How to Solve Word Problems Programmatically






Chapter 3.  Abstraction






Real Things


Abstract Things


Dimensions of Real Things


Dimensions of Abstract Things


Constructing Shapes


Relating Shapes


Relating Angles


Activities and Explorations




Real Things



Real Things


The concept of a "thing" comes from that which we experience in our lives.  Things include not only physical objects, but other concepts as well such as events in time, monetary transactions, software programs, etc.  In this book, we will focus on things that we can measure. 




Abstract Things



Abstract Things Describe

Real Things


Abstract things help us to describe and relate real things.  The abstract concept of a circle can describe the shape of a coin or the hole in scissor handles,

When we associate an abstract thing like a "sphere" to a real thing like a "baseball", then the dimensions (like radius, diameter, surface area, volume, etc.) of that abstract thing apply to the real thing "baseball".  We can develop general relationships between the dimensions of abstract things and apply them to all real things associated with them.



Dimensions of Real Things
















Scaled Dimensions of A Real Thing


Things usually have many properties, and the properties that we can measure (with a unit scale) we call dimensions.  This includes not only geometric dimensions such as width, height, and depth; but also such dimensions as weight, temperature, cost, surface area, volume, speed, time, density, surface area, etc.  Note how the width, height, and depth dimensions of an appliance (like a washer machine) correspond to an abstract rectangular solid (a box) with minimal size to contain the appliance.




Dimensions of Abstract Things




Dimensions of a Right










Scaled Dimensions Associated with a Triangle


A triangle is an example of an abstract thing.  It has many related dimensions.  The dimensions listed here appear a basic algebra course.  But with a little bit of curiosity, browse through many other properties of triangles at  It may surprise you how many dimensions you will find.  Similarly for dimensions of circles, check out




Constructing Shapes










Right Angle


Parallel Lines

The Fundamental Building Blocks of Classical Geometry

In the 4-th century BC, a brilliant teacher Euclid set up a system of geometric objects in his famous work, The Elements.  In essence, he began with some basic shapes and construction concepts (points, lines, rays, circles, right angles, parallel lines) and then constructed and proved the field of classical geometry.  Used for over two millenniums as the primary resource for math education, The Elements presents logical arguments to construct many of the abstract things and relationships found in mathematics.  (


The proof of Proposition 1 illustrates Euclid's construction approach.  From his five fundamental postulates, one can construct a line between two points and a circle around a point.  From these he shows how to construct an equilateral triangle using a line between two points that are the centers of two circles with that line as their radius.  See more details at

Proposition 1.

Construction of Equilateral Triangle


It is amazing how Euclid began with 23 definitions and 5 postulates in Book I and then proceeded to prove over 465 propositions beginning with such simple concepts of point, line, circle, and angle.   Indeed, geometry gives us many abstract shapes and relationships between those shapes that help us describe the world around us.  These basic shapes allow us to create many interesting geometric figures and properties.  Explore the interactive examples of constructions at the website:


Even today, geometry textbooks teach the concept of "proof", and build propositions and theorems based on assumed definitions and postulates.



Relating Shapes













Related Geometric Figures


Certain words in the mathematical vocabulary describe shapes (figures) and the relationships between those geometric shapes.  In the above diagram, use the listed words to describe patterns that you see.  For resources to do this visit:




Relating Angles










Alternate interior

Alternate exterior



Angle Vocabulary Words


Certain words in the mathematical vocabulary describe types of angles and the relationships between angles.  The above diagram contains two horizontal parallel lines with two other lines, one perpendicular and one oblique (having a slant).  Use the listed words to describe relationships and patterns in this diagram.  For resources to do this visit:




Activities and Explorations





Play “Reality” game

Either draw or find examples of a real things associated with each word in a given mathematical vocabulary list.  Such a list can come from the other team or you may want to start with a list of vocabulary words needed for Math Competency and SAT exams in Basic Algebra Vocabulary Words.


Play “Abstract Scavenger Hunt” game

Have each team search a particular area and record as many abstract things and dimensions that they can apply to real objects they find.  The team with the most things and dimensions in their list wins.


Play “Clueless” game -

Be the last person to state a new clue (a unified quantity) for a known object within a specific time.  (No dimension can be repeated).





Review Basic Geometry Terms and Concepts


Construct Geometric Shapes from Other Shapes


Explore Euclid's Elements





Do Related Problems In Your Textbook


Using Meaning Table

Use a Meaning Table to solve word problems in

Word Problem Proportional Exercises (Set 1).


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Copyright © 2004  Dr. Ranel E. Erickson