Frogs Investigation

mrchapmanmaths.com/algebra/frogs

This interactive tool provides a digital environment for the classic "Frogs" investigation, allowing pupils to explore logical sequencing and algebraic generalisation. By manipulating two sets of frogs across a central void, pupils can systematically record data to uncover the underlying patterns governing the minimum number of moves required.

How the tool works

The interface provides a clean, visual representation of the problem, removing the physical fiddle of using counters or Unifix cubes in the classroom.

• Customisable Setup: Teachers or pupils can use the controls to set the number of frogs on the left (\(n\)) and the right (\(m\)), allowing for both symmetrical (\(n \times n\)) and asymmetrical (\(n \times m\)) investigations.
• Interactive Movement: Frogs move via a simple click; the tool enforces the rules (sliding to an adjacent space or jumping over one frog of a different colour), preventing illegal moves that often confuse the investigation on paper.
• Live Counters: The tool automatically tracks the number of Slides, Jumps, and Total Moves, allowing pupils to focus on the mathematical structure rather than manual bookkeeping.
• Reset Functionality: Instant resets encourage a "low-stakes" trial-and-improvement approach, essential for pupils struggling to find a consistent movement strategy.

Classroom Uses

Generating and Tabulating Sequences

The tool is excellent for KS3 pupils to practice systematic recording. By starting with \(1\) frog on each side and moving up to \(5\) or more, pupils can generate a sequence of total moves (\(3, 8, 15, 24, \dots\)).

Strategy: Ask pupils to predict the moves for a \(4 \times 4\) setup before using the tool to verify. This builds the habit of searching for a term-to-term rule before jumping to the \(n^{th}\) term.

Deriving Quadratic Generalisations

For higher-tier GCSE pupils, the tool facilitates the derivation of the formula \(n^2 + 2n\). Because the tool separates "Slides" and "Jumps," pupils can see that the number of slides is always \(2n\) and the number of jumps is \(n^2\).

Example: In a \(3 \times 3\) game, pupils will observe \(6\) slides and \(9\) jumps. This makes the structural generalisation \(M = n^2 + 2n\) or \(M = (n+1)^2 - 1\) far more intuitive than through a table of values alone.

Multi-variable Investigation

The tool allows for \(n \times m\) investigations (e.g., \(3\) frogs on the left and \(2\) on the right). This pushes more able pupils toward complex generalisation involving two variables.

Strategy: Challenge pupils to find a rule for the total moves when \(n \neq m\). They should eventually discover that Total Moves \(= nm + n + m\).

Teaching Strategy

1. Demonstration: Project the tool and show a \(1 \times 1\) game. Ask the class to define the "rules" of movement based on what they see.
2. Strategy Hunt: Give pupils 5 minutes to find a "no-fail" strategy for a \(3 \times 3\) game. Discuss why "gridlock" happens (usually when two frogs of the same colour end up side-by-side).
3. Data Collection: Pupils use the tool to complete a table for \(n = 1, 2, 3, 4, 5\), recording Jumps (\(J\)), Slides (\(S\)), and Total Moves (\(M\)).
4. Algebraic Transition: Ask pupils to find the relationship between \(n\) and \(S\), then \(n\) and \(J\).
5. Verification: Use the tool to test their resulting formula with a larger value, such as \(n = 6\).

Pedagogical Value

The digital Frogs tool significantly reduces the "clutter" of the investigation. In a traditional "pen and paper" or counter-based lesson, pupils often lose track of their count or make an illegal move halfway through, forcing a total restart that can lead to frustration and disengagement. By automating the counting of jumps and slides, the tool makes the mathematical structure visible. It allows pupils to "see" that jumps increase quadratically while slides increase linearly, providing a concrete foundation for the abstract quadratic generalisation required at KS4.