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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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## Partial Pies

A series of tasks encourage students to reflect on the reasonableness of their number sense and use benchmarks to refine their estimations.

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## Global Connections through Mathematical Problem Solving

Teachers from two countries designed a model-eliciting activity about the global issue of wind energy. They share teaching and student outcomes from a cross-border engagement in the task with students from Indonesia and the United States through synchronous video conference.

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## One Task, Multiple Proportional Reasoning Strategies

Tasks and materials that allow for different approaches can help teachers incorporate student reasoning and can promote connections across different mathematical ideas.

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## Extending Choral Counting

This instructional activity gives teachers access to student thinking that can be leveraged to extend and connect their ideas.

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## Unit Fractions as Superheroes for Instruction

The complexity of understanding unit fractions is often underappreciated in instruction. We introduce a continuum of children's understanding of unit fractions to explore this complexity and to help teachers make sense of children's strategies and recognize milestones in the development of unit-fraction understanding. Suggestions for developing this understanding are provided.

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## Using Scratch Programming to Explore Coordinates

We present a Scratch task we designed and implemented for teaching and learning coordinates in a dynamic and engaging way. We use the 5Es framework to describe the students' interactions with the task and offer suggestions of how other teachers may adopt it to successfully implement Scratch tasks.

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## Exploring the Mathematics of Gravity

We provide an example from our integrated math and science curriculum where students explore the mathematical relationships underlying various science phenomena. We present the tasks we designed for exploring the covariation relationships that underlie the concept of gravity and discuss the generalizations students made as they interacted with those tasks.

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## Is the Last Banana Game Fair?

Is the “Last Banana” game fair? Engaging in this exploration provides students with the mathematical power to answer the question and the mathematical opportunity to explore important statistical ideas. Students engage in simulations to calculate experimental probabilities and confirm those results by examining theoretical probabilities