# Lesson 1

Let’s Make a Box

### Lesson Narrative

The work of this lesson asks students to investigate the volumes of open-top boxes made from single sheets of paper. This work provides students with a path into polynomial functions, because the volume of the box is a function of the side length of the square cut from each corner. The size of the square cutouts affects all three dimensions of the box, which in turn leads students to writing a cubic expression for the volume in the form of length times width times height. This example can be a visual reference for the meaning of polynomial functions that students can refer back to throughout the unit.

Students graph their volume functions and interpret different input-output pairs as they use the graph to estimate the input with the greatest output (MP2). In future lessons, students will learn more precise language, like relative maximum and degree, to describe graphs and equations of polynomials with precision. Throughout the unit, students will make connections between equivalent forms of polynomial functions and the graph of the function.

In the last activity of this lesson, students are introduced to the word polynomial and told they will learn about other situations polynomials can model in future lessons. A polynomial function of $$x$$ is a function given by a sum of terms, each of which is a constant times a whole number power of $$x$$. The word polynomial is used to refer both to the function and to the expression defining it. More vocabulary about polynomials is established in future lessons. Students will use these concepts to better understand the meaning of polynomial functions in context and to draw conclusions about them.

### Learning Goals

Teacher Facing

• Generalize from specific calculations to create a polynomial to model the volume of a constructed box.
• Interpret features of a graph of a polynomial.

### Student Facing

• Let’s investigate volumes of different boxes.

### Required Preparation

Have rulers, scissors, sheets of paper, and tape available for groups of 2 to use to make their boxes. Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

### Student Facing

• I can create and interpret a polynomial that models the volume of a box.

### CCSS Standards

A polynomial function of $$x$$  is a function given by a sum of terms, each of which is a constant times a whole number power of $$x$$. The word polynomial is used to refer both to the function and to the expression defining it.