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.

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.

Mishpatim- Calculating Interest

"When you will lend money to My people, to the poor person who is with you, do not act toward him as a creditor; do not place interest upon him." ~Shemot 22;24

Rashi on 22;24--
Interest (root word is the same as "snake")- This means increase, for it is like the bite of a snake, for a snake bites a small wound in one's foot and [the victim] does not feel it, but suddenly it causes puffiness and swelling up to the crown of his head. So it is with interest. He does not feel it and is not aware of it, until the interest accumulates and causes him a loss of much money.

Definitions:
Principal: An original sum of money
Interest (or Simple Interest): A calculated percentage value of the principal amount
Compound Interest: A method of calculating interest, where the previous interest is added to the principal amount for future interest calculations

In my experience, interest is a particularly confusing concept for students. I believe that this stems from the somewhat complex aspects of the calculation. In order to calculate interest, you need to first convert the percentage (interest rate) into a decimal or fraction in order to make the calculation. Once you've made the calculation, you have only actually calculated the interest amount. When using interest, the interest is paid on top of the original principal amount. So, in order to calculate the full amount, you need to then add the interest amount together with the original principal amount. The combination of a calculation requiring conversion within a two-step calculation problem, tends to lead to a lot of confusion for many students. Breaking down the steps into a logical progression and making each step meaningful to students can help them make sense of the process, rather than forcing them to memorize a seemingly complex rote process.

Connection to Parsha:
Rashi explains that we are commanded not to charge interest on a loan because the interest builds up without the borrower realizing and suddenly the borrower owes a great amount of money to the lender. Let's use an example to understand Rashi's explanation of how interest accrues and how it can surreptitiously overtake a borrower.

Let's assume that I borrow $1,000 from a friend of mine, and he charges me just 1% interest per week on my loan. Since the parsha doesn't specify simple or compound interest, let's see how they compare side-by-side over the course of 3 months or a 13 week period.

Since we are calculating 1%, we change it into a fraction or decimal (1/100 or .01) in order to calculate with the number (See if you can use the information from Parshat Vayigash to work backwards to create the fractions/decimals for calculation with a percent). Then, we multiply the fraction or decimal times the principal amount. Once we have the weekly interest, we add the interest to the principal to see what is owed each week that the loan goes unpaid. For Simple Interest, we just have $10 interest that adds to the total each week; for Compound Interest, we need to recalculate 1% of each week's new total to calculate the new interest for the following week, and the new interest is added to the previous sum that was owed.

	Simple Interest	Compound Interest
Initial Loan	$1,000	$1,000
Week 1- interest	$10	$10
Week 1- total due	$1,010	$1,010
Week 2- interest	$10	$10
Week 2- total due	$1,020	$1,020
Week 3- interest	$10	$10
Week 3- total due	$1,030	$1,030
Week 4- interest	$10	$10
Week 4- total due	$1,040	$1,041
Week 5- interest	$10	$10
Week 5- total due	$1,050	$1,051
Week 6- interest	$10	$11
Week 6- total due	$1,060	$1,062
Week 7- interest	$10	$11
Week 7- total due	$1,070	$1,072
Week 8- interest	$10	$11
Week 8- total due	$1,080	$1,083
Week 9- interest	$10	$11
Week 9- total due	$1,090	$1,094
Week 10- interest	$10	$11
Week 10- total due	$1,100	$1,105
Week 11- interest	$10	$11
Week 11- total due	$1,110	$1,116
Week 12- interest	$10	$11
Week 12- total due	$1,120	$1,127
Week 13- interest	$10	$11
Week 13- total due	$1,130	$1,138

You'll notice in the calculations above that the compound interest doesn't seem much different from the simple interest for almost the full first half of the loan period. This is because we used such a small interest rate.

Let's say that my friend is not as nice to me, and charges me just a little bit more, say 5% interest. Below we'll see a much bigger difference.

	Simple Interest	Compound Interest
Initial Loan	$1,000	$1,000
Week 1- interest	$50	$50
Week 1- total due	$1,050	$1,050
Week 2- interest	$50	$53
Week 2- total due	$1,100	$1,103
Week 3- interest	$50	$55
Week 3- total due	$1,150	$1,158
Week 4- interest	$50	$58
Week 4- total due	$1,200	$1,216
Week 5- interest	$50	$61
Week 5- total due	$1,250	$1,276
Week 6- interest	$50	$64
Week 6- total due	$1,300	$1,340
Week 7- interest	$50	$67
Week 7- total due	$1,350	$1,407
Week 8- interest	$50	$70
Week 8- total due	$1,400	$1,477
Week 9- interest	$50	$74
Week 9- total due	$1,450	$1,551
Week 10- interest	$50	$78
Week 10- total due	$1,500	$1,629
Week 11- interest	$50	$81
Week 11- total due	$1,550	$1,710
Week 12- interest	$50	$86
Week 12- total due	$1,600	$1,796
Week 13- interest	$50	$90
Week 13- total due	$1,650	$1,886

From these two examples, we can really see what Rashi is referring to when he talks about interest accumulating quickly without the borrower realizing what he has entered into.