Word Problem & Modeling Academy

Math Lab™

Represent real situations mathematically, state your assumptions, solve, and — critically — interpret and validate the result against reality.

The Modeling Cycle

Real situation→Identify variables→State assumptions→Formulate model→Solve→Interpret→Validate against reality→Refine

Interactive mastery laboratories

Hands-on mathematical laboratories

Not passive textbook pages — these are interactive stations where you discover, classify, apply, verify, and explain the mathematics yourself.

Mastery Laboratory

Parallel Lines + Transversal Lab™

Discover, classify, apply, verify, and explain every transversal angle relationship — the foundation for triangle similarity.

Open laboratory Mastery Laboratory

Parallel Lines + Transversals Academy™

Elite guided teaching studio: 18 sequential missions — discover, manipulate, classify, solve, prove, and transfer. Professor Live mode included.

Open laboratory Mastery Laboratory

Triangle Similarity Academy™

From angle relationships → similarity → proportionality → proof. 21 sequential missions — discover, match, justify, prove, solve, and transfer AA, SAS, and SSS similarity. Bridges directly from the transversal engine.

Open laboratory Mastery Laboratory

Elite Interactive Master Lab™

See it · Touch it · Move it · Classify it · Predict it · Prove it · Transfer it. The unified interactive teaching + adaptive mastery engine spanning parallel lines, transversals, and triangle similarity with Socratic professor mode.

Open laboratory Mastery Laboratory

Geometry Discovery Lab™

Sixteen hands-on missions — inside/outside, same/opposite, F/Z/C/X, theorem selection, one-angle-unlocks-all, converse reasoning, algebra from geometry, proof-building, similarity detective, vertex matching, proportions, and the parallel-lines → AA similarity bridge. Professor Live mode + Mission Matrix.

Open laboratory Mastery Laboratory

Geometry Reasoning Academy™

One connected system — parallel lines → angle relationships → triangle sum & exterior angles → similarity → proportions → proof → mastery. Grand Master angle reference, theorem table, proof dependency graph, the Geometry Reasoning Gauntlet™, and the Professor Teaching Console with live / board / student modes.

Open laboratory Mastery Laboratory

Intelligent Geometry Practice Lab™

Practice the reasoning, not merely the answer. Six modes — guided theorem-first scaffolds, independent classification, challenge algebra bridge, misconception repair, cumulative grand mastery challenge, and the Professor Live Console. Angle Theorem Library + Grand Master Table included.

Open laboratory

Represent

Choose the representation that exposes the structure — diagram, table, graph, equation, or simulation.

Assumptions

Every model assumes something. Name your assumptions explicitly so the model's limits are visible.

Validate & Refine

Does the answer make sense in the real world? If not, revisit assumptions and refine the model.

Lab problem — the cell phone plan

A modeling problem demonstrating the GUS framework and multiple representations.

Given

•Plan A: $20/month + $0.10/text
•Plan B: $40/month + $0.05/text

Unknown

•The number of texts t where Plan B becomes cheaper

Strategy

Write cost functions C_A(t) and C_B(t), set them equal to find the break-even point, then test values on either side.

Multiple representations

C_A(t) = 20 + 0.10t
C_B(t) = 40 + 0.05t

Hint ladder

0 hints used

Try independently first. Climb only as far as you need.

Where this leads next

Linear cost models→Optimization & systems

Add constraints and multiple plans — the model becomes a linear programming problem.