Elementary Pull-Out: Only the Shadows Know

The Elementary Mathematician, COMAP's award-winning K-6 newsletter is available for the first time in electronic format. Each issue contains a collection of K-6 mathematic lessons that are fun to teach and exciting to learn.

Each issue of The Elementary Mathematician contains a Pull-Out section which includes thematic lessons based on the national census, wildlife, other cultures, technology, and provides a practical way to bring the concepts and pedagogy of the NCTM Standards into your classroom. 

Note: The information below was created with the assistance of AI.

Level of Mathematics
This activity is most appropriate for upper elementary to early middle school students, likely Grades 4–7.

Mathematical Complexity: Moderate

While no equations are directly used, the concepts involve geometric reasoning, proportional thinking, and spatial visualization.

Ideal for enrichment or pull-out programs to deepen understanding of light, perspective, and geometry.

Prerequisite Skills
Students should be familiar with:

Basic geometry (angles, parallel lines, points, line segments)

Measurement and scaling

Use of rulers and drawing tools to model projections

Shadow formation concepts from science (basic light travel and obstruction)

Logical reasoning to infer locations and shapes based on indirect data

No calculator or computation-heavy work is required, but visual reasoning is essential.

Application Areas

1. Geometry and Spatial Reasoning
Core to this activity is the use of geometric intuition to interpret and project the direction and shape of shadows.

Students explore relationships between objects, light sources, and the shapes/sizes of resulting shadows.

2. Physics (Light and Optics)
Integrates science by applying concepts of light propagation, obstruction, and shadow casting.

Encourages students to think about how shadows form depending on the position of the light source.

3. Art and Perspective
Visual interpretation and drawing of light effects mimics techniques used in art and design, particularly with perspective and shading.

4. Critical Thinking
Students must deduce hidden information (e.g., the location of the lamp or the shadow origin point) from the visible effects (shadows).

Encourages multiple problem-solving strategies and visual estimation.

Subject Matter Overview
Key Mathematical and Conceptual Topics:

Light and shadow relationships: How angles from a light source create varying lengths and directions of shadows.

Symmetry and parallelism: Recognizing consistent patterns of projection from similar objects.

Proportionality: Though not computed directly, students engage with proportional reasoning when shadows lengthen or shorten with distance or object height.

Coordinate/planar reasoning: Many problems are posed with top or side views, requiring students to interpret a 2D diagram spatially.

Types of Problems Included:

Drawing shadows in different lighting contexts (e.g., lamp vs. sunlight)

Inferring light source location based on object shadows

Finding entry points into light/shadow zones for moving objects (e.g., bird, rat)

Constructing and matching shadows for knotted ropes based on light angle

Correlation to Mathematics Standards
Common Core State Standards (CCSS):

CCSS.MATH.CONTENT.4.G.A.1: Draw points, lines, line segments, rays, angles, and identify these in two-dimensional figures.

CCSS.MATH.CONTENT.5.G.B.4: Classify two-dimensional figures based on properties of lines and angles.

CCSS.MATH.CONTENT.6.G.A.1: Solve real-world problems involving area, surface area, and volume.

CCSS.MATH.PRACTICE.MP4: Model with mathematics—students use diagrams to model real-life shadow projections.

CCSS.MATH.PRACTICE.MP5: Use appropriate tools strategically—rulers, diagrams, spatial tools.

CCSS.MATH.PRACTICE.MP7: Look for and make use of structure—e.g., equal spacing, projection lines.

NCTM Process and Content Standards:

Geometry: Use visualization, spatial reasoning, and geometric modeling to solve problems.

Connections: Recognize and apply mathematics in contexts outside of mathematics.

Problem Solving: Build new mathematical knowledge through problem solving and apply it in varied contexts.