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.

Outlines

00:00

📚 Introduction to Polynomials and Box Modeling

The video begins with an introduction to a lesson on polynomials in the context of modeling the volume of a box formed by cutting squares from the corners of a sheet of paper. The instructor sets a learning goal to understand how to write and interpret a polynomial that models this volume. The dimensions of the paper are arbitrary, with the example using 10 by 15 units. The concept of a polynomial is explained as a function involving a sum of terms, each being a constant multiplied by a whole number power of x. Examples of polynomial functions are given, and the importance of not having negative or fractional exponents is highlighted. The lesson transitions into a warm-up exercise where the task is to identify which of several boxes does not belong based on given criteria, such as dimensions and volume.

05:02

📏 Constructing an Open Top Box from Paper

The instructor demonstrates how to construct an open-top box from a standard sheet of paper with dimensions 8.5 inches by 11 inches. The process involves cutting squares of side length x from each corner and then folding up the sides along dotted lines to form the box. The video explains how the dimensions of the box change as the size of the cut squares (x) increases, affecting the length, width, and height of the box. The volume of the box is calculated for different values of x, and the results are plotted to visualize the relationship between the side length of the cutouts and the volume of the box. The instructor emphasizes the practical limitations, such as not cutting squares larger than half the smallest side of the paper, to ensure that a box can still be formed.

10:07

📉 Maximizing Volume with Polynomial Functions

The video continues with an exploration of how to maximize the volume of the box by adjusting the size of the cutouts (x). The instructor uses graphing technology to plot the volume function and identify the maximum volume point, which corresponds to an optimal side length for the cutouts. The lesson discusses the importance of understanding the relationship between the paper's dimensions and the size of the cutouts to maximize the box's volume. The instructor also explains how to calculate the volume for different paper sizes, using an 8.5 by 11-inch sheet as an example, and how to adjust the calculations for other paper dimensions.

15:17

📊 Graphing and Finding the Maximum Volume

The instructor uses graphing technology to create a graph representing the volume function V(x) and approximates the value of x that allows for the construction of an open-top box with the largest volume. The graph helps visualize the maximum volume point and the corresponding side length of the cutouts. The lesson also discusses the strategy used to answer the question, which includes drawing a picture, labeling, and using knowledge about volume. The instructor emphasizes the importance of considering practical constraints, such as not cutting squares larger than half the smallest side of the paper, to ensure that enough material remains to form the box.

20:18

📐 Applying the Concept to Different Paper Sizes

The video concludes with an application of the concept to different paper sizes, including A4 paper, which is commonly used outside the United States. The instructor demonstrates how to calculate the volume function for an A4 sheet by cutting out squares of side length x from each corner. The domain for the side length of the cutouts is discussed, emphasizing that it must be less than half of the smallest side of the paper to ensure the box can be formed. The lesson provides a comprehensive understanding of how to apply polynomial functions to real-world problems like maximizing the volume of a box made from a sheet of paper.

Keywords

💡Algebra 2

Algebra 2 is a course that typically follows Algebra 1 and covers more advanced topics in algebra. It is part of the high school mathematics curriculum. In the context of the video, Algebra 2 is the subject area where the lesson on creating a box using polynomials is situated.

💡Polynomial

A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation. In the video, polynomials are used to model the volume of a box created by cutting squares out of a sheet of paper.

💡Volume

Volume refers to the amount of space occupied by an object. In the video, the concept of volume is central as it is calculated for a box formed by cutting squares from a sheet of paper. The formula for volume used is length times width times height.

💡Box

In the video, a box is created by cutting squares from the corners of a sheet of paper and folding up the sides. The dimensions of the box are determined by the size of the squares cut out and the original dimensions of the paper.

💡Function

A function in mathematics is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In the video, a polynomial function is used to represent the volume of the box as a function of the side length of the squares cut out.

💡Variable

A variable is a symbol, often a letter, used to represent an unknown value in mathematics. In the video, 'X' is used as a variable to represent the side length of the squares cut from the paper to form the box.

💡Graphing Technology

Graphing technology refers to tools or software that can plot functions or relations. In the video, graphing technology is suggested to be used to create a graph of the volume function to find the maximum volume of the box.

💡Domain

The domain of a function is the set of all possible input values (the 'X' values) which will produce a valid output from the function. In the video, the domain is discussed in terms of the possible values for 'X' that make sense given the physical constraints of the paper size.

💡Maximum Volume

Maximum volume refers to the largest possible volume that the box can have. In the video, the goal is to find the side length of the squares to cut out that will result in a box with the maximum volume.

💡A4 Paper

A4 is a standard size of paper used in many parts of the world. In the video, an example is given where the dimensions of an A4 sheet of paper are used to calculate the volume of a box that can be formed from it.

💡Unit

In the context of the video, a unit refers to a section or chapter in a course or textbook. 'Algebra 2 unit two lesson one' indicates the specific part of the curriculum where the lesson on making a box is located.

Highlights

Introduction to the concept of creating a box by cutting squares from the corner of a sheet of paper.

Definition of a polynomial function and its relation to the volume of the box.

Explanation of how the dimensions of the paper sheet are irrelevant for the initial setup.

Demonstration of cutting squares (X) from the corners of the paper to form a box.

Description of the process to fold the sides of the paper to create the box.

Introduction of the polynomial equation that models the volume of the box.

Warm-up exercise to identify which box dimensions do not belong based on volume calculations.

Explanation of how to calculate the volume of the box using length, width, and height.

Activity to construct an open-top box from a standard sheet of paper by cutting out squares.

Calculation of the box's dimensions after cutting out squares of side length one inch.

Incremental increase in the side length of the cut squares and the effect on the box volume.

Graphical representation of the volume as a function of the side length of the cut squares.

Identification of the maximum volume and the corresponding side length of the cut squares.

Discussion on the practical limitations of the side length of the cut squares based on the paper size.

Application of the concept to different paper sizes, such as A4, and determining a reasonable domain for the side length.

Final activity involving a random paper size and the calculation of the volume function.

Conclusion summarizing the lesson's key points and the importance of understanding polynomial functions in relation to real-world applications.

Transcripts

okay Algebra 2 unit two lesson one is

called let's make a

box all right so our learning goal today

is I can write and interpret a polom

that models the volume of a box that is

created by cutting squares out of the

corner of a sheet of paper so basically

what is going to happen here is we're

going to have a sheet of paper here and

you know the dimensions on this sheet of

paper don't really matter um let's say

I'm just going to use whole numbers here

I'm gonna say this side is

10 and let's say this side is I don't

know

15 so if I cut squares

out all right and I call these X that

would mean they all have a length of

X we're going to basically cut these out

so we're gonna you know rip those out

and we're g to make a box by folding

these dotted lines up okay we're going

to fold these sides up fold all these

sides up and tape it all together and

it'll form a

box so we're going to be able to come up

with a polom from that

situation all right so moving

on so first of all what is a polinomial

it's part of our learning goal

polom is a function of X polinomial

function of X is a function given by a

sum of terms Each of which is a constant

times the whole number power of X the

word polom is used to refer both to the

function and to the expression defining

it so an

example I call F ofx my

function 3x^2 + 2x -

one I got another one g of x 4x 3r + 2x

- 1 all right those are some examples

notice you

have um a constant

times x to a

power constant times x to a power a

constant really it's times x to the zero

but we don't need to put the times x to

the 0 but it's

there all right so that's a polinomial

there notice don't have negative

exponents notice you don't have fraction

exponents um and you add up all the

terms all right so let's start with our

warm-up which one doesn't belong so try

to come up with a reason for all of them

um a would be obvious because a does not

have a picture so if you pick a we'd say

that's because there's no

picture um if you check the volume all

of them it's length times width times

height so if you go ahead and multiply

them all out let's see here 4 * 8 * 10

this one has a volume of

320 this one would be 10 * 2 * 8 this

volume equals 160 cubic

cm this one here is 320 as well

and then this one they tell you the

volume so I can't really multiply it out

because I don't have numbers there but

right away I could say B is the only

one with

volume not

equal to

320 Cub

cm

if I go to C uh you could probably say

since this one's oh you know what I was

wrong that's not centimeters that's

inches cubic inches so right away there

on that one you could say oh C's the

only one in

inches and then this one you could say

it's the only

one with

variables for the

sides now luckily we only have one

variable the only variable there is X

now this is

4X and then this one here is X and then

this one's X plus one but luckily we

don't have x y z we don't have multiple

variables so it's kind of

nice all right so on to the first

activity so if you were hearing class we

would be doing this with like a piece of

paper and scissors and a ruler and stuff

like that but I'm going to try my best

to do it here without that so um so

basically we're going to construct an

open top box from a sheet of paper by

cutting out a square from each corner

and then folding up the sides so um on a

standard sheet of

paper um this side is 8.5

in and this side

here is 11 in okay so if you were in

class I'd be giving you a sheet a paper

that was 8 and2 by 11 and what we're

going to do is we're going to cut out

squares in the corner okay so I'm going

to make a little square there let's see

what I'm

doing

same oh here it is

follow so I cut these squares

out and basically we're going to start

with the side length of one if you look

right there it says one to start so I'm

going to basically label this as

one by

one all right so what happens is when I

go to make this

box you're G

to cut out those corners and these

dotted lines are are going to be the

folding lines and these pieces right

here are going to all get folded up so

I'll use some colors here these get

folded

up these get folded

up okay and then the little squares that

we drew get

um get thrown out so we get a box down

here that looks like looks like this

okay um so what happens is when you fold

it up if you look here this side is this

side right here is going to go with that

so what is that side well it was 11 to

start but you're taking off one and one

so this side's going to have a length of

nine okay and then this side over

here all right this side had a length of

eight and a half but you're cutting out

one and one so this side is going to be

six at half right there

then if you look at the height when I

fold this flap up really you fold all

these flaps up here okay when you fold

them up that's going to have a height of

one so what's the length the width the

height so length will be nine the width

will be

6.5 and the height will be

one so will be the volume we got to

multiply those

together

9 * 6.5 * 1 that's

58.5 so I'm going to just change it I'm

gonna make this a

two right so all I'm going to do here is

I'm going to just change this these

numbers here to two I'm not going to

redraw the picture I'm just going to

change the numbers and we can kind of

mentally just do it so these are two by

two

boxes remember this was 11 right here

okay so I'm basically I'm subtracting

two on the top and two on the bottom so

that blue side

there is now going to be 11 - 2 - 2 so

that's going to be

seven the green side right here is going

to be 82 - 2 -

2 so 8 and half minus 4

is

1.5 and then when you fold up the flap

that's going to be two

now right there so I can fill everything

in here height is

two CH

blue go length is

seven this is 4.5 so let's multiply

those together for volume length time

width time height

2 *

63 and let's go ahead let's try

three so I'm going to just change these

twos to

three these little flaps are three

now pull them

up

this side here with the blue this side

of the box now becomes 11 minus 3 -

3 11 - 6 would be

5 the green

side over here was 8 and a half up top

so the bottom is 8 and 1 half as well 8

and2 - 3 -

3 because it's really the whole thing is

8.5 all the way across but you're you're

basically this is 8.5 right here it's

like you cut off this piece which is

three and this piece which is three so

it's like you subtracted

six so that would be

2.5 and then your flap that you fold up

would

be oops to be

so your volume is going to be length

time width time

height so go ahead and multiply

those gives

me

37.5

okay so if you kind of plot those if you

look your side length of the

cutout we got

one as a volume of 50

8.5 two has a volume of

63 and three has a volume of

37.5 so it looks something like that

okay now we were in class we might do

some more numbers I could give different

groups different sizes but basically

what's happening is you get a shape

somewhat like that okay so the key is

since volume is on the Y AIS we want the

maximum volume so the maximum volume is

going to be right around at that point

which would have a side cut out of

like

1.75

roughly all

right all right so the volume v of X in

cubic inches of an open top box is a

function of the side length x inches in

inches of the square cutouts make a to

figure out how to construct the box with

the largest

value okay

so basically what happens here

is what you need to

know is you need to

know the length of the sheet of paper so

we're going to keep the same numbers

we're going to go eight and a half by

11 and when you start cutting these

squares out

here of the

corners we could just call these

x x byx x byx x byx okay and what

happens is the volume when you make your

box your

volume it's always going to be you know

the side that's 11 over here this is

going to be 11 minus two x's because

it's like you cut these X's off and got

rid of them the bottom side is going to

be eight and a half but then you're

minusing 2x because you get rid of two

of them and then when you always fold up

the flap the flap is always going to be

X

so I guess I kind of did one and two

together um and and it'll always kind of

be the same exact thing anytime you go

to make a box um so you

know let's just for the heck of

it let's say that I

had different sides here okay let's say

that this paper was different dimensions

let's say this side was 20 and this side

was 15 if that was the case your volume

B of X is going to be 20 -

2x 15 - 2x

X time x it's pretty much always going

to follow that pattern with a sheet of

paper sheet of cardboard or

whatever

okay all right let's go on the third

question use graphing technology to

create a graph representing V ofx so I

went back to the original numbers 11 and

the 8 and

A2 right um approximate the value of x

to allow to construct an open top box

with the largest value so we want to

find the part on the graph that is a

maximum so I plotted my function that I

had on

decimos and my maximum is right here

that point so this axis is

volume and this

is um oops this

is side length of a

square okay so obviously my maximum

volume is

66 and the side length that would would

work for us would be 1

585 all right so you're looking for that

maximum why did I pick that point

because it's right at the top you see we

go up and then we come down that's a Max

all right now you might say Well it goes

higher up here it does but six inches is

too big of a square to cut out you won't

have enough paper left over to make make

a box when you fold it

up all right so what strategy strategy

strategy did you use to answer the

question um I would say we drew a

picture

label

use knowledge about

volume right what are some side lengths

for the square cutouts that don't make

sense so that was kind what I was

talking about earlier so you got your

picture here and basically

whoops you know this side's

8.5 this side's

11 you know I can't go over half of the

smallest side because look if what's

eight and a half divided by

two 8.5 / two is what that

4.25 I can't make a square that's bigger

than

4.25 because you

know if my Square

here is

4.25 and I cut

them there's not going to be any box

left to fold up it's gonna basically I'm

G to cut it

into right

here I cut it right there there won't be

a side left on that

um on that piece to make a box so

basically your smallest

side I should

say we can

only use

less than half

of the small s side okay because we just

won't have any left to make a box now

this was four you know this is eight and

a half

oops you know if I made these little

smaller if I made

these you know four

and

four we would have you know right here

we'd have 05 left because that if you

add up all those sides you add up this

plus this plus that it'd be 8.5 but you

got to have less than half of the small

side

okay all right let's take a look at this

one outside the UN United States the

common paper size is called A4 it

measures 21 by 29.7 CM let V of x equal

21- 2x 29.7 - 2x and X or times 21 - 2x

* 29.7 - 2x x x be the volume in cubic

centimeters of the

Box made from A4 paper by cutting out

squares of side length x and centimeters

from each corner and then p in theze

what is a reasonable domain so the

smallest side here so this is kind of

the same thing that we just did the

smallest side would be

21 whereas the longer side would be

29.7

so half of the smallest side

so you could say here small

side equals 21

in long

side

equals 29.7 in so half of the smallest

side would be 10.5 so we can only

cut and only

cut squares

less than 10.5

in okay it's got to be half of the small

side or

less all

right so you know let's do one more

thing just to make sure we got this

so if I give

you my sheet of paper looks like this

I'm going to make up some random numbers

here let's say this side is

[Music]

32

in let's say this side is

18

in so you cut out your

boxes x byx x

[Music]

byx x

byx x byx so your function volume

function is going to be 32 - 2x * 18 -

2x * X all right it's always going to be

side length minus 2X and

then times X for the

height so your

domain so take your smaller side which

is

18 so X has to be less than half of that

so less than

nine okay

18 / two is 9 so your side lengths have

to be smaller than than

n all right so there you

go hope you enjoyed the lesson