Have your parents ever found you munching on candy and asked you, "How much candy did you eat?" Instead of saying, "I do not know?" and getting in trouble, maybe you would rather say, "I ate precisely 10.7 cubic centimeters of candy, Mom." Make your parents proud of their candy-eating genius child (you) with this simple science project.
Objective
Investigate which formula is the most accurate for estimating the volume of an M&M'S® candy.
Geometry is the study of how to use math to describe and investigate different points, lines, and shapes. The way that a shape is described in geometry is with a formula, which is simply a mathematical way to calculate different properties of a shape like size, area, or volume. Volume is a unique property of three-dimensional shapes because three-dimensional shapes take up space in three different directions. Most real-world objects are three dimensional: balls, cars, food, etc.
The problem with geometric formulas is that they describe "perfect" or "ideal" shapes. A sphere is an "ideal" three-dimensional shape that is perfectly circular in all directions. Even though a ball is spherical in shape, it is not a perfect sphere. If geometric formulas describe "ideal" shapes and not "real" shapes, then how are they useful in the "real" world?
Most real-world shapes are not simple shapes and use complex geometry to be calculated. The properties of real-world shapes can also be approximated, or estimated, to the best possible measure with a geometric formula. This is called making a geometric model, and the most important part of making a good geometric model is choosing the formula that best describes the object. Even the most irregular objects can be modeled by using geometry: cars, airplanes, electronics, plastics, food, etc. Geometric modeling is very important for manufacturing because a product needs to have the same shape, made the same way, every time.
In this mathematics science project you will use geometry to produce a mathematical model of an M&M'S candy. If you look closely, you will see that the volume of an M&M'S candy is a bit irregular - it is not quite perfectly round. It looks like a ball shape (sphere) that has been squished on one side, as shown in Figure 1 below. You will test three different formulas (one for a sphere, one for a cylinder, and one for an ellipsoid) to see which formula makes the best geometric model of an M&M'S candy. You will test each formula by using it to calculate the volume of an M&M'S candy and then you will compare your result to the actual volume of a single piece of candy.
Figure 1.
An M&M'S candy looks like a sphere that has been flattened on one side. The blue M&M'S; on the left is shown from the top, while the orange M&M'S; on the right is shown from the side.
Terms and Concepts
Geometry
Formula
Volume
Model
Radius
Diameter
Height
Sphere
Cylinder
Ellipsoid
Questions
What is a geometric model? Why can it be useful?
Which formula do you think will calculate the most accurate volume of an M&M'S® candy? Why?
How are geometric formulas different from each other?
What other ways can you use geometric formulas to measure real-world objects?
Bibliography
You can do further research by visiting the following websites, which give information about geometry and calculating areas and volumes:
Mississippi State University. (n.d.). Agricultural and Biological Engineering: Tools - Unit-Free Volume Calculators. College of Agriculture and Life Sciences, Agricultural & Biological Engineering Department. Retrieved December 3, 2012, from
http://www.abe.msstate.edu/Tools/Volume/index.php
First measure the actual volume of an M&M'S candy with a water displacement test.
In your lab notebook, make a data table like Table 1 below. You will be recording your measurements in it.
Fill the metric measuring glass or cup with 100 milliliters (mL) of water.
Make sure it has exactly 100 mL. You can do this by looking at where the top of the water is when your eyes are level with it.
Add 100 M&M'S to the water.
Why do you think you are using 100 M&M'S instead of just one? Dropping just one M&M'S into a glass of water will not change the water level by much. By using 100 M&M'S you will be able to more easily see a larger change in the water level that will be easier to measure. You can then divide the change you see for a hundred M&M'S by the number 100 to calculate the volume of a single M&M'S candy.
In the data table in your lab notebook, record the new, final volume of water.
Estimate the new volume as closely as you can based on the marks on the glass. For example, if it is right between a mark that says "150" and one that says "175," then you can estimate that it is at about 163 mL.
Subtract the beginning volume of water (100 mL) from the new volume of water (that you just measured) to calculate the actual volume of the 100 M&M'S. Write this in your data table.
To continue the example above, if the volume for 100 M&M'S is 163 mL, then you do this calculation: 163 mL - 100 mL = 63 mL. Meaning that 100 M&M'S have a volume of 63 mL.
Divide your answer by 100. This is the actual volume of a single M&M'S candy in milliliters. Write this answer in your data table. You will be referring to this value later.
In our example you would do this calculation: 63 mL / 100 = 0.63 mL. Meaning that each M&M'S candy has a volume of 0.63 mL. Remember, this is just an example of the calculations. You will have to do the experiment yourself to see what the real volume is!
Actual Volume (mL)
Starting Volume (mL)
100 mL
Final Volume (mL)
Actual Volume of 100 M&M'S (mL)
Actual Volume of 1 M&M'S candy (mL)
Table 1.
In your lab notebook, make a data table like this one. You will be recording your water displacement measurements in it, which you will be using to figure out the actual volume of 1 M&M'S candy.
Next you will test different mathematical formulas to see which one is the best geometric model of an M&M'S® candy. Before doing this, make sure you do your background research and know what the terms radius, diameter, height, sphere, cylinder, and ellipsoid mean.
You will be making some careful measurements with (fresh!) M&M'S candies to use in the different formulas. In your lab notebook, make a data table like Table 2 below to record your measurements in.
Long Side (cm)
Short Side (cm)
Diameter of 10 M&M'S
Diameter of 1 M&M'S candy (divide by 10)
Radius of 1 M&M'S candy (divide by 2)
Table 2.
In your lab notebook, make a data table like this one. You will be recording your measurements in it, which you will be using to do calculations with different formulas.
Measure the long side of 10 fresh M&M'S lined up in a row. (Do not use any of the M&M'S that you used in the water displacement test!) Do this by using the following neat little trick:
Place a piece of paper on a clean table or countertop.
On top of the paper, place a small amount of clay or Play-Doh.
Flatten it and stretch it out into a little line. Make it run along the length of the ruler.
Line up 10 fresh M&M'S on their flat side, end-to-end, as shown in Figure 2 below. Poke them into some clay to keep them in a neat row with each M&M'S touching the next and no gaps in between them.
Figure 2.
To measure the long sides of 10 M&M'S, line them up length-wise in clay or Play-Doh, next to the ruler. This image only shows 4 M&M'S lined up, but you will be using 10 M&M'S.
Measure the whole line of 10 M&M'S from end-to-end. Write this measurement in the new data table in your lab notebook. Write it in the "Long Side" column as the "Diameter of 10 M&M'S."
Divide your answer by 10. This is the long diameter of a single M&M'S candy. Write the data in your data table.
Divide your answer by 2. This is the long radius of a single M&M'S candy. Write the data in your data table.
Remove the M&M'S from the clay.
On the clay, line up the 10 M&M'S on their side so that you are measuring across the short side, or short diameter, as shown in Figure 3 below.
Again, use the clay to hold the M&M'S in place in a neat row with each M&M'S® touching the next.
Figure 3.
To measure the short sides of 10 M&M'S, line them up on their side in clay or Play-Doh, next to the ruler. This image only shows 4 M&M'S lined up, but you will be using 10 M&M'S.
Repeat steps 5-8, but this time measure and do calculations for the short side of the M&M'S.
In the data table in your lab notebook, record your measurements in the "Short Side" column.
Next you will be making some calculations of volume using different formulas to see which one best calculates the volume of the M&M'S. In your lab notebook, make a data table like Table 3 below to record your results for each formula.
For the "Actual Volume" use the value you determined in step 1f. Note: Milliliters is the same as cubic centimeters (cm^{3}). So, even though you determined the actual volume of one M&M'S candy in milliliters, you can write this value in cubic centimeters instead.
Calculated Volume (cm^{3})
Actual Volume
Sphere - Long Radius
Sphere - Short Radius
Cylinder
Ellipsoid
Table 3.
In your lab notebook, make a data table like this one to record your results for each formula.
First you will calculate the volume of a sphere. Click on the "Full Sphere" link and you will see the calculator shown in Figure 4 below.
Figure 4.
Use the Full Sphere Volume Calculator (from Dept. of A&BE at MSU, 2006) to determine the volume an M&M'S candy would have if it were a perfect sphere.
You will be making the sphere calculation in two ways, with the long radius and the short radius.
Type the long radius into the box under "Radius" and click "CALCULATE." Write the answer in your data table next to "Sphere - Long Radius."
Note: Since your measurements are in centimeters, the answers will be in cubic centimeters.
Type the short radius into the box under "Radius" and click "CALCULATE." Write the answer in your data table next to "Sphere - Short Radius."
Next you will calculate the volume of a cylinder. Click on the "Cylinder" link and you will see the calculator shown in Figure 5 below.
Figure 5.
Use the Cylinder Volume Calculator (from Dept. of A&BE at MSU, 2006) to determine the volume an M&M'S candy would have if it were a perfect cylinder.
To use the cylinder volume calculator, do the following:
Type the long radius into the box under "Outer Radius."
Type a zero into the box under "Inner Radius" (since the M&M'S candy is not hollow).
Type the short diameter into the box under "Height."
Click "CALCULATE." Write the answer in your data table next to "Cylinder."
Next we will calculate the volume of an ellipsoid. Click on the "Ellipsoid" link and you will see the calculator shown in Figure 6 below.
Figure 6.
Use the Ellipsoid Volume Calculator (from Dept. of A&BE at MSU, 2006) to determine the volume an M&M'S candy would have if it were a perfect cylinder.
To use the ellipsoid volume calculator, do the following:
Type the long diameter into the box under "Major Axis" and into the box under "Minor Axis" because in the case of M&M'S® they are actually the same.
Then type the short diameter into the box under "Vertical Axis."
Click "CALCULATE." Write the answer in your data table next to "Ellipsoid."
Now you are ready to make a bar graph of your data. You can make one by hand or you can try using the Create a Graph website for kids from the National Center for Education Statistics.
Along the x-axis (the horizontal axis), make one bar for each type of volume calculation you did, such as a sphere using the long radius, sphere using the short radius, cylinder, and ellipsoid. Also include a bar for the actual volume that you determined in step 1f.
On the y-axis (the vertical axis) put the volume measurements in cubic centimeters (cm^{3}).
How do each of the different calculated volumes compare to the actual volume that you measured? Which ones were more and which ones were less? Why do you think this is? Which calculation came the closest? Which formula do you think is the best one to use for an M&M'S candy?
Another way to look at your data is to calculate the difference between each calculation and the actual volume measurement. You can do this by subtracting the actual volume from the calculated volume for each formula. A bigger number is more different from the actual volume than a smaller number. You can also calculate something called the percent difference by dividing your answer by the actual volume. If you make another graph comparing the percent difference of each method, what does it show?
You can use this same experiment to find the best formula to calculate any other volume. Try using it for an egg, a football, an apple, a bar of soap, or any other irregular shaped object. Just make sure that you choose an object that can safely be submerged in water!
Which formula is the best?
For a more advanced science project, you can try to investigate how the shape of a candy affects how well many of those candies pack together. Use the water displacement test on a couple differently shaped candies to determine the actual volume of a single candy. Then fill a measuring glass with a certain amount of each type of candy, one type at a time (without water). See how high this filled the glass and divide this total volume by the number of candies to determine how much space one candy took, on average, when taking packing into account. How much space does each type of candy take up in the measuring glass (when packing is taken into account) compared to the actual volume of one candy? In other words, which types of candies pack together the best?
How do you think their shape affects this?
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