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

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.

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

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.

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## GPS: Interpreting Whole-Number Remainders

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.

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## Supporting Understanding Using Representations

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

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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.