Major features of the CCSS include the idea that students should learn mathematics with *focus*, *coherence*, and *rigor*. We included activities throughout the project for teachers to develop their own understanding of focus, coherence, and rigor, and to support their understanding of how to incorporate focus, coherence and rigor in planning and implementation of lessons throughout Algebra 1.

The writers of the CCSS describe each of these features:

**Coherence** means attending to the structure of mathematics and the natural pathways through that structure, where “natural” means taking into account both the imperatives of logic and the imperatives of cognitive development in designing the sequence of ideas. Since these two imperatives are sometimes in conflict, attaining coherence is a complex exercise in judgment, requiring a certain amount of professional craft and wisdom of practice not easily obtained from any one source.

**Focus** means attending to fewer topics in greater depth at any given grade level, giving teachers and students time to complete that grade’s learning.

**Rigor** means balancing conceptual understanding, procedural fluency, and meaningful applications of mathematics. Here the word rigor is used not in the way that mathematicians use it, to indicate a correct and complete chain of logical reasoning, but in the sense of a rigorous preparation for a sport or profession: one that exercises all the necessary proficiencies in a balanced way.

### Coherence of Rate of Change

Teachers think about how students’ understanding of rates develops over three grades and how that prepares them to learn about average rates of change in contexts and multiple representations.

Rate_of_changeCoherence(FacilitatorNotes)

### Finding Coherence in Functions

This activity consists of two parts. In the first part, the goal is to better understand what is meant by coherence of the standards. Teachers work together to examine pairs of standards from the CCSS across grades to discuss how the function standards in one grade compare to those in the next grade, and to compare how standards in one grade compare to each other. The second part uses the idea of coherence as using prior knowledge and making connections between concepts and procedures as teachers examine three tasks related to the function standards to describe how students’ understanding of the standards may be developed in each of the tasks.

Finding_Coherence_in_Functions(Facilitator notes)

Finding_Coherence_in_Functions(Comparisons)

Finding coherence in functions(Answers)

### Justifying perpendicular lines

The goals of this activity are to expand the ideas of coherence to include connecting ideas across topics usually treated separately (like geometry and algebra) and to include the idea of ** logical necessity** in the context of justifying why the slopes of perpendicular lines are opposite reciprocals. This activity also incorporates the idea of inquiry into finding and developing an idea with coherence. Teachers heighten their awareness of coherence and finding coherence in the CCSS, and also start thinking about the inquiry mode of planning.