## Design Principles

### Developing Conceptual Understanding and Procedural Fluency

Each unit begins with a pre-assessment that helps teachers gauge what students know about both prerequisite and upcoming concepts and skills, so that teachers can gauge where students are and make adjustments accordingly. The initial lesson in a unit is designed to activate prior knowledge and provide an entry point to new concepts, so that students at different levels of both mathematical and English language proficiency can engage productively in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. Practice problems, when assigned in a distributed manner, give students ongoing practice, which also supports developing procedural proficiency.

### Applying Mathematics

Students have opportunities to make connections to real-world contexts throughout the materials. Frequently, carefully-chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. Many units include a real-world application lesson at the end. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on mathematical contexts. Additionally, a set of mathematical modeling prompts provide students opportunities to engage in authentic, grade-level appropriate mathematical modeling.

### Use of Digital Tools

These curriculum materials empower high school teachers and students to become fluent users of widely-accessible mathematical digital tools to produce representations to support their understanding, solve problems, and communicate their reasoning.

Digital tools are included when they are required by the standard being addressed and when they make better learning possible. For example, when a student can use a graphing calculator instead of graphing by hand, use a spreadsheet instead of repeating calculations, or create dynamic geometry drawings instead of making multiple hand-drawn sketches, they can attend to the structure of the mathematics or the meaning of the representation.

Lessons are written with three anticipated levels of digital interaction: some activities require digital tools, some activities suggest digital tools, and some activities allow digital tools. In a few cases, activities may prohibit digital tools if they interfere with concept development.

In most cases, instead of being given a pre-made applet to explore, students have access to a suite of linked applications, such as graphing tools, synthetic and analytic geometry tools, and spreadsheets. Students (and teachers) are taught how to use the tools, but not always told when to use them, and student choice in problem-solving approach is valued.

When appropriate, pre-made applets may be included to allow for students to practice many iterations of a skill with error checking, to shorten the amount of time it takes students to create a representation, or to help students see many examples of a relationship in a short amount of time.

### The Five Practices

Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014). These activities include a presentation of a task or problem (may be print or other media) where student approaches are anticipated ahead of time. Students first engage in independent think-time followed by partner or small-group work on the problem. The teacher circulates as students are working and notes groups using different approaches. Groups or individuals are selected in a specific, recommended sequence to share their approach with the class, and finally the teacher leads a whole-class discussion to make connections and highlight important ideas.