Illustrative Math | Alegbra 2 | 2.1 Lesson
Summary
TLDRThis 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
📚 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.
📏 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.
📉 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.
📊 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.
📐 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.
Mindmap
Keywords
💡Algebra 2
💡Polynomial
💡Volume
💡Box
💡Function
💡Variable
💡Graphing Technology
💡Domain
💡Maximum Volume
💡A4 Paper
💡Unit
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
تصفح المزيد من مقاطع الفيديو ذات الصلة
5.0 / 5 (0 votes)