Activity 5: Making Cents of DensityFebruary 6, 2008 through February 13, 2008 |
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In this activity we will:
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Collect the following materials:
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Make three lists in your calculator. Press   to get to the List Editor. |
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| Move your cursor to the header of the fist list. Notice the 1 in the upper right hand corner. | 
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Now press  until you get tired. That is, when all the lists are gone. |
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Enter your lists COINS, VOL, and MASS. While in the header of the first list press  [TEST] to access the alphabet. Move around with the cursor keys     to locate the letters you need. Press  to select a letter. If you make a mistake use the to remove back to your error. Highlight the Done and press   to accept the name and move to the next column. Repeat the process until finished. |
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| You will measure 7, 14, 21, and 28 coins at a minimum. Move into the list for COINS and enter these numbers. If you have more than 28 coins, keep adding the sets of 7 until you are finished. | 
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Determine the volume of your coins by measuring the diameter of one of your coins. Then make a stack of 7 coins to make a cylinder. Measure its height. Now calculate the volume using the following formula: where d is the diameter in cm, h is the height in cm, and V is the volume in cubic centimeters.Optional: Get the volume of the stacks of 7 coins by displacement. To do this, after you mass them, put 50 ml of water in a graduated cylinder and then drop the 7 coins in. Record the new water level and subtract to get the number of milliliters of water the coins occupy. This is the volume in cubic centimeters. |
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To get the volume we will go to the Solver. Press  to get to there. |
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If you don’t get a screen like this, move up to the top of the screen and you will be in the equation editor area. Press  to remove the old equation. Note: Each line/variable will need a value on it if you have an old equation so you can’t clear out values and then move off of the variable line. |
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Key in the formula above. Find the = sign in the Text Editor and pi is the key just above the  key. |
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| Press the key so we get this view. We will enter the values for D (diameter) and H (height of the stack of coins) and then solve for V. |
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| To solve for volume, you will need to put a guess in for V and then move down to the V at the bottom and press to get the answer. |
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| Notice the bullet that indicates you have the answer. Enter this volume in your list. You probably want to round off the answer. In this case the volume would be 4.28 cm3. Change the value for the height (H) and repeat the process. Remember that the diameter (D) stays the same. Try to guess the volume each time. | 
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| Now place your coins on the massing device and get the mass of 7 coins. Then add 7 more coins and get the mass for 14 coins, then add 7 each time to get the mass for 21 coins and 28 coins. If you have additional sets of 7 coins, collect those masses as well. Record the mass in grams in your list. | 
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| Do you see a relationship between the numbers? A pattern? What do you think the mass of one coin is? The volume of one coin? How about 1 dollar? | |
To help determine the pattern in another way we will need to set up a plot. Press [PLOT] and set up the plot as shown. Recall that you can access the list names by pressing  [STAT]. |
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Now check to see if you have any equations or plots on. Press  and turn off or clear your equations and see you only have Plot1 on. |
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Set the window with a ZoomStat. Press   and explore the graph with  . |
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We want to guess the equation that fits the data. Press  and start with Y1=1*X. Make it Bubble Baby by moving to the left of the Y1 and press until you get it. |
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Press  and see how close you are. |
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| If you didn’t hit it, keep trying. Recall that [INS] to insert rather than overstrike. |
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| This number, the one in front of the X, is the slope. The slope is the ratio of the change in Y and the change in X. In this situation that is the change in mass divided by the change in volume, or grams per cubic centimeters. This is the Density! | 
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| Check on the Internets or with your teacher to get the True density for your coin. The U.S. Mint would be a good place to look. To compare your answer with the Truth we can do a percent error. To get a better value for the density of your coin we can have the computer do the calculation. Return to the List Editor and make a new list named DENSE. Use the techniques used earlier. Press to get to the lists. |
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| In the header of this new list we will divide the mass by the volume. But since the math people have a rule about dividing by zero we need to delete the zero values in each list. To do this move to the place where the zero is and press then move to the next one until you have trashed all the zeros. |
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| These numbers should vary so we want to get the average. Do you see that these densities are close to the value you used in the Y= area? | 
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| To get the mean of a list go to the Home Screen and clean up. Press [QUIT] . |
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Press [STAT PLOT]   to get the mean( command. |
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| Now we say the name of the list we want the mean of. Press [STAT] to get the list names. Press to get the value. When you round this off is it better that what you guessed? You can change your value in Y= if you wish. |
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Now we are ready to do a percent error. The formula is: Key this in at the Home Screen using the value you picked for Y= as the Observed and the Truth will be what they report is the density of your type of coin. This should be less than 10%. In the sample it is 1.74%! |
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