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.