Illustrative Math | Alegbra 2 | 2.1 Lesson

Summary

This Algebra 2 lesson focuses on creating a polynomial function to model the volume of a box made by cutting squares from the corners of a sheet of paper and folding it up. The instructor explains the concept of a polynomial and provides examples. They then guide through a hands-on activity using an 8.5 by 11-inch sheet to demonstrate how cutting squares of side length 'x' affects the box's dimensions and volume. The lesson explores the relationship between the side length of the cutouts and the box volume, aiming to find the optimal 'x' for maximum volume. The instructor also discusses the application of this concept to different paper sizes, including A4, emphasizing the importance of the domain in polynomial functions.

Takeaways

• 📐 The lesson focuses on creating a polynomial that models the volume of a box made by cutting squares from the corners of a sheet of paper and folding it up.
• 🔢 A polynomial is defined as a function of X, which is a sum of terms, each being a constant times the whole number power of X.
• 📚 The lesson includes examples of polynomial functions, such as f(x) = 3x^2 + 2x - 1 and g(x) = 4x^3 + 2x - 1, emphasizing that polynomials do not have negative or fractional exponents.
• 📏 The process of making a box involves cutting squares of side length 'x' from each corner of a sheet of paper and then folding the sides to form the box.
• 📊 The volume of the box is calculated by multiplying the length, width, and height of the box, which are determined by the original dimensions of the paper and the size of the squares cut out.
• 📉 The lesson includes a warm-up activity where students identify which of several boxes does not belong based on volume calculations and other criteria.
• 📈 The instructor demonstrates how to graph the volume function and find the maximum volume by identifying the peak on the graph, which corresponds to the optimal size of the squares to cut out.
• 🔍 The lesson discusses the importance of considering the domain of the function, which is limited by the size of the paper and the requirement that the squares cut out must be less than half the length of the smallest side of the paper.
• 🌐 The lesson also touches on the practical application of the concept by considering different paper sizes, such as A4 paper, and how it affects the volume calculation and the domain of possible 'x' values.
• 📝 The lesson concludes with a reminder to consider the practical constraints when applying the mathematical model to real-world situations, such as ensuring enough paper remains to form the box after cutting out the squares.

Q & A

What is the learning goal of the Algebra 2 unit two lesson one?

The learning goal is to write and interpret a polynomial that models the volume of a box created by cutting squares out of the corner of a sheet of paper.

What is a polynomial function according to the lesson?

A polynomial function is a function given by a sum of terms, each of which is a constant times the whole number power of x.

What are the characteristics of a polynomial function?

A polynomial function does not have negative exponents or fraction exponents, and it is the sum of terms where each term is a constant multiplied by x raised to a whole number power.

How does the lesson differentiate between different types of boxes in the warm-up activity?

The lesson differentiates boxes by their dimensions, units (cubic cm vs. cubic inches), and whether they have variables for the sides.

What is the process of creating a box from a sheet of paper as described in the lesson?

The process involves cutting squares out from each corner of a sheet of paper, folding up the sides along dotted lines, and taping them together to form a box.

What are the dimensions of the standard sheet of paper used in the lesson for creating a box?

The standard sheet of paper used is 8.5 inches by 11 inches.

How does the lesson determine the volume of the box created from the sheet of paper?

The volume of the box is determined by multiplying the length, width, and height of the box, which are calculated based on the side length of the squares cut out from the corners.

What is the maximum volume found in the lesson for the box created from an 8.5 by 11-inch sheet of paper?

The maximum volume found is 66 cubic inches, which occurs when the side length of the square cutouts is approximately 1.585 inches.

Why can't the side length of the square cutouts be more than half of the smallest side of the paper?

If the side length of the square cutouts is more than half of the smallest side of the paper, there won't be enough paper left to fold up and form a box.

What is the domain for the side length of the square cutouts when using A4 paper?

For A4 paper, which measures 21 cm by 29.7 cm, the domain for the side length of the square cutouts is less than half of the smallest side, which is 10.5 cm.