4.2 Number Analogies: Compound Rules and Ambiguity

Break a compound rule into operations

A compound rule uses more than one operation. For a teaching example with rule $y=3x-4$, multiply the input by three, then subtract four. Inputs four and seven give eight and seventeen; input ten gives twenty-six. Checking both known examples ensures that a tempting single-operation rule does not slip through.

Another rule, $y=(x-4)\times3$, subtracts first. For input ten it gives eighteen. The same numbers and operations can produce different answers when their order changes. Write the stages in words or use parentheses to keep the transformation unambiguous.

This section teaches mathematical reasoning using original examples. The public CogAT format description does not publish a mandatory hierarchy for choosing linear, quadratic, digit-based, or other rules. Do not attribute such a hierarchy to the publisher.

Use a linear model when the problem supports it

A linear model has the form $y=ax+b$. If a problem explicitly states that the mapping is linear and supplies two distinct inputs, those pairs determine the slope and constant.

For $(4,8)$ and $(7,17)$:

$$a=\frac{17-8}{7-4}=\frac{9}{3}=3,\qquad b=8-3(4)=-4.$$

Thus the linear model is $y=3x-4$. The computation is a useful derivation, not evidence that every analogy must be linear. If the inputs are identical but the outputs differ, no function of the input alone can reproduce both pairs. If both input and output pairs are repeated, they provide only one distinct constraint.

A noninteger slope does not prove a linear rule is invalid. For $(2,5)$ and $(6,7)$, the linear rule is $y=x/2+4$. The arithmetic fits both. Rejecting it solely because the slope is one-half would be an invented restriction.

Understand what two pairs cannot prove

Consider $(3,14)$ and $(6,41)$. Two simple models fit:

Both models satisfy the evidence. Therefore, the two pairs alone do not establish whether the next output is fifty-nine or sixty-nine. An exercise can resolve this by explicitly specifying a quadratic rule of the form $x^2+c$, adding a discriminating pair, or offering choices that do not include competing plausible outputs.

For example, adding $(4,21)$ rejects the linear candidate, which would predict twenty-three. It is consistent with $x^2+5$. Even three finite data points do not establish uniqueness among all possible functions, but they distinguish these two candidates. In ordinary practice, the goal is a supported relationship among reasonable candidates, not proof about infinitely many functions.

Verify powers and constants independently

If the rule is $x^2+5$, first compute the square and then add five. For input eight, $8^2=64$ and $64+5=69$. Adding before squaring would give $(8+5)^2=169$, a different expression.

If the rule is $x^3-2$, input three produces $27-2=25$ and input four produces $64-2=62$. The difference between outputs changes as inputs increase. That can suggest a power relationship, but changing differences alone do not prove one. Check the actual candidate rule.

Maintain a small reference set of squares and cubes during learning if it helps your arithmetic. Such a list is a home practice aid, not material you may bring to an official session. Do not infer that every memorized power appears on the test.

Treat digit patterns as hypotheses

A practice pair might use the sum of a number's digits: twelve maps to three and twenty-one maps to three. That relationship is different from the ordinary arithmetic expression $x+3$. A digit operation depends on how the number is represented, so describe it explicitly.

Do not declare that CogAT never uses a particular kind of relationship unless the owner publishes that restriction. Likewise, do not prefer a digit hypothesis simply because one pair fits. Test the second pair and the answer options. A supported rule must explain all given examples under the same interpretation.

Use candidate comparisons efficiently

Write a compact table with rule, first-pair result, and second-pair result. Discard a candidate as soon as a calculation fails. This is especially useful during untimed learning, when the aim is to discover which error caused a wrong prediction.

During actual testing, scratch paper is permitted for Number Analogies under the current proctoring guidance, but you should follow the administrator's material instructions. No calculator is allowed. Use short notation only if it helps you reason; elaborate algebra on every straightforward multiplication item can waste time.

Repair ambiguous original questions

When creating or reviewing independent practice, state assumptions necessary for a unique key. “Under the rule square then add a constant” legitimately narrows the model. “The publisher always prefers powers” does not. Changing the answer options may also remove a competing plausible result, but the explanation should still say what was established rather than claim mathematical uniqueness.

The lesson is to separate consistency, selection, and uniqueness. A rule may be consistent with the givens; the choices may support selecting its result; uniqueness requires enough constraints for the claimed model. Keeping those ideas distinct makes both your solutions and your practice questions more accurate.