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UNPACKING the CONFUSIONPARRC Mathematics 6-12
NJPSA/FEA conference October 2015 presentation
Room: Oceanport North10:45 A.M. – 12:15 P.M.
by Judith T. Brendel, Ed.M.educational consultant
a 3-minute video (for parents or students)LEARN ABOUT THE COMMON CORE IN THREE MINUTES
http://www.corestandards.org/other-resources/key-shifts-in-mathematics/
• Common Core Content vs. Math Practice Standards—a quick review of shifts and focus at each grade and in each high school course
• Lesson planning and instruction to help students become more independent math learners
• A review of new resources for grades 6-12
AGENDA
3 PRINCIPALS guided the STANDARDS
1. Knowledge, skills and understandings for all students to be CAREER and COLLEGE READY
2. Standards must be based on EVIDENCE not just what people feel students need to succeed.
3. Allow TIME for teachers to teach and TIME for students to practice.
An Extra Video Resource
From EngageNY info for parents/students:
Common Core in Mathematics: An Overview
This 14-minute video provides an overview of the Common Core State Standards in Mathematics.
NYS Commissioner of Education John B. King Jr. and contributing author David Coleman discuss the background of the Common Core State Standards, their value in the state, the principles of their development, and the changes required of schools during this transition.
WHAT are the SHIFTS?
6 SHIFTS
1. FOCUS on the math that really matters2. COHERENCEY relates grade-to-grade3. FLUENCY really matters4. deep UNDERSTANDING5. APPLICATION in new situations6. DUAL INTENSITY (both: procedures with
practice and meaning and application with rich set of problems)
1. Greater FOCUS on FEWER TOPICSNOT racing to cover many topics in a mile-wide, inch deep curriculum. YES, focus on the major work of each grade.
• Grades K-2 + - Concept, skills, and problem solving related to addition and subtraction.
• Grades 3-5 X ÷ Concept, skills and problem solving related to multiplication and division of whole numbers and fractions. Grade-5 (decimals) 5.6 ÷ 9.04 = price w/tax = (1.07)($38.00) =
• Grade 6 4/8=2/4 a+2(a+3)= x+6=12 Ratios and proportional relationships, and early algebraic expressions and equations
• Grade 7 5/8+2/3= ( ¼ )( ½ =) 3/4 ÷ 2/3= Ratios and proportional relationships, and arithmetic of rational numbers
• Grade 8 y = mx+b f(x) = 3x-2 Linear algebra and linear functions (parallel lines, perpendicular lines, systems of equations, …. )
9
K 12
Number and Operations
Measurement and Geometry
Algebra and Functions
Statistics and Probability
Traditional U.S. Approach
10
Focusing Attention Within Number and Operations
Operations and Algebraic Thinking
Expressions and Equations
Algebra
→ →
Number and Operations—Base Ten →
The Number System
→
Number and Operations—Fractions
→
K 1 2 3 4 5 6 7 8 High School
11
ALG. - 1Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding
UNIT-1 Relationships Between Quantities and Reasoning with Equations
UNIT-2 Linear Relationships
UNIT-3 Expressions and Equations
UNIT-4 Quadratic Functions and *Modeling
UNIT-5 Functions and Descriptive Statistics
Focus Areas in Mathematics (CCSS)- MS/HS
f(x) = x2
12
GEOMETRYFocus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding
UNIT-1 Congruencey (translations), Proof, and Constructions
UNIT-2 Similarity, Proof, and Trigonometry
UNIT-3 Extending to Three Dimensions
UNIT-4 Connecting Algebra and Geometry Through Coordinates
UNIT-5 Circles With and Without Coordinates
UNIT-6Applications of Probability
Focus Areas in Mathematics (CCSS) - HS
13
ALG. - 2Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding
UNIT-1 Polynomial, Rational, and Radical Relationships
UNIT-2 Trigonometric Functions
UNIT-3 Modeling with Functions
UNIT-4 Inferences and Conclusions from Data (statistics)**
Focus Areas in Mathematics (CCSS) – HS
3 CRITICAL ASPECTS
• Fluency• Understanding• Application
3. FLUENCIES expected (without a calculator)• K 1.OA.5 Add/subtract within 5• 1 1.OA.6 Add/subtract within 10• 2 2.OA.2 Add/subtract within 20 (know single-digit sums from
memory) 2.NBT.5 Add/subtract within 100
• 3. 3.OA.7 Multiply/divide within 100 (know single-digit products from memory)
3.NBT.2 Add/subtract within 1000• 4. 4.NBT.4 Add/subtract within 1,000,000
• 5. 5.NBT.5 Multi-digit multiplication• 6. 6.NS.2,.3 Multi-digit division
Multi-digit decimal operations
What strategies and/or resources have you used to help your students become fluent in required skills for your grade? (in school? At home?) Have 100% of your students become fluent?
3. FLUENCIES and 4. UNDERSTANDINGSignificant Shifts grades 3-5
How fractions are taught, understood and assessed:
*Activity: Do one and Pass left
Gr.3 Compare 2 fractions w/same denominator
Gr.4 Compare 2 fractions w/different denominators
Gr.5 Add or subtract 2 fractions with unlike denominators.
4. UnderstandingPrevious and Newer Type Questions
*Activity: Compare style, expectations (page 1)
Understanding:The CCSS Difference: Grade 8 Mathematics
(what the NJCCCS vs CCSS say)
(2004) Before NJCCCS:1. Understand and apply the Pythagorean Theorem.
(2010) After CCSS 2. Explain a proof of the Pythagorean Theorem and its converse.
3. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
4. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
The CCSS Difference: Grade HS Mathematics
5. APPLICATIONS to NEW SITUATIONS (modeling?)• Estimate how much water and food is needed for
emergency relief in a devastated city of 3 million people, and how it might be distributed.
• Plan a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player.
• Design the layout of the stalls in a school fair so as to raise as much money as possible.
• Analyzing stopping distance for a car.• Modeling savings account balance, bacterial colony
growth, or investment growth.• Engaging in critical path analysis, e.g., applied to
turnaround of an aircraft at an airport.
ONLINE TEST CHALLENGESWHAT do Common Core
and online PARCC questions look like?
Where do you see difficulties?
*Check off list (workbook page 2 )• √ Vocabulary in directions; within the task• √ Complex text: Persevering • √ Manipulating on the screen• √ Organizing work on/off the screen• √ Diagrams: Re-drawing/labeling/details• √ Writing explanations• √ Other?
Screen Shot: Traditional SCR (grade-3 EOY)
Student does computation on scrap paper.
Student types in answer.
Student knows to use ADDITION and ADDS CORRECTLY
Screen Shot: *Traditional SCR but …? (grade-3 EOY)
Multiplication and division with whole numbers. (FLUENCY) Different types of equations in one question.
Screen Shot: *table Part-A Part-B(grade-4 EOY test)
What computation is needed? Where does student do the computation?
APPLICATION of ADDITION and DIVISION in multi-step real-life situation.
Screen Shot: *table Part-A Part-B(grade-3 EOY test) Part MC and Part SCR
Traditional MC and SCR in one question. Partial credit; B not dependent on A answer.
Part A. Read the bar graph (between markings) then easy addition.
Part B. Know what and when to ADD and SUBTRACT
Screen Shot: Complete Picture GraphDrag and Drop (grade-3 EOY)
Each star = 5 minutes
Experience READING and USING a variety of graphs is essential.
sbac a GRADE-11 Practice Test examplew/solutions and rubrics DRAG tick marks
2 POINT TASK Experience CREATING and USING a variety of graphs is essential.
Screen Shot: *Multiple correct answers(grade-3 EOY test)
Notice “square-like shape” of the “bubble-in” form when more than one correct answer.
DEEP UNDERSTANDING of concept of multiplication
No computation required.
Screen Shot: *Multiple correct answers(grade-3 EOY test)
Notice “square-like shape” of the “bubble-in” form when more than one correct answer.
D. ( D and E are below C in the same
format.)
E.
Screen Shot: *Three correct answers(grade-4 EOY test)
Select the three choices that are factor pairs for the number 28.
VOCABULARY and MULTIPLE ANSWERS.
Screen Shot: *Two Correct Answers(canot shade-in on screen) (grade-4 EOY)
Notice that the student CANNOT actually shade-in on the screen.
Screen Shot: More than one correct answers
(high school)
From grades 6-HS the student is NOT told how many correct answers to select.
– Select all that apply.– How many show that … ?– Which ones match … ?
Screen Shot: *Multiple correct answers(All graphics not given to students)
Note: Beginning with grade-6 the questions do NOT specify “Select the two … or three … correct choices.”
Screen Shot: Tools to measure(grade-4 EOY)
Notice “circle shape” of “bubble-in” form when there is only “one” correct answer.
Screen Shot: Tools to measure(grade-4 EOY)
170˚ or 11˚ ?
Screen Shot: Plotting on Grid(grade-5 EOY)
Point value could be: 2 points for 3 correct answers1 point for 2 correct answers0 points for 1 or no correct answers.
Screen Shot: *Tools to Graph
Line t: y = -x + 5 Line s: y = 1/3x - 3
sbac GRADE-11 Practice Test w/solutions and rubrics DRAG-DROP
2-POINT TASK
Screen Shot: *Click/Drag or Type (one correct answer) (grade-4 EOY test)
Notice the “fraction” and “mixed
number” forms.
Also note: no “work” is scored,
only the final answer.
Acceptable answers might be:
Screen Shot: *Use Symbols or Type (one correct answer; answer forms)
Scrap paper work:27 – 18 x = 20 – 16x + 18 x + 18x
27 = 20 + 2x-20 -20
28 = 2x 7/2 = x
Acceptable answers:
7/2 or x = 7/2
3 ½ or x = 3 ½ 3.5 or x = 3.5
Screen Shot: Drag/Drop Part A Part B (grade-4)
Part A: drag and drop
Part B: fraction symbol + drag-and-drop or type.
or 7/10
Screen Shot: Drag/Drop (grade-3 EOY)
Screen Shot: Drag/Drop (grade-3 EOY)
VOCABULARY from grade-2
Screen Shot: Check-off Table (grade-4 EOY)
Scrolling is necessary to see the entire table.
Screen Shot: USING “EXHIBITS” (Reference sheet Grade-5)
Yes, right now, the “exhibit” sheet covers the question(s). It cannot be moved. What will students need to do?
See what online looks like! HS Teachers outside of math
use grade-level-appropriate math
See what online looks like! HS Teachers outside of math
use grade-level-appropriate math
960
1920
3840
7680
Part B
Understanding VOCABULARY
Part C
MULTIPLE correct answers.
Part D
Explain
MODELING: applying in real life
sbac GRADE-11 Practice Test w/solutions and rubrics (2 point task)
http://sbac.portal.airast.org/wp-content/uploads/2014/10/
Grade11Math.pdf
(click or copy/paste)Performance Tasks
Writing Rubrics (see rubric ex. 666) Select grade 6, 7, 8 or 11.
How should our students be learning differently?
What are “new” skills our students need to be successful?
STUDENT learning strategies
Teaching student learning strategies that THEY can use to become more successful learners … more responsible for their own learning.
2. COHERENCY and 4. UNDERSTANDINGlinking topics and thinking across grades
COHERENCY
3 + 5 = 5 + 3
1 dog + 3 cats + 6 dogs = 1 dog + 6 dogs + 3 cats
3a + 5b + a = 5b + a + 3a
ORDER doesn’t matter in ADDITION
COHERENCYrefers to the fact that each year students learn something that relates and continues from the prior year; topics are related; all is NOT new!
3 x 5 = 5 x 3 or (8)(9) = (9)(8)
3a(2a) =6a2 and 2a(3a) = 6a2
2 x 3 x 5 = 2 x 5 x 3 2 x 3 x 5 = 2 x 5 x 3 6 x 5 = 30 10 x 3 = 30
4a(3a)(-2b) = -24a2b or 3a(-2b)(4a) = -24a2b
ORDER doesn’t matter in MULTIPLICATION
COHERENCY
refers to the fact that each year students learn something that relates and continues from the prior year; topics are related; all is NOT new!
= 5 + 3 not 8
3 cats + 2 cats + 4 dogs = 5 cats + 4 dogs not 9cdgs
3a + 2a + 3b = 5a + 3b not 8abs
COMBINE “LIKE” TERMS
COHERENCY
refers to the fact that each year students learn something that relates and continues from the prior year; topics are related; all is NOT new!
2’x3’ = 24” x 36” = 863sq.” 20” x 38” = 760sq.”
1) Which area is larger? 2’x3’ or 20”x38” Why?
2) Put in order: 3.2 6/7 0.33 2/3 π 0.33 2/3=0.66 6/7=.857 π=3.14 3.2
3) Which has the greatest rate of change?equation table of x/y values a graphed line
Use “same format” to compare
Which function has the greatest rate-of-change (the greatest slope)? (A) (B) (C)
Here, “I” decided to write each as an equation and compare them.y = 3x+4 y = 1x+1 y = 2x -1 Correct answer: A (slope = 3)
PARENT FUNCTIONS and … y = x y = |x| y = x2
linear function absolute value function quadratic function
y = -x y = -|x| y = -x2
PARENT FUNCTIONS and … y = x y = |x|
y = x2
y = -x y = -|x| y = -x2
y = x + 2 y = |x| + 2 y = x2 + 2
Pre Algebra Algebra Algebra-I and II
VOLUME of basic SOLIDS
V = b x h x l V = s3
V = πr2h
V = (area base)(height) V = (area base)(height) V= (area base) (height) V = Bh V = Bh
V = Bh
A CUBE
CORRESPONDING ANGLES are EQUAL
similar congruent parallel lines cut by triangles triangles transversals
equilateral triangles? ? ?
1
2
5
3
3
4
4
Recap: RULES and STRATEGIES that DON’T CHANGE K-12
• The ORDER of numbers, variables or terms, does not matter in ADDITION or in MULTIPLICATION.
• COMBINE LIKE-TERMS (or LIKE-SHAPES) as a first step in solving problems.
• When COMPARING put all in the SAME FORMAT first.• See what is the SAME when certain PARENT functions are
modified• See what is the SAME about selected VOLUME formulas.• Remember CHARACTERISTICS that are the same in different
polygons.
Look for patterns; look for what you already know!
Differentiated Tasks for Understanding
CONCRETE – Circle fold
PICTORIAL – Geometry find area
SYMBOLIC – Create equations to represent …. X + 1.07x = $2000
ABSTRACT – compare f(x) = x2 with f(x) = 3(x-2)2+1
CIRCLE FOLD (CONCRETE)
CIRCLE-FOLD ACTIVITY (2D – to – 3D)
INTERACTIVE ONLINE RESOURCES (NCTM)http://www.nctm.org/Classroom-Resources/Interactives/Geometric-Solids/
1) "Do you agree? Disagree?”
The area of this rectilinear figure is 66.75 sq. in. 12.3”
3.5”1.5”
15.8”
3.5”
(12.3)(3.5) = 43.05
(15.8)(1.5) = 23.7 (12.3)(5) = 61.5
(3.5)(1.5) = 5.25
2) "Does anyone have the same answer but a different way to explain it?"
12.3”3.5”
1.5”3.5”
12.3”
3.5”1.5”
15.8”
5”
3.5”
(15.8)(5) = 7912.25
Area = 79 – 12.25 = 66.75
PICTORIAL
a2 + b2 = c2
a
b
c
3
3
9
4 16
4
a = 4
b = 3
c = ?
5
25
5
Still PICTORIAL, not concrete
SYMBOLICThe souvenir shop at …. sells balls, caps, and jerseys ….. • Samantha bought a cap and five balls for $51. • The four caps Carlos bought cost $12 more than the jersey his brother bought.• Mr. Kurowski spent $177 on three balls and three jerseys for his grandchildren. How much does each
item cost? (Assume sales tax is included.)
• First, list the unknown quantities and assign a variable to each. Let b represent the cost of a ball. Let c represent the cost of a cap. Let j represent the cost of a jersey.
• Second, use the information from the problem to write equations.
(1) C + 5b = 51 (2) 4c – j = 12 (3) 3b + 3j = 177
• Equation (1) Samantha’s purchases translated into an algebraic equation. Equation (2) Information about Carlos’s and his brother’s purchases.
Equation (3) Mr. Kurowski’s purchases.
• Third, solve the system of equations to find the values for the variables.
• Finally, interpret your solution. A ball costs $7, a cap costs $16, and a jersey costs $52.
ABSTRACT
abstraction (noun): the process of formulating a generalized concept of a common property by disregarding the differences between a number of particular instances …
What are “new” non-math skills our students need to be successful?
Workbook page 3
More then one right answerMORE RIGOR
ACTIVITY Student pairs
GEOMETRY • Same perimeter different areas• Same area different perimeters
AREA with PERIMETER
Activity:
FIND THE AREA: Draw 3-4 different rectangles that have a perimeter of 36. Record the area of each. (Use whole numbers only.)
• Which shapes have the largest & smallest area?
• What do you observe?
Perimeter(s)1 +1 + 17 + 17 = 362 + 2 + 16 + 16 = 363 + 3 + 15 + 15 = 364 + 4 + 14 + 14 = 365 + 5 + 13 + 13 = 366 + 6 + 12 + 12 = 367 + 7 + 11 + 11 = 368 + 8 + 10 + 10 = 369 + 9 + 9 + 9 = 36
Areas:(1)(17) = 17 square units (9)(9) = 81 square units
PERIMETER with AREA
Activity:
FIND THE PERIMETER: Draw 4-5 different rectangles that have a area of 36. Record the perimeter of each. (Use whole numbers only.)
• Which has the largest & smallest perimeter?
Area = 1 x 36 = 36 (p=74) Area = 2 x 18 = 36 (p=38) Area = 3 x 12 = 36 (p=30) Area = 4 x 9 = 36 (p=26)Area = 6 x 6 = 36 (p=24)
What you SHOULD NOT see !
ZN
x
y
>
<
Slope =
slope =
(3,6)
(3,2)
What should I see in Lesson Plans?
Math Practices Standards K-12(workbook page 4)
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the
reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure8. Look for and express regularity in repeated reasoning
See in Lesson Plans(workbook pages 5-6 and on FEA website)
Standards of Math Practices and Student Learning Strategies
ScreenShots of PARCC examples/MP and /grade 3-Algebra II
Links: MP1 - Make Sense & Persevere in problem solving
• Gr.4: Bus, Vans and Cars (we solved this one) http://ccsstoolbox.agilemind.com/parcc/elementary_3775_1.html
• Link: Gr.7: Annie’s Family Trip ** Do a & b http://ccsstoolbox.agilemind.com/parcc/about_middle_3808.html
• Math Practices Examples– Workbook pg.7: Gr.5 “Deb has a board that
measures ….” (EngageNY grade 5 test 2014)
– Workbook pg.7: Gr.8 “The combined volume ….”
Links: MP2 Reason Abstractly and Quantitatively
Grade 6• Link: Inches and Centimeters
http://ccsstoolbox.agilemind.com/parcc/about_middle_3789.html (math practices 2 and 6)
MP3 - Construct viable arguments and critique the reasoning of others.
• Extra Math Practices Examples:– Workbook pg.8: Gr.5 “Alice draws a triangle ….”– Workbook pg.8: Gr.8 “Does the equation …
define a linear ….”
• Link: Go to http://schools.nyc.gov/Academics/CommonCoreLibrary/TasksUnitsStudentWork/default.htm Select [grade 9], [Math], Scroll down and select [COMPANY LOGO]. See pages 4, 5, and 9.
MP4 - Model with mathematics
• Math Practices Examples:– Workbook pg. 9: Grade 8 “The population
growth of two towns ….”
Link: MP5 – Use appropriate tools strategically
The Library of Virtual Manipulativeshttp://nlvm.usu.edu/ennav/vlibrary.html
MP6 - Attend to precision.
• Math Practices Examples:– Workbook pg.10: Grade 5 “A race car ….”
• Link: Geometry: The Inheritance (mp # 1, 6) go to this link and select [math] [grade 10] and locate this geometry task: http://schools.nyc.gov/Academics/CommonCoreLibrary/TasksUnitsStudentWork/default.htm
• Link: Algebra-II: Isabella’s Credit Card: *Link and see complexity of each/all parts, a, b, chttp://ccsstoolbox.agilemind.com/parcc/about_highschool_3829_align.html
MP7 - Look for and make use of structure
• Math Practices Examples:– Workbook pg. 11 Grade 8: “Four tables ….” – Workbook pg. 12 Grade-8: “A box contains ….”
MP8 -Look for and express regularity in repeated reasoning
• Math Practices Examples:– Workbook pg. 13: Grade 5 “Roberto used ….”– Workbook pg.14: Grade 8 Using (4-3 )(42) ….
WHAT HAVE they TRIED?WHAT HAVE they DONE DIFFERENTLY?
• Tell a neighbor• Share with a group
What should I see in the classroom?
Videos• Illustrative Math (all grades: collaboration) a
Smarter Balanced projecthttps://www.teachingchannel.org/videos/illustrative-mathematics-sbac
*Activity: (workbook pages 1-17)List of differentiated strategies: How often do you see these being used in elementary, middle school or high school classes? (Frequently/sometimes/rarely/never)
Plan High-Level, Open-Ended Questions
Plan out the questions you are going to ask prior to your lesson.
The best types of questions are high-level questions; they require thought processes beyond basic rote memory.
Higher-level questions compel learners to synthesize, analyze, interpret or evaluate data. The most thought-provoking questions focus not on simple recall of facts but require engagement in open problem solving and investigation.
LOW-LEVEL vs HIGH LEVEL QUESTION
• Round the number 2.175 to the nearest hundredth.
• Think of three numbers that produce 2.18 when rounded to the nearest hundredth.
• Other types of questions in this genre might begin with,– “What happens if you…” – “How many ways can…” – “What can you make from…." – Still others might include asking students to “name a
counterexample” or – determine why an incorrect solution is indeed incorrect.
These types of probing questions encourage logical thought by emboldening students to mull over multiple related ideas.
The Professional Standards propose five categories of
questions that teachers should ask:Category 1 questions focus on helping students work together to make sense of mathematics.
1) "Do you agree? Disagree?”
The area of this rectilinear figure is 66.75 sq. in. 12.3”
3.5”1.5”
15.8”
3.5”
(12.3)(3.5) = 43.05
(15.8)(1.5) = 23.7 (12.3)(5) = 61.5
(3.5)(1.5) = 5.25
2) "Does anyone have the same answer but a different way to explain it?"
12.3”3.5”
1.5”3.5”
12.3”
3.5”1.5”
15.8”
5”
3.5”
(15.8)(5) = 7912.25
Area = 79 – 12.25 = 66.75
Category 2 contains questions that help students rely more on themselves to determine whether something is mathematically correct.
10.25 > 6.12 + 4.20 True or False?
1. "Does that make sense?” 2. "How do you know? ”3. "What model shows that?"
Category 3 questions seek to help students learn to reason mathematically. 1. "Does that always work?”2. "How could we prove that?
The area of a triangle is always one-half the base times the height.
Category 4 questions focus on helping students learn to conjecture, invent, and solve problems.
1. "What would happen if...?” The sides of a rectangle are 5 and 5.
a. What would happen to the perimeter if we change the sides to 3 and 7?
b. What would happen to the area if we change the sides to 3 and 7?
2. “What pattern do you see?” 1, 4, 9, 16, 25 ….
Category 5 questions relate to helping students connect mathematics, its ideas, and its applications.
1. "Have we solved a problem that is similar to this one?” How is this similar to above? 3a + 4a = ?
2. "How does this relate to ...?
3. ”How does it relate to
How to Make Sure a Butterfly
Doesn’t Fly
When the butterfly is ready, it starts to break through the cocoon.
First a hole appears. Then the butterfly struggles to come out through the hole. This can take a few hours.
If you try to “help” the butterfly by cutting the cocoon, the butterfly will come out easily but it will never fly. Your “help” has destroyed the butterfly.
The butterfly can fly because it has to struggle to come out.
The ‘pushing’ forces lots of enzymes from the body to the wing tips. This strengthens the muscles, and reduces the body weight. In this way, the butterfly will be able to fly the moment it comes out of the cocoon. Otherwise it will simply fall to the ground, crawl around with a swollen body and shrunken wings, and soon die.
If the butterfly is not left to struggle to come out of the cocoon, it will never fly.
We can learn an important lesson from the butterfly.
If we do not have struggles and challenges in our work, we will never grow strong and capable. If life has no difficulties, we will become weak and helpless.
-- Lim Siong Guan, Former Secretary, Singapore’s Ministry of Education
Links to helpful ResourcesKey Shifts (Scholastic)http://www.scholastic.com/teachers/top-teaching/2013/03/common-core-key-shifts-mathematics
Common Core Standards_Mathematicshttp://www.corestandards.org/Math/Practice/
PowerPoint: William McCallum and Jason Zimba (two lead writers of the Common Core State Standards for Mathematics) on the background of writing the Standards.http://www.youtube.com/watch?v=dnjbwJdcPjE
Sample Assessments by gradehttp://www.achievethecore.corg/
Common Core Practice Testshttp://parcc.pearson.com (sample PARCC tests and tutorials)https://sbacot.tds.airast.org/student/login.aspz?c=SBAC.PThttp://sbac.portal.airast.org/practice-test/
Common Core Resources to use with studentshttp://www.illustrativemathematics.org
Dana Center Resources http://www.ccsstoolbox.org/http://ccsstoolbox.agilemind.com/pdf/DanaCenter_YAG_HS.pdf
Common Core and Special Education Studentshttp://www.ode.state.or.us/search/page/?id=3741
IN CLOSING ….
Thank you for your participation.
JUDITH T. BRENDEL, Ed.M.
