This task sequence for adding and subtracting like terms—grounded in the concepts of equivalence and algebra as generalized arithmetic—helps students see connections between concepts and procedures in algebra.

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## Building Algebraic Procedures From Concepts: Like Terms

### Leah M. Frazee and Adam R. Scharfenberger

## Characterizing Secondary Teachers’ Structural Reasoning

### Stacy Musgrave, Cameron Byerley, Neil Hatfield, Surani Joshua, and Hyunkyoung Yoon

The Common Core State Standards for Mathematical Practice asks students to look for and make use of structure. Hence, mathematics teacher educators need to prepare teachers to support students’ structural reasoning. In this article, we present tasks and rubrics designed and validated to characterize teachers’ structural reasoning for the purposes of professional development. Initially, tasks were designed and improved using interviews and small pilot studies. Next, we gave written structure tasks to over 600 teachers in two countries and developed and validated rubrics to categorize responses. Our work contributes to the preparation and support of mathematics teachers as they develop their own structural reasoning and their ability to help students develop structural reasoning.

## Understanding Algebraic Expressions Through Figural Patterns

### Casey Hawthorne and John Gruver

This instructional sequence develops your students’ meaningful understanding of algebraic expressions.

## GPS: Making Meaningful Use of Structure in PK–12

### Katherine Ariemma Marin and Natasha E. Gerstenschlager

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.

## Delving Deeper: What Else Comes after 1 + 2 = 3?

### Charles F. Marion

The simplest of prekindergarten equations, 1 + 2 = 3, is the basis for an investigation involving much of high school mathematics, including triangular numbers, arithmetic sequences, and algebraic proofs.

## GPS: Interpreting Whole-Number Remainders

### Michelle T. Chamberlin and Robert A. Powers

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.

## Supporting Understanding Using Representations

### Eric Cordero-Siy and Hala Ghousseini

Three deliberate teaching practices can help students strengthen multiple connections to a unifying concept.

## Area of a Changing Triangle: Piecing It Together

### Blake Peterson

Examining the covariation of triangle dimensions and area offers a geometric context that makes analyzing a piecewise function easier for students.

## “Counting” on Quantitative Reasoning for Algebra

### Lori Burch, Erik S. Tillema, and Andrew M. Gatza

Use this approach to developing algebraic identities as a generalization of combinatorial and quantitative reasoning. Secondary school students reason about important ideas in the instructional sequence, and teachers consider newfound implications for and extensions of this generalization in secondary algebra curricula.

## The Structure of the Quadratic Formula

### Kristin Frank

Lessons that focus on a conceptual understanding offer an opportunity for students to learn about mathematical structure, not just computation.