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Part 1: Finding Patterns and
Relationships
Open
the spreadsheet file named Ratio.xls.
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Use the scroll
bars to explore several common fractions such as 1/4, 1/2 , 3/4. Discuss
the relationship between the four representations (fraction, decimal,
percent, pie graph) as well as the relationship between 1/4, 1/2,
and 3/4.
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Change the fraction
values to 1/2. Find several other fractions that will also shade 50%
of the pie. Justify why the other fractions are also equivalent to
50%. (Hint: Relate this to money as 50 cents is half of 1 dollar.)
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Do the same type
of exploration with fractions such as 1/4, 1/5, 3/4, 1/3, 2/5, etc.
Starting with the fraction in simplest form, use the scroll bars to
change the numerator and denominator to obtain the next fraction in
the equivalence class. Connect the number of clicks used with the
scroll bars to the simplest fraction.
What do you notice?
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Use the patterns found in the previous
explorations to solve the following task without
the use of the spreadsheet.
If the following
list of equivalent fractions:
1/5, 2/10, 3/15, 4/20, 5/25, 6/30 . . .
were to continue, what would the numerator be if the denominator is
215? Justify your answer numerically.
THE REMAINING TASKS
IN PART 1 ARE AN EXTENSION FOR ALGEBRA STUDENTS.
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Justify your answer to the above problem
algebraically. Write the
pattern you used as a function of the numerator.
f(numerator)=denominator.
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Open a clean worksheet and explore the function
both numerically and graphically. First make a two-column table
containing the numerators and denominators as ordered pairs.
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Create an XY scatterplot of the ordered pairs.
What do you notice about the points on the graph?
To make
a XY scatterplot:
- Highlight the cells containing the labels and the
cells containing the data to be graphed.
- Click on the ChartWizard then click in the spreadsheet at the point where you
want your graph to appear. The first pop-up window shows all the different types of
graphs that Excel can create. Choose the XYScatter icon and click on Next.
- In the next pop-up window, verify the cell range
reference for the data you have selected. If this is correct,
click Next.
- In the final step, give the graph a title, label
the X and Y axes, and click on
Finish.
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To add a trendline to a graph:
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Select the graph of the data. Under the Chart menu, choose Add Trendline…
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Select the type of regression (in this case
choose linear).
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Click on the Options tab. Be sure to select the option to display the equation.
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Click OK.
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- What is the slope of the trendline?
How does the slope of this line connect with the pattern found
in the original list of equivalent fractions?
1/5, 2/10, 3/15,
4/20, 5/25, 6/30 . . .
Part
2: Connecting Fractions, Decimals, and Percents
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Set the fraction
to 1/20 and incrementally change the numerator to explore the following
pattern: 1/20, 2/20, 3/20, 4/20,…19/20, 20/20. Consider the following
questions:
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Why is 1/20 equivalent
to 0.05 and 5% ?
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Why does the pattern increment by 5% with each
new fraction?
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What is the relationship between the numerator
and the percent value?
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Do the same exploration with patterns such as:
1/5, 2/5, ….5/5
1/10, 2/10,….9/10,
10/10
1/50, 2/50, 3/50,
4/50,…49/50, 50/50
EXTENSION
FOR PRE-CALCULUS
STUDENTS:
1/9, 2/9, 3/9,
. . . ,8/9, 9/9
What do you notice about the decimal
representations of these fractions?
Think about the following statement:
0.999999… = 1
Is this statement true or false? Justify your
reasoning numerically and demonstrate the proof algebraically.
Part
3: But I only missed one!
Sarah took a spelling
test with 20 words on it, misspelled one word, and received a score
of 19/20. James took a different spelling test with 16 words on it,
misspelled one word, and received a score of 15/16.
Since both students
missed one word, are these scores equivalent? Why or why not?
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Use the scroll
bars to display the fraction 1/2. Then increment both the numerator
and denominator by one to model the pattern 2/3, 3/4, 4/5, 5/6, .
. . , 99/100. What happens to the area shaded in the pie graph as
both the numerator and denominator increase by 1? Will the pie ever
be completely shaded? Why or why not?
EXTENSION FOR ALGEBRA
II AND ABOVE STUDENTS:
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Think about the above pattern as a function of x
(e.g., if x=1, f(x)=1/2). Open a clean worksheet and create two
columns labeled x and f(x). In
the x column, enter the values 1 to 100 (Hint: Use the Fill
Series option). If x
represents the term in the sequence of fractions, write a function
rule to determine each fraction in the sequence. In the cell adjacent
to x=1, in the f(x) column, use an appropriate formula to calculate
f(x). Fill this formula
down to complete your table of values.
What do you notice about the values in the f(x) column?
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Create an XY scatterplot of your table of values.
What do the numerical and graphical representations suggest
about
lim(as x approaches infinity) x/x+1?
Part
4: Creating the Template
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Open a new Excel workbook.
To create the fraction, decimal, and percent displays, you need
4 cells. Cell B2 will
contain the value of the numerator, cell B3 the value of the
denominator. Cell C2 will display the decimal representation, and cell
D2 will display the percent equivalent. Select all 4 of these cells
and change the background color of these cells.
To select
non-adjacent cells:
- Select the first desired cell, then while holding
down the COMMAND (MAC) or CTRL (PC) key, highlight the
non-adjacent cells.
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To change
the background color of a cell:
- Select the cell or cells that you wish to color.
- Click on the down arrow on the Color
Bucket button (upper right hand corner on toolbar) and
choose the desired color.
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To get us started, let B2=1 and B3=2 to represent
1/2
. Use a formula in cell
C2 to calculate the decimal equivalent of this fraction. Let D2=C2 and
use the percent button on the toolbar to display cell D2 as a percent.
Try changing the values in cells B2 and B3 to be sure the other cells
change accordingly.
To display
a value as a percent:
- Click on the cell containing the value you wish
to display as a percent.
- Click on the
key on the toolbar. The decimal place of the current value should
move two places to the right and the percent symbol should be
displayed.
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Note:
Objects such as scroll bars and buttons can only be accessed by first
adding the Forms toolbar to the standard Excel toolbar.
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To add the Forms toolbar:
- The Forms toolbar will then appear either as part
of the existing toolbars at the top of the document or as a
floating box within the spreadsheet.
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To add a Scroll Bar to a spreadsheet:
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Once the Forms toolbar appears, click on the
Scroll Bar button
 , and then click and drag to form a horizontal
narrow rectangle where you want the scroll bar placed in the
spreadsheet.
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If the scroll bar is not the desired size or
orientation, the scroll bar can be resized with the corner
handles and will float over the spreadsheet so it can be moved
to the desired position. To select the scroll bar, either do a
right mouse click (PC) or a Control mouse click (MAC).
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Once the scroll bars are in place, you must
format each scroll bar to define which cell it will control.
When formatting the scroll bars, recall that you want to avoid
having the denominator display a 0.
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To Format a Scroll Bar:
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To select the scroll bar, either do a right mouse
click (PC) or a Control mouse click (MAC). A pop-up menu should
appear. Select the Format Control option.
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Once Format Control is selected, a pop-up window
will appear.
- Within this widow, define the maximum
and minimum values for the scroll bar, as well how the numbers will increment
when you click on the left or right arrows of the scroll bar.
Leave the page change
value as 10.
- In the box labeled Cell Link, type the cell
reference for the cell you want the scroll bar to control.
- When finished, click OK to close this pop-up
window.
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The final representation displayed in the
template is a pie graph. How
can you create a pie graph so that it displays the fraction as a part
of the whole (100%)? Recall that the pie graph displays two
quantities, the fraction and its complement. Use what you know about
pie graphs and percents to create the pie graph display.
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To make a pie graph:
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Select the cells containing the data to be graphed.
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Click on the ChartWizard then click in the spreadsheet at the point where you
want your graph to appear.
The first pop-up window shows all the different types of
graphs that Excel can create.
Choose the Pie Graph icon and click on Next.
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In the next pop-up window, verify the cell range
reference for the data you have selected. If this is correct,
click Next.
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In the final step, click on Finish.
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To change the colors in a graph:
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Double-click on the chart in order to edit it.
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Double-click on the desired pie slice until only that slice is
selected.
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Select the Patterns tab and select the appropriate color.
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Repeat this process for each slice.
SHORTCUT:
Once the chart is selected for editing, only do a single-click
on the slice you want to change.
Once that slice is selected, use the Color
Bucket button on the toolbar to choose the desired color.
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To change the display options in the pie graph:
- Double-click on the pie graph.
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Select the Data Labels tab and select the display percents
option.
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Select the Options tab and change the degree setting to
90.
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Click OK to update the pie graph
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Part
5: Conceptual and Pedagogical Discussion
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Did this activity improve your own understanding
of the relationships and patterns in rational numbers?
If so, how?
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Discuss how the concept of ratio is a precursor
to conceptual understanding of slope and limits.
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Discuss the benefits and/or drawbacks of using
multiple representations of rational numbers.
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How could you use parts of this activity with
students in various grade levels and mathematics courses?
What other types of activities could you use prior to,
concurrent with, or following this type of exploration that would
further enhance students' conceptual understanding?
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