Fractions As Result of Fair Sharing

The JavaBars Microworld:

Sample Activities

Dr. Heide G. Wiegel

The University of Georgia

2/9/2000

(updated 2/15/2010 by Dr. John Olive)

Note: The microworlds have been designed as part of the research project "Children's Construction of the Rational Numbers of Arithmetic," NSF-Grant No. RED-8954678. Many of the activities are based on activities designed by the members of the Fraction Project research team (principal investigators Dr. Leslie Steffe and Dr. John Olive, University of Georgia). The programmer for JavaBars is Barry Biddlecomb.

Multiplication and Division

Activity Group 1: Practicing Facts

JavaBars can provide some motivation for students who still have to work on their multiplication facts. In addition, the work in JavaBars will reinforce the array model of the operations.

I suggest that students work as pairs and pose problems to each other. Have them start with a rather small unit bar. After students have decided which multiplication or division facts they want to work on, they make a bar of that length. In Figure 1, I have chosen “eight’s” as the facts to practice. Next, make a big cover and copy the 8-bar. While one of the partners closes the eyes the other repeats the 8-bar under the cover. Both multiplication and division can be practiced with this set-up. For multiplication, the student posing the task tells how many rows there he/she made. The partner finds the result and can check by measuring the array of bars. For division, the student posing the task measures the array of bars, and the partner finds how many rows are. I modeled both operations in Figure 1. You can produce and save the set-up page for the students and load it before the students begin the session.

Figure 1: Practicing eight’s.

Unit Fractions as Fair Shares

The advantage of commercially produced fraction manipulatives is that the parts for a particular fraction automatically have the correct size. That is, all the 1/8 pieces, for example, are congruent. This way we can be assured that the students’ work will be accurate.

The disadvantage of commercially produced fraction manipulatives is that the parts for a particular fraction automatically have the correct size. That is, all the 1/8 pieces, for example, are congruent. This way, students may miss important learning opportunities.

It does matter that students experiment with the concept of fair shares as they divide a given shape manually, fold papers into equal parts, or cut and measure; they need to have those experiences. Even if we assume that your students experimented with equal shares in elementary school, the concept of fractions as equal shares might still be shaky or absent in certain contexts. For example, during one of my visits to summer school, I intended to demonstrate how 1/2 of a pizza could be divided into two fourths (Figure A). A student spontaneously said, “Now we have thirds” (Justin, 7/1/99). Justin saw a whole divided into three pieces; he disregarded the size of the pieces. Later another student suggested to divide one of the fourths into two eighths by dividing the radius of the circle—rather than the central angle—into two equal parts (Figure B). This student focused on length rather than area.

Figure AFigure B

Activity Group 2: Divide a candy bar into two, three, … , n equal shares.

The Cut function of JavaBars as well as the functions in the Pieces menu offer opportunities to reinforce the concept of equal shares. In the examples below, I used the functions of the Pieces menu.

Draw a bar, designate it as the unit bar, and make a copy of the unit bar. Activate the Up/Down or the Left/Right function and mark the bar into, for example three, equal shares. Break the bar and compare the pieces visually (Figure 2a). Join the pieces back together and make a new estimate. In Figure 2b, I used the PullOut function, disembedded the first part, and repeated it twice. As you can see, my estimate has improved, and the first piece is a good estimate of 1/3 because three of these pieces make up the whole. The yellow line that was left after joining the bars served as a guide for my new estimate.

Of course, students do not need to use the Repeat function in order to produce the whole bar (Figure 2b). The Copy and Join functions would work in the same way.

Figure 2a. Visual comparison of pieces.Figure 2b. Improvement of the estimate.

Unit Fractions as Building Blocks for the Whole

Activity Group 3: Share a candy bar among two, three, … people. Mark your share and check how close you are.

When we look at unit fractions as fair shares, we start with the whole and define the fraction in terms of the whole. The checking procedure depicted in Figure 2b represents a second perspective: We can start with a piece and attempt to produce the whole by reproducing the piece several times. For example, a bar represents one third of a given unit bar (=whole) if three of the bars make up the whole.

Draw a bar, designate it as unit bar, and make a copy of the unit bar. Activate the Up/Down or the Left/Right function and mark your share. Activate the Pullout function and disembed the marked share. Repeat the disembedded share—in this case, two times—and compare the new bar to the unit bar. Improve your estimate if necessary. Students can clear the mark from the bar and start from scratch, or students can make another copy of the unit bar and use the previous estimate as guide for the next try.

In Figure 3a, I modeled a strategy that students frequently use: I used the difference between the unit bar and the bar that was the result of the first estimate for my new estimate without distributing that difference (mentally) across the three shares.

In Figure 3b, I modeled two other strategies students often use. Because the estimate turned out to be too small, students often simply adjust the unit bar. Notice that when you manipulate the unit way, the label “unit bar” disappears; I inserted the label “1. lose unit bar” via the Label function. Alternately, students may add a small bar to the last piece in order to make up for the difference. Now the third piece will be larger than the other two.

Figure 3a. Construction of a whole bar from a share.

Figure 3b. Student strategies for equalizing bars.

Activity Group 4. Given a part of a bar, make the whole bar.

Figure 4a shows the initial screen for a session (5/26/99) with Joshua and Josh. Their task was to pick a part and produce the whole from that part. Note that all fractional parts are labeled, so from the adult point of view the task may seem trivial. Note also, that the unit bar is partially hidden, so the students could not use the unit bar for visual comparison, at least not for the first solution.

Figure 4a. Make the whole from a part—beginning screen.

Joshua started. He chose the 1/3-bar and repeated it twice. I don’t know whether my face gave me away or what other clue he used to stop. However, my initial hypothesis that the task was indeed trivial for him proved wrong immediately. When asked why he made three parts for the whole he said something like, “It looked right.” (See Figure 4b for the screen after completion of the first two examples.) In his next turn, Joshua chose the 1/8-bar. He moved it up to the 3/3-bar and said, “Three fit.” He then repeated the 1/8-bar eight times (resulting in nine parts altogether). He made a visual comparison of the 9/8-bar (my language) and the 3/3-bar and found his bar to be too large. He broke the bar and joined eight of the nine parts for an 8/8-bar.

Figure 4b. Two bars are completed.

In this session, Joshua never used the label on the part or the verbal description (one sixth) in his solution. He compared each part he chose to the1/3-part, and from that comparison he estimated how many parts he would need to make the whole. He then compared the newly produced bar to the 3/3-bar and made his corrections based on this visual comparison.

Note: All fractions were produced with functions from the Parts menu rather than the Pieces menu. For advanced work with fractions, the Parts functions are more appropriate than the Pieces functions.

In a more difficult version of this task, students are given fractional parts that are not unit fractions. Figures 4c and 4d show two possible set-ups. Notice that in Figure 4d the partitions inside the fractional parts are erased (use the Combine function). Generally, I would use the set-up in Figure 4c for elementary students and the set-up in Figure 4d for middle school students.

Figure 4c. Make the whole from a part – beginning screen

Figure 4d. Make the whole from a part – beginning screen.

Activating Multiplicative Knowledge in the Context of Fractions

Activity Group 5. Splitting a unit fraction into smaller, equal parts.

Given a unit fraction 1/n, make the whole bar. Pull out any one part of the n/n bar and divide it into two (three, … ) equal parts. How large is one of the newly created small parts? What is its fraction name? How many of these small parts would it take to make the original whole bar?

Joshua (6/17/99) chose the 1/5-bar. Without problems, he completed the whole bar (Repeat). When I asked him how he knew when to stop, he said he stopped when the bars were of the same size. He was still visually comparing rather than knowing when to stop. He divided the first part into two smaller, equal parts. Asked for the fraction name of one small part, he gave “one half” as his first answer and “two fifths” next. He needed to make the whole bar to find “one tenth” as the name for the new part (Figure 5a). I tried the language “one half of one fifth” once, but did not pursue it. I only used it because Joshua himself had used “one tenth of a tenth” in a previous session (without knowing, however, what that fraction would be).

Figure 5a. Finding half of one fifth.

Next, Joshua chose the 1/3-part. After he had partitioned the first third, he guessed that the smaller part would be “one sixth.” He pointed to the other two parts and extended the partition over these parts (2, 4, 6). Nevertheless, he had to make the whole bar to be sure.

Joshua solved the next task in a similar way. Of course, he got more efficient as he went on. He predicted correctly how many of the smaller parts would make a whole and what the fraction name would be. He even omitted the checking step several times, but not always.

Of course, any part can be divided by using the Left/Right button rather than the Up/Down button. After several sessions of splitting parts I was curious whether Joshua would see the different set-up as a different problem. But he identified 1/2 of 1/2 (1/3, 1/4, 1/5, 1/6) as 1/4 (1/6, 1/8, 1/10, 1/12) without a moment of hesitation (Figure 5b, not original).

Figure 5b. Finding 1/2 of 1/n.

Activity Group 6. Splitting unit fractions into equal smaller parts—finding the second division.

Given the unit fractions 1/2 and 1/3, make the whole bar. Pull out any 1/2-part and 1/3-part. Divide the 1/2-bar and the 1/3-bar into smaller, equal parts such that each part will measure 1/12. Vary the new denominators as well as the given unit fractions. Figure 6 shows Joshua’s work with halves and thirds. Joshua has partitioned 1/2 and created 1/10, 1/14, 1/18, 1/24, and 1/30. He has partitioned 1/3 and created the fractions 1/12, 1/15, and 1/24.

Figure 6. Joshua’s work with halves and thirds.

Activity Group 7. Sharing fractional parts (common fractions).

Given fractional parts of a unit bar. Students can find the size of the parts by either measuring them directly or by comparing them to the unit bar. Share each fractional part among two (three, …, ) friends. How much of a whole bar does each friend get?

Figure 7. Possible set-up for a more difficult sharing task.

Activity Group 8. Exchanging equal shares from fractional parts.

Draw a unit bar and two copies of the unit bar; divide one of the bars into fifth, the other into thirds. Use PullParts to disembed 1/3 and 1/5, respectively. The task is to divide the two fractional parts in such a way that we can exchange equal pieces.

Joshua has worked on several of these tasks: 1/3 and 1/5; 1/2 and 1/3; 1/4 and 1/3; and 1/5 and 1/7. In the beginning he used a trial-and-error approach. When he worked with the fractions 1/3 and 1/4, he knew intuitively that somehow 12 would be involved, but he did not know how. He divided both fractions into 12 parts and was surprised when the resulting shares were not equal (Figure 8a, bottom right). He finally arrived at two equal shares to exchange, 1/48 and 1/48 (Figure 8a, on the left).

Joshua chose two pairs of fractions himself: 1/4 and 1/12; and 1/4 and 1/8. He chose these particular numbers because he knew the outcomes ahead of time.

Figure 8a. Joshua’s work with 1/3 and 1/4—exchanging equal parts.

In the meantime, Joshua has made considerable progress. He knows that you always get equal parts if you “reverse the numbers.” For example, if you have the fractional bars 1/5 and 1/7, you can divide the 1/5-bar into seven equal parts and the 1/7-bar into five equal parts and you will get equal pieces to exchange. He does not always know how large the resulting pieces will be.

Joshua also knows that numbers “that are similar” are more complicated. For example, on 7/13/99 he chose to work with 1/6 and 1/9. He knew that he would have solutions in addition to those he would get when “reversing the numbers.” He found that he could partition both the 1/6- and the 1/9-bar into pieces of 1/36. As a second solution he found that pieces of 1/18 would also work (Figure 8b).

Figure 8b. Joshua’s work with 1/6 and 1/9—exchanging equal parts.