CCGPS Mathematics
Unit-by-Unit Grade Level Webinar
Analytic Geometry
Unit 3: Circles and Volume
August 13, 2013
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CCGPS Mathematics
Unit-by-Unit Grade Level Webinar
Analytic Geometry
Unit 3: Circles and Volume
August 13, 2013
James Pratt – jpratt@doe.k12.ga.us
Brooke Kline – bkline@doe.k12.ga.us
Secondary Mathematics Specialists
These materials are for nonprofit educational purposes
only. Any other use may constitute copyright infringement.
Expectations and clearing up confusion
• Intent and focus of Unit 3 webinar
• Framework tasks
• GPB sessions on Georgiastandards.org
• Standards for Mathematical Practice
• Resources
• http://ccgpsmathematics9-10.wikispaces.com/
• CCGPS is taught and assessed from 2013-2014 and beyond
Welcome!
• The big idea of Unit 3
• Resources available from GaDOE Assessment Division
• Incorporating SMPs into circle & volume tasks
• Resources
Lighting the Way
The Georgia Formative Item Bank Sheds Light
on Student Performance Expectations and
Achievement
A Presentation by
Dr. Dawn H. Souter, Project Manager
Georgia Department of Education
Assessment and Accountability
5
Major reviews of research on the
effects of formative assessment
indicate that it might be one of the
more powerful weapons in a
teacher‘s arsenal.”
(Robert Marzano, 2006)
6
Purpose of Georgia’s Assessment Initiatives
• To provide assessment resources that reflect the rigor of
Georgia’s state-mandated content standards
• To balance the use of formative and summative
assessments in the classroom
• To promote student learning
• To sustain implementation of Georgia’s rigorous content
standards
7
Inside the Formative Assessment Toolbox
• Development of a three-prong toolkit to support teachers and leaders in
promoting student learning
– An expansive bank of formative instructional assessment items/tasks
based on CCGPS in ELA and Mathematics as a teacher resource Phase I Release Fall 2012; Phase II release Fall 2013
– A set of benchmark assessments in ELA, mathematics – Initial Pilot
Fall 2013
– An assessment literacy professional learning opportunity that focuses
on implementation of research-based formative instructional practices
(FIP) – Initial Pilot January/February 2013 with statewide launch
2013-2014
8
Formative Assessment Initiatives
Bringing a Balanced Assessment Focus to the Classroom
Phase I items
released into
OAS fall 2012;
Phase II items
to be released
in fall 2013
Formative
Item Bank
Assessment
Literacy
Professional
Learning
Benchmark
Assessments
Pilot in winter
2013; Statewide
launch in
summer 2013
Phase I item pilot in
fall 2013; Phase II pilot
in winter 2014
9
Using Formative Assessment
• Conducted during instruction (lesson, unit, etc.)
• Identifies student strengths and weaknesses
• Helps teacher determine next steps
– Review
– Differentiation
– Continuation
• Offers teachers immediate feedback to provide students with detailed
feedback
• Assessment for the purpose of improving achievement
• LOW stakes – no student performance reports generated
10
Formative Instructional Practices—
Formative Assessment in Action
Formative
Assessment/
Diagnostic
Summative
Assessment
• Re-Design
• Teach
Formative
Assessment
• Re-Design
• Teach
State-Mandated
Content
Standards
Formative
Assessment
• Design
• Teach
Formative
Assessment
• Re-Design
• Teach
Re-Design might involve changing activities, instructional techniques, assessment
methods or content, and/or differentiation based upon student needs.
11
The Georgia Formative Item
Bank
12
Purpose of the Formative Item Bank
The purpose of the Formative Item Bank is to
provide items and tasks used to assess students’
knowledge while they are learning the statemandated standards. The items will provide an
opportunity for students to show what they know
and show teachers what students do and do not
understand.
13
The Georgia Formative Item Bank
• Bank of over 1600+ classroom assessment items aligned with the
state’s content standards in ELA and Mathematics
• Created for exclusive use in Georgia classrooms
• Piloted with Georgia students
• Reviewed by Georgia educators
• Housed in the Georgia Online Assessment System (OAS)
• Preponderance of items at DOK 3 and 4
• Item, rubric and scored student sample papers provided
• Available to ALL Georgia Teachers!
– 700+ items piloted in spring 2012; currently in OAS
– 900+ new items piloted in spring 2013; available in OAS fall of 2013
14
Formative Item Bank Use
The Georgia Formative Item Bank can be used in order to:
• Prepare students for the more rigorous expectations of the
state-mandated standards.
– To show these expectations, students must engage with a variety
of item formats beyond multiple-choice
• Provide students practice with open- and constructedresponse items
• Provide educators insight and access to evaluating openended response items
15
The Nature of Georgia’s Formative Item Bank
• Items created for Georgia educators as an instructional
resource to be used formatively during instruction
– Provide information about student performance throughout the
academic year to inform instruction and interventions
– Reteach, remediate, move forward, enrich
– Low stakes; grading discouraged
• Support Georgia educators foster learning with an informative tool
that keeps students, parents, administrators and educators
themselves informed of students’ current position on the pathway to
proficiency.
16
The Nature of Georgia’s Formative Item
Bank (continued)
• Aligned with state-mandated content standards in
English Language Arts (ELA) and Mathematics,
grades 3 – HS
• Various formats, but primarily constructed response, in order to measure the full expectations
of what students need to know and be able to do to
be on the trajectory of exiting high school collegeand career-ready
17
Item Content
• Georgia’s Content Standards
– Mathematics: Grades 3 – 8; high school Coordinate
Algebra, Analytic Geometry and Advanced Algebra
– English/Language Arts (including Reading): Grades 3 –
8; high school 9th and 10th grade literature and American
Literature
• Items aligned to multiple standards
– One primary standard
– One or more secondary standards
• Alignment verified by Georgia educators
18
Valuable Features of Formative
Items – Mathematics
• Items include intentional focus on assessing processes
used by students as well as the required content
• Items applied in real-world context
• Writing requirements, such as explanations and
reasoning
• Student responses on constructed-response items/tasks
− make student knowledge and skills transparent to teachers
− illuminate student misconceptions
19
Item Formats
• Multiple Choice
• Primarily Constructed-response
– Extended Response
– Scaffolded
• Constructed-response items require students to provide
explanations/rationales, provide evidence, and/or to
show work
• Preponderance of items at DOK 3 and 4
• Provide teachers with evidence of true student
understanding of content and process
20
Multiple Choice Items
• Have four answer options
• Distractors (incorrect answers) should be believable
and represent common conceptual and/or
application errors
• Distractor rationales should assist teachers to
identify specific student misconceptions to inform
instruction
21
21
Extended Response Items
• May address multiple standards, multiple domains,
and/or multiple areas of the curriculum
• May allow for multiple correct responses and/or
varying methods of arriving at a correct answer
• Scored through use of a rubric and associated
student exemplars
22
Example of Extended Response Item
Math—Advanced Algebra
STEM: If the stem is to contain art, complete the art page provided. In the stimulus
Advanced
Algebra,
A.REI.2; A.REI.4; A.APR.6, A.REI.1; DOK 3
box below
place the artStandards
where applicable
Sreeja and Brandon solved the equation shown in different ways.
x 2  3x  2
 2x
x 1
Part A
Before solving the equation, what solution could Sreeja and Brandon identify as
extraneous? Explain your reasoning.
Part B
x 2  3x  2
2x

.
x 1
1
Demonstrate how Sreeja used the proportion to solve the equation. In each
step, explain the properties she used to determine the solution.
Sreeja solved the equation by creating the proportion
Part C
Brandon solved the equation by simplifying the left side of the equation first.
Demonstrate how Brandon simplified the expression on the left and then solved
the equation. In each step, explain the properties he used to determine the
solution.
Be sure to complete ALL parts of the task.
Write your answer and show your work on the paper provided.
Do NOT type your answer in the text box below.
23
Scaffolded Items
Include a sequence of items or tasks
Designed to demonstrate deeper understanding
May be multi-standard and multi-domain
May guide a student to mapping out a response to a
more extended task
• Scored through use of a rubric and associated
student exemplars
•
•
•
•
24
Example of Scaffolded Item
Math—Analytic Geometry
Analystic STEM:
Geometry
Standards
G.SRT.6;
G.SRT.5;
DOK 4
If the stem is to
contain art, complete
the art page provided.
In the stimulus
box below place the art where applicable
In the figures shown,
KLM is similar to
PQR, with measurements as shown.
Part A
Write an equation to determine tan  L. Solve for the tangent. Explain your
answer.
Part B
Determine m L and m Q. Round your answers to the nearest tenth of a
degree. Show your work and explain your answer.
Part C
Use the Pythagorean theorem and corresponding sides of similar triangles to
determine the lengths of KL, PR, and PQ. Show your work.
Part D
Prove the measures of KL, PR, and PQ you found in Part C are correct by
using the trigonometric ratios relating to  L and  Q. Show your work or
explain your answer.
Be sure to complete ALL parts of the task.
Write your answer and show your work on the paper provided.
Do NOT type your answer in the text box below.
25
Rubrics
• Holistic
• 5-point scale (0 – 4)
– 4: Thoroughly Demonstrated
– 3: Clearly Demonstrated
– 2: Basically Demonstrated
– 1: Minimally Demonstrated
– 0: Incorrect or Irrelevant
26
Rubric
Score
4
Designation
Thoroughly
Demonstrated
3
Clearly
Demonstrated
2
Basically
Demonstrated
1
Minimally
Demonstrated
0
Incorrect or
irrelevant
Rubric
Description
The student successfully completes all elements of the
item by demonstrating knowledge and application of
the relation between the trigonometric function of an
angle and the side lengths of the triangle (G.SRT.6),
the use of trigonometric ratios and the Pythagorean
Theorem to solve right triangle problems (G.SRT.8),
and the similarity criteria for triangles (G.SRT.5). All
four parts are completed correctly with sufficient
explanations or work shown.
The student demonstrates clear understanding of the
standards listed above by completing all four parts, but
makes one or two minor calculation errors or omissions
Or
The student completes all four parts, but the
explanation or work shown in one part is insufficient
Or
The student correctly completes three of the four parts
with sufficient explanations or work shown.
The student demonstrates basic understanding of the
standards listed above by completing two of the four
parts with at least one sufficient explanation or work
shown.
The student demonstrates minimal understanding of
the standards listed above by completing one of the
four parts with or without sufficient explanation or work
shown.
The response is incorrect or irrelevant to the skill or
concept being measured.
27
Exemplar Papers
• Prototype answer – the “ideal” response
• Set of responses from actual Georgia students, collected
during item pilots
• Samples scored by trained raters using rubric
• Papers allow teachers to review and compare their own
students’ work to the sample responses for each score
point
– Helps standardize expectations of the standards
• Score point and annotations provided for each sample item
response
Note: The pilot was conducted using standard administration procedures in order to ensure that results
were comparable across the state. When items/tasks are used during instruction, these administration
rules do not have to apply and student results may vary; thus, teachers may want to modify the rubrics
and even raise expectations. Rubrics and exemplars should remain focused on high expectations.
28
Exemplar Paper--Excerpt
Exemplar
Part A
Step 2: f(1)  0, so x  1 is a factor, not x  1.
Step 3: (x  1) 6 x 2  x  12  6 x 3  5x 2  11x  12 , not 6 x 3  7 x 2  11x  12 .
Step 4: (3x  4)(2x  3)  6 x 2  17 x  12 , not 6 x 2  x  12 .


Part B
x  1 is a factor since f(1)  0. Student may use synthetic division or long
division. Work should be shown.
6x 2  x
 12
x  1 6x 3  7x 2  11x  12
6x 2  x  12  (2x  3)(3x  4), so
6x 3  7x 2  11x  12  (2x  3)(3x  4)(x  1).
Set each factor equal to 0 to determine the zeros/intercepts.
3
2
4
3x  4  0  3x  4  x 
3
2x  3  0  2x  3  x 
x  1  0  x  1
Part C
29
6 > 0, so as x  , f (x)   and as x  , f (x)  
Formative Item Bank
Pilot I Item Examples and Results
Mathematics
30
Grade 6 Mathematics Item
Item from FIB Phase I
31
Grade 6 Mathematics
Exemplar and Sample Student Response
32
Grade 6 Mathematics Sample
Student Responses
33
Grade 6 Mathematics Item
•
•
•
•
Performance Data
Item Type: Extended Response
DOK: 3
Number of Students in Pilot: 87
Response Data
– 4’s:
– 3’s:
– 2’s:
– 1’s:
– 0’s:
0 for 0%
5 for 5.75%
34 for 39.08%
39 for 44.83%
9 for 10.34%
34
Grade 3 Mathematics Item
35
Grade 3 Mathematics Item
Performance Data
•
•
•
•
Item Type: Scaffolded
DOK: 3
Number of Students in Pilot: 44
Response Data
– 4’s:
– 3’s:
– 2’s:
– 1’s:
– 0’s:
5 for 11.36%
3 for 6.82%
18 for 40.19%
15 for 34.09%
3 for 6.82%
36
Overall Mathematics Pilot I (Spring 2012)
Summary Data
Grade
3
4
5
6
7
8
9-10
11-12
Number of students and percent falling into each score point
Total student
N/ %
0
1
2
3
4
771
667
373
81
36
1928
40.00%
34.60%
19.30%
4.20%
1.90%
100%
795
800
360
87
58
2100
37.90%
38.10%
17.10%
4.10%
2.80%
100%
548
513
252
124
44
1481
37.00%
34.60%
17.00%
8.40%
3.00%
100%
927
768
269
65
14
2043
45.40%
37.60%
13.20%
3.20%
0.70%
100%
896
632
243
62
11
1844
48.60%
34.30%
13.20%
3.40%
0.60%
100%
984
791
314
100
51
2240
43.90%
35.30%
14.00%
4.50%
2.30%
100%
798
697
186
45
27
1753
45.50%
39.80%
10.60%
2.60%
1.50%
100%
690
602
178
63
9
1542
44.70%
39.00%
11.50%
4.10%
0.60%
100%
37
Key Findings from Phase I Pilot
(Spring 2012)
• Overall performance shortfalls
– Many students lacked organization and neatness
– Don’t seem to understand what question was requiring
– Don’t follow directions well
– Didn’t answer all parts of questions
– Don’t follow through with “units” in Mathematics answers
– Don’t have keyboarding skills – future of testing is online and
there are ELA standards (W6) for keyboarding/word
processing in grades 4, 5, and 6.
38
Implications for Phase II Pilot
and for Classrooms
• Clearer directions for students with more specificity for student
responses
— FOR ELA: Be sure to complete ALL parts of the task. Use details from the text to
support your answer. Answer with complete sentences, and use correct
punctuation and grammar.
— FOR Mathematics: Be sure to complete ALL parts of the task. Write your answer
and show your work on the paper provided. Do NOT type your answer in the text
box below.
• Checklists to assist students in assessing their own responses
• Reinforce instructional recommendations to teachers
– Classroom instruction and assessments aligned with state-mandated content
standards
– Implementation of lessons and assessments that address multiple standards and
domains.
39
Formative Item Bank
Phase II Examples
Mathematics
40
Grade 6
Mathematics
Item
Item from FIB Phase II
41
Coordinate Algebra Mathematics Task
42
Advanced Algebra Mathematics
Task
STEM: If the stem is to contain art, complete the art page provided. In the stimulus
box below place the art where applicable
Jocelyn is given 2 polynomial functions. They are
f(x)  6 x 3  7x 2  11x  12 and g(x)  2x 4  2x 3  6 x  8. Jocelyn needs
to find zeros of the function f(x). The work Jocelyn did is shown in the four
steps below.
Step 1: f(1)  6(1)3  7(1)2  11(1)  12  0  10  0
Step 2: f(1)  6(1) 3  7(1) 2  11(1)  12  0  0  0, so x  1 is a
factor
Step 3: f(x)  (x  1)(6x 2  x  12)
Step 4: f(x)  (x  1)(3x  4)(2x  3)
Part A
Identify any steps in which Jocelyn made an error. Explain your answer.
Part B
Find the zeros of the function f(x)  6 x 3  7x 2  11x  12. Show your work.
Part C
Sketch a graph of f(x). Include all intercepts. Show your work or explain your
answer.
Part D
Find the difference when f(x) is subtracted from g(x).
Be sure to complete ALL parts of the task.
Write your answer and show your work on the paper provided.
Do NOT type your answer in the text box below.
43
Advanced Algebra Task
STEM: If the stem is to contain art, complete the art page provided. In the stimulus
box below place the art where applicable
Jocelyn is given 2 polynomial functions. They are
f(x)  6 x 3  7x 2  11x  12 and g(x)  2x 4  2x 3  6 x  8. Jocelyn needs
to find zeros of the function f(x). The work Jocelyn did is shown in the four
steps below.
Step 1: f(1)  6(1)3  7(1)2  11(1)  12  0  10  0
Step 2: f(1)  6(1) 3  7(1) 2  11(1)  12  0  0  0, so x  1 is a
factor
Step 3: f(x)  (x  1)(6x 2  x  12)
Step 4: f(x)  (x  1)(3x  4)(2x  3)
Part A
Identify any steps in which Jocelyn made an error. Explain your answer.
Part B
Find the zeros of the function f(x)  6 x 3  7x 2  11x  12. Show your work.
Part C
Sketch a graph of f(x). Include all intercepts. Show your work or explain your
answer.
Part D
Find the difference when f(x) is subtracted from g(x).
Be sure to complete ALL parts of the task.
Write your answer and show your work on the paper provided.
Do NOT type your answer in the text box below.
44
FIB Items in OAS
• 700+ Phase I items already in Georgia Online
Assessment System (OAS)
• 900+ Phase II items will be loaded into OAS in fall
of 2013
• Same OAS log-in as used in the past
– If you need an OAS log-in access code,
• Contact your School Administrator for OAS log-in access code
• School Administrators should contact their system test
coordinator for assistance if needed
45
Where do you Find the Items?
www.georgiaoas.org
46
Searching in the OAS
You create
test name
and ID that
are
meaningful
to you.
Naming Idea:
“Formative” and
Domain Name, such
as literary
comprehension
47
Searching in the OAS
48
Searching the OAS for Formative Items
Example Search
The name you created
in previous steps
Drop down to select subject and grade
level
Drop down to select domain and
standard (or select all)
All of the formative items are in Level 2 of the OAS which means all teachers have access.
49
Searching the OAS for Formative Items
Search Steps
1. Create Test (Test Name and Test Identifier, submit)
2. Item Level (Select Level 2, which provides access to all
Georgia teachers)
3. Subject (Select Language Arts or Mathematics)
4. Grade Level (Select grade 3 – 8)
5. Domain ( Select Formative)
6. Domain (Select Appropriate Domain or ALL, i.e. Information
and Media Literacy)
7. Standard (Select Standard or ALL)
8. Show Items
50
Finding the Rubrics and Sample Student
Papers
Once you create a test in the previous steps:
• Assign the test to a class
• Assign the test to a student in the class
• Take the test as if you were a student
• Click on “Score the Test”
• At that point, you will be able to see the rubrics and
student work samples
(Screen shots are unavailable at this time due to annual repairs being made to the
OAS.)
51
Formative Item Bank Information
On-line
For more
information
about the
Formative Item
Bank Project
www.georgiaoas.org
52
Formative Item Bank Information On-line
http://www.gadoe.org/Curriculum-Instruction-and-Assessment/Assessment/Pages/OAS-Resources.aspx
Includes:
• About the Formative Item Bank
(document)
• About the Formative Item Bank
(presentation)
• Student Checklist for ELA
• Students Checklist for
Mathematics
• Link to the OAS
• Link to Georgia Standards.org
53
Upcoming Benchmark Assessments
54
Purpose of the Benchmark Assessments
The purpose of the benchmark assessments is to
provide a tool for teachers to use throughout the year
in support of classroom instruction and frequent or
continual evaluation of student proficiency, keeping
students, parents, administrators and educators
informed of students’ progress on the pathway to
proficiency.
55
Interim Benchmark Assessment
Item and Benchmark Development
• Benchmark Assessments will be created for:
– ELA: Grades 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11
– Mathematics: Grades 1, 2, 3, 4, 5, 6, 7, 8, Coordinate Algebra,
Analytic Geometry and Advanced Algebra
– Social Studies: U.S. History
– Science: Biology
• Additional items that will be housed in an item bank will be
created for:
– Science: Grades 3-8
– Social Studies: Grades 3-8
56
Benchmark Assessment
Production Schedule
Phase I
Fall 2013 pilot
Available Winter 2014
• ELA
– Grades 1, 2, 3, 6, 7, 8, and
10
• Mathematics
– Grades 1, 2, 3, and
Coordinate Algebra
• U.S. History
Phase II
Winter 2014 pilot
Available Fall 2014
• ELA
– Grades 4, 5, 9, and 11
• Mathematics
– Grades 4, 5, 6, 7, 8,
Analytic Geometry, and
Advanced Algebra
• Biology
57
Comparison to Formative Item Bank
•
•
•
•
•
Formative Item Bank
ELA and Mathematics
Grades 3 – High School
Item Bank only
Housed in OAS Level 2 (for
teacher-use)
Variety of item types
– Mostly constructed response
Benchmark Assessments
• ELA, Mathematics, Science and
Social Studies
• Grades 1 – High School
• Item Bank and one benchmark test
form per grade/subject
• Housed in OAS Level 3 (for districtuse)
• Variety of item types
– Mostly multiple-choice and shortanswer
58
On the Horizon for Georgia’s Formative
Assessment Initiatives
• Phase II Formative Item Bank loaded into OAS—
Fall 2013
• Phase I and Phase II Interim Benchmark
Assessments loaded into OAS, projected Spring
and Fall 2014
• Formative Instructional Practices (FIP) professional
learning, will be available summer 2013 and
implementation in school systems TBD by each
school system
59
“Quality assessment is a system of
assessing what students know and
are able to do in a manner that
garners accurate information from
students for the purpose of
improving learning.” (Rick Stiggins,
2008)
60
Questions
61
Georgia Department of Education
Assessment and Accountability
Dr. Melissa Fincher
Associate Superintendent
Assessment and Accountability
404.651.9405
mfincher@doe.k12.ga.us
Dr. Melodee Davis
Director
Assessment Research and Development
404.657.0312
medavis@doe.k12.ga.us
Michael Huneke
Assessment Specialist, OAS Manager
404.232.1208
mhuneke@doe.k12.ga.us
Dr. Dawn Souter
Project Manager, RT3
404.463.6667
dsouter@doe.k12.ga.us
Dr. Jan Reyes
Assessment Specialist, RT3
Interim Benchmark Assessments
404.463.6665
jreyes@doe.k12.ga.us
Kelli Harris-Wright, Ed.S.
Assessment Specialist, RT3
Assessment Literacy Professional Learning
404.463.5047
kharris-wright@doe.k12.ga.us
62
Wiki/Email Questions
 Question: Will students have to calculate arc length (and get a
numeric answer) or just have an understanding that arc length
is part of the circumference?
Wiki/Email Questions
 Question: The standards do not specify which notation should
be used when writing an interval. Should we use inequality
notation, set-builder notation, or interval notation?
What’s the big idea?
•Develop understanding with theorems
related to circles.
•Determine arc length and area of
sectors of circles.
•Develop understanding of volume
formulas and apply them to solve
problems.
What’s the big idea?
Standards for Mathematical Practice
Coherence and Focus
• K-8th
 Identifying geometric figures
 Area of polygons and circles
 Cross sections
 Volume of prisms, cylinders, cones,
and spheres
• 10th-12th
 Geometric concepts in modeling
situations
 Trigonometric functions
Examples & Explanations
You have been asked to place a fire hydrant so that it is an
equal distance from three locations indicated below.
Adapted from Illustrative Mathematics G.C Placing a Fire Hydrant
Examples & Explanations
You have been asked to place a fire hydrant so that it is an
equal distance from three locations indicated below.
A
B
C
Adapted from Illustrative Mathematics G.C Placing a Fire Hydrant
Examples & Explanations
You have been asked to place a fire hydrant so that it is an
equal distance from three locations indicated below.
A
B
C
Adapted from Illustrative Mathematics G.C Placing a Fire Hydrant
Examples & Explanations
You have been asked to place a fire hydrant so that it is an
equal distance from three locations indicated below.
A
B
C
Adapted from Illustrative Mathematics G.C Placing a Fire Hydrant
Examples & Explanations
The Georgia Aquarium wants to put in a fish tank in a new
exhibit. They either want to use a cylinder that would fit into a
pillar in the center of the exhibit room, or a hemi-sphere that
would protrude from the ceiling. Both have a diameter of 10 feet.
The cylinder tank is 5-foot tall and contains an “underwater
mountain” in the shape of a 5-foot-tall right cone.
5 ft
10 ft
10 ft
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
To compare the volumes of the tanks Cavalier’s Principle can be
used, by comparing the surface area of the water at varying
levels for both tanks.
5 ft
10 ft
10 ft
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
To compare the volumes of the tanks Cavalier’s Principle can be
used, by comparing the surface area of the water at varying
levels for both tanks.
Determine the surface area of each tank filled at height h from
the top.
5 ft
10 ft
10 ft
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
The radius and height of the cylinder and cone are both 5 feet, so
the vertical cross-section through the vertex of the cone shown
below shows that the cross section of the water forms an
isosceles right triangle.
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
The radius and height of the cylinder and cone are both 5 feet, so
the vertical cross-section through the vertex of the cone shown
below shows that the cross section of the water forms an
isosceles right triangle.
h
5-h
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
The radius and height of the cylinder and cone are both 5 feet, so
the vertical cross-section through the vertex of the cone shown
below shows that the cross section of the water forms an
isosceles right triangle.
5-h
h
h
5-h
5-h
h
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
The surface area of the water can be found by subtracting the
area of the inner circle with radius h from the area of the larger
circle with radius 5.
𝐴 = 52 𝜋 − ℎ 2 𝜋
𝐴 = 25𝜋 − ℎ2 𝜋
h
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
The horizontal distance between the curved wall and the
sphere’s center is denoted by x. But x can be expressed in terms
of h, since x and h both represent lengths of a right triangle with
hypotenuse 5.
h
5-h
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
The horizontal distance between the curved wall and the
sphere’s center is denoted by x. But x can be expressed in terms
of h, since x and h both represent lengths of a right triangle with
hypotenuse 5.
h
h
x
5-h
5-h
𝑥 2 + ℎ2 = 25
𝑥 = 25 − ℎ2
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
The surface area of the water can be found by determining the
area of a circle whose radius is 25 − ℎ2 .
𝐴 = 𝜋( 25 − ℎ2 )2
𝐴 = 𝜋(25 − ℎ2 )
𝐴 = 25𝜋 − ℎ2 𝜋
25 − ℎ2
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
To compare the volumes of the tanks Cavalier’s Principle can be
used, by comparing the surface area of the water at varying
levels for both tanks.
Determine the surface area of each tank filled at height h from
the top.
5 ft
10 ft
𝐴 = 25𝜋 − ℎ2 𝜋
10 ft
𝐴 = 25𝜋 − ℎ2 𝜋
Adapted from Illustrative Mathematics G.MD Use Cavalier’s Principle to Compare Aquarium Volumes
Examples & Explanations
Selina wants to install motion detectors for her 9 meter by 12 meter
backyard. The 12 meter side of her backyard is adjacent to her house.
She has two motion detectors, each of which can detect motion within an
8.4-meter radius. Her insurance company offers a discounted rate on her
homeowners insurance policy if the motion detectors can detect motion in
at least 90% of her lawn. Selina can either place the two motion detectors
on the back corners of her house or she can place one on the back left
corner of her house and the other on a pole which is located in the back
right corner of her yard. Which placement of the detectors would detect
motion in more of her yard? Would both of these placements allow her to
receive the discount from her insurance company?
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
Examples & Explanations
Selina wants to install motion detectors for her 9 meter by 12 meter backyard.
She has two motion detectors, each of which can detect motion within an 8.4meter radius. Selina can either place the two motion detectors on the back
corners of her house or she can place one on the back left corner of her house
and the other on a pole which is located in the back right corner of her yard.
Which placement of the detectors would detect motion in more of her yard?
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
Examples & Explanations
Selina’s insurance company offers a discounted rate on her homeowners
insurance policy if the motion detectors can detect motion in at least 90%
of her lawn. Would both of these placements allow her to receive the
discount from her insurance company?
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
Examples & Explanations
Computing the area detected if Selina places the two motion detectors on
the back corners of her house.
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
Examples & Explanations
Computing the area detected if Selina places the two motion detectors
on the back corners of her house.
X
The detected area is comprised of the area of
isosceles triangle AXB and two circular sectors.
A
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
M
B
Examples & Explanations
Computing the area detected if Selina places the two motion detectors
on the back corners of her house.
X
The detected area is comprised of the area of
isosceles triangle AXB and two circular sectors.
A
𝐴∆ = 12∙𝑋𝑀∙𝐴𝐵
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
M
B
𝑋𝑀2 = 𝐴𝑋 2 − 𝐴𝑀2
𝑋𝑀2 = 8.42 − 62
𝑋𝑀2 = 34.56
𝑋𝑀 ≈ 5.88
Examples & Explanations
Computing the area detected if Selina places the two motion detectors
on the back corners of her house.
X
The detected area is comprised of the area of
isosceles triangle AXB and two circular sectors.
A
𝐴∆ = 12∙𝑋𝑀∙𝐴𝐵
𝐴∆ ≈ 12∙5.88∙12
𝐴∆ ≈ 35.28
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
M
B
𝑋𝑀2 = 𝐴𝑋 2 − 𝐴𝑀2
𝑋𝑀2 = 8.42 − 62
𝑋𝑀2 = 34.56
𝑋𝑀 ≈ 5.88
Examples & Explanations
Computing the area detected if Selina places the two motion detectors
on the back corners of her house.
X
The detected area is comprised of the area of
isosceles triangle AXB and two circular sectors.
A
𝐴∆ = 12∙𝑋𝑀∙𝐴𝐵
𝐴∆ ≈ 12∙5.88∙12
𝐴∆ ≈ 35.28
𝑚∠
𝐴↶ =
∙ 𝜋 ∙ 8.42
360
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
M
B
Examples & Explanations
Computing the area detected if Selina places the two motion detectors
on the back corners of her house.
X
The detected area is comprised of the area of
isosceles triangle AXB and two circular sectors.
A
𝐴∆ = 12∙𝑋𝑀∙𝐴𝐵
𝐴∆ ≈ 12∙5.88∙12
𝐴∆ ≈ 35.28
𝑚∠
𝐴↶ =
∙ 𝜋 ∙ 8.42
360
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
M
B
𝑚∠ = 90 − 𝑚∠XAM
6
cos ∠𝑋𝐴𝑀 = 8.4
𝑚∠𝑋𝐴𝑀 ≈ 44.4
𝑚∠ ≈ 45.6
Examples & Explanations
Computing the area detected if Selina places the two motion detectors
on the back corners of her house.
X
The detected area is comprised of the area of
isosceles triangle AXB and two circular sectors.
A
𝐴∆ = 12∙𝑋𝑀∙𝐴𝐵
𝐴∆ ≈ 12∙5.88∙12
𝐴∆ ≈ 35.28
𝑚∠
𝐴↶ =
∙ 𝜋 ∙ 8.42
360
45.6
𝐴↶ ≈
∙ 𝜋 ∙ 8.42
360
𝐴↶ ≈ 28.08
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
M
B
𝑚∠ = 90 − 𝑚∠XAM
6
cos ∠𝑋𝐴𝑀 = 8.4
𝑚∠𝑋𝐴𝑀 ≈ 44.4
𝑚∠ ≈ 45.6
Examples & Explanations
Computing the area detected if Selina places the two motion detectors
on the back corners of her house.
X
The detected area is comprised of the area of
isosceles triangle AXB and two circular sectors.
A
𝐴∆ = 12∙𝑋𝑀∙𝐴𝐵
𝐴∆ ≈ 12∙5.88∙12
𝐴∆ ≈ 35.28
𝑚∠
𝐴↶ =
∙ 𝜋 ∙ 8.42
360
45.6
𝐴↶ ≈
∙ 𝜋 ∙ 8.42
360
𝐴↶ ≈ 28.08
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
M
B
𝐴 ≈ 35.28 + 2 28.08
𝐴 ≈ 91.44 𝑚 2
Examples & Explanations
Selina’s insurance company offers a discounted rate on her homeowners
insurance policy if the motion detectors can detect motion in at least 90%
of her lawn. Would both of these placements allow her to receive the
discount from her insurance company?
𝐴 𝑜𝑓 𝑏𝑎𝑐𝑘𝑦𝑎𝑟𝑑 𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑 ≈ 91.44 𝑚 2
𝐴 𝑜𝑓 𝑏𝑎𝑐𝑘𝑦𝑎𝑟𝑑 = 9𝑚 12𝑚 = 108𝑚2
91.44𝑚2
% 𝑜𝑓 𝑏𝑎𝑐𝑘𝑦𝑎𝑟𝑑 𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑 =
108𝑚2
≈ 85%
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
X
A
M
B
Examples & Explanations
Computing the area detected if Selina places one of the motion detectors
on the back left corner of her house and the other on a pole which is
located in the back right corner of the backyard.
B
P
X
Q
A
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
Examples & Explanations
Computing the area detected if Selina places one of the motion detectors on the
back left corner of her house and the other on a pole which is located in the back
B
right corner of the backyard.
P
The detected area is comprised of the area of
rhombus APBQ and four circular sectors.
X
Q
A
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
Examples & Explanations
Computing the area detected if Selina places one of the motion detectors on the
back left corner of her house and the other on a pole which is located in the back
B
right corner of the backyard.
P
The detected area is comprised of the area of
rhombus APBQ and four circular sectors.
𝐴𝐴𝑃𝐵𝑄 =
1
2∙𝑃𝑄∙𝐴𝐵
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
X
Q
A
𝑃𝑋 2 = 𝐴𝑃2 − 𝐴𝑋 2
𝑃𝑋 2 = 8.42 − 7.52
𝑃𝑋 2 = 14.31
𝑃𝑋 ≈ 3.78
𝑃𝑄 = 2𝑃𝑋
𝑃𝑄 ≈ 7.56
Examples & Explanations
Computing the area detected if Selina places one of the motion detectors on the
back left corner of her house and the other on a pole which is located in the back
B
right corner of the backyard.
P
The detected area is comprised of the area of
rhombus APBQ and four circular sectors.
1
2∙𝑃𝑄∙𝐴𝐵
1
2∙7.56∙15
𝐴𝐴𝑃𝐵𝑄 =
𝐴𝐴𝑃𝐵𝑄 ≈
𝐴𝐴𝑃𝐵𝑄 ≈ 56.7
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
X
Q
A
𝑃𝑋 2 = 𝐴𝑃2 − 𝐴𝑋 2
𝑃𝑋 2 = 8.42 − 7.52
𝑃𝑋 2 = 14.31
𝑃𝑋 ≈ 3.78
𝑃𝑄 = 2𝑃𝑋
𝑃𝑄 ≈ 7.56
Examples & Explanations
Computing the area detected if Selina places one of the motion detectors on the
back left corner of her house and the other on a pole which is located in the back
B
right corner of the backyard.
P
The detected area is comprised of the area of
rhombus APBQ and four circular sectors.
1
2∙𝑃𝑄∙𝐴𝐵
1
2∙7.56∙15
𝐴𝐴𝑃𝐵𝑄 =
𝐴𝐴𝑃𝐵𝑄 ≈
𝐴𝐴𝑃𝐵𝑄 ≈ 56.7
𝑚∠
𝐴↶ =
∙ 𝜋 ∙ 8.42
360
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
X
Q
A
Examples & Explanations
Computing the area detected if Selina places one of the motion detectors on the
back left corner of her house and the other on a pole which is located in the back
B
right corner of the backyard.
P
The detected area is comprised of the area of
rhombus APBQ and four circular sectors.
1
2∙𝑃𝑄∙𝐴𝐵
1
2∙7.56∙15
𝐴𝐴𝑃𝐵𝑄 =
𝐴𝐴𝑃𝐵𝑄 ≈
𝐴𝐴𝑃𝐵𝑄 ≈ 56.7
𝑚∠
𝐴↶ =
∙ 𝜋 ∙ 8.42
360
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
X
Q
A
𝑚∠ = 180 − 4𝑚∠PAX
7.5
cos ∠𝑃𝐴𝑋 =
8.4
𝑚∠𝑃𝐴𝑋 ≈ 26.77
𝑚∠ ≈ 180 − 4 ∙ 26.77
𝑚∠ ≈ 72.92
Examples & Explanations
Computing the area detected if Selina places one of the motion detectors on the
back left corner of her house and the other on a pole which is located in the back
B
right corner of the backyard.
P
The detected area is comprised of the area of
rhombus APBQ and four circular sectors.
1
2∙𝑃𝑄∙𝐴𝐵
1
2∙7.56∙15
𝐴𝐴𝑃𝐵𝑄 =
𝐴𝐴𝑃𝐵𝑄 ≈
𝐴𝐴𝑃𝐵𝑄 ≈ 56.7
X
Q
A
𝑚∠
𝐴↶ =
∙ 𝜋 ∙ 8.42 𝑚∠ = 180 − 4𝑚∠PAX
360
7.5
72.92
cos ∠𝑃𝐴𝑋 =
𝐴↶ ≈
∙ 𝜋 ∙ 8.42
8.4
360
𝑚∠𝑃𝐴𝑋 ≈ 26.77
𝐴↶ ≈ 44.90
𝑚∠ ≈ 180 − 4 ∙ 26.77
𝑚∠ ≈ 72.92
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
Examples & Explanations
Computing the area detected if Selina places one of the motion detectors on the
back left corner of her house and the other on a pole which is located in the back
B
right corner of the backyard.
P
The detected area is comprised of the area of
rhombus APBQ and four circular sectors.
1
2∙𝑃𝑄∙𝐴𝐵
1
2∙7.56∙15
𝐴𝐴𝑃𝐵𝑄 =
𝐴𝐴𝑃𝐵𝑄 ≈
𝐴𝐴𝑃𝐵𝑄 ≈ 56.7
𝑚∠
𝐴↶ =
∙ 𝜋 ∙ 8.42
360
72.92
𝐴↶ ≈
∙ 𝜋 ∙ 8.42
360
𝐴↶ ≈ 44.90
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
X
Q
A
𝐴 ≈ 56.7 + 44.9
𝐴 ≈ 101.6 𝑚 2
Examples & Explanations
Selina’s insurance company offers a discounted rate on her homeowners
insurance policy if the motion detectors can detect motion in at least 90%
of her lawn. Would both of these placements allow her to receive the
discount from her insurance company?
B
𝐴 𝑜𝑓 𝑏𝑎𝑐𝑘𝑦𝑎𝑟𝑑 𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑 ≈ 101.6 𝑚
𝐴 𝑜𝑓 𝑏𝑎𝑐𝑘𝑦𝑎𝑟𝑑 = 9𝑚 12𝑚 = 108𝑚2
2
% 𝑜𝑓 𝑏𝑎𝑐𝑘𝑦𝑎𝑟𝑑 𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑 =
≈ 94%
91.44𝑚2
108𝑚2
Adapted from Illustrative Mathematics G.C, G.SRT Setting up Sprinklers
P
X
Q
A
If four circles, each with diameter of 10 cm, are placed around a central
circle, as pictured below, what is the perimeter of the formed figure?
10 cm
Adapted from Illustrative Mathematics G.SRT Seven Circle III
If four circles are placed around a central circle with diameter of 10 cm, as
pictured below, what is the perimeter of the formed figure?
Adapted from Illustrative Mathematics G.SRT Seven Circle III
x°
10
(360 – x)°
Adapted from Illustrative Mathematics G.SRT Seven Circle III
1
(𝑥−(360−𝑥)
2
1
(2𝑥−360)
2
90 =
90 =
90 = 𝑥 − 180
270 = 𝑥
Adapted from Illustrative Mathematics G.SRT Seven Circle III
x°
10
(360 – x)°
270
𝑙=(
)(2𝜋 ∙ 5)
360
3
𝑙 = (10𝜋)
4
𝑙 = 7.5𝜋
Adapted from Illustrative Mathematics G.SRT Seven Circle III
270°
10
90°
7.5π
𝑃 = 4 7.5𝜋
𝑃 = 30𝜋
Adapted from Illustrative Mathematics G.SRT Seven Circle III
10
Examples & Explanations
A bicycle has two gears, one of radius 3 centimeters and the
other of radius 7 centimeters. The gears are attached so that the
centers are 37 centimeters apart. The manufacturer needs to
produce a chain that will fit around the two gears. How long
should the chain be?
3 cm
37 cm
Adapted from Illustrative Mathematics G.C Two Wheels and a Belt
7 cm
Examples & Explanations
The length of the chain consists of four parts: two segments
tangent to the two circles, and two circular arcs.
3 cm
37 cm
Adapted from Illustrative Mathematics G.C Two Wheels and a Belt
7 cm
Examples & Explanations
The two tangent segments are lengths of sides of a rectangle
formed by the radius of the small gear. They are also the leg of a
right triangle with hypotenuse 37 and leg 4.
𝑙 = 372 − 42
𝑙 = 1353
𝑙 ≈ 36.78
3 cm
3 cm
Adapted from Illustrative Mathematics G.C Two Wheels and a Belt
θ
37 cm
4 cm
Examples & Explanations
The two arcs are determined by the central angle θ.
4
cos 𝜃 =
37
𝜃 = 1.462
3 cm
3 cm
θ
37 cm
Adapted from Illustrative Mathematics G.C Two Wheels and a Belt
4 cm
Examples & Explanations
The two arcs are determined by the central angle θ.
Small gear arc = 2(1 .462) ∙ 3 ≈ 8.77cm
3 cm
3 cm
θ
37 cm
Large gear arc = 2π − 2(1.462) ∙ 7 ≈ 23.51cm
Adapted from Illustrative Mathematics G.C Two Wheels and a Belt
4 cm
Examples & Explanations
The total length is the sum of all four sections.
3 cm
3 cm
θ
37 cm
Total Length ≈ 2 ∙ 36.78 + 8.77 + 23.51 ≈ 105.84cm
Adapted from Illustrative Mathematics G.C Two Wheels and a Belt
4 cm
Resource List
The following list is provided as a
sample of available resources and
is for informational purposes only.
It is your responsibility to
investigate them to determine
their value and appropriateness
for your district. GaDOE does not
endorse or recommend the
purchase of or use of any
particular resource.
• CCGPS Resources
Resources
 SEDL videos - http://bit.ly/RwWTdc
or http://bit.ly/yyhvtc
 Illustrative Mathematics - http://www.illustrativemathematics.org/
 Mathematics Vision Project - http://www.mathematicsvisionproject.org/index.html
 Dana Center's CCSS Toolbox - http://www.ccsstoolbox.com/
 Common Core Standards - http://www.corestandards.org/
 Tools for the Common Core Standards - http://commoncoretools.me/
 LearnZillion - http://learnzillion.com/
• Assessment Resources
MAP - http://www.map.mathshell.org.uk/materials/index.php
 Illustrative Mathematics - http://illustrativemathematics.org/
 CCSS Toolbox: PARCC Prototyping Project - http://www.ccsstoolbox.org/
 Smarter Balanced - http://www.smarterbalanced.org/smarter-balanced-assessments/
 PARCC - http://www.parcconline.org/
Online Assessment System - http://bit.ly/OoyaK5

Resources
• Professional Learning Resources
 Inside Mathematics- http://www.insidemathematics.org/
 Annenberg Learner - http://www.learner.org/index.html
 Edutopia – http://www.edutopia.org
 Teaching Channel - http://www.teachingchannel.org
 Ontario Ministry of Education - http://bit.ly/cGZlce
Achieve - http://www.achieve.org/
• Blogs
 Dan Meyer – http://blog.mrmeyer.com/
 Robert Kaplinsky - http://robertkaplinsky.com/
• Books
 Van De Walle & Lovin, Teaching Student-Centered Mathematics, Grades 5-8
Resources
http://www.mathematicsvisionproject.org/
Resources
http://www.mathematicsvisionproject.org/
Feedback
http://ccgpsmathematics9-10.wikispaces.com/
James Pratt – jpratt@doe.k12.ga.us
Brooke Kline – bkline@doe.k12.ga.us
Secondary Mathematics Specialists
Thank You!
Please visit http://ccgpsmathematics9-12.wikispaces.com/ to share your feedback, ask
questions, and share your ideas and resources!
Please visit https://www.georgiastandards.org/Common-Core/Pages/Math.aspx
to join the 9-12 Mathematics email listserve.
Follow on Twitter!
Follow @GaDOEMath
Brooke Kline
Program Specialist (6‐12)
bkline@doe.k12.ga.us
James Pratt
Program Specialist (6-12)
jpratt@doe.k12.ga.us
These materials are for nonprofit educational purposes only.
Any other use may constitute copyright infringement.