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Session I: Unpacking the Learning Progressions Time Allotment 90 minutes

Outcomes(s)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

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

1 minute

Welcome participants to 2nd Quarterly Meeting for 2014-2015 school year.

5 minutes

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

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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.”

2 minutes

Emphasize “locally”

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)

1 minute

Good Morning, the outcomes for the QM #2 is: (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,

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school, and students.

1 minute

Read slide.

15 minutes Have participants reflect first and then share out at table.

Give participants some time to write individually, then facilitate a discussion around some or all of the questions listed on the slide.

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

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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

Facilitator should monitor when participants are sharing at their table (step 3), select participants to share out (making participants aware of what they said that you want them to share whole group). Facilitator should sequence the order in which participants should share out. The facilitator should make the connection of what each person shared.

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(*) in the CCRS.

4 minutes

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 by the logical structure of mathematics.

1 min

Because of structural differences between the disciplines of mathematics and English Language Arts, the mathematics standards do not support such easy analysis of the progression of standards across grades.

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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.

5 minutes

1st) Read slide.

2nd) Have participants record their definition of Operations and Algebraic Thinking on Journal Reflection hand-out.

3rd) Have participants share their definition with members at their table (participants should record this on their K-W-L handout under the “K”).

Facilitator should monitor when participants are sharing at their table (step 3), select participants to share out (making participants aware of what they said that you want them to share whole group). Facilitator should sequence the order in which participants should share out. The facilitator should make the connection of what each person shared.

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4th) Have each person selected while monitoring share out whole group

5 minutes

1st) Say, “We are about to watch a video (video is 1:50 in length) to connect our background Knowledge of Operations and Algebraic Thinking to what one of the writers of the standards, Jason Zimba, says about it.

2nd) Jot down any information you learn while Watching the video.” (participants should record this on their K-W-L handout under the “W”).

3rd) Have participants share any new knowledge gained with members at their table.

4th) Have each person selected while monitoring share out whole group

2 minutes

Say, “We have a whole group

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overarching definition of Operation and Algebraic Thinking.”

We’ve heard what Jason Zimba, one of the writers of the CCRS, has said about Operations and Algebraic Thinking.

Now, we will go deeper. You will deepen your content knowledge of the specified grade as you read the Learning Progression for Operations and Algebraic Thinking.”

Say, “As you are reading your section, keep in mind the fluencies required for this domain, conceptual understandings student’s need, and different ways operations and algebraic thinking can be applied.”

Say, “Turn to the section of the progressions document that addresses the grade level you are going deeper in.”

Say, “You have a purpose for reading collectively: fluencies, student’s conceptual understanding, and application of this domain. Because there is a wealth of information in this document, because our conceptual understanding of this

(Divide the participants into 6 groups: K, 1st, 2nd, 3rd,4th, 5th)

Set a purpose for reading:

(have a participant restate the purpose for reading)

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domain is at different levels, as you read you will code the text.

15 minutes:

Go over slide of how participants should code the text as they read.

10 mins

1st) Say, “Now that you have read the Operations and Algebraic Thinking Progression. You will share with members at your table and chart what you Learned as it relates to our purpose for reading the progression document (3 purposes on this slide) on chart paper.

2nd) Have 1 person from each group share out what they learned. (Starting with K, then 1st, 2nd, etc.) Participants can record this on there K-W-L handout.

Facilitator sums up the 3 purposes for reading by saying,

Once participants have finished reading independently, have them record what they learned about the grade level they just read about. Have them record this on their K-W-L handout under the “L” column.

Allow 15 minutes for participants to read the learning progression independently.

(Even though 4th and 5th reads both progressions, have the teachers chart only for the grade they are sitting with.)

Have them discuss with their group and record and chart out the fluencies that are required for their grade level, and conceptual understanding students need and how this learning can be applied (situations, word problems, other domains like geometry and measurement, etc.)

(Note: Emphasize that OA can be applied to Word problems if the participants don’t bring out this point.)

Facilitator should monitor when

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“these attend to the rigor in the standards.”

1 min

One way students can apply their knowledge in Order of Algebraic Thinking is through word problems.

3 minutes

1st) Have participants write a word problem for each equation within the grade-band they read on a different post-it.

9 minutes

1st) Have participants turn to page 7 (if they read K-2) or 23 (if they read 3-5) of the progression document and identify the type of problem they wrote.

participants are sharing at their table. The facilitator should make the connection of what each group shared.

Possible Questions to ask as groups share: 1.) How do the learning progressions develop within this domain?2.) How will/can students use this knowledge in the next grade?

It is important that students learn to solve all these different types of problem as this will demonstrate a full understanding of the meaning of the addition and subtraction operations. Practice with lots of examples is needed but this should be done after starting with plenty of hands-on activities with concrete materials.

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(While they are writing, facilitate and ask questions.)

2nd) Have each table post their problem on the K-2 or 3-5 story problem poster.

3rd) Have participants take a moment to look at the poster and reflect on what they notice, then share at their table, and finally whole group discussion.

Possible Questions to ask:

Were there any problem types not shown?

Say, “The same story problem handout that you are looking at is found in the appendix section of ALCOS.

3 minutes

Say, “However that handout isn’t as explicit as the handout in the learning progression document.”

Have participants turn to page 9 of the OA progression document.

Go over which grade level requires mastery of which problem types/subtypes.

Say, “Van de Walle describes four types of

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addition/subtraction problems & four categories of multiplication/division problems.” (Noted in pink vertically)

Say, “Each of these main types/categories have two or three subtypes/subcategories.” (noted in pink horizontally)

Say, “Kindergarten has some Problem types to be mastered by the end of the Kindergarten year (yellow).

Ask, “What is the difference in the Add To/Take From and Put Together/Take Apart problem types?”

Say, “ Add To/ Take From problem types require movement and Put Together/Take Apart problem types require no movement (total items mentioned in word problem are already visable).

1st: Problem types to be mastered by the end of the First Grade year (yellow and white). However, First Grade students should have experiences with all 12 problem types.

2nd: Problem types to be mastered by the end of the Second Grade year (yellow, white and green).

The green shaded problems are

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the four difficult subtypes that students should work with in Grade 1 but need not master until Grade 2.

Say, “So when you look at the story problems in your curriculum, think about the rigor in the problems.

It depends on the content in the problem to determine the type of problem.

2 minutes

Have participants turn to page 23 of the OA progression document.

Go over which grade level requires mastery of which problem categories/subcategories.

3rd Grade has some problem types that must be mastered by the end of the 3rd grade, they are highlighted in yellow. They include Equal Groups and Arrays (Multiplicative)

4th Grade has some problem types that must be mastered by the end of the 4th grade, they are the yellow and white sections on the chart. They include Equal

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Groups, Arrays, Area, and Multiplicative Comparison problems using whole numbers.

5th Grade has some problem types that must be mastered by the end of the 5th grade, they are the yellow and white sections on the chart. They include Equal Groups, Arrays, Area, and Multiplicative Comparison problems using whole numbers and fractions.

3 minutes

Have participants turn to this Journal Reflection Hand-Out in their Participants Packet.

Read Slide

Give participants some time to write individually, turn and talk, then facilitate a discussion.

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

While you are waiting for participants to return: Recommend that participants collect their thoughts from this

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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.

1 min

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 mins

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.

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3 min

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 mins

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.

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2 mins

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.

2 mins

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

**Note: Refer to the learning progression discussed this morning. The learning progression graphic is in their participant packet.

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is only one interpretation.

10 mins

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

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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.

12 mins

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

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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/(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

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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/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.

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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, 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

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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.

50 mins

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, table/person discusses how their standard and task is a building block toward the high school standard (and task) shown –

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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

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without evaluating them

6 –Identify when two equations are equivalent (Sixth grade also learns order of operations)

7 – rewriting expressions in different forms

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.

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. 

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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).

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.

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

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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

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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.

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=

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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

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Common Core in line with other high-achieving countries.

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.

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.

The purpose of this instructional task is to illustrate how different, but equivalent, algebraic expressions can reveal different

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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.

5 mins (for next 8 slides)

Summarize the big ideas discovered during the whole

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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

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

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understanding.

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.

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

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

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.

With your district team think about your next steps.

Record your thoughts on this template and share with the rest

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of your team when you join them in a few minutes.