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## Area of a Changing Triangle: Piecing It Together

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

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## “Counting” on Quantitative Reasoning for Algebra

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.

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## The Structure of the Quadratic Formula

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

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## Structuring Boardspace to Facilitate Repeated Reasoning

The planful use of boardspace can help move the structure and regularity to the visual realm and make it more readily perceivable by students.

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## Improving Understanding of Mathematical Equivalence

Modify arithmetic problem formats to make the relational equation structure more transparent. We describe this practice and three additional evidence-based practices: (1) introducing the equal sign outside of arithmetic, (2) concreteness fading activities, and (3) comparing and explaining different problem formats and problem-solving strategies.

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## GPS: Pattern Recognition to Explore Mathematical Structure

In this month’s Growing Problem Solvers, we aimed to help students explore patterns where they pay attention to the mathematical structures behind those patterns.

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## Decision Tree

This poem starts with the question in the trunk of the tree, where we imagine that we are deciding to do or not to do something. Each level represents steps in making the decision, with the top indicating a resolution in the future. Phrases wander and change direction, leading to different results. How many paths to a resolution do you see?