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Session I: Unpacking the Learning Progressions Time Allotment: 90 minutes

Outcomes(s)Participants will:

Reflect on year one and year two implementation of CCRS Mathematics

Review and deepen understanding of Learning Progressions and how they are sequenced within and across the grades

Discuss how the progressions in the standards can be used to inform teaching and learning

Slides Lesson Flow Research/Helpful HintsFor facilitators only (Do not read to participants!)

Slide 1:Welcome the participants to the 2nd Quarterly Meeting for 2014-2015 school year.1 minSlide 2: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.”

1 min

Purpose of the ContinuumBased 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

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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

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careerThe 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

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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.

Slide 3:Emphasize “locally”Support examples (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)1 minSlide 4: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.1 min

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Slide 5:Participants 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

1 minSlide 6:Have participants reflect first 5 minutes and then share out the last 5 minutes. Give participants some time to write individually, then facilitate a discussion around some or all of the questions listed on the slide.Wrap up with a whole group discussion. Possible comments: When teachers begin with a possible learning path in mind they…

• Consider strategies for instructional scaffolding to get students to the next stage of learning

• Use formative & summative assessments “strategically” and more frequently; they value “uncovering student thinking”

• Collaboratively analyze student work creating a deeper understanding of how learning develops

• Uncover “flawed assessments” they have been using

• Use smaller, more targeted assessment and pre-assessments (of pre-requisite

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skills) at the start of a unit=better information about learning

• Adjust instruction according to what students CAN do, not what they CANNOT do

• Shift perceptions, especially of their lower performing students & what to do next to support learning

(K. Hess, Center for Assessment 2012)

What questions can learning progressions help us to answer?

How do students’ understandings and abilities to apply core ideas develop over time?

How can a sequence of instructional experiences & core learnings be identified to promote optimal progress for most students?

How can students’ progress towards targeted understandings and abilities be monitored and addressed with targeted instruction?

10 minSlide 7: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

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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.1 minSlide 8:Today we are focusing on the Operation and Algebraic Thinking, Expressions and Equations, and Algebra Progressions which can be found at http://ime.math.arizona.edu/progressions/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

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by the logical structure of mathematics.2 minSlide 9: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.3 minSlide 10: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!!!.

Slides 10 – 14, 69 minutes

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Slide 11:The progressions have been divided into sections. Everyone should read the Overview pages 2 and 3If 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.

Slide 12: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

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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).

Slide 13: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 QuestionsWhat 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?

Slide 14:Give participants some time to write individually, then facilitate a discussion.

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Slide 15:Tell participants to enjoy lunch and you will see them after lunch.

Slide 16:While you are waiting for participants to return: Recommend that participants collect their thoughts from this morning and record them on the “Professional Development Transition Plan” that they will use later on in their district planning meeting.Welcome participants back from lunch.

Slide 17: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?1 minSlide 18: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.2 minSlide 19: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. BE SURE TO START AT 16 SECONDS!!3 min

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Slide 20: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.3 minSlide 21:To be even more specific, these are cluster headings from the CCRS: Mathematics in grades K - 8. 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.2 min

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Slide 22:Our next step in understanding the algebra progression and its effect on student learning, leads us to explore these high-level tasks from K - 12. 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.2 minSlide 23: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, 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.10 min

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Slide 24: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

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fraction equals 100/(100+150)=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 (100+150)/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/2In 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/(100+150)=0.4.PQ and QP can be interpreted in two different ways.PQ can be interpreted as a unit rate,

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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 100150=23, so there would be 23 of an animal in population P for every 1 animal in population Q, while 150100=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 100150=23, so there are 23 as many animals in population P as there are in population Q, while 150100=32, so there are 32 as many animals in population Q as there are in population P.12 minSlide 25: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

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standard and illustration, 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 equations1 – meaning of equal sign (does not mean output or “give me an answer”)2 - Begin using, <, >3 - Properties of operations – commutative, associative, distributive4 – four operations with remainders , equations with letters5 – simple expressions, interpret without evaluating them6 –Identify when two equations are equivalent (Sixth grade also learns order of operations)7 – rewriting expressions in different forms50 minSlide 26: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.

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Any number between 2 and 10 can be used in place of 9 to address K.OA.3.Slide 27: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 form4+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).

Slide 28: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.

Slide 29:This task is a follow-up task to a first grade task: http://www.illustrativemathematics.org/illustrations/466.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

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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 6 + 6 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.

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Slide 30: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+40Sometimes students are tempted to do something similar when multiplication is also involved; however this will get them into trouble since20×(5+40)≠(20+5)×40This 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

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adopted the Common Core in line with other high-achieving countries.

Slide 31: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 on3.OA.5 Apply properties of operations as strategies to multiply and divideand4.OA.1 Interpret a multiplication equation as a comparison.

Slide 32: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.

Slide 33: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

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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.

Slide 34: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 gradesTeachers 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 gradesIf teachers’ own knowledge of the content and how mathematical ideas are developed over time in stronger, their instruction can be stronger

Slides 34 – 41, 5 minutes

Slide 35: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.

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Slide 36: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.

Slide 37:Ask participants to bring student work from their next steps assignment to the 3rd quarterly meeting.

Slide 38:Have participants read the slide. The Special Education curriculum is hyperlinked to the symbol.

Slide 39: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.

Slide 40:With your district team think about your next steps.Record your thoughts on this template and share with the rest of your team when you join them in a few minutes.

Slide 41: