Korach- Equally Shared Surface Area

"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?

Tetzaveh- Mixed Bag of Elementary Geometric Problem Solving

"You shall make the Choshen of Judgement the work of an artist, like the work of the Eiphod shall you make it, of gold, turquoise wool, and purple wool, and scarlet wool, and twisted linen shall you make it. Square shall it be, folded, a zeret its length and a zeret its width. You shall fill it with stone filling, four rows of stone: a row of odem, pit'dah, and barekes- the one row; the second row: nofech, sapir, and yahalom; the third row: leshem, shevo, and achlamah; and the fourth row: tarshish, shoham, and yashfeh; they shall be of golden settings with their fillings. The stones shall be according to the names of the sons of Israel, twelve according to their names, like the engraving of a signet ring, each man by his name shall they be, for the twelve tribes." ~Shemot 28; 15-21

This week's parsha gives us a description of the structure of the Choshen, or the breastplate, that the Cohen Gadol (High Priest) wore. The mathematical parameters that we are given are:

--1 zeret x 1 zeret square, when folded (this is following Rashi's interpretation of the text; also according to Rashi, 1 zeret = 1/2 amah or approx 9-12 inches)

--12 stones, representative of the 12 tribes

--the 12 stones should be organized as 4 rows of 3 stones each

I'll provide here some problem solving questions and answers, but the true way to learn through these questions is by making models or diagraming the problems to sketch out a way of calculating the answers.

Geometric problem solving:

If we know that it needs to make a 1 zeret x 1 zeret square when it's folded once, what are the dimensions of the original shape before it is folded? What is the name of the original shape?

*Answer- In order to be folded once and become a square, we know that, on the original shape, one side is 1 zeret long. The second side, in order to be 1 zeret when folded, would have to be twice as long originally, or 2 zerets (zratim? zratot?) long. Thus, the original shape, before folding, was 1 zeret x 2 zrat__, which would give us a rectangle.

Follow-up for higher levels could be to calculate the area of the Choshen (the original rectangle, and the folded square)- What's the area measured in zrat__, amot, and modern day measures? Do you see a pattern in the rectangular areas compared to the square areas? (hint: you should!)

Representing Multiplication Arrays as Rectangles:

We are told that the 12 tribes were represented in a 4 x 3 arrangement on the Choshen. What other possible organizations could have been used to lay out the 12 stones (assuming complete rows and columns)?

*Answer- When organizing a given number of items into evenly divided rows and columns, we are essentially looking for the possible combinations of factor pairs for the given number. Laying items out in rows and columns (or drawing them out) is a common technique for diagraming factor pairs when students are first processing multiplication facts. This both helps them come up with factor pairs for a given number and also helps set a basis for areas of rectangular shapes when they move into geometry. These rectangular diagrams of factor pairs are called arrays. 

Here we are looking for all possible factor pairs that will make 12. The Torah gives us 4 x 3 (4 rows, with 3 columns in each). What else can we find?

1 x 12

2 x 6

3 x 4

4 x 3 (Torah's description)

6 x 2

12 x 1

In total, there are 6 possible arrays for 12 items. Half of them are repetitions (1 x 12 can be the same as 12 x 1), but it's important for students to understand that swapping the length and width results in the same total number of items (or area)- an internalization of the Commutative Property: changing the order in multiplication or addition facts does not change the final product or sum.

What shapes could the stones be in order to fit into the Choshen as it is described?

*Answer- They could have had round or square stones, which would have left uneven spaces between rows and columns. Alternatively, if they were oval or rectangular, it would have made it possible to allow for equal spacing between rows and columns on the square plate. Try it and see.

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?