should be about
the same.
Summarizing Strategy:
As an exit ticket, ask students:
1. What are two useful properties of a triangle’s centroid?
2. What STEM professions would need to use centroids?
10
I can prove
Geogebra
theorems about
Compass
triangles.
Patty paper
I can prove
theorems about
Ruler (or
lines and
straight edge)
angles.
“Incenter and
Circumcenter”
Activity Sheet
(Appendix H)
Essential Question(s):
Remediation:
Peer Tutoring
1. How can I prove theorems about triangles?
2. How can I prove theorems about lines and angles?
½ Project Day – See Unit Plan
Colonial News Network – Filming
Set:
Ask students to being by writing down 3 facts or properties they remember about
circumcenters. Help to jog their memory if necessary. Discuss as a class all of the different
facts and properties mentioned. Fill in any gaps and clear-up any misconceptions.
Teaching Strategy:
1. Explain to students that they will be investigating two different points of
concurrency in this activity.
2. Provide students with a copy of the “Incenter and Circumcenter” Activity Sheet and
have them work through the activity in pairs.
3. Give students about 20 minutes to complete the activity. As a class, discuss the
following questions:
a. Where does the incenter occur in a right triangle, an acute, and an obtuse
triangle?
b. What conjectures did you develop and test about the incenter?
Heterogeneous
Grouping
Use of
GeoGebra can
assist students
in taking
precise
measurements
easily
Prompting
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
The use of
GeoGebra
software allows
students in
Formative
Assessment:
Teacher
observations of
opening activity
Teacher
observations of
students
conjectures
Performance
Assessment:
Discussions
about
circumcenters
and their
properties
Discussions
about
GeoGebra
findings
Discussions
about incenters
c. If the incenter is equidistant from all of the angles of a triangle, what other
features do you think it has? Give students an opportunity to construct this
with their triangles.
d. Where does the circumcenter occur in a right triangle, acute triangle, and
obtuse triangle?
e. What conjectures did you develop and test about the circumcenter?
f. If the circumcenter is equidistant from the sides of a triangle, what other
features do you think it has? (You can use it to inscribe a circle inside a
triangle.) Give students an opportunity to construct this with their
triangles.
4. As a class, have students review the theorems about the incenter and circumcenter
of a triangle and discuss them as a group.
need of a
challenge a tool
to further
explore and
deepen their
understanding
GeoGebra
figures
Final triangle
Summative
Assessment:
Exit Ticket
Summarizing Strategy:
As an exit ticket, ask the students:
1. What is the difference between the incenter and the circumcenter?
2. Name 2 properties of an incenter.
11
Patty paper
I can prove
theorems about
parallelograms. Ruler (straight
edge)
Essential Question(s):
How can I prove theorems about parallelograms?
½ Project Day – See Unit Plan
Compass
Colonial News Network – Filming
“Parallelograms” Set:
Let the students know that today they will apply the definition of a parallelogram for
Activity
geometric construction. Have them write down all the properties they know about
(Appendix G)
parallelograms. Discuss with the class what they have written, and clear up any
misconceptions.
iPads
Teaching Strategy:
1. Place the students in heterogeneous groups of 3-4.
2. Give students the “Parallelograms” Activity
3. Once students have completed the activity create a gallery walk for students to see
all of the solutions
4. Discuss the different solutions, and talk about how they are different. How are they
alike?
5. Have students work independently and look up 4 ways parallelograms are used in
Remediation:
Peer Tutoring
Heterogeneous
Grouping
Use of
laptops/iPads
and websites to
provide support
Formative
Assessment:
Teacher
observations of
opening activity
Performance
Assessment:
Discussion
about
parallelograms
Prompting
Final figure
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Use of
laptops/iPads
Final findings of
using
parallelograms
in the real
world
Summative
the real world. Come back together and share as a class.
12
I can prove
“Running up the
theorems about Scoreboard”
triangles
(Appendix H)
Meter or yard
stick
Mirror
Calculator
Assessment:
Exit Ticket
Summarizing Strategy:
As an exit ticket, have students answer:
1. What are the properties of parallelograms?
2. Why are these properties useful, and how do we use them?
and websites to
provide a
enrichment for
deeper
investigation at
the student’s
pace
Essential Question(s):
How can I prove theorems about triangles?
Remediation:
Peer Tutoring
Formative
Assessment:
Teacher
observations
early estimates
and methods
½ Project Day – See Unit Plan
Colonial News Network – Filming
Set:
Present the “Running up the Scoreboard” activity to the class. Have groups write an initial
guess of how high they think the scoreboard is. Share the guesses with the class and ask
how they came to that estimate.
Teaching Strategy:
1. Assign students to heterogeneous groups of 3-4. Give each group a meter or yard
stick, a small mirror, and a calculator.
2. Have students complete the “Running up the Scoreboard” activity.
Summarizing Strategy:
As an exit ticket, ask students to answer:
1. What other considerations should be included in choosing the height of the ladder?
2. What danger do you think O.S.H.A. feels might be present if the ladder is set too
close to the wall? What danger do you think O.S.H.A. feels might be present if the
ladder is set too far away from the wall?
3. Are there any other methods you can think of to find the height of the scoreboard?
Homework:
1. Find the heights of other objects. Have students find others in the school or assign
Heterogeneous
Grouping
Prompting
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Teacher
observations of
students
finding the
height of the
scoreboard and
ladder
Performance
Assessment:
Discussion
about
estimates and
methods
Final
calculations
Summative
Assessment:
Exit Ticket
Homework
2.
3.
4.
5.
13
I can find the
point on a
directed line
segment
between two
given points
that partitions
the segment in
a given ratio.
Graph paper
Calculators
Envelopes
them to find the height of something at home.
Introduce the idea of clinometers to find angle of elevation. If it did not come up in
question 3 on the exit ticket, remind students of the idea of trig ratios to solve right
triangles.
Take the students to the gym to find the height of the scoreboard. Have groups
compare their guess with the height they found.
After students have successfully found the height of the scoreboard in your gym,
bring them back to class and have them use that height and the ladder diagram
(also included in the “Running up the Scoreboard” Activity) to find the proper
length of the ladder.
Have groups compare answers and make any necessary revisions.
Essential Question:
How can find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
½ Project Day – See Unit Plan
Colonial News Network – Filming and Editing
Set:
Scott and Misty are starting a tutoring business called Math Made Easy. To open their
business, they must come up with $1,500 to cover the fees for starting the business and
legal fees. Because Misty will be the principal owner, they have both decided that she will
invest five times the amount of money that Scott will invest. How much will each owner
invest? What is the ratio of their investments? Have students discuss and compare their
methods of solving the given problem. Use this as a chance to identify any misconceptions
and provide clarity.
Teaching Strategy:
1. Assign students to heterogeneous pairs. Give each pair multiple pieces of graph paper,
calculators, and the following choices:
POINTS: (6, 6), (10, 10), (12, 12)
Remediation:
Peer Tutoring
This can serve
as a formative
assessment of
Heterogeneous whole class or
Grouping
individual
understanding.
Prompting
If the goal is to
assess whole
Enrichment:
class
Peer Tutoring
understanding,
student
Heterogeneous identifiers are
Grouping
unnecessary. If
the aim is to
Students in
assess
need of a
individual
challenge may, learning,
as they
consider giving
complete the
each student a
envelope
number, letter,
activity. Give
color, or unique
students the
sticker to apply
following pair
to his/her
of points (-2, -3) answers. As a
and (6, 13), and class, discuss
have them find
the point which
Tell the pairs of students to choose one of the points and two of the given ratios. Have divides the
segment
each pair draw a directed segment from the origin to their given point and have them
locate the ratios they have in common. Have groups share their methods with the class connecting
these points
and discuss which points were easier with which ratios and why.
into a 3:1 ratio.
(ANSWER – (4,
2. At this point, each pair should have two segments, separated into given ratios. They
9))
may wish to re-draw clean copies of each of their segments in order to proceed with
the lesson. Have students draw segments perpendicular to the x-axis in order to
construct similar right triangles.
3. Have the pairs compare the triangles they have created and make conjectures about
similarity within the triangles. As a class, discuss the conjectures made. Students
should recognize that the triangles are proportional. Have students justify this using
proportional reasoning. If necessary, have students choose a different point and ratio,
repeat the process, and justify their conjectures again. If students are not convinced,
have them attempt to generate a counterexample. As an extension, have students
explore the relationship between the areas of the two triangles.
4. Give each pair more graph paper, if necessary, and their choice of the following points:
POINTS: (8, 1), (1, 7), (9, 7)
RATIOS: 2:1, 3:1, 1:1, 1:3
Have students draw a new directed segment and choose one of the original ratios.
Instruct them to attempt to find the point at which the segment is partitioned into the
ratio they chose. Discuss how this process is different from what they have already
done, and different types of methods they could use.
Have students create boxes labeled with thee points:
(8, 8) and (20, 20), (6, 6) and (14, 14), (7.5, 9) and (25, 30), (10, 15) and 4:1, (18, 27) and
4:5, (16, 24) and 3:5
5. Have pairs classify each set of points and place them in the appropriate box. For the
stations with two points listed they must figure out the ratio at which the first point
partitions the directed segment that begins at the origin and ends at the second point.
6. When everyone has completed each problem, have students share their answers and
how they got them. Allow students to discuss their methods and decide which methods
worked best and why. Use this as an opportunity to clarify and misconceptions
how the
process might
change if the
directed
segment did
not begin at the
origin. If there
is time,
attempt the
extension
problem in
small group
and discuss it
as a class.
students may have.
Summarizing Strategy:
As an exit ticket, ask the students:
Find the points that would partition the directed segment connecting the origin to these
points into a 2:3 ratio.
1. (15, 15)
2. (25, 40)
As you collect the exit tickets, have students name one thing they learned for the day and
one Standard for Math Practice they feel they used.
Homework:
Create two directed segments and find the points at which those segments are partitioned
into three ratios of their choosing.
Adapted from tasks available through the Common Core Website (2013).
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STANDARDS
Identify what you want to teach. Reference State, Common Core, ACT
College Readiness Standards and/or State Competencies.
G.CO.A.2- Represent transformations in the plane, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs
and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
G.CO.C.9-Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are
congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
G.CO.C.10- Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the
segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.CO.C.11-Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect
each other, and conversely, rectangles are parallelograms with congruent diagonals.
G.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.
G.SRT.B.4 Prove theorems about triangles.
G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G.GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.