Several times in this course, you will be asked to write or record a good explanation of something mathematical.  Sometimes you are explaining a solution, and sometimes you are explaining an algorithm.

When explaining an algorithm, you should keep in mind:

• if you are supposed to choose your own numbers for the explanation, make sure you choose ones that will let you illustrate well the algorithm you are explaining. 

• Thus, if your algorithm involves finding a common denominator, make sure that the numbers you choose will let you show the details about that process that you should show.
• Don't use unit fractions: in all of the standard algorithms there are steps that you need if the fraction is not a unit fraction (like 2/3) that you don't need if your fraction is a unit fraction (1/3).  Make sure you choose numbers that let you show all of the steps.
• If your task includes modeling the algorithm with manipulatives, make sure that you choose numbers that you have appropriate manipulatives to demonstrate with.
• As much as possible within the guidelines above, make sure that all of the digits are different--this makes it easier to keep track of what's going on (if there's more than one 4x5 in your problem, there's a confusion about which is which, and indeed, you should be aware of which numbers have common factors that could simplify in either useful or distracting ways--useful being something you should put in, distracting being something you should avoid--it depends on what you are explaining how to do).

• Use correct language:
  
• Numerator, denominator, factor, multiple, and unit fraction are key terms for fractions that you should know and use correctly.
  
• parts of an operation: know which is the divisor, and which is the quotient, know what a product and a factor are, and use those words correctly
• In operations (subtraction and division) make sure that you are reading the problem in the right order: if you are saying divide __ by ___ or subtract ___ from __, make sure you have the numbers in the right order.
• Explain the renaming as an equivalent fraction steps explicitly. Explain what you do to get the equivalent fraction, and what it means to be an equivalent fraction (6/8 is equivalent and equal to 3/4 because they identify the same part of a whole)
  
• Go from concrete to abstract
• If you have both a manipulative and a number way of showing the algorithm, do each step first with the manipulative, and then with the numbers, and explain each step in the number work as a way of recording what is happening with the manipulatives
• Make the connections step by step--don't wait until you have done the whole problem with the manipulatives and then make the connections: do a step with the manipulatives, and then the same thing with the numbers. 
  
• Think about what your audience knows:
• If you are showing how to do a particular algorithm with manipulatives or diagrams, that implies that your audience does not yet know how to do that same algorithm without those manipulatives and diagrams.
  
• You may assume for now, for fraction work, that your audence knows the basic computations for whole numbers (though when teaching the same thing to actual students, you will find that one of your major stumbling blocks is that they do not know their multiplication facts well enough)
  
• You may assume that your audience is learning fraction operations in the order: addition, subtraction, multiplication and division.
  
• Do not assume that your audience knows what should be recorded where or why.
  
• Feel free to copy me.  That's what the examples are there for.  You don't need to reinvent the wheel, you just need to understand it well enough that you can do it right.

Good explanations are your stock in trade as a teacher.  Even if you have a very student-directed class, you need to be able to make good, concise, complete explanations at the point where you are summarizing, and there will be students for which nothing else works (you will also find students for which this does not work.  If you find something that works all the time, patent it.)

I insist that you practice giving good explanations.  You should practice your explanations 2-3 times before you record them.  You will be glad later, when you are teaching, if you have the words practiced so your explanations come out right the first time. If your explanations are missing important things, I will insist that you redo them before I am willing to grade them.