Download CCRS Quarterly Meeting # 2 Unpacking the Learning

CCRS Quarterly Meeting # 2
Unpacking the Learning
Progressions
http://alex.state.al.us/ccrs/
ALABAMA QUALITY
TEACHING STANDARDS
1.4-Designs instructional activities based on
state content standards
1.4
2.7-Creates learning activities that optimize
each individual’s growth and achievement
within a supportive environment
5.3
5.3-Participates as a teacher leader and
professional learning community member to
advance school improvement initiatives
2.7
The Five Absolutes + A Balanced Instructional Core
= A Prepared Graduate
Five Absolutes
•
Teach to the standards (Alabama College- and
Career-Ready Standards – Math Course of
Study)
•
A clearly articulated and “locally” aligned K-12
curriculum
•
Aligned resources, support, and professional
development
•
Regular formative, interim/benchmark
assessments to inform the effectiveness of the
instruction and continued learning needs of
individual and groups of students
•
Each student graduates from high school with
the knowledge and skills to succeed in post-high
school education and the workforce
The Instructional Core
1.4
5.3
2.7
Outcomes
Participants will:
 Reflect on Next Steps from QM #1
 Review and deepen understanding of the Algebra
Learning Progression and how the content is sequenced
within and across the grades (coherence)
 Illustrate, using tasks, how math content develops over
time
 Discuss how the progressions in the standards can be
used to inform planning, teaching, and learning
from CCRS QM #1
Participants should have done three things for the CCRS
QM #2 :
• Decide which task you will implement in your class, solve
task, and anticipate possible student solutions.
• Implement task in the classroom ( monitor, select, and
sequence)
• Bring student work samples (student’s solution path) to
share with your group.
Journal Reflection
CCRS-Mathematics
Learning Progressions
Flows Leading to Algebra
Operations and Algebraic Thinking
VIDEO: Operations and Algebraic Thinking
Mathematical Learning Progression:
Operations and Algebraic Thinking
Purpose for Reading
• What fluencies are required for this domain?
• What conceptual understandings do students
need?
• How can this be applied?
Code the Text
+ ~ Fluencies
-
~ Conceptual Understandings
x ~ Application
Mathematical Learning Progression:
Operations and Algebraic Thinking
• What fluencies are required for this domain?
• What conceptual understandings do students
need?
• How can this be applied?
Rigor
Operations and Algebraic Thinking
Operations and Algebraic Thinking
K-2
3-5
8 + _?_ = 43
6 x 4 = _?_
_?_ - 14 = 21
27 ÷ _?_ = 3
Write 1 problem for each of the equations.
Operations and Algebraic Thinking
K-2 Table (page 7)
3-5 Table (page 23)
1.) At your table identify the type of problem you
wrote.
2.) Post your problem on the K-2 or 3-5 story
problem poster.
3.) Be prepared to share what you noticed from
poster.
Addition and Subtraction Problem Situations
Common Multiplication and Division Situations¹
Reflection
How did reading the Progression document deepen
your understanding of the flow of the CCRS math
standards?
How might understanding a mathematical progression
impact instruction? Give specific examples with respect
to:
• planning lessons
• helping students make mathematical connections,
• working with struggling students, and
• using formative assessment and revising
instruction
LUNCH
How might the idea of learning
progressions connect to student
experience, learning, misconceptions
and common mistakes?
How might the idea of learning
progressions connect to the tasks a
teacher selects to guide student
learning?
The progression of student understanding of Algebra begins with Counting
and Cardinality, moves through Operations and Algebraic Thinking, to
Expressions and Equations, and finally to Algebra.
How do you connect standards
to standards so that children are
equipped to think mathematically?
How do you
work as a
team across
grades to
ensure student
growth in
algebraic
reasoning?
Major Work of the Grade: A Progression to Algebra
Using Tasks from Illustrative Mathematics
for Algebraic Development
• These tasks are not meant to be considered in
isolation. When taken together as a set of tasks,
they illustrate a particular standard.
• These tasks were grouped together to represent
one interpretation of the algebra learning
progression.
• This representation illustrates how mathematical
knowledge and skills develop over time.
Tracking the Algebra Progression
Toward a High School Standard
A-SSE.A.1
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its
context.★
a. Interpret parts of an expression, such as terms, factors, and
coefficients.
b. Interpret complicated expressions by viewing one or more
of their parts as a single entity. For example, interpret
P(1+r)n as the product of P and a factor not depending on P.
Sample Illustration of A-SSE.A.1
•
Tracking the Algebra Progression
Toward a High School Standard
1. Read the task.
2. Discuss the concepts that are involved in your particular
task that are necessary for students to connect their
learning to algebra.
3. Discuss the concepts that students will build upon from
the previous grade and the concepts which will lead to in
the next grade.
4. Relate the concepts from the task to the original high
school task.
Tracking the Algebra Progression
Toward a High School Standard
K.OA.A.3
Decompose numbers less than or equal to 10 into pairs in more than
one way, e.g., by using objects or drawings, and record each
decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 +
1).
Sample Illustration
Make 9 in as many ways as you can by adding two numbers between
0 and 9.
http://www.illustrativemathematics.org/illustrations/177
Tracking the Algebra Progression
Toward a High School Standard
1.OA.D.7
Understand the meaning of the equal sign, and determine if
equations involving addition and subtraction are true or false. For
example, which of the following equations are true and which are
false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
Sample Illustration
Decide if the equations are true or false. Explain your answer.
• 2+5=6
• 3+4=2+5
• 8=4+4
• 3+4+2=4+5
• 5+3=8+1
• 1+2=12
• 12=10+2
• 3+2=2+3
• 32=23
https://www.illustrativemathematics.org/illustrations/466
Tracking the Algebra Progression
Toward a High School Standard
2.NBT.A.4
Compare two three-digit numbers based on meanings of the
hundreds, tens, and ones digits, using >, =, and < symbols to
record the results of comparisons.
Sample Illustration
Are these comparisons true or false?
A) 2 hundreds + 3 ones > 5 tens + 9 ones
B) 9 tens + 2 hundreds + 4 ones < 924
C) 456 < 5 hundreds
D) 4 hundreds + 9 ones + 3 ones < 491
E) 3 hundreds + 4 tens < 7 tens + 9 ones + 2 hundred
F) 7 ones + 3 hundreds > 370
G) 2 hundreds + 7 tens = 3 hundreds - 2 tens
http://www.illustrativemathematics.org/illustrations/111
Tracking the Algebra Progression
Toward a High School Standard
3.OA.B.5
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4
= 24 is known, then 4 × 6 = 24 is also known. (Commutative property of
multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 =
10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40
and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56.
(Distributive property.)
Sample Illustration
Decide if the equations are true or false. Explain your answer.
4 x 5 = 20
6 x 9 = 5 x 10
34 = 7 x 5
2 x (3 x 4) = 8 x 3
3x6=9x2
8x6=7x6+6
5 x 8 = 10 x 4
4 x (10 + 2) = 40 + 2
Tracking the Algebra Progression
Toward a High School Standard
4.OA.A.3
Solve multistep word problems posed with whole numbers and having wholenumber answers using the four operations, including problems in which
remainders must be interpreted. Represent these problems using equations with
a letter standing for the unknown quantity. Assess the reasonableness of
answers using mental computation and estimation strategies including
rounding.
Sample Illustration
Karl's rectangular vegetable garden is 20 feet by 45 feet, and Makenna's is 25
feet by 40 feet. Whose garden is larger in area?
http://www.illustrativemathematics.org/illustrations/876
Tracking the Algebra Progression
Toward a High School Standard
5.OA.A.2
Write simple expressions that record calculations with numbers, and interpret
numerical expressions without evaluating them. For example, express the
calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3
× (18932 + 921) is three times as large as 18932 + 921, without having to
calculate the indicated sum or product.
Sample Illustration
Leo and Silvia are looking at the following problem:
• How does the product of 60 × 225 compare to the product of 30 × 225?
• Silvia says she can compare these products without multiplying the
numbers out. Explain how she might do this. Draw pictures to illustrate
your explanation.
https://www.illustrativemathematics.org/illustrations/139
Tracking the Algebra Progression
Toward a High School Standard
6.EE.A.4
Identify when two expressions are equivalent (i.e., when the two
expressions name the same number regardless of which value is
substituted into them). For example, the expressions y + y + y and 3y
are equivalent because they name the same number regardless of which
number y stands for.
Sample Illustration
Which of the following expressions are equivalent? Why? If an
expression has no match, write 2 equivalent expressions to match it.
• 2(x+4)
• 8+2x
• 2x+4
• 3(x+4)−(4+x)
• x+4
http://www.illustrativemathematics.org/illustrations/177
Tracking the Algebra Progression
Toward a High School Standard
7.EE.A.2
Understand that rewriting an expression in different forms in a problem
context can shed light on the problem and how the quantities in it
are related. For example, a + 0.05a = 1.05a means that “increase by
5%” is the same as “multiply by 1.05.”
Sample Illustration
Malia is at an amusement park. She bought 14 tickets, and each ride
requires 2 tickets.
• Write an expression that gives the number of tickets Malia has left in
terms of x, the number of rides she has already gone on. Find at
least one other expression that is equivalent to it.
• 14−2x represents the number of tickets Malia has left after she has
gone on x rides. How can the 14, -2, and 2x be interpreted in terms
of tickets and rides?
• 2(7−x) also represents the number of tickets Malia has left after she
has gone on x rides. How can the 7, (7 – x), and 2 be interpreted in
terms of tickets and rides?
Learning Progressions for Learning
• How does algebra progress from kindergarten to high
school?
• What are some ways that understanding the learning
progressions can strengthen grade level instruction?
• Why do you believe it is important to understand
mathematical trajectories and how knowledge is built
over time?
Toward Greater Coherence
“The Standards are designed around coherent progressions from grade to
grade. Principals and teachers carefully connect the learning across grades
so that students can build new understanding onto foundations built in
previous years. Teachers can begin to count on deep conceptual
understanding of core content and build on it. Each standard is not a new
event, but an extension of previous learning.”
Student Achievement Partners, 2011
Step Back – Reflection Questions
• What are the benefits of considering coherence when
designing learning experiences (lesson planning) for
students?
• How can understanding learning progressions support
increased focus of grade level instruction?
• How do the learning progressions allow teachers to
support students with unfinished learning (struggling
students)? **
Next Steps
• Identify standards and select a high level task.
• Plan a lesson with colleagues.
• Anticipate student responses, errors, and misconceptions.
• Write assessing and advancing questions related to student responses.
Keep copies of planning notes.
• Teach the lesson. When you are in the Explore phase of the lesson, tape
your questions and the students responses, or ask a colleague to scribe
them.
• Following the lesson, reflect on the kinds of assessing and advancing
questions you asked and how they supported students to learn the
mathematics.
Survey
• Are there any aspects of your own thinking and/or practice that our work
today has caused you to consider or reconsider? Explain.
• How will today’s learning help you as you work with:
– Collaborative Planning
– Struggling students (Special Ed and ELL, etc)
– Formative assessment?
• With respect to CCRS Math, what would you like more
information/learning on?
…. The Teacher Leader (AQTS 5.3)
• How can today’s learning of
the progressions be used to
inform your teaching and
learning?
• How can today’s learning of
the progressions be used to
inform your professional
learning community?
Wrapping up…..
Prepare for District Team Planning
References
• “The Structure is the Standards” Daro, McCallum, Zimba (2012)
http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/
• www.illustrativemathematics.org
• K–8 Publishers’ Criteria for the Common Core State Standards for Mathematics
(2013)