Common Core @ Home: Math - What's the rectangular fraction model, and how are we supposed to use it to prove an answer?



Part Four of a continuing series


Dear Jeff, What's the rectangular fraction model, and how are we
supposed to use it to prove an answer?     


Fractions are the mathematical equivalent of taxes: If want to enjoy the benefits of living in a society, you have to deal with them from time to time.  The good news is that working with fractions is a lot easier than trying to fill out a Form 1040, even with problems like:

Penny used 2/5 lb of flour to bake a cake. She used another 3/4 lb of flour to bake another cake.  How much flour did she use altogether?  Draw a picture and/or write the number sentence that proves the answer. Simplify the answer.

Problems like these are introduced in the fifth grade, though fractions themselves are introduced towards the end of third grade.

The rectangular fraction model is one of the more insightful ways to represent a fraction.  We begin with a rectangle that represents the whole amount, and divide it into equal parts. Each part is a unit fraction. For example, if our rectangle represented one pound of flour, we could divide it into five equal vertical parts (on the left) or four equal horizontal parts :

 

2/5 pound 3/4 pound

On the left, each of the small rectangles is 1/5 pound and, since the shaded part is 2 of those rectangles, it represents 2/5 pounds:  the amount used for the first cake.  On the right, each of the rectangles is 1/4 pound and, since the shaded part is 3 of those rectangles, it represents 3/4 pound:  the amount used for the second cake.

The total amount of flour used is just the sum of the two portions: 2/5 + 3/4.  While we have five  shaded pieces altogether, they're of different size and shape.  But notice that if we divide the first rectangle horizontally into four parts and the second rectangle vertically into five parts (keeping the original partitions and shadings), we obtain:

  

2/5 = 8/20 3/4 = 15/20

We haven't changed the size or shape of either of the shaded regions, so the first still represent 2/5 pounds and the second 3/4 pounds.  But we have broken both into smaller pieces, each 1/20 of the original rectangle.  On the left, our shaded region now consists of 8 pieces, each representing 1/20 of a pound: thus it represents 8/20 of a pound, and we've drawn the fractional equivalence 2/5 = 8/20By a similar analysis, the rectangle on the right shows the fractional equivalence 3/4 = 15/20. 

Now, because all our pieces are the same size, we can just count:  there are 8 + 15 = 23 pieces altogether, each of which is 1/20 of the pound.  Or, since 20 of these pieces make up one pound, we can rearrange them to see that we have 1 3/20 pounds altogether:

 and    

2/5 pound + 3/4 pound


          gives us   



1 3/20 pounds

As with other aspects of the Common Core, the goal is to promote conceptual understanding, which helps learners avoid common errors and pitfalls. As a single example, the algorithm for producing a fraction equivalent to a/b is to multiply numerator and denominator by n:  in this case, 2/5 = 2x4/5x4.  A common error is to only multiply the denominator and obtain the “equivalent” fraction 2/20.  A common pitfall is to obtain the correct equivalent fraction 8/20 but, since students have been trained to reduce fractions to lowest terms, to immediately reduce this fraction back to 2/5. 

The rectangular fraction model avoids both error and pitfall.

Common Core material is available at EngageNY under a Creative Commons non-commercial license: in effect, school districts can use the materials free of charge, as long as they attribute the material correctly.

Jeff Suzuki teaches mathematics at Brooklyn College, and is one of the founders of the Mid-Hudson Valley Math Teachers Circle, a group of teachers, professors, and mathematics aficionados working to promote mathematics education in the Hudson Valley.

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