Play
• For players of your maze game, the goal is to move from the starting point to
the ending point using the shortest possible path—the path that requires
traveling the least total distance.
• Players can only move through the maze in horizontal or vertical directions.
They cannot move diagonally or through polygons.
• Show and code two trial games. Use colored pencils or markers to draw two
different paths through the maze. Use coordinates to describe each path.
• The device will use the coordinates for each path to mathematically calculate
the total distance. Show how the device will calculate the total distance for
each trial path that you drew.
• Is either trial path the shortest possible path? Explain.
Game 1:
Let's assume our maze is 5x5 grids, and the starting point A is at (0,0), and the ending point B is at (4,4). Here are two possible paths:
Path 1 (marked in blue): A (0,0) - P (4,0) - Q (4,2) - R (2,2) - B (4,4)
Coordinates: (0,0) - (4,0) - (4,2) - (2,2) - (4,4)
Path 2 (marked in red): A (0,0) - X (2,0) - Y (2,4) - B (4,4)
Coordinates: (0,0) - (2,0) - (2,4) - (4,4)
To calculate the total distance, we can use the Manhattan distance formula which is simply the sum of horizontal and vertical distances between two points.
For Path 1:
The distance from A to P = 4; P to Q = 2; Q to R = 2; R to B = 4.
Total distance for Path 1 = 4 + 2 + 2 + 4 = 12 units
For Path 2:
The distance from A to X = 2; X to Y = 4; Y to B = 2.
Total distance for Path 2 = 2 + 4 + 2 = 8 units
Is either path the shortest possible path?
Path 2 is the shorter of the two paths, but to determine if Path 2 is the shortest possible, let's check the minimum required distance without the maze. The Manhattan distance from A to B without any obstacles is |4-0|+|4-0|=8. Since our Path 2 is equal to the minimum required distance from A to B, we can conclude that Path 2 is the shortest possible path.
To play the maze game and calculate the total distance for each trial path, follow these steps:
1. Start by drawing two different mazes with a starting point and an ending point. Make sure the mazes have multiple paths to choose from.
2. For the first trial game, use colored pencils or markers to draw one path through the maze. Let's call this path A.
3. Assign coordinates to each point along path A. You can use letters for the horizontal axis and numbers for the vertical axis. For example, the starting point can be A1, and the ending point can be D5.
4. To calculate the total distance for path A, you need to find the distance between each pair of consecutive coordinates and sum them up. To do this, measure the distance between each point using a ruler or any suitable measuring tool. Add these distances together to find the total distance.
5. Repeat steps 2-4 for the second trial game, drawing a different path B through the maze and assigning coordinates to each point along path B.
6. Compare the total distances of paths A and B. If the total distance of path A is smaller than path B, then path A is the shortest possible path. Similarly, if the total distance of path B is smaller, then path B is the shortest possible path.
7. Explain why the shortest path is the one with the least total distance. In this maze game, players can only move horizontally or vertically, so diagonal movements or traversing through polygons are not allowed. Therefore, the shortest path would be the one that maximizes the use of horizontal and vertical movements, minimizing any unnecessary detour.
By following these steps, you can play the maze game, calculate the total distances for each trial path, and determine if either path is the shortest possible path.
To explain how to calculate the total distance for each trial path, we need to understand the principles mentioned in the given information. Let's break it down step by step:
1. Visualization of the maze: Draw the maze using colored pencils or markers. Keep in mind that players can only move horizontally or vertically, not diagonally. The maze should have a starting point and an ending point.
2. Trial Game 1: Draw a path from the starting point to the ending point. Use coordinates to describe this path. For example, you can assign letters to the rows and numbers to the columns, and then note the coordinates as you draw the path. For instance, (A1, A2, A3) represents a horizontal path along the first row.
3. Trial Game 2: Draw a different path from the starting point to the ending point using the same coordinate system as in Trial Game 1. Again, note the coordinates as you draw the path.
4. Calculating the total distance: To mathematically calculate the total distance for each trial path, first, assign a distance value to each line segment of the path. Assuming the line segments are of equal length, you can assign a value of 1 to each segment. Count the number of line segments on the path and that will give you the total distance.
Example: If Trial Game 1 has five line segments, then the total distance for Trial Game 1 would be 5.
Repeat the same process for Trial Game 2 to find its total distance.
5. Determining the shortest possible path: Compare the total distances of both trial paths. The path with the lower total distance will be the shortest possible path. Explain this in terms of the number of line segments or distance values assigned.
Note: It's important to mention that the total distance calculation assumes equal length line segments and is a simplified approach. In a real maze game, distances may vary, and more precise calculations may be needed.
By following these steps, you can show how the device would calculate the total distance for each trial path and determine which path is the shortest.