teacher road map for lesson 1: modeling with polynomials ... · pdf filemodeling – in...

Hart Interactive – Algebra 2 M1 Lesson 1 ALGEBRA II

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Teacher Road Map for Lesson 1: Modeling with Polynomials – The Cardboard Box Problem

Objective in Student Friendly Language:

Today I am: creating an open box and analyzing the effect the height has on the volume

So that I can: graph a mathematical model for the volume.

I’ll know I have it when I can: analyze other mathematical graph models.

Flow: o Students discuss what are important characteristics of cardboard boxes. o In groups, students create an open box by cutting out squares from each corner of a

rectangular grid. o The class collates their data to determine the box with the greatest volume. o Students graph the height versus volume on a grid and analyze the graph.

Big Ideas: o This is an introduction to polynomial functions. Although students may describe the graph as

parabolic in shape, don’t be too quick to point out that the graph is not symmetric. In the next lesson, students will determine the equation of the graph and see that it could not be quadratic.

Materials Needed: o Each group needs one Grid Sheet Handout, one pair of scissors, and tape.

CCS Standards:

o Modeling – In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs of observations are a familiar descriptive model—for example, graphs of global temperature and atmospheric CO2 over time.

Lesson 1: Modeling with Polynomials—The Cardboard Box Problem

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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US





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Student Outcomes Students transition between verbal, numerical, algebraic, and graphical thinking in applying a real

world polynomial problem.

Lesson Notes

Creating an open-topped box of maximum volume is a very common problem seen in calculus. The goal is to optimize resources by enclosing the most volume possible given the constraint of the size of the construction material; here, students use grid paper. You may want to copy the grid handout onto cardstock for a sturdier box. This is the first part of a two-day lesson on modeling. Lesson 1 focuses more on students analyzing the graph of the function.

Classwork

Opening (5 minutes)

We start with an open questions about the ordinary cardboard box. You may wish to have students watch a 7 minute video on the history of packaging design on YouTube at https://www.youtube.com/watch?v=7Shwzu0VpQg.

Discussion (5 minutes)

1. What is the most important characteristic of a cardboard box?

Possible answers include: Durability, volume, amount of materials, weight, dimensions (ease of holding), esthetics

Mathematical Modeling Exercise (30 minutes)

Suppose your company, EcoBox, is looking to create an open topped box with the greatest volume possible. As lead designers, your group is tasked with determining what the dimensions should be. The grid you are starting with is 25 by 13.

The net, pattern for the open box, is shown on the right. Each group will be assigned a specific size square to cut out of each corner. Then we’ll gather the data from each group to determine the dimensions for the box with the greatest volume.

25

13


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https://www.youtube.com/watch?v=7Shwzu0VpQg


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2. Make a prediction about the dimensions that will give the greatest volume. Remember we are limited to a 25 by 13 grid.

Answers will vary.

My dimensions guess: _______ length; _______ width; _______ height; _______ volume

B. Why would a company want the greatest volume of a box?

Answers may vary. Sample response: They have a specific amount of materials they want to box up.

Each group has a handout of a grid that measures 25 x 13 units. Students will cut out congruent squares from each corner, and fold the sides in order to create an open-topped box. The goal is to create a box with the maximum possible volume.

Your group will need: Grid Sheet Handout, scissors, tape

3. A. With your group, create a box from the Grid Sheet Handout with your assigned size of square to use. Cut out the 25 x 13 grid on the handout, and then cut out congruent squares from each corner. Fold the sides to create an open-topped box as shown in the figure below. Pro Tip: It is easier to determine the dimensions if the grid lines are on the outside of the box. Finally, tape the corners so that the box can stand alone.

B. Determine the volume of your group’s box.

MP.3

Length

Heig

ht

MP.3


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4. Record your box’s data and the data from all other groups in your class. Don’t worry about the last row for now.

There is no need to give out the square sizes in a particular order. A handout is included at the end of this lesson for you to randomly distribute the box sizes. The grid handout is also at the end of this lesson.

The last row will be filled in during Exercise 12. Tell students to leave it blank for now.

Group Size of Cut Out Square

Length Width Height Volume

1 1 x 1 23 11 1 253

2 1.5 x 1.5 22 10 1.5 330

3 2 x 2 21 9 2 378

4 2.5 x 2.5 20 8 2.5 400 5 3 x 3 19 7 3 399 6 3.5 x 3.5 18 6 3.5 378 7 4 x 4 17 5 4 340 8 4.5 x 4.5 16 4 4.5 288 9 5 x 5 15 3 5 225

10 6 x 6 13 1 6 78 n x n 25 – 2n 13 – 2n n (25 – 2n)(13 – 2n)(n)

5. A. Can you tell from the table which dimensions give the greatest volume? Explain your thinking.

From the table is looks like the box with dimensions 20 x 8 x 2.5 gives the greatest volume, but it is possible there is one that is close to these dimensions that gives a greater volume. B. How close is your prediction from Exercise 2 to the results from the table?


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To better see the patterns in the data, we’ll create a scatterplot and see if our prediction of the box with greatest volume is correct. 6. Use the grid below to plot the height of the box versus the volume of the box. Note: The 0.5 x 0.5 data point has been added. Students do this in Exercise 8. And the 0 x 0 data point has been added. Students add this point in Exercise 9.

7. Do you believe your prediction from Exercise 5 is correct based on the graph? Explain your thinking.

Answers will vary.

The Invention of the Cardboard Box

The Chinese invented cardboard in the 1600s. The English played off that invention and created the first commercial cardboard box in 1817. Pleated paper, an early form of corrugated board, initially served as lining for men's hats. By the 1870s, corrugated cardboard cushioned delicate glassware during shipment. Stronger, lined corrugated cardboard soon followed. American Robert Gair produced the first really efficient cardboard box in 1879. His die-cut and scored box could be stored flat and then easily folded for use. Refinements followed, enabling cardboard cartons to substitute for labor-intensive, space-consuming, and weighty wooden boxes and crates. Since then, cardboard boxes have been widely appreciated for being strong, light, inexpensive, and recyclable. [source: http://www.toyhalloffame.org/toys/cardboard-box]


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http://www.toyhalloffame.org/toys/cardboard-box

http://www.toyhalloffame.org/toys/cardboard-box


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8. Discussion - What type of function is this? Explain your reasoning.

Answers will vary. Many students may insist that this a quadratic function, but others may point out that the graph is not symmetrical. Allow students to ponder this idea.

9. A. What would the volume of the open box be if the square cut out was 0.5 x 0.5? Use the graph to

make a prediction and then verify your prediction by thinking about the patterns in the table.

The volume for the open box with the dimensions 24 x 12 x 0.5 is 144 cubic units.

B. Mark the height and volume of the 0.5 x 0.5 square on the graph.

10. What would be the volume of an open box if the height was 0? Mark this point on the graph.

There is no box possible since the height would be 0. The volume would be 0 cubic units. 11. At what other point would the volume be 0? Mark this point on the graph.

There is no box possible when the square cut out was 6.5 x 6.5. 12. With your group, determine a rule that could be used to determine the dimensions of any size square

that was cut out of the grid. Enter your rule in the last row of the table from Exercise 4. You’ll use these rules in the next lesson.


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Lesson Summary Mathematical modeling is when you

use ________________,

___________________, and/or

________________ to represent

real world situations.

Since the modeling of devices and phenomena is essential to both engineering and science, engineers and scientists have very practical reasons for doing mathematical modeling.

[source: http://www.sfu.ca/~vdabbagh/Chap1-modeling.pdf]

13. Read over the Modeling Principles flowchart above. Then highlight all of steps you completed

in this lesson.

Closing (5 minutes)

Have students share responses.

Why would our goal be to maximize the volume? Maximizing resources, enclosing as much volume as possible using the least amount of

material Is constructing a box in such a way that its volume is maximized always the best option?

No, a box may need to have particular dimensions (such as a shoe box). In some cases, the base of the box may need to be stronger, so the material is more expensive. Minimizing cost may be different than maximizing the volume.

Exit Ticket (5 minutes)


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http://www.sfu.ca/%7Evdabbagh/Chap1-modeling.pdf

http://www.sfu.ca/%7Evdabbagh/Chap1-modeling.pdf


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Name Date


Exit Ticket

Jeannie wishes to construct a cylinder closed at both ends. The figure below shows the graph of a cubic polynomial function used to model the volume of the cylinder as a function of the radius if the cylinder is constructed using 150𝜋𝜋 cm2 of material. Use the graph to answer the questions below. Estimate values to the nearest half unit on the horizontal axis and to the nearest 50 units on the vertical axis.

1. What is the domain of the volume function? Explain.

2. What is the most volume that Jeannie’s cylinder can enclose?

3. What radius yields the maximum volume?

4. The volume of a cylinder is given by the formula 𝑉𝑉 = 𝜋𝜋𝑟𝑟2ℎ. Calculate the height of the cylinder that maximizes the volume.


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Exit Ticket Sample Solutions

Jeannie wishes to construct a cylinder closed at both ends. The figure below shows the graph of a cubic polynomial function used to model the volume of the cylinder as a function of the radius if the cylinder is constructed using

𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 𝐜𝐜𝐜𝐜2 of material. Use the graph to answer the questions below. Estimate values to the nearest half unit on the horizontal axis and the nearest 𝟏𝟏𝟏𝟏 units on the vertical axis.

1. What is the domain of the volume function? Explain.

The domain is approximately 𝟏𝟏 ≤ 𝒓𝒓 ≤ 𝟖𝟖.𝟏𝟏 because a negative radius does not make sense, and a radius larger than 𝟖𝟖.𝟏𝟏 gives a negative volume, which also does not make sense.

2. What is the most volume that Jeannie’s cylinder can enclose?

Approximately 𝟖𝟖𝟏𝟏𝟏𝟏 𝐜𝐜𝐜𝐜3

3. What radius yields the maximum volume?

Approximately 𝟏𝟏 𝐜𝐜𝐜𝐜

4. The volume of a cylinder is given by the formula 𝑽𝑽 = 𝟏𝟏𝒓𝒓𝟐𝟐𝒉𝒉. Calculate the height of the cylinder that maximizes the volume.

Approximately 𝟏𝟏𝟏𝟏.𝟐𝟐 𝐜𝐜𝐜𝐜


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Homework Problem Set Sample Solutions

Bonus: Ask students what is meant by the caption on the t-shirt. (Hint that they can do a web search to find out.) Note: The data in Homework Problem Set #1 is discrete, while the data in the activity was continuous.

1. For a fundraiser, members of the math club decide to make and sell “Pythagoras may have been Fermat’s first problem but not his last!” t-shirts. They are trying to decide how many t-shirts to make and sell at a fixed price. They surveyed the level of interest of students around school and made a scatterplot of the number of t-shirts sold (𝒙𝒙) versus profit shown below.

a. Identify the 𝒚𝒚-intercept. Interpret its meaning within the

context of this problem.

The 𝒚𝒚-intercept is approximately −𝟏𝟏𝟐𝟐𝟏𝟏. The −𝟏𝟏𝟐𝟐𝟏𝟏 represents the money that they must spend on supplies in order to start making t-shirts. That is, they will lose $𝟏𝟏𝟐𝟐𝟏𝟏 if they sell 𝟏𝟏 t-shirts.

b. If we model this data with a function, what point on the graph of that function represents the number of t-shirts they need to sell in order to break even? Why?

The break-even point is the first 𝒙𝒙-intercept of the graph of the function because at this point profit changes from negative to positive. When profit is 𝟏𝟏, the club is breaking even.

c. What is the smallest number of t-shirts they can sell and still make a profit?

Approximately 𝟏𝟏𝟐𝟐 or 𝟏𝟏𝟏𝟏 t-shirts

d. How many t-shirts should they sell in order to maximize the profit? Approximately 𝟏𝟏𝟏𝟏 t-shirts

e. What is the maximum profit? Approximately $𝟐𝟐𝟖𝟖𝟏𝟏

f. What factors would affect the profit? The price of the t-shirts, the cost of supplies, the number of people who are willing to purchase a t-shirt

g. What would cause the profit to start decreasing? Making more t-shirts than can be sold


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2. The following graph shows the temperature in Aspen, Colorado during a 48-hour period beginning at midnight on Thursday, January 21, 2014. (Source: National Weather Service)

a. Let 𝑻𝑻 be the function that represents the temperature, in degrees Fahrenheit, as a function of time 𝒕𝒕, in hours. If we let 𝒕𝒕 = 𝟏𝟏 correspond to midnight on Thursday, interpret the meaning of 𝑻𝑻(𝟏𝟏). What is 𝑻𝑻(𝟏𝟏)?

The value 𝑻𝑻(𝟏𝟏) represents the temperature at 𝟏𝟏 a.m. on Thursday. From the graph, 𝑻𝑻(𝟏𝟏) =𝟏𝟏𝟏𝟏.

b. What are the relative maximum values? Interpret their meanings.

The relative maximum values are approximately 𝐓𝐓(𝟏𝟏𝟏𝟏) = 𝟐𝟐𝟖𝟖 and 𝐓𝐓(𝟏𝟏𝟑𝟑) = 𝟏𝟏𝟑𝟑. These points represent the high temperature on Thursday and Friday and the times at which they occurred. The high on Thursday occurred at 𝟏𝟏:𝟏𝟏𝟏𝟏 (when 𝐭𝐭 = 𝟏𝟏𝟏𝟏) and was 𝟐𝟐𝟖𝟖°𝐅𝐅. The high on Friday occurred at 𝟏𝟏:𝟏𝟏𝟏𝟏 (when 𝐭𝐭 = 𝟏𝟏𝟑𝟑) and was 𝟏𝟏𝟑𝟑°𝐅𝐅.

Review 3. Graph each function on the grid provided. You may use what you know about the functions or you can

create a table of values to complete the graphs. Then identify the type of graph each one is.

A. 2( 3) 1y x= − + B. 2

23

y x= − + C. 1y x= +

Quadratic Function Linear Function Square Root Function


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Rule in Symbols Example

Product Rule m n m nx x x += 7 3 10x x x=

Quotient Rule m

m nn

xx

x−=

74

3

xx

x=

Power Rule ( )nm m nx x= ( )37 21x x=

Negative Exponent Rule 1mmx

x− = 7

7

1x

x− =

Zero Power Rule 0 1x = ( )0 1xyz =

4. Use the rules above to simplify each expression.

A. 6 2 3 11x x x x= B. 2 2 0 1y y y− = = C. ( )42 3 8 12m n m n=

D. ( )223 162

4h

g hg

−− = E. 6 2

33

t ut u

t u= F.

2 3 4 3 4

1 0 3 3 2

a b cd a da b c b c

−

− =

5. Write an integer in the empty space to make a true equation.

A. 23 2

9 57

a aa a

a−

=

B. 13 4 2 3

0 2

b b b bb b

−− − =

C. ( )2 22 yx y xc c c+ =


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Box Sizes Handout

Cut out the squares below and have teams randomly select one from a bag.

1 x 1 2 x 2 3 x 3 4 x 4 5 x 5

1.5 x 1.5 2.5 x 2.5 3.5 x 3.5 4.5 x 4.5 6 x 6


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