Operations in Fractions

BatBoy understands fractions in the concrete sense, as equal parts of a whole.  We haven't really touched on concepts like finding common denominators or multiplying fractions.  Instead, he has spent time and energy really getting to know a few basic fractions and how they "work together."  As he has learned to add and subtract, and then multiply and divide, he has been able to apply these operations to these basic fractions (halves, thirds, quarters, eighths, and occassionally, sixths).  Using his visual understanding, he has solved problems involving adding or subtracting fractions of the same "family" (halves, quarters, and eighths; or thirds and sixths) including their mixed numeral relatives.  He is able to find answers with negative values, such as the solution to 1/8 - 1/4.  Today, he was playing with the sequence of dividing numbers by two.  Beginning with 8, he divided by 2, again and again, until he got to 1/16 and couldn't (his words) "multiply 16 by 2" without paper.

SpiderGirl plays with concepts less and so, seems less drawn to mathematics than BatBoy.  But given problems, she solves them quickly.  She gets mental blocks when she is anxious.  If she thinks that there is a "right" way to solve a problem (a way that she is less than confident with), or that I expect a particular answer, solved in a particular way, she freezes and claims she has no idea.  But, if the problem is presented in a low pressure environment, she can shine brightly indeed.  SpiderGirl understands fractions visually.  Even when fractions are not of the same "family," she can puzzle it out using manipulatives.  She used manipulatives to figure out to find common denominators in order to add or subtract.  When she gets more comfortable with multiplication, factors, and multiples, I have every confidence she will gain a more methodical way of finding common denominators.   When manipulatives are not available, SpiderGirl is also to present reasonable guesses to problems involving fractions, decimals (to hundredths), and percents.  And she is learning that there is value in her ability to estimate.