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In this meeting, participants will reflect on the previous meeting, deepen their understanding of Algebra Learning Progressions, and discuss how the progressions can inform planning, teaching, and learning.
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CCRS Quarterly Meeting # 2Unpacking the Learning Progressions http://alex.state.al.us/ccrs/
Alabama Quality Teaching Standards 1.4 2.7 5.3 1.4-Designs instructional activities based on state content standards 2.7-Creates learning activities that optimizeeach 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
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) • Aclearly 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 2.7 5.3
Outcomes Participants will: • Reflect on Next Steps from QM #1 • Review and deepen understanding of the Algebra Learning Progression and how the content is sequenced within and across the grades (coherence) • Illustrate, using tasks, how math content develops over time • Discuss how the progressions in the standards can be used to inform planning, teaching, and learning
Next Steps Participants should have done three things for the CCRS QM #2 : Decide which task you will implement in your class, solve task, and anticipate possible student solutions. Implement task in the classroom ( monitor, select, and sequence) Bring student work samples (student’s solution path) to share with your group.
What are the big ideas explored in the progressions Expressions and Equations and Algebra for grades 6 – 12?
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)
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)
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?
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
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)
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?
Using Tasks from Illustrative Mathematics for Algebraic Development • These tasks are not meant to be considered in isolation. When taken together as a set of tasks, they illustrate a particular standard. • These tasks were grouped together to represent one interpretation of the algebra learning progression. • This representation illustrates how mathematical knowledge and skills develop over time.
Tracking the Algebra Progression Toward a High School Standard A-SSE.A.1 Interpret the structure of expressions 1. Interpret expressions that represent a quantity in terms of its context.★ a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)nas the product of P and a factor not depending on P.
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.
Tracking the Algebra Progression Toward a High School Standard K.OA.A.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). Sample Illustration Make 9 in as many ways as you can by adding two numbers between 0 and 9. http://www.illustrativemathematics.org/illustrations/177
Tracking the Algebra Progression Toward a High School Standard 1.OA.D.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. Sample Illustration Decide if the equations are true or false. Explain your answer. • 2+5=6 • 3+4=2+5 • 8=4+4 • 3+4+2=4+5 • 5+3=8+1 • 1+2=12 • 12=10+2 • 3+2=2+3 • 32=23 https://www.illustrativemathematics.org/illustrations/466
Tracking the Algebra Progression Toward a High School Standard 2.NBT.A.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons. Sample Illustration Are these comparisons true or false? A) 2 hundreds + 3 ones > 5 tens + 9 ones B) 9 tens + 2 hundreds + 4 ones < 924 C) 456 < 5 hundreds D) 4 hundreds + 9 ones + 3 ones < 491 E) 3 hundreds + 4 tens < 7 tens + 9 ones + 2 hundred F) 7 ones + 3 hundreds > 370 G) 2 hundreds + 7 tens = 3 hundreds - 2 tens http://www.illustrativemathematics.org/illustrations/111
Tracking the Algebra Progression Toward a High School Standard 3.OA.B.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) Sample Illustration Decide if the equations are true or false. Explain your answer. 4 x 5 = 20 6 x 9 = 5 x 10 34 = 7 x 5 2 x (3 x 4) = 8 x 3 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
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? http://www.illustrativemathematics.org/illustrations/876
Tracking the Algebra Progression Toward a High School Standard 5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. Sample Illustration Leo and Silvia are looking at the following problem: • How does the product of 60 × 225 compare to the product of 30 × 225? • Silvia says she can compare these products without multiplying the numbers out. Explain how she might do this. Draw pictures to illustrate your explanation. https://www.illustrativemathematics.org/illustrations/139
Tracking the Algebra Progression Toward a High School Standard 6.EE.A.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Sample Illustration Which of the following expressions are equivalent? Why? If an expression has no match, write 2 equivalent expressions to match it. • 2(x+4) • 8+2x • 2x+4 • 3(x+4)−(4+x) • x+4 http://www.illustrativemathematics.org/illustrations/177
Tracking the Algebra Progression Toward a High School Standard • 7.EE.A.2 • Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” • Sample Illustration • Malia is at an amusement park. She bought 14 tickets, and each ride requires 2 tickets. • Write an expression that gives the number of tickets Malia has left in terms of x, the number of rides she has already gone on. Find at least one other expression that is equivalent to it. • 14−2x represents the number of tickets Malia has left after she has gone on x rides. How can the 14, -2, and 2x be interpreted in terms of tickets and rides? • 2(7−x) also represents the number of tickets Malia has left after she has gone on x rides. How can the 7, (7 – x), and 2 be interpreted in terms of tickets and rides? • https://www.illustrativemathematics.org/illustrations/1450
Learning Progressions for Learning • How does algebra progress from kindergarten to high school? • What are some ways that understanding the learning progressions can strengthen grade level instruction? • Why do you believe it is important to understand mathematical trajectories and how knowledge is built over time?
Toward Greater Coherence “The Standards are designed around coherent progressions from grade to grade. Principals and teachers carefully connect the learning across grades so that students can build new understanding onto foundations built in previous years. Teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.” Student Achievement Partners, 2011
Step Back – Reflection Questions • What are the benefits of considering coherence when designing learning experiences (lesson planning) for students? • How can understanding learning progressions support increased focus of grade level instruction? • How do the learning progressions allow teachers to support students with unfinished learning (struggling students)? **
Next Steps Identify standards and select a high level task. Plan a lesson with colleagues. Anticipate student responses, errors, and misconceptions. Write assessing and advancing questions related to student responses. Keep copies of planning notes. Teach the lesson. When you are in the Explore phase of the lesson, tape your questions and the students responses, or ask a colleague to scribe them. Following the lesson, reflect on the kinds of assessing and advancing questions you asked and how they supported students to learn the mathematics.
Survey • Are there any aspects of your own thinking and/or practice that our work today has caused you to consider or reconsider? Explain. • How will today’s learning help you as you work with: • Collaborative Planning • Struggling students (Special Ed and ELL, etc) • Formative assessment? • With respect to CCRS Math, what would you like more information/learning on?
…. The Teacher Leader (AQTS 5.3) • How can today’s learning of the progressions be used to inform your teaching and learning? • How can today’s learning of the progressions be used to inform your professional learning community?
Wrapping up….. Prepare for District Team Planning
References • “The Structure is the Standards” Daro, McCallum, Zimba (2012) http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/ • www.illustrativemathematics.org • K–8 Publishers’ Criteria for the Common Core State Standards for Mathematics (2013)
