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Solving the Rabbit Problem and Pythagorean Theorem Application

In this lesson, students will learn to solve a real-world problem using the Pythagorean Theorem, determining the distance a hot air balloon would be from a standing point after rising vertically. The class will also explore recursive equations through the Rabbit Problem, following Fibonacci’s sequence, to calculate rabbit populations over months. Students will write rules for these sequences, engage in hands-on activities with index cards, and collaborate with peers to reinforce their understanding of mathematical patterns and relationships.

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Solving the Rabbit Problem and Pythagorean Theorem Application

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  1. Warm-Up 1. Solve: 2. Solve: 3. Use the Pythagorean Theorem to solve the problem. Draw a model to represent the situation described. The launching pad for a hot air balloon is 1.2 miles away from where you're standing. If the balloon rises vertically 3000 feet into the air, how far (in feet) will it be from where you’re standing to its position in the sky?

  2. Unit 3: Functions Day 1: Next/Now (recursive) Equations

  3. The Rabbit Problem (Fibonacci’sProblem)

  4. a. What pattern do you notice?b. How many rabbits are there after 11 months have passed?C. How many rabbits are there after 12 months have passed?D. Write a rule to find out how many rabbits are in any given month, based on the number of rabbits from previous months. What is the START value

  5. What you have created is called a Sequence A sequence a list of numbers where each term is based on the previous term or a combination of previous terms using a set pattern or rule. More specific: A recursive equation is an equation that shows how to calculate the value of the next term in a sequence from the value of the current term (or a combination of previous terms)

  6. When writing a rule you must include: • The start Value • The Next/Now Equation

  7. “Guess My Rule” – Back to sequences a) 5, 10, 15, 20, . . . START = 5, NEXT = NOW + 5 b) 2, 4, 8, 16, 32, . . . START = 2, NEXT = NOW•2

  8. C. D. The term means the placement of that value. If the term is 1, it is the first value in that sequence.

  9. Using the Calculator n = 1 n = 2 n = 3 n = 4 A = 1 A = 3 A = 5 A = 7 Use the calculator to create the term n = 10. • Enter your initial value (start), then take advantage of the ANS key to create a function to iterate simply by pressing ENTER repeatedly. For example, to iterate NEXT = NOW + 2, starting at an initial value of 1: Hit 1, ENTER. Enter Ans+2 and then hit ENTER repeatedly.

  10. Use the calculator to create the table of the function : Next = 5Now + 10

  11. a)Write the function for the sequence: b) find the 10th term of the sequence

  12. a)Write the function for the sequence: b) find the 10th term of the sequence

  13. a)Write the function for the sequence: b) find the 10th term of the sequence

  14. On the lined side of the index card: • Create a sequence that can be represented by a now/next function. • Give at least 4 terms in your sequence. • Write your name on the lined side of the index card.

  15. Find a partner NOT at your table. • Write the partners name on the white side of the index card. • Have your partner create the now/next function for your sequence. • Partners turn in the index card to the bin at the front of the room.

  16. HW. 3.1

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