Arithmetic and Geometric Sequences

After working with patterns and attempting to organize what we see and act on it, we are formalizing our experience through a couple of weeks of analyzing arithmetic and geometric sequences both in their explicit and recursive form.  This formalization has a great deal of notation to adjust to, but is ultimately just a shorthand way of writing how the pattern works.  Can you write an explicit and recursive equations for the sequences above?

All of our homework for the next three weeks will be from It's All Connected and while I cannot upload the file here, if you are a student I can email you the file if you lose your it!

I have embedded three videos to help those of you who want to review some of the big concepts at home:

1.  Here is a video from Khan Academy on explicit and recursive definitions

Explicit and recursive definitions of sequences:

2.  Writing Recursive Formulas from Arithmetic Sequences

 

3.  Writing Recursive Formulas form Geometric Sequences

Online Practice and Videos from Khan Academy available here: https://www.khanacademy.org/math/trigonometry/seq_induction/seq_and_series/v/explicit-and-recursive-definitions-of-sequences

Trying Simpler Cases and Understanding Relationships

Adapted from Burkhardt's The Real World and Mathematics.

Last week we were working with contexts that emphasized numerical relationships in order to develop strategies to solve non-routine problems. This week we are beginning to focus more on relationships between quantities and how graphs, verbal descriptions, pictures, and tables convey meaning about a situation or context.  As the image above shows the representation of these relationships and the ability to translate from one to another is crucial in our development of problem-solving skills. Here are the goals for translation fluency that we will cover over the coming two weeks: 

1. Interpreting mathematical representations using words or pictures.
2. Translating words or pictures into mathematical representations.
3. Translating between mathematical representations.
4. Describing functional relationships using words or pictures.
5. Combining information presented in various ways, and drawing inferences where appropriate.
6. Using mathematical representations to solve problems arising from realistic situations.
7. Describing or explaining the methods used and the results obtained.
(from The Language of Functions and Graphs)

The problem sets we will be investigating are below.  They focus on interpreting points within a context (A1) and sketching graphs from words (A3). As we work through these problem sets I hope that we develop a fluency in using the mathematical language of graphs, tables and algebra to describe and analyse situations from the real world.  We will do this by engaging in thoughtful discussion as a whole class, between partners, and in groups in order to comprehend as well as communicate information presented in a mathematical form.

A1 Interpreting Points
A3 Sketching Graphs from Words

While we investigate and solve problems in class, our opening problems and homework will be reviewing the work on systems of equations you did in Algebra 1.  The homework for this week is at the link below:

Pet Sitters, Part 1

Here is to a great week!