Look for What Changes—and What Stays the Same
Suppose a table records the cost of renting a bicycle:
| Hours, \(h\) | Cost, \(C\) |
|---|---|
| 0 | 12 |
| 1 | 20 |
| 2 | 28 |
| 3 | 36 |
The cost begins at ($12) and increases by ($8) for each hour. Those two observations become the rule
\[ C=12+8h. \]The table, the verbal description, and the equation express the same relationship in different forms. Translation means preserving that relationship while changing its representation.
\[ \boxed{\text{organize}\rightarrow\text{compare}\rightarrow\text{model}\rightarrow\text{verify}} \]A Reliable Translation Method
- Identify the quantities. Determine what is counted, measured, chosen, or produced.
- Define the variables. State what each row, column, label, dimension, or stage number represents.
- Choose an input and an output. Ask which quantity can be selected and which quantity responds.
- Compare corresponding values. Examine differences, ratios, totals, repeated parts, dimensions, and constraints.
- Describe the structure in words. Explain the starting amount and how the output changes.
- Write a symbolic rule. Use an expression, equation, formula, recurrence, inequality, or system as appropriate.
- Test the rule. Substitute several known values and compare the results with the original representation.
- State assumptions. Record units, domains, and any decision to continue a visible pattern in a particular way.
The method is deliberately slower than guessing a formula. A rule is useful only when we can explain why its parts correspond to the original structure.
Reading the Language of a Table
A table organizes corresponding values. Its labels provide essential mathematical information.
| Number of Notebooks, \(n\) | Total Cost, \(C\) in Dollars |
|---|---|
| 1 | 3 |
| 2 | 6 |
| 3 | 9 |
| 4 | 12 |
Here (n) is the input and (C) is the output. Each output is three times its input:
\[ C=3n. \]The coefficient (3) has units of dollars per notebook. The rule says more than “multiply by three”; it says that total cost is proportional to the number of notebooks at a constant unit price.
Before analyzing any table, ask:
- What do the headings mean?
- Which units are used?
- Are the input values equally spaced?
- Does each input have exactly one listed output?
- Are the values exact, measured, rounded, or estimated?
Additive Change and Linear Rules
When equal changes in the input produce equal changes in the output, a linear rule may be appropriate.
| Input, \(x\) | Output, \(y\) | First Difference in \(y\) |
|---|---|---|
| 0 | 5 | — |
| 1 | 8 | \(+3\) |
| 2 | 11 | \(+3\) |
| 3 | 14 | \(+3\) |
The output increases by (3) whenever the input increases by (1). The value at (x=0) is (5). Therefore,
\[ y=3x+5. \]In the general linear form
\[ y=mx+b, \]- (m) is the change in output per unit change in input, and
- (b) is the output when the input is zero.
If the input changes by amounts other than (1), calculate the rate of change rather than merely reading the raw output difference:
\[ m=\frac{\text{change in output}}{\text{change in input}} =\frac{\Delta y}{\Delta x}. \]Multiplicative Change and Exponential Rules
Some patterns change by a constant factor rather than a constant amount.
| Stage, \(n\) | Quantity, \(Q\) | Ratio to Previous Quantity |
|---|---|---|
| 0 | 4 | — |
| 1 | 12 | \(3\) |
| 2 | 36 | \(3\) |
| 3 | 108 | \(3\) |
The quantity begins at (4) when (n=0) and is multiplied by (3) at every stage:
\[ Q=4\cdot3^n. \]This is multiplicative, or exponential, change. Compare the two structures:
| Structure | Repeated Action | Typical Rule |
|---|---|---|
| Additive | Add the same amount | \(y=mn+b\) |
| Multiplicative | Multiply by the same factor | \(y=ab^n\) |
A constant ratio is meaningful only when the relevant previous values are nonzero and the inputs advance by equal steps.
Changing Differences and Quadratic Patterns
If first differences are not constant, compare the differences themselves.
| \(n\) | \(P(n)\) | First Difference | Second Difference |
|---|---|---|---|
| 0 | 1 | — | — |
| 1 | 4 | 3 | — |
| 2 | 9 | 5 | 2 |
| 3 | 16 | 7 | 2 |
| 4 | 25 | 9 | 2 |
The outputs are
\[ 1,4,9,16,25, \]which follow the rule
\[ P(n)=(n+1)^2 \]for the stage numbering shown. Constant second differences are characteristic of quadratic polynomial patterns when the inputs are equally spaced.
This test helps classify a proposed model; it does not prove that a short table must continue quadratically. The intended context or construction still matters.
Recursive and Explicit Descriptions
A pattern can often be described in two complementary ways.
For the sequence
\[ 7,11,15,19,\ldots, \]a recursive description gives a starting value and tells how to obtain the next term:
\[ a_1=7, \qquad a_n=a_{n-1}+4\quad(n\ge2). \]In words: begin with seven and add four to obtain each new term.
An explicit description gives any term directly from its position:
\[ a_n=7+4(n-1)=4n+3. \]In words: the (n)th term is seven plus four for each step after the first.
| Description | Emphasizes | Useful For |
|---|---|---|
| Recursive | How the pattern grows | Generating the next term |
| Explicit | How position determines value | Finding a distant term directly |
Translating Diagrams into Relationships
A diagram may encode lengths, areas, counts, adjacency, motion, or another relationship. Begin by naming the variable parts.
The area is the product of the two dimensions:
\[ A=x(x+3)=x^2+3x. \]The perimeter is the sum of all four sides:
\[ P=2x+2(x+3)=4x+6. \]The same diagram supports different formulas because area and perimeter measure different attributes. Before translating, identify what the question asks the symbols to represent.
Growing Visual Patterns
Consider a row of adjacent squares made from matchsticks. A new square shares one side with the preceding square.
The counts form the table:
| Number of Squares, \(n\) | Number of Matchsticks, \(M\) |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 10 |
| 4 | 13 |
There are two useful ways to see the rule.
- Start-and-grow view: The first square uses four matchsticks, and each additional square adds three.
- Repeated-parts view: Count three matchsticks for each square, then add the one unmatched side at the end.
The two formulas are equivalent:
\[ 4+3(n-1)=3n+1. \]Counting the Same Diagram in Different Ways
A good visual pattern often supports more than one correct count. Different counts can reveal algebraic equivalence.
Imagine a rectangular arrangement with (n) rows and (n+2) columns. Counting all objects by rows gives
\[ n(n+2). \]We could instead view the arrangement as an (n\)-by-(n) square plus two additional columns of (n) objects each:
\[ n^2+2n. \]Because both expressions count the same objects,
\[ n(n+2)=n^2+2n. \]This is not merely a symbolic manipulation. The diagram explains why the distributive law works in this case.
When translating a diagram, ask whether the whole can be:
- built by adding stages,
- split into familiar regions,
- counted by rows or columns,
- counted as a total minus missing pieces, or
- paired or rearranged without changing the quantity.
Context, Units, and Reasonable Domains
A rule should preserve the meaning of the quantities, not only reproduce the numbers.
Suppose a table records the water in a tank that begins with (120) liters and drains at (6) liters per minute:
\[ V=120-6t. \]The formula alone could produce negative values, but the physical model applies only while the tank contains water:
\[ 0\le t\le20. \]The variables and units are:
- (t): elapsed time in minutes,
- (V): volume in liters,
- (120): initial volume in liters,
- (-6): rate of change in liters per minute.
For a pattern of physical tiles or matchsticks, the stage number usually belongs to the positive integers. For a measurement, a continuous interval may be appropriate. The context determines the domain.
What a Pattern Does Not Prove
A finite list of values does not determine one unique rule unless additional structure or assumptions are supplied.
For example, the values
\[ 2,4,6 \]suggest “add two,” but many other rules could begin with the same three terms and then behave differently. A deliberately unusual pattern could continue
\[ 2,4,6,100,\ldots \]without contradicting the first three entries.
In ordinary pattern exercises, we usually seek the simplest rule consistent with the visible construction or context. State that assumption rather than treating it as a logical certainty.
Similarly:
- a table can support a model without proving that the model remains valid outside the observed range,
- a few constant first differences suggest a linear continuation but do not compel it,
- a diagram may be drawn schematically and not to scale, and
- measured data may vary because of rounding or experimental error.
Worked Example: From Table to Rule
A delivery company's charges are shown below.
| Distance, \(d\) in Miles | Cost, \(C\) in Dollars |
|---|---|
| 2 | 11 |
| 5 | 17 |
| 8 | 23 |
| 11 | 29 |
Compare Changes
The distance increases by (3) miles while the cost increases by ($6). The rate of change is
\[ m=\frac{6\text{ dollars}}{3\text{ miles}} =2\frac{\text{dollars}}{\text{mile}}. \]Find the Fixed Amount
Use the form (C=2d+b) and one table pair, such as ((2,11)):
\[ 11=2(2)+b, \] \[ b=7. \]Write and Interpret the Rule
\[ \boxed{C=7+2d}. \]The company charges a fixed ($7) fee plus ($2) per mile.
Verify
For (d=8),
\[ C=7+2(8)=23, \]which matches the table.
Worked Example: From Diagram to Rule
A staircase pattern is built from unit squares. Stage (1) has one row of length (1). Stage (2) has rows of lengths (1) and (2). Stage (3) has rows of lengths (1), (2), and (3).
Organize the Counts
| Stage, \(n\) | Total Squares, \(S_n\) |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 6 |
| 4 | 10 |
Describe the Growth
Stage (n) adds a new row containing (n) squares:
\[ S_n=S_{n-1}+n, \qquad S_1=1. \]Find an Explicit Rule
Two copies of the staircase can be paired to form an (n\)-by-((n+1)) rectangle. One staircase therefore contains half as many squares:
\[ \boxed{S_n=\frac{n(n+1)}{2}}. \]Verify
For stage (4),
\[ S_4=\frac{4(5)}{2}=10, \]which matches the table and the described construction.
Checking Across Representations
A strong translation survives movement among several representations.
| Check | Question |
|---|---|
| Table check | Does substitution reproduce every displayed input-output pair? |
| Difference or ratio check | Does the rule change at the observed rate or factor? |
| Diagram check | Can each term of the formula be connected to a visible part? |
| Unit check | Do the units combine into the stated output unit? |
| Boundary check | Does the rule make sense at the first stage, at zero, or at another meaningful endpoint? |
| Verbal check | Can the rule be explained clearly without relying on unexplained symbols? |
For the matchstick rule (M=3n+1):
- stage (1) gives (M=4),
- stage (2) gives (M=7),
- each additional stage increases (M) by (3), and
- the (3n) and (1) can both be identified in the diagram.
Agreement across representations provides stronger evidence than matching a single data point.
Common Translation Errors
Ignoring Labels and Units
Numbers without their headings may be interpreted backward or assigned the wrong meaning.
Assuming Inputs Increase by One
A constant output difference represents a constant rate only after the corresponding input change is considered.
Confusing Addition with Multiplication
A constant difference suggests additive change; a constant ratio suggests multiplicative change.
Using the Term Number Incorrectly
A pattern beginning at (n=0) can require a different formula from the same values indexed beginning at (n=1).
Counting Shared Parts Twice
Adjacent shapes may share sides, vertices, faces, or other components.
Fitting Values Without Explaining the Structure
A formula that matches a table is not yet a good translation if none of its parts can be connected to the context or construction.
Extending a Pattern as Though It Were Certain
Finite evidence permits many continuations. State the assumed pattern or use the diagram's construction rule.
Ignoring the Model's Domain
Stage numbers, counts, time intervals, and physical quantities may permit different kinds of values.
Practice
- A table contains ((0,6),(1,10),(2,14),(3,18)). Describe the change and write a rule for (y) in terms of (x).
- A table contains ((0,5),(1,10),(2,20),(3,40)). Describe the change and write a rule.
- For the values (3,8,15,24,35), calculate first and second differences and propose a rule for the (n)th term when (n) begins at (1).
- A gym charges a ($25) registration fee and ($18) per month. Write a rule for total cost (C) after (m) months and state a reasonable domain.
- A row of (n) adjacent equilateral triangles is made with sticks. The first triangle uses (3) sticks, and each additional triangle shares one side and adds (2) sticks. Write a recursive and an explicit rule.
- A rectangle has side lengths (x+2) and (x+5). Write formulas for its area and perimeter.
- A pattern contains (2,5,8,11,\ldots) tiles in stages (1,2,3,4,\ldots). Write the rule in start-and-grow form and simplified explicit form.
- A tank contains (90) liters and drains at (5) liters per minute. Write a volume rule and the physically meaningful time interval.
- Explain why the table ((1,2),(2,4),(3,6)) does not logically force the rule (y=2x) for every possible input.
- The first four staircase counts are (1,3,6,10). Use the explicit rule to find the number of squares at stage (20).
Selected Answers
- The output increases by (4) per input unit; (y=4x+6).
- The output doubles at every stage; (y=5\cdot2^x).
- First differences are (5,7,9,11); second differences are (2,2,2). A matching rule is (a_n=n^2+2n).
- (C=25+18m), with (m) a nonnegative integer if charges occur only by complete billing months.
- Recursive: (S_1=3) and (S_n=S_{n-1}+2). Explicit: (S_n=3+2(n-1)=2n+1).
- (A=(x+2)(x+5)) and (P=2(x+2)+2(x+5)=4x+14).
- (T_n=2+3(n-1)=3n-1).
- (V=90-5t), with (0\le t\le18).
- Many rules can match three data pairs and differ elsewhere. The linear rule is a simple proposed continuation, not the only mathematically possible one.
- (S_{20}=\frac{20(21)}{2}=210).
Summary
- Tables, diagrams, patterns, words, and formulas can represent the same relationship.
- Define the quantities, inputs, outputs, stage numbers, and units before searching for a rule.
- Constant first differences with equally spaced inputs suggest a linear model.
- Constant ratios with equally spaced inputs suggest a multiplicative or exponential model.
- Constant second differences are characteristic of quadratic polynomial patterns.
- Recursive rules describe growth from one stage to the next; explicit rules locate a term directly.
- Diagrams can reveal shared parts, repeated parts, decompositions, and equivalent expressions.
- A finite table or partial pattern does not uniquely determine how the relationship must continue.
- Verify a proposed rule using several values, visible structure, units, and meaningful boundaries.