# Illustrating the Standards for Mathematical Practice

## Content at a Glance

#### Introduction to CCSS Standards for Mathematical Practice, K-12

Participants explore both the language of the Practice Standards and sample tasks from the national assessment projects as contexts though which they can consider implications that the Practices will have on their classrooms and students. The antecedents of the Standards for Mathematical Practice are described and next steps are explored.

#### Problem Solving and Precision, K-8

Teachers and students across grades K-8 engage in a set of related, leveled, non-routine, and challenging geometric tasks. This module highlights ways in which Problem of the Month (POM) can be used as an instructional strategy to create authentic opportunities for students to learn to solve problems accurately and with precision.

#### Problem Solving and Precision, 9-12

Cathy Humphreys facilitates a geometric design task intended to deepen students' understanding of properties of quadrilaterals and scaffold their use of reasoning and proof. Central to the students' exploration of the mathematics is a poster describing an "Investigative Process." The poster is used as a tool by the teacher and students to support problem solving and perseverance.

#### Reasoning and Explaining, K-5

Classroom artifacts are used in this module to explore strategies that support students' early development of reasoning and explaining. Participants experience a way in which properties of operations can serve as a vehicle to help students learn to look for regularities and construct viable arguments.

#### Reasoning and Explaining, 3-8

The teacher engages students with the Button Pattern task over the course of two class periods. During students' second experience, the teacher uses a powerful formative assessment strategy called "reengagement" to prompt advances in students' mathematical thinking and reasoning. The teacher focuses the class on work samples strategically selected from the previous period.

#### Reasoning and Explaining, 6-12

This module prompts participants to examine the meaning of defining congruence and similarity through transformations as articulated in the Common Core State Standards. To do this, participants are asked to compare and contrast static definitions of congruence and similarity with dynamic definitions of congruence and similarity. They are also prompted to consider implications for instruction that the dynamic definitions have on teaching and learning mathematics.

#### Reasoning and Explaining, 6-12

Participants will unpack the connection between similarity, slope, and the graphs of linear functions by doing mathematical tasks, analyzing student thinking, and exploring a computer-based applet. Together, these activities serve as the context for participants to consider strategies to support students to reason mathematically and learn to use precise language in their explanations.

#### Modeling and Using Tools, K-5

This module explores ways in which the practice of reasoning and explaining can be used to lay a foundation for mathematical modeling. Participants consider student thinking as they engage in modeling with the Penny Jar task using physical models, numbers, tables, graphs, and formulas.

#### Modeling and Using Tools, 3-5

The teacher implements a typical algebraic thinking task that asks students to represent a number sentence. However, the teacher launches this task in a non-traditional manner by focusing first on pictorial representations of the context with the intent of creating greater access. This unique manner of setting up the task appears to create substantive opportunities for student learning.

#### Modeling and Using Tools, 6-8

This module integrates a series of linear function tasks that focuses on mathematical modeling. A primary goal of the lesson is to help students make sense of mathematical situations and model them appropriately. This is achieved, in part, by prompting students to move beyond the notion that a particular representation is either right or wrong when analyzing real-world contexts.

#### Modeling and Using Tools, 9-12

Cathy Humphreys facilitates a geometric design task intended to deepen students' understanding of properties of quadrilaterals and scaffold their use of reasoning and proof. Students work in small groups discussing their strategies and reasoning while using a variety of tools that appear to support their engagement with and access to the mathematical ideas.

#### Seeing Structure and Generalizing, K-5, 6-8, and 9-12

This set of three complementary modules uses student work samples from patterning tasks (Patterns with Walls K-5, Odd Number Patterns 6-8, and Sidewalk Patterns 9-12) as a stimulus for teacher conversations focused on mathematical structure. These modules are designed to help teachers learn to analyze student work without focusing solely on correct/incorrect solutions and provide a context for teachers to learn to recognize mathematical structures being used by students in their own classrooms.