Foundational Standards in Mathematics
Algebra 2
Session 1: The Duck Principle
The Duck Principle
Introductions
Sarah Sword
Kevin Waterman
The Duck Principle
Plan for this Session
Foundational Standard
Number and Quantity: The Real Number System
AII-N.RN. Exploring meaning of rational exponents following from properties of integer exponents and converting expressions with rational exponents.
Note: Standards for Mathematical Practice
3: Construct viable arguments and critique the reasoning of others
7: Look for and make use of structure
8: Look for and express regularity in repeated reasoning
The Duck Principle
Leveraging Sequences
The duck principle: If it walks like a duck, and quacks like a duck, then it probably is a duck.
The Duck Principle
Leveraging Sequences
An arithmetic sequence is a list of numbers in which the difference between consecutive terms is constant.
Each term in an arithmetic sequence is the sum of the previous term and �the common difference.
Examples:
(common difference of 4)
(common difference of 6)
The Duck Principle
Leveraging Sequences
A geometric sequence is a list of numbers in which the ratio �between consecutive terms is constant.
Each term in a geometric sequence is the product of the previous term and the common ratio.
Examples:
(common ratio of 2)
(common ratio of -3)
Work Time
Put on your “student hat”!
We will pause later for your “Teacher hat” discussion.
What we would look for if we were with you: �What’s on your mind as you do these problems?
Pause to work on problems 1–5 from the handout for Session 1.
Making Sense of Exponents
Just as multiplication can be described as repeated addition:
Exponentiation can be described as repeated multiplication:
But that definition only works if the exponent is a positive integer. (It’s even tricky if the exponent is 1).
The Laws of Exponents
Let a ≠ 0.
The Fundamental Law of Exponents
ab • ac = ab+c
Corollaries
(ab)c = abc
= ab–c
Using the basic understanding we just reviewed, these can be shown relatively simply when b and c are positive integers.
ab |
ac |
Extending Exponents
Consider this geometric sequence:
16807, 2401, 343, 49, 7
The initial term is 16807, and the common ratio is .
Using the “repeated multiplication” definition, you can rewrite the first four terms as
75, 74, 73, 72, 7?
The pattern of exponents would make the last term 71, and for the two sequences to match, 71 = 7.
Extending Exponents
Continue the sequence by multiplying each term by .
And extend the pattern with the matching sequence of exponents:
For the two sequences to match,
Extending Exponents
Consider the original geometric sequence in Problem 4.
1, 27, 729, 19683, . . .
You can write this as
270, 271, 272, 273, . . .
Notice that the exponents form an arithmetic sequence:
0, 1, 2, 3, 4
Extending Exponents
Now consider the expanded geometric sequence in Problem 4a.
1, 3, 9, 27, 81, 243, 729, . . .
Let’s write this as:
270, 27☐, 27☐, 271, 27☐, 27☐, 272, . . .
For the exponents to still form an arithmetic sequence, you have to fill in the gaps like you did in problem 2a.
0, __, __, 1, __, __, 2, . . .
Extending Exponents
The arithmetic sequence filled in would be
And so the geometric sequence would be
These entries would have to match the geometric sequence
And so you have
The Laws of Exponents
Let a ≠ 0.
The Fundamental Law of Exponents
ab • ac = ab+c
Corollaries
(ab)c = abc
= ab–c
Now we’re pretty confident these apply for any rational b and c.
ab |
ac |
Extending Exponents
What is the value of 2π ?
Can we extend exponents to include all real numbers?
Reasoning by Continuity �(a little hand-waving!)
Input, x | Output, f(x) = 2x |
3.14 | 8.81524 |
3.141 | 8.82135 |
3.1415 | 8.82441 |
3.14159 | 8.82496 |
3.1416 | 8.82502 |
Work Time
Put on BOTH your “student hat” and “teacher hat”!
Take 15 minutes to work on problems 6–9 in the handout. We encourage you to work together!
Pause to work on problems 6–9 from the handout for Session 1.
Defining Square Roots
The Duck Principle in Action
How do we make meaning of this:
Other Duck Principles
How do we make meaning of this:
Other Duck Principles
How do we make meaning of
log39
Perspective from Dr. Anne Marie Marshall
Final Reflection
How do you help your students make sense of definitions that might at first seem obscure, abstract, arbitrary, or just very unfamiliar?
Use whatever time you have left to share your
practices with each other.