Tuesday, May 24, 2011 Fraction Division - third example

Dear Alexander Morning Math Families:


For this example, if I laid out three of the fraction “half” pie pieces, then say, “I want to share three halves of a pizza it with four friends – what do I do?” the children respond, “cut each half in four equal pieces.”

So we took out four little green “skittles” (shaped to resemble people, with little round heads) then found the four fraction pie pieces that together equaled each of the halves. Altogether, we “exchanged” the three halves for 12 smaller pieces.

The Montessori pie pieces have the value printed on them, and the child can assemble them into a whole to “check” that (for example) 8 of the eighths pieces will equal one whole.

We shared the pieces equally with each of the four skittles, and found that each skittle (person) got 3/8 of a whole pie.

So I invited the students to LOOK at the materials as they worked with them, and consider how to write statements that are OBSERVATIONS of what they see, then to copy my diagram.

When a child learns to DRAW the picture of the materials, we see that some “process” the example more deeply than if they did not.

They actually traced the fraction pieces, OR simply drew the picture as a copy of my own, and labeled the diagram.

I also wrote the number sentence that was true:  3/2 ÷ 4 = 3/8.

Here’s the part I did NOT explain to the children:

In school, “years ago”, we adults (as students) might have been taught to “set it up and solve it” like this, using the algorithm:

Step one:

3    ÷      4   = ?

2

Step two:  “invert the divisor and multiply across”

3    x    1   =    3

2          4        8

When we carry out step two, in fact we are saying, “if we take one fourth of three halves, we get three eighths (of a whole).

However, the FIRST number sentence, stating the division, is the correct description of the operation in the word problem.

The SECOND number sentence simply shows the relationship of the reciprocal, same as the “facts family” for a multiplication and division fact.

The discovery that this is true, led to mathematicians using the “step two” process to solve such fraction division process.

This method is still taught today, but how many students can understand and explain the concept?

That’s all for now!