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## Learning Trajectories for Vertical Coherence

Three learning trajectories and their connections show how to promote vertical coherence in PK–12 mathematics education.

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## Using TileFarm to Support Emerging Multiplication

Symmetric dot patterns are a particularly powerful object for investigation, providing opportunities for foundational learning across PK–5. We found that second-grade students naturally used repeated addends to count symmetric dot patterns created using the new software TileFarm.

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## Using Artwork to Explore Proportional Reasoning

By examining ratios in paintings and using a free educational app, students can size up artists' use of proportional reasoning in their creations.

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## Chivalry, at Least “Math Chivalry,” Is Not Dead!

A cartoon exploring a problem about order of operations is coupled with a full-page activity sheet.

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## Questioning the Order of Operations

Abandon mnemonics and make stronger connections between the operations and properties of arithmetic.

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## Selecting Systems of Linear Equations

Students make strategic choices–and justify them–to solve a system of two linear equations.

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## Early Algebra: Expressing Covariation and Correspondence

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## Exploring Geometric Sequences

Use popular culture to draw students' attention to mathematical topics.

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## Factor and Remainder Theorems: An Appreciation

Core content provides opportunities to focus on the structure of mathematical theory, proof, and anticipation of subsequent topics.

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## Mystery Fractions

This activity engages students in problem solving while exploring key concepts of number theory, such as divisibility and divisibility tests, place value, fractions, and scale factors.