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:
!
!
!
Student encounter with the aspect of mathematical modeling.
Development of group cooperation, problem solving, and communication skills.
Student produced algorithms for changing rounded decimals to simple fractions.
(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
Funded in part by the National Science Foundation and Indiana University 1995
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.
20% 1. Keep a complete record of your attempt to solve the problem. Include
calculator work.
25% 2. Organize your notes. Include any tables, charts, equations, arithmetic,
diagrams, etc. that you used.
25% 3. Write a paragraph explaining your notes and thought processes.
20% 4. Write an algorithm (a set of directions) for finding the least number of
at bats for any given batting average.
10% 5. Use a batting average of your choice to demonstrate that your
algorithm works.
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
Funded in part by the National Science Foundation and Indiana University 1995
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
Funded in part by the National Science Foundation and Indiana University 1995
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.
Funded in part by the National Science Foundation and Indiana University 1995
Here's my work:
2/5
3/5
3/6
3/7
3/8
4/8
4/9
4/10
5/10
5/11
5/12
5/13
6/13
6/14
6/15
6/16
7/16
7/17
7/18
8/18
8/19
8/20
9/20
9/21
9/22
< .409
> .409
> .409
> .409
< .409
> .409
> .409
< .409
> .409
> .409
> .409
< .409
> .409
> .409
> .409
< .409
> .409
> .409
> .409
> .409
> .409
< .409
> .409
> .409
 .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
Funded in part by the National Science Foundation and Indiana University 1995
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.
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.
Funded in part by the National Science Foundation and Indiana University 1995
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:
Funded in part by the National Science Foundation and Indiana University 1995
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:
Funded in part by the National Science Foundation and Indiana University 1995
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:
Funded in part by the National Science Foundation and Indiana University 1995
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