Eikev- An activity in comparisons and estimated measures

Last year for Parshat Eikev, I wrote about Rashi and Tanchuma's commentaries regarding the size and weight of the stone tablets that Moshe brought down with the 10 commandments inscribed on them. Their commentaries explained that despite there being differences between the two tablets, the size and weights of the two tablets were actually identical.

Thoughts on a follow-up activity:

• Last week in Parshat Va'etchanan (Devarim 5:6-18) we read the 10 commandments. These commandments were divided into the first 5, which are between man and Hashem and inscribed on the first tablet, and the second 5, which are between man and man and inscribed on the second tablet.
• If you compare, side by side, the wording, number of letters, number of characters, etc. between the two subsets of commandments, what are the actual differences between them? Which has more words, letters, characters, etc. and which has fewer? 
• Think about how you might compare the inscriptions proportionally to each other? 
• Is there a way to calculate- if I start with two 25 lb. tablets, and then inscribe in each one, is there a way to estimate how much each one should have weighed at the end? 
• Could you estimate surface area if you take a standard font and printed onto a sheet of paper- what's the ratio of print to total surface area for each tablet? What might the proportional volume differences be? What about weight differences? 
• There is a discussion in Gemara Bava Batra 14a wherein it is determined that the size of each tablet was 6 tefachim (handbreadths) high, 6 tefachim wide, and 3 tefachim deep. 
• Based on these measurements, what might the original volume of each tablet have been?
• Based on these measurements, what might the original weight of each tablet have been?
• Using estimated proportionally carved out sections from each tablet, what would one have assumed the new volumes/weights of each tablet to have been after the inscription of the commandments?
• Older students can also work through the measurement calculations from the source in Bava Batra related to the ark (where the tablets were kept) and the tablets, to see how the measurements were determined.

Beshalach- Measurement related activities

"This is the thing that Hashem has commanded, 'Gather from it, for every man according to what he eats- an omer per person- according to the number of your souls, everyone according to whomever is in his tent shall you take." ~Shemot 16:16

"They measured in an omer and whoever took more had nothing extra and whoever took less was not lacking; everyone according to what he eats had they gathered." ~Shemot 16:18

"It happened on the sixth day that they gathered double bread, two omers for each; and all the princes of the assembly came and they told Moshe." ~Shemot 16:22 

Rashi on Shemot 16:22 explains that the Israelites actually collected the same amount when they went out on Fridays, but when they came home to measure how much they had, they found that they had 2 omers.

"The omer is a tenth of an ephah." ~Shemot 16:36

Rashi on Shemot 16:36 explains- 
1 ephah = 3 se'in
1 se'ah = 6 kabin
1 kav = 4 lugin
1 log = 6 beitzim
1 beitzah = the volume of 1 egg
Therefore, a tenth of an ephah is 43 beitzim & 1/5 of a beitzah

(Chart from Artscroll Series, The Sapirstein Edition; Student size edition; Vol 2- Shemos/Exodus pg 198)

Activity Connections:
Activity ideas in increasing conceptual difficulty:

• [Pre-activity 1] Younger students begin learning about measurement by playing with standard and non-standard measurements. Young students may begin by playing around with how many paper-clips long different items are, how many blocks tall each of their friends is (using standard size blocks), or how many classroom cups of water or rice it takes to fill a container. By playing around with these non-standard measurements, they get a sense of how to compare lengths of items without needing to worry about how to read rulers or measuring cups. Following-up on a non-standard activity, students can have a discussion about how you might be able to talk about your non-standard activities with students in other classes. What if their paperclips/blocks/cups are a different size? How could you express your measurements to those students? This helps illustrate for them where standard measurements are needed.
• [Pre-activity 2] Younger students can also have an opportunity to investigate different containers or items that have been pre-measured with standard units for them to compare quart/liter/gallon, inch/foot/yard- or metric equivalents. 
• Connecting to this week's parsha, a pre-measured container could also show the estimated volume of an omer (based on our understanding of the measurement- see below), so that students can see how much food each Israelite was apportioned for each day in the desert.
• Using a pre-measured "Omer Container" students could play with the measurement and see how it compares to our current day measurements. Using investigation- which is bigger- an omer or a quart? an omer or a gallon? How many of the smaller measure does it take to make up the larger measure? Can you express the smaller measure as a fractional amount of the larger measure? These are two different ways of expressing the same equivalency based on the measure you use for a frame of reference (ie 2 cups = 1 pint; 1/2 pint = 1 cup).
• Older students can work on calculating conversions based on the information provided by Rashi in 16:36. How do the different Biblical measures compare to each other? A modern day understanding of the volume of a beitzah is indicated here as 2.53 fl. oz. (or 75 ml). Using this understanding of beitzah, what would each of the other volume measurements be in Rashi's explanation? Ultimately, working through the calculations, what would the volume of an omer have actually been? What would that look like in current day measures? What was the volume of food that the Israelites had for each day's sustenance? What did the volume collected on Fridays (in preparation for Shabbat) look like?

Terumah- Volume (again) and Surface Area

"They shall make an Aron of shittim wood, two and a half amot its length; an amah and a half its width; and an amah and a half its height. You shall cover it with pure gold, from inside and from outside you shall cover it, and you shall make on it a golden diadem all around...You shall make a lid of pure gold, two and a half amot its length, and an amah and a half its width" ~Shemot 25;10-11, 17

In Parshat Terumah, we are essentially given the blueprints for building the mishkan, or traveling tabernacle, that the Jewish people built while traveling in the dessert. This parsha is filled with so many mathematical possibilities, but I'd like to focus on a follow-up topic from back in Parshat Vayeishev. 

When investigating with 3-dimensional figures, the major calculations that students investigate are volume and surface area. Volume, as discussed in Parshat Vayeishev, is the space, in cubic units, that an object occupies or the amount of space inside the shell of a shape that can be filled. We learned that the calculation for the volume of a rectangular prism (or box shape) is to multiply the length x width x height of the prism.

Surface area is the calculation of the 2-dimensional area that covers the entire shape. For a rectangular prism, that would be the total area of all 6 sides of the prism added together. While volume is measured in cubic units, surface area is calculated in square units (the difference between a 3-D measurement for volume and a 2-D measurement for area). Lateral surface area, a variant of surface area, is the calculation of the area that covers the middle section of the prism, but does not include the top and the bottom. As a reminder, the area of a rectangular 2-dimensional shape is calculated by multiplying the length x width.

An example that I like to use to help understand the meaning of volume, surface area, and lateral surface area:

If I have a rectangular shaped room...
...volume is the amount of space inside the room if I filled it completely from floor to ceiling  and wall to wall.
...surface area is the amount of wall space that I would cover if I wanted to paint the entire room- all four walls, the floor, and the ceiling.
...lateral surface area is the amount of wall space that I would cover if I wanted to paint or wallpaper just around the four walls.

Connection to the parsha:
Given the information that we are given for building the mishkan, can we calculate volume, surface area, and lateral surface area? If so, what are each of these three measurements?

We are given the 3 necessary dimensions for accurately building the mishkan, so we do have enough information to make our calculations.

Length- 2.5 amot
Width- 1.5 amot
Height- 1.5 amot

We also know that the mishkan had a lid made for it as well, with a length of 2.5 amot and a width of 1.5 amot, which matches perfectly to fit right on top of the mishkan and close it to create a closed rectangular prism.

Volume- how much space was inside the mishkan?
2.5 x 1.5 x 1.5 = 5.625 cubic amot

Surface area- the parsha states that the mishkan was to be covered in gold, both the box and the lid. So, how much gold was needed to cover the mishkan on all sides?
The area of the top and bottom of the mishkan was 2.5 x 1.5 = 3.75 square amot for each (or 3.75 x 2 = 7.5 square amot total for those two sides).
The area of the front and back of the mishkan were 2.5 x 1.5 = 3.75 square amot for each (or 3.75 x 2 = 7.5 square amot total for those two sides).
The area of the left and right sides of the mishkan were 1.5 x 1.5 = 2.25 square amot for each (or 2.25 x 2 = 4.5 square amot total for those two sides).

When you add the square area of all six sides together, you get 7.5 + 7.5 + 4.5 = 19.5 square amot of gold to cover the entire mishkan.

Lateral surface area would be the same calculation as above, but we would leave out the area of the top and bottom, which would leave us with 7.5 + 4.5 = 12 square amot. (Note that another way to calculate lateral surface area is to calculate the perimeter, or length around the outside of the base shape, and multiply that length times the height of the prism.)

An extra question- The parsha states that the bottom of the mishkan was made out of shittim wood before being covered in gold, but the top is just made of gold. How much wood was necessary to build the wooden framework of the mishkan?

Answer- This is a combination of the calculation for surface area and lateral surface area. Really, the calculation is for lateral surface area with the addition of just the bottom area (but not the top area). From our lateral surface area calculations, we know that the area of the four sides is 12 square amot. We also know that the area of just the bottom part was 3.75 square amot. So, we add 12 + 3.75 = 15.75 square amot of shittim wood for the base framework of the mishkan.

Vayeishev- Volume

"Joseph, at the age of seventeen years, was a shepherd with his brothers by the flock, and he was a youth with the sons of Bilhah and the sons of Zilpah, his father's wives; and Joseph would bring evil reports of them to their father." ~Bereishit 37;3

"Then [Joseph's brothers] took [Joseph], and cast him into the pit; and the pit was empty, no water was in it" ~Bereishit 37;24


This week we learn that Joseph was 17 when his brothers had enough of his reporting on their behavior and they decided to throw him into a pit before, eventually, pulling him out and selling him off to a passing caravan. So our question is, what would be a reasonable estimated minimum size for a pit from which a 17 year old man would be unable to get out?

Volume: the amount of space, measured in cubic units, that an object or substance occupies (ref dictionary.com)

For our purposes, when we think about volume, it's going to be the amount of empty space (technically, air filled space; this happened in Canaan, not in the vacuum of Space) inside the dug out pit. Volume, being a measurement of amount of space that is occupied or contained in an area, is calculated by multiplying the length x width x height (Let's keep it to simple shapes, for right now). This calculation works nicely with a space that is a box or a cube, but not, say, a cylinder. A more general calculation rule, which will also work for cylinders and some other shapes, is to calculate the area of the bottom shape and then multiply the bottom area times the height of the object. For a cylinder, this bottom shape would be the circle. Area of a circle = (radius)² x π 
**radius is the distance from the center of the circle to any point along the edge or circumference of the circle. From the exact center, this distance will be the same for any point along the circumference.
**π- read pi- is a constant number that, for our purposes, can be rounded to 3.14)

Since we don't have records of Joseph's size and no Biblical growth charts remain intact for us to be able to determine the size of the average Biblical 17 year old male, we will make our calculations using the assumption that they were the same size as we are today. Any variations from this assumption will cause the answer to be scaled proportionally.

Height- If we use a growth chart, we can see that, nowadays, an average 17 year old man would be approximately 70 inches tall (about 5 ft 10 in).

Width- Shoulder to shoulder would be the widest part of the body. I'm having trouble finding a shoulder to shoulder measurement from a reliable source, but an average men's suit is approximately 18.25 inches across at the shoulders. Let's assume, given that Joseph was out working as a shepherd, that he was broader than that- let's say 20 inches.

Length- Just to establish an easier set of numbers, let's just say that, whether circular or square, he would have fit into the pit in any direction in which he was put, so the length would also need to be more than 20 inches for him to fit inside.

Okay, so if we assume that this pit was a square at the bottom, and he had to fit deep inside the pit in order to be stuck and not pull himself out, and it had to be wide enough for him to actually fit in when they threw him down, we could estimate the dimensions to be:

Height- 120 inches. Human arm span is typically equivalent to a person's height, which is 70 for Joseph. We already assumed his shoulder span to be 20 inches. If we subtract his shoulder span from his arm span we will get the distance from his shoulders to his fingertips (70 - 20 = 50). Divide that by 2 and we get the distance from each shoulder to fingertips. (50/2 = 25). The average man's fingers are approximately 4 inches, and if we subtract that from his shoulder to fingertip length, we get the distance from his shoulders to knuckles - just enough for him to grab on the the edge of the pit and pull himself up, assuming sufficient upper body strength. That number is 25 - 4 = 21 inches. Most online sources of head size only give dimensions for head circumference, which is relevant for hat sizes but not the distance from shoulder to head. Based upon observation we can assume that when a person's arms are outstretched above their head, the head reaches halfway up the arms. This would reach 10.5 inches on Josephs arms, let's make it 11 to simplify the numbers. 21 inches of shoulder to knuckle length - 11 inches of shoulder to top of head height = 10 inches that his arms reach above his head. Add that to his height: 70 + 10 = 80 inches is the height of his knuckles with his arms outstretched. The average vertical leap for an NBA player is 28 inches. Adding this to his outstretched knuckle height and we get 80 + 28 = 108 inches, or 9 feet. That means that if Joseph had explosive jumping power and a cliff hanger's upper body strength, he could hoist himself out of a pit 9 feet deep. There are records of NBA players with vertical jumps ranging from 40 inches to 60 inches, but we're going to stick with the average for this and say that a 10 foot pit (120 inches) is sufficiently deep enough for Joseph to be unable to rescue himself.

Width and length- 40 inches in each direction (that would give him a little less than an extra foot on either side of him to move around inside the pit)

Now for the calculation:
in inches: 40 x 40 x 120 = 192,000 cubic inches 
in feet: 3 1/3 x 3 1/3 x 10= approximately 111 cubic feet

If we wanted to make it more challenging, we could try the calculation assuming that the pit was a cylinder. In that case, we could still use the 120 in or 10 ft for the height, and we would use 20 in or 1 2/3 ft for the radius (if we assume that it's 40 inches all the way across, then 20 inches would be from the center to the side of the pit).
The calculation:
in inches: (20)² x 3.14 x 120 = 150,720 cubic inches
in feet: (1 2/3)² x 3.14 x 10 = approximately 87 cubic feet

If you compare the volume of the square pit to the volume of the cylindrical pit, you can see how much extra space is cut out when you have a comparable sized circle cut into a square. That extra corner space really adds up!

So, how big would your pit be?