Meadows or Malls?

The Central Problem

The title problem of this unit concerns a decision that must be made about land use. This problem can be expressed using a system of linear inequalities, which lends itself to a solution by means of linear programming, a topic introduced in the Year 2 unit Cookies. Building on their work in that unit, students see that a key step in solving the system of linear inequalities is to find various points of intersection of the graphs of the corresponding equations. This, in turn, leads to the need to solve systems of linear equations. Along the way, students learn about graphing equations in three variables, see that the graph of a linear equation in three variables is a plane, and study the possible intersections of planes in space.

Because graphing calculators allow students to find inverses of square matrices (when the inverses exist), matrices are a good tool for solving systems of linear equations with several variables. So, in addition to strengthening their skills with traditional methods, students learn to express linear systems in terms of matrices and develop the matrix operations required to understand the role of matrices in the solution process.

Meadows or Malls?

Topic                                                 Page in your book

Recreation Versus Development:

A Complex Problem . . . . . . . . . . . . . . . . . . . . . . . . . .155

Meadows or Malls? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156

Homework 1: Meadows, Malls, and Variables . . . . . . . . .159

POW 5:That’s Entertainment! . . . . . . . . . . . . . . . . . . . . .161

Homework 2: Heavy Flying . . . . . . . . . . . . . . . . . . . . . . .163

A Strategy for Linear Programming . . . . .165

To see a quick review of Linear Programming go here

Homework 3: Programming and Algebra Reflections . . .166

• Linear Programming Grapher
  (Two Variables)

Ideas for Solving Systems . . . . . . . . . . . . . . . . . . . . . . . . .167

Programming Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . .169

Homework 4: Donovan Meets the Beatles . . . . . . . . . . . .172

Homework 5: Finding Corners Without the Graph . . . . . .173

What Wood Would Woody Want? . . . . . . . . . . . . . . . . . . . .174

Homework 6:Widening Woody’s Woodwork . . . . . . . . . . .175

Homework 7: More Equations . . . . . . . . . . . . . . . . . . . . .176

Equations, Points, Lines, and Planes . . . . .. . . . . . . . .177

Being Determined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .178

Homework 8: How Much After How Long? . . . . . . . . . . .179

Homework 9:The Points and the Equations . . . . . . . . . .181

The Three-Variable Coordinate System . . . . . . . . . . . . . . .182

Homework 10:What Do They Have in Common? . . . . . .184

Homework 11:Trying Out Triples . . . . . . . . . . . . . . . . . .185

Homework 12: More Cookies . . . . . . . . . . . . . . . . . . . . . .186

Just the Plane Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .188

Homework 13: Solving with Systems . . . . . . . . . . . . . . . .189

Homework 14: Fitting a Line . . . . . . . . . . . . . . . . . . . . . .190

More Cookies . . . . . . . . . . . . . . . . . . .192

POW 6: SubDivvy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193

Homework 15:The “More Cookies” Region

and Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196

Homework 16: Finishing Off the Cookies . . . . . . . . . . . . .197

Equations, Equations, Equations . .198

Homework 17: Easy Does It! . . . . . . . . . . . . . . . . . . . . . .199

Get Rid of Those Variables! . . . . . . . . . . . . . . . . . . . . . . . .201

Homework 18: Eliminating More Variables . . . . . . . . . . .202

Homework 19: Gardener’s Dilemma . . . . . . . . . . . . . . . .203

Elimination in Three Variables . . . . . . . . . . . . . . . . . . . .205

Homework 20: More Equation Elimination . . . . . . . . . . .206

Equations and More Variables

in Linear Programming . . . . . . . . . . . . . . . . . . . . . . . .207

Ming’s New Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . .208

Homework 21: Let Me Count the Ways . . . . . . . . . . . . . . .209

Homework 22:Three Variables, Continued . . . . . . . . . . .211

Homework 23: Grind It Out . . . . . . . . . . . . . . . . . . . . . . .212

POW 7: Crack the Code! . . . . . . . . . . . . . . . . . . . . . . . . . .213

Homework 24: Constraints Without a Context . . . . . . . . .216

Eastside Westside Story . . . . . . . . . . . . . . . . . . . . . . . . . . .217

Homework 25: Fitting More Lines . . . . . . . . . . . . . . . . . .219

Homework 26:Ages, Coins, and Fund-Raising . . . . . . . . .221

Saved by the Matrices! . . . . . . . . . .222

Matrix Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .223

Inventing an Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . .224

Homework 27: Fitting Quadratics . . . . . . . . . . . . . . . . . .227

Flying Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .229

Homework 28: Matrices in the Oven . . . . . . . . . . . . . . . .231

Homework 29: Fresh Ingredients . . . . . . . . . . . . . . . . . . .233

Calculators to the Rescue . . . . . . . . . . . . . . . . . . . . . . . . .235

Homework 30: Make It Simple . . . . . . . . . . . . . . . . . . . . .237

Back and Forth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .238

Matrices and Linear Systems . . . . . . . . . . . . . . . . . . . . . .240

Homework 31: Solving the Simplest . . . . . . . . . . . . . . . . .241

Homework 32:Things We Take for Granted . . . . . . . . . . .242

Finding an Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243

Homework 33: Inverses and Equations . . . . . . . . . . . . . .244

Calculators Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245

Homework 34: Fitting Mia’s Bird Houses . . . . . . . . . . . . .246

Solving “Meadows or Malls?” . . .247

Homework 35: Getting Ready for “Meadows or Malls?” . . .248

“Meadows or Malls?” Revisited . . . . . . . . . . . . . . . . . . . . .250

Homework 36: Beginning Portfolios—Part I . . . . . . . . . .251

Homework 37: Beginning Portfolios—Part II . . . . . . . . . .252

Homework 38:“Meadows or Malls?” Portfolio . . . . . . . . .253

Appendix:

Supplemental Problems . . . . . . . . . . .255

How Many Regions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .256

The Eternal Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . .258

The Music Business . . . . . . . . . . . . . . . . . . . . . . . . . . . . .260

Special Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261

Embellishing the Plane Facts . . . . . . . . . . . . . . . . . . . . . .262

A Linear Medley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .263

The General Two-Variable System . . . . . . . . . . . . . . . . . .265

Playing Forever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .266

The Shortest Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .267

Producing Programming Problems—More Variables . . . .268

Your Own Three-Variable Problem . . . . . . . . . . . . . . . . . .269

Fitting a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .270

Surfer’s Shirts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272

An Associative Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . .274

When Can You Find an Inverse? . . . . . . . . . . . . . . . . . . .275

Determining the Determinant . . . . . . . . . . . . . . . . . . . . .276

Cracking Another Code . . . . . . . . . . . . . . . . . . . . . . . . . .278

I am Under Construction