Instructions
Solve the following problems that connect the worlds of Wobbledogs, Lego robotics, and algebraic thinking. Show your work where necessary to receive full credit. Calculators are permitted, but try to solve what you can without one.
Part 1: Wobbledog Combinatorics
1. In the world of Wobbledogs, a dog's physical form is determined by its genes. Imagine a simplified set of genes that control a dog's core features.
- Body Shape (3 options): Round, Square, Tube
- Leg Count (4 options): Two, Four, Six, Eight
- Tail Style (5 options): Nub, Curly, Pointy, Forked, Spade
a) If a Wobbledog must have one and only one option from each of these three gene categories, how many unique Wobbledog combinations are possible?
b) You've just discovered a new gene for Wings, which has three possibilities: None, Feathered, or Bat-like. Factoring in this new gene category, what is the new total number of unique Wobbledog combinations?
Part 2: Pythagorean Robotics with Lego Spike Prime
2. You've programmed your Spike Prime robot to navigate a course on a large grid. The robot starts at coordinate (0, 0). Its program instructs it to move 15 units directly North, and then 8 units directly West to retrieve a Lego piece.
What is the straight-line distance from the robot's starting point (0, 0) to the final location of the Lego piece? (This is the distance "as the crow flies.")
3. You are building a support structure for your robot's lifting arm and need to create a perfect right-angled triangle using Lego beams. The longest beam (the hypotenuse) is 25 Lego units long. One of the other beams (a leg) is 7 Lego units long. What must the length of the third beam be to complete the right triangle?
Part 3: The Algebraic Challenge
4. You are designing an advanced rectangular pen for your prized Wobbledog. You've determined that the total perimeter of the pen must be 46 feet. To ensure the dog has enough room to run diagonally, the diagonal distance from one corner to the opposite corner must be 17 feet.
Let the length of the pen be L and the width be W.
a) Based on the information above, write two distinct equations using the variables L and W. One equation should represent the perimeter, and the other should represent the diagonal relationship.
b) Solve the system of equations you created in part (a) to find the specific length and width of the pen.
Answer Key
1. a) To find the total combinations, multiply the number of options in each category: 3 Body Shapes × 4 Leg Counts × 5 Tail Styles = 60 unique combinations.
1. b) Multiply the previous result by the number of options in the new category: 60 combinations × 3 Wing options = 180 unique combinations.
2. This forms a right triangle with legs of length 15 and 8. Using the Pythagorean theorem (a² + b² = c²):
15² + 8² = c²
225 + 64 = c²
289 = c²
c = √289
c = 17 units.
3. The hypotenuse (c) is 25 and one leg (a) is 7. We need to find the other leg (b).
a² + b² = c²
7² + b² = 25²
49 + b² = 625
b² = 625 - 49
b² = 576
b = √576
b = 24 Lego units.
4. a) The two equations are:
Perimeter: 2L + 2W = 46 (which simplifies to L + W = 23)
Diagonal: L² + W² = 17² (which is L² + W² = 289)
4. b) Solution using substitution:
- From the perimeter equation:
L = 23 - W - Substitute this into the diagonal equation:
(23 - W)² + W² = 289 - Expand the equation:
(529 - 46W + W²) + W² = 289 - Combine like terms and set to zero:
2W² - 46W + 240 = 0 - Divide by 2 to simplify:
W² - 23W + 120 = 0 - Factor the quadratic equation:
(W - 8)(W - 15) = 0 - The possible values for W are 8 and 15. If W = 8, then L = 15. If W = 15, then L = 8.
The dimensions of the pen are 15 feet by 8 feet.