Introductions to Fraction Bars

Activities Journal (The portions in colored print are my observations of the sessions)

March 2, 2005

Introduction to the topic

We will spend time discussing what would be covered in our study sessions, and
organizing the paperwork and contracts.
We will look at the different methods that we will use during the study sessions.
Fraction bar manipulitives
Computer Applet
Class-time Lecture

Things are off to a good start. It seems as though our study group is co operative and
willing to work together.

March 4, 2005

Introductions to Fraction Bars

Initially, the study group will organize the fraction bars according to color. This will put
the ½, 1/3, ¼, etc. . . bars in the appropriate categories. We will look at the number of
divisions on each of the pieces and the categories.

It seemed as though Crystal and Shawn were a bit bored with this activity, but Bryan took
this activity in with attention to the subtleties. He asked what the shaded region
represented. When it was explained to him, he commented that the un-shaded region also
re presented a fraction . It was apparent that he hadn’t considered what a fraction really
meant. I was impressed by his interest in the material. Crystal and Shawn both were
bored because their understanding of the material was beyond this exercise.

After organizing the fraction bars into their individual categories, the study group will
look at a specific fraction bar. (I.e. 3/5) This will be the center of a discussion regarding
what the fraction bar represents.

We spent time discussing what the shaded versus un-shaded portions of the fraction bars
meant. We looked at the fraction represented by the shaded region and its compliment,
represented by the un-shaded area. Crystal commented that the shaded region translates
to 4 parts out of 7 shaded for the 4/7 - fraction bar. When Bryan asked what the fraction
bar with 8 shaded parts out of 8 divisions meant, Shawn said that this was 8/8 which
meant that the whole piece was shaded, therefore was equal to 1 whole unit. Crystal also
explained that since none of the piece was shaded, the 0/5 fraction bar was equivalent to 0.

The study group will spend time discussing the relationships between fraction bars and
fraction termino logy . We will compare the meaning of a fraction, (½ for instance) and
the fraction bar that represents the same.

Shawn pointed out that the fraction bar with 1 of 2 parts shaded was the visual equivalent
of ½. Bryan observed that a fraction was part of a whole thing. He made a comparison
to 4 slices of an 8-slice pizza remaining. While discussed the meaning of the fraction
bars, we started using fraction terminology. Crystal observed that a fraction is a
comparison of two numbers . Shawn added that it’s simply a ratio .

Next, the study group will discuss the meaning of numerators and denominators and how
they are related to the fraction bars.

In our discussion on terminology, numerator and denominator came up. Shawn
explained that the top number of a fraction is called a numerator, and the bottom number
is the denominator. (Or as Crystal calls it, the dominator) Crystal went on to explain that
the numerator is represented by the number of shaded parts of the fraction bar, while the
number of divisions on the fraction bar is the denominator.

March 9, 2005

Finding equivalent fractions using Fraction Bars

The study group will organize the fraction bars according to equivalent areas shaded from
one fraction bar to the next. The study group will then make a table that has all the
fraction bar equivalents as fractions. (An example of the table is shown below.)

When the students have found all the fractions that are equivalent using the fraction bars,
they will organize the equivalent fractions into the appropriate categories on the table.
The study group will then look at the fractions that are the same. The discussion will
center on what it means when two fractions are the same, or equivalent.

The study group didn’t like making the table for this activity. When they would get the
table set up, they would find that they needed another column for a fraction that is not
equivalent to any other. This was tedious, but when they saw the table of fractions set up,
both Shawn and Crystal had “eureka” moments. They both had an instant understanding
of why equivalent fractions were the same. They were able to visualize this because of
the way the table “lined-up” equivalent fractions. Bryan was less clear, but Crystal
pointed out with the fraction bars, that 2/3 and 8/12 were the same. To illustrate this, she
divided the numerator and denominator of 8/12 by 4 to illustrate that it was the same as 2/3.

The study group now needs to be able to find equivalent fractions. A list of fractions will
be written on the board. The study group will make a list of equivalent fractions (At least
two for each fraction written on the board.)

When Crystal heard the rules for this activity , she just giggled and got to work. Shawn
and Bryan had to talk through what Crystal had said at the end of the previous activity,
regarding the numerator and denominator being divided by the same number. Shawn
remembered that he could multiply with the correct results. Bryan added that this was
“cross multiplication”. . . right? I asked what he meant by “cross multiplication” and he
explained that he needed to multiply both top and bottom of the fraction by the same
number. He gave an example of ½ being multiplied by 4 to get 4/8. This led into a
discussion of how to find the numerator or a fraction when you are given another fraction
and the denominator. (I.e.: ) Shawn pointed out that you could use “cross
multiplication” to solve the proportion for the missing value. He was able to illustrate
using several different equivalent fractions. Crystal was a bit unclear as to why “cross
multiplication” worked, so we discussed that it was really some simple algebra using the
multiplication and division properties of equality to isolate the variable. (The missing
numerator: Mr. Anderberg would be so happy with us.) One other thing that came about
with our discussions was the concept of reducing fractions to lowest terms . After a few
minutes, the study group had agreed upon the lowest term of a fraction . They defined it
as when a numerator and denominator cannot be reduced by dividing by a common
number. (Relatively prime.) We went on to discuss the term relatively prime. As I was
describing the concept with an example (4/5) Bryan and Crystal pointed out that 4 was
not prime. As I attempted to continue the explanation, I could see their eyes glaze over.
The study group had hit its wall. I abandoned this explanation.

March 11, 2005

Adding fractions and mixed numbers with common denominators

The study group will find the 4/7 bar and the 2/7 bar. Next, it will be demonstrated that
the sum of the two fractions is 6/7 and therefore, less than 1. The two added fractions
will be placed next to the rest of the 1/7-fraction bars to find what the total would be.
This will be repeated with several other sums to demonstrate what fractions with
common denominators total.

All three of the members of the study group were able to visualize this concept at first
glance. It seems that the fraction bars give a great visual for adding simple fractions.
Bryan, who has had the most difficulty, up to this point, explained that the 0/7 or 7/7 bar
can be used to count out how many segments are shaded when the two fractions are
added, instead of using all the sevenths fraction bars to guess and check to find the
appropriate answer.

As this progresses, the value of the sum of the fractions will increase. This activity will
be repeated by varying the addends according to size/color of the fraction bar.

Well, for every success, there is a challenge. Just when I thought Bryan was gaining
understanding, he ran into a wall, when the total of the two fractions was greater than 1.
For 6/7 + 5/7 he was able to see the 4/7, but he didn’t understand where the 1 came from.
Crystal explained using another whole bar to show that the total was 1⁴/7. He seemed to
catch on, but later it became an issue. It seems that he will be able to come up with the
fractional part of the answer, but when the answer is greater than 1, he will have
difficulty. Well, maybe not. After looking at the initial answer, 1-4/7 he looked at the
initial problem and saw how the fractions were added instead of how the fraction bars
combined to give an answer. Bryan then knew that since the answer was greater than 1,
that it would be a mixed number. Crystal then started explaining that there 7 could go in
to 11, once with 4 left over. Bryan started to catch on better at that point. He was able to
repeat the process several times. Shawn, by this time, was distracted and exploring with
the fraction bars to solve more complex problems.

March 16, 2005

Adding fractions with different denominators

The study group will add fraction bars that have different denominators, like 1/3 and 1/2.
They will show that the total isn’t in either thirds or halves. The students will use the
other fraction bars to show an equivalent answer when adding the two fractions. The
students will demonstrate that the total can be found using the sixths fraction bars. This
will be repeated with a variety of other sums.

I enjoy seeing kids when they discover something I am trying to get them to work
toward. Shawn picked up the 1/3 and 1/2 fraction bars and looked at them compared to
the other fraction bars. He got frustrated but kept looking. He understood that the 1/2
and 1/3 categories didn’t match. He looked at the 1/5 category, expecting to see the 2/5-
fraction bar to match, only to be disappointed. Crystal just laughed and told Shawn that
he needed to find the common denominator. She kept looking and happened across the
10/12-fraction bar. That was a match. She had figured out the concept I was planning on
explaining. Her answer wasn’t perfect (5/6) but still equivalent. I asked her about
simplifying the fraction and she said that this was a perfect match , but Shawn stepped in
and pointed out that the answer was 5/6. He searched for the 5/6 fraction bar to prove it
to Crystal and Bryan. When he found the correct piece, they both agreed after having
matched it up to the 10/12 fraction bar. I am starting to think that the more these kids
understand equivalent fractions, the simpler the process of adding fractions gets.

March 22, 2005

Adding fractions and mixed numbers with different denominators

As the study group becomes proficient with adding the fractions with different
denominators, I will increase the value of the fractions to include mixed numbers and
sums that are in excess of 1. The students will use extra whole unit pieces to represent
the whole number portion of the mixed numbers. They will then build upon the lesson
that we covered today in class.

Well, there’s not much to say about this activity. The study group picked up where they
had left off last week. They were able to associate the concept of adding mixed numbers
and improper fractions with common denominators to the concept of adding fractions
with different denominators. Shawn led the way on this activity for the first few minutes.
Once he had mastered the process, he handed over the reigns to Crystal, who worked
with Bryan to test each other. Bryan is becoming less involved in the activities. He has
become more cynical and sarcastic toward the process. Crystal appears to have a better
comprehension than at the beginning of the study sessions. Shawn goes back and forth
from being very attentive to being very distracted. The dynamic of the study group is
still positive, but Bryan is being excluded due to his behavior.

March 23, 2005

Using the computer applet to add fractions and mixed numbers with varying
denominators

The students will use my classroom computer to access the following web applet.
At this site, they will experiment with the fraction bars to add fractions with various
denominators. This applet is interactive and lets them know when they have an answer
correct or incorrect.

I had the students use this applet, because it most closely resembled the fraction bar
manipulatives. I saved this activity until the last couple of sessions, because I wanted to
center my research around using the actual hands on apparatus rather than computer
applet.

One thing I rediscovered was that different kids learn very differently. Each member of
the study group had a different attitude toward using the applet. Shawn really (and I
mean really) got into using the computer applet. He pointed out how the fraction bars
related to the fraction and how the varying denominators don’t match up on the fraction
bars, so they can’t just be added and once again the term common denominator was the
center of his conversation. Crystal did a very good job of making connections from the
fraction bar manipulatives to the fraction bar applet. For her, this helped her deepen her
comprehension of the concept of adding fractions. Bryan was very indifferent. He kept
referring back to the classroom lectures and to the fraction bars. Again, he is good for
about 5 minutes, before he checked out.

March 24, 2005

Comparing the computer applet to using the Fraction Bar manipulatives

We will spend a few more minutes working on the fraction bar applet followed by some
discussion on how each student feels about the methods we used to learn the concept of
adding fractions. Hopefully, there will be varied opinions from the study group.

Shawn was still very excited about the computer applet, while Crystal went back and
forth between the applet and the fraction bar manipulatives. Bryan emphasized that he
benefited from the classroom sessions the most. He felt he had a greater understanding
from the lectures than either of the study group methods. I’m glad to see he was more
involved today. I hope he will succeed on the upcoming post-test for the fraction unit.