In this unit, students are formally introduced to the concept of a function as a relationship between “inputs” and “outputs” in which each allowable input determines exactly one output. Due to the ordering of units in IM 6–8 Math Accelerated v.360, students may have been exposed informally to function terminology earlier in this course.

First, students work with relationships that are familiar from previous grades or units (perimeter formulas, proportional relationships, linear relationships), expressing them as functions. They study the different ways functions can be represented, making connections between the representations and interpreting what they mean in context. The use of function notation is left for a future course.

Next, students analyze and describe cross-sections of prisms, pyramids, and polyhedra. They understand and use the formula for the volume of a right rectangular prism and solve problems involving area, surface area, and volume. Students should have access to their geometry toolkits so that they have an opportunity to select and use appropriate tools strategically.

Students build on their knowledge of the formula for the volume of a right rectangular prism, learning formulas for volumes of cylinders, cones, and spheres. Students express functional relationships described by these formulas as equations, focusing on situations involving proportional relationships. They use these relationships to reason about how the volume of a figure changes as one of its dimensions changes, transforming algebraic expressions to get the information they need. In future courses, students will continue this thinking as they study nonlinear relationships and question how, for example, the volume of a sphere changes as the radius increases.

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Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as comparing, explaining, and generalizing. 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:

Compare

• Different representations of functions (Lesson 3).
• Features of graphs, equations, and situations (Lesson 4).
• Features of a situation with features of a graph (Lesson 5).
• Temperatures shown on a graph with different temperatures given in a table (Lesson 6).
• Cross-sections of figures (Lesson 8).
• The volumes of cones with the volumes of cylinders (Lesson 16).
• Methods for finding and approximating the volume of a sphere as a function of its radius (Lesson 20).
• Characteristics of triangles and prisms (Lesson 22).

Explain

• How to find the volume of prisms (Lessons 9 and 10).
• Reasoning about finding the volume of a cylinder (Lesson 11)
• How to find the surface area of prisms (Lesson 13).
• Reasoning about the relationship between volumes of hemispheres and volumes of boxes, cylinders, and cones (Lesson 19)
• How to determine characteristics of triangles and prisms (Lesson 22)

Generalize

• About what happens to inputs for each rule (Lesson 1).
• About categories for cross-sections (Lesson 8).
• About dimensions of cylinders (Lesson 12).
• About the relationship between the volumes of cylinders and cones (Lesson 15).
• About dimensions of cones (Lesson 16).
• About volumes of spheres, cones, and cylinders as functions of their radii (Lesson 21).

In addition, students are expected to interpret representations of volume functions of cylinders, cones, and spheres. Students are also expected to describe the following: quantities in a situation, volume measurements and features of three-dimensional figures, the effects of varying dimensions of rectangular prisms and cones on their volumes, approximately linear relationships, and cross-sections of prisms and pyramids. Students are also expected to use language to represent relationships between volume and variable side length of a rectangular prism and relationships between volume and variable height of a cylinder.

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.