Transcript Cup - K-12 Alliance

Making Sense
Of The New Standards
K-12 Alliance Staff Developer
Training
January 25, 2013
New Opportunities for All
Learners
California
Common Core
State
Standards
(ELA and
Math)
Next Generation
Science
Standards
21st Century
Skills
Prior Knowlege
• With your group discuss what you know
about 21st Century Skills, CCSS, and
NGSS
• What connections do you see?
• Chart your ideas. You will return to your
thinking on Sunday.
In The Classroom
Math
Science
ELA
21st Century
Skills
Activity
•
•
•
•
•
•
•
Choose a secret number (don’t say it aloud)
add 5 to that number
double that sum
subtract 4 from that product
divide the results by 2
subtract your secret number from the quotient
you have 3.
How Did That Happen?
• What is behind this trick?
• Is it foolproof?
• Discuss your thoughts at your table.
Explain this trick
Use the tools at your table to explain what
you think is happening.
Remember:
•You may all have had different secret
numbers.
•No matter what number you selected, you all
ended up with 3.
Transfer to a Whiteboard
• Think about how to use the tools to
demonstrate how the trick was done.
• Use the white board to depict your
demonstration.
Connect action to a direct model
with a specific “Secret #” choice
Instructions
What I did
My model
Choose a secret number
I chose 2 beans
•• (2 beans)
Add 5
I added 5 more
•• •••••
Double that sum
I multiplied by two
•• •••••
•• •••••
Subtract 4
Subtracted 4 beans
•••••
•••••
Divide by 2
I pulled half away
•••••
Subtract secret number
I pulled off 2 beans
You have three
I was astonished!
•••
Connect action to a model for any
choice (A set of all secret numbers)
Instructions
What I did
My model
Choose a secret number
I chose a cup to
represent “any”
Cup
Add 5
I added 5 beans
Cup •••••
Double that sum
I multiplied by two
Cup •••••
Cup •••••
Subtract 4
Subtracted 4 beans
Cup •••
Cup •••
Divide by 2
I pulled half away
Cup •••
Subtract secret number
I pulled off the cup
You have three
Voila!
•••
Words
Manipulatives
Algebra
Expressions/Result
Choose a secret number
Cup
n
Add 5
Double that sum
Subtract 4
Divide by 2
Subtract secret number
You have three
Words
Manipulatives
Algebra
Expressions/Result
Choose a secret number
Cup
n
Add 5
Cup • • • • •
N+5
Double that sum
Subtract 4
Divide by 2
Subtract secret number
You have three
Words
Manipulatives
Algebra
Expressions/Result
Choose a secret number
Cup
n
Add 5
Cup • • • • •
N+5
Double that sum
Cup •••••
Cup •••••
2(n+5) = 2n + 10
Subtract 4
Divide by 2
Subtract secret number
You have three
Words
Manipulatives
Algebra
Expressions/Result
Choose a secret number
Cup
n
Add 5
Cup • • • • •
N+5
Double that sum
Cup •••••
Cup •••••
2(n+5) = 2n + 10
Subtract 4
Cup•••
Cup•••
2n + 10 – 4= 2n+6
Divide by 2
Subtract secret number
You have three
Words
Manipulatives
Algebra
Expressions/Result
Choose a secret number
Cup
n
Add 5
Cup • • • • •
N+5
Double that sum
Cup •••••
Cup •••••
2(n+5) = 2n + 10
Subtract 4
Cup•••
Cup•••
2n + 10 – 4= 2n+6
Divide by 2
Cup•••
(2n+6)/2 = n + 3
Subtract secret number
You have three
Words
Manipulatives
Algebra
Expressions/Result
Choose a secret number
Cup
n
Add 5
Cup • • • • •
N+5
Double that sum
Cup •••••
Cup •••••
2(n+5) = 2n + 10
Subtract 4
Cup•••
Cup•••
2n + 10 – 4= 2n+6
Divide by 2
Cup•••
(2n+6)/2 = n + 3
Subtract secret number
•••
N+3–n=3
You have three
Words
Manipulatives
Algebra
Expressions/Result
Choose a secret number
Cup
n
Add 5
Cup • • • • •
N+5
Double that sum
Cup •••••
Cup •••••
2(n+5) = 2n + 10
Subtract 4
Cup•••
Cup•••
2n + 10 – 4= 2n+6
Divide by 2
Cup•••
(2n+6)/2 = n + 3
Subtract secret number
•••
N+3–n=3
You have three
Yes, I do.
3
8th Grade Expectation
Words
Results
Cumulative Expression
Choose a secret number
n
n
Add 5
N+5
N+5
Double that sum
2(n+5) = 2n+10
2(n+5)
Subtract 4
2n + 10 – 4 = 2n+6
2(n + 5) – 4
Divide by 2
(2n+6)/2 = n + 3
2(n + 5) – 4
2
Subtract secret number
N+3–n=3
You have three
3
2(n + 5) – 4
2
2(n + 5) – 4
2
-n
-n = 3
Language Of Doing Math
Which of these words best describes what I asked you to
do:
•Represent
•Solve
•Evaluate
•Simplify
•Determine
•Compare
•Find
•Explain
Instructional Words =
implication for process
• Represent – To portray in another abstract way.
• Solve – Finding an unknown quantity or expression.
• Evaluate – To find the value of by substituting values
in an expression or function.
• Simplify – To make a math expression simpler.
• Determine – Find or choose
• Compare – Relate values of 2 or more expressions
• Explain – Describe a process.
• Justify – Support the logic of a process or solution
Standards for
Mathematical Content K-8
How the grade level standards are organized
• Standards • Clusters • Domains
7th Grade Common Core
Standard Connection
7.EE.4. Use variables to represent
quantities in a real-world or mathematical
problem, and construct simple equations and
inequalities to solve problems by reasoning
about the quantities.
Common Core Connections
7 ------> 6
• 7th Grade (p. 31) 7.EE.4. Use variables to
represent quantities in a real-world or mathematical
problem, and construct simple equations and
inequalities to solve problems by reasoning about the
quantities.
• 6TH: (P. 26) 6.EE.6. Use variables to represent
numbers and write expressions when solving a realworld or mathematical problem; understand that a
variable can represent an unknown number, or,
depending on the purpose at hand, any number in a
specified set.
Follow the Standard Back
5—4—3—2—1—Kinder!
• Keep track of the standards you determine to
be in the line from Kindergarten to 5th grade
and on to Middle school.
• Remember the key idea is representing
(modeling) a mathematical idea in a manner
chosen by the student.
Common Core Standards
for Mathematics
Two Types of Standards
•Mathematical Practice (recurring throughout
the grades)
•Mathematical Content (different at each
grade level)
Standards for
Mathematical Practice
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
Which of These Describe your work on the “Trick”?
Common Core Math
Standards
1997 CA. Math Standards
CCSS Requires Three Shifts in Mathematics
1.
Focus: Focus strongly
where the standards focus.
2.
Coherence: Think across
grades, and link to major
topics
3.
Rigor: In major topics, pursue
conceptual understanding,
procedural skill and fluency,
and application
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Shift #1: Focus Strongly where the
Standards Focus
• Significantly narrow the scope of content and
deepen how time and energy is spent in the math
classroom.
• Focus deeply on what is emphasized in the
standards, so that students gain strong
foundations.
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Key Areas of Focus in Mathematics
Focus Areas in Support of Rich Instruction and
Grade Expectations of Fluency and Conceptual
Understanding
Addition and subtraction - concepts, skills, and problem
K–2
solving and place value
Multiplication and division of whole numbers and fractions
3–5
– concepts, skills, and problem solving
Ratios and proportional reasoning; early expressions and
6
equations
Ratios and proportional reasoning; arithmetic of rational
7
numbers
8
Linear algebra
31
Grade Shifts: Examples
Concept
Compose simple shapes to form
larger shapes (e.g., 2 triangles to
form a rectangle)
Introduction to Probability
Introduction of fractions as
numbers
Add and subtract simple
fractions
1997
Standards
CCSS
Grade
2
K
Grade
3
Grade
2
Grade
3
Grade
7
Grade
3
Grade
4
Shift #2: Coherence: Think Across Grades, and
Link to Major Math Topics Within Grades
•
Carefully connect the learning within and across
grades so that students can build new understanding
on foundations built in previous years.
•
Begin to count on solid conceptual understanding
of core content and build on it. Each standard is not
a new event, but an extension of previous learning.
33
Shift #3: Rigor: Expect fluency, deep
understanding, and application
•
The CCSSM require a balance of:
 Solid conceptual understanding
 Procedural skill and fluency
 Application of skills in problem solving
situations
•
Pursuit of all threes requires equal intensity in time,
activities, and resources.
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Solid Conceptual Understanding
• Teach more than “how to get the answer” and
instead support students’ ability to access
concepts from a number of perspectives
• Students are able to see math as more than a set
of mnemonics or discrete procedures
• Conceptual understanding supports the other
aspects of rigor (fluency and application)
35
Fluency
• The standards require speed and accuracy in
calculation.
• Teachers structure class time and/or homework
time for students to practice core functions such as
single-digit multiplication so that they are more able
to understand and manipulate more complex
concepts
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Required Fluencies in K-6
Grade
K
1
2
3
4
5
6
Standard Required Fluency
K.OA.5 Add/subtract within 5
1.OA.6 Add/subtract within 10
Add/subtract within 20 (know single-digit sums from
2.OA.2
memory)
2.NBT.5
Add/subtract within 100
Multiply/divide within 100 (know single-digit
3.OA.7
products from memory)
3.NBT.2
Add/subtract within 1000
4.NBT.4 Add/subtract within 1,000,000
5.NBT.5 Multi-digit multiplication
Multi-digit division
6.NS.2,3
Multi-digit decimal operations
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Application
• Students can use appropriate concepts and procedures for
application even when not prompted to do so.
• Teachers provide opportunities at all grade levels for
students to apply math concepts in “real world” situations,
recognizing this means different things in K-5, 6-8, and HS.
• Teachers in content areas outside of math, particularly
science, ensure that students are using grade-levelappropriate math to make meaning of and access science
content.
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