1. Jackson County 6-12 Math CCRS Quarterly Meeting # 2 Unpacking the Learning Progressions August 10, 2015
1 minute
Welcome participants to 2nd Quarterly Meeting for school year.

2. Alabama Quality Teaching Standards
1.4-Designs instructional activities based on state content standards
2.7-Creates learning activities that optimize each individual’s growth and achievement within a supportive environment
5.3-Participates as a teacher leader and professional learning community member to advance school improvement initiatives
1.4
5.3
2.7
Say, “you are all here today for specific professional learning. Let’s not forget that today’s learning aligns with the Alabama Quality Teaching Standards.
Grounded on the five Alabama Quality Teaching Standards, the Continuum is based on two assumptions: (1) that growth in professional practice comes from intentional reflection and engagement in appropriate professional learning opportunities and (2) that a teacher develops expertise and leadership as a member of a community of learners focused on high achievement for all students, which we are doing in the CCRS quarterly meetings.”
Put in notes section:
Purpose of the Continuum
Based on the five Alabama Quality Teaching Standards (AQTS), which are listed elsewhere in this document, the Continuum articulates a shared vision and common language of teaching excellence to guide an individual’s career-long development within an environment of collegial support. It is a tool for guiding and supporting teachers in the use of reflection, self-assessment, and goal setting for professional learning and growth.
Specifically, the Continuum is intended to support meaningful reflective conversations among teachers, mentors, coaches, and administrators. It supports teachers in setting professional goals and pursuing professional development to reach those goals. It also serves as a focus for teacher preparation institutions and pre-service candidates.
The Continuum is one component of a comprehensive program of support for the ongoing development of teaching practice. While it provides guidance in the gathering of formative data upon which to reflect, it is not intended as an evaluation or observation instrument. The Continuum presents a holistic view of teaching and was developed to do the following:
• Delineate the diversity of knowledge and skills needed to meet the changing needs of Alabama’s students
• Support the reflective practice and ongoing learning of all teachers
• Support an ongoing process of formative assessment of beginning and experienced teachers’ practice based on standards, criteria, and evidence
• Help educators set goals for professional development over time
• Describe the development of high-quality, effective teaching practices throughout a teacher’s career
The Continuum is organized to describe five increasingly complex and sophisticated levels of development of practice: Pre-Service and Beginning, Emerging, Applying, Integrating, and Innovating. The indicators at each level describe what a teacher should know and be able to do at that level; these indicators are cumulative and include those stated in previous levels. While the “Pre-Service and Beginning” and “Emerging” columns describe the skills and abilities that novice teachers aim to develop during their induction period, it is not assumed that beginning teachers will necessarily enter the profession at this level of practice for every standard indicator.
The levels do not represent a chronological sequence in a teacher’s growth; rather, each describes a developmental level of performance. A teacher may be at an Emerging or Applying level of practice for some indicators on the Continuum and at an Integrating or Innovating level for other indicators, regardless of how many years she or he has been in the profession. In fact, it is not uncommon for accomplished teachers to self-assess and find themselves moving from right to left on the continuum in response to new teaching contexts and challenges.
The Continuum is based on two assumptions: (1) that growth in professional practice comes from intentional reflection and engagement in appropriate professional learning opportunities and (2) that a teacher develops expertise and leadership as a member of a community of learners focused on high achievement for all students.

3. The Instructional Core
The Five Absolutes + A Balanced Instructional Core = A Prepared Graduate
Five Absolutes
The Instructional Core
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
1.4
5.3
2.7
If we integrate Dr. Bice’s Five Absolutes with a balanced instructional Core we will experience prepared graduates.
Emphasize “locally” in the second bullet.
Support samples (Sample lesson plans and supporting resources found on ALEX, differentiated support through ALSDE Regional Support Teams and ALSDE Initiatives, etc.)
Assessments - (GlobalScholar, QualityCore Benchmarks, and other locally determined assessments)

4. Outcomes Participants will:
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
Good Morning, the outcomes for the QM #2 are: (read slide). As always, the CCRS-Implementation Team is representing the administrators and teachers that are not able to receive this training, and will think about ways in which the information, strategies, and resources from QM #2 can be taken back to benefit the system, school, and students.

5. CCRS-Mathematics Learning Progressions
2 minutes
In this table, related domains are grouped together. Each “colored row” identifies how domains at the earlier grades progress and lead to domains at the middle and high school levels. The right side of the chart lists the five conceptual categories for high school: Number and Quantity, Algebra, Functions, Geometry, and Statistics and Probability. You will need to emphasize the Algebra conceptual category because it is the main idea of this CCRS meeting. If you select one conceptual category and move left along the row, you’ll find the domains at the middle and elementary school levels from which this concept builds.”
Say, “Notice that the K-8 horizontal organization demonstrates grade level progressions of mathematics content. In High School, the mathematics content is organized into five conceptual categories which progress over multiple high school courses.
Notice in K-8 that the domains change as students move through their school years. These domains provide foundational knowledge for each high school conceptual category. The new emphasis on “college and career readiness” for all students implies that it is everyone’s responsibility to help prepare students for mastery of foundational mathematics content.
There is a sixth conceptual category of Modeling which does not have separated standards, but there are specific standards designated throughout these five conceptual categories as modeling standards. These standards are identified with a (*) in the CCRS.

6. Today we are focusing on the Operation and Algebraic Thinking, Expressions and Equations, and Algebra Progressions which can be found at
The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics.

7. Video Flows Leading to Algebra
Show video. This diagram depicts some of the structural features of the mathematics standards, where several different domains from grades K-8 converge toward algebra in high school. This diagram does not include other “flows,” such as from Number and Operations—Fractions in grades 3-5, to Ratios and Proportional Relationships in grades 6 and 7, to Functions in grade 8 and high school, with connections to geometry and probability.

8. What are the big ideas explored in the progressions Expressions and Equations and Algebra for grades 6 – 12?
Everyone should sit in their grade-level. Each grade should chart their ideas about the answers to the question. Facilitator should not comment or question participants. You will be assessing their knowledge. Let the participants discuss and argue. Let them put ALL ideas on the chart, even if it appears in two grades and even if they cannot agree. When completed, you should have a piece of chart paper with 6th and their ideas, 7th and their ideas, 8th and their ideas, and HS with their ideas. Four (4) pieces of chart paper on the wall. After a whole group discussion(but no comments from the facilitator) about what belongs in each grade, allow participants to change, cross through and/or add their ideas. Again, remember that you the facilitator has NO OPINION!!!.

9. A Study of the Expressions and Equations Progression and the Algebra Progression
Everyone reads the Overview pages 2 and 3
6th Grade
Apply and extend previous understandings of arithmetic to algebraic expressions (pages 4,5,6)
Reason about and solve one-variable equations and inequalities AND Represent and analyze quantitative relationships between dependent and independent variables (pages 6 and 7)
7th Grade
Use properties of operations to generate equivalent expressions AND Solve real-life and mathematical problems using numerical and algebraic expressions and equations (pages 8, 9, and 10)
8th Grade
Work with radicals and integer exponents AND Understand the connections between proportional relationships , line, and linear equations (pages 11, 12, and 13)
The progressions have been divided into sections. Everyone should read the Overview pages 2 and 3
If possible divide the 6th grade into two groups. The 1st group will read :Apply and extend previous understandings of arithmetic to algebraic expressions (pages 4,5,6). The second 6th grade group will read:
Reason about and solve one-variable equations and inequalities AND Represent and analyze quantitative relationships between dependent and independent variables (pages 6 and 7).
The 7th grade will only need to be one group and they should read: Use properties of operations to generate equivalent expressions AND Solve real-life and mathematical problems using numerical and algebraic expressions and equations (pages 8, 9, and 10)
The 8th grade will only need to be one group and they will read: Work with radicals and integer exponents AND Understand the connections between proportional relationships , line, and linear equations (pages 11, 12, and 13).. Continue with assignments in the high school. .

10. High School Everyone reads the Overview (pages 2 and 3)
Seeing Structure in Expressions (page 4, 5, and 6)
Arithmetic with Polynomials and Rational Expressions (pages 7, 8, and 9)
Creating Equations AND Variables, parameters, and constants (pages 10 and 11)
Modeling with Equations (pages 11,12)
Reasoning with Equations and Inequalities
Equations in one variable (pages 13 and 14)
Systems of equations AND Visualizing solutions graphically (pages 14 and 15)
The high school group should be separated into 6 groups. The section labeled “Reasoning with Equations and Inequalities” needs two groups. Everyone will read the Overview (pages 2 and 3). The first group will read: Seeing Structure in Expressions (page 4, 5, and 6). The second group will read: Arithmetic with Polynomials and Rational Expressions (pages 7, 8, and 9). The 3rd group will read: Creating Equations AND Variables, parameters, and constants (pages 10 and 11). The 4th group will read: Modeling with Equations (pages 11,12). The 5th group will read: Reasoning with Equations and Inequalities, Equations in one variable (pages 13 and 14). The 6th group will read: Systems of equations AND Visualizing solutions graphically (pages 14 and 15).

11. Discussion
Consider how the learning progression develops within and across grade levels. Discuss key points from your reading.
Discussion Questions
What are the big mathematical ideas for this domain?
How does the learning progression develop within this domain?
Are there any changes that need to be made to your chart paper?
Consider how the learning progressions develop within and across grade levels. Discuss key points from your reading that you want to remember. Use the guiding questions to advise your reflection. Allow participants time to make changes to their chart paper. Participants may not agree with the contents of the progression, but make sure that whatever is on the chart paper at this time is from the progression. Let participants know that their opinion is valued, but the progression document is what guides the information that is written on the chart paper.
Guiding Questions
What are the “big” mathematical ideas for this domain?
How do the learning progressions develop within this domain?
Are there any changes that need to be made to your chart paper?

12. 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
Give participants some time to write individually, then facilitate a discussion.

13. Why is professional peer discussion about progressions important for the teaching and learning process?
“Analysis and related discussion with your team is critical to develop mutual understanding of and support for consistent curricular priorities, pacing, lesson design, and the development of grade-level common assessments.” Together you can develop a greater understanding of the intent of each content standard cluster and how the standards are connected within and across grades. (Common Core Mathematics in a PLC at Work, Kanold, 2012, pg. 67)
Take a moment to read the question and quote and reflect on the implications for your role as a CCRS Team member. How has discussing and reflecting on the Number progression from the last CCRS meeting impacted your practice?
2 minutes

14. 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?
This is just a transition slide to frame the next activity. Give them time to read it and advance to the next slide. Now that you have reflected on the effect of progressions on your practice, discuss these two questions and give specific examples about how student learning in your classroom was impacted.

15. BE SURE TO START AT 16 SECONDS!

16. 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?
Give them time to read the slide.
Give them the handout containing Zimba’s Wire Diagram for their grade level. Ask them to read the handout and think through the questions and comments on the handout. Say, “In the morning session, you read through how algebra concepts are connected across grades. This afternoon, you will see a diagram designed by Jason Zimba, one of the original writers of the College and Career Ready Standards that also shows how standards progress.”
Allow time for each person to have a discussion with someone at or adjacent to their grade level about what kind of conversations a team should have to organize Algebra instruction within and across years.

17. Major Work of the Grade: A Progression to Algebra
To be even more specific, these are cluster headings from the CCRS: Mathematics in grades K These cluster headings are the foundational topics for each of the grades that lead to the conceptual category of algebra. Note that in the middle grades, there are more clusters that begin with “apply and extend” as students build on what has been previously learned. Today’s discussion is not about an Algebra 1 class, but is about algebra as a critical strand of mathematical thinking and reasoning.
**Note: Refer to the learning progression discussed this morning. The learning progression graphic is in their participant packet.

18. 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.
Our next step in understanding the algebra progression and its effect on student learning, leads us to explore these high-level tasks from K The tasks chosen for this activity were grouped together to represent one interpretation of a learning progression. There are other pathways that are different, this is only one interpretation.

19. 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.
In this section, you will track the progression of how an idea develops from kindergarten to high school. View a high school standard and then see how we start preparing for high school in kindergarten. Have a whole group discussion about what the standard encompasses. Student expectations should be included in this discussion. Participants should discuss topics such as:
Vocabulary including the word “complicated”, simplifying expressions, evaluating expressions, exponential expressions, etc. Allow participants to share, but don’t spend too much time dissecting the standard. Focus on the general big ideas.
Participants have the tasks in their packet. You may hide them if you would like. Directions on how to distribute the tasks:
Each table (or person) gets one or two (depending on group size/structure). After studying the standard and illustration (2-3 minutes), table/person discusses how their standard and task is a building block toward the high school standard (and task) shown – structure, parts of an expression, context, interpreting in context, quantities, etc.

20. Sample Illustration of A-SSE.A.1
In this task students have to interpret expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations. For example, PP+Q   is the fraction that population P  makes up of the combined population P+Q  .
Although the context is quite thin, posing the question in terms of populations rather than bare numbers encourages students to think about the variables as numbers and provides avenues for them to use their common sense in explaining their reasoning. This encourages them to see expressions as having meaning in terms of operations, rather than seeing them as abstract arrangements of symbols.
This is for reference only:
Solution: Comparing expressions The expression P+Q is larger.
The expression P+Q gives the total size of the two populations put together.
The expression 2P gives the size of a population twice as large as P.
Putting the smaller population together with the larger yields more animals than merely doubling the smaller.
Another way to see this is to notice that 2P=P+P, which is smaller than P+Q because adding P to P is less than adding Q to P.
The expression P+Q2 is larger.
The total size of the two populations put together is P+Q, so the expression PP+Q gives the fraction of this total belonging to P. Since P<P+Q, this will be a number less than 1. For instance, if P=100 and Q=150, this fraction equals 100/( )=0.4=40%.
The average or mean size of the two populations is their sum divided by two, or P+Q2. This will be a number between P and Q, so it is larger than 1 (since P and Q describe animal populations). For instance, if P=100 and Q=150, the average is ( )/2=125.
The expression Q−P/2 is larger.
The expression (Q−P)/2 gives half the difference between P and Q. For instance, if Q=150 and P=100, half the difference is (150−100)/2=25.
The expression Q−P/2 gives the difference between Q and a population half the size of P. For instance, if Q=150 and P=100, this difference equals 150−100/2=100.
To see why the second of these is bigger, write
(Q−P)/2=Q/2−P/2
In the expression Q−P/2, we subtract P/2 from Q. But in (Q−P)/2, we subtract the same value, P/2, from a smaller amount, Q/2.
The expression Q+50t is larger.
In both expressions, the same value, 50t, is added to the population.
Since P<Q, adding 50t to P results in a smaller value than adding the same amount to Q.
The expression 0.5 is larger.
The total size of the two populations put together is P+Q, so the expression PP+Q gives the fraction of this total population belonging to P. Since there are fewer animals in population P than Q, this fraction is less than 12. For instance, if P=100 and Q=150, this fraction equals 100/( )=0.4.
PQ and QP can be interpreted in two different ways.
PQ can be interpreted as a unit rate, namely, the number of animals in population P for every 1 animal in population Q. Similarly, QP can be interpreted as the number of animals in population Q for every 1 animal in population P. Since there are more animals in population Q, the unit rate QP will be greater than the unit rate PQ.
For example, if P=100 and Q=150, then =23, so there would be 23 of an animal in population P for every 1 animal in population Q, while =32, so there would be 32 of an animal in population Q for every 1 animal in population P.
Some people think it is awkward to talk about fractions of animals, so here is another way to think about it:
PQ can also be interpreted as the fraction that population P is of population Q. Since there are fewer animals in population P, as a fraction of the population of Q it will be less than 1. Similarly, QP can also be interpreted as the fraction that population Q is of population P. Since there are more animals in population Q, as a fraction of the population of P it will be greater than 1.
For example, if P=100 and Q=150, this fraction equals =23, so there are 23 as many animals in population P as there are in population Q, while =32, so there are 32 as many animals in population Q as there are in population P.

21. Tracking the Algebra Progression Toward a High School Standard
Read the task.
Discuss the concepts that are involved in your particular task that are necessary for students to connect their learning to algebra.
Discuss the concepts that students will build upon from the previous grade and the concepts which will lead to in the next grade.
Relate the concepts from the task to the original high school task.
DIRECTIONS FOR ACTIVITY: Each table (or person) gets one or two grade level standards and task illustrations (depending on group size/structure). After studying the standard and illustration (2-3 minutes), table/person discusses how their standard and task is a building block toward the high school standard (and task) shown – structure, parts of an expression, context, interpreting in context, quantities, etc. Make sure each group has a quality discussion about the task before you give them half of a piece of chart paper. Each grade should chart the discussion points from the slide. After posting the chart papers, bring the group together as a whole. Have each group share their discussions about the task. Be sure to connect the groups’ discussions as they present. The big picture should be how each grade builds to develop this algebra progression as seen in the documents in the morning session.
Below are sample responses:
K – decomposes numbers using drawings or equations
1 – meaning of equal sign (does not mean output or “give me an answer”)
2 - Begin using, <, >
3 - Properties of operations – commutative, associative, distributive
4 – four operations with remainders , equations with letters
5 – simple expressions, interpret without evaluating them
6 –Identify when two equations are equivalent (Sixth grade also learns order of operations)
7 – rewriting expressions in different forms

22. 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 = and 5 = 4 + 1). Sample Illustration Make 9 in as many ways as you can by adding two numbers between 0 and 9.
Because of the limited reading skills of kindergarten students, this task should be introduced by the teacher, followed by the students carrying out the activity. Teachers should have counters on hand for students to use. Any number between 2 and 10 can be used in place of 9 to address K.OA.3. Some students may notice or be ready to appreciate the observation that each possibility has a "companion" where the order of numbers is switched. Students who make this observation are engaging in Standard for Mathematical Practice 7 Look for and make use of structure; the structure they are detecting will later be called the commutative property of addition. Although it is not necessary to meet this standard, listing the possible pairs of numbers in a systematic way might help the student show that s/he has found all of the possible number pairs that make 9

23. 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, = 2 + 5, =
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
The purpose of this task is to help broaden and deepen students understanding of the equals sign and equality. For some students, an equals sign means "compute" because they only see equations of the form
4+3=7. 
In this task, students must attend to the meaning of the equal sign by determining whether or not the left-hand expression and the right hand expression are equal. This task helps students attend to precision (as in Standard for Mathematical Practice 6).

24. 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
This task requires students to compare numbers that are identified by word names and not just digits. The order of the numbers described in words are intentionally placed in a different order than their base-ten counterparts so that students need to think carefully about the value of the numbers. Some students might need to write the equivalent numeral as an intermediate step to solving the problem.

25. 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) = = 56. (Distributive property.)
Sample Illustration
Decide if the equations are true or false. Explain your answer.
4 x 5 = x 9 = 5 x 10
34 = 7 x 5 2 x (3 x 4) = 8 x 3
3 x 6 = 9 x 2 8 x 6 = 7 x 6 + 6
5 x 8 = 10 x 4 4 x (10 + 2) =
This task is a follow-up task to a first grade task:
On the surface, both tasks can be completed with sound procedural fluency in addition and multiplication. However, these tasks present the opportunity to delve much more deeply into equivalence and strategic use of mathematical properties. These tasks add clarity to the often misunderstood or neglected concept of equivalence. Students often understand the equal sign as the precursor to writing the answer. Class discussion should be carefully guided to ensure that students come to the understanding that the equal sign indicates equivalence between two expressions. Though these tasks can be completed by evaluating each expression on either side of the equal sign, they present deliberate next levels of reasoning that invite students to look for different approaches.
Anyone facilitating a conversation about this task should constantly ask, "Is there another way to know whether this equation is true?" Consider 5 x 8 = 10 x 4. Students will likely know these facts relatively quickly and come to the conclusion that both sides are equal to 40, thus this equation is true. When pressed to see other options, students may reason that the 8 can be broken down into 4 x 2. The equation becomes 5 x (2 x 4) = 10 x 4. Through the associative property, this becomes (5 x 2) x 4 = 10 x 4. We can see that these expressions are equivalent because we know that 5 x 2 has the same value as 10. The same opportunity presents itself in part f. Part g presents an opportunity for students to think critically about the meaning of multiplication.
Third graders interpret multiplication as equal sized groups. Students might reason that 8 x 6 means 8 groups of 6. Thus 7 x would mean 7 groups of 6 with another group of 6. Students might recognize that extra 6 as the "8th group of 6," thereby making the two expressions equivalent.

26. Tracking the Algebra Progression Toward a High School Standard
4.OA.A.3
Solve multistep word problems posed with whole numbers and having whole-number 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?
The purpose of the task is for students to solve a multi-step multiplication problem in a context that involves area. In addition, the numbers were chosen to determine if students have a common misconception related to multiplication. Since addition is both commutative and associative, we can reorder or regroup addends any way we like. So for example,
20+45 =20+(5+40)=(20+5)+40=25+40  
Sometimes students are tempted to do something similar when multiplication is also involved; however this will get them into trouble since
20×(5+40)≠(20+5)×40 
This task was adapted from problem #20 on the 2011 American Mathematics Competition (AMC) 8 Test. Observers might be surprised that a task that was historically considered to be appropriate for middle school aligns to an elementary standard in the Common Core. In fact, if the factors were smaller (since in third grade students are limited to multiplication with 100; see 3.OA.3), this task would be appropriate for third grade: "3.MD.7.b Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning." For example, we could use a 5 ft by 12 ft garden, and a 7 ft by 10 ft garden to make this appropriate for a (challenging) third grade task. This earlier introduction to the connection between multiplication and area brings states who have adopted the Common Core in line with other high-achieving countries.

27. 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 × ( ) is three times as large as , 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.
The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on
3.OA.5 Apply properties of operations as strategies to multiply and divide
and
4.OA.1 Interpret a multiplication equation as a comparison.

28. 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
In this problem we have to transform expressions using the distributive, commutative and associative properties to decide which expressions are equivalent. Common mistakes are addressed, such as not distributing the 2 correctly. This task also addresses 6.EE.3.

29. 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 a = 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?
The purpose of this instructional task is to illustrate how different, but equivalent, algebraic expressions can reveal different information about a situation represented by those expressions. This task can be used to motivate working with equivalent expressions, which is an important skill for solving linear equations and interpreting them in contexts. The task also helps lay the foundation for students' understanding of the different forms of linear equations they will encounter in 8th grade. In part (b), the task asks students to interpret pieces of the expression that arise by parsing the expression from different algebraic perspectives. In particular, it requires students to think about the difference between interpreting −2x  as −2  times x  vs. subtracting 2x  from 14. Note that the meaning of the 2  in the expression 2(7−x)  is slightly different than the meaning given in the problem statement because of the role it plays in the expression. The class will probably need to have a whole-group conversation to grasp this subtlety.

30. 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?
Summarize the big ideas discovered during the whole group discussion of both the morning and afternoon sessions. Note: Remember to refer to the equip rubric in past CCRS meetings.
Some sample responses:
Supports remediation and differentiation – teachers can know better how to identify and address gaps in unfinished learning from previous grades
Teachers build on previous understandings – this will result in greater focus because teachers can spend less time reviewing.
Teachers can understand how their grade level content fits into the larger picture of a student’s mathematical trajectory and help ensure success in future grades
If teachers’ own knowledge of the content and how mathematical ideas are developed over time in stronger, their instruction can be stronger

31. 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
To summarize the session, allow participants to read the slide. Ask the participants if they are truly connecting the progressions in their practice in order to develop deep conceptual understanding.

32. 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)? **
At our last CCRS meeting, we explored the three instructional shifts. Tell participants to look in their packet for the page that contains the instructional shifts.

33. …. 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?
Plan with your table group on how you will use today’s learning to inform your teaching and learning. Be sure to share these ideas with the group, your colleagues, and your administrators.

34. References
“The Structure is the Standards” Daro, McCallum, Zimba (2012)
K–8 Publishers’ Criteria for the Common Core State Standards for Mathematics (2013)

35. acos2010.wikispaces.com

36. Sustainability
ongoing research, support, and validation of the system to reflect changes in college and career readiness standards
flexible professional development on the state, district or school levels
If the standards change, the test will change? PD!!!! Haven’t seen any of this yet.

37. http://www. discoveractaspire
Content Specifications – Technical Bulletin #1
“The ACT Aspire mathematics assessments emphasize quantitative reasoning frequently applied to real-world contexts rather than memorization of formulas or computational skills. “ (p. 26)
Some items give the formula(s) they need, but others do not.
“Students are allowed and expected to strategically use acceptable calculators on the ACT Aspire mathematics assessments for Grade 6 and above.” (p. 27)
Paper-and-pencil tests test do not have technology-enhanced items. Multiple choice items are used in their place. (p. 27)
I didn’t copy this bulletin for them, but the link is on our website. I pulled out info that I thought they might find interesting.

38. Score Scale
The high score for 3rd grade is higher than the benchmark for 10th grade. This tells me that there should be some questions on each test that a proficient student may not be able to answer.

39. Constructed response items are worth 4 points each.
I found the information on point values in the Technical Bulletin. This chart and the times came from the link at the top of the screen.
Selected response and technology enhanced items are worth 1 point each.
Constructed response items are worth 4 points each.

40. Reporting Categories All questions are either measuring
Grade Level Progress – mathematical topics new to the grade
Foundation – topics learned in previous grades
Some questions are also categorized as
Modeling – questions that assess understanding of mathematical models and their creation, interpretation, evaluation, and improvement
Justification and Explanation – giving reasons for why things work as they do, where students create a mathematical argument to justify (constructed response)

41. Points by category and grade.

42. How does ASPIRE match the CCSSM?
“Through grade 7 the two are the same.” (page 5)
“Across all parts of the test, students can apply Mathematical Practices to help them demonstrate their mathematical achievement.” (page 2)
This is from the Exemplar packet. I copied these pages for the handout.

43. Justification and Explanation
Level 1 – students should have a fluent command of these skills
Level 2 – most closely aligned with grade level focus
Level 3 – more advanced
As students progress from grade to grade, expectations increase according to which JE skill belongs to which level. Some level 3 JE skills will become level 2, and some level 2 will become level 1.
A full-credit response shows evidence of the required level of JE skills needed to solve the problem and applies these skills to complete the task.
Evaluated by trained scorers.
From exemplar packet. Page 2. See pages 3-4 for a detailed progression of JE skills.

44. Depth of Knowledge
“…assessing new topics for the grade and whether students continue to strengthen their mathematical core. Within this structure of content comes a level of rigor represented in part by a distribution of depth of knowledge (DOK) through Webb’s level 3. The Foundation component includes only DOK level 2 and level 3 because the component is about assessing how well students have continued to strengthen their mathematical core. Across all parts of the test, students can apply Mathematical Practices to help them demonstrate their mathematical achievement.”
This information is also included in Quality Core training. This seems to be new for a lot of teachers.

45. Webb’s Depth of Knowledge
Recall and Reproduction
Skills and Concepts
Strategic Thinking / Reasoning
Extended Thinking
Handout. There is also another handout about DOK in the packet and more links on the website.

46. Percentage of Points by DOK
6th Grade
7th Grade
8th Grade
DOK 1
7 – 15 %
8 – 15 %
DOK 2
33 – 41 %
30 – 38 %
DOK 3
48 – 57 %
51 – 58 %
Not many Level 1’s!!!!
Foundation questions are DOK 2 and 3.

47. Practice Test http://www.discoveractaspire.org/assessments/test-items/
UN: math
PW: actaspire
This is a 6-8 test with questions from multiple grade levels. I have copied selected questions for the handout. The Exemplar packet also contains these questions with a detailed explanation of correct response.

48. 5.NBT.B MP3 N(13-15) 6-8 Foundation JE Level 3 DOK Level 3
Explain the different pieces of information given for each question.
5.NBT.B – the B at the end refers to the 2nd cluster in the 5th grade NBT standard. Grade level progress question for 5th grade.
Math Practice 3 – Construct viable arguments and critique the reasoning of others.
N(13-15) – ACT CCRStandards. The numbers refer to a specific score range. This is something that a student who scored are likely to know and be able to do.
6-8 Foundation question for 6-8
Justification and explanation level
DOK level

49. 8.F.A MP -- F(20-23) 8 Grade Level Progress JE Level -- DOK Level 2
This question does not exhibit a math practice or justification and explanation.

50. 6.G.A MP3 G(20-23) 6 Grade Level Progress JE Level 3 DOK Level 3
7-8 Foundation JE Level DOK Level 3

51. Multiple levels Multiple problems with common information
Questions are independent of each other. It is not necessary to get one correct in order to correctly answer the others.
Students must extract only the information needed for a particular question.

52. 7.EE.B MP4 A(24-27) 7 Grade Level Progress JE Level -- DOK Level 3
8 Foundation JE Level DOK Level 2
MP4 – I’m thinking that this will also count for modeling
Different DOK levels for different grades.

53. 6.SP.B MP-- S(16-19) 6 Grade Level Progress JE Level -- DOK Level 3
7-8 Foundation
JE Level DOK Level 2
6th grade example

54. More Practice Items www.illustrativemathematics.org
More places to get rigorous problems based on the standards. These are explained and linked to on the wiki.

55. mathpractices.edc.org
Interpreting the SMP Course – Session 1

56. Mathematics Task Exploration: Postage Stamp Problem Solve Mathematics Task
Individual work on task.
Track the twists and turns in your thinking.
Small group work on task.
Interpreting the SMP Course – Session 1

57. Mathematics Task Exploration: Share Strategies – Part I
Share and discuss:
How did you start out thinking about the task?
How did your thinking change, and what prompted that change?
What conclusions, hypotheses, and questions have you generated about possible and impossible postage amounts?
Interpreting the SMP Course – Session 1

58. Mathematics Task Exploration: Share Strategies – Part II
Share and discuss:
Examples of thinking from your own or your colleagues’ work on the task that illustrate the Standards for Mathematical Practice.
MP 1: Make sense of problems and persevere in solving them.
MP 2: Reason abstractly and quantitatively.
MP 3: Construct viable arguments and critique the reasoning of others.
MP 4: Model with mathematics.
MP 5: Use appropriate tools strategically.
MP 6: Attend to precision.
MP 7: Look for and make use of structure.
MP 8: Look for and express regularity in repeated reasoning.
Interpreting the SMP Course – Session 1

59. Standards for Mathematical Practice
MP1 – Make sense of problems and persevere in solving them.
MP3 – Construct viable arguments and critique the reasoning of others.
Interpreting the SMP Course – Session 2

60. Student Dialogue Exploration: Introduce Student Dialogues – Part I
Dialogue between three fictitious high school characters (Chris, Lee, and Matei) working on a mathematics task
The dialogues are intended to:
Clarify the meaning of particular SMP by showing what student discourse could be
Illustrate key ideas about the Standards for Mathematical Practice (SMP) in context using specific mathematical content
Serve as an artifact to promote discussion among educators about the SMP, about mathematics, and about issues of teaching practice.
Change back to just middle school students after facilitator institute
Interpreting the SMP Course – Session 1

61. Student Dialogue Exploration: Introduce Student Dialogues – Part II
Given the intention to illustrate the meaning and key ideas of the SMP:
Plausible student thinking, but the discourse may not always sound realistic.
The student characters are “caricatures,” intended to illustrate particular types of thinking and discussion.
A teacher voice is intentionally not included.
Discussion of whether or how a teacher might intervene, or of how to promote similar thinking with your own students, are productive avenues for discussion.
Interpreting the SMP Course – Session 1

62. Student Dialogue Exploration: Read Student Dialogue
Read the Student Dialogue out loud.
Read the Student Dialogue individually – focus on mathematical thinking used by the students.
Interpreting the SMP Course – Session 1

63. Student Dialogue Exploration: Discuss Teacher Reflection Questions
Small groups: Discuss Questions #1-4 on the Teacher Reflection Questions handout.
Refer to the Standards for Mathematical Practice handout.
Be specific about what evidence you see in the dialogue.
Time permitting: Discuss any or all of Questions #5-6.
Whole group: Discuss Question #2.
Resource for later: Mathematical Overview
Interpreting the SMP Course – Session 1

64. mathpractices.edc.org
Interpreting the SMP Course – Session 1

65. Feedback
Today I learned….
A question I still have is….

66. Contact Information
acos2010.wikispaces.com