* Graphics and algorithms, part 2

I made three points in yesterday’s post on this topic:

  1. Students should be allowed access to manipulatives far longer than is typically expected.
  2. Students should not simply memorize algorithms, but should understand the mathematical reasoning behind them.
  3. Visual representations of math processes can be helpful, but can often be more confusing than a standard algorithm.  I have concerns that kids struggle to understand the area model of multiplying decimals and suggested a number line.

Today’s post deals with a terrific process and graphic for teaching division.  It is featured in this month’s “Teaching Children Mathematics,” published by the National Council of Teachers of Mathematics.  Called “division quilts,” it represents a measurement model of division as opposed to “fair shares.”  The fair share model is like dealing cards into equal hands.  The measurement model begins with the dividend, created on graph paper in a rectangular representation.  This is one way to draw 35 ÷ 4.

Division quilts_0001

Next, color groups of 4, using a different color for each adjoining set.  A good question to ask kids is why they should use different colors.Division quilts_0002

By counting the groups of 4, it’s easy to see that there are 8 equal groups.  Finally, number the left-over spaces, which is the remainder.Division quilts_0003

There is a strong relationship between this graphic and the long division algorithm.  For kids struggling with math, that relationship must be explicitly taught.  We want kids to use algorithms, but this process helps explain not only how we got a quotient of 8, but what a remainder means.  Depending upon the context of the problem, kids must decide what to do with that remainder.  I love to use field trip math problems with remainders because the imagery is so vivid.  “If we had parent drivers and placed 35 students in groups of 4 per car, how many cars would we need?”  At first, most kids will say 8.  Then we imagine those last three kids running behind the cars to the museum.  After they exhaust the possible ways for those kids to get smushed on the freeway, most students never forget that remainders are important!