4.NF 4a: Bean-o Fractions & Mixed Numbers (#264)


A Year in the Life: Ambient Math Wins the Race to the Top!
Day 264

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful.  The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
4.  Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b.  For example, use a visual fraction model to represent 5/4 as the product of 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4).

In my usual search for Common Core examples, I found the usual scenarios, with repetition and worksheets as a mode of instruction.  Conversely, if concepts are taught clearly and economically, there’s no need to bore and browbeat students with demeaning tasks.  One such task I found was: to find the product of 7 x 2/5, use repetitive addition by drawing objects.  In this case, 14 objects would need to be drawn.  This sort of activity is demeaning to a fourth grader.

Math By Hand uses beans to teach fraction concepts, with kidney beans representing whole numbers and black beans representing fractions’ numerators and denominators.  Similarly to how beans are used to represent place values in Grade 2 (10 black beans/ones are cashed in for 1 kidney bean/ten), mixed numbers are generated by cashing in fractions (with remainders).  This example taken from the Math By Hand Grade 4 binder shows an addition problem, but the same method is used for the other 3 processes.




Students, using a graph paper workbook to facilitate correct placement, learn the basics of the 4 processes with fractions in a concentrated way.  The trust that’s implicit in this sort of teaching is empowering and and builds self-confidence, because once the method is learned there’s no need for repetition and boring, demeaning worksheets.  The knowledge gained can be applied where needed.

Less time on repetitive tasks leaves more time for joyful learning, like deconstructing the woven pattern in this Longobardian Knot, from waldorftoday,com.  This knot is #4 in complexity, see the page for increasingly complex knots up to #8.  Just another example of how knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

Grade 04 - Longobardian Knot 4

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