Course Name: Geometry/Math II
Unit 8
Unit Title: Modeling with Geometry
Enduring understanding (Big Idea): Students will be able to derive the formula of a circle from the Pythagorean
Theorem. Student will be able to visualize the relationships between two- dimensional and three- dimensional objects.
Student will be able to use properties of distance and volume to model real life problems.
Essential Questions:
How do you determine the intersection of a solid and a plane?
How do you find the equation of a circle in a coordinate plane?
How do you partition a line segment into a given ratio?
How do you use geometric solids to model real life figures?




BY THE END OF THIS UNIT:
Students will know…
 (x - h)2 + (y – k)2 = r2

Density =
mass
volume


Formulas for area
Formulas for volume
Vocabulary:
Cross section, prism, cylinder, cone, pyramid, sphere, slant
height, Pythagorean Theorem, ratio, segment, trapezoid,
hexagon, density
Unit Resources:
Test Specification Weights for the Common Exams in
Common Core Math II:
Standards
G-GPE
G-GMD
G-MG
%
Constructed
Response
0%
0%
3% to 7%
%
MultipleChoice
7% to 10%
Category
Percentage
(Geometry)
27% to 30%
Putting it together:
It is suggested that the unit be started with the formula a circle
(G.GPE.1). Finding the midpoint of the diameter of a circle can lead
in to finding ratios of a given segment (G-GPE.6). Starting with the
area and volume of a circle and sphere continue reviewing area and
volume of other geometric shapes (G-GMD.3). The emphasis of this
review it the real world applications of area and volume (G-MG.1, GMG.2, G-MG.3). The unit is wrapped up with a discussion of twodimensional cross-sections of solid object and three-dimensional
objects created by the rotation of two-dimensional objects and their
applications (G-GMD.4)
Suggested Pacing:
Equations of circles : Geometry BK: Section 12-5
Ratios of line segments : Geometry BK: CC-1
Modeling with Volume : Geometry BK: 10-1, 10-2, 10-3,10-5, 11-2, 113, 11-4, 11-5, 11-6, 11-7
Students will be able to:
 Derive the equation of a circle given center and
radius using Pythagorean Theorem
 Find the point on a directed line segment between
two given points that divides the segment in a
given ratio
 Identify the shapes of 2-dimensional crosssections of 3-dimensional shape
 Identify 3-dimensional objects generated by
rotations of 2-dimensional objects
 Describe objects using their properties
 Apply concepts of density based on area and
volume in modeling situations
 Apply geometric methods to solve design
problems.
Mathematical Practices in Focus:
1. Make sense of problems and persevere in solving
them
2. Reason abstractly and quantitatively
4. Model with mathematics
6. Attend to precision
7. Look for and make use of structure
CCSS-M Included: G-GPE.1, G-GPE.6, G-GMD.4, GGMD.3,
G-MG.1, G-MG.2, G-MG.3
Abbreviation Key:
CC – Common Core Additional Lessons found in the Pearson
online materials.
CB- Concept Bytes found in between lessons in the Pearson
textbook.
ER – Enrichment worksheets found in teacher resources per
chapter
Visualizing relationships between 2-dimemsional and three
dimensional objects : Geometry BK: CC-20
Geometry Sections 11-1 and 12-6
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 1
Course Name: Geometry/Math II
Unit 8
Unit Title: Modeling with Geometry
CORE CONTENT
Cluster Title: Translate between the geometric description and the equation of a conic section
Standard: G.GPE.1 Derive the equation of a circle given a center and radius using the Pythagorean Theorem: complete
the square to find the center and radius of a circle given by an equation.
Concepts and Skills to Master:
 Write the equation of a circle and apply it given a graph or a circle’s center and radius.
 Find the center and radius of a circle using the coordinate plane or the general form of the equation of a circle.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
 Distance Formula
 Sketching graphs on a coordinate plane (x-y axis).
Academic Vocabulary:
standard form of an equation of a circle, center of a circle on the coordinate plane (h, k), radius (r)
Suggested Instructional Strategies:
Starting Resources:
 Arrange students into pairs of mixed abilities.
 Equation of Circle Interactive Applet
On the board, draw a circle on a coordinate
http://www.mathwarehouse.com/geometry/circle/equationplane. One student will write an equation of the
of-a-circle.php
circle using the center and the radius, and the
 Equations of Circles Powerpoint
other student will use the center and one point.
(Including Completing the Square)
Tell them to share their equations and discuss
www.mathxtc.com/Downloads/MeasureGeo/files/Circles.ppt
any discrepancies. You may vary this activity
 Online Teacher Resource Center
by having one student draw a circle on a
www.pearsonsuccessnet.com- Geometry
coordinate plane and the other write the
Dynamic Activity 12-5: Circles in the Coordinate Plane
equation. The drawings can be done on graph
 Completing the square is not covered in the Pearson
paper in a page protector so that the paper can
Geometry text. However, online resources from
be cleaned and reused.
Chapter 10 of the Pearson Algebra 2 text can be used
 Emphasize that writing the equation for a circle
as a resource to teach or review completing the square.
in standard form makes it easier to identify the
www.pearsonsuccessnet.com - Algebra 2 p.633 –
center (h, k).
Problem 4
 http://www.shmoop.com/common-core-standards/ccssRemind students to take the square root of the value r2
hs-g-gpe-6.html
in order to find the radius.
Sample Assessment Tasks
Skill-based task:
Problem Task:
What is the standard equation of each circle?
1. Suppose you know the center of a circle and a point on the
1. center(2,3);radius = 5
circle. How do you determine the equation of the circle?
2. center(0,1);radius = 7
2. A student says that the center of a circle with equation:
What is the center and radius of each circle?
(x – 2)2 + (y + 3)2 = 16 is (-2, 3). What is the student’s error?
How should the equation read in order to make the student
3. (x4)2+(y–3)3 = 16 4. (x+7)2+y2 = 10
correct?
Teacher Created Argumentation Task (W1-MP3&6)
THINK ABOUT A PLAN:
Find the circumference and area of the circle whose equation is (x – 9)2 + (y – 3)2 = 64.
Leave your answers in terms of pi.
Include in your answer the following: What essential information do you need? What formulas will you use? (Taken from PH
Geometry, p.802 #44)
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 2
Course Name: Geometry/Math II
Unit 8
Unit Title: Modeling with Geometry
CORE CONTENT
Cluster Title: Translate between the geometric description and the equation of a conic section
Standard: G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a
given ratio
Concepts and Skills to Master:
 Use coordinate geometry to divide a segment into a given ratio
 Determine the midpoint of a segment given two endpoints
 Determine the endpoint of a segment given a midpoint and one endpoint
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
solving equations, distance formula
Academic Vocabulary:
 Coordinates, a:b ratio, directed line segment, midpoint, endpoint
Suggested Instructional Strategies:
Starting Resources:
Geometry Textbook Correlation: CC-1
After using the midpoint formula to find the midpoint of the
http://www.shmoop.com/common-core-standards/ccss-hs-gdiameter of a circle, lead in to finding other ratios of a line
gpe-6.html
segment.
Sample Assessment Tasks
Skill-based task:
Problem Task:
A segment with endpoints A(3,2) and B(6,11) is partitioned
A point B(4, 2) on a segment with endpoints A(2, -1) and
by a point C such that AC and CB form a 2:1 ratio. Find C.
C(x, y) partitions the segment in a 1:3 ratio. Find x and y.
If general points N at (a,b) and P at (c,d) are given. Why are
the coordinates of point Q (a,d)? Can you find the
coordinates of point M?
If you are given the midpoint of a segment and one
endpoint. Find the other endpoint.
a. midpoint: (6, 2) endpoint: (1, 3)
b. midpoint: (-1, -2) endpoint: (3.5, -7)
If Jennifer and Jane are best friends. They placed a map of
their town on a coordinate grid and found the point at which
each of their house lies. If Jennifer’s house lies at (9, 7)
and Jane’s house is at (15, 9) and they wanted to meet in
the middle, what are the coordinates of the place they
should meet?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 3
Course Name: Geometry/Math II
Unit 8
Unit Title: Modeling with Geometry
CORE CONTENT
Cluster Title: Visualize relationships between two-dimensional and three- dimensional objects
Standard: G-GMD.4 Identify the shapes of two-dimensional cross sections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects.
Concepts and Skills to Master:
 Identify the shapes of two-dimensional cross-sections of three-dimensional objects
 Identify three-dimensional objects generated by rotations of two-dimensional shapes
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
Recognize basic two-dimensional shapes
Recognize basic three- dimensional shapes
Academic Vocabulary:
Cross section, square, circle, concentric circles, trapezoid, rectangle, sphere, rectangular, prism, cone, cylinder
Suggested Instructional Strategies:
Starting Resources:
Geometry Textbook Correlation: CC-20
Geometry Chapter 11-1, 12-6
 Have students tape cut-out to a straw, spin the
Enrichment Activity 11-3
straw and describe the three-dimensional shape
Geometry pg 706 problems 33-36
created
Geometry pg 731 problems 33-36
 Students can slice objects shaped in clay to
http://www.shmoop.com/common-corediscover the cross-section
standards/handouts/g-gmd_worksheet_4.pdf
Sample Assessment Tasks
Skill-based task:



Problem Task:
A circle has a radius of 15cm. What is the volume
of the sphere made by rotating this circle?
Given a cylinder with radius 7 in and height 10 in,
find the area of a cross section that is parallel to its
base.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 4

Given a cylinder with height 60mm and radius
20mm, find the area of the rectangle formed by a
perpendicular cross-section right down the
cylinder’s center.

If an equilateral triangle with perimeter 24 cm is
rotated, find the volume of the cone that is formed.
Course Name: Geometry/Math II
Unit 8
Unit Title: Modeling with Geometry
CORE CONTENT
Cluster Title: Geometric Measurement and Dimensions
Standard: G-GMD.3 – Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Concepts and Skills to Master:
Find the volume of cylinders, pyramids, cones, and spheres in contextual problems.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
Formulas for the volumes of cones, cylinders, and spheres (8.G.9).
Academic Vocabulary:
pyramid, cylinder, cone, sphere, volume, length, width, height, base, radius, π.
Suggested Instructional Strategies:
Starting Resources:
 Have students bring household objects with the
Geometry Textbook Correlation: Pearson Chapter 11.2, 3,
given characteristics. Provide opportunities for
4, 5, 6
students to measure with rulers or tape measures to
gather needed information. Compute the volume of http://www.shmoop.com/common-core-standards/ccss-hs-gthe objects.
gmd-3.html
 Make connections between metric measurements.
(For example, using rice to fill a cylinder compare
the liquid volume in liters and the geometric volume
in cubic meters.)
Sample Assessment Tasks
Skill-based task:
Problem Task:

Find the volume of a cylindrical oatmeal box

Calculate the volume of helium needed to inflate a
spherical latex balloon with a diameter of 18 inches.

Given a cube with an edge of 8in and a sphere with
a diameter of 8 in. calculate the volume remaining
in the cube if the sphere in inserted into the cube.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 5

Given a three-dimensional object, compute the
effect on volume of doubling or tripling one or more
dimension(s). (For example, how is the volume of a
cone affected by doubling the height?)

A cone is enclosed inside a cylinder. The cone and
the cylinder have equal bases and equal heights. If
the volume of the cone is 30 cm3, what is the
volume of the cylinder
Course Name: Geometry/Math II
Unit 8
Unit Title: Modeling with Geometry
CORE CONTENT
Cluster Title: Apply geometric concepts in modeling situations
Standard: G-MG.1 Use Geometric shapes, their measures and their properties to describe objects (e.g modeling a tree
trunk or a human torso as a cylinder)
This is a Math I topic. It is being reviewed to support Standards G-MG.1, G-MG.2 and G-MG.3
Concepts and Skills to Master:
 Use geometric shapes, their measures, and properties to describe objects
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
 Formulas for area and volume
 Names and properties of geometric shapes
Academic Vocabulary:
Circle, rectangle parallelogram, trapezoid, hexagon, triangle, prism, cone, cylinder, pyramid
Suggested Instructional Strategies:
Resources:
Geometry Textbook Correlation: 10-1, 10-2, 10-3, 10-4
During the review of area and volume of geometric objects
11-3, 11-4, 11-5, 11-6, and 11-7
the emphasis should be placed on the modeling and using
http://www.shmoop.com/common-core-standards/ccss-hs-gthese properties to solve real life problems.
mg-1.html
Sample Assessment Tasks
Skill-based task:
Problem Task:
Consider a rectangular swimming pool 30 feet long and 20
feet wide. The shallow end is 3½ feet deep and
 A fountain takes the shape of a half sphere with
extends for 5 feet. Then for 15 feet (horizontally) there is a
diameter 10 feet that sits, flat side down, in the
center of a round brick-paved patio of diameter 100 constant slope downwards to the 10 foot-deep end.
feet. What is the area of the portion of the patio that a. Sketch the pool and indicate all measures on the sketch.
b. How much water is needed to fill the pool to the top? To a
is left uncovered?
level 6 inches below the top?
c. One gallon of pool paint covers approximately 75 sq feet
Explanation:
of surface. How many gallons of paint are needed
to paint the inside walls of the pool? If the pool paint comes
The area of the entire patio is 2500π square feet,
2
2
in 5-gallon cans, how many cans are needed?
since A = πr = π(50) = 2500π. The base of the
d. How much material is needed to make a rectangular pool
fountain has an area of 25π square feet, since,
2
2
cover that extends 2 feet beyond the pool on all sides?
again, A = πr = π(5) . The portion left uncovered
will be the difference of the two numbers, or 2500π e. How many 6-inch square ceramic tiles are needed to tile
the top 18 inches of the inside faces of the pool? If the
– 25π = 2475π square feet.
lowest line of tiles is to be in a contrasting color, how many
of each tile are needed?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 6
Course Name: Geometry/Math II
Unit 8
Unit Title: Modeling with Geometry
CORE CONTENT
Cluster Title: Apply geometric concepts in modeling situations
Standard: G.MG.2- Apply concepts of density based on area and volume in modeling situations (e.g. persons per square mile, BTU’s
per cubic foot)
Concepts and Skills to Master:
 Students will be able to use the concept of density when referring to situations involving area and volume models.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
 Areas of Rectangles, Parallelograms, Circles, Triangles, Rhombuses, Kites, and Trapezoids
 Ability to choose the correct formula, substitute in values, and solve.
Academic Vocabulary:
Density, volume, modeling
Suggested Instructional Strategies:
Resources:
 When covering area you can discuss population
Geometry Textbook Correlation: Pearson Chapter 11.2, 3, 4, 5, 6
density
http://www.shmoop.com/common-core-standards/ccss-hs-g-mg When covering volume you can discuss density of
2.html
Pearson Additional Problems
m
solid objects d 
v
Sample Assessment Tasks
Skill-based task:
Problem Task:
• A hot air balloon holds 74,000 cubic meters of helium, a very noble gas with
the density of 0.1785 kilograms per cubic meter. How many kilograms of helium
does the balloon contain?
Explanation:
Density is calculated as
, or mass divided by volume. Fill in the two
parts we know and solve for m = Vd = 74,000 m3 × 0.1785 kg⁄m3 = 13,209 kg.
• The Mom & Pop Coffee Shop wants to open new locations, either downtown
or uptown. They will open a new location wherever the ratio of existing coffee
shops per person is less than 0.01. The population density of the 20-city-block
downtown area is 225 people per city block, but there are already 48 coffee
shops in the area. The population of the 30-block uptown area is 125 people
per block, and there are 16 coffee shops around. Where, if anywhere, should
they open their new location(s)?
Explanation:
(A)
Downtown only
(B)
Uptown only
(C)
Both downtown and uptown
(D)
Neither downtown nor uptown
First, we should find the total number of people in the uptown and downtown
areas. The 225 people per block × 20 blocks downtown = 4,500 people
downtown, while the uptown area has 125 people per block × 30 blocks uptown
= 3,750 people uptown. Now, we can divide these by the number of coffee
shops to get the ratios we need. Downtown, we have a ratio
of
, but uptown, the ratio is
only
. They can open a new
uptown shop because it's less than the specified 0.01 ratio.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 7

Over a 24-hour period, brown bears
counted 840 salmon swimming upstream,
and they safely assumed that they only
counted 30% of the total number of fish
going by. The salmon-spawning haven
along Copper River measures
approximately 90,000 cubic meters of
water. Given that the haven was empty
before this week, that the salmon swim
upstream at a constant rate, and that once
they reach the haven, the salmon hang out
there indefinitely, what will the population
density of salmon in the Copper River
spawning haven be after one week (to the
nearest thousandth fish/m3)?
Explanation:
The bears only counted 30%, which
means the salmon are swimming
upstream at a rate of 2,800 fish per day,
or 19,600 fish per week. The population
density will
be
.
Course Name: Geometry/Math II
Unit 8
Unit Title: Modeling with Geometry
CORE CONTENT
Cluster Title: Apply geometric concepts in modeling situations
Standard: G.MG.3- Apply geometric methods to solve design problems (e.g. designing an object or structure to satisfy
physical constraints or minimize cost; working with typographic grid systems based on ratios).
Concepts and Skills to Master:
Students will be able to solve design problems by designing an object or structure that satisfies certain constraints.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
Parts of a right triangle and congruent corresponding parts
Academic Vocabulary:
congruent triangles, hypotenuse, legs of a right triangle
Suggested Instructional Strategies:
Resources:
Geometry Textbook Correlation: Pearson Chapter 11-2, 11-3,
11-4, 11-5, 11-6
The Best Container Task
Perfume Packaging –Dana Center chapter 5 pg 1
Pearson Additional Problems
Shmoops Resources for teachers:
http://www.shmoop.com/common-core-standards/ccss-hs-g-mg3.html
Sample Assessment Tasks
Skill-based task:
Problem Task:
Wine Glass Task
A cereal company is redesigning its cereal boxes to
http://www.insidemathematics.org/pdfs/geometry/glasses/task.p
make them stand out on the supermarket shelves. For
some unknown reason, they went with a triangular prism df
for the shape of their box. (It certainly stands out from
the crowd, but it's so hard to pour cereal out of!) The
front and back are isosceles triangles with base 10
inches and height 12 inches. The surface area of the
entire box is 384 square inches. What is the depth of the
box?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 8
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MATH II Unit 8 MODELING WITH GEOMETRY