33.2 Proportional Table Analysis

Proportional Table Analysis is the procedure for determining whether a table of input-output values represents a direct variation relationship, by calculating the output-to-input ratio for several nonzero rows, comparing those ratios for agreement, and separately checking that any zero-input row pairs correctly with a zero output. This procedure applies the definitional requirements established under direct proportionality scope directly to the tabular representation, turning the abstract conditions for direct variation into a concrete, step-by-step check that can be carried out on any given table.

Because a table only ever shows a finite selection of a relationship's values, this analysis relies on checking every available row rather than assuming that agreement at one or two rows is enough to confirm that the entire table reflects a proportional relationship.

Nonzero Input Row Selection

Identifying Rows With a Nonzero Input

Analysis begins by identifying every row of the table whose input value is not zero, since the output-to-input ratio calculation performed in the next step cannot be carried out for a row where the input is zero.

Setting Aside the Zero-Input Row for Later

Any row with a zero input is deliberately set aside at this stage rather than discarded, since it will be examined separately under zero-input row inspection using a different check specific to that particular value.

Working With the Full Set of Nonzero Rows

All nonzero-input rows identified in this step are carried forward into the ratio calculation, since a proportional table analysis should account for every available nonzero row rather than selecting only a convenient subset.

Output-to-Input Ratio Calculation

Calculating the Ratio for Each Selected Row

For every nonzero-input row identified in the previous step, the output value is divided by the input value, producing a ratio for that specific row following the constant output-to-input ratio definition established under direct proportionality scope.

Recording Each Calculated Ratio

Each calculated ratio is recorded alongside the row it was calculated from, preserving a clear record of which ratio corresponds to which specific input-output pair for use during the comparison step that follows.

Simplifying Ratios for Easier Comparison

Where a calculated ratio results in a fraction, simplifying that fraction to its lowest terms makes it easier to compare directly against the ratios calculated from other rows, reducing the chance that two equal ratios are overlooked simply because they were left in different unsimplified forms.

Ratio Comparison across Table Rows

Comparing Every Calculated Ratio

Once ratios have been calculated for every nonzero-input row, they are compared against one another to check whether all of them are equal, following the requirement that a directly proportional relationship must have the same ratio across every pair of values.

Confirming Full Agreement

Agreement is only confirmed once every calculated ratio has been checked and found equal to the others, mirroring the every table row requirement described under candidate function rule verification, since checking only some of the ratios could overlook a disagreement present elsewhere in the table.

Identifying a Disagreement Among Ratios

If even one calculated ratio differs from the others, this comparison step reveals that the table does not exhibit the constant ratio required for direct proportionality, providing specific evidence for that conclusion.

Shared Variation Constant

Naming the Common Ratio

Once every ratio has been confirmed equal, the shared value is identified as the constant of proportionality for the table, matching the same constant referred to as $$k$$ in the direct variation rule form.

Using the Shared Constant to State the Rule

The shared variation constant, once identified, allows the entire table to be summarized by a single algebraic rule, such as $$y=3x$$ if the shared ratio was found to be $$3$$, condensing every row of the table into one compact statement.

Verifying the Rule Against the Original Table

The resulting rule can be checked against the original table using candidate function rule verification, substituting each table input into the rule and confirming that the predicted outputs match the table's recorded outputs.

Zero-Input Row Inspection

Checking the Output Paired With a Zero Input

If the table includes a row with an input of zero, that row is examined separately to confirm that its output is also zero, following the required pairing described under zero input paired with zero output.

Why This Check Is Handled Separately

Because the ratio calculation used for nonzero rows cannot be applied to a zero input without dividing by zero, this specific row must be checked using a different, more direct comparison, simply confirming whether the paired output is itself zero.

Consequences of a Nonzero Output Paired With a Zero Input

If the table pairs an input of zero with any output other than zero, this immediately disqualifies the table from representing a direct variation relationship, regardless of how consistently the other rows' ratios might agree with one another.

Proportional Table Confirmation

Confirming the Table Represents Direct Variation

A table is confirmed as representing a direct variation relationship once every nonzero-input row has produced the same ratio, and any zero-input row, if present, pairs correctly with a zero output.

Stating the Confirmed Result

A confirmed proportional table is reported together with its identified constant of proportionality and the corresponding rule form, providing both the classification and the specific quantitative relationship the table describes.

Using the Confirmation for Further Work

Once a table has been confirmed proportional and its constant identified, that constant and rule can be used to predict outputs for inputs not directly listed in the original table, extending the confirmed relationship beyond the specific rows that were checked.

Nonproportional Table Rejection

Recognizing a Rejected Table

A table is rejected as not representing direct variation as soon as either a ratio disagreement is found among the nonzero-input rows or a zero-input row is found paired with a nonzero output.

Reporting the Specific Reason for Rejection

A rejected table is reported together with the specific evidence responsible for the rejection, whether the two disagreeing ratios and the rows they came from, or the zero-input row paired with its nonzero output, rather than a bare statement that the table failed the check.

Recognizing That Rejection Does Not Rule Out Other Relationships

A table rejected as nonproportional may still represent some other type of algebraic relationship, such as one involving an additive offset as discussed under additive offset exclusion, meaning rejection from direct proportionality is not itself a claim that the table describes no relationship at all.