CCGPS Mathematics
Unit-by-Unit Grade Level Webinar
Analytic Geometry
Unit 4: Extending the Number System
September 3, 2013
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CCGPS Mathematics
Unit-by-Unit Grade Level Webinar
Analytic Geometry
Unit 4: Extending the Number System
September 3, 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 4
• Incorporating Standards for Mathematical Practice
• Resources
Wiki/Email Questions
• Multiply Polynomials
When do students multiply polynomials such as (𝑥 + 1)(𝑥 + 2)?
Wiki/Email Questions
• Imaginary numbers
I see that we teach imaginary numbers prior to teaching students about
quadratic equations, factoring, etc. Should this be reversed?
Complete the following tables.
Adapted from Illustrative Mathematics N.RN Operations with Rational and Irrational Numbers
Based on the information from your chart, conjecture which of the
statements is ALWAYS true, which is SOMETIMES true, and which is
NEVER true.
• The sum of a rational number and a rational number is rational.
• The sum of a rational number and an irrational number is irrational.
• The sum of an irrational number and an irrational number is irrational.
• The product of a rational number and a rational number is rational.
• The product of a rational number and an irrational number is irrational.
• The product of an irrational number and an irrational number is
irrational.
Adapted from Illustrative Mathematics N.RN Operations with Rational and Irrational Numbers
Based on the information from your chart, conjecture which of the
statements is ALWAYS true, which is SOMETIMES true, and which is
NEVER true.
• The sum of a rational number and a rational number is rational.
ALWAYS true.
• The sum of a rational number and an irrational number is irrational.
ALWAYS true.
• The sum of an irrational number and an irrational number is irrational.
SOMETIMES true ( 2 + (− 2) = 0)
Adapted from Illustrative Mathematics N.RN Operations with Rational and Irrational Numbers
Based on the information from your chart, conjecture which of the
statements is ALWAYS true, which is SOMETIMES true, and which is
NEVER true.
• The product of a rational number and a rational number is rational.
ALWAYS true
• The product of a rational number and an irrational number is irrational.
SOMETIMES true ( 2 × 0 = 0)
• The product of an irrational number and an irrational number is
irrational.
SOMETIMES true ( 2 × 1
2
= 1)
Adapted from Illustrative Mathematics N.RN Operations with Rational and Irrational Numbers
What’s the big idea?
• Extend the properties of exponents
to rational exponents.
• Use properties of rational and
irrational numbers.
• Perform arithmetic operations on
polynomials.
• Perform arithmetic operations with
complex numbers.
What’s the big idea?
Standards for Mathematical Practice
What’s the big idea?
• SMP 1 – Make sense of problems
and persevere in solving them
• SMP 2 – Reason abstractly and
quantitatively
• SMP 3 – Construct viable
arguments and critique the
reasoning of others
• SMP 4 – Model with mathematics
• SMP 5 – Use appropriate tools
http://blog.mrmeyer.com/
strategically
http://bit.ly/17QDmw9
http://www.schooltube.com/video/81f35b2779ef8d4727fd/
http://www.youtube.com/watch?v=jRMVjHjYB6w
Resources
http://robertkaplinsky.com/
Coherence and Focus
• K-9th
 Algebraic expressions
 Properties of operations
 Rational and irrational numbers
 Radicals and integer exponents with
numerical expressions
• 11th-12th
 Polynomial identities and equations
 Polynomial, square root, and cube root
functions
Examples & Explanations
A biology student is studying bacterial growth, She was
surprised to find that the population of the bacteria doubled
every hour.
Complete the following table:
Adapted from Illustrative Mathematics N-RN Extending the Definitions of Exponents, Variation 2
Examples & Explanations
The student conducting the study wants to create a table with
more entries; specifically, she wants to fill in the population at
each half hour. However she forgot to make these
measurements so she want s to estimate the values.
Complete the following table:
Adapted from Illustrative Mathematics N-RN Extending the Definitions of Exponents, Variation 2
Examples & Explanations
What if the student wanted to estimate the population every 20
minutes instead of every 30 minutes. What multiplier would be
necessary to be consistent with the population doubling every
hour?
Complete the following table:
Adapted from Illustrative Mathematics N-RN Extending the Definitions of Exponents, Variation 2
Examples & Explanations
Use the population context to explain why it makes sense that
1
2
1
3
3
we define 2 to be 2 and 2 to be 2
Adapted from Illustrative Mathematics N-RN Extending the Definitions of Exponents, Variation 2
Examples & Explanations
Use the population context to explain why it makes sense that
1
3
1
2
3
we define 2 to be 2 and 2 to be 2
The equation for the population
is 𝑃 = 4(2)𝑡 .
𝑃
1
2
=
1
4(2)2 =
4∙ 2
1
2
∴ (2) = 2
Adapted from Illustrative Mathematics N-RN Extending the Definitions of Exponents, Variation 2
Examples & Explanations
Use the population context to explain why it makes sense that
1
3
1
2
3
we define 2 to be 2 and 2 to be 2
The equation for the population
is 𝑃 = 4(2)𝑡 .
𝑃
1
2
=
1
2
1
4(2)2 =
∴ (2) = 2
4∙ 2
𝑃
1
3
=
1
4(2)3 =
1
3
∴ (2) =
3
3
4∙ 2
2
Adapted from Illustrative Mathematics N-RN Extending the Definitions of Exponents, Variation 2
Examples & Explanations
A garden is created so that the garden bed is seven yards more
than twice the width of the bed. A walkway is created around
the garden that is 2 yards wide. Write an expression that
represents the area of the walkway surrounding the garden.
Adapted from Sophia.org Adding and Subtracting Polynomials in the Real World
Examples & Explanations
A garden is created so that the garden bed is seven yards more than twice the width of
the bed. A walkway is created around the garden that is 2 yards wide. Write an
expression that represents the area of the walkway surrounding the garden.
Adapted from Sophia.org Adding and Subtracting Polynomials in the Real World
Examples & Explanations
A garden is created so that the garden bed is seven yards more than twice the width of
the bed. A walkway is created around the garden that is 2 yards wide. Write an
expression that represents the area of the walkway surrounding the garden.
Area of large rectangle – Area of small rectangle
𝑥 + 4 2𝑥 + 11 − 𝑥 2𝑥 + 7 =
2𝑥 2 + 11𝑥 + 8𝑥 + 44 − 2𝑥 2 + 7𝑥 =
2𝑥 2 + 11𝑥 + 8𝑥 + 44 − 2𝑥 2 − 7𝑥 =
2𝑥 2 − 2𝑥 2 + 11𝑥 + 8𝑥 − 7𝑥 + 44 =
12𝑥 + 44
Adapted from Sophia.org Adding and Subtracting Polynomials in the Real World
Examples & Explanations
Rewrite each of the following expressions involving complex
numbers in the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are real numbers.
3 + 2𝑖 2 − 5𝑖
5 + 4𝑖 17 − 13𝑖 − 5 + 3𝑖 17 − 13𝑖
2 −(5+ 𝑖 )2
(52+7𝑖
)
2
2 2
1 + 𝑖 13 − 4𝑖 1 − 𝑖
1 + 𝑖 + 𝑖2 + 𝑖3
Adapted from Illustrative Mathematics N-CN, A-SSE Computations With Complex Numbers
Examples & Explanations
Rewrite each of the following expressions involving complex
numbers in the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are real numbers.
3 + 2𝑖 2 − 5𝑖 =
3 2 − 5𝑖 + 2𝑖 2 − 5𝑖 =
6 − 15𝑖 + 4𝑖 − 10𝑖 2 =
6 − 15𝑖 + 4𝑖 + 10 =
16 − 11𝑖
Adapted from Illustrative Mathematics N-CN, A-SSE Computations With Complex Numbers
Examples & Explanations
Rewrite each of the following expressions involving complex
numbers in the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are real numbers.
5 + 4𝑖 17 − 13𝑖 − 5 + 3𝑖 17 − 13𝑖 =
17 − 13𝑖
5 + 4𝑖 − 5 + 3𝑖
=
17 − 13𝑖 𝑖 =
17𝑖 − 13𝑖 2 =
13 + 17𝑖
Adapted from Illustrative Mathematics N-CN, A-SSE Computations With Complex Numbers
Examples & Explanations
Rewrite each of the following expressions involving complex
numbers in the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are real numbers.
2 −(5+ 𝑖 )2
(52+7𝑖
)
2
2 2
(25
4
+
70𝑖
4
49𝑖 2
+ 4 ) − (25
4
2
60𝑖
48𝑖
+
4
4
15𝑖 + 12𝑖 2
+
10𝑖
4
+
𝑖2
4)
−12 + 15𝑖
Adapted from Illustrative Mathematics N-CN, A-SSE Computations With Complex Numbers
Examples & Explanations
Rewrite each of the following expressions involving complex
numbers in the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are real numbers.
1 + 𝑖 13 − 4𝑖 1 − 𝑖 =
1 + 𝑖 1 − 𝑖 13 − 4𝑖 =
1 − 𝑖 2 13 − 4𝑖 =
2 13 − 4𝑖 =
26 − 8𝑖
Adapted from Illustrative Mathematics N-CN, A-SSE Computations With Complex Numbers
Examples & Explanations
Rewrite each of the following expressions involving complex
numbers in the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are real numbers.
1 + 𝑖 + 𝑖2 + 𝑖3 =
1 + 𝑖 + −1 + −𝑖 =
0
Adapted from Illustrative Mathematics N-CN, A-SSE Computations With Complex Numbers
Assessment – Released Items
We have posted a set of released […] EOCT items to the GaDOE
website. In addition to the item booklet itself, you will find commentary and
field test performance data. […] The items are posted on the EOCT
webpage, under the link 'EOCT Resources.' A direct link to this webpage is
provided below. Please scroll down the page and look under the heading
'Other Documents and Resources.' […]
http://www.gadoe.org/Curriculum-Instruction-and-Assessment/Assessment/Pages/EOCTResources.aspx
~ Dr. Melissa Fincher, Associate Superintendent for Assessment and Accountability
(excerpt from an email sent to K-12 Assessment Directors from Dr. Fincher)
Assessment – Released Items
Which statement is true about this expression?
4
5 3+ 4 +
5
A
It is rational because it is the sum of two irrational numbers.
B
It is irrational because it is the sum of two irrational numbers.
C
It is rational because it is the sum of a rational number and an irrational number.
D
It is irrational because it is the sum of a rational number and an irrational number.
Assessment – Released Items
Which statement is true about this expression?
4
5 3+ 4 +
5
A
It is rational because it is the sum of two irrational numbers. 15.52%
B
It is irrational because it is the sum of two irrational numbers. 28.14%
C
It is rational because it is the sum of a rational number and an irrational number. 26.26%
D
It is irrational because it is the sum of a rational number and an irrational number. 29.38%
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://learnzillion.com/
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.