Grade 10: Space

Warm up with a few quick space warm-ups on the board while the class settles, so every child starts thinking about the skill.

A network (or graph) models connections between things using vertices (points) and edges (connecting lines). Euler's formula, V - E + F = 2, relates the number of vertices, edges and faces of any connected planar network. Model the method clearly, thinking aloud:

• Introduce networks with a real example (a map of train stations, a social network of friendships) before any formula.
• Count vertices, edges and faces together on a simple drawn network, including the outer face, before stating Euler's formula.
• Verify the formula on several small examples (a triangle, a square, a network with one internal division) to build confidence it always holds.
• Practise rearranging the formula to find whichever one value (V, E or F) is missing, given the other two.
• Connect networks to real situations: interpreting what a network diagram represents, not just calculating with it.

Work this through step by step on the board, then have the class talk you through a second one.

• A connected planar network has 6 vertices and 9 edges. Find the number of faces
• F = 2 - V + E = 2 - 6 + 9 = 5

Do the first few questions of the practice worksheet together, one child explaining each step. Check for understanding before releasing the class to work alone.

Students complete the practice worksheet independently while you circulate and support.

Circulate and look for these, they are the usual sticking points:

• Forgetting to count the outer (unbounded) region as one of the faces.
• Rearranging Euler's formula incorrectly, e.g. adding instead of subtracting a term.
• Miscounting vertices or edges on a busy diagram, especially where edges cross without an actual vertex.
• Applying Euler's formula to a network that is not connected or not planar, where it does not directly hold.

Pull the class back together. Ask one child to explain space in their own words, pose a single check question everyone answers on a mini whiteboard, and name what you will build on next lesson.

Use the independent worksheet as the evidence. A child who can complete it accurately and explain one answer has met the objective; anyone who cannot needs the easier level and a short reteach next session.