CCGPS Mathematics
Unit-by-Unit Grade Level Webinar
Analytic Geometry
Unit 6: Modeling Geometry
October 22, 2013
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CCGPS Mathematics
Unit-by-Unit Grade Level Webinar
Analytic Geometry
Unit 6: Modeling Geometry
October 22, 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.

Welcome!
• The big idea of Unit 6
• Incorporating SMPs into geometric modeling
• Resources

Wiki/Email Questions
 Question: Why is MCC9-12.A.REI.7 addressed in both Unit 5 and Unit 6?

Use the Pythagorean theorem to find an equation in x and y whose solutions are the points on the circle of radius 2 with center (1,1) and explain why it works.
Adapted from Illustrative Mathematics G-GPE Explaining the Equation of a Circle

What’s the big idea?
•
•
•

What’s the big idea?
Standards for Mathematical Practice

The big idea!

 Minimal student explanations
 Passive/receptive

The big idea!

Coherence and Focus
• K-9 th
 Write equivalent expressions
 Solving equations for variables on interest
 Pythagorean Theorem
 Quadratic functions
 Completing the square
• 11th-12th
 Modeling with geometry
 Equations of ellipses and hyperbolas

Examples & Explanations
A flashlight mirror has the shape of a paraboloid of diameter 4 inches and depth 2 inches. Where should the bulb be placed so that the emitted light rays are parallel to the axis of the paraboloid?
Adapted from http://www.uwlax.edu/faculty/hasenbank/archived/mth151su10/notes_old/12.02%20-
%20Conics%20and%20Parabolas.pdf

Examples & Explanations
A flashlight mirror has the shape of a paraboloid of diameter
4 inches and depth 2 inches. Where should the bulb be placed so that the emitted light rays are parallel to the axis of the paraboloid?
Adapted from http://www.uwlax.edu/faculty/hasenbank/archived/mth151su10/notes_old/12.02%20-
%20Conics%20and%20Parabolas.pdf

Examples & Explanations
A flashlight mirror has the shape of a paraboloid of diameter
4 inches and depth 2 inches. Where should the bulb be placed so that the emitted light rays are parallel to the axis of the paraboloid?
Adapted from http://www.uwlax.edu/faculty/hasenbank/archived/mth151su10/notes_old/12.02%20-
%20Conics%20and%20Parabolas.pdf

Examples & Explanations
A flashlight mirror has the shape of a paraboloid of diameter
4 inches and depth 2 inches. Where should the bulb be placed so that the emitted light rays are parallel to the axis of the paraboloid?
Adapted from http://www.uwlax.edu/faculty/hasenbank/archived/mth151su10/notes_old/12.02%20-
%20Conics%20and%20Parabolas.pdf

Examples & Explanations
A flashlight mirror has the shape of a paraboloid of diameter
4 inches and depth 2 inches. Where should the bulb be placed so that the emitted light rays are parallel to the axis of the paraboloid?
𝑦 =
1
4𝑝𝑥
2
2 =
1
4𝑝(2)
2
8𝑝 = 4
1 𝑝 =
2
Adapted from http://www.uwlax.edu/faculty/hasenbank/archived/mth151su10/notes_old/12.02%20-
%20Conics%20and%20Parabolas.pdf

Examples & Explanations
A flashlight mirror has the shape of a paraboloid of diameter 4 inches and depth 2 inches. Where should the bulb be placed so that the emitted light rays are parallel to the axis of the paraboloid?
The bulb would need to be placed ½ inch from the rear of the flashlight mirror.
Adapted from http://www.uwlax.edu/faculty/hasenbank/archived/mth151su10/notes_old/12.02%20-
%20Conics%20and%20Parabolas.pdf

Examples & Explanations
Annika wonders why we are suddenly thinking about parabolas in a completely different way than when we did quadratic functions. She wonders how these different ways of thinking match up. For instance, when we talked about quadratic functions earlier we started with 𝑦 = 𝑥 2 . “Hmmmm. …. I wonder where the focus and directrix would be on this function,” she thought. Help Annika find the focus and directrix for 𝑦 = 𝑥 2 .
Adapted from Functioning with Parabolas Mathematics Vision Project http://www.mathematicsvisionproject.org/index.html

Examples & Explanations
Annika wonders why we are suddenly thinking about parabolas in a completely different way than when we did quadratic functions. She wonders how these different ways of thinking match up. For instance, when we talked about quadratic functions earlier we started with 𝑦 = 𝑥 2 . “Hmmmm. …. I wonder where the focus and directrix would be on this function,” she thought. Help Annika find the focus and directrix for 𝑦 = 𝑥 2 .
The vertex is at the origin, so the focus will be located at (0, p) and the directrix will be located at 𝑦 = −𝑝
Adapted from Functioning with Parabolas Mathematics Vision Project

Examples & Explanations
Annika wonders why we are suddenly thinking about parabolas in a completely different way than when we did quadratic functions. She wonders how these different ways of thinking match up. For instance, when we talked about quadratic functions earlier we started with 𝑦 = 𝑥 2 . “Hmmmm. …. I wonder where the focus and directrix would be on this function,” she thought. Help Annika find the focus and directrix for 𝑦 = 𝑥 2 .
The vertex is at the origin, so the focus will be located at (0, p) and the directrix will be located at 𝑦 = −𝑝 𝑥 2 =
1
→ 4p = 1
4𝑝 𝑥 2
1 𝑝 =
4
Adapted from Functioning with Parabolas Mathematics Vision Project

Examples & Explanations
Annika wonders why we are suddenly thinking about parabolas in a completely different way than when we did quadratic functions. She wonders how these different ways of thinking match up. For instance, when we talked about quadratic functions earlier we started with 𝑦 = 𝑥 2 .
“Hmmmm. …. I wonder where the focus and directrix would be on this function,” she thought. Help Annika find the focus and directrix for 𝑦 = 𝑥 2 .
The vertex is at the origin, so the focus will be located at (0, ¼ ) and the directrix will be located at 𝑦 = −
1
4
Adapted from Functioning with Parabolas Mathematics Vision Project

Examples & Explanations
You probably know that the smaller |a| in the standard form
equation of a parabola , the wider the parabola . In other words y = .1x² is a wider parabola than y = .2x². How does this relate to the directrix and focus?
Adapted from mathwarehouse.com
Focus and Directrix of Parabola explained with pictures and diagrams

Examples & Explanations
You probably know that the smaller |a| in the standard form equation of
a parabola , the wider the parabola . In other words y = .1x² is a wider parabola than y = .2x². How does this relate to the directrix and focus?
𝑦 = 2𝑥 2 + 4𝑥 + 1 → 𝑦 + 1 = 2(𝑥 + 1) 2 → 𝑝 = 1/8
Adapted from mathwarehouse.com
Focus and Directrix of Parabola explained with pictures and diagrams

Examples & Explanations
You probably know that the smaller |a| in the standard form equation of
a parabola , the wider the parabola . In other words y = .1x² is a wider parabola than y = .2x². How does this relate to the directrix and focus?
𝑦 = 4𝑥 2 + 4𝑥 + 1 → y = 4(𝑥 +
2
) 2 → 𝑝 = 1/16
Adapted from mathwarehouse.com
Focus and Directrix of Parabola explained with pictures and diagrams

Examples & Explanations
You probably know that the smaller |a| in the standard form equation of
a parabola , the wider the parabola . In other words y = .1x² is a wider parabola than y = .2x². How does this relate to the directrix and focus?
𝑦 = 𝑥 2 + 4𝑥 + 1 → 𝑦 + 3 = (𝑥 + 2) 2 → 𝑝 = 1/4
Adapted from mathwarehouse.com
Focus and Directrix of Parabola explained with pictures and diagrams

Examples & Explanations
You probably know that the smaller |a| in the standard form equation of
a parabola , the wider the parabola . In other words y = .1x² is a wider parabola than y = .2x². How does this relate to the directrix and focus?
𝑦 = .4𝑥 2 + 4𝑥 + 1 → 𝑦 + 9 = .4(𝑥 + 5) 2 → 𝑝 = 5/8
Adapted from mathwarehouse.com
Focus and Directrix of Parabola explained with pictures and diagrams

Examples & Explanations
You probably know that the smaller |a| in the standard form
equation of a parabola , the wider the parabola . In other words y = .1x² is a wider parabola than y = .2x². How does this relate to the directrix and focus?
As |a| decreases p increases, ie. the distance between the vertex and directrix and vertex and focus increases.
Adapted from mathwarehouse.com
Focus and Directrix of Parabola explained with pictures and diagrams

Use the Pythagorean theorem to find an equation in x and y whose solutions are the points on the circle of radius 2 with center (1,1) and explain why it works.
Adapted from Illustrative Mathematics G-GPE Explaining the Equation of a Circle

Adapted from Illustrative Mathematics G-GPE Explaining the Equation of a Circle

For any point (x, y) on the circle.
The horizontal length is |x – 1| and the vertical length is |y – 1|. The hypotenuse is 2.
Adapted from Illustrative Mathematics G-GPE Explaining the Equation of a Circle

For any point (x, y) on the circle.
The horizontal length is |x – 1| and the vertical length is |y – 1|. The hypotenuse is 2.
(𝑥 − 1) 2 +(𝑦 − 1) 2 = 2 2
Adapted from Illustrative Mathematics G-GPE Explaining the Equation of a Circle

Released Items: Housed in the
Assessment Division of GADOE http://www.gadoe.org/Curriculum-Instruction-and-
Assessment/Assessment/Documents/Analytic%20Geometry%
20Released%20Items%20Booklet%20Revised%208-27-
13.pdf
Released Items Commentary: Housed in the Assessment Division of GADOE http://www.gadoe.org/Curriculum-Instruction-and-
Assessment/Assessment/Documents/Analytic%20Geometry%2
0Released%20Items%20-%20Commentary%20Revised%208-
27-13.pdf
There are 20 released items and commentary about each item available for
Analytic Geometry

•
•
•

http://www.gadoe.org/Curriculum-Instruction-and-
Assessment/Assessment/Documents/EOCT%20Anal ytic%20Geometry%20Study%20Guide%20FINAL%2
08.27.13.pdf

http://www.gadoe.org/Curriculum-Instruction-and-
Assessment/Assessment/Documents/EOCT%20Anal ytic%20Geometry%20Study%20Guide%20FINAL%2
08.27.13.pdf

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.

Resources
• CCGPS Resources
 Georgia Virtual Learning http://www.gavirtuallearning.org/Resources.aspx
 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/
 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

Georgia Virtual Learning www.gavirtuallearning.org/Resources.aspx
No password required!
Click here to access shared content

Georgia Virtual Learning

Georgia Virtual Learning

Georgia Virtual Learning

Georgia Virtual Learning

Resources
• Professional Learning Resources
 Inside Mathematicshttp://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

Feedback http://www.surveymonkey.com/s/WZKG5G2
James Pratt – jpratt@doe.k12.ga.us
Brooke Kline – bkline@doe.k12.ga.us

Thank You!
Please visit http://ccgpsmathematics9-10.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 us on Twitter
@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.