Length Ratio
a
b
Surface Area Ratio
2
a
b2
Volume Ratio
a3
b3
Similarity
Context: This unit is designed for a high school Geometry classroom. Our school is on a block schedule, so I
see my students for roughly100 minutes two times a week and 60 minutes on Friday. This will be the second to
the last unit of the year. We will have just finished units on area and volume, and earlier in the year the students
have already learned properties of triangles and how to prove triangles are congruent. This unit will lead into
the next unit on basic trigonometric ratios.
For the creation of this unit, I looked at the state and district standards first to help develop the enduring
understandings and essential questions for the unit. In the curriculum map, the state standard designated by a G
(G.3.A, G.3.B, etc.) and the district standards are designated by a PS (PS3, PS10, etc). The primary method of
assessment in this unit is from observations of the students work during investigations and guided practice and
on homework. There are additional formal assessments listed in the curriculum map.
Standards:
G.3 Core Content: Two- and three-dimensional figures

PS3
Students are able to determine, prove, and apply triangle congruence, triangle similarity, and the sum of angles property for triangles.

G.3.A Know, explain, and apply basic postulates and theorems about triangles and the special lines, line segments, and rays associated with a triangle.

G.3.B Determine and prove triangle congruence, triangle similarity, and other properties of triangles
G.6 Additional Key Content

PS10
Students are able to solve problems by applying formulas for surface area and volume of three-dimensional figures (prisms, cylinders, pyramids, cones, and spheres) and are able
to predict and verify the effect of changing one or more linear dimensions on the surface area and volume of a figure.

G.6.C Apply formulas for surface area and volume of three-dimensional figures to solve problems.

G.6.D Predict and verify the effect that changing one, two, or three linear dimensions has on perimeter, area, volume, or surface area of two- and three- dimensional figures.
G.7 Core Processes: Reasoning, problem solving, and communication

PS11
Students are able analyze and represent problem situations mathematically; select and apply appropriate strategies to solve the problem, verify their solutions, and communicate
their methods and solutions efficiently using appropriate mathematical representations, symbols, and language.

G.7.A Analyze a problem situation and represent it mathematically

G.7.B Select and apply strategies to solve problems.

G.7.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.

G.7.F Summarize mathematical ideas with precision and efficiency for a given audience and purpose.
Essential
Questions
Concepts
Skills
Assessments
Methods
1. Proportions
1. Solve a
proportion
regardless of the
location of the
missing
information
Formative
Assessments:
1. Unit PreAssessment given
after last test.
2. Homework
3. Classroom
observations
Anticipatory Set – The students will
work on a percentage problem which
they have had experience with in past
years.
Model – Show the students the steps for
solving 2 different proportions where the
missing part is in a different location for
each
Guided Practice – Students solve 4
proportion problems while working with
their partner.
Independent Practice – Students will
work on problem from the book
P.561 #1-18
Investigation – Students will measure
the side lengths and angles of two
similar shapes in order to form a
definition of similarity
Group Discussion – Students will
propose their definitions of similarity
Modeling – Show students how
proportions can be used to solve for
missing side lengths
Guided Practice – Have students find
missing side lengths and angles of
similar shapes.
Independent Practice – Students will
work on problem from the book
P.568 #3, 6-14
Day 1
Enduring Understandings
Day 2
Week 1
The
students will
understand
a constant ratio
exists between
the
corresponding
lengths of
similar figures.
1. What is the
difference
between
similarity and
congruence?
2. What is the
relationship
between similar
figures and
proportions?
2. Similar
Polygons
2. Solve for
missing lengths
of similar shapes
by applying the
properties of
proportional
segments
Materials
1. Proportions ActivBoard
Flipchart
2. Proportions worksheet
3. Book
1. Similarity ActivBoard
Flipchart
2. Handout with pairs of similar
shapes for investigation
3. Patty Paper
4. Rulers
5. Book
3. Similar
Triangles
3. Determine if
two triangle are
similar and then
solve for missing
sides and angles
3. Similar
Triangles
3. Determine if
two triangles are
similar and then
solve for missing
lengths of
altitudes,
medians, and
angle bisectors
4. Area and
Volume of similar
shapes
4. Find the ratio
of area and
volume for
similar shapes
when given the
ratio of the side
lengths and vice
versa.
Day 1
Day 3
3. How is
similarity of
geometric
figures applied
and verified?
Day 3
Day 2
Week 2
The students
will understand
the change in
lengths between
similar figures
has a
predictable
change on the
area and/or
volume of those
figures.
4. How does the
change of
lengths between
similar figures
affect the area
and volume of
the figures?
Formative
Assessments:
1. Homework
2. Classroom
observations
3. Review Quiz
over content
covered during the
first five days of
the unit
Investigation – Jigsaw activity. Students
divide into groups of 3. Each student
gets a similarity shortcut to investigate.
Students with similar shortcuts get
together compare results, and then go
back to share with original group.
Group Discussion – Have students
share findings and write on the board so
students have common notes.
Model – Use the shortcuts that were just
discovered to show students how to
determine if two triangles are similar
Guided Practice – Students work in
pairs to determine if pairs of triangles are
similar
Independent Practice – Students will
work on problem from the book
P. 574 #2-7, 9, 10, 11
Investigation – Students will construct
two similar triangles and measure the
corresponding altitudes, medians, and
angle bisectors.
Group Discussion - Have students share
findings and the class will decide on a
common theorem for their findings.
Guided Practice – Students missing
altitudes, medians, and angle bisectors
between similar triangle pairs
Independent Practice – Students will
work on problem from the book
P. 588 #1-10
Investigation – Students will double and
triple the dimensions of a square and
compare the ratio of side lengths to the
ratio of areas. Then they will do the
same for a cube and compare the ratio of
side lengths to the ratio of volume.
Group Discussion - Have students share
findings and the class will decide on a
common theorem for their findings.
Guided Practice – Students will find the
area and volume ratios of a cube and
cylinder when given the length ratios.
Independent Practice – Students will
work on problem from the book
P. 595 #1-8, 10
Today will be a review day. The students
will take a review quiz over proportions,
similar polygons, similar triangles, and
volume and area of similar shapes.
1. Similar Triangle ActivBoard
Flipchart
2. Jigsaw activity worksheet
3.Compasses
4. Straightedges
5. Protractors
6. Book
1. Similar Triangle ActivBoard
Flipchart
2. Compasses
3. Rulers
4. Book
1. Area and Volume ActivBoard
Flipchart
2. Square Tiles
3. Stacking Blocks
4. Book
1. Review Quiz
5. Indirect
Measurement
Day 1
5. When can you
use proportions
to solve
problems?
Formative
Assessments:
1. Homework
2. Classroom
observations
Summative
Assessments:
1. Similarity
Station Lab
2. Unit Test
Station Lab – The students will spend
the entire class working on different labs
that are set up around the room and in
the hall. Each lab deals with a from of
indirect measure or other similarity idea.
Day 3
Day 2
Week 3
The students
will understand
the properties of
similarity can
be used to solve
applied
problems.
5. Find a height
or distance
without
measuring by
using the
properties of
similar triangles.
Anticipatory Set – The students will try
to determine the height of a boy standing
next to my friend, who is 6’5”, in a
photograph.
Guided Practice – Students will break
into groups and work on a problem from
the book that uses shadow lengths and
the height of a pole to determine the
height of a building.
Independent Practice – Students will
work on problem from the book
P.582 #1-3, 5, 11-13
Unit Test – The students will take a unit
test on proportions, similar polygons,
similar triangles, area and volume ratios,
and indirect measurement.
Reference:
Washington State 6-12 Mathematics Standards
http://www.k12.wa.us/mathematics/Standards.aspx
Lake Washington School District Geometry Power Standard
http://www.lwsd.org/PARENTS/TEACHING-CURRICULUM/Pages/Power-Standards.aspx
Greenwich Public Schools Math Standards
http://www.greenwichschools.org/uploaded/district/curriculum/math/High_Objectives/MathGeometry.pdf
1. Indirect measurement
ActivBoard Flipchart
2. Book
1.Tape Measures
2.Meter Sticks
3. Rulers
4. String
5. Circle Templates
6. Projector with slide of smiley
face and tripod with cardboard
sheet to move in front of screen
7. Plastic Mirrors
8. Stacking Blocks
1. Similarity Unit Test