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Rational Number Relationships

 Activity Description Activity Guide

Part 1: Finding Patterns and Relationships

Open the spreadsheet file named Ratio.xls.

• 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.

• 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.)

• 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?

• 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.

• Justify your answer to the above problem algebraically.  Write the pattern you used as a function of the numerator.
f(numerator)=denominator.

• 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.

• 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.
• Excel can add a trendline to a graph that will give the line of best fit as well as the equation of that line. Add a linear trendline to the scatterplot.

 To add a trendline to a graph:   Select the graph of the data. Under the Chart menu, choose Add Trendline… Select the type of regression (in this case choose linear). Click on the Options tab. Be sure to select the option to display the equation. Click OK.
• 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 . . .

• 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:

• Why is 1/20 equivalent to 0.05 and 5% ?

• Why does the pattern increment by 5% with each new fraction?

• What is the relationship between the numerator and the percent value?

• 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:

• Explore the following fractions in this pattern:

1/9, 2/9, 3/9, . . . ,8/9, 9/9

What do you notice about the decimal representations of these fractions?
0.999999… = 1
Is this statement true or false? Justify your reasoning numerically and demonstrate the proof algebraically.

• Think about the following situation:

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?

• 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:

• 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?

• 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?

• 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.

 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.
• 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.
• In order to add the tactile interactivity to the spreadsheet, you need to insert two scroll bars that will control the values of the numerator and denominator.

Note: Objects such as scroll bars and buttons can only be accessed by first adding the Forms toolbar to the standard Excel toolbar.

 To add the Forms toolbar: Under the View menu, choose Toolbars…, and then choose Forms. 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.

 To add a Scroll Bar to a spreadsheet: 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.  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).
• 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.

 To Format a Scroll Bar:   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. 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.
• 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.

 To make a pie graph:   Select 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 Pie Graph 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, click on Finish.

 To change the colors in a graph:  Double-click on the chart in order to edit it.  Double-click on the desired pie slice until only that slice is selected.  Select the Patterns tab and select the appropriate color.  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.

 To change the display options in the pie graph: Double-click on the pie graph.  Select the Data Labels tab and select the display percents option.  Select the Options tab and change the degree setting to 90.  Click OK to update the pie graph
• Use the sliders to change the value of the fraction and be sure that all the displays are accurate.

• Did this activity improve your own understanding of the relationships and patterns in rational numbers?  If so, how?

• Discuss how the concept of ratio is a precursor to conceptual understanding of slope and limits.

• Discuss the benefits and/or drawbacks of using multiple representations of rational numbers.

• 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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