Download Day 2 Activity - High School Math Teachers

Day 2 Activity
Name ___________________________
Heron’s method of approximating square roots can be used as a basis for a computer program
to find square roots. In programming, a process of sequential approximations such as this is
called iteration. When a function undergoes iteration, the initial input is called the seed, and
the sequence of outputs created by iterating (repeating) the function is called an orbit. For
example, the iteration of
𝑓 (𝑥 ) =
𝑥+
21
𝑥
2
can be used to find an approximation of √21. Start by choosing any number for the seed. Then
substitute the seed for x to find the first orbit.
Suppose you choose 4 as the seed. The first orbit is
4+
21
4
2
= 4.625.
1. Use the first orbit as your new seed, and find the second orbit.
2. Use the second orbit as your new seed, and find the third orbit.
3. Repeat the process two more times. What is your final orbit?
4. Describe any patters you notice. How is the final orbit related to √21.
5. Find the first four orbits of 𝑔(𝑥) =
What happens to the orbits?
𝑥+
16
𝑥
2
𝑥+
by choosing an initial seed that is less than zero.
7
6. Find the first four orbits of ℎ(𝑥) = 2 𝑥 by starting with 2 as the initial seed. Explain the
relationship between the orbits and the square root of 7.
Algebra1Teachers @ 2015
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Day 2 Activity
Name ___________________________
Answer Key:
1. 4.58277027
2. 4.582575699
3. 4.58257555; 4.582575695; 4.582575696
4. Possible answer: the orbits get closer and closer to the same number; they begin to
repeat. The final orbit is an approximation for the square root of 21.
5. Answers will vary. For example, if the seed used is -4, the approximation of the square
root of 16 is -4. The orbits are all negative numbers.
6. 2.75; 2.6477272; 2.645752048; 2.645751311; the orbits approximate the square root of
7.
Algebra1Teachers @ 2015
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