packaging - megan pacheco

8/3/2019 Packaging - Megan Pacheco

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Problem OverviewProblem Title: The Popcorn Tin

Course: Algebra 2

Author(s): Megan Pacheco and Kevin Gant

Facilitation Notes

Phase Anticipated Student

Action

Notes/tips including time

for phase

Assessment

Roll out (k/ntk/next

steps)

Students will likely identify these

KNOWs:

box is a rectangular prismsquare base

second box is twice as high,but same base

And if they have prior

knowledge about efficiency,

they might KNOW:

efficiency relates surface areato volume

efficiency is a ratio

Total Roll-out time: 25 minutes

Beware: if someone says

efficiency is a ratio of surface

Problem

A manufacturing company wants to measure the efficiency of its packaging. Currently, they produce a

popcorn tin that is a rectangular prism with a square base. They are considering designing a new tin with

the same base and twice the height of the old tin. Determine which tin is more efficient. Be sure to justify

mathematically.

Standards/Big Ideas Addressed

Rational functions: The student formulates equations and inequalities based on rational functions, uses a variety

of methods to solve them, and analyzes the solutions in terms of the situation.

The student analyzes various representations of rational functions with respect to problem situations.

Likely units/big ideas that came before this

problem

Likely units/big ideas that come after this problem

Most Typically done soon after quadratics. Graphing Rational Functions, asymptotes, inequalities.Thereafter, exponential functions and logarithms.

Assumptions about Student Prior Knowledge

Students should already know

How to distribute and un-distribute How to find the area of rectangles and squares, and volume of regular prisms (though this is easily

researchable)

Exponent rules (like xn/xm = x(n-m) , etc.)


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efficiency is a ratio of volumeto surface area

Likely NTKs:

What is efficiency?How do we find efficiency?What is a rectangular prism?Likely Next Steps

Look up what efficiency mightmean.

Look up rectangular prism(If they know what the

efficiency might be)find

surface area or volume of box.

area to volume.

thats incorrect (check out

solution to see why), so you might

need to ask the question, Do you

know that?or how do you

know?

It is a good idea to publicly elicit

from every group what their next

step will be, so that you know

what everyone is going to do, and

so that it normalizes that everyone

is to start working on a next step.

Student work time Some students will go forwriting out variables for an

expression of the volume and

area.

Others may assume actual

dimensions of the box, so that

they can actually calculate

volume and surface area.

Likely online searches:

rectangular prism

efficiency

Time for Student work: 25

minutes

Recommend different studentswith different colors of pen, all

working on same large piece of

paper.

Doesnt really matter which

approach students take (variable

or definite dimensions) they

should get the same conclusion.

People may get stuck, some may

not. To groups that arent making

progress, especially about

efficiency, you might want tooffer hints to some that are stuck,

as you circulate around.

Have you noticed that mostof the definitions for

efficiency are ratios?

Do you know of gasefficiency? Miles per Gallon

ie Miles:Gallon ie

miles

gallon

That is a ratio of (somethinguseful)/(input of energy or

material)

What is the use of a box?What purpose does it serve?

If you are manufacturing abox, what is the energy or

material input?

Informal/formative

assessment of both

mathematical skill,and group

collaboration.

Sharing out At least 2 groups present theirwork so far. Most will likely not

Group 1 Sharing out: 10 min

Group 2 Sharing out: 10 min

Great opportunity for

students to receive


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have the problem done.

However, they should be able to

state assumptions that they

made, and describe the

problem-solving process that

they used up to this point.

Groups that are not presentingshould be listening, and can be

invited to provide questions and

comments to those that are

presenting.

Follow up questions: 10 min

Best to select the groups that are

presenting, including at least one

group that is definitely on the right

track.

Its ok if the work is not complete.This is a way to have students

provide other groups with

instruction.

Socratic questioning can be used

to get groups on the right track.

informal feedback from

their peers (thumbs

up/down/sideways, or

something similar) on

Oral Communication.

Work time* Students start to incorporatethe ideas that they saw in the

sharing out that just

happenedand complete the

problem.

Groups that finish early can get

started on follow-up

assignment.

Time can vary20 40 minutes

If a group finishes early, but had a

largely heuristic approach, then

they may not be able to make

much progress on the follow up

problems. You might decide to

have a workshop for some of

those students, or direct them to

try the problem anew using

variables

Direct instruction *

(this can occur at

different times, as

long as students work

on the problem first)

Discussion of canceling out both

single variables and polynomials

in a rational expression, with

practice factoring.

This will probably be necessary

especially for those students who

did not create expressions for the

efficiency with variables.

Final Action on

Problem

2 new groups present.

Groups convert work on largepiece of paper to formal write-

up, according to problem write-

up rubric.

Group 1: 10 minutes

Group 2: 10 minutes

Note that not all groups must

present. You might want to have

only the groups that did NOT

present turn in their work as a

formal write-up.

All groups receive work

ethic grade according to

the extent to whichthey finish the work in

class.

Those that turn in

formal write up receive

Work Ethic, Written

Communication, and

Content grades.

*These are typically optional, depending upon the nature/quality of the initial sharing out of students.

Follow UpExtension Here are two possibilities:

1. What would be more efficient a rectangular prism with square base (sidelength = x) or a triangular prism with right triangle base (each leg lengths = x)?

Assume that both tins are the same height.

2. Graph the efficiency as a function of the size of x (side of the base), of eithertin. Discuss what happens to the function when x = 0. Explain the physical


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meaning of this.

Practice

Problems 1. For the given figure, write and simplify a rational expression for the ratio of the

perimeter to its area:

Rectangle: Square:

2. Simplify the rational expression, if possible:

a.

48a2

16ab.

12y 4

12y2 18yc.

6z2 24z

2z2 8z

3. A bookseller uses shipping cartons in the shape of rectangular prisms. The cartons

have the same size base but vary in heights. Describe how the efficiency of the carton

changes as the height increases. Justify your answer.

4. Standardized Test Question:

The expression

a

2x2 5x 6simplifies to

2x 5

x 6. What is a?

A.

2x2 7x 5 B.

2x2 5x 1 C.

2x2 3x 5 D.

2x2 7x 5

Solution to Problem

Step 1: Define Efficiency

26cm24cm

h cm

x + 6

2x5x


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Efficiency = ratio of volume held/material of box (ie surface area)

This way, if a box holds a larger volume, with the same amount of surface area, then the denominator

gets larger, making the whole ratio larger, which means the efficiency is higher.

Step 2: Determine volume and surface area of the first box

Area of Base = Length Width = xx = x2

Height of original box = h

Volume = Height Length Width = h x2

= hx2

Surface area of one side = xh

Entire surface area = surface area of base + surface area of top (which is the same as the base) + surface

of 4 sides.

S = x2 + x2 + 4(xh) = 2x2 + 4xh

Step 3: Create an expression for the efficiency of the first box and simplify

So efficiency of first box =

volume

surface.area

hx2

2x2

4xh

Because there is the variable x in every term, you can simplify this by dividing every term by x:

hx2

x2x2

x 4xh

x

hx2x 4h

x

x

h


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(Another way to look at this is to factor out x from both denominator and numerator, and then cancel

xs)

hx2

2x 2 4xh

x(hx)

x(2x 4h)

hx

2x 4h

\

Step 4: Determine volume, surface area and efficiency of box 2

Lets take a look at the second package, that has the same base, but is twice as tall.

Reminder: First boxs height = h

Second boxs height = h2 = 2h

Reminder: First boxs surface area = S1 = x2 + x2 + 4xh = 2x

2 + 4xh

Second boxs surface area = S2 = x2

+ x2 + 4x2h = 2x

2+ 8xh

Reminder: First boxs volume = h x2

= hx2

Second boxs volume = 2h x2 = 2hx2

Efficiency of Second box =

volume

surface.area

2hx2

2x2 8xh

In every term this time, there is a common factor of 2x. Lets simplify by dividing all terms by 2x:

2hx2

2x2x2

2x 8xh

2x

hx

x 4hor by factoring a 2x out:

2hx 2

2x 2 8xh

2x(hx)

2x(x 4h)

hx

x 4h

Step 5: Compare efficiency of the two boxes

Now, lets compare expressions for efficiency for both boxes:


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Box 1 efficiency =

hx

2x 4h

Box 2 efficiency =

hx

x 4h

Since both fractions have the same numerator, but box 1 has a larger denominator, the overall ratio (ie

fraction) for box 1 is smaller, i.e. its efficiency is smaller.

Therefore, box 2 is more efficient.

packaging - megan pacheco

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