Grade 3 Cognitive Skills (CogAT): 10 Free Practice Questions

Answer: 12
The correct answer is 12, since the rule is 'multiply by 2, then add 2.' Step 1: Compare 3 and 8. Test multiply by 2: 3 × 2 = 6, then add 2: 6 + 2 = 8. This confirms the two-step rule 'multiply by 2, then add 2.' Step 2: Apply the same rule to 5: 5 × 2 = 10, then 10 + 2 = 12. Students often apply only one step of the rule, such as multiplying by 2 alone (5 × 2 = 10), and forget to add 2, incorrectly choosing 10.

Answer: <svg xmlns="http://www.w3.org/2000/svg" width="150" height="232" viewBox="0 0 150 232"><rect x="0" y="0" width="150" height="232" fill="#ffffff"/><rect x="1" y="1" width="148" height="188" fill="#ffffff" stroke="#111827" stroke-width="2"/><polygon points="75.0,64.4 101.5,79.7 101.5,110.3 75.0,125.6 48.5,110.3 48.5,79.7" fill="#1f2937" stroke="#111827" stroke-width="2"/><rect x="0" y="188.0" width="150" height="44" fill="#111827"/><text x="75.0" y="219.0" font-size="22" font-weight="bold" font-family="Arial, sans-serif" text-anchor="middle" fill="#ffffff">C</text></svg>
Rule: the hexagon's shading (darkness) increases by a constant step of 1 unit(s) per panel (shading-scale index). Panel 1 = shading level 3 of 6 (darker = higher), panel 2 = shading level 4 of 6 (darker = higher), panel 3 = shading level 5 of 6 (darker = higher). Adding the same step once more gives the missing panel: shading level 6 of 6 (darker = higher) — this is option C. Option B applies a step of 2 unit(s) instead of 1, over- or under-shooting the established rate of change. Option A shows the same shading (darkness) as panel 2, reversing the progression instead of continuing it forward. Option D has the correct shading (darkness) value but substitutes a diamond for the hexagon, breaking the requirement that the same shape appear in every panel.

Answer: <svg xmlns="http://www.w3.org/2000/svg" width="62" height="62" viewBox="0 0 62 62"><rect x="0" y="0" width="62" height="62" fill="#efeade" stroke="#211d16" stroke-width="1.4" rx="4"/><polygon points="31.0,6.0 52.7,18.5 52.7,43.5 31.0,56.0 9.3,43.5 9.3,18.5" fill="#fffdf7" stroke="#3a4a63" stroke-width="1.8"/><circle cx="52.7" cy="18.5" r="4" fill="#8a3b2b" stroke="#1c1a15" stroke-width="1.1"/><circle cx="31.0" cy="56.0" r="4" fill="#8a3b2b" stroke="#1c1a15" stroke-width="1.1"/><circle cx="9.3" cy="43.5" r="4" fill="#8a3b2b" stroke="#1c1a15" stroke-width="1.1"/><circle cx="9.3" cy="18.5" r="4" fill="#8a3b2b" stroke="#1c1a15" stroke-width="1.1"/></svg>
FRAME C must contain every dot that appears in FRAME A or FRAME B — a logical OR of the two source frames, not an overlap-only AND. Example 1 shows A={1,2,5} combined with B={1,3,4} to make C={1,2,3,4,5}: every dot from either source survives. Example 2 confirms the same rule with A={2,3,4} and B={2,3,4} making C={2,3,4}. For the final row, A={1,3,5} and B={3,4,5}, so the correct sum is every corner in either set: {1,3,4,5} — choice D. Verify by listing A3's corners, then B3's corners, then merging the two lists and removing repeats; that merged list should match choice D exactly. Each wrong choice fails for a specific, checkable reason: one keeps only the corners common to both frames (an AND instead of an OR), one is missing a dot that should have carried over from one of the source frames, and one has an extra dot at a corner that never appeared in A3 or B3 at all.

Answer: 7
The correct answer is 7, since the rule is divide by 3. Step 1: Verify the rule with 12: 12 ÷ 3 = 4, which matches. Step 2: Apply the rule to 21: 21 ÷ 3 = 7. Students may divide by 3 incorrectly using estimation, or subtract 8 (the difference between 12 and 4) instead, incorrectly choosing 13, mismatched here to distractor 8 pattern-based errors.

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Rule: each row is assigned a fixed shape (parallelogram for row 1, hexagon for row 2, pentagon for row 3) that stays constant across all three columns, and each column is assigned a fixed size (small for column 1, medium for column 2, large for column 3) that stays constant across all three rows. The missing cell is row 3, column 3, so it must combine row 3's shape (pentagon) with column 3's size (large) — this is option A. Option B uses the correct row-3 shape (pentagon) but the wrong column size (medium instead of large), violating the column rule. Option D uses the correct column-3 size (large) but the wrong shape (hexagon instead of pentagon), violating the row rule. Option C uses the correct size but substitutes a diamond, a shape that does not appear anywhere in the grid, violating the row rule entirely.

Answer: <svg xmlns="http://www.w3.org/2000/svg" width="170" height="262" viewBox="0 0 170 262"><rect x="0" y="0" width="170" height="262" fill="#ffffff"/><rect x="1" y="1" width="168" height="218" fill="#ffffff" stroke="#111827" stroke-width="2"/><polygon points="85.0,21.2 106.6,58.7 63.4,58.7" fill="#6b7280" stroke="#111827" stroke-width="2"/><polygon points="85.0,86.1 108.9,110.0 85.0,133.9 61.1,110.0" fill="#6b7280" stroke="#111827" stroke-width="2"/><circle cx="85.0" cy="173.8" r="21.7" fill="#6b7280" stroke="#111827" stroke-width="2"/><rect x="0" y="218.0" width="170" height="44" fill="#111827"/><text x="85.0" y="249.0" font-size="22" font-weight="bold" font-family="Arial, sans-serif" text-anchor="middle" fill="#ffffff">A</text></svg>
Rule: each successive box is generated by a single-step upward cyclic rotation of the stack — the shape that occupies the top position moves to the bottom, and the other two shapes shift down to fill the remaining slots. Box 1 shows triangle (top), diamond (middle), circle (bottom). Applying the rotation gives Box 2: diamond (top), circle (middle), triangle (bottom). Applying it again gives Box 3: circle (top), triangle (middle), diamond (bottom). Applying the same rotation a third time to Box 3 reproduces the missing stack: triangle (top), diamond (middle), circle (bottom) — this is option A. Option B rotates the stack in the downward direction instead, which incorrectly reconstructs Box 2's arrangement rather than continuing the established rotation. Option C applies an incomplete transformation, swapping only the top and bottom shapes (leaving the middle shape fixed) of Box 3 rather than performing the full three-position rotation, so it fails to follow the rule consistently. Option D preserves the correct rotated order but replaces the shape in the bottom position with a square, a shape outside the shared set of circle, triangle, diamond, violating the requirement that every box use only the same three shapes.

Answer: 4
A square has 4 lines of symmetry: two through the midpoints of opposite sides and two through opposite corners. Using a fold test, a square can be folded in half vertically, horizontally, and along both diagonals, with each fold producing two matching halves, giving 4 lines of symmetry total. Students count only the horizontal and vertical lines, forgetting the two diagonal lines, and incorrectly answer 2.

Answer: <svg xmlns="http://www.w3.org/2000/svg" width="170" height="262" viewBox="0 0 170 262"><rect x="0" y="0" width="170" height="262" fill="#ffffff"/><rect x="1" y="1" width="168" height="218" fill="#ffffff" stroke="#111827" stroke-width="2"/><polygon points="85.0,24.1 104.2,35.1 104.2,57.3 85.0,68.3 65.8,57.3 65.8,35.1" fill="#6b7280" stroke="#111827" stroke-width="2"/><polygon points="101.1,93.9 101.1,126.1 68.9,126.1 68.9,93.9" fill="#6b7280" stroke="#111827" stroke-width="2"/><polygon points="85.0,149.9 108.9,173.8 85.0,197.7 61.1,173.8" fill="#6b7280" stroke="#111827" stroke-width="2"/><rect x="0" y="218.0" width="170" height="44" fill="#111827"/><text x="85.0" y="249.0" font-size="22" font-weight="bold" font-family="Arial, sans-serif" text-anchor="middle" fill="#ffffff">A</text></svg>
Rule: each successive box is generated by a single-step downward cyclic rotation of the stack — the shape that occupies the bottom position moves to the top, and the other two shapes shift down to fill the remaining slots. Box 1 shows hexagon (top), square (middle), diamond (bottom). Applying the rotation gives Box 2: diamond (top), hexagon (middle), square (bottom). Applying it again gives Box 3: square (top), diamond (middle), hexagon (bottom). Applying the same rotation a third time to Box 3 reproduces the missing stack: hexagon (top), square (middle), diamond (bottom) — this is option A. Option B rotates the stack in the upward direction instead, which incorrectly reconstructs Box 2's arrangement rather than continuing the established rotation. Option C applies an incomplete transformation, swapping only the top and bottom shapes (leaving the middle shape fixed) of Box 3 rather than performing the full three-position rotation, so it fails to follow the rule consistently. Option D preserves the correct rotated order but replaces the shape in the bottom position with a parallelogram, a shape outside the shared set of square, hexagon, diamond, violating the requirement that every box use only the same three shapes.

Answer: <svg xmlns="http://www.w3.org/2000/svg" width="150" height="232" viewBox="0 0 150 232"><rect x="0" y="0" width="150" height="232" fill="#ffffff"/><rect x="1" y="1" width="148" height="188" fill="#ffffff" stroke="#111827" stroke-width="2"/><polygon points="75.0,66.8 103.2,95.0 75.0,123.2 46.8,95.0" fill="#6b7280" stroke="#111827" stroke-width="2"/><rect x="0" y="188.0" width="150" height="44" fill="#111827"/><text x="75.0" y="219.0" font-size="22" font-weight="bold" font-family="Arial, sans-serif" text-anchor="middle" fill="#ffffff">C</text></svg>
Rule: each row is assigned a fixed shape (square for row 1, hexagon for row 2, diamond for row 3) that stays constant across all three columns, and each column is assigned a fixed size (small for column 1, medium for column 2, large for column 3) that stays constant across all three rows. The missing cell is row 3, column 3, so it must combine row 3's shape (diamond) with column 3's size (large) — this is option C. Option A uses the correct row-3 shape (diamond) but the wrong column size (medium instead of large), violating the column rule. Option B uses the correct column-3 size (large) but the wrong shape (hexagon instead of diamond), violating the row rule. Option D uses the correct size but substitutes a triangle, a shape that does not appear anywhere in the grid, violating the row rule entirely.

Answer: <svg xmlns="http://www.w3.org/2000/svg" width="150" height="232" viewBox="0 0 150 232"><rect x="0" y="0" width="150" height="232" fill="#ffffff"/><rect x="1" y="1" width="148" height="188" fill="#ffffff" stroke="#111827" stroke-width="2"/><polygon points="94.0,76.0 94.0,114.0 56.0,114.0 56.0,76.0" fill="#6b7280" stroke="#111827" stroke-width="2"/><rect x="0" y="188.0" width="150" height="44" fill="#111827"/><text x="75.0" y="219.0" font-size="22" font-weight="bold" font-family="Arial, sans-serif" text-anchor="middle" fill="#ffffff">C</text></svg>
Rule: each row is assigned a fixed shape (hexagon for row 1, pentagon for row 2, square for row 3) that stays constant across all three columns, and each column is assigned a fixed size (small for column 1, medium for column 2, large for column 3) that stays constant across all three rows. The missing cell is row 3, column 3, so it must combine row 3's shape (square) with column 3's size (large) — this is option C. Option B uses the correct row-3 shape (square) but the wrong column size (medium instead of large), violating the column rule. Option A uses the correct column-3 size (large) but the wrong shape (pentagon instead of square), violating the row rule. Option D uses the correct size but substitutes a parallelogram, a shape that does not appear anywhere in the grid, violating the row rule entirely.