This optional unit consists of eleven lessons. The first section has four lessons about exploring our world. These lessons are independent of each other and explore working with estimation and decimals or large numbers. These lessons should be taught after Unit 3. The second section has five lessons about different systems of voting. These lessons build on each other and should be completed in order after Unit 3 has been taught. The final section has two lessons making connections between algebraic and geometric representations of topics in earlier units. This section should be taught after Unit 8 and the lessons should be completed in order.

All related standards in this unit have been addressed in prior units. These sections provide an optional opportunity for students to go more deeply and make connections between domains.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as critiquing, justifying, and “comparing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Critique

• Reasoning about Fermi problems (Lesson 1).
• Claims about percentages (Lesson 5).
• Reasoning about the fairness of voting systems (Lessons 8 and 9).

Justify

• Reasoning about Fermi problems (Lesson 1).
• Reasoning about the fairness of voting systems (Lessons 6, 7, 8, and 9).

Compare

• Sources of energy (Lessons 2 and 3).
• Rectangles and fractions (Lesson 10).
• Voting systems (Lessons 6 and 7).

In addition, students are expected to interpret and represent characteristics of the world population, describe distributions of voters, and generalize about decomposition of area and numbers.

The table shows lessons where new terminology is first introduced in this course, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.