Shoftim- Division and Fractional Segments

"When Hashem, your G-d, will cut down the nations whose land Hashem, your G-d, gives you, and you will possess them, and you will settle in their cities and in their houses, you shall separate three cities for yourselves in the midst of your land, which Hashem, your G-d, gives you to take possession of it. Prepare the way for yourself, and divide three times the boundary of your land that Hashem, your G-d, causes you to inherit; and it shall be for any killer to flee there." ~Devarim 19;1-3

Rashi on 19;3:
"And divide three times the boundary of your land"- That there should be from the beginning of the boundary until the first city of refuge like the amount of travel, i.e., like the distance, that there is from it until the second one. And so, too, from the second one to the third one. And, so too, from the third one until the second, i.e., the opposite, the border of the Land of Israel.

The Gemara in Makkot 9b talks about the word ושלשת, which was translated above as "divide three times". The Gemara brings up the idea that this is unclear because it could either be understood as dividing by three or multiplying by three. Rashi explains that it means division, but not just division into three parts. Rather, it means creating three lines of division, which actually creates 4 sections.

Division and the Concept of Physically Dividing:
When we talk about division, we are dealing with separating a unit with a certain measurement or a group with a number of contents into a given number of smaller equal pieces or groups. Some examples:
*A 12 ft. piece of rope divided amongst 3 people. We would calculate 12 ft. + 3 people = 4 ft. per person
*A bag with 30 marbles divided amongst 5 students. We would calculate 30 marbles + 5 students = 6 marbles per student

Now, let's think about if we are asked to cut a loaf of bread into 8 slices. How many cuts would we need to make in order to get 8 slices? The answer is that we would need to make 7 slices. So, when we are asked to cut a certain number of pieces, how can we know how many cuts we will need? 
*If your cuts are not intersecting with each other at all (think of cutting a loaf), then you will always need one cut less than the number of slices that you want:
1 cut = 2 pieces

2 cuts = 3 pieces

3 cuts = 4 pieces

How does this work? When you make your first cut, you are creating 2 pieces out of the 1 original. Once you continue cutting, however, you are only cutting off 1 more smaller section with each cut.

* If your cuts are going to intersect with each other (think of cutting a rectangular cake), then it helps to think of the area as a multiplication grid. If you need 30 pieces, you could make it 6 pieces by 5 pieces- then you would need 5 cuts in one direction and 4 cuts in the other direction. 
To think about grid cuts:

2 cuts = 4 pieces

3 cuts = 6 pieces

4 cuts = 8 pieces or 9 pieces, depending on where you make the cuts

With cutting on a grid, you can adjust your number of pieces depending on how you choose to place your cuts lengthwise and widthwise, as you can see with our example of 4 cuts above. Trying to keep the lengthwise cuts and widthwise cuts as close to the same number as possible will always result in the most number of pieces. Consider that for the loaf cuts, we needed 3 cuts to get 4 pieces, but for the "cake" cut we only needed 2 cuts to get the same 4 pieces.

Investigating with how many pieces you can get from the fewest cuts is an interesting investigation for students to play with. The more intersections of cuts you make, the more pieces you can get from each individual cut. 

Parsha Connection:
In this week's parsha, it seems that we are told to divide the land in three sections to create cities of refuge. However, Rashi and the Gemara explain that it's actually telling us to make 3 divisions (or 3 cuts) in the land, which will actually separate the land into 4 pieces. With the 3 cities of refuge built on each of the 3 division lines, anyone in each of the 4 settled sections of land will have approximately the same distance to travel to get to the nearest city of refuge. In this way, no one living in any of the settled areas will be closer or farther than anyone else from refuge.

The subtlety in the language is so slight- the difference between dividing by 3 (making 3 cities) and making 3 divisions (making 4 settlements, equidistant to the 3 cities).

Everyday Connection:
How often are you cutting something to share with others? Can you maximize your cuts so that you use the fewest number of cuts possible to get the number of pieces that you want?

Further thoughts: How do intersecting cuts differ when you're dealing with cutting circles into wedges? Is it like a loaf? Is it like a cake? Is it completely different?

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.