# Lesson 12

Filling Containers

### Lesson Narrative

This lesson is the beginning of a sequence of lessons that interweaves the development of the function concept with the development of formulas for volumes of cylinders and cones. Because students have not yet learned these formulas, the context of filling a cylindrical container with water is useful for developing the abstract concept of function. It makes physical sense that the height of the water is a function of its volume even if we cannot write down an equation for the function. At the same time, considering how changing the diameter of the cylinder changes the graph of the function helps students develop a geometric understanding of how the volume is related to the height and the diameter. In later lessons they will learn a formula for that relation.

In this lesson, students fill a graduated cylinder with different amounts of water and draw the graph of the height as a function of the volume. They next consider how their data and graph would change if their cylinder had a different diameter. The following activity turns the situation around: when given a graph showing the height of water in a container as a function of the volume of water in the container, can students create a sketch of what the container must look like?

Teacher Notes for IM 6–8 Math Accelerated
Due to the arrangement of lessons in IM 6–8 Math Accelerated, students will develop the formulas for volume and surface area of prisms prior to their work on cylinders and cones as stated in the lesson narrative.

### Learning Goals

Teacher Facing

• Create a graph of a function from collected data, and interpret (in writing) a point on the graph.
• Draw a container for which the height of water as a function of volume would be represented as a piecewise linear function, and explain (orally) the reasoning.
• Interpret (orally and in writing) a graph of heights of certain cylinders as a function of volume, and compare the rates of change of the functions.

### Student Facing

Let’s fill containers with water.

### Required Preparation

Students work in groups of 3–4 for the activity Height and Volume. Each group needs 1 graduated cylinder and water.

### Student Facing

• I can collect data about a function and represent it as a graph.
• I can describe the graph of a function in words.

Building Towards

### Glossary Entries

• base (of a prism or pyramid)

The word base can also refer to a face of a polyhedron.

A prism has two identical bases that are parallel. A pyramid has one base.

A prism or pyramid is named for the shape of its base.

• cross section

A cross section is the new face you see when you slice through a three-dimensional figure.

For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.

• cylinder

A cylinder is a three-dimensional figure like a prism, but with bases that are circles.

• prism

A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.

Here are some drawings of prisms.

• pyramid

A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.

Here are some drawings of pyramids.