Selecting a Representation

The Same Problem Can Look Completely Different

Suppose a taxi charges a fixed \$4 starting fee plus \$2 for each mile.

In words:

“The total cost is four dollars plus two dollars for every mile traveled.”

As an equation:

As a table:

As a graph, the same relationship becomes a straight line with vertical intercept \(4\) and slope \(2\).

Each form is mathematically equivalent, but each makes something different easier to see.

What Is a Representation?

A representation is a way of expressing or organizing a mathematical object, relationship, or process.

Representations include:

• words,
• symbols,
• equations,
• tables,
• diagrams,
• graphs,
• coordinate systems,
• matrices,
• algorithms.

A representation is not merely a picture of mathematics. It is a mathematical tool.

Representations Have Purposes

Different representations are good at different jobs.

The best representation is the one that supports the task you are trying to perform.

Natural Language

Words are useful for:

• describing context,
• explaining meaning,
• stating assumptions,
• interpreting results.

But natural language can be verbose or ambiguous.

For example:

“Three more than twice a number equals eleven.”

is clearer for many calculations as:

Symbols and Equations

Symbolic representation compresses relationships.

For example:

captures the relationship among distance, rate, and time.

Equations are especially useful when:

• an unknown must be isolated,
• relationships must be transformed,
• general formulas matter,
• exact reasoning is important.

Tables and Lists

Tables are useful when a problem involves repeated values or paired quantities.

For example, for

a table may reveal the pattern:

Tables are often excellent intermediate representations between formulas and graphs.

Diagrams and Sketches

A diagram is often the fastest way to organize spatial information.

For geometry, a useful sketch can reveal:

• which lengths correspond,
• where angles are located,
• which objects intersect,
• which quantities are missing.

A sketch does not need to be perfectly to scale unless scale itself matters. Its purpose is to organize relationships.

Number Lines and Intervals

Number lines are especially useful for:

• inequalities,
• intervals,
• distance,
• absolute value,
• ordering,
• one-dimensional motion.

For example:

is often easier to interpret when seen as an interval on a number line.

Graphs and Coordinate Representations

Graphs make relationships visible as shapes.

They are especially useful for:

• comparing functions,
• finding intersections,
• seeing increasing and decreasing behavior,
• recognizing periodicity,
• estimating roots,
• visualizing rate of change.

For a system such as

the solution can be represented as the intersection of two lines.

Matrices, Arrays, and Structured Data

Matrices and arrays are useful when information naturally forms rows and columns.

For example:

is more compact than repeatedly describing four values and their positions in words.

Matrices are especially useful for:

• systems of equations,
• transformations,
• data tables,
• networks of relationships,
• repeated linear operations.

Algorithms and Pseudocode

When the problem is about process rather than only relationship, pseudocode may be the clearest representation.

For example:

total = 0

FOR each value in data
    total = total + value
END FOR

RETURN total

A formula may describe the final relationship, but pseudocode makes the procedure explicit.

A Representation Selection Map

Criteria for Choosing a Representation

Ask:

• What structure do I need to see?
• What information is repeated?
• Is the problem mainly spatial, symbolic, numerical, or procedural?
• Do I need exact values or visual estimates?
• Will the representation support the operations I need next?
• Can another reader understand the representation easily?

A representation is efficient when it reduces unnecessary work without hiding important conditions.

Switching Representations

If a problem feels stuck, change its form.

For example:

• turn words into an equation,
• turn an equation into a graph,
• turn a graph into a table,
• turn a geometric description into a diagram,
• turn repeated arithmetic into an algorithm.

Many problems are difficult only in their current representation.

Sometimes it means:

Using More Than One Representation

Strong problem solving often combines representations.

Example

A function may be studied with:

• an equation for exact algebra,
• a table for sample values,
• a graph for overall behavior,
• words for interpretation.

These are not redundant. They reveal different aspects of the same object.

What a Representation Can Hide

Every representation emphasizes some features and suppresses others.

A graph may show the shape clearly but hide exact coordinates.

A decimal approximation may show magnitude but hide exact structure.

A formula may be exact but obscure how a process unfolds step by step.

So ask not only:

“What does this representation show?”

but also:

“What does it make harder to see?”

Representation as Verification

Changing representation can provide an independent check.

Example

Suppose algebra gives the solution to

as

A graph of

and

should intersect at

Agreement between different representations strengthens confidence in the solution.

Worked Examples

Example 1: Word Statement to Equation

“Five less than three times a number equals sixteen.”

Choose a symbolic representation:

Then solve:

Example 2: Repeated Data to Table

A machine produces \(12\) items per minute.

The table makes the proportional pattern visible.

Example 3: Spatial Information to Diagram

A ladder leans against a wall. The foot of the ladder is \(6\text{ ft}\) from the wall, and the ladder is \(10\text{ ft}\) long.

A right-triangle diagram immediately suggests:

So:

Example 4: Inequality to Number Line

The solution

is compact symbolically, but a number line makes the direction of the solution set visually immediate.

Example 5: System to Graph

Set them equal:

The graph represents this same solution as the intersection point

Example 6: Process to Pseudocode

To sum the first \(n\) positive integers procedurally:

total = 0

FOR i = 1 TO n
    total = total + i
END FOR

RETURN total

The pseudocode reveals the repeated process more clearly than the compact formula

The two representations answer different questions about the same mathematical structure.

Common Errors

1. Staying with the First Representation

A poor initial choice can make an otherwise simple problem unnecessarily hard.

2. Choosing a Familiar Representation Instead of a Useful One

Use the form that exposes the needed structure, not merely the one used most recently in class.

3. Treating a Sketch as Exact Measurement

A diagram may organize relationships without being drawn to scale.

4. Using a Graph When Exact Values Are Required

Graphs are excellent for visual structure, but algebra may be needed for exact coordinates.

5. Using Symbols before Meaning Is Clear

Define what each variable represents.

6. Forgetting What a Representation Hides

A compact form may suppress conditions, units, or intermediate structure.

7. Assuming One Representation Must Do Everything

Combining representations is often more efficient than forcing one form to carry the entire solution.

Practice

Choose a Representation

A problem describes how the temperature changes every hour for twelve hours. Which representation is especially useful for organizing the recorded values?

Spatial Problem

A problem describes distances and angles in a triangle. Which representation should usually be considered early?

Process Problem

A task asks how to repeatedly update a balance until it exceeds a target. Which representation is especially natural?

Exact versus Visual

If a graph suggests that two lines intersect near \(x=2.1\), but an exact answer is required, what should you do next?

When Stuck

What is one productive response when a problem feels difficult in its current form?

The Deeper Idea

A representation is a lens on a mathematical object.

The object does not change, but what becomes easy to see can change dramatically.

This connects directly to the Malone Math problem-solving framework:

• known and unknown information tells us what must be represented,
• objects and restrictions tell us what structures must be preserved,
• representation choice prepares the strategy,
• switching representations can unlock a stuck problem,
• multiple representations can verify one another.

Representation is therefore not a decorative step between reading and solving. It is one of the main ways mathematical thinking reorganizes a problem into a form that can be solved.

Summary

• A representation is a way of expressing or organizing a mathematical object, relationship, or process.
• Natural language explains meaning and context.
• Equations expose symbolic relationships and support algebraic transformation.
• Tables organize repeated values and paired quantities.
• Diagrams reveal spatial relationships.
• Number lines clarify order, intervals, and inequalities.
• Graphs show shape, trend, intersections, and rates of change.
• Matrices and arrays organize structured row-and-column information.
• Algorithms and pseudocode expose ordered procedures.
• The best representation depends on what structure the problem needs to make visible.
• Changing representation is often a productive response to being stuck.
• Multiple representations can complement and verify one another.
• Every representation reveals some features while hiding others.

Do not ask only, “How do I solve this?” Ask, “What form would make this problem easier to think about?” Very often, choosing the right representation is the step that makes the strategy visible.

Problem Solving and Verification

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