4.1 Number Analogies: Testing a Common Rule

Use both given pairs

Riverside's public description of Number Analogies presents two pairs of numbers and asks the student to apply their relationship to a third pair. The instruction is not to extend one long number sequence. Read each pair as an input and an output, then look for a rule that works for both known pairs.

These original examples use ordinary arithmetic to teach testing and checking. They do not claim that Riverside publishes a list of operation frequencies or a fixed high-school difficulty progression.

For (3, 8), (5, 12), (7, ?), the rule “double the first number, then add two” works on both pairs: $2(3)+2=8$ and $2(5)+2=12$. Applying it to seven gives $2(7)+2=16$. “Add five” works on the first pair but fails on the second, where $5+5=10$ rather than twelve.

Translate a verbal rule into arithmetic

Writing $y=2x+2$ is useful, but you do not need to use algebra notation to reason correctly. You can say “multiply the input by two, then add two.” What matters is carrying out the same operations in the same order for each pair.

A failed test eliminates that candidate for this prompt. A successful test establishes consistency with the given evidence; it does not prove the rule is the only mathematical function that could pass through two points.

Try simple families systematically

Begin with addition, subtraction, multiplication, and division. If none works, consider a repeated combination such as multiply then add, or a power with a constant adjustment. This order is a practical search method for original exercises, not an official scoring rule that “simpler” must always be correct.

For (4, 12), (7, 21), (9, ?), multiplication by three works on both pairs and predicts twenty-seven. For (2, 7), (4, 19), (6, ?), squaring and adding three works: $2^2+3=7$, $4^2+3=19$, and $6^2+3=39$.

Keep the operations identical. Do not use “add five” for the first pair and “multiply by three” for the second without a justified common rule connecting those choices. An answer explanation should show the shared relationship.

Respect order of operations

“Multiply by three, then subtract four” means $3x-4$. “Subtract four, then multiply by three” means $3(x-4)$. For input six, these give fourteen and six respectively. Parentheses make the difference visible.

To check a predicted output, recalculate using the original verbal instruction. If you wrote the expression incorrectly, repeating the same expression will repeat the error. A second route, such as substituting the candidate answer into a relationship sentence, can catch it.

For fractional inputs, maintain the same order. If the rule is double then add one, input $3/4$ gives $2(3/4)+1=3/2+1=5/2$. Fractions do not license changing the rule. Keep intermediate fractions exact until the final comparison.

Read zero and negative numbers without shortcuts

A zero output does not automatically mean subtraction or multiplication by zero. Squaring then subtracting one maps both one and negative one to zero. Adding a constant can also produce zero for a particular input. Infer the operation from both pairs rather than treating one special value as a diagnostic sign.

A negative input requires attention to signs. Under $x^2+3$, input negative two gives seven because $(-2)^2=4$. Under $2x+3$, the same input gives negative one. Do not confuse squaring a negative input with putting a negative sign in front of a square.

Keep paired data separate from a series

In an analogy, the first pair and the second pair provide examples of one mapping. You should not automatically subtract the output of the first pair from the input of the second as if every printed number belonged to a single sequence. The layout and directions identify how the numbers are grouped.

In practice, label the input and output positions. If you copied the pair backward, a correct arithmetic rule may predict the wrong quantity. Direction matters here just as it does in verbal analogies.

Check choices and acknowledge evidence limits

After finding a consistent rule, calculate the predicted answer before reading nearby distractors. Then compare every option. A distractor may reflect a reversed order, a sign mistake, or a rule that fits only one known pair.

If two straightforward rules fit the givens but predict different offered answers, examine whether the prompt supplies an additional constraint. If it does not, an independent practice item may be ambiguous. Do not invent a publisher preference to settle it. The next section explains this evidence limit and how to build clearer worked examples.

Official quantitative format description.