5.1 Number Puzzles: Missing Numbers, Shape Substitution & Bottom-Up Solving

5.1 Number Puzzles: Missing Numbers, Shape Substitution & Bottom-Up Solving

The Number Puzzles subtest is the second part of the Quantitative Battery on CogAT Level 9 (Grade 3). At the primary levels (Levels 5/6–8), this subtest shows two trains, and children choose the picture that makes the second train carry the same number of objects as the first. At Level 9, the trains are replaced by short written equations. Riverside's product guide describes the format: students see one or more equations in which at least one number is missing. If one element is missing, they choose the missing number. If two or three elements are missing, they substitute the numbers provided for those elements and then solve the equation.

With 16 items in 10 minutes, students have about 37.5 seconds per question, tied with Paper Folding for the most generous pace on the test. Riverside's practice guide says this subtest measures understanding of mathematical identities: the idea that both sides of an equals sign name the same amount. Students may use scratch paper on the actual test.

---

What Level 9 Number Puzzles Look Like

The missing number is always shown as a square with a question mark (written as ? below). Harder items add other shapes, such as a diamond (◆) or a circle (●), whose values are given in extra rows. The answer is always the value of the question-mark square, never the value of a helper shape.

Riverside's Level 9 practice items use whole numbers with addition and subtraction. Quick recall of basic facts helps, but the core skill is keeping both sides equal and tracking which shape stands for which number.

---

The Equals Sign Means "Same Amount"

Many children first learn the equals sign as a command: "write the answer here." When they see $7 + 5 = \_\_$, they compute 12 and move on. That habit breaks down on Number Puzzles, where the unknown can sit on either side and both sides can hold several numbers.

Consider this original practice item:

$9 + 4 = ? + 6$

A student using the "write the answer" habit computes $9 + 4 = 13$ and picks 13. But then the right side would be $13 + 6 = 19$, and 13 does not equal 19. A student who treats the equals sign as "same amount" reasons in three steps:

1. Total the complete side: $9 + 4 = 13$.
2. Set the target: the other side must also total 13.
3. Solve for the unknown: $? + 6 = 13$, so $? = 13 - 6 = 7$.

A balance scale is a helpful picture for this idea at home: whatever sits on one pan must weigh the same as what sits on the other. The test itself shows equations, not scales, so practice should move quickly from the picture to written equations.

---

Bottom-Up Solving for Shape Rows

Riverside's practice guide lists working from the top down as a common mistake. In a multi-row item, the top row contains the question mark but usually cannot be solved first, because it still contains an unknown helper shape. The bottom row states or nearly states a helper value, so that is where to begin.

Worked example (original practice item):

• Row 1: ? − ◆ = 8
• Row 2: ◆ + 2 = 5

1. Solve the bottom row first: ◆ + 2 = 5, so ◆ = 3.
2. Substitute upward: replace the diamond in Row 1 with 3, giving ? − 3 = 8.
3. Solve for the question mark: ? = 8 + 3 = 11.
4. Check every row: 11 − 3 = 8 and 3 + 2 = 5. Both rows work, so the answer is 11.

Three-row items follow the same chain, one link longer: the bottom row gives one shape, the middle row uses it to find the next shape, and the top row uses that to find the question mark. Writing each shape's value on scratch paper as soon as it is found keeps working memory free for the next step.

---

The 4-Step Solving Protocol

---

Distractor Traps and Common Third-Grade Slips

Riverside's practice guide names three common mistakes on this subtest, and practice distractors often match them:

1. Solving for the wrong shape: In a two-row item, the student finds the diamond and chooses that number instead of the question mark. In the worked example above, 3 is the trap; the answer is 11.
2. Working from the top down: Trying to solve Row 1 while it still contains an unknown helper shape leads to guessing. Start at the bottom.
3. Using the wrong operation: Adding when the item requires subtracting, or the reverse. In $20 - ? = 14$, a student who adds gets 34; the answer is $20 - 14 = 6$.

One more trap appears constantly in practice:

1. The "first total" lure: In $9 + 4 = ? + 6$, the total of the complete side (13) is a tempting choice. It is the target for the other side, not the missing number.