One goal for my sixth-graders is for them to think about fractions most days throughout the school year. That thinking is not simply solving problems, but exploring the relationships of the numbers within the problems. An example is teaching kids why they multiply by the reciprocal to solving a “dividing by fraction” problem.

In teaching this algorithm and why it works (multiply by the reciprocal), I started with some simple examples of dividing a whole number by a unit fraction. The problems were:
(1)     3 ÷ 1/4, and
(2)     3 ÷ 1/5

The answers (quotients) are simple (12 and 15) and easy to draw as students begin to develop an understanding of the algorithm. I had been talking to my kids about what happens to the quotient when the divisor is doubled/halved, tripled/a third, etc., for example, divide 24 by 6, the quotient equals 4. When the divisor is doubled to 12, the quotient is halved to 2.

Likewise, divide 24 by one-third of the divisor (1-third of 6 = 2), the quotient is tripled to 12.  So, there must be a similar relationship between 1/4 and 1/5, and 12 and 15. There is: 1/5 is 4-fifths of 1/4, and, 15 is 5-fourths of 12. The relationship is the reciprocal of the change. Whatever the divisor is increased/decrease by, the corresponding quotient is increased/decreased by the reciprocal.

What follows is an example of how sixth-graders can explore the relationship between the divisors and the quotients.

Questions #1:
What is the relationship, in this case, the difference, between 1/4 and 1/5?
Usually, my students will default to the additive relationship when asked to compare two values.

1/4 = 5/20          (1•5/4•5)
1/5 = 4/20          (1•4/5•4)

The difference is 1/20:  5/20 – 4/20 = 1/20;
1/4 is 1/20 greater than 1/5 or 1/5 is 1/20 less than 1/4.
This is the relationship “additively”, using addition or subtraction.
What is the relationship “multiplicatively”, using multiplication or division?
In other words, 1/5 is what fraction of 1/4 (because 1/5 is less than 1/4)?
x 1/4 = 1/5: what fraction of 1/4 = 1/5?
This equation will be written as 1/4 x = 1/5, with 1/4 as the coefficient even though the question is “What fraction of 1/4 = 1/5”.

There are several ways to solve this problem, the easiest of which is algebraically:
1/4 x = 1/5
(1/4 •Ÿ 4)x = 1/5 •Ÿ 4
4/4 x = 4/5
1x = 4/5
x = 4/5
So, 4/5 of 1/4 = 1/5; 1/5 is 4/5 of 1/4.
The value of x, 4/5, must be a proper fraction because 1/4 is “reduced” in size to 1/5.

Two Drawings:
(1) A linear drawing of 4/5^th of 1/4^th, called a tape diagram:
Note that the “tape” will be divided into twentieths.

In the drawing, 1/4^th is divided into five equal parts to get twentieths of the whole rectangle and, at the same time, to get fifths of fourths. Four-fifths of one-fourth has the purple dots. The four parts equal 4-twentieths of the whole rectangle. Four-twentieths can be simplified to one-fifth. Also shown in the diagram is that one-fifth is one-twentieth smaller than one-fourth.

(2) An array drawing of 4/5^th of 1/4^th;
4 columns x 5 rows to equal twentieths:
Note that the array will be divided into twentieths.

In the drawing, the rectangle is divided into four equal parts (fourths) by drawing columns vertically. The rectangle is then divided into five equal parts (fifths) by drawing rows horizontally, constructing a 4 x 5 array of twenty cells (twentieths). One-fourth is divided into five equal parts. Four-fifths of 1-fourth is shown by the four red dots. These red dots equal 4-twentieths of the rectangle that simplify to 1-fifth of the rectangle. As above, 1-fifth is shown to be 1-twentieth less than 1-fourth.

The above discussion describes 1/5 as a fraction of 1/4; specifically 4/5 of 1/4. The corresponding question is what fraction of 1/5 is 1/4. In this case, the value of 1/5 is raised through multiplication to 1/4. This is an odd question because the multiplier is an improper fraction (greater than one) in order to raise the value. This is a great discussion to have with sixth-graders – how to raise and lower values using multiplication.

As with the question above, 1/5 is what fraction of 1/4, 1-fourth is what fraction of 1-fifth is solved easily using algebra:

1/5 x = 1/4
(1/5 • 5)x = 1/4 • 5
1x = 5/4
x = 5/4

So, 5/4 of 1/5 = 1/4; 1/4 is 5/4 of 1/5.
The value for x, 5/4, must be an improper fraction because 1/5 is “increased” in size to 1/4.

Two Drawings – again:
(1) A tape diagram of 5/4^th of 1/5^th:
Note that the tape will be divided into twentieths, again.

• draw fifths (green)
• divide fifths into four parts (red)
• total number of parts equals twenty; twentieths of the whole rectangle
• 1-fifth of the rectangle equals 4-twentieths
• 1-fourth of the rectangle equals 5-twentieths
• shows that 1/4^th is 1/20^th greater than 1/5^th
• 1/4^th of 1/5^th equals 1/20^th; one of the small parts
• 5/4^th of 1/5^th is more than 1/5^th; it is 1/4^th of 1/5^th more than 1/5^th
• increase the value of 1-fifth to 1-fourth; multiply by an improper fraction (5-fourths)

(2) An array drawing of 5/4^th of 1/5^th;
5 columns x 4 rows to equal twentieths:
Note that the array will be divided into twentieths.

• draw fifths (5 columns, tan)
• divide fifths into four parts (4 rows, green)
• total number of parts equals twenty; twentieths of the whole rectangle
• 1-fifth of the rectangle equals 4-twentieths
• 1-fourth of the rectangle equals 5-twentieths
• shows that 1/4^th is 1/20^th greater than 1/5^th
• 1/4^th of 1/5^th equals 1/20^th; one of the small parts
• 5/4^th of 1/5^th is more than 1/5^th; it is 1/4^th of 1/5^th more than 1/5^th
• increase the value of 1-fifth to 1-fourth; multiply by an improper fraction (5-fourths)

From above, 1/5 is 4-fifths of 1/4; the genesis of this exploration was comparing the two division problems:

(1)     3 ÷ 1/4 = 12, and
(2)     3 ÷ 1/5 = 15

Now show that 15 is 5/4 of 12 to complete the comparison that the relationship of the quotients is the reciprocal of the relationship of the divisors.

The last part of this exploration is to show that 12 is 4/5 of 15 which is coupled with the comparison of the divisors: 1-fourth is 5/4 of 1-fifth.