Post # 39 Questions and THAT Fraction Page

I loved the questions posed in our recent video conference and wanted to post the fraction image from a recent presentation.

Remember, the language of instruction for fractions is comparable to the language of place value instruction.  In terms of conceptual horizons in math, fractions are incredibly important. 

Many teachers tell me that they rely almost exclusively on procedural steps when teaching multiplication, division and the use of/creation of common denominators.  Yet, we know that students who use manipulatives retain information longer and perform better on follow up tests including standards based assessments.  Know the meaning behind the math allows students to construct mental math supports for their computations and applications. 

The discussion we had about fraction division led us to this image from my IDA presentation this year:  5 ÷ ¾ with a remainder of 2/3 …meaning two of the three fourths needed to create another “group.” 

The picture shows a colored image of “creating groups of the quantity ¾” and then a number line version showing division as repeated subtraction.  Remember that students with LD many not have sufficient fine motor skills to navigate small number lines so using those as the exclusive representation may not yield substantial results. 

For students with LD, “blow it up.”  Create large representations which can use whole body/ gross motor movements.  Use near point references at each student’s desk or small group table.  Students enjoy “coloring” fraction solutions but students who only color fractional parts are not reported to retain those concepts so include precise mathematically accurate language as demonstrated in class.  

Our other interesting question related to the traditional division algorithm and why the “FIRST NUMBER FOR DIVISION”  box might be a two digit number.  We linked the algorithm to the array model in which the dividend must align to the divisor in terms of place value.  If the divisor is smaller than the first digit of the dividend, the “tens place digit” would need to be deconstructed into units for proper alignment.  It is why we do not use base ten blocks to divide a quantity greater than one hundred by say “9”.  The hundreds flat would need to be deconstructed into single unit cubes to create “groups of 9.”  This would be an inefficient use of the manipulatives which we avoid.  It is not that it cannot be done, but it is cumbersome and time consuming. 

So, if we are dividing 27 by 9 and we start with the representation of 27 as two ten rods and seven unit cubes, we would need to “unbundle” or deconstruct the two ten rods into single cubes to align them with our divisor of nine.  Not a good use of time or manipulatives.  If you need to do that once to demonstrate the concept, fine; but don’t over use larger numbers for demonstrations. A better manipulative for this concept would be Unifix cubes and making groups of nine. Remember manipulatives are used to illustrate the concept, then you move on to other representations and finally calculations at the abstract level.