Pikudei- Mixed Measures

"All the gold that was used for the work- for all the labor of the Sanctuary- the offered-up gold was twenty-nine kikar and seven hundred thirty shekel, in the shekel of the Sanctuary. The silver of the accountings of the assembly, a hundred kikar, one thousand seven hundred seventy-five shekel, in the shekel of the Sanctuary." ~Shemot 38;24-25

"The offered-up copper was seventy kikar and two thousand four hundred shekel." ~Shemot 38;29

Rashi on 38;24 explains that a kikar of the Sanctuary is three thousand shekalim, and, therefore, the individual shekalim that are listed are mentioned because together they don't add up to one kikar.

What is a "mixed measure"?:
Understanding how to measure with standard measurements is a basic skill that students begin practicing with from an early age. How many feet long is the room? How many cups of water are in the pitcher? But, what happens when you don't have enough to complete an exact measurement with the measure that you're using? What if the room is between 10 and 11 feet long? What if the pitcher has between 5 and 6 cups of water? Sometimes we use fractions to indicate these middle measures. However, we also have other, smaller measurements that can be used to measure these partial portions. We have a standard equivalency that 12 inches are equivalent to 1 foot, for example. So, if my room is between 10 and 11 feet long, I can measure the "extra" length using inches. Maybe my room is 10 feet and 3 inches. This type of measurement is known as a mixed measure. To simplify a measurement out of a mixed measure, we could convert the mixed measure into the larger measurement using a mixed number (a whole number with a fraction) or decimal, or we could convert the mixed measure into the smaller measurement. With our room example, 10 ft 3 in would be 10 1/4 ft or 10.25 ft when measured only in feet, or 123 inches when measured only in inches. For younger students, the calculation for converting the mixed measure into the smaller unit (from feet and inches to just inches) is an easier one and also helps to give them a better sense of the full meaning of the number, where mixed measures and mixed numbers are often harder for them to visualize and fully understand.

Let's calculate:
So how do we convert to the smaller measure unit? Let's use our 10 ft 3 in room example. We know that 1 foot is made up of 12 inches. So, we multiply 10 x 12 (10 feet, with 12 inches in each) = 120 inches. Then, we add the extra 3 inches that were left over, so 120 + 3 = 123 inches.

What does this have to do with the parsha?:
In this week's parsha, Rashi helps us understand that the weight measurements that we're given for the gold, silver, and copper that were donated to the mishkan are mixed measures using kikar and shekalim. Rashi explains that 1 kikar = 3,000 shekalim.

The amounts that we are given in the parsha are:
Gold: 29 kikar and 730 shekel
Silver: 100 kikar and 1,775 shekel
Copper: 70 kikar and 2,400 shekel

How can we convert these to all shekalim measures?
We know that 1 kikar has 3,000 shekalim, so:
Gold: 29 x 3,000 (29 kikar, with 3,000 shekalim in each) = 87,000 + 730 shekel = 87,730 shekel
Silver: 100 x 3,000 (100 kikar, with 3,000 shekalim in each) = 300,000 + 1,775 shekel = 301,775 shekel
Copper: 70 x 3,000 (70 kikar, with 3,000 shekalim in each) = 210,000 + 2,400 shekel = 212,400 shekel

So, our final calculations after converting from mixed measures to just shekalim are:
Gold: 87,730 shekel
Silver: 301,775 shekel
Copper: 212,400 shekel

A further thought:
The silver calculation is particularly meaningful, since we are told "a beka for every head, a half-shekel in the shekel of the Sanctuary for everyone who passed the counters, from twenty years of age and up, for the six hundred three thousand, five hundred fifty." (28;26) So, our silver calculation should be accurate for 603,550 men 20 years of age and older having each given a 1/2 shekel. Does this work? 
301,775 shekel of silver x 2 (two 1/2-shekel pieces for each shekel) = 603,550 1/2-shekel pieces.
So, it seems that there isn't even estimation here, but rather the calculations work exactly!

Vayakhel- A multi-level hands-on activity

"'Every wise-hearted person among you shall come and make everything that Hashem has commanded: the Tabernacle-spread, its Tent-spread, and its Cover, its hooks, its beams, its bars, its pillars, and its sockets; the Aron and its staves, the Lid, the Parochet screen; the Shulchan, and its staves, and all its implements, and the Bread of Surfaces; and the Menorah of illumination, and its implements, and its lamps, and the oil of illumination; and the Mizbe'ach for the Incense and its staves; and the anointing oil and the incense spices; and the entrance screen for the entrance of the Mishkan; the Mizbe'ach of the olah-offering and its copper lattice, its staves, and all its implements; the Kiyyor and its pedestal; the curtains of the Courtyard, its pillars, and its sockets, and the screen of the gate of the Courtyard; the pegs of the Mishkan, and the pegs of the Courtyard, and their cords; the mesh garments of the holy for Aaron the Kohen and the garments of his sons, to be Kohanim.'" ~Shemot 35;10-19

The above passage gives the list of items which the Jewish people were instructed to build as part of the Mishkan for worship in the desert. What follows afterwards in Shemot 36;8-38;20 are detailed descriptions for the building of the Mishkan and all its contents.

Note: Whereas most weeks I try to provide some added insight or understanding into the parsha through my mathematical investigation, this week's topic is an overview for an academic unit based on the information provided in the parsha. The students completing such a unit would certainly gain a deeper insight and understanding of the mishkan and it's contents. 

The major substantive part of this week's parsha is very specific blueprints for building the mishkan and it's contents- the aron, mizbeach, kiyyor, menorah, etc. 

An outline for a hands-on learning opportunity for multiple levels:
Older students can use the details to draw a scale blueprint. This would include:
--conversion from Torah measurements to modern day measurements
--adjusting measurements to a scale proportion
--accuracy of measurement and drawing in their blueprint
--attention to detailed descriptions provided in the text
Older students can create a list of required supplies and materials to build a scale model. This would include:
--accounting for amounts of each material needed based on length, width, height, area, etc. of where and how each material is used
--computation of conversions between the blueprint and model sizes
--being mindful of economic use of materials and trying to calculate to take only what's needed and use materials in such a way that minimal scrap is left over
Younger students can use the blueprint and/or text to help create materials lists. This would include:
--listing the different parts that are to be created
--listing the different materials that are used/needed
--counting out items, such as rings for attaching panels, to determine how many are needed all together throughout the project
Students of all ages can work together on building the model. This would include:
--measuring out and cutting materials to size
--following the blueprints of the text and that created by the student to accurately build the mishkan and it's contents.

Ki Tisa- Proportions

"Hashem spoke to Moshe, saying: 'Now you, take for yourself prime spices: five hundred shekel-weights of pure myrrh; and aromatic cinnamon, half of it, two hundred and fifty; and cane of aromatic spice, two hundred and fifty; and kiddah, five hundred- in the sacred shekel-weight, and a hin of olive oil. Of it you shall make oil of sacred anointment, a blend of mixture, the work of a spice-blender; it shall be oil of sacred anointment...'" ~Shemot 30;22-25

"To the Children of Israel You shall speak, saying: 'This shall be for Me oil of sacred anointment for your generations. He shall not smear on the flesh of man and in its measure you shall not make anything like it; it is holy, it shall be holy for you...'" ~Shemot 30;31-32

"Hashem said to Moshe: 'Take yourself spices- stacte, onycha and galbanum- spices and pure frankincense: they shall be equal one to another. You shall make it into incense, the work of a spice-blender, thoroughly mixed, pure and holy...'" ~Shemot 30;34-35

"'...The incense that you shall make- in its measure you shall not make for yourselves; it shall be holy to you for Hashem...'" ~Shemot 30;37

In this week's parsha, we are given 2 recipes, one for anointing oil and one for incense - both for use in the Mishkan, and given explicit instruction about whether or not they can be replicated in any form. There is an interesting commentary by the Nachalat Yaakov explaining Shemot 30;32 and 30;37. The commentary explains that while the recipe for the anointing oil can be recreated in proportion to the recipe in the Torah (but not in the exact recipe itself), the recipe for incense may not be recreated itself or in any proportional recipe, greater or lesser.

Definitions:
Proportion- comparative relation between things or magnitudes as to size, quantity, number, etc. (dictionary.com)
What does this mean? Back in Parshat Vayishlach we took a look at ratios and compared how quantities of items in a group compare to each other. The idea of proportion is maintaining these existing ratios within the whole group. Here we'll look at what that means.

Baking is always a fun, hands-on way of incorporating math skills into an activity. In lower grades, just measuring out ingredients and following the directions is an important skill. As students get older, calculating proportionally larger or smaller recipes gets them working with multiplication, division, ratios, and fractions. When working with recipes, the traditional form of a recipe (3 eggs, 1 C flour,...) is good for multiplying and dividing to make it larger or smaller, but a more basic, generic recipe that is solely based on proportions (1 part water, 2 parts flour,...) actually fits much more easily into lessons on proportionality, since the nature of the recipe is based on comparison from the start.

Let's take a look, now, at the recipes in this week's parsha.
Anointing Oil:
*500 shekel-weights pure myrrh
*half of it, 250 shekel-weights aromatic cinnamon (commentary explains that it's actually 500 total, but weighed out in half-size portions)
*250 shekel-weights cane of aromatic spice
*500 shekel-weights kiddah
*1 hin of olive oil

Incense:
*stacte
*onycha
*galbanum
*additional spices (7 other spices, according to Rashi citing the Mishna)
*pure frankincense
they shall be equal one to another.
Rashi's interpretation here is that the spices specifically mentioned (stacte, onycha, galbanum, and pure frankincense) were all equal amounts (1 to 1 ratio for all 4). Rashi explains that the amount for each was 70 maneh in weight. Rashi does not explain the possible ratio for the other additional spices.

It's interesting to note, that the recipe for the anointing oil, which can be replicated in larger or smaller quantities, is given in specific weight amounts, and the incense, which may not be replicated in any proportion, is given in a proportional recipe, without any measurements. Perhaps the missing proportions are missing to keep us from actually attempting to replicate the recipe?

So what if we wanted to make some nice tiny vials of anointing oil to use as a perfume? How could we make a smaller recipe? Let's divide the whole recipe by 5 to get our smaller proportionally similar recipe:

*500 ÷ 5 = 100 shekel-weights pure myrrh
*half of it, 250 ÷ 5 = 50 shekel-weights aromatic cinnamon (commentary explains that it's actually 500 ÷ 5 = 100 total, but weighed out in half-size portions)
*250 ÷ 5 = 50 shekel-weights cane of aromatic spice
*500 ÷ 5 = 100 shekel-weights kiddah
*1 ÷ 5 = 1/5 hin of olive oil

Therefore, our new, smaller recipe would be:
*100 shekel-weights pure myrrh
*two batches of 50 shekel-weights aromatic cinnamon (100 total weight)
*50 shekel-weights cane of aromatic spice
*100 shekel-weights kiddah
*1/5 hin of olive oil

Want to make it larger, rather than smaller? Just multiply each ingredient by the same number instead of dividing.

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.