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Using Arrays in Reasoning with Fractions

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					                        Using Arrays to Reason with Fractions

                                        Loren Pitt
                   University of Virginia, Department of Mathematics


Introduction: The mathematics of fractions, rational numbers, and proportional reasoning
is perhaps the most sophisticated, complex, and abstract mathematics in the K-8 school
curriculum. Mastery of this material requires that students have facility with
multiplicative thinking at several different cognitive levels of. They must be able to use
the mental constructs associated with different interpretations of rational numbers, and
they need a set of representations that will enable them to easily interpret fractions as
numbers, to compare fractions, and to perform arithmetic operations with these numbers.
For students it may suffice if this knowledge is subconscious, but for practicing
mathematics specialists we would hope the knowledge and understanding will be explicit.
We want their knowledge to be at least sufficient for assessing other’s understanding and
discussing these matters with other teachers.

Our rational numbers course aims at providing teachers with a set of experiences that will
enable teacher to become flexible users and reflective users of this mathematics. Among
the most important activities in our course are those which develop representations for
use in thinking about rational numbers and which can facilitate the growth of flexibility
and reflective understanding. Because of the complexity of the subject, different
representations or manipulatives play different roles. One representation may be highly
effective in one context and less effective in another.

Let me give one example. An important conceptual step in the student’s growth is the
realization that fractions (or rational numbers) are numbers. This is not trivial and does
not happen automatically. If a student’s conception of ¾ is the elementary “part –
whole” concept: Perhaps that of a pie that has been cut into four quarters and one quarter
is missing,




then almost surely the student will not have conceptualized ¾ as a number. They will not
be ready to think about the computational arithmetic of fractions, and they will certainly
                                                             y
not possess a foundation for understanding the meaning of        . Another class of
                                                             x
students (and teachers) are well represented by the statement, “I find decimals much
easier than fractions, so I always convert fractions to decimals.” These students are also
among those unprepared to deal with the calculus of derivatives. My point here is that all


                                            1
representations are limited. While the part – whole picture does have potential for
developing initial fraction concepts and in thinking about equivalent fractions, comparing
fractions, or constructing fractions between two given fractions;




it is inadequate to support most of the mathematics that we carry in our Rational
Numbers and Proportional Reasoning bucket.

For a second example, in the context of understanding that fractions are numbers, the
number line is, for many of us, the most compelling and powerful representation, but the
number line does not help much with understanding multiplication of fractions and is of
no use in understanding the invert and multiply procedure for dividing fractions.



              0     1        1       3      2             3             4
                    3                2


One model that is very useful in the arena is that of rectangular arrays. Rectangular
arrays offer a visual structure that can be a powerful aid in understanding and fraction
comparisons and operations. The exercises in today’s activity explore some ways that
arrays may be used to reason about fractions and especially to understand basic fraction
operations.

The origin of these activities may be of some interest. Last summer, I participated in
teaching a two week residential institute on rational numbers and proportional reasoning.
We have now taught this class seven times in Virginia. It was the second time I had
taught this material. A little more than halfway through the class the four of us on the
instruction team realized that we believed the participants needed some additional on
operations with fractions, and I was given the task of preparing an activity for the next
day to address this need. So I typed up a couple pages of activities on arrays to address
fraction operations. The next day I was scheduled to facilitate this work starting, if I
remember correctly, around 10 AM and going until noon. But something remarkable
happened. The activities took much longer than anticipated and the students really
immersed themselves in the activities. The discussions were so rich that we kept
extending the time and we finally called an end to the activity about noon of the next day.

Before we start I want to provide two bits of vocabulary that may be new to some of us
and which occur in the activities.



                                             2
1. In this arena, students often get confused about what the fractions are referring to.
Does the answer ¾ refer to ¾ of a pizza, ¾ of the pizza order, or ¾ of a slice? For this
reason there is lots of talk about “What is the unit?” or “What is the whole?”

2. The second is that there are two kinds of division. One is about partitioning or doling
out equal shares (partative) and the other is about measuring out partitions of a fixed size
(quotative).




                                              3
                Rectangular Arrays and Fraction Operation Activities

I. Comparisons. Consider the question: Which fraction 3 or 2 is largest? How much is
                                                           4   3
the difference? Use rectangular arrays as an area model to explain your reasoning.

This can be discussed in at least two ways. One approach which follows the directions in
a strict sense essentially treats the rectangles as fraction strips and divides things up as
illustrated here:




The second approach, which is richer because it exhibits the common divisor of 3 and 4,
again starts with two congruent rectangles and displays ¾ on one with shading a partition
of vertical strips but then shows 2 3 in the other by shading a partition of horizontal strips.


                      3/4                          2/3




Comparing directly is somewhat difficult, but if we partition both rectangles by adding
the horizontal strips to one and the vertical strips to the other,

                      3/4                          2/3




we see they are both the ¾ and the 2 3 are represented by rectangles each of which can be
decomposed into small rectangles each with area          1     of the original rectangle and all that
                                                          12
is required to find the difference of ¾ and 2 3 is to count these two sets of small rectangles
and then multiply the difference by the area of one of the small rectangles.

If you are lucky, or if you are a skilled classroom facilitator, three additional items will
come up in your classroom discussion:




                                               4
   Someone will be lazy and do both drawings on one array and will then notice that
    they did not, in fact, need to count all of the little squares to decide which fraction
    was largest.




   There will be a discussion about the relationship between the thirds, fourths, and
    twelfths. Maybe even the common denominator.
   Someone will find a connection between the original comparison problem and a
    subtraction problem.

Problem 1: a) Use arrays to compare the fractions 3/4 and 2/3; 7/9 and 8/10; and 5/7 and
       3/5. Explain your reasoning.

    b) Calculate 3/4 - 2/3, 7/9 - 8/10, and 5/7 - 3/5.

    c) Is this a method that will always work to subtract fractions? Explain.

    d) A method that works for subtraction must surely also work for addition (with
       appropriate modifications). Use arrays to find the following sums:
       3/4 + 2/3, 7/9 + 8/10, and 5/7 + 3/5. Explain!

    e) What do these arrays have to do with finding common denominators? Explain.

    f) What needs to be added to these ideas to add and subtract fractions that may be
       greater than 1?

    g) (For the instructors) What would be the benefits, weaknesses, and dangers that
       you might encounter if you used this model to introduce addition and subtraction
       of rational numbers; both positive and negative.

II. Multiplication

If arrays can be used to illustrate common multiples we might be able to use arrays to
multiply fractions.

Problem 2. Try to find an interpretation of the picture that could be used to explain why
2/3 x 3/4 = 1/2.




                                                          .



                                               5
What is the whole in your explanation? More precisely, in the picture if the answer is 1/2
it must be 1/2 of some unit. What is this unit?

Problem 3. a) Interpret the picture


                               7/4




                               1




                                0              1                3/2




in a manner that helps explain why 1½ x 1¾ = 2 5 8 . What is the whole in your
explanation?

b) Interpret the picture




                               1
                              4/5




                                0                  1              5/3

                                      4
in a manner that helps explain why    5   x1 2 =
                                             3
                                                       4
                                                           3   . What is the whole here?

c) Can an array/area model always be used to calculate and interrupt the product of
fractions?

3.     About Division.

Problem 4. Interpret the picture that helps you see the answers to the two division
problems 2/3  3/4 and 3/4  2/3. What is your whole?




                                                   6
4.     Interpreting “Invert and Multiply”.

Problem 5. I am moseying along at 3/2 miles per hour.
   a) How long does it take me mosey a mile?
   b) If I need to mosey 4/5 of a mile, how long will this take? Is this division? Do
       you see invert and multiply in your answer?

Problem 6. Jane is sewing shirts that require 5/3 yards of cloth.
   a) What fraction of a shirt can she sew with 1 yard of material?
   b) If Jane has 11½ yards of cloth, now many shirts can she sew? Is this division?
       Do you see invert and multiply?

Problem 7. Billy’s Burgers has a special on Wow burger meals at 3 Wows for $7.
   a) What part of a Wow can you buy with a dollar?
   b) How many Wows can you purchase with $21?

Problem 8.
   a) Use your work in the problems above to explain why we invert and multiply when
       dividing fractions.
   b) Use fraction strips or arrays to act out a division problem of your choice?

Partitive Division Problems with Fractions

Problem 9. When Grandma Judy died she left $100,000 to be divided between her 4
children and 7 grandchildren. The children were meant to receive equal shares. The
grandchildren were meant to receive half a share each. How was the money to be
divided?

Problem 10. At Sunday dinner Grandma Judy baked two cherry pies and cut them into 5
equal sized adult pieces and 7 half sized pieces for her grandchildren. Between her 4
children and 7 grandchildren, how large were the adult pieces?




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