Decimals to Fractions: How Many Times At Bat?

Why is it that Middle School teachers teach students how to change fractions to decimals but you

don't teach them to change decimals to fractions? -- Henry Pollack, Summer 1991(paraphrase)

Van Campbell, Central Middle School, Columbus, IN.

Thomas Keys, Kenwood Academy High School, Chicago, IL.

Charles Loeffler, Jones Middle School, Marion, IN.

Class: First year Algebra and above, Gifted Middle School

Materials: Calculators required, and, due to the nature of this open ended problem, any other tool ( graphing calculator, spreadsheets, etc...) that the student feels is useful is fine. Student warm-up sheets are included as well as several different approaches to the proposed problem.

Goals:

(Note: Simple fractions are not to be confused with simplified fractions. The simplest fraction of a set of fractions with integral numerators and denominators is the one that has the least denominator. Using this definition and the set {2/4, 2/5, 2/7}, 2/5 and 2/7 are in simplest form but 2/4 is the simplest fraction of the three. Of course, when 2/4 is reduced to lowest terms an even simpler fraction is obtained.)

Time Required: Minimum of 4 to 5 days at one 40-50 minute class per day.

Background: Practice and conceptualization of ordering fractions. Familiarity with converting common fractions to decimal fraction equivalents.

Setting: Often a finite set of data leads to a ratio that is reported as a decimal fraction with a given precision. A common example is a baseball batting average. Consider the following cases:

HITS

AT BATS

HITS/ AT BATS

AVERAGE

3

6

0.5

0.500

3

7

0.42857143

0.429

3

8

0.375

0.375

3

9

0.33333333

0.333


More often than not, as in the case of 3 hits in 7 at bats, computation of batting averages yields a decimal that must be rounded to the thousandths. This means that reversing the process, going from the rounded decimal to a simple fraction, is not simply a matter of converting the batting average to a common fraction with a denominator of 1000 and simplifying.

Designing an algorithm that searches and finds simple fractions that produce a given rounded decimal is within reach of early adolescents. The process of discovering such an algorithm is an excellent problem solving activity, provides insight into student conceptualization of fractions, and is a good vehicle for introducing mathematical modeling.

Problem: Describe how to find the least number of times at bat necessary for a batting average of .409. Generalize to an algorithm for converting a given decimal fraction, x, that is precise to n-decimal places to the simplest fraction that will yield x when converted and rounded to an n-place decimal fraction.

Sample Solutions: The final student product is expected to communicate the cooperative group's approaches to solving the problem, including false starts. The heart of each of the written and oral presentations is to be a clear, concise algorithm for converting any batting average to its corresponding simplest fraction. To illustrate the variety of possible students responses, four approaches to the ".409" problem have been appended. The four methods are not meant to be exemplary. They are given to provide the teacher with food for thought for planning assessment and supporting activities to precede and/or follow this modeling project.

Evaluation:

Evaluation will be made on teacher observation of group processes, interaction and communication.

Extensions:

A sampling simulation in which the students try to determine the ratio of a mixture.

A class discussion on efficient and inefficient algorithms.

A spreadsheet activity that demonstrates student's algorithm.

A newspaper activity that searches for rounded decimals in the news.


Teacher Notes:

Groups of two to four students are suggested. The following problems will be given to set the stage for the primary task of developing algorithms for converting a rounded decimal to its corresponding fraction.

Warm-up 1:

Carlos' current batting average is .375. It's very early in the season. How many times do you think Carlos has been to bat? You must come up with a reasonable answer that you can support with a clear and precise explanation.

Warm-up 2:

Reggie is an 85% free throw shooter for the playoff series. Assuming the reporter did not err, what is the least number of free throws that Reggie could have made to be an 85% free throw shooter? (Hint: Consider what the reporter would say about a player that shot only 3 free throws and made 2 of them.)

The numerical content of Warm-up 1 is intentionally simple, as the main intent is to introduce an ambiguous real world situation that needs to be clarified before creating a real world model. The problem is presented with questions that help student focus. (See Appendix.) There are several reasonable responses, however the separation between reasonable and unreasonable is fuzzy. This gives the students the opportunity to list aspects of real world situations that must be considered when translating the real world situation to a real world model before translating to a mathematical model. In this problem, changing the decimal fraction to simplified equivalent common fraction yields the least number of times that Carlos could have been at bat. In the subsequent problem the same strategy does not yield the least number of times for the particular event.

The students are given warm-up 2 without the hint. As a result, most students will arrive at a premature conclusion of 17 free throws by converting 85% to 85/100 and simplifying to 17/20. At this time the hint about a player going 2 for 3 can be used to help them see the effect of rounding to the nearest percent.


Appendix

Sample approaches to the .409 problem

Method 1:

In thinking about two whole numbers whose quotient is .409, I used the idea of inverse operations to conclude that .409 times the number of at bats would equal the number of hits.

So, I multiplied .409 by 1,2,3,... until I found a product that was close to a whole number. I realized that 409 divided by 1000 would give exactly .409, so I knew since 409/1000 could not be reduced that no other product using whole numbers less than 1000 would be exactly a whole number.

This was going to be a lot of work, so I decided to use a spreadsheet. In cell A1 I entered .409 and copied this value down to row A50. In cell B1 I entered 1; in cell B2 I entered the formula =B1+1 and filled this formula down to B50. In cell C1 I entered the formula =A1*B1 and filled down to row C50.

I could continue this process if needed.

Looking at the entries in column C for values that were close to being a whole number, I realized that the solution had to be slightly more or slightly less than a whole number. It was immediately apparent that I needed to know the answer to the question " How close is close?" or its equivalent "Just what does slightly mean?"

Looking at my products, by far the best candidate for an answer was .409*22 which equaled 8.998. So I tried dividing 9 by 22 which equaled .4090909.... which did round to .409. Success! The answer was 9 divided by 22. In thinking about this method I am still wondering about "How close is close?" and "Just what does >slightly= mean?".

Method 2:

I was supposed to find the least number of times a batter would have to go to bat in order to have a .409 batting average. I first noticed that .409 is a little larger than .4 which is 2/5. I also noticed that .409 is less than .5 or 1/2. That is .409 is between 2/5 and 1/2. I decided to average 2/5 and 1/2. This was easy with my calculator. I got 9/20 which I then converted to a decimal and got .45. Too large! But, this told me to look between 2/5 and 9/20. Averaging again I got 17/40, which converts to .425. Too large again. The average of 2/5 and 33/40 gave me 31/80 or .4125. I noticed a pattern that told me my next average would be .40625 which is too small. Sure enough, when I averaged 2/5 and 31/80 the answer was 65/160 or .40625 which is too small. I also noticed that 65/160 simplifies. My calculator quickly told me that 65/160 reduces to 13/32. Since 13/32 is too small I now had to average 13/32 and 31/80.

Bingo! .409325  .409, so the batter had to go to bat at least 80 times to bat .409. However, I saw that Julie got an answer of 22 times. She said that 9 out of 22 works. I checked and it does. I don't understand where I went wrong!


Method 3:

If the numerator is increased, the fraction becomes larger. If the denominator is increased the fraction becomes smaller. I bumped either the numerator or denominator up as needed until I found the answer which is 9/22 = .40909...  .409. Also if I start with 0/1 my method will work with any rounded decimal.

Here's my work:

2/5 < .409

3/5 > .409

3/6 > .409

3/7 > .409

3/8 < .409

4/8 > .409

4/9 > .409

4/10 < .409

5/10 > .409

5/11 > .409

5/12 > .409

5/13 < .409

6/13 > .409

6/14 > .409

6/15 > .409

6/16 < .409

7/16 > .409

7/17 > .409

7/18 > .409

8/18 > .409

8/19 > .409

8/20 < .409

9/20 > .409

9/21 > .409

9/22 .409


Method 4:

I know .409 is slightly larger than 2/5. Lets think graphically!

A batting average of .400 is possible with 2 hits in 5 at bats, 4 hits in 10 at bats, 6 hits in 15 at bats, 8 hits in 20 at bats and 10 hits in 25 at bats. These data points are represented by the circles in the graph below. A batting average of .409 must be slightly above the line connecting these points. Looking for grid points slightly above the line, I find 3/7, 5/12, 7/17, 9/22, and 11/27. These points are marked with X's in the graph below. Using the calculator to check these possible solution points, we find that 3/7.429, 5/12 .417, 7/17 .412, 9/22 .409. So the least number of at bats that gives a .409 batting average is 9 hits in 22 at bats.

0x01 graphic


Method 5:

An Efficient Algorithm

Any rounded decimal can be converted to its corresponding simplest fraction by following these six steps.

1. The desired fraction is between 0/1 and 1/1.

Call 0/1 the Lower Bound.

Call 1/1 the Upper Bound.

2. Add the numerators AND the denominators of the Upper and Lower Bounds to obtain a new fraction.

3. Convert this new fraction as a decimal.

4. Round this decimal to the desired precision.

5. Compare this rounded decimal with the problem decimal.

6A. If the new fraction is greater than the problem decimal, it becomes the new Upper Bound, and you return to step 2.

6B. If the new fraction is less than the problem decimal, it becomes the new Lower Bound, and you return to step 2.

6C. If the new fraction is equal to the problem decimal, you are finished. The last new fraction is the answer.

This algorithm generates the sequence

0/1, 1/1, 1/2, 1/3, 2/5, 3/7, 5/12, 7/17, and 9/22

for the problem decimal 0.409. This sequence is embedded in sample methods 3 and 4. Revisiting methods 3 and 4 may lead to student discovery of this efficient algorithm.


Student Sheet 1:

0x01 graphic

Warm-up 1:

Carlos' current batting average is .375. It's very early in the season. How many times do you think Carlos has been to bat? You must come up with a reasonable answer that you can support with a clear and precise explanation.


STUDENT SHEET 2:

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Warm-up 2:

Reggie is an 85% free throw shooter for the playoff series. Assuming the reporter did not err, what is the least number of free throws that Reggie could have made to be an 85% free throw shooter?


STUDENT SHEET 3:

0x01 graphic

If Carlos was leading the American League in hitting when the league released its first statistics of the season, find the number of hits and at bats he had to have in order to have a .409 batting average.

Funded in part by the National Science Foundation and Indiana University 1995