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CCGPS MathematicsUnit-by-Unit Grade Level Webinar
8th GradeUnit 3: Geometric Applications of Exponents
September 4, 2012
Session will be begin at 8:00 amWhile you are waiting, please do the following:
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CCGPS MathematicsUnit-by-Unit Grade Level Webinar
8th GradeUnit 3: Geometric Applications of Exponents
September 4, 2012
James Pratt – jpratt@doe.k12.ga.usBrooke Kline – bkline@doe.k12.ga.usSecondary Mathematics Specialists
These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.
Expectations and clearing up confusion• This webinar focuses on CCGPS content specific to Unit 3, 8th Grade. • For information about CCGPS across a single grade span, please access the list of recorded GPB sessions on Georgiastandards.org.• For information on the Standards for Mathematical Practice, please access the list of recorded Blackboard sessions from Fall 2011 on GeorgiaStandards.org.• CCGPS is taught and assessed from 2012-2013 and beyond. • A list of resources will be provided at the end of this webinar and these documents are posted in the 6-8 wiki.
http://ccgpsmathematics6-8.wikispaces.com/
Expectations and clearing up confusion
• The intent of this webinar is to bring awareness to: the types of tasks that are contained within the unit. your conceptual understanding of the mathematics in this unit. approaches to the tasks which provide deeper learning situations for your students.
We will not be working through each task during this webinar.
Welcome!• Thank you for taking the time to join us in this discussion of Unit 3.• At the end of today’s session you should have at least 3 takeaways:
the big idea of Unit 3 something to think about…some food for thought
how might I support student problem solving? what is my conceptual understanding of the material in this unit?
a list of resources and support available for CCGPS mathematics
Welcome!• Please provide feedback at the end of today’s session.
Feedback helps us become better teachers and learners.Feedback helps as we develop the remaining unit-by-unit webinars. Please visit http://ccgpsmathematics6-8.wikispaces.com/ to share your feedback..
• After reviewing the remaining units, please contact us with content area focus/format suggestions for future webinars.
James Pratt – jpratt@doe.k12.ga.us Brooke Kline – bkline@doe.k12.ga.usSecondary Mathematics Specialists
Misconception?
Richard Benson: The Very Best Totally Wrong Answers
Misconception?
Richard Benson: The Very Best Totally Wrong Answers
Welcome!• For today’s session have you:
read the mathematics CCGPS? read the unit and worked through the tasks in the unit? downloaded and saved the documents from this session?
• Ask questions and share resources/ideas for the common good.• Bookmark and become active in the 6-8 wiki. If you are still wondering what a wiki is, we will discuss this near the end of the session.
Activate your Brain Use the Pythagorean Theorem to find the length of
orange line segment inside the cube.
5 in
Activate your Brain Use the Pythagorean Theorem to find the length of
orange line segment inside the cube.
5 in
• Do you like this question?
• Does this question require students to reveal their understanding of when to apply the Pythagorean Theorem to determine lengths in 3-D figures?
• Could you improve this question in order to assist in revealing student misconceptions with the Pythagorean Theorem and 3-D figures?
Misconceptions
It is important to realize that inevitably students will develop misconceptions…
Askew and Wiliam 1995; Leinwand, 2010; NCTM, 1995; Shulman, 1996
Misconception – Invented Rule?
Richard Benson: The Very Best Totally Wrong Answers
Misconception – Invented Rule?
Richard Benson: The Very Best Totally Wrong Answers
Misconceptions
Therefore it is important to have strategies for identifying, remedying, as well as for avoiding misconceptions.
Leinwand, 2010; Swan 2001; NBPTS, 1998; NCTM, 1995; Shulman, 1986;
Activate your Brain Use the Pythagorean Theorem to find the length of
orange line segment inside the cube.
5 in
Activate your Brain
How do you find the length of the orange line segment inside of the cube?
Adapted from Learnzillion.com 8.G.7 & 8.EE.2
5 in
Activate your Brain
How do you find the length of the orange line segment inside of the cube?
Adapted from Learnzillion.com 8.G.7 & 8.EE.2
5 in
Activate your Brain
How do you find the length of the orange line segment inside of the cube?
Adapted from Learnzillion.com 8.G.7 & 8.EE.2
5 in
5 in
5 in
5 in
Activate your Brain
How do you find the length of the orange line segment inside of the cube?
Adapted from Learnzillion.com 8.G.7 & 8.EE.2
5 in
5 in
5 in
5 in
leg2 + leg2 = hypotenuse2
52 + 52 = x2
25 + 25 = x2
50 = x2
Activate your Brain
How do you find the length of the orange line segment inside of the cube?
Adapted from Learnzillion.com 8.G.7 & 8.EE.2
5 in
5 in
5 in
7.1 in
5 in
leg2 + leg2 = hypotenuse2
52 + 52 = x2
25 + 25 = x2
50 = x2
=7.1 = x
Activate your Brain
How do you find the length of the orange line segment inside of the cube?
Adapted from Learnzillion.com 8.G.7 & 8.EE.2
leg2 + leg2 = hypotenuse2
5 in
7.1 in
x
Activate your Brain
How do you find the length of the orange line segment inside of the cube?
Adapted from Learnzillion.com 8.G.7 & 8.EE.2
leg2 + leg2 = hypotenuse2
5 in
7.1 in
x
52 + 7.12 = x2
25 + 50.4 = x2
75.4 = x2
Activate your Brain
How do you find the length of the orange line segment inside of the cube?
Adapted from Learnzillion.com 8.G.7 & 8.EE.2
leg2 + leg2 = hypotenuse2
5 in
7.1 in
x
52 + 7.12 = x2
25 + 50.4 = x2
75.4 = x2
8.68… = x
8.7 = x
=
Activate your Brain Learnzillion.com
• Review• Common Mistakes• Core Lesson• Guided Practice• Extension Activities• Quick Quiz
What’s the big idea?
• Overview • Key Standards• Enduring Understandings• Essential Questions• Strategies for Teaching & Learning
What’s the big idea?• Deepen understanding with evaluating square roots and cube roots. • Develop deep understanding with using square root and cube root symbols to represent solutions of simple quadratic and cubic equations.•Develop deep understanding with applications of the Pythagorean Theorem.•Deepen understanding with volume. • Develop deep understanding with using and applying the volume formulas for a cone, cylinder and sphere.
What’s the big idea?Standards for Mathematical Practice
Education Week Webinar – Bristol CT School District
What’s the big idea?Standards for Mathematical Practice
Education Week Webinar – Math Practices and the Common Core
Questions that arose•Converse of the Pythagorean Theorem•“Small” perfect squares and cubes•Operations with radicals
Questions that arose•Essential Questions
Questions that arose•Enduring Understandings
estimate
Questions that arose•Acting Out Task – Essential Questions
Questions that arose•Angry Bird Task – Essential Questions
Questions that arose•Angry Bird Task Extension – Distance Formula
Coherence and Focus – Unit 3
Education Week Webinar – Jason Zimba, lead writer of the CCSM
Coherence and Focus – Unit 3What are students coming with?
Coherence and Focus – Unit 3What foundation is being built?
Where does this understanding lead students?
•Enduring Understandings•Evidence of Learning
Coherence and Focus – Unit 3View across grade bands
• K-7th 3-D shapes & volumeRational/Irrational numbers, square roots & cube rootsSolving Equations
• 9th-12th Distance FormulaSolving quadratic and cubic equationsTrigonometry
Misconception?
Richard Benson: The Very Best Totally Wrong Answers
Examples & ExplanationsUse the right triangle with side lengths of 3 cm, 4cm, and 5 cm to
explain a proof of the Pythagorean Theorem?
4 cm
3 cm 5 cm
Examples & ExplanationsHow can you explain a proof of the Pythagorean Theorem using
the diagram below?
Adapted from Learnzillion.com 8.G.6
Examples & Explanations
Adapted from Learnzillion.com 8.G.6
4 cm
3 cm
Examples & Explanations
Adapted from Learnzillion.com 8.G.6
9 cm2
16 cm2
Examples & Explanations
Adapted from Learnzillion.com 8.G.6
9 cm2
16 cm2
Examples & Explanations
Adapted from Learnzillion.com 8.G.6
9 cm2
16 cm2
Examples & Explanations
Adapted from Learnzillion.com 8.G.6
9 cm2
16 cm2
6
66
6
1
Examples & Explanations
Adapted from Learnzillion.com 8.G.6
9 cm2
16 cm2
6
66
6
1 25 cm2
Examples & Explanations
Adapted from Learnzillion.com 8.G.6
9 cm2
16 cm2
6
66
6
1 25 cm2
32 + 42 = 52
9 cm2 +
16 cm2 =
25 cm2
Examples & Explanations
Adapted from Learnzillion.com 8.G.6
a c
leg hypotenuse
bleg
32 + 42 = 52
9 cm2 +
16 cm2 =
25 cm2
a2 + b2 = c2
Pythagorean Theorem
leg2 + leg2 = hypotenuse2
Examples & ExplanationsUse the Pythagorean Theorem to determine if the triangle is a
right triangle.
Examples & ExplanationsIs it possible to determine if the triangle is a right triangle? If so,
explain how you could prove whether it is or is not a right triangle. If not possible, explain why you can not make this determination.
8.G.6, 8.G.8, & 8.EE.2
Examples & Explanations
8.G.6, 8.G.8, & 8.EE.2
Examples & Explanations
8.G.6, 8.G.8, & 8.EE.2
8
6
Examples & Explanations
8.G.6, 8.G.8, & 8.EE.2
8
6
xx
x
x
x
x
10
100
100
6436
86
2
2
2
222
Examples & Explanations
8.G.6, 8.G.8, & 8.EE.2
x
x
x
x
x
10
100
100
6436
86
2
2
2
222
8
6
x
4
4
4
10
y
z
Examples & Explanations
8.G.6, 8.G.8, & 8.EE.2
x
x
x
x
x
10
100
100
6436
86
2
2
2
222
8
6
x
4
4
4
10
y
z
y
y
y
y
y
7.5
32
32
1616
44
2
2
2
222
z
z
z
z
z
8.10
116
116
10016
104
2
2
2
222
Examples & Explanations
8.G.6, 8.G.8, & 8.EE.2
8
6
x
4
4
4
10
y
z
If the triangle is a right triangle, then the squares of the two shorter sides must equal the square of the longest side,
or x² + y² = z².
Examples & Explanations
8.G.6, 8.G.8, & 8.EE.2
8
6
x
4
4
4
10
y
z
If the triangle is a right triangle, then the squares of the two shorter sides must equal the square of the longest side,
or x² + y² = z².
x² = 100, y² = 32, z² = 116100 + 32 ? 116
132 ≠ 116Therefore, the triangle is not a
right triangle.
AssessmentHow might it look?
• Mathematics Assessment Project - http://map.mathshell.org/materials/tests.php
• Illustrative Mathematics - http://illustrativemathematics.org/
• Dana Center’s CCSS Toolbox: PARCC Prototype Project - http://www.ccsstoolbox.org/
• Online Assessment System - http://www.gadoe.org/Curriculum-Instruction-and-Assessment/Assessment/Pages/OAS.aspx
Race to the TopAssessment Toolbox
Update Fall 2012
RT3 Assessment Initiatives
• Purpose – To support teachers in preparing the students for the
Common Core Assessment that is to occur in spring 2015– To provide assessment resources that reflect the rigor of
the CCGPS– To balance the use of formative and summative
assessments in the classroom
62
RT3 Assessment Initiatives• Development of a three-prong toolkit to support
teachers and districts and to promote student learning– A professional development opportunity that focuses on
assessment literacy– A set of benchmarks in ELA, Math, and selected grades for
Science and Social Studies– An expansive bank of formative assessment
items/tasks based on CCGPS in ELA and Math as a teacher resource
63
Formative Assessment• Conducted during instruction (lesson, unit, etc.)• Identifies student strengths and weaknesses• Helps teacher determine next steps
– Review– Differentiation– Continuation
• Supplies information to provide students with detailed feedback• Assessment for the purpose of improving achievement• LOW stakes
64
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 curriculum. The items will provide an opportunity for students to show what they know and show teachers what students do not understand.
65
Formative Item Bank Assessments
• Aligned to CCGPS• Format aligned with Common Core Assessments• Open-ended and constructed response items as
well as multiple choice items• Holistic Rubrics• Anchor Papers• Student Exemplars• 750+ Items Available in OAS by late September
66
Formative Item Bank Availability
• All items that pass data review will be uploaded to the Georgia OAS at Level 2.
• Formative Item Bank will be ready for use by all Georgia educators mid-September, 2012.
67
68
Item Types
–Multiple Choice (MC)
–Extended Response (ER)
–Scaffolded Item (SC)
Extended Response Items• Performance-based tasks• 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
69
Mathematics Sample Item – Grade HSan extended response item
70
Example Rubric
71
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
72
Mathematics Sample Item – Grade 3a scaffolded item
73
Mathematics Items
• Assess students’ conceptual and computational understanding
• Tasks require students to– Apply the mathematics they know to real world problems– Express mathematical reasoning by showing their work or
explaining their answer
74
Where do you Find the Items?
75
rt1234567890
student
Georgia Department of Education Assessment and Accountability
Melissa FincherAssociate SuperintendentAssessment and Accountabilitymfincher@doe.k12.ga.us
Dr. Melodee Davis Director Assessment Research and Developmentmedavis@doe.k12.ga.us
Robert Anthony Assessment SpecialistFormative Item BankRace to the Topranthony@doe.k12.ga.us
Jan ReyesAssessment SpecialistInterim Benchmark AssessmentsRace to the Topjreyes@doe.k12.ga.us
Dr. Dawn SouterProject ManagerRace to the Topdsouter@doe.k12.ga.us
Suggestions for getting started:• Read the unit and work through the tasks with your colleagues.
The only way to gain deep understanding is to work through each task.
• Make note of where, when, and what the big ideas are.• Discuss the focus and coherence of the unit.• Make note of where, when, and what the pitfalls might be. • Look for additional tools/ideas you want to use.• Determine any changes which might need to be made to make
this work for your students.• Share, ask, and collaborate on the wiki.
http://ccgpsmathematics6-8.wikispaces.com/
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.
What is a Wiki?
Resources• Common Core Resources
SEDL videos - https://www.georgiastandards.org/Common-Core/Pages/Math.aspx or http://secc.sedl.org/common_core_videos/ Illustrative Mathematics - http://www.illustrativemathematics.org/ Dana Center's CCSS Toolbox - http://www.ccsstoolbox.com/ Arizona DOE - http://www.azed.gov/standards-practices/mathematics-standards/ Ohio DOE - http://www.ode.state.oh.us/GD/Templates/Pages/ODE/ODEPrimary.aspx?page=2&TopicRelationID=1704Common Core Standards - http://www.corestandards.org/ Tools for the Common Core Standards - http://commoncoretools.me/ Phil Daro talks about the Common Core Mathematics Standards - http://serpmedia.org/daro-talks/index.html
•BooksVan DeWalle and Lovin, Teaching Student-Centered Mathematics, 6-8
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
• Assessment Resources MAP - http://www.map.mathshell.org.uk/materials/index.php CCSS Toolbox: PARCC Prototyping Project - http://www.ccsstoolbox.org/ PARCC - http://www.parcconline.org/parcc-states
• BlogsDan Meyer – http://blog.mrmeyer.com/Timon Piccini – http://mrpiccmath.weebly.com/3-acts.htmlDan Anderson – http://blog.recursiveprocess.com/tag/wcydwt/
Resources• Dana Center’s CCSS Toolbox - PARCC Prototyping Project
http://www.ccsstoolbox.com/
Resources• Dan Meyer’s Three-Act Math Tasks
https://docs.google.com/spreadsheet/lv?key=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE
As you start your day tomorrow…
…the standards are not units of instruction; you don’t always “teach a standard” in one chunk, whatever the order…The standards describe achievements we want students to have. As my colleague Jason Zimba likes
to say, you don’t teach standards, you teach mathematics.
Bill McCallum – lead writer of the CCSM
Thank You! Please visit http://ccgpsmathematics6-8.wikispaces.com/ to share your feedback, ask
questions, and share your ideas and resources!
Please visit https://www.georgiastandards.org/Common-Core/Pages/Math.aspxto join the 6-8 Mathematics email listserve.
Brooke KlineProgram Specialist (6‐12)
bkline@doe.k12.ga.us
James PrattProgram Specialist (6-12)
jpratt@doe.k12.ga.us
These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.
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