UNPACKING the CONFUSION PARRC Mathematics 6 12 NJPSAFEA

UNPACKING the CONFUSION PARRC Mathematics 6 -12 NJPSA/FEA conference October 2015 presentation Room: Oceanport North 10: 45 A. M. – 12: 15 P. M. by Judith T. Brendel, Ed. M. educational consultant edleaderk 12@hushmail. com 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/

AGENDA • 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

3 PRINCIPALS guided the STANDARDS 1. Knowledge, skills and understandings for all students to be CAREER and COLLEGE READY 1. Standards must be based on EVIDENCE not just what people feel students need to succeed. 1. Allow TIME for teachers to teach and TIME for students to practice.

An Extra Video Resource From Engage. NY 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. 2. 3. 4. 5. 6. FOCUS on the math that really matters COHERENCEY relates grade-to-grade FLUENCY really matters deep UNDERSTANDING APPLICATION in new situations DUAL INTENSITY (both: procedures with practice and meaning and application with rich set of problems)

1. Greater FOCUS on FEWER TOPICS NOT 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) = 3 x-2 Linear algebra and linear functions (parallel lines, perpendicular lines, systems of equations, …. )

Traditional U. S. Approach K Number and Operations Measurement and Geometry Algebra and Functions Statistics and Probability 9 12

Focusing Attention Within Number and Operations and Algebraic Thinking Expressions → and Equations Number and Operations— Base Ten → K 1 2 3 4 Algebra The Number System Number and Operations— Fractions → → → 5 6 7 8 High School 10

Focus Areas in Mathematics (CCSS)- MS/HS ALG. - 1 Focus 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 f(x) = x 2 UNIT-5 Functions and Descriptive Statistics and *Modeling 11

Focus Areas in Mathematics (CCSS) - HS GEOMETRY Focus 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-6 Applications of Probability 12

Focus Areas in Mathematics (CCSS) – HS ALG. - 2 Focus 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)** 13

3 CRITICAL ASPECTS • Fluency • Understanding • Application

3. FLUENCIES expected (without a calculator) • K • 1 • 2 1. OA. 5 1. OA. 6 2. OA. 2 2. NBT. 5 • 3. OA. 7 • 4. 3. NBT. 2 4. NBT. 4 • 5. • 6. 5. NBT. 5 6. NS. 2, . 3 Add/subtract within 5 Add/subtract within 10 Add/subtract within 20 (know single-digit sums from memory) Add/subtract within 100 Multiply/divide within 100 (know single-digit products from memory) Add/subtract within 1000 Add/subtract within 1, 000 Multi-digit multiplication 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. UNDERSTANDING Significant 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. Understanding Previous 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 apply the Pythagorean Theorem. (2010) After CCSS 1. Explain a proof of the Pythagorean Theorem and its converse. 1. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 1. 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 CHALLENGES WHAT 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 knows to use ADDITION and ADDS CORRECTLY Student does computation on scrap paper. Student types in answer.

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) APPLICATION of ADDITION and DIVISION in multi-step real-life situation. What computation is needed? Where does student do the computation?

Screen Shot: *table Part-A Part-B (grade-3 EOY test) Part MC and Part SCR Part A. Read the bar graph (between markings) then easy addition. Part B. Know what and when to ADD and SUBTRACT Traditional MC and SCR in one question. Partial credit; B not dependent on A answer.

Screen Shot: Complete Picture Graph Drag 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 example w/solutions and rubrics DRAG tick marks 2 POINT TASK Experience CREATING and USING a variety of graphs is essential.

DEEPcorrect UNDERSTANDING of Screen Shot: *Multiple answers concept (grade-3 EOY test) of multiplication No computation required. Notice “square-like shape” of the “bubble-in” form when more than one correct answer.

Screen Shot: *Multiple correct answers (grade-3 EOY test) D. ( D and E are below C in the same format. ) E. Notice “square-like shape” of the “bubble-in” form when more than one correct answer.

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 answers 1 point for 2 correct answers 0 points for 1 or no correct answers.

Screen Shot: *Tools to Graph Line t: y = -x + 5 Line s: y = 1/3 x - 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) d e x i m “ d n” an o i t c a fr “ , d e e h r t. o e s c c s i m t s r i o f No ” k ” r r e o b w “ m. o r nu n e : w s te n o a n inal Also he f. Acceptable answers might be: t only

Screen Shot: *Use Symbols or Type (one correct answer; answer forms) Scrap paper work: 27 – 18 x = 20 – 16 x + 18 x 27 -20 7 = 20 + 2 x -20 = 7/2 = x 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 MODELING: applying in real life Explain

sbac GRADE-11 Practice Test w/solutions and rubrics (2 point task) http: //sbac. portal. airast. org/wpcontent/uploads/2014/10/Grade 11 Math. 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. UNDERSTANDING linking topics and thinking across grades

ORDER doesn’t matter in ADDITION COHERENCY 3 + 5 = 5 +3 1 dog + 3 cats + 6 dogs = 1 dog + 6 dogs + 3 cats 3 a + 5 b + a = 5 b + a + 3 a

ORDER doesn’t. COHERENCY matter in MULTIPLICATION 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! 3 x 5 = 5 x 3 or (8)(9) = (9)(8) 3 a(2 a) =6 a 2 and 2 a(3 a) = 6 a 2 2 x 3 x 5=2 x 5 x 3 4 a(3 a)(-2 b) = -24 a 2 b 2 x 3 x 5 = 2 x 5 x 3 6 x 5 = 30 10 x 3 = 30 or 3 a(-2 b)(4 a) = -24 a 2 b

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! =5 +3 not 8 3 cats + 2 cats + 4 dogs = 5 cats + 4 dogs not 9 cdgs 3 a + 2 a + 3 b = 5 a + 3 b not 8 abs

Use “same format” to compare 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’x 3’ = 24” x 36” = 863 sq. ” 20” x 38” = 760 sq. ” 1) Which area is larger? 2’x 3’ or 20”x 38” Why? 1) 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

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 = 3 x+4 y = 1 x+1 y = 2 x -1 Correct answer: A (slope = 3)

PARENT FUNCTIONS and … y=x y = |x| linear function absolute value function y = -x y = -|x| y = x 2 quadratic function y = -x 2

PARENT FUNCTIONS and … y=x y = |x| y = x 2 y = -x y = -|x| y = -x 2 y=x+2 y = |x| + 2 y = x 2 + 2 Pre Algebra-I and II

VOLUME of basic SOLIDS V=bxhxl πr 2 h V = s 3 V= A CUBE V = (area base)(height) V = Bh V= (area base) (height) V = Bh

CORRESPONDING ANGLES are EQUAL similar triangles congruent triangles parallel lines cut by transversals 3 4 1 5 3 2 4 equilateral triangles ? ? ?

Recap: RULES and STRATEGIES that DON’T CHANGE K-12 Look for patterns; look for what you already know! • 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.

Differentiated Tasks for Understanding CONCRETE – Circle fold PICTORIAL – Geometry find area SYMBOLIC – Create equations to represent …. X + 1. 07 x = $2000 ABSTRACT – compare f(x) = x 2 with f(x) = 3(x-2)2+1

CIRCLE FOLD (CONCRETE) CIRCLE-FOLD ACTIVITY (2 D – to – 3 D) INTERACTIVE ONLINE RESOURCES (NCTM) http: //www. nctm. org/Classroom. Resources/Interactives/Geometric-Solids/

1) "Do you agree? Disagree? ” 2) "Does anyone have the same answer but a different way to explain it? " The area of this rectilinear figure is 66. 75 sq. in. 12. 3” 3. 5” 15. 8” 12. 3” (12. 3)(3. 5) = 43. 05 1. 5” 12. 3” 3. 5” (15. 8)(1. 5) = 23. 7 3. 5” 1. 5” (12. 3)(5) = 61. 5 PICTORIAL 15. 8” Area = 79 – 12. 25 = 66. 75 (15. 8)(5) = 79 3. 5” (3. 5)(1. 5) = 5. 25 1. 5” 12. 25 5”

Still PICTORIAL, not concrete a 2 + b 2 = c 2 5 5 25 a=4 c=? 4 16 c a 4 b 3 9 3 b=3

SYMBOLIC The 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 + 5 b = 51 (2) 4 c – j = 12 (3) 3 b + 3 j = 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 answer MORE RIGOR ACTIVITY Student pairs GEOMETRY • Same perimeter different areas • Same area different perimeters

Activity: FIND THE AREA: Draw 3 -4 different rectangles that have a perimeter of 36. Perimeter(s) Record the area of each. (Use whole 1 +1 + 17 = 36 numbers only. ) 2 + 16 = 36 3 + 15 = 36 • Which shapes have the 4 + 14 = 36 largest & smallest area? 5 + 13 = 36 6 + 12 = 36 • What do you observe? 7 + 11 = 36 8 + 10 = 36 Areas: AREA with 9 + 9 PERIMETER + 9 = 36 (1)(17) = 17 square units (9)(9) = 81 square units

Activity: FIND THE PERIMETER: Draw 4 -5 different rectangles that have a area of 36. Record the perimeter of each. (Use whole numbers only. ) Area = 1 x 36 = 36 (p=74) • Area = 2 x 18 = 36 Area = 3 x 12 = 36 Area = 4 x 9 = 36 Which has the largest &Area smallest = 6 x 6 = 36 perimeter? (p=38) (p=30) (p=26) (p=24) PERIMETER with AREA

What you SHOULD NOT see !

y N Z slope = > (3, 6) Slope = (3, 2) x <

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 structure 8. 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

Screen. Shots of PARCC examples/MP and /grade 3 -Algebra II

Links: MP 1 - 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 …. ” (Engage. NY grade 5 test 2014) – Workbook pg. 7: Gr. 8 “The combined volume …. ”

MP 2 Reason Abstractly and Quantitatively Links: Grade 6 • Link: Inches and Centimeters http: //ccsstoolbox. agilemind. com/parcc/about_middle_3789. html (math practices 2 and 6)

MP 3 - 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/Common. Core. Library/Task s. Units. Student. Work/default. htm Select [grade 9], [Math], Scroll down and select [COMPANY LOGO]. See pages 4, 5, and 9.

MP 4 - Model with mathematics • Math Practices Examples: – Workbook pg. 9: Grade 8 “The population growth of two towns …. ”

Link: MP 5 – Use appropriate tools strategically The Library of Virtual Manipulatives http: //nlvm. usu. edu/ennav/vlibrary. html

MP 6 - 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/Common. Core. Library/Tasks. Units. Student. Work/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

MP 7 - 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 …. ”

MP 8 -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 project https: //www. teachingchannel. org/videos/illustrativ e-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? ” 2) "Does anyone have the same answer but a different way to explain it? " The area of this rectilinear figure is 66. 75 sq. in. 12. 3” 3. 5” 15. 8” 12. 3” (12. 3)(3. 5) = 43. 05 1. 5” 12. 3” 3. 5” (15. 8)(1. 5) = 23. 7 3. 5” 1. 5” (12. 3)(5) = 61. 5 15. 8” Area = 79 – 12. 25 = 66. 75 (15. 8)(5) = 79 3. 5” (3. 5)(1. 5) = 5. 25 1. 5” 12. 25 5”

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? 3 a + 4 a = ? 1. "How does this relate to. . . ? 1. ”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 Resources Key Shifts (Scholastic) http: //www. scholastic. com/teachers/top-teaching/2013/03/common-core-key-shifts-mathematics Common Core Standards_Mathematics http: //www. corestandards. org/Math/Practice/ Power. Point: William Mc. Callum 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=dnjbw. Jdc. Pj. E Sample Assessments by grade http: //www. achievethecore. corg/ Common Core Practice Tests http: //parcc. pearson. com (sample PARCC tests and tutorials) https: //sbacot. tds. airast. org/student/login. aspz? c=SBAC. PT http: //sbac. portal. airast. org/practice-test/ Common Core Resources to use with students http: //www. illustrativemathematics. org Dana Center Resources http: //www. ccsstoolbox. org/ http: //ccsstoolbox. agilemind. com/pdf/Dana. Center_YAG_HS. pdf Common Core and Special Education Students http: //www. ode. state. or. us/search/page/? id=3741

IN CLOSING ….

JUDITH T. BRENDEL, Ed. M. edleaderk 12@hushmail. com Thank you for your participation.