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.
"Hashem spoke to Moshe, saying, "Say to Elazar son of Aaron the pans from amid the blaze- and he should move away the fire- for they have become holy. As for the fire-pans of these sinners against their souls- they shall make them thinned-out sheets as a covering for the Altar, for they offered them before Hashem, so they became holy; they shall be for a sign to the Children of Israel." ~Bamidbar 17;1-3
In this week's parsha, we learn that Elazar is told to collect the fire-pans belonging to Korach and his followers, thin out the pans, and make a covering for the Copper Alter out of the thinned fire-pans. On reading this directive, I wondered how much area each pan actually covered once it was thinned out.
If we look back in Shemot 27;1-2, we can remember the description of the Copper Altar:
"You shall make the Mizbe'ach of shittim wood, five amot long and five amot wide- the Mizbe'ach shall be square- and three amot its height. You shall make its horns on its four corners, from it shall its horns be; and you shall cover it with copper."
Back in Parshat Terumah we looked at how to calculate surface area. Multiplying the length x width gives us the measurement of square area, telling us how much flat space is covered. Using this calculation, and knowing from Shemot that the Copper Altar is 5 amot long and 5 amot wide, we can calculate that the area of the altar was 25 square amot.
The next step is to figure out how big a section from the 25 square amot came from each fire-pan. To find this out, we need to divide the area by the number of fire-pans. There were 250 men with Korach, which means that Elazar had to thin out 250 fire-pans in order to cover 25 square amot. So, we divide 25 square amot ÷ 250 pans = 1/10 square amah per fire-pan. In other words, each thinned out fire-pan covered an area of .1 square amot.
Everyday Connections:
For our reference, how big is that? An Amah is estimated between 18-24 inches. We can repeat our calculations with these two estimated lengths to have an idea of a range of the size of the fire-pans.
Assuming 1 Amah = 18 in
The length and width of the altar were each 90 in
90 x 90 = 8,100 square inches for the area of the top of the altar
8100 ÷ 250 = 32.4 square inches per fire-pan
Assuming 1 Amah = 24 in
The length and width of the altar were each 120 in
120 x 120 = 14,400 square inches for the area of the top of the altar
14,400 ÷ 250 = 57.6 square inches per fire-pan
So, from these calculations, each fire-pan was able to be thinned out to between 32.4-57.6 square inches. This means that each fire pan was only able to be thinned out to between 5.5-7.5 inches along each side if they were flattened into perfect squares. So, the final thinned out pieces were smaller than the bottom of an 8 x 8 baking pan. Imagine how small they must have been before they were thinned out!
An extra thought- What if the covering that Hashem required was supposed to cover the entire altar rather than just the top? Using surface area calculations from Parshat Terumah, how large would the surface area have been? How much bigger would each fire-pan have been with these new calculations?
"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.
