4.1 Number Analogies: Mathematical Relationships & Single-Step Rules

4.1 Number Analogies: Mathematical Relationships & Single-Step Rules

The Number Analogies subtest forms the opening component of the Quantitative Battery on the Cognitive Abilities Test™ (CogAT®) Level 9. Administered to Grade 3 students (typically ages 8 to 9), this subtest evaluates mathematical reasoning, inductive pattern recognition, and quantitative transformation skills. Unlike routine classroom computation tests that assess procedural fluency through drills, Number Analogies measures a student's ability to discover unstated mathematical relationships and apply those rules systematically to new numerical pairs.

At Level 9, the subtest consists of 18 multiple-choice questions to be completed within a strict 10-minute time window. These timing parameters yield an average of 33.3 seconds per question. Succeeding under these conditions requires rapid pattern recognition, a disciplined problem-solving routine, and a firm grasp of fundamental third-grade arithmetic operations.

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The Landmark Grade 3 Transition: Moving from Pictorial Sets to Symbolic Numerals

To appreciate the cognitive demands of Level 9 Number Analogies, it is helpful to examine how the subtest evolves from the primary battery levels:

• Levels 5/6, 7, and 8 (Kindergarten through Grade 2): Primary-level Number Analogies use a $2 \times 2$ matrix of pictures, following the same process as Picture Analogies but with quantities. A practice-style question might show a plate with 2 cookies beside a plate with 4 cookies, then a plate with 3 cookies beside an empty cell. The teacher reads the directions aloud, there is no time limit, and children can count individual objects instead of reading numerals.
• Level 9 (Grade 3): Pictures give way to numerals. Each question shows two number pairs joined by arrows that follow the same rule, plus a third pair with a missing number. Riverside's product guide notes that some Level 9 questions still use the matrix format, with pairs arranged vertically, while the rest use a row of number pairs; its practice script tells students that the arrows mean the same thing in both layouts. The student must work out the rule and apply it to the target number. Scratch paper is allowed on the actual test.

This transition from concrete objects to abstract symbols requires third graders to possess strong number sense, flexible mental math strategies, and instant recall of basic arithmetic facts.

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Structural Anatomy of a Number Analogy Prompt

Every Number Analogy question on CogAT Level 9 follows an invariant three-part architecture. The prompt presents two complete reference pairs that embody an identical mathematical rule, followed by a third incomplete pair that contains the target number and a question mark:

$\mathbf{[\text{Pair 1: } x \to y]} \quad \mathbf{[\text{Pair 2: } a \to b]} \quad \mathbf{[\text{Target Pair: } m \to ?]}$

Consider this foundational Grade 3 prompt:

$\mathbf{[3 \to 12]} \quad \mathbf{[5 \to 20]} \quad \mathbf{[8 \to ?]}$

Why Two Reference Pairs Are Provided

Students often wonder why the test provides two reference pairs instead of just one. The reason is rooted in mathematical ambiguity. A single numerical pair almost always supports multiple valid rules:

• If a student looks only at $[3 \to 12]$, they might hypothesize that the rule is "add 9" ($3 + 9 = 12$).
• Alternatively, they might hypothesize that the rule is "multiply by 4" ($3 \times 4 = 12$).
• Both operations are completely valid for the first pair alone!

This is where Pair 2 serves as the indispensable verification gate. When the student checks $[5 \to 20]$:

• Testing the addition rule: $5 + 9 = 14 \neq 20$ (Fails!).
• Testing the multiplication rule: $5 \times 4 = 20$ (Matches perfectly!).

Pair 2 eliminates competing interpretations and confirms the single true mathematical law governing the problem. Once confirmed, that exact law is applied to the third pair: $8 \times 4 = 32$.

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The Power of the Verbal Bridging Phrase

Riverside's Level 9 practice guide encourages students to think of a rule that describes how to change the first number into the second, for example "the second number is 8 less than the first." This guide calls that sentence the Bridging Phrase.

A bridging phrase describes the exact journey from the first number (the input) to the second number (the output). It always takes the standard form:

$\text{"The second number is } [\text{mathematical operation}] \text{ the first number."}$

Formulating Effective Bridging Phrases

• For $[4 \to 16]$: "The second number is 4 times the first number." (Multiplication)
• For $[18 \to 11]$: "The second number is 7 less than the first number." (Subtraction)
• For $[7 \to 15]$: "The second number is 8 more than the first number." (Addition)
• For $[24 \to 6]$: "The second number is the first number divided by 4." (Division)

By silently speaking the bridging phrase, the student anchors the operational direction in working memory. This verbal anchor prevents the common mistake of reversing the operation (e.g., dividing instead of multiplying) when calculating the final answer.

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The Four Single-Step Operational Transformations

Riverside's Level 9 practice guide says that most Number Analogies questions require only one rule, while some use two rules, such as adding and then doubling. Its advice is to try the simplest rule first. One-step rules fall into four families:

1. Addition Transformations ($x + c$)

• Visual Indicator: The output number is moderately larger than the input number.
• Cognitive Hallmark: The difference between output and input remains constant across both pairs ($y - x = b - a = c$).
• Practice Tip: Include sums that cross a tens boundary (e.g., $17 + 8 = 25$), where most slips happen.

2. Subtraction Transformations ($x - c$)

• Visual Indicator: The output number is moderately smaller than the input number.
• Cognitive Hallmark: The output is obtained by taking away a fixed constant ($x - y = a - b = c$).
• Practice Tip: Include differences that cross a tens boundary (e.g., $32 - 7 = 25$).

3. Multiplication Transformations ($x \times c$)

• Visual Indicator: The output number increases substantially relative to the input number.
• Cognitive Hallmark: The ratio between output and input is identical across pairs ($y \div x = b \div a = c$).
• Practice Tip: Riverside's Level 9 practice items include doubling; fluent recall of facts through $10 \times 10$ covers most other multiplication rules.

4. Division Transformations ($x \div c$)

• Visual Indicator: The output number decreases sharply relative to the input number.
• Cognitive Hallmark: The input divided by the output yields a constant whole-number divisor ($x \div y = a \div b = c$).
• Practice Tip: Division rules are the inverse of multiplication facts (e.g., $54 \div 6 = 9$, $56 \div 8 = 7$, $72 \div 9 = 8$); Riverside's practice items include halving.

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Comprehensive Guide to Single-Step Rules

The following reference table summarizes the four single-step rule families with original practice examples:

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Worked Step-by-Step Problem Models

To master single-step analogies, let us examine four distinct worked problem models representing each arithmetic operation.

Model 1: Addition Transformation

Prompt: $[6 \to 15] \quad [11 \to 20] \quad [18 \to ?]$

1. Analyze Pair 1: The number grows from 6 to 15. The change is an increase of 9 ($15 - 6 = 9$). We hypothesize: Add 9.
2. Verify on Pair 2: We apply the hypothesis to 11. Does $11 + 9 = 20$? Yes, exactly 20. The addition rule ($+9$) is verified.
3. Formulate Bridging Phrase: "The second number is 9 more than the first number."
4. Calculate Target: Apply the verified rule to 18: $18 + 9 = 27$.
5. Confirm Solution: The correct missing value is 27.

Model 2: Subtraction Transformation

Prompt: $[22 \to 14] \quad [31 \to 23] \quad [19 \to ?]$

1. Analyze Pair 1: The number decreases from 22 to 14. The drop is 8 units ($22 - 14 = 8$). We hypothesize: Subtract 8.
2. Verify on Pair 2: We apply the hypothesis to 31. Does $31 - 8 = 23$? Yes, $31 - 8 = 23$. The subtraction rule ($-8$) is verified.
3. Formulate Bridging Phrase: "The second number is 8 less than the first number."
4. Calculate Target: Apply the rule to 19: $19 - 8 = 11$.
5. Confirm Solution: The correct missing value is 11.

Model 3: Multiplication Transformation

Prompt: $[5 \to 35] \quad [8 \to 56] \quad [6 \to ?]$

1. Analyze Pair 1: The number jumps from 5 to 35. Comparing growth rates, adding 30 is possible, but 35 is a direct multiple of 5 ($5 \times 7 = 35$). We hypothesize: Multiply by 7.
2. Verify on Pair 2: Check 8 against the rule. Does $8 \times 7 = 56$? Recalling standard third-grade multiplication facts confirms $8 \times 7 = 56$. The additive rule $+30$ completely fails ($8 + 30 = 38 \neq 56$). Multiplication by 7 is firmly confirmed.
3. Formulate Bridging Phrase: "The second number is 7 times the first number."
4. Calculate Target: Apply the rule to 6: $6 \times 7 = 42$.
5. Confirm Solution: The correct missing value is 42.

Model 4: Division Transformation

Prompt: $[28 \to 7] \quad [36 \to 9] \quad [44 \to ?]$

1. Analyze Pair 1: The number contracts from 28 to 7. A difference of $-21$ is possible, but 28 divided by 4 equals 7 ($28 \div 4 = 7$). We hypothesize: Divide by 4.
2. Verify on Pair 2: Test 36 against the rule. Does $36 \div 4 = 9$? Yes, $9 \times 4 = 36$. Testing the subtraction hypothesis yields $36 - 21 = 15 \neq 9$. Division by 4 is verified.
3. Formulate Bridging Phrase: "The second number is the first number divided by 4."
4. Calculate Target: Apply the rule to 44: $44 \div 4 = 11$.
5. Confirm Solution: The correct missing value is 11.

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Practical Solving Guidelines for Single-Step Items

When practicing single-step analogies, students should apply three golden habits:

1. Check Growth Velocity First: If the second number is only slightly larger or smaller (less than double), test addition or subtraction first. If the second number is three, four, or five times larger, test multiplication immediately.
2. Never Skip Pair 2 Verification: Riverside's practice guide lists finding a rule that works for only one pair as a common mistake, along with using the wrong operation. A student who looks only at $[4 \to 8]$ will guess $+4$ when the actual rule might be $\times 2$.
3. Re-Read the Bridging Phrase Aloud Silently: Before marking the answer sheet, repeat: "If 6 times 7 is 42, then 9 times 7 is 63." This mental audio check catches careless multiplication slips instantly.