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Candy in Jars

CTSE 5040 Multimedia Project

Kelly Graham

1. Introduction

“How many pieces of candy can fit in the jar? How many pieces will fit in bigger jars?”

https://www.youtube.com/watch?v=Cz0D0r0gXhk&feature=youtu.be

The mason jar shown in the video has a height of 9 centimeters. The width of the jar is 5.5

centimeters. Each piece of candy has a diameter of 2 centimeters. The task for this problem is to

figure out how many pieces of candy can fit inside the jar. The second question in the problem

asks the same for bigger jars, lets say one with a height of 15 centimeters and width of 10

centimeters and another with a height of 25 centimeters and a width of 16 centimeters.

To investigate this problem, I will find the volumes of the jars and candies in order to calculate

how many pieces of candy can fit inside the jar. I will do this for multiple jar sizes and gather my

results in a spreadsheet.

2. Student Solution

First, I will find the volume of the mason jar. The mason jar has a cylindrical shape, so I will use

the volume formula for a cylinder. The volume of a cylinder is 𝜋𝑟!ℎ, where 𝑟 is the radius of the

base of the cylinder and ℎ is the height. We know that the height, ℎ, is 9 centimeters. We also

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know the diameter is 5.5 centimeters, but we need the radius for the formula. I know that the

radius is half of the diameter so 𝑟 = !.!!= 2.75. Now I have all the information I need to find the

volume of the jar. 𝑉 = 𝜋𝑟!ℎ = 𝜋 ∗ 2.75 ! ∗ 9 = 𝜋 ∗ 7.5625 ∗ 9 = 68.0625𝜋 centimeters3.

Next, I need to find the volume of a piece of candy. The candy has a spherical shape, so I will

use the volume formula for a sphere. The volume of a sphere is !!𝜋𝑟!, where 𝑟 is the radius of

the sphere. Since the diameter of the candy is 2 centimeters, its radius is 1 centimeter. Thus the

volume of each piece of candy is 𝑉 = !!𝜋𝑟! = !

!∗ 𝜋 ∗ 1 ! = !

!𝜋 centimeters3.

Now I have to figure out how many pieces of the candy will fit in the jar. I know that each piece

of candy has a volume of !!𝜋 centimeters3 and the mason jar has a volume of 68.0625𝜋

centimeters3, I can set up an equation to solve the problem. I will use the variable 𝑥 to signify the

number of pieces of candy. My equation is 𝑥 ∗ !!𝜋  centimeters3 =  68.0625𝜋 centimeters3. Now

I have to solve for 𝑥, so 𝑥 = !".!"#$!  !"#$%&"$"'(!!!!  !"#$%&"$"'(!

. Centimeters3 and 𝜋 will cancel, leaving

𝑥 = 68.0625× !!= 51.046875. This number represents the approximate amount of candies that

can fit inside the mason jar. To account for errors in candy production and air space left in the

jar, I am going to round 𝑥 down to 50 pieces of candy.

The second question of the problem asks how many pieces of candy will fit in bigger jars. To

investigate this, I am going to create a spreadsheet in Microsoft Excel. Columns A and B contain

heights and widths of mason jars in centimeters. Column C contains a formula for the volume of

the mason jar in centimeters3, inputted as the product of 𝜋,  column A, and column B/2 squared.

Colum D contains the formula for the volume of a piece of candy in centimeters3, the product of

𝜋 and !!. Column E contains the formula column C/column D. I got the same answer for the

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original mason jar using the spreadsheet as I did with my calculations. Next I put other values for

the height and width into my spreadsheet, and then filled the formulas for the rest of the

columns. The formulas were able to compute the approximate number of candies that would fit

in jars with those dimensions, as shown below.

The results of the spreadsheet indicate that a jar with height 15 centimeters and width 10

centimeters would fit about 280 pieces of candy. A jar with a height of 25 centimeters and width

of 16 centimeters would fit about 1,200 pieces of candy.

3. Teacher Discussion

This problem addresses 8th grade and High School Geometry standards in the Alabama College

and Career-Ready Standards. The problem allows students to use mathematics to solve a real-

world problem. Problems like this are often seen in guessing contests. The standard addressed in

the 8th grade geometry category is “Know the formulas for the volumes of cones, cylinders, and

spheres, and use them to solve real-world and mathematical problems.” [8-G9] The standard in

High School Geometry, under the category geometric measurement and dimension, is very

similar: “Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.”*

[G-GMD3] As the asterisk following this standard indicates, the standard is also a modeling

standard. In this problem, students have to know the formulas for the volumes of cylinders and

spheres, and then apply then to solve the problem. Students then have to create an equation to

model relationships between the volumes in the situation. They then have to solve the equation

and analyze and interpret the solution. Then students may use technology, as shown in the form

of a spreadsheet, to expand upon the solution.

11

22

33

44

AA BB CC DD EE

Mason/Jar/Height/in/cm Mason/Jar/Width/in/cm Volume/of/Mason/Jar/in/cm^3 Volume/of/Candy/in/cm^3 Approx/Number/of/Candies

9 5.5 213.82465 4.188790205 51.046875

15 10 1178.097245 4.188790205 281.25

25 16 5026.548246 4.188790205 1200

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In the classroom, this problem would be a good activity for either 8th grade or high school

geometry students. In both courses, students should know how to calculate the volumes of

cylinders and cones. This problem allows them to apply their knowledge to a real-world problem

that they are likely familiar with. This problem could be done as either an in-class assignment or

an at-home project. It would be beneficial for all students to have access to technology such as

iPads or computers so that they can create a spreadsheet to display their findings. I enjoyed

coming up with and working through this problem, especially being able to see the real world

implications of the problem. I think that students would also enjoy doing this problem since they

have probably seen contests of guessing how much of something is in a container. This problem

also introduces an opportunity to discuss error, in measurements, rounding, and space in the jar.

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References

Alabama course of study: Mathematics. (2013). Montgomery, Ala.: Alabama State Dept. of

Education.