Modeling- The instructor models the creation and use of algorithms, including various algorithms for one problem.  The instructor stresses that algorithms may be unique while still conveying the same underlying concept.  The instructor also notes that algorithms are not a “cookbook” approach to mathematics; algorithms are a product of observation, inductive reasoning, and investigation among others. 

Discuss scientific and standard notation.  Show examples of scientific notation and standard notation and discuss how to get from one to the other.  Derive the algorithms using their conjectures.  Discuss the reversibility of these algorithms.  Apply the algorithms to real-world practice problems.

Practice- The instructor provides several opportunities for the students to get guided practice in creating and implementing algorithms.  This includes group work, board work, and handouts.

Discuss distance of a number from zero using a number line.  Introduce absolute value notation.  Use whole-class discussion to investigate these types of absolute value equations and decide on an algorithm.  Apply the algorithm to real-world practice problems.  Pose the question of representing distance between any two numbers on the number line and use it to fuel an additional lesson similar in structure.

Transition- The instructor continuously relinquishes responsibility to the student.  The instructor’s role becomes that of a facilitator.  Feedback is still provided but not until students produce a final product or at least make several attempts at creating and implementing algorithms.  Group work is an important element of this phase.

Students form four groups.  Each group is assigned a specific probability problem that provides a shell for inductive reasoning that results in the development of an algorithm.  One member from each group forms a new group; each member of the new group presents his/her problem, explains his/her reasoning, and discusses the algorithm.  Apply the algorithms to real-world practice problems.

Independence- The instructor provides the students with many opportunities to illustrate their algorithmic capabilities.  The instructor provides a supportive environment.  All aspects of problem solving such as strategies, real-world applicability, and an integrated curriculum are heavily emphasized. 

From real-world problems involving addition, subtraction, multiplication, and division, calculate both an exact and estimated solution.  Discuss the relationship between both results.  Apply rounding to the exact solution.  Compare and contrast estimating and rounding.  Develop strategies for identifying when and how to use previously developed and used estimation algorithms including term identification such as “about” and “approximate” and types such as using nice numbers, and using compatible numbers, averaging, and front-ending.