Unit 5

Functions and Volume

Unit Narrative

In this unit, students are introduced to the concept of a function as a relationship between “inputs” and “outputs” in which each allowable input determines exactly one output.

In the first three sections of the unit, 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. Linear functions are a focus of the third section, and students...

to access Pre-Unit Assessments.

Section A

Section Goals

Section Narrative

This section introduces students to functions and develops the idea of a function as a rule that assigns to each allowable input exactly one output.

A focus of this section is on examples of “input–output rules,” such as “Divide by 3” or “If even, then . . . . If odd,
then . . . .” In the starting examples,...

Let’s make some rules.

Let’s learn what a function is.

to access Section Checkpoints.

Section A

Let’s make some rules.

Let’s learn what a function is.

Section B

Section Goals

Section Narrative

In this section, students connect the terms “independent variable” and “dependent variable” (which they learned in grade 6) with the inputs and outputs of a function. They use equations to express a dependent variable as a function of an independent variable, viewing formulas from earlier grades (for example, ) as determining functions. They work with tables, graphs, and equations for...

Let’s find outputs from equations.

Let’s connect equations and graphs of functions.

Let’s interpret graphs of functions.

Let’s draw a graph from a story.

Let’s connect tables, equations, graphs, and stories of functions.

to access Section Checkpoints.

Section B

Let’s find outputs from equations.

Let’s connect equations and graphs of functions.

Let’s interpret graphs of functions.

Let’s draw a graph from a story.

Let’s connect tables, equations, graphs, and stories of functions.

Section C

Section Goals

Section Narrative

In this section, students use linear and piecewise linear functions to model relationships between quantities in real-world situations, interpreting information from graphs and other representations in terms of the situations. The lessons provide an opportunity for students to coordinate and synthesize their understanding of new and old terms that describe aspects of linear and piecewise linear functions.

Students begin the...

Let’s investigate linear functions.

Let’s model situations with linear functions.

Let’s explore functions built out of linear pieces.

to access Section Checkpoints.

to access Mid-Unit Assessments.

Section C

Let’s investigate linear functions.

Let’s model situations with linear functions.

Let’s explore functions built out of linear pieces.

Section D

Section Goals

Section Narrative

In this section, students extend their understanding of volume from right prisms to right cylinders and right cones.

Students begin by investigating the volume of water in a graduated cylinder as a function of the height of the water, and vice versa. They examine depictions of cylinders, prisms, spheres, and cones in order to develop their abilities to identify radii,...

Let’s fill containers with water.

Let’s reason about the volume of different shapes.

Let’s explore cylinders and their volumes.

Let’s figure out the dimensions of cylinders.

Let’s explore cones and their volumes.

Let’s figure out the dimensions of cones.

to access Section Checkpoints.

Section D

Let’s fill containers with water.

Let’s reason about the volume of different shapes.

Let’s explore cylinders and their volumes.

Let’s figure out the dimensions of cylinders.

Let’s explore cones and their volumes.

Let’s figure out the dimensions of cones.

Section E

Section Goals

Section Narrative

The focus of this section is establishing the formula for the volume of a sphere. The first lesson invites students to use equations and graphs of proportional relationships to make sense of what happens to the volumes of familiar shapes when one dimension changes. A goal of this work is for students to reason about volume using equations, which will...

Let’s see how changing one dimension changes the volume of a shape.

Let’s change more dimensions of shapes.

Let’s estimate volumes of hemispheres with figures we know.

Let’s explore spheres and their volumes.

Let’s find the volume of shapes.

to access Section Checkpoints.

Section E

Let’s see how changing one dimension changes the volume of a shape.

Let’s change more dimensions of shapes.

Let’s estimate volumes of hemispheres with figures we know.

Let’s explore spheres and their volumes.

Let’s find the volume of shapes.

Section F

Section Narrative

The lesson in this section is optional because it offers additional opportunities to practice standards that are not a focus of the grade.

Let’s compare water heights in different containers.

to access Section Checkpoints.

Section F

Let’s compare water heights in different containers.

to access End-of-Unit Assessments.

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Functions and Volume

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