Download SHIFTS in MATH

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
[email protected]
a 3-minute video (for parents or students)
LEARN ABOUT THE COMMON CORE IN THREE MINUTES
http://www.corestandards.org/other-resources/key-shifts-inmathematics/
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 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.
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) = 3x-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
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) = x2
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.
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 1,000,000
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 and 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)
2
D.
3
2
6
( D and E are below C
in the same format.)
1
E.
6
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 = 1/3x - 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)
2
1
Acceptable answers might be: 4 or 4
4
2
Screen Shot: *Use Symbols or Type
(one correct answer; answer forms)
Scrap paper work:
27 – 18 x = 20 – 16x
+ 18 x
+ 18x
Acceptable answers:
7/2 or x = 7/2
27
-20
7
= 20 + 2x
-20
=
7/2 = x
2x
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.
7
10
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
11Math.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
3a + 5b + a = 5b + a + 3a
1 2 3 6
2 1 3 6
+ + =
and
+ + =
8 8 8 8
8 8 8 8
ORDER doesn’tCOHERENCY
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)
3a(2a) =6a2 and 2a(3a) = 6a2
2x3x5=2x5x3
4a(3a)(-2b) = -24a2b
2x3x5 =
2x5x3
6 x 5 = 30 10 x 3 = 30
or
3a(-2b)(4a) = -24a2b
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 9cdgs
3a + 2a + 3b = 5a + 3b not 8abs
1 3 3 1 7 1
+ + + = +
8 8 8 2 8 2
2 6 + 6 +5 4 = 3 6 + 5 4
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’x3’ = 24” x 36” = 863sq.”
20” x 38” = 760sq.”
1) Which area is larger? 2’x3’ or 20”x38” 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 = 3x+4
y = 1x+1 y = 2x -1
Correct answer: A (slope = 3)
PARENT FUNCTIONS and …
y=x
y = |x|
linear function
absolute value function
y = -x
y = -|x|
y = x2
quadratic function
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=bxhxl
πr2h
V = s3
V=
A CUBE
V = (area base)(height)
V = Bh
V = (area base)(height)
V= (area base) (height)
V = Bh
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 + 1.07x = $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/ClassroomResources/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”
3.5”
1.5”
15.8”
12.3”
(12.3)(3.5) = 43.05
12.3”
3.5”
(15.8)(1.5) = 23.7
3.5”
1.5”
(12.3)(5) = 61.5
PICTORIAL
3.5”
3.5”
(3.5)(1.5) = 5.25 1.5”
15.8”
12.25
Area = 79 – 12.25 = 66.75
(15.8)(5) = 79
5”
Still PICTORIAL, not concrete
a2 + b2
= c2
5
5
25
a=4
c=?
4
16
a
4
c
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 + 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 …
(x - h) + (y - k) = r
2
2
2
x
y
+ = 1
2
2
a
b
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.
Record the area of each. (Use
whole
Perimeter(s)
1 +1 + 17 + 17 = 36
numbers only.)
2 + 2 + 16 + 16 = 36
3 + 3 + 15 + 15 = 36
• Which shapes have the 4 + 4 + 14 + 14 = 36
largest & smallest area? 5 + 5 + 13 + 13 = 36
6 + 6 + 12 + 12 = 36
• What do you observe? 7 + 7 + 11 + 11 = 36
8 + 8 + 10 + 10 = 36
Areas:
AREA with
9 + 9PERIMETER
+ 9 + 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 !
y1 - y 6 - 2 4
m = = =
x1 - x2 3-3 0
y
(3,6)
N
Z
0
slope =
4
Slope =
(3,2)
>
4
0
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
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 ….”
MP2
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)
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/Task
sUnitsStudentWork/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 Manipulatives
http://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 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”
3.5”
1.5”
15.8”
12.3”
(12.3)(3.5) = 43.05
12.3”
3.5”
(15.8)(1.5) = 23.7
3.5”
1.5”
(12.3)(5) = 61.5
3.5”
3.5”
(3.5)(1.5) = 5.25 1.5”
15.8”
12.25
Area = 79 – 12.25 = 66.75
(15.8)(5) = 79
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. 3 4
7
8
+
8
= 8
1. "Have we solved a problem that is similar to this
one?” How is this similar to above? 3a + 4a = ?
1. "How does this relate to ...?
1. ”How does it relate to
3 5 + 4 5 = ?
4x + 6x = ?
3
3
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-shiftsmathematics
Common Core Standards_Mathematics
http://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 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/DanaCenter_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.
[email protected]
Thank you for your
participation.