PRESENTATION TITLE

Maximizing Critical Thinking via AlgebraCMC – South 2015

Roberto Soto

November 7, 2015

“Real World” Problem

A recent school play at Desert High School broke attendance and revenue records. There were 75 tickets sold for a total of $585. If parent tickets were sold for $12 and student tickets were sold for $5, how many parents and how many students attended the play?

Standards for Mathematical Practice

• Make sense of problems and persevere in solving them• Reason abstractly and quantitatively• Construct viable arguments and critique the reasoning

of others• Model with mathematics• Use appropriate tools strategically• Attend to precision• Look for and make use of structure• Look for and express regularity in repeated reasoning.

Critical Thinking via Linear Programming

• Critical ThinkingTo analyze and evaluate information in order to make a decision and/or form an opinion.

• Linear ProgrammingA method of solving optimization problems using linear algebra.

Sample Linear Programming Problem

Company A has determined that the cost of producing a smartphone is $350 and the cost of producing a tablet is $450. Company A has also determined that it takes 1 labor-hour to produce a smartphone and 2 labor-hours to produce each tablet. The profit on a smartphone is $100 and $150 for each tablet. If their factory has a capacity of 120 labor hours per day and Company A would like to keep their costs per day under $31,500, how many of each product should it produce in order to maximize profit?

The Setup

Let x = the number of smartphones that Company A will produce

Let y = the number of tablets that Company A will produce

Solution

Graph created by Desmos.com

Problem Needing 3 VariablesA department store has up to $20,000 to spend on television advertising for a sale. All ads will be placed with one television station. A 30-second ad costs $1000 on daytime TV and is viewed by 14,000 potential customers, $2000 on prime-time TV and is viewed by 24,000 potential customers, and $1500 on late-night TV and is viewed by 18,000 potential customers. The television station will not accept a total of more than 15 ads in all three time periods. How many ads should be placed in each time period in order to maximize the number of potential customers who will see the ads? How many potential customers will see the ads? (Ignore repeated viewings of the ad by the same potential customer.) from Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences, 12th edition, by Raymond Barnett, Michael Ziegler, and Karl Byleen.

Solution

Let x = the number of daytime adsLet y = the number of primetime adsLet z = the number of late night ads

3D-Model

Plot created using Mathematica

The Table Method

The Table Methodx y z s t V = 14000x + 24000y + 18000z0 0 00 0 00 0 00 0 00 0 00 0 0

0 0 00 0 00 0 0

0 0 0

The Table Methodx y z s t V = 14000x + 24000y + 18000z0 0 0 2000 15 00 0 40/3 0 5/3 240,0000 0 15 -2500 0 ----0 10 0 0 5 240,0000 15 0 -10,000 0 ----0 -5 20 0 0 ----

20 0 0 0 -5 ----15 0 0 5000 0 210,00010 0 5 0 0 230,00010 5 0 0 0 260,000

The Simplex Method

• Simplex Method – Standard Maximization Problem by Brian Veitchhttps://youtu.be/XK26I9eoSl8

• The Simplex Method in Excel– Video– Adding Solver to Excel

Other Resources

• Handouts for this talk

• Scaffolding to this topic

• Sample written work

Possible Extension

• Class Project – “Bake” Sale– Determine what to sell in order to maximize profit.

– Use surveys to determine appropriate prices.

Contact Info

Roberto C. Soto, PhDCSU Fullertonrcsoto@fullerton.edu(657) 278-2743http://math.fullerton.edu

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