jackson county 6-12 math ccrs quarterly meeting # 2 jackson county 6-12 math ccrs quarterly meeting...

Jackson County 6-12 MathJackson County 6-12 MathCCRS Quarterly Meeting # 2CCRS Quarterly Meeting # 2

Unpacking the Learning Progressions

August 10, 2015

http://alex.state.al.us/ccrs/

ALABAMA QUALITY ALABAMA QUALITY TEACHING STANDARDSTEACHING 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

The Five Absolutes + A Balanced Instructional Core = A Prepared Graduate

Five Absolutes• Teach to the standards (Alabama

College- and Career-Ready Standards – Math Course of Study)

• A clearly articulated and “locally” aligned K-12 curriculum

• Aligned resources, support, and professional development

• Regular formative, interim/benchmark assessments to inform the effectiveness of the instruction and continued learning needs of individual and groups of students

• Each student graduates from high school with the knowledge and skills to succeed in post-high school education and the workforce

The Instructional Core

1.4

5.3 2.7

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

Outcomes

CCRS-Mathematics Learning Progressions

VideoFlows Leading to Algebra

What are the big ideas explored in the progressions Expressions and Equations and Algebra for grades 6 – 12?

• 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)

A Study of the Expressions and Equations Progression and the Algebra Progression

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)

DiscussionConsider 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?

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ReflectionHow 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

“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)

Why is professional peer discussion about progressions important for the teaching

and learning process?

How might the idea of learning progressions connect to student

experience, learning, misconceptions and common mistakes?

How might the idea of learning progressions connect to the tasks a

teacher selects to guide student learning?

The progression of student understanding of Algebra begins with Counting and Cardinality, moves through Operations and

Algebraic Thinking, to Expressions and Equations, and finally to Algebra.

How do you connect standards to standards so that children are

equipped to think mathematically?

How do you work as a

team across grades to

ensure student

growth in algebraic

reasoning?

Major Work of the Grade: A Progression to Algebra

Using Tasks from Illustrative Mathematics for Algebraic Development

• These tasks are not meant to be considered in isolation. When taken together as a set of tasks, they illustrate a particular standard.

• These tasks were grouped together to represent one interpretation of the algebra learning progression.

• This representation illustrates how mathematical knowledge and skills develop over time.

Tracking the Algebra Progression Toward a High School Standard

A-SSE.A.1Interpret the structure of expressions1. Interpret expressions that represent a quantity in terms of its context.★

a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

Sample Illustration of A-SSE.A.1

Tracking the Algebra Progression Toward a High School Standard1. Read the task.

2. Discuss the concepts that are involved in your particular task that are necessary for students to connect their learning to algebra.

3. Discuss the concepts that students will build upon from the previous grade and the concepts which will lead to in the next grade.

4. Relate the concepts from the task to the original high school task.


K.OA.A.3Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

Sample IllustrationMake 9 in as many ways as you can by adding two numbers between 0 and 9.

http://www.illustrativemathematics.org/illustrations/177

Tracking the Algebra Progression Toward a High School Standard1.OA.D.7Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

Sample IllustrationDecide if the equations are true or false. Explain your answer.•2+5=6 •3+4=2+5 •8=4+4 •3+4+2=4+5 •5+3=8+1 •1+2=12 •12=10+2 •3+2=2+3 •32=23

https://www.illustrativemathematics.org/illustrations/466


2.NBT.A.4Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Sample IllustrationAre these comparisons true or false?A) 2 hundreds + 3 ones > 5 tens + 9 onesB) 9 tens + 2 hundreds + 4 ones < 924C) 456 < 5 hundredsD) 4 hundreds + 9 ones + 3 ones < 491E) 3 hundreds + 4 tens < 7 tens + 9 ones + 2 hundredF) 7 ones + 3 hundreds > 370G) 2 hundreds + 7 tens = 3 hundreds - 2 tens



3.OA.B.5Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Sample Illustration

Decide if the equations are true or false. Explain your answer.

4 x 5 = 20 6 x 9 = 5 x 10

34 = 7 x 5 2 x (3 x 4) = 8 x 3

3 x 6 = 9 x 2 8 x 6 = 7 x 6 + 6

5 x 8 = 10 x 4 4 x (10 + 2) = 40 + 2


4.OA.A.3Solve 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 IllustrationKarl'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?



5.OA.A.2Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Sample IllustrationLeo 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.



6.EE.A.4Identify 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 IllustrationWhich 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


Tracking the Algebra Progression Toward a High School Standard7.EE.A.2Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

Sample IllustrationMalia is at an amusement park. She bought 14 tickets, and each ride requires 2 tickets.•Write an expression that gives the number of tickets Malia has left in terms of x, the number of rides she has already gone on. Find at least one other expression that is equivalent to it.•14−2x represents the number of tickets Malia has left after she has gone on x rides. How can the 14, -2, and 2x be interpreted in terms of tickets and rides?

•2(7−x) also represents the number of tickets Malia has left after she has gone on x rides. How can the 7, (7 – x), and 2 be

interpreted in terms of tickets and rides?


Learning Progressions for Learning• How does algebra progress from kindergarten to high

school?

• What are some ways that understanding the learning progressions can strengthen grade level instruction?

• Why do you believe it is important to understand mathematical trajectories and how knowledge is built over time?

Toward Greater Coherence

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)? **

• 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?

…. The Teacher Leader (AQTS 5.3)

References• “The Structure is the Standards” Daro, McCallum, Zimba (2012)

http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/

• www.illustrativemathematics.org

• K–8 Publishers’ Criteria for the Common Core State Standards for Mathematics (2013)

acos2010.wikispaces.com

Sustainabilityongoing research, support, and validation of the system to reflect changes in college and career readiness standardsflexible professional development on the state, district or school levels

http://www.discoveractaspire.org/assessments/summative/

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)

Score Scale

Selected response and technology enhanced items are worth 1 point each. Constructed response items are worth 4 points each.

Reporting CategoriesAll 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)

Points by category

and grade.

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)

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.

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.”

Webb’s Depth of Knowledge

1. Recall and Reproduction

2. Skills and Concepts3. Strategic Thinking /

Reasoning4. Extended Thinking

Percentage of Points by DOK

6th Grade 7th Grade 8th GradeDOK 1 7 – 15 % 7 – 15 % 8 – 15 %DOK 2 33 – 41 % 33 – 41 % 30 – 38 %DOK 3 48 – 57 % 48 – 57 % 51 – 58 %

Foundation questions are DOK 2 and 3.

Practice Test

http://www.discoveractaspire.org/assessments/test-items/

UN: math PW: actaspire

5.NBT.B MP3 N(13-15)

6-8 Foundation JE Level 3 DOK Level 3

8.F.A MP -- F(20-23)

8 Grade Level Progress JE Level -- DOK Level 2

6.G.A MP3 G(20-23)

6 Grade Level Progress JE Level 3 DOK Level 37-8 Foundation JE Level 3 DOK Level 3

Multiple levelsMultiple 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.

7.EE.B MP4

A(24-27)

7 Grade Level Progress JE Level --DOK Level 38 Foundation

JE Level -- DOK Level 2

6.SP.B MP--

S(16-19)

6 Grade Level Progress JE Level --DOK Level 37-8 Foundation JE Level --

DOK Level 2

More Practice Itemswww.illustrativemathematics.org

http://www.parcconline.org/samples/item-task-prototypes

http://www.smarterbalanced.org/smarter-balanced-assessments/

https://www.engageny.org/resource/new-york-state-common-core-sample-questions

edc.org

mathpractices.edc.org

Interpreting the SMP Course – Session 1 55

edc.org

Mathematics Task Exploration: Postage Stamp ProblemSolve Mathematics Task

• Individual work on task.– Track the twists and turns in your thinking.

• Small group work on task.


edc.org

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?


Mathematics Task Exploration:Share Strategies – Part IIShare and discuss:

•Examples of thinking from your own or your colleagues’ work on the task that illustrate the Standards for Mathematical Practice.

MP 1MP 1:: Make sense of problems and persevere Make sense of problems and persevere in solving them.in solving them.MP 2MP 2:: Reason abstractly and quantitatively.Reason abstractly and quantitatively.MP 3MP 3:: Construct viable arguments and critique Construct viable arguments and critique the reasoning of others.the reasoning of others.MP 4MP 4:: Model with mathematics.Model with mathematics.MP 5MP 5:: Use appropriate tools strategically.Use appropriate tools strategically.MP 6MP 6:: Attend to precision.Attend to precision.MP 7MP 7:: Look for and make use of structure.Look for and make use of structure.MP 8MP 8:: Look for and express regularity in Look for and express regularity in repeated reasoning.repeated reasoning.


edc.org

Standards for Mathematical Practice

• MP1 – Make sense of problems and persevere in solving them.

• MP3 – Construct viable arguments and critique the reasoning of others.


edc.org

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.


edc.org

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.


edc.org

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.


edc.org

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


edc.org

mathpractices.edc.org


Feedback Today I learned….

A question I still have is….

Contact Information

[email protected] [email protected] acos2010.wikispaces.com

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jackson county 6-12 math ccrs quarterly meeting # 2 jackson county 6-12 math ccrs quarterly meeting...

Documents

linear equations pages

equations progression

algebraic expressions

variable pages

overview pages

rational expressions

constants pages

variable equations