Download 8th Math Leadership

Leadership
Two Weeks
Geometry
Lesson Plan
Teacher:
Grade:
8th Grade Math Teacher
8th Grade
Lesson Title:
Experimenting with Perimeter, Area, Transformations, Congruency and Similarity for Future STEM Career Leaders.
STRANDS
Similarity, Right Triangles, and Trigonometry
Expressing Geometric Properties with Equations
Congruence
LESSON OVERVIEW
Summary of the task, challenge, investigation, career-related scenario, problem, or community link.
The unit will start with a STEM career professional speaking to students about their competitive advantage entering the workplace given their STEM education.
Students will take information from their readings in English class and write an essay with central ideas on the changing face of education since the 1900s to STEM in the
workplace. Students will review finding perimeter and area of polygons in the different forms in which they can be represented (coordinates, figures, and equations) by
using various methods such as the distance formula and Pythagorean’s Theorem. Students will then design a floor plan of their dream home and backyard, scale the
drawing, and estimate the costs of putting up fencing and laying down sod given a price list for each of those items. This not only reinforces important Common Core
Standards, but also prepares students in planning for their future homes and showing how to estimate costs before purchasing items. This will also prepare students for
their scale drawing that will be required with the catapult project. The video clip about the Mercedes Bens Logo and it’s design will lead into the lessons on
transformations and tie into the final math component for the State of Stem culminating presentation while tying in the history of mathematics used in business.
Students will then experiment with transformations in the plane. Students have had experience with transformations: translations, reflections, rotations, and dilations
from Algebra I and Pre-Algebra, and they will be reviewed. Comparisons of transformations will provide the foundation for understanding similarity and congruence,
which will also play a part in the culminating presentation requirements. Students will then perform various transformations as requested. Similarity and Congruence
will then be visited, first with definitions and following up with discussion questions. These questions are intended to formatively assess prior knowledge and to begin
student discussion of similarity versus congruence. Students should be encouraged to answer in their own words and to critique each other’s assessments. This allows
students to practice communication of knowledge using language rather than algebraic expressions to demonstrate definitions and the importance of math concepts.
Hook for the week unit or supplemental resources used throughout the week. (PBL scenarios, video clips, websites,
literature)
MOTIVATOR
Day 1 "Home Quick Planner":
This motivator will utilize the following video clip – “Home Quick Planner” (Appendix A). The students will then discuss what formulas are used to calculate distance,
and how these could be used in real world applications.
Day 3 "Motion Geometry":
This motivator will utilize the following video clip – “Motion Geometry” (Appendix F). The students will then discuss how transformations are used in today’s society
after watching the informative video. Things will be brought up such as games and digital animation. The students will then debate what kinds of skills and education
would be necessary to be a video game designer or creator of digital animation. Teacher will lead the discussion by offering how mathematics is a leading component
in programming and animation.
Day 6 “The History of the Mercedes Logo”:
This motivator will utilize the following video clip – “The History of the Mercedes Logo” (Appendix I). The students will discuss pre-image, and how the pre-image of
the Mercedes Logo would compare to a rotated image of the same figure. Since they would appear the same, the teacher would explain this gives the Mercedes Logo
the property of rotational symmetry. The students will need this knowledge to design their own logo with rotational symmetry for the “State of STEM Presentation”
as well as lead into further investigation of transformed figures and their pre-images.
DAY
Objectives
(I can….)
1
I can use
coordinates to
compute
perimeters of
polygons and
areas of triangles
and rectangles.
I can know
precise
definitions of
angle,
perpendicular
line, parallel line,
and line segment,
Materials &
Resources
Instructional Procedures
“Home Quick
Planner”
(Appendix A)
Essential Question(s):
1. What are polygons?
2. How to find the area and perimeter and area of a polygon?
3. How to define angle, perpendicular line, parallel line and line segment
based on undefined notions of point, line and distance along a line?
Set:
Teacher will begin by showing the “Home Quick Planner” video clip, then
asking students to write down the distance formula, Pythagorean’s Theorem,
and name one situation where you may use each formula. How are they
similar and can they be used for the same purpose? Teacher will activate
discussion regarding distance formulas and their real world applications such
as floor plans for homes, determining how to estimate supplies needed for
construction, and how knowing perimeter and area of a given location could
change the approach to a design.
Graph paper
Station Work
(Appendix B)
Ruler (or
straight edge)
iPad
Calculator
“Need More
Teaching Strategy:
Differentiated Assessment
Instruction
Remediation:
Peer Tutoring
Heterogeneous
Grouping
“Need More
Support”
Stations
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
“Need More
Challenge”
Stations
Formative
Assessment:
Opening Writing
Assignment
Teacher
Observations
Performance
Assessment:
Exit Ticket
Summative
Assessment:
Work Station
calculations with
graphs where
based on the
undefined
notions of point,
line, and distance
along a line.
2
I can use
coordinates to
compute
perimeters of
polygons and
areas of triangles
and rectangles.
Support” Station
Work
(Appendix C)
“Need More
Challenge”
Station Work
(Appendix D)
Graph paper
“Dream Home
Design”
Handout
(Appendix E)
Ruler (or
straight edge)
I can use
geometric shapes,
Protractor
their measures,
and their
iPad
properties to
Calculator
describe objects.
I can apply
geometric
methods to solve
design problems.
1. Teacher will assign students to heterogeneous groups of 2-4.
2. Each group will travel to Stations 1-4 (Appendix B) and complete the
directions given at each station. All stations are concerning perimeter
and area of differently shaped polygons presented in different forms.
Groups can pick from three folders at the station: Assigned Work
(Appendix B), Need More Challenge, (Appendix D) and Need More
Support (Appendix C). Teacher will direct students to the folders that
are appropriate given the group ability level.
3. Have the students complete the stations and regroup for a discussion.
Were any of the stations harder than the others? What technique
was most effective for calculating the length of each polygon’s side?
Would the order in which you complete the stations make a
difference?
Summarizing Strategy:
As an exit ticket, have students summarize their findings. Ask what they think
their strengths and weaknesses are for finding perimeter and area for given
polygons.
Essential Question(s):
1. How to use geometric shapes, their measurements, and their properties to
describe objects?
2. How to apply geometric methods to solve design problems?
Set:
Begin by asking students about their dream home floor plan. How many
bedrooms and bathrooms would you like? Explain that they get to design their
own one floor, dream home design. All of this must be put into a plan first,
and perimeter and area must be calculated to estimate the total costs in order
to estimate the amount of materials to buy and what they may cost.
Teaching Strategy:
1. Have students design their floor plan design on a coordinate
plane. Students should use each of the quadrants and make sure that all
vertices are plotted on integer coordinates. Students may have to modify
their design slightly in order to transfer the diagram successfully. Allow
students access to graph paper, rulers, and protractors for this task.
2. Ask students to calculate the length of each of the walls and the total
perimeter their home’s floor plan design. Have students keep each of
these measurements on their “Dream Home Design” handout.
appropriate
Remediation:
Have students
create floor
plans with only
90º angles.
Enrichment:
Have students
create at least
six angles that
are not 90º
angles.
Formative
Assessment:
Teacher
observations of
methods used to
find area and
perimeter.
Performance
Assessment:
Ending discussion
of methods used
by students
Summative
Assessment:
Floor Plan drawing
with perimeters
and areas, along
with cost
3. Within the design, have students add a backyard that includes a pool
(with straight sides) behind their dream home design. Have students
calculate the perimeter of the backyard, the perimeter of their dream
pool, and the area of their backyard. Using the area, have students
calculate the number of square feet of sod that would need to be
purchased.
4. Ask students to use the provided prices of fence, stone and sod to
determine the total cost of all the landscaping materials on the “Dream
Home Design” handout.
estimates for sod,
brick and fencing.
Summarizing Strategy:
Ask the class to compare costs, and see what caused some floor plans to be
more or less expensive than others. Ask students to share their techniques for
finding area and perimeter of the spaces they created, and let the class
discuss which would be easiest to use and why.
Adapted from: Leinwand, S. (2009). Accessible Mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
3
I can use
geometric
descriptions of
rigid motions to
predict the effect
of a given rigid
motion on a given
figure.
Video “Motion
Geometry”
(Appendix F)
Essential Question(s):
How can I use geometric descriptions of rigid motions to predict the effect of
a given rigid motion on a given figure?
½ Project Day – See Unit Plan
The Catapult Project - Writing
Set:
Teacher will begin by showing “Motion Geometry” a video clip on use of
transformations in video animation.
Teaching Strategy:
Students and teacher will discuss how transformations are used in today’s
society after watching the informative video. Things will be brought up such
as games and digital animation. Ask students if they realized that it is all
transformations that are programmed that create the look of movement on
Remediation:
None
Enrichment:
none
Formative
Assessment:
Teacher
observations of
current knowledge
on the subject.
Summative
Assessment:
Exit ticket.
screen. Spark a debate on what kinds of skills and education would be
necessary to be a video game designer or creator of digital animation. Lead
the discussion by offering how mathematics is a leading component in
programming and animation.”
Summarizing Strategy:
Tell students that after the following project days, they will begin diving into
transformations and their properties. As an exit ticket, have them write down
3 things they know about transformations and 2 things they would like to
learn.
4
Project Day – See Unit Plan
The Catapult Project – Traditional Design and STEM Design
5
Project Day – See Unit Plan
The Catapult Project – STEM Design
6
I can articulate
the definitions of
the
transformations:
reflection,
rotation, and
translation.
I can recognize
the difference
between the
examples and
non-examples of
reflections,
rotations, and
translations
“Transformation
Examples and
Non-Examples”
(Appendix G)
“Transformation
Identification”
(Appendix H)
“The History of
the Mercedes
Logo” video clip
(Appendix I)
Graph Paper
Calculator
Essential Question(s):
1.What are the definitions of rotations, reflections, and translations in terms
of angles, circles, perpendicular lines, parallel lines, and line segments?
2.How to recognize the difference between examples and non-examples of
reflections, rotations, and translations?
Set:
The teacher will ask students to make a list of the different types of
transformations that can occur in a plane. The teacher will then show “The
History of the Mercedes Logo” (Appendix I) video clip. The students will
discuss pre-image, and how the pre-image (the original figure before the
transformation) of the Mercedes Logo would compare to a rotated image of
the same figure. Since they would appear the same, the teacher would explain
this gives the Mercedes Logo the property of rotational symmetry. The
students will need this knowledge to design their own logo with rotational
symmetry for the “State of STEM Presentation” (See Unit Plan) as well as lead
into further investigation of transformed figures and their pre-images.
Teaching Strategy:
Determining the definitions for transformations activity
Create three transformation stations: reflection, rotation, and translation,
from the Transformation Examples and Non-examples.
1. Divide the class into heterogeneous groups of 3-4 and have students
look at different examples provided for rotation, reflections, and
translations and write their own definitions for those types of
translations.
2. Go through each group and have them provide their definition for
each. As a group discuss the definitions, and decide as a class on the
Remediation:
- Peer Tutoring
Heterogeneous
Grouping
Enrichment:
- Peer Tutoring
Heterogeneous
Grouping
Formative
Assessment:
Teacher
observations of
definitions
developed by
students for
transformations
Performance
Assessment:
-Ending discussion
of determining the
best definitions
- Homework: Ask
students to create
their own
examples of a
rotation,
reflection, and
translation using
the definitions
they developed in
class
Summative
Assessment:
-Exit Ticket
nest one. Use this as an opportunity to clear up any misconceptions
students may have.
Identifying transformations activity
1. Have student access the “Transformation Identification” activity.
2. Have the groups go through each of the transformations and classify
them into the appropriate group. Some of the transformations may
be classified as more than one transformation.
3. After completing the “Transformation Identification” activity have
students reexamine their definitions. Discuss as a group if the
definition is well written or needs to be changed to fit their new
findings.
4. Discuss with students each group’s definitions. Decide which are the
best and why.
Summarizing Strategy:
As an exit ticket, ask students how does a reflection, rotation, or translation
affect the lines and angles of a transformed figure? Where do you find
examples of reflection, rotation, and translation in the real world?
Adapted from: Howard County Public Schools (HCPSS) Secondary Mathematics Office (v2.1)
7
I can accurately
identify and verify
images that are
the product of
dilation.
Graph paper
I can create
representations
that preserve
angle measure
and double all
lengths.
Protractors
Rulers
Compass
Calculator
Essential Question(s):
1. How to accurately identify and verify images that are the product of
dilation?
2. Can I create representations that preserve angle measure and double all
lengths?
Set:
Ask the students to write down the types of transformations they learned
about in the previous lesson. Ask students to identify the only transformation
they did not define yesterday (dilation), and have them attempt to define this
once they have identified it on their paper. Explain to the class that today’s
lesson will be investigating the properties of dilation.
Teaching Strategy:
Divide the class into groups of 3-4. Groups will work together to verify the
properties of dilation. Have the groups follow the steps below for the
verification.
Remediation:
Peer Tutoring
Heterogeneous
Grouping
Rather than
recreating the
dilated figure
for
investigation,
give students in
need of support
the already
dilated figure,
and have
complete step
10 from the
Formative
Assessment:
Teacher
observations of
students verifying
dilation
Performance
Assessment:
Ending discussion
of determining
how numbers less
than one would
effect the dilated
figure
Dilated figures
Using graph paper, rulers and protractors:
1. Draw a line segment.
2. Select a point not on the line for the center of dilation.
3. Extend a ray from the center of dilation through each endpoint of the
segment.
4. Choose a scale factor that will double the size of the segment (k = 2).
5. Determine the lengths of the segments from the center to each
endpoint along each ray.
6. Multiply the lengths by the scale factor to determine the dilated
distance.
7. Measure the dilated distance along the appropriate ray from the
center to the new endpoint.
8. Connect the dilated endpoints.
9. Determine the length of the original and dilated segments.
(Measure the corresponding angles formed by the intersection of the
ray, original segment, and dilated segment.”
10. Verify the following properties associated with similar figures:
a. The dilated distance is twice as large as the original distance.
b. Corresponding angles are congruent.
c. The distance along the ray from the center to the dilated
endpoint is twice as large as the distance from the center to
the original endpoint.
11. Without dilating the figure, describe how the properties of dilation
would affect a size change with a scale factor less than 1 (k = 0.75).
Discuss as a class how this would affect the figure.
Summarizing Strategy:
As an exit ticket, ask the students to write down 3 careers that use dilation
while working. How does dilation relate to the scale drawing of the floor plan
they created in the previous lesson?
Homework:
Fill in a dilation table and determine whether or not figures have been dilated
teaching
strategy.
Enrichment:
Peer Tutoring
Heterogeneous
Grouping
For students in
need of a
challenge, have
them actually
create the figure
discussed in
Step 11 of the
teaching
strategy.
Summative
Assessment:
Exit Ticket
8
Project Day – See Unit Plan
The State of STEM – Research
9
I can determine if
figures are similar
by comparing
their sides and
angles.
Rulers
I can create
similar figures by
maintaining angle
measure and
creating
proportional
sides.
Calculator
I can find lengths
of sides of
proportional
figures given
enough
information
about their
proportionality.
Protractors
Graph Paper
“Transformed
Figure Handout”
(Appendix J)
Essential Question(s):
1. What are the similarities in angles and sides in congruence?
2. How to create similar figures by maintaining angles and creating
proportional sides?
3. How to find lengths of proportional figures given enough information about
their proportionality?
Set:
Ask what needs to be true for proportions (ratios) to be equal. Have students
draw a geometric figure in which they think they could create a similar and
proportional figure. Recommend a triangle or quadrilateral.
Remediation:
Peer Tutoring
Heterogeneous
Grouping
Rather than
recreating the
similar figure for
investigation,
give students in
need of support
Teaching Strategy:
the
1. Place students in groups of 3-4. Have students exchange their drawings
“Transformed
with another group of students.
Figure Handout”
2. Have each group of students draw a “similar” version of the figure they
and have them
were given from the other group, either twice or half as large (the option
complete step 5
will avoid some problems with size limitations).
from the
3. Once they have completed their new figures, have students discuss and
teaching
share how they created their similar figure with the entire class. During
whole group sharing, highlight the necessity for angles to be the same and strategy.
all three pairs of sides to be proportional by 2:1 or 1:2.
Enrichment:
4. Ask, “What must be the same and what does not for the figures to be
Peer Tutoring
similar?” Once it is established that the angles must be equal in measure
Formative
Assessment:
Teacher
observations of
student
conjectures and
use of those
conjectures.
Performance
Assessment:
Discussion about
creating similar
figures.
Student
transformed
similar figures
Summative
Assessment:
Exit Ticket
(start using “congruent”.)
5. Have them measure with a ruler and record the lengths of the sides they
have drawn. Have students discuss the criteria for the corresponding
sides. Have students write their conjecture and explain how the
conjecture is verified. Refer to the opening set for guidance, if necessary.
Summarizing Strategy:
As an exit ticket, ask students what happens to the angles when
transformations are applied to create similar figures? What happens to the
sides when transformations are applied? Do these things always happen?
10
Presentation Day – See Unit Plan
State of STEM
Heterogeneous
Grouping
For students in
need of a
challenge,
rather than
having them
complete step 5,
have them
measure one
side of the
transformed
figure and
calculate the
lengths of the
remaining sides
of the figure
given the
information
received from
the first side.
STANDARDS
Identify what you want to teach. Reference State, Common Core, ACT
College Readiness Standards and/or State Competencies.
G.GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
G.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line,
and distance around a circular arc.
G.CO.A.2 Represent transformations in the plane using 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.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line,
and distance around a circular arc.
G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.SRT.A.1.b Verify experimentally the properties of dilations given by a center and a scale factor. The dilation of a line segment is longer or shorter in the ratio given
by the scale factor.
G.SRT.A.2 Given two figures use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the
meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.