Spatial Aptitude

Spatial reasoning improves when a shape is represented with coordinates, a grid, or named faces. State the viewing direction and rotation convention before moving a shape. The text descriptions and tables below serve as figure descriptions: each gives enough information to reconstruct the arrangement without an image.

Translation, rotation, reflection, and scaling

Idea. A translation moves every point by the same vector and preserves size and orientation. A rotation turns a figure about a stated center; a reflection reverses handedness across an axis. A scale factor $k$ multiplies lengths by $k$ and areas by $k^2$; for $k>0$, it preserves shape and orientation.

Solved examples.

1. Question. Triangle vertices are $(1,1),(4,1),(2,3)$. Translate by vector $(2,-1)$. Solution. (1) Add 2 to each x-coordinate. (2) Subtract 1 from each y-coordinate. (3) Vertices become $(3,0),(6,0),(4,2)$. Answer. Those three image vertices. Trap. A translation moves every vertex equally.
2. Question. Rotate point $(3,1)$ by 90 degrees counterclockwise about origin. Solution. (1) Use $(x,y)\mapsto(-y,x)$. (2) Substitute: $(3,1)\mapsto(-1,3)$. Answer. $(-1,3)$. Trap. Clockwise rotation instead uses $(y,-x)$.
3. Question. Reflect point $(-2,5)$ across the y-axis. Solution. (1) A y-axis reflection changes the sign of x only. (2) y remains 5. Answer. $(2,5)$. Trap. Reflection across x-axis would instead give $(-2,-5)$.
4. Question. A rectangle is 3 by 5 units. Scale every length by 2 about one corner. State image dimensions and area factor. Solution. (1) Side lengths become 6 and 10. (2) Original area is 15; new area is 60. (3) The area factor is $2^2=4$. Answer. 6 by 10; area factor 4. Trap. Area does not scale by the length factor.
5. Question. A right-pointing arrow is reflected across a vertical mirror line. Which way does its tip point? Solution. (1) Reflection reverses left-right position. (2) The direction vector $(1,0)$ becomes $(-1,0)$. Answer. Left. Trap. Reflection is not a 180-degree rotation for shapes with asymmetric details.

Assembling, grouping, and 2-D patterns

Idea. To assemble pieces, track boundary lengths, orientation, and whether edges align. A grouping rule should be stated as a property shared by all members, such as equal area or rotational symmetry. For a pattern sequence, compare one feature at a time: count, orientation, position, or shading.

Solved examples.

1. Question. Two identical unit squares touch along a full side. What is the perimeter of the combined rectangle? Solution. (1) Each square contributes perimeter 4, total 8 before joining. (2) The shared edge of length 1 is internal and counted twice. (3) Subtract 2: perimeter 6. Answer. 6 units. Trap. A shared boundary disappears from the outside perimeter.
2. Question. Three unit squares are placed in a straight row. Describe the result and perimeter. Solution. (1) The outline is a 3-by-1 rectangle. (2) Perimeter is $2(3+1)=8$. Answer. 3-by-1 rectangle, perimeter 8. Trap. Count only exposed edges.
3. Question. A shape has vertices $(0,0),(2,0),(2,1),(0,1)$. Is it unchanged by a 180-degree rotation about its center? Solution. (1) Its center is $(1,0.5)$. (2) A half-turn maps opposite corners to one another. (3) The rectangle's set of points coincides with itself. Answer. Yes. Trap. A rectangle has half-turn symmetry even if it is not a square.
4. Question. Four figures have shaded corner counts 1, 2, 3, 4, while all other features match. Predict the fifth if the count increases by one. Solution. (1) Counts form +1 increments. (2) Continue the stated feature only. Answer. Five shaded corners, if the figure has enough corners or repeats the pattern. Trap. A pattern rule must respect the shape's available positions.
5. Question. Pieces A and B are rectangles 2-by-1 and 1-by-1. Can they tile a 3-by-1 rectangle without overlap or gaps? Solution. (1) Their total area is 2+1=3. (2) Place the long side of A along two units of the target. (3) Place B in the remaining unit. Answer. Yes. Trap. Equal area is necessary but not always sufficient; also check edge geometry.

Paper folding, cutting, and 3-D transformations

Idea. A fold reflects material across the crease. A cut made through multiple layers appears at every layer after unfolding, mirrored across each crease. For a cube, record face adjacencies and orientation; opposite faces never share an edge. Rotating a solid preserves adjacency, while mirroring reverses handedness.