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MVP Year 2 (Mod 3): 3.1

Exploring Rational Exponents

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Quote to discuss:  "The greatest shortcoming of the human race is our inability to understand the exponential function." - Prof. Al Bartlett (Physicist)  http://www.albartlett.org/

 

Common Core Focus: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents (N.RN.1)

 

Homework Resources:

 

Ready -

This is a review of material learned back in the 1st quarter this year.  Note that the instructions give a lot of clues by pointing out that we're dealing with discrete SEQUENCES instead of continuous FUNCTIONS.  Also, the instructions remind us of the initial two types of sequences we studied; (1) arithmetic [a growth pattern where the previous term is built upon by a common term using addition or subtraction], and (2) geometric [a growth pattern where the previous term is built upon by a common number using multiplication or division].

 

A possible strategy for problems 1-5:  

  1.  Study the numbers going across the bottom row of each problem and try to determine the pattern.  Fill in the rest of the empty cells based on the pattern you've discovered.  You'll likely be finding a common difference (arithmetic) or a common multiple (geometric).
  2. Clearly state which of those two types of sequences is contained in the given table. (Quick Check: There are more geometric sequences shown in this set of problems than there are arithmetic.)
  3. Finally, go back through each problem and build an explicit equation to define each table's pattern.

 

Arithmetic:       an = d(n-1) + a1  where d is the common difference and a1 is the first term

 

Geometric:          an = a1(r)n-1 where a1 is the first term and r is the common multiple

 

 

A possible strategy for problems 6-11:

  1. This time you're being asked to investigate a graph instead of a table in order to discover an explicit equation that will define the function.  Use the sample point shown below to answer similar problems in 6-9.  Recall that the function notation shows the input value inside the parentheses and the output value after the equals sign:    f(input) = output