Download 8th Math Agriculture

Agriculture
Two Weeks
Geometry
Lesson Plan
Teacher:
Grade:
8th Grade Math Teacher
8th Grade
Lesson Title:
Congruence and Right Triangles: Their Properties and How They Lend to Trigonometric Ratios and Real World Relevance
STRANDS
Similarity, Right Triangles, and Trigonometry
Congruence
LESSON OVERVIEW
Summary of the task, challenge, investigation, career-related scenario, problem, or community link.
The unit will start with an investigation of congruence and the real world relevance of using congruence in order to be time and cost efficient. Students will then
investigate how different transformations affect the congruency of triangles. Students will also learn the minimum information needed in order to recreate a triangle
that is congruent to the original. This will lead into student investigating right triangles and how their similarities make way into the trigonometric ratios that allow
humans to solve large-scale measurement problems. Social Studies will be examining agricultural practices used in the past, how they have changed over time, and how
agriculture has shaped the United States. Language Arts will be exploring agriculture through literature as well as practicing technical writing at the conclusion of the
lab. As well as examining physical and chemical changes students will also develop a working definition for matter, gain an understanding of density, and investigate
states of matter. This will culminate in a raised bed vegetable garden project that will integrate the finding from all subjects. Ben Hunter, with the Agriculture Extension
Office, will come during the final 2 project days to assist and educate students on the construction and maintenance of the raised bed vegetable gardens.
MOTIVATOR
Hook for the week unit or supplemental resources used throughout the week. (PBL scenarios, video clips, websites,
literature)
Day 1 -"Modular Efficiency":
This motivator will utilize the following video clip – “Modular Efficiency” (Appendix A). The students will then discuss the pros and cons of modular construction.
What type of figures are these modules used to build (congruent figures), and how does having modules (congruent figures) make a company more efficient? The
teacher will lead the discussion of how modular construction is utilized in STEM fields in order to be cost and time efficient.
Day 4 -"Distance to the Stars":
This motivator will utilize the following video clip – “Distance to the Stars” (Appendix F). The students will then discuss how trigonometry is used in today’s society
after watching the informative video. Things will be brought up such as GPS and satellite location. The students will then debate Other types of STEM careers and
technology that involve the use of triangles and trigonometry. Teacher will lead the discussion by offering how trigonometry is a leading component in structural
analysis, as a measuring device for all things that are too large to physically measure, and sailing.
DAY
Objectives
(I can….)
1
I can use the
definition of
congruence in
terms of rigid
motions to show
that two triangles
are congruent if
and only if
corresponding
pairs of sides and
corresponding
pairs of angles are
congruent.
Materials &
Resources
Instructional Procedures
“Modular
Efficiency”
(Appendix A)
Essential Question(s):
How can I use the definition of congruence in terms of rigid motions to show
that two triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent?
“Verifying
Triangle
Congruence”
(Appendix B)
Ruler (or
straight edge)
Calculator
“Need More
Support”
Verifying
Triangle
Congruence Task
(Appendix C)
“Need More
Challenge”
Triangle
Congruence Task
(Appendix D)
Differentiated Assessment
Instruction
Remediation:
Peer Tutoring
Heterogeneous
Grouping
Set:
Teacher will begin by showing the “Modular Efficiency” video clip, The
students will then discuss the pros and cons of modular construction. What
type of figures are these modules used to build (congruent figures), and how
does having modules (congruent figures) make a company more efficient? The
teacher will lead the discussion of how modular construction is utilized in
STEM fields in order to be cost and time efficient. The teacher will then ask
the students how they may prove that these figures are congruent (sides,
angles.)
Teaching Strategy:
1. Place students in pairs and hand out “Verifying Triangle Congruence” Task. Also
have available the “Need More Support” Triangle Congruence Task and “Need
More Challenge” Triangle Congruence Task for students in need of differentiated
instruction.
2. Have the students work together to transform the figure, and determine if the
new figure is congruent to the original figure.
3. Come back together as a class and conduct a class discussion using the following
discussion questions:

“How does a reflection, rotation, or translation affect the corresponding
parts of a transformed figure?”
 “Is there a transformation that does not create a congruent figure?”
“Using your prior knowledge of angles, side lengths, and triangles is there an
alternative approach to verifying the triangles are congruent besides showing
“Need More
Support”
Triangle
Congruence
Task. This gives
the graphed
figure with the
angles and side
measurements
already
completed. It
also gives the
students the
different types
of
transformations
they are
investigating.
Enrichment:
Peer Tutoring
Formative
Assessment:
Teacher
Observations
Performance
Assessment:
Exit Ticket
Summative
Assessment:
Verifying Triangle
Congruence Task
graphs and Results
that all of the corresponding sides lengths are congruent and all of the
corresponding angles are congruent.”
2
I can explain how
the criteria for
triangle
congruence (ASA,
SAS, and SSS)
follow from the
definition of
congruence in
terms of rigid
motions.
Construction
paper
“Bulletin Board
Congruence”
Task (Appendix
E)
Rulers
Protractor
Markers
Heterogeneous
Grouping
Summarizing Strategy:
As an exit ticket, have students summarize their findings. What needs to be true in
order for triangles to maintain congruency (sides and angles)? What is the only
transformation that does not maintain congruency?
“Need More
Challenge”
Triangle
Congruence
Task. This
deepens
students
understanding
of congruence
by questioning
what
transformations
do not conserve
congruence, but
do conserve
similarity within
the transformed
figures.
Essential Question(s):
Remediation:
Peer Tutoring
How can I explain the criteria for triangle congruence (ASA, SAS, and
SSS) follow from the definition of congruence in terms of rigid motions?
Set:
Begin by asking students to jot down what it means for two or more triangles
to be congruent to one another. Ask them to think about and write down how
they would go about creating congruent triangles. Come together as a group
and discuss answers from the students
Teaching Strategy:
1. Divide the class into groups of 2-3. Provide the students with a copy
of the “Bulletin Board Congruence” Task and a large piece of
construction paper. The paper will be used as the size of the bulletin
board. Allow students a few minutes to read over task and make any
additional notes or jot down any questions.
Heterogeneous
Grouping
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Formative
Assessment:
Teacher
observations of
methods used to
create congruent
triangles
Performance
Assessment:
Ending discussion
of methods used
by students
Summative
Assessment:
2. Allow students time to work on the task and create solutions. Walk
around and observe the students. Ask them about their techniques
and how they are meeting the goal of the task.
3. Allow the class to complete a gallery walk to see the different
solutions from around the classroom.
Completed
Bulletin Boards
Exit Ticket
Summarizing Strategy:
As an exit ticket, have students summarize their findings. What is the smallest
number of measurements needed (sides and/or angles) to recreate congruent
triangles?
3
Project Day – See Unit Plan
Feeding America: Exploring Raised Bed Gardening –
Research and Planning
4
I can understand
that by similarity,
side ratios in right
triangles are
Printer Paper
“Distance to the
Stars” Video Clip
(Appendix F)
Essential Question(s):
How can I understand that by similarity, side ratios in right triangles are
properties of the angles in the triangle, leading to definitions of trigonometric
ratios for acute angles?
Remediation:
Peer Tutoring
Give students
Formative
Assessment:
Teacher
observations of
properties of the
angles in the
triangle, leading
to definitions of
trigonometric
ratios for acute
angles.
“Graphic Chart”
(Appendix G)
“Trigonometric
Table Spread
Sheet.”
(Appendix H)
Graphing
Calculator
Ruler
Protractor
Ruler
“Pre-designed
Right Triangle”
(Appendix I)
“Need More
Challenge” Task
(Appendix J)
Set:
Begin the class by showing the video clip: “Distance to the Stars.” The students will
then discuss how trigonometry is used in today’s society after watching the
informative video. Things will be brought up such as GPS and satellite location. The
students will then debate Other types of STEM careers and technology that involve
the use of triangles and trigonometry. Teacher will lead the discussion by offering
how trigonometry is a leading component in structural analysis, as a measuring device
for all things that are too large to physically measure, and sailing.
Teaching Strategy:
1. Give each student a sheet of unlined paper a protractor and a ruler. Instruct
students to draw, as accurately as possible, a right triangle with one of its acute
angles measuring 30o. Ask students, in pairs, to compare triangles. Ask students
to share their observations. (Students should recognize that, while some of the
triangles may be approximately congruent, all the triangles in the room are
approximately similar.)
2. Have students form trios by finding two partners that created triangles that are
“different” than theirs. Ask, “What are the differences that made you choose
your group members? What is still the same about your triangles?”
3. Have groups measure, as accurately as possible, the lengths of all three sides of
their triangles and find the measure of the unknown angle. (They may find the
angle by measuring, but some may use the fact that the sum of the measures of
the angles of a triangle is 180o to find that the remaining angle measures 60 o.)
Explain to students a system of identifying the sides of their triangles. (One side
is the “hypotenuse” and will be denoted as “H.” The hypotenuse will be a
previously learned concept. It may be necessary to review it in the context of the
Pythagorean theorem. The legs can now be identified relative to one of the
acute angles called the “reference angle.” The leg across from the reference
angle is called the “opposite leg,” which will be denoted as “O,” and the leg which
makes up one side of the reference angle is called the “adjacent leg” which will
be denoted as “A.”) Instruct students to use the 30o angle as the reference
angle. Have each group choose one pair of sides and create a ratio of those sides
for each triangle in the group and find a decimal approximation of the ratio.
(Groups should find that the ratio they find is equal for all three triangles.) Have
groups report their results. Record the results on a board or projector (using the
graphic organizer). Give each group the “Graphic Chart” to record the results for
any of the ratios found by the class. If all six ratios have not been chosen, have
the class find the remaining ratios in groups and report them to complete the
table. Have groups add these labels to three of the ratios on the chart: O/H =
sine 30o, A/H = cosine 30o, O/A = tangent 30o. Point out abbreviations “sin, cos,
tan.” (Anticipate the possibility of questions as to why the other three ratios,
the “Predesigned Right
Triangle”
with the 30o
angle,
hypotenuse,
adjacent side
and opposite
side clearly
identified.
Enrichment:
Peer Tutoring
student
conclusions
Performance
Assessment:
Discussions
throughout task
Summative
Assessment:
Completed
“Graphic Chart”
Triangle Designs
Give students
the option of
the “Need More
Challenge” Task
in which
students
deepen their
understanding
of the
relationship
between the
acute angle,
length of sides,
and their
trigonometric
properties.
Exit Ticket
4.
5.
6.
cosecant or csc and secant or sec and cotangent or cot, are not being named at
this point. In this case, point out that these ratios are merely reciprocals of the
first three and not necessary for relating the sides at this point. Feel free to
identify them by name to satisfy curiosity.) Ask,
 In any right triangle with a 30o angle, what do we now know about the
ratio of the leg opposite the 30o reference angle to the hypotenuse
(O/H)? (It is always the same and equal to .5.)
 In any right triangle with a 30o angle, what do we now know about the
ratio of the leg adjacent to the 30o reference angle to the hypotenuse
(A/H)? (It is always the same and equal to approximately .8660.)
 In any right triangle with a 30o angle, what do we now know about the
ratio of the leg opposite to the 30o reference angle to the leg adjacent to
the 30o reference angle (O/A)? (It is always the same and equal to
approximately .5774.)
 What is the measure of the other acute angle? What would happen to
the trig ratios if we used this angle as the reference angle?
Ask, “What would happen to the trig ratios if we used a different angle? What,
more specifically, do you think would happen to the sin and cos of the angle if we
reshaped the triangle so that the reference angle was changed from 30o angle to
40o? Be prepared to explain why you think this will be true.” Have students
discuss the accuracy of the answers. Elicit responses from groups who found
specific answers gleaned from ratios. Note these answers as with 30 o before.
Look for or make a suggestion that there are other ways than construction of a
triangle to find the ratios.
Make sure each group has a scientific or graphing calculator. (One calculator per
student is preferred.) Ask them to explore and find the sin30o, cos 30o, sin 40o,
and cos 40o on their calculators. Tell them they will know they have found it
correctly, without using a constructed right triangle, when they get the same
approximate answers as discovered earlier. Ask them to find tan of the two
reference angles as well.
Give each student a copy of the “Trigonometric Table Spread Sheet.” Ask
students to find the sin, cos, and tan of 30o and 40o on the table and compare the
answer to that on the calculator. (Note that the answers are approximate and
that the decimals for the ratios non-terminating and non-repeating. Ask, “What
generally happens to the sine as the reference angle gets bigger? What happens
to the cosine and tangent?” Other than, looking at answers on the table or
calculator, why do you think this is true?
Summarizing Strategy:
As an exit ticket, have students summarize their findings. How do acute angles lend to
the trigonometric functions? How are these functions useful in STEM professions?
5
I can explain and
use the
relationship
between the sine
and cosine of
complementary
angles.
“Trigonometric
Table Spread
Sheet.”
(Appendix H)
Ruler (or
straight edge)
Essential Question(s):
How can I explain and use the relationship between the sine and cosine of
complementary angles?
Remediation:
Peer Tutoring
Set:
Begin by asking students to jot down what it means for two angles to be
complementary. Ask them to then write down what observations they can
make about complementary angles in a triangle.
Heterogeneous
Grouping
Teaching Strategy:
1. The goal of this lesson is to assist the learner in seeing that the sine and
cosine of complementary angles are the same. There are a variety of ways to
initially establish this. Some options include:
a) Have students explore on the accompanying trigonometry table
spreadsheet for answers that are the same. Ask them make a
conjecture about what sorts of angles and trig functions produce
equal trig ratio.
b) Have students explore on a calculator for answers that are the same.
Ask them make a conjecture about what sorts of angles and trig
functions produce equal trig ratio.
2. Have students write a conjecture as to the equality of sine and cosine of
complementary angles.
3. After a conjecture has been established of the equality of sine and cosine of
complementary angles, have students work in groups and produce an
explanation (or proof) of why these ratios would be equal. Have students
draw a triangle and label parts.
Note: Here, students will be asked to discover a rationale for the equality.
These hints, given judiciously, will help the process along:
a) Note that the two acute angles of a right triangle are complementary
b) Note that the designations of the sides (opposite leg, adjacent leg)
will change depending on the reference angle.
c) On the right triangle, label the legs “a” and “b” and then find the sine
and cosine relative to each reference angle.
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Formative
Assessment:
Teacher
observations of
conclusions as to
why sine and
cosine of
complementary
angles is equal
Performance
Assessment:
Proof by group
work of the given
conjecture
Summative
Assessment:
Exit Ticket
Summarizing Strategy:
As an exit ticket, have students summarize their findings. What do we know
about the sine and cosine of complementary angles? Why is this always true?
6
I can use
trigonometric
ratios and the
Pythagorean
Theorem to solve
right triangles in
applied problems.
I can understand
and apply the Law
of Sines and the
Law of Cosines to
find unknown
measurements in
right and nonright triangles
“Constructing a
Clinometer” –
Video Clip
(Appendix K)
“Checklist for
the
Presentation”
(Appendix L)
Protractors
String
Straws
3x5 Notecards
Paper Clips
Tape
Pens
iPad
A basket for
Each Pair
Containing the
Items for the
Clinometer
Measuring Tape
Essential Question(s):
1. How can I use trigonometric ratios and the Pythagorean theorem to
solve right triangles in applied problems?
2. How can I understand and apply the Law of Sines and the Law of
Cosines to find unknown measurements in right and non-right
triangles?
Set:
Ask students to guess the height of the school building, school flagpole, or the
trees outside of your school building? Ask each of them to give justification to
support their guess. After hearing their guesses, tell them they will learn to
measure the height of these objects using their knowledge of trigonometric
functions and a simply constructed tool known as a clinometer.
Teaching Strategy:
Place the students in pairs. Let them know that together they will determine
the height of the school building using indirect measure and a clinometer that
they will design and construct. They will organize their results into a
presentation such as a poster, or power point, or iMovie. Be prepared to
explain step-by-step how you determined your answer to your classmates.
*Note: Students need to understand the indirect measurement and solving
right triangles before completing this task.
1. View the video “Constructing a Clinometer.”
2. Allow students 8-10 minutes to construct their clinometer and
determine how to use the instrument along with their
mathematical knowledge of trigonometric functions to find the
height of the objects
3. Come together as a class and discuss the length of time available
for the measurements, as well exactly what data will need to be
obtained to solve for the height of the object.
4. Develop a list of objects to measure around your school and
create a tour of the school using the clinometers and indirect
Remediation:
Peer Tutoring
Heterogeneous
Grouping
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Formative
Assessment:
Teacher
observations of
clinometer
construction
Teacher
observation of
techniques used
for solving the
problem
Teacher
observation of
solutions to the
problem.
Performance
Assessment:
Discussion of data
required in order
to find the height
of the object.
Students using
clinometers as a
tool for
measurement
Summative
Assessment:
Exit ticket
Calculators
5.
6.
7.
8.
measurement to determine the height of objects.
Take a walk around the school and obtain data that will help
determine the height of the objects listed.
Come back to the classroom and allow the students some time to
calculate their solutions.
Pass out a “Checklist for the Presentation” and discuss what will
be expected for the presentation. Presentations could be posters,
power points, iMovies, etc.
Let the students know they are to bring in the supplies tomorrow
to work on their presentations.
Summarizing Strategy:
As an exit ticket, ask students to summarize their findings. How did the
clinometer work? What type of trigonometric function was used to solve for
the missing side?
Homework:
Ask students to plan their presentation and bring in the needed supplies to
complete the presentation
7
I can use
trigonometric
ratios and the
Pythagorean
Theorem to solve
right triangles in
applied problems.
I can understand
and apply the Law
of Sines and the
Law of Cosines to
find unknown
measurements in
right and nonright triangles
Laptops
iPads
Materials
Students are
Bringing for
Their
Presentation
Essential Question(s):
1. How can I use trigonometric ratios and the Pythagorean theorem to
solve right triangles in applied problems?
2. How can I understand and apply the Law of Sines and the Law of
Cosines to find unknown measurements in right and non-right
triangles?
Set:
Begin the class by discussing the various results from yesterday’s findings? Are
the solutions found equal or approximately equal? What techniques were
used? What steps did you take as a group to ensure the measurements are
accurate?
Teaching Strategy:
1. Have students get into their pairs and begin working on their
presentations from yesterday’s findings.
2. Make sure each pair of students has the “Checklist for the
Presentation.”
3. Observe the students as they are putting the presentation together.
Make sure they are displaying correct information. Ask questions to
Remediation:
Peer Tutoring
Heterogeneous
Grouping
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Formative
Assessment:
Teacher
observations of
justifications used
in presentations
for group
solutions.
Performance
Assessment:
Discussion of
solutions found by
various groups
Summative
Assessment:
Final Presentation
guide the students in the correct direction.
4. Ask the students to turn in their presentations with about ten minutes
left in the class period.
5. Take a gallery walk of the presentations.
Summarizing Strategy:
As an exit ticket, ask students decide which presentation best explained how
to use trigonometric ratios to solve the height problem? Why was this
presentation method best?
8
Project Day – See Unit Plan
Feeding America: Exploring Raised Bed Gardening – Building
and Planting
9
Project Day – See Unit Plan
Feeding America: Exploring Raised Bed Gardening – Building
and Planting
Exit ticket
10
I can derive the
formula A = ½ ab
sin(C) for the area
of a triangle by
drawing an
auxiliary line from
a vertex
perpendicular to
the opposite side.
iPads
Rulers
Protractors
Graph Paper
Calculator
“Superman’s
Shopping Center
Site Map and
Scenario.”
(Appendix M)
“Triangles
Resource” Sheet
(Appendix N)
Essential Question(s):
How can I derive the formula A = ½ ab sin(C) for the area of a triangle by
drawing an auxiliary line from a vertex perpendicular to the opposite side?
Remediation:
Peer Tutoring
Set:
Tell students Superman’s Shopping Center wants to build a new store in the
space bounded by three roads. This would mean they would have to develop
on an area that was currently a forest. The Department of Natural Resources
(DNR) has restrictions in this area that no more than 120,000 square feet of
forest may be taken down during a single building project. You and your team
have been hired by Superman’s Shopping Center to advise them on whether
or not their building project will exceed these limits. Because much of the
land has not been developed you only know certain distances. You do know
the distance of Road A from points B to C measures 650 feet and the distance
of Road B from points A to C measures 800 feet. You also know that Roads A
and B intersect at a 35 degree angle. Using these measurements, determine if
the area exceeds the 120,000 square feet.
Heterogeneous
Grouping
Teaching Strategy:
1. Before beginning the task, access students’ prior knowledge of the proof
of the area of a triangle. If students need a review of this topic, provide
students with the virtual manipulates link below to explain why this
formula works. http://math.kendallhunt.com/x19469.html
2. After accessing students’ prior knowledge of the area of a triangle, divide
the class into small groups of 3-4 students. Then, provide them with the
“Superman’s Shopping Center Site Map and Scenario.” Provide students
with rulers, graph paper, and protractors in order to create a model of the
store.
3. Allow students a few minutes to ask questions before they begin. Remind
them after the questioning period is over; the teacher role will be to
observe and not be to answer questions or help with the activity.
*Note: This allows students time to try the problem on their own without
interference and gives them appropriate time to struggle while encouraging
them to use what they have previously learned. Students who are still
struggling can use the “Need More Support” task.
4. As groups are finishing, have them summarize their solutions on their
graph paper. Groups can then post their solutions with any necessary
diagrams around the room.
5. Have students participate in a gallery walk, while thinking about the
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
Require
students submit
a scale model of
the store. The
scale must be
included with
the model and
clearly visible.
Formative
Assessment:
Teacher
observations
student prior
knowledge of the
Triangle Proof.
Performance
Assessment:
Discussion about
creating similar
figures.
Student
transformed
similar figures
Summative
Assessment:
Exit Ticket
following questions:
a. How did other groups’ solve the problem?
b. Which group had the most efficient method?
c. Did everyone come to the same conclusion? Why or why not?
Summarizing Strategy:
As an exit ticket, ask students what technique their group used to solve the
problem. Whose group had the most effective technique and why?
STANDARDS
Identify what you want to teach. Reference State, Common Core, ACT
College Readiness Standards and/or State Competencies.
G.CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
G.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G.SRT.B.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute
angles.
G.SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G.SRT.D.9 Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
G.SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.
G.SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems,
resultant forces).