Korach- Currency Equivalency Activities

"Every first issue of the womb of any flesh that they offer to Hashem, whether of man or of animal, shall be yours; but you shall surely redeem the firstborn of man, and the firstborn of an impure animal shall you redeem. And those that are to be redeemed- from one month shall you redeem according to the valuation, five silver shekalim by the sacred shekel; it is twenty geirah." ~Bamidbar 18:15-16

Towards the end of this week's parsha,  we read of what items are set aside to be given to Aaron and the Kohanim. Within this section, we learn that firstborn male infants and firstborn animals are part of what is to be given to the Kohanim. However, we also learn that firstborn children and impure firstborn animals are to be redeemed at the age of one month. The value for redeeming them is 5 silver shekalim, and we are told that one silver shekel is equivalent to 20 geirah.

Activity Suggestions related to Monetary Equivalence:
Learning equivalencies of monetary values can be confusing for young children. Opportunities for hands-on acting out with manipulatives, repetition, and practice help them become proficient in working with monetary conversions.

• Students can convert between geirahs and shekalim.                        
  Given that 1 shekel = 20 geirah:
• Representing the conversion-
• Younger students could calculate how many geirahs would be in 5 shekalim, by using manipulatives for representation. They could have cups to represent 1 shekel, and small chips or counters representing a geirah. Students could work together to create shekel value cups, each filled with 20 geirah. Once they have the cups set up, they can then go back and count out 5 shekalim. 5 cups would represent 5 shekalim. 
• Students who are ready could draw representative circles or boxes for shekalim and show 20 tally marks within each one to represent the geirahs in each. In this case, 5 circles or boxes would represent the 5 shekalim. 
• Making the calculations-
• Youngest students can work together (or as a class) to count out how many geirahs there are all together in the 5 shekalim. 
• Slightly older children could use their knowledge of "skip counting" or addition of simple large numbers to add the twenties together and make the calculation.
• Older students can use their knowledge of multiplication to problem solve and calculate how many geirahs would be in 5 shekalim.
• Follow-up calculations:
• Can students (young and old) figure out how many shekalim would be needed to redeem multiple firstborns? If 7 families redeem their firstborns, how many shekalim would be given to the Kohanim all together? How many geirahs would be given to the Kohanim all together?
• If you chart the redemption values in shekalim and geirahs for different numbers of firstborns, can you identify any patterns?
• For upper elementary or middle schoolers- can they convert the patterns that they find into functions? How would these functions look when graphed?
• Older students can also research conversion values to current day currencies. Based on the researched conversion rates, what is the value today of a geirah? of a shekel? How much would families need to pay to redeem firstborns in different countries today? Can they write a function to calculate redemption values for multiple firstborns using current currencies?

Shelach- Activities with One-to-One Correspondence and Measurements

For younger students:

"Hashem spoke to Moshe, saying, 'Send forth for yourself men, and let them spy out the Land of Canaan that I give to the Children of Israel; one man each from his fathers' tribe shall you send, every one a leader among them.'" ~Bamidbar 13:1-2

"...Your children will roam in the wilderness for forty years and bear your guilt, until your carcasses shall cease to be, in the wilderness. Like the number of the days that you spied out the land, forty days, a day for a year, a day for a year, shall you bear your iniquities- forty years- and you shall know what parting from me [is]." ~Bamidbar 14:33-34

 I've addressed the idea of one-to-one correspondence a couple of times in the past (see Parshat Haazinu and Parshat Ki Tisa). In this week's parsha, we have 2 instances that deal directly with this idea. 

At the beginning of the parsha, Hashem instructs Moshe to have one leader from each tribe assigned to go out and visit the land of Canaan to see what the Children of Israel were to expect upon entering the land. 

• If there is one man from each tribe, how many men were sent? Since we know that there are 12 tribes, if each tribe sends one man, there will also be 12 men. 
• Acting it out- Students can be separated into 4 or 5 groups ("tribes"). Within each group, a "leader" can be chosen. Students can count how many groups/tribes there are. Based on this, ask students how many leaders there will be. Have the leaders come to a separate area in the room and have the class count the leaders. Were there predictions correct? Are the students all making the one-to-one connection?

In the middle of the parsha, Hashem becomes upset with the spies' report of Canaan, which led to fear-mongering amongst the Children of Israel. As punishment, Hashem decides that the Children of Israel will wander in the desert one year for each day that the spies were out in Canaan.

• If we know that the spies were in Canaan for 40 days, how many years would the Children of Israel wander the desert? If the spies were there for 40 days, then 1 year for each day would mean that they would wander the desert for 40 years.
• Extrapolation discussion- [Sample questions] If you received one present for every birthday that you've had, how many presents would you have now? (5yo would have 5, 6yo would have 6, etc.); If there are 10 classes in school, and each class makes a poster to put in the hallway, how many posters will be in the hallway? (10 classes = 10 posters)

*****

Multi-level: 

"Hashem spoke to Moshe, saying, 'Speak to the Children of Israel and say to them: When you will come to the land of your dwelling places that I give you, and you perform a fire-offering service to Hashem- an olah-offering or a [peace-offering] sacrifice by articulating a vow or a free-will offering, or on your festivals, to produce a pleasing fragrance to Hashem, from the cattle or from the flock- the one who brings his offering to Hashem shall bring a meal-offering of a tenth [of an ephah] fine flour, mixed with a quarter-hin of oil; and a quarter-hin of wine for a libation shall you prepare for the olah-offering or the [peace-offering] sacrifice for each sheep. If it is a ram- you shall prepare a meal-offering, two-tenths [of an ephah] fine flour mixed with a third-hin of oil; and a third-hin of wine for a libation shall you bring as a satisfying aroma to Hashem. When you prepare a young bull as an olah-offering or a [peace-offering] sacrifice, because of an articulated vow, or a peace-offering to Hashem, one shall bring with the young bull a meal-offering: three tenths [of an ephah] fine flour mixed with a half-hin of oil. You shall bring a half-hin of wine for a libation, a fire-offering, a pleasing fragrance to Hashem...'" ~Bamidbar 15:1-10

At the end of this week's parsha there is a list of information regarding different sacrifices. In this section, depending on the animal offered as part of the sacrifice, the specifications of the meal-offering ingredients differ. There are a number of activities that can be arranged around this information.

• The first step is to organize the information- I recommend charting the information based on the animal that each different meal-offering is assigned to. For younger students, you can have the information pre-charted for them, including pictures of ingredients for pre-readers.
• Comparing information: 
• Which sacrifice has the most flour? Which has the most oil? Which has the most wine? 
• Order them from least to greatest (or greatest to least) amounts of flour. Order by amounts of oil. Order by amounts of wine. 
• Are there any patterns that you see? Between the different sacrifices? When they are ordered by amounts?
• Conversions- Using current day equivalencies for ephah and hin, how much of each ingredient would be used for each sacrifice? 
• Note: For conversion reference, I posted a chart with conversions including ephah for Parshat Beshalach and Wikipedia has a page with Biblical measurements, including hin under liquid measures; as I wrote in Parshat Shelach last year, a se'ah is usually calculated between 2.25 and 4 gallons in current measurements.
• Interesting conceptual thought:
•  For younger students- When you convert from Biblical units to modern day units, does the amount that you have change? Another way of thinking about it- if you measure a table using inches and then using feet, does the size of the table change, or just the measurement you're using to think about the item?
• For older students- When you convert the measurements, will they still compare in the same ways when you order them? What about patterns? If there is a relationship between the measurements, will the relationships remain the same when you have converted into modern day measurements?

Bamidbar- Reading, Comparing, and Calculating with large numbers

Note that the section of quotes from this week's parsha is quite long in order to show the context for Levite and firstborn census numbers. As always, I organize all of the information for you beginning below the quotes. If your time is short, feel free to scroll past the citations.

"Hashem spoke to Moshe in the Wilderness of Sinai, saying, 'Count the sons of Levi according to their fathers' house, according to their families, every male from one month of age and up shall you count them.' Moshe counted them according to the word of Hashem, as he had been commanded.

These were the sons of Levi, by their names: Gershon, Kohath, and Merari. These were the names of the sons of Gershon according to their families: Livni and Shimei. The sons of Kohath according to their families: Amram and Izhar, Hevron and Uzziel. The sons of Merari according to their families: Mahli and Mushi. These are the families of the Levites, according to their fathers' house.

For Gershon- the Livnite family and the Shimeite family; these are the Gershonite families- their counted ones according to the number of every male, from one month of age and up: their counted ones [were] seven thousand, five hundred." ~Bamidbar 3:14-22

"And for Kohath- the Amramite family, the Izharite family, the Hevronite family, and the Uzzielite family; these were the Kohathite families- the number of every male from one month of age and up [was] eight thousand, six hundred; the guardians of the charge of the sanctity." ~Bamidbar 3:27-28

"For Merari- the Mahlite family and the Mushite family; these were the Merarite families- their counted ones according to the number of every male from one month of age and up [was] six thousand, two hundred." ~Bamidbar 3:33-34

"All the counted ones of the Levites, which Moshe and Aaron counted by the word of Hashem according to their families, every male from one month of age and up, were twenty-two thousand.

Hashem said to Moshe, 'Count every firstborn male of the Children of Israel from one month of age and up, and take a census of their names. You shall take the Levites for Me- I am Hashem- in place of every firstborn of the Children of Israel, and the animals of the Levites in place of every firstborn of the animals of the Children of Israel.' Moshe counted- as Hashem had commanded him- every firstborn of the Children of Israel. Every firstborn male according to the number of their names, from one month of age and up, according to their countings, was twenty-two thousand, two hundred and seventy-three.

Hashem spoke to Moshe, saying, 'Take the Levites in place of every firstborn of the Children of Israel, and the animals of the Levites in place of their animals, and the Levites shall be Mine, I am Hashem. And as for those redeemed of [the firstborn], the two hundred and seventy-three of the firstborn of the Children of Israel who are in excess over the Levites, you shall take five shekels each according to the head count, in the sacred shekel shall you take; the shekel is twenty geirah. You shall give the money to Aaron and his sons, of those redeemed in excess among them.'" ~Bamidbar 3:39-48

In this week's parsha, we learn of multiple censuses that were taken in the desert. Last year, I looked at the census information for the 12 Israelite tribes. This census was taken for men from the age of 20 years old and older. In my post last year, I analyzed that information from a basic statistical averages viewpoint- exploring the mean, median, and mode for the data set of the census. 

This year, I would like to look at the census information for the Levites. Since we're dealing with a lot of information here, let's first focus and pull out the numerical information that we're trying to look at. The census information for the Levites is first broken down into smaller census numbers from the individual Levite families, and then summed up at the end. Then, following that information, there is another census of the firstborn Israelites. We will look at all of these sets of information.

Let's begin by weeding through the information above to organize the pertinent information that we need:

Gershon- 7,500 males from 1 month of age and older

Kohath- 8,600 males from 1 month of age and older

Merari- 6,200 males from 1 month of age and older

Then, the total sum listed for all Levites is 22,000 males from 1 month of age and older

Stop for a minute and test out this calculation. 

• Does 7,500 + 8,600 + 6,200 = 22,000? In fact, it actually adds to 22,300. 

So what's going on here? Why does the Torah omit 300 Levites in it's calculation? Rashi on 3:39 deals with this exact question. He explains that the total number of Levites was needed in order to redeem the firstborn Israelites. For every 1 Levite, 1 firstborn Israelite would be redeemed. Rashi says that the 300 Levites who are not included in the total sum were actually firstborns, themselves, and therefore would not count towards being able to redeem other Israelite firstborns, since they would be redeeming themselves.

Let's look on further- We are told that the firstborn Israelite males from 1 month of age and older totaled 22,273. 

So, if we replace one-for-one each firstborn Israelite with a non-firstborn Levite, we have 22,273 firstborns & 22,000 Levites available for redeeming:

22,273 - 22,000 = 273

Now, it seems that we have 273 firstborns who do not have a Levite to redeem them. If we continue reading, though, we see that Hashem has told Moshe that there are, in fact, 273 firstborns "in excess" of the Levites, and these 273 should be redeemed by collecting 5 shekels per person for each of them and that money will go to the Kohanim. 

Some other thoughts & activities:

• It my be interesting to compare the census for the Levites with the census of the rest of the Israelites that's listed in the first part of the parsha. In making a comparison of the numbers, it would be important to keep in mind that the census for the 12 Israelite tribes was taken from 20 years and older, while the Levite census was taken from 1 month and older. Even more so, what interesting findings do you see when comparing the numbers of the two groups?
• An interesting discussion topic may be to think about whether there is a way to compare the information given for the general Israelite census to the firstborn Israelite census to figure out any further information regarding numbers of Israelites between certain ages? Is there enough information given to create comparisons of smaller age groups- either from each tribe or from the whole group?
• For younger students, it might be interesting to just think about place value and reading large numbers. Students could have a matching activity in which they have to identify the numbers of the censuses and match them to the appropriate tribes. Students could also talk about how to think about comparing which numbers are larger and which numbers are smaller- place values and how you can use place values to compare numbers to each other to figure out which is larger and which is smaller. Students who catch on quickly to this concept could think about ordering the numbers from smallest to largest or largest to smallest.
• Follow-up activity- values are given for how many shekels were collected for the 273 "excess" firstborns who did not have Levite redeemers (Bamidbar 3:50). Older students could check the calculations for the values to see if it matches the amount that they calculate should have been donated. We are also told (Bamidbar 3:47) that a shekel was worth 20 geirah. Students can calculate a conversion of the value to also figure out the value of the donation in geirahs. 

Emor- Calculation Activities with Fractions & Unit Conversions

"From your settled places you shall bring bread of elevation, two loaves made of two tenth-ephahs, they shall be fine flour,  they shall be baked leavened; first fruits to Hashem." ~Vayikra 23:17

"You shall take fine flour and bake it into twelve loaves; each loaf shall be two tenth-ephahs. You shall place them in two stacks, six the stack, upon the pure Table, before Hashem." ~Vayikra 24:5-6 

In this week's parsha, we find two instances of descriptions of loaves of bread. In our first quote (23:17), there are two loaves of bread that were offered for the meal offering. In our second quote (24:5-6), there are twelve loaves of bread that were put in the Mishkan and replaced each week on Shabbat. 

If you read these passages, you'll notice that in both cases, the size of each loaf of bread was standard- each loaf was to be made using two tenth-ephahs of fine flour.

Some activity thoughts based on this information:
*Note that I have posted information related to relevant, applicable calculations in my previous posts on Parshat Beshalach (equivalency table for biblical measurements equivalent to an ephah, including a se'ah) and Parshat Shelach (unit conversion calculations from se'ah to modern measurements).

**For younger students, they could be given representative 1/10 ephah pieces (representative weights) to think about the quantities necessary for each question below. A more developmentally concrete class activity could also be done using a pre-measured modern day equivalency of 1/10 ephah and then having students measure out the amounts necessary for individual loaves and each scenario listed below. While the first activity offers students the opportunity to think about how to count multiple fractions of weighted measures, the second activity offers students the opportunity to visualize and think about how much flour would have actually gone into each recipe.

**For older students, they could work through the paper and pencil calculations. Students could then take their calculations and, using baking flour, compare how their measurements compare to some of their own challah recipes from home or collected as a class from a quick recipe search ahead of time. They can think about how many loaves of challah the parsha recipes would make based on the way their own families make challah. Are their challahs bigger or smaller than the estimated size of the loaves in the parsha?

Questions:
*If you were batch baking the bread for the meal offering, how much flour would be needed to make the two loaves?
*If you were batch baking the bread for the Mishkan, how much flour would be needed to make the two loaves?
*How much flour was needed for the combined loaves of each stack, the way they were divided for storing in the Mishkan each week?
*If you were baking today, how much flour would be needed for each loaf of bread?
*If you were baking today, how much flour would be needed to batch bake the bread for the meal offering?
*If you were baking today, how much flour would be needed to batch bake the bread for the Mishkan?
*If you were baking today, how much flour would be needed for the combined loaves of each stack as they were divided in the Mishkan?

Terumah- Scale Representation & Unit Conversion Acitivities

In Parshat Terumah, we are given blueprints for multiple sections of the Mishkan (the Holy Tabernacle that traveled with the Israelites in the desert). Together with Rashi and other commentaries, this parsha offers a wealth of information for understanding the specifics of how the Mishkan was put together. Below is a breakdown of the basic blueprint sections that are described in the parsha:

• blueprints for the Aron & its lid ~Shemot 25:10-22
• blueprints for the Shulchan (table) & information about the vessels that were stored in the Shulchan ~Shemot 25:23-30
• blueprints for the Menorah ~Shemot 25:31-40
• blueprints for the Tabernacle-spread ~Shemot 26:1-6
• blueprints for the Tent-spread (laid over the Tabernacle-spread) ~Shemot 26:7-13
• blueprints for the Tent-spread Cover (a roof covering) ~Shemot 26:14
• blueprints for the Beams & Bars of the Mishkan ~Shemot 26:15-30
• blueprints for the Parochet (a curtain/screen that enclosed the Aron as a separation) ~Shemot 26:31-33
• instructions regarding locations of each item in the Mishkan ~Shemot 26:34-35
• blueprints for the screen at the entrance of the Mishkan ~Shemot 26:36-37
• blueprints for the Mizbe'ach (altar) ~Shemot 27:1-8
• blueprints for the Courtyard area of the Mishkan ~Shemot 27:9-19

Each individual section listed above offers specific dimensions, information about materials that were used, and instructions about positioning of aspects or details pertaining to each section. With these basic blueprints, there is a plethora of activity options that can be developed. Below are a few suggestions specifically connected to the concepts of scale representation and unit conversion.

Activity Connections:

Activity ideas in increasing conceptual difficulty:

Activity suggestions can be used on a small scale for individual sections of the parsha or on a larger scale as a more extensive project recreating multiple or all sections of the Mishkan.

Younger students rely very heavily on visual learning. Even for older students, when learning about 3-Dimensional figures, it's always clearer for them if they can view or build a model for reference. 

Scale Representation:

• For younger students, providing them with a preset block size to represent the scale conversion will allow them to just focus on the construction aspect of the activity. For example, choosing a unifix cube or snap cube to be equivalent to 1 amah in each direction. Be careful in choosing a representative cube- students will need to measure length, width, and height. Keep in mind that in order to give a true representation the height of the cube will need to be the same measure as both the length and the width (ie, it must be a true cube in order to represent an amah in every direction). With a good number of the chosen block and the directions for the chosen Mishkan item to be built, students can work on building a scale model of the chosen section(s) of the Mishkan.
• Younger students will need a lot of guidance in this project. Some options are either working cooperatively in small groups with teacher guidance, or having a teacher run an activity station where students can each rotate in and work on a portion of the building activity. Students with more skill, can work individually or in groups with less teacher support to follow the blueprints and build the chosen model(s).
• (3-5) Elementary students who have familiarity with measuring using rulers and yardsticks can use other representative materials to build a model by setting a scale of, for example, 1 inch = 1 amah. Using this equivalency, students can measure out pieces of poster, cardboard, etc. to create pieces to fit together to build chosen items from the Mishkan. 
• (6-8+) As students develop their fine motor dexterity, they can draw scale representations of the items in the Mishkan, layout of the Mishkan, etc. 

Unit Conversions:

(See posts on Parshat Chayei Sarah and Parshat Shelach for understanding unit conversions.)

• Understanding different measurement systems within the parameters of familiar measurements gives students a better understanding and frame of reference for size of items. Older students can use the blueprints given and calculate what the measurements for different items would be if measured in US/Imperial or metric units. (for reference, 1 amah = between 18-24 inches)
• With blueprints converted into familiar measurements, students can model aspects of the items (height, length) for visual reference, or even build life-size models of sections of the Mishkan to give a full perspective of, for example, what the Shulchan looked like, or what the Aron looked like if you were standing next to it.

Bo- Using Unit Conversions in Problem Solving

"The Children of Israel journeyed from Rameses to Succoth, about six hundred thousand on foot, the men, aside from the children." ~Shemot 12:37

Rashi here explains that the distance from Rameses to Succoth was 120 mils, and that Hashem made their travel there as if in an instant.

To gain a better understanding of this explanation in today's measurements, 1 mil = 2,000 amot, and 1 amah is estimated between 18-24 inches. Based on this estimate, 1 mil would be anywhere from 3,000-4,000 ft.

Rashi also explains that the estimated number that is given here is men from twenty years and older, possibly draft age, which would be men 20-60 years old.

As I was reading this, I wondered how much distance a group of this size would cover when they're traveling as a whole group (ie, how much land would the whole group cover at once?). If Hashem hadn't brought them immediately through a miracle, would the first person have reached Succoth before the last people left Rameses?

To begin to figure this out, we have 2 calculations to consider:
1) What would be today's equivalent of the distance they needed to travel?
2) What would the full population estimate be for the group?

Distance Traveled:
Based on Rashi's information above, they traveled 120 mils, where each mil is anywhere from 3,000-4,000 ft. For an easier calculation, you can choose either the lower estimate or the upper estimate. I'm going to work through both calculations and offer a range.
To calculate, we will multiply the distance in feet for 1 mil times the number of mils that was their travel distance (120).
For the lower estimate, that means 3,000 (ft) x 120 (mils) = 360,000 ft
For the upper estimate, that means 4,000 (ft) x 120 (mils) = 480,000 ft

So, the distance from Rameses to Succoth was anywhere from 360,000 - 480,000 ft.

Another level of unit conversion to consider:
If we know that 5,280 ft = 1 mile, we can convert this range into miles. Do we multiply or divide here? This question will stump many students. Let's think, we are trying to convert a number of feet into miles. We know that every group of 5,280 ft together make 1 mile. We are taking big chunks of feet and changing them each out for 1 mile. To make this conversion, we are finding out how many chunks of 5,280 ft can be fit into, say, 360,000 ft. This means division.

So, 360,000 ft ÷ 5,280 ft = approximately 68.18 miles 
and 480,000 ft ÷ 5,280 ft = approximately 90.91 miles

So, the distance from Rameses to Succoth was anywhere from 68.18 - 90.91 miles.

Population Estimate:
If we use the estimate that there were approximately 600,000 men between the ages of 20-60, it's probably a fair estimate to say that this is 1/3 of the male population (men younger than 20 and men older than 60 making up the other 2/3 of the male population). We can also say that the male population is only half of the total population, assuming that the population was roughly 1/2 male and 1/2 female. 

To calculate how many men there were, we can multiply 600,000 x 3 = 1,800,000 men.
To calculate the total population, we can double this number: 1,800,000 x 2 = 3,600,000 total estimated population. 

So, we can safely estimate that there were more than 3.5 million people who left Egypt and traveled from Rameses to Succoth.

Final Calculations:
Although the numbers are larger to calculate with, I'd like to use the distance measured in feet to finalize our calculations. One important aspect of understanding the information and numbers that you're working with is recognizing which numbers make the most sense to use for a calculation. If we want to estimate how much space a person needs, it's easier to estimate this number using a measurement of feet rather than miles. 
If we estimate, conservatively, that each person needs 2 feet of space, then with a distance of 360,000 ft, you could have a line of 360,000 ÷ 2 = 180,000 people, at most.

With the same conservative estimate, then with a distance of 480,000 ft, you could have a line of 480,000 ÷ 2 = 240,000 people, at most.

So, with little space in front or back of them, you could have a line of between 180,000 - 240,000 people reaching all the way from Rameses to Succoth. 

But we have an estimate of 3,600,000 people!?
3,600,000 (population) ÷ 180,000 (people in each line) = 20 lines of people 
or
3,600,000 (population) ÷ 240,000 (people in each line) = 15 lines of people 

So, without consideration for all the animals and possessions that the Israelites left with, if they lined up almost front to back with each other, there would have been 15-20 lines of people spanning the entire distance from Rameses to Succoth.

An Extra Look:
In Judaism, we often speak of a person's "daled amot" or 4 amot of personal space. With this measurement in mind, we could use the distance from Rameses to Succoth and calculate how much space each person would have needed if we allowed each person 4 amot.

Let's go back to the first equivalency above: 1 mil = 2,000 amot
We know that there were 120 mil, so that would be 
120 x 2,000 = 240,000 amot as the distance between Rameses and Succoth.

If we want to allow 4 amot each to 3,600,000 people, we need to calculate 
3,600,000 x 4 = 14,400,000 amot needed.

Ok, so now we need to figure out how many times the distance from Rameses to Succoth would be needed in order to accommodate all the space needed by the population.

14,400,000 amot (population) ÷ 240,000 amot (distance) = 60 lines of people.

Another way to look at the same calculation is to follow the same calculation process that we did above for feet.
The distance was 240,000 amot ÷ 4 amot for each person = 60,000 people would fill a line reaching from Rameses to Succoth.

Our population of 3,600,000 ÷ 60,000 people in each line = 60 lines of people.

We just confirmed the same answer through 2 different calculation approaches!

So, again, if each person had 4 amot of space to travel (not accounting for animals or possessions), they would have had 60 lines of people lined all the way from Rameses to Succoth. 

Shelach- More Unit Conversions

"[The spies] arrived at the Valley of Eshcol and cut from there a vine with one cluster of grapes, and they carried it on a pole, by two, and of the pomegranates and of the figs." ~Bamidbar 13;23 

Rashi on 13;23 says:
And they carried it on a pole, by two: From the implication of that which has been said, "And they carried it on a pole," am I not aware that it is carried by two people? Why does the Torah say "by two"? It means that it was carried by two poles, not by two people. How was this done? Eight of the spies took a cluster; one took a fig; and one a pomegranate. Joshua and Caleb did not take anything, for with [the spies'] entire being, i.e., they had no goal in bringing the fruit other than that they intended to spread slander against the Land of Israel, by saying, "Just as its fruit is unusual in its size, so are its people unusual." And if you wish to know how much the burden of one of them was, i.e., how much one of them was able to carry, you can go and derive this from the stones which they set up in Gilgal. They lifted for themselves, each individual, one stone from the Jordan onto his shoulder, and set it up at Gilgal. Our Rabbis weighed [these stones]. The weight of each one was forty se'ah. And we learned that the load that a person can raise onto his own shoulder is nothing but a third of the load of the entire load that can be raised when they assist him to raise.

Before I begin, I want to make note that I first got this idea from 2 of my students who used this topic as the basis for their Math 'n Torah Fair project.

Back in Parshat Chayei Sarah we converted between two Biblical units to be able to compare weights that were measured with different Biblical measurements. What if we want to figure out, here, the weight of each fruit? Or what about how much weight each person was carrying? Again, we'll need to look to unit conversion to find out answer. In order to figure out these weights, we'll need to know the equivalent weight of a se'ah.

A se'ah is usually used for measuring volume. However, the reference which Rashi is using here is from Gemara Sotah (34a), where they chose to use the weight measure of se'ah. Most modern day estimates for the weight of a se'ah are between 2.25 gallons and 4 gallons. For our purposes, let's use the 2.25 gallon reference, as this will tell us the smallest possible estimate for the weight of the fruit. 1 gallon of water is just over 8 lbs, so the weight of the fruit, if we use the weight of water, would be 2.25 x 8 = 18 lbs. 

Let's think about:
1) What was the weight of each fruit?
2) How much weight was each person carrying?

To begin, let's organize the information that we have from Rashi.
*The fig was carried by one person
*The pomegranate was carried by one person
*The cluster of grapes was carried by 8 people

According to Rashi, each of the spies could carry a weight of 40 se'ah. This would mean that the spies carrying the fig and pomegranate were each carrying 40 se'ah x 18 lbs each, which is 720 lbs.

Now, for the grapes, Rashi says that the load that a person carries on his own is only 1/3 of the load that a person can carry with assistance. By this reasoning, 40 se'ah was only 1/3 of what each spy could carry; in other words, each spy carrying the grapes would be able to carry 40 x 3= 120 se'ah. This is the same as 120 x 18 = 2,160 lbs.

By this reasoning, the 8 spies carrying the grapes together should have been able to carry a total of:
120 x 8 = 960 se'ah or 
2,160 x 8 = 17,280 lbs

So, by our calculations the weights were:
pomegranate = 40 se'ah = minimum of 720 lbs
fig = 40 se'ah = minimum of 720 lbs
cluster of grapes = 960 se'ah = minimum of 17,280 lbs

The parsha does say that the fruit was big...

To give a little perspective, 
*Arnold Schwarzenegger is recorded as deadlifting 710 lbs. This would mean that each of the spies was approximately as strong as Arnold Schwarzenegger.
*A large elephant is approximately 15,000 lbs, which means that, based on the estimates here, 8 Arnold Schwarzeneggers could possibly carry 1 reasonably patient large elephant.

Everyday Connection:
Let's test the premise- The idea that each spy carrying with a group was able to carry 3x the amount that they could carry individually is an interesting one. I wonder how this would test out in everyday life. How much weight can you lift on your own? How much weight can you lift with one other person? Two other people? Is there are pattern that emerges in the ratio of [# of people] to [weight lifted]?

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.

Chayei Sarah - Unit Conversion

"And it was, when the camels had finished drinking, the man took a golden nose ring, its weight a beka, and two bracelets on her arms, ten gold shekels their weight." ~Bereishit 24;22

Unit Conversion:

Unit (or Unit of Measure): a standard amount of a physical quantity (ref. dictionary.com)

In Math and Science, units are a critical aspect of calculation. Units, in simple terms, are the base quantity that we choose to use to measure physical properties- a length, a weight, a volume, etc. You can measure with standard units- units that are already established by society and have specific value assigned to them (e.g inch, cm, pound, gram, gallon, liter, etc.); You can also measure with non-standard units- units that you choose for you convenience, but are not standardized to all places and situations (e.g. the paperclips that I have in a box on my desk, or the length of my right foot).

In order to compare measurements of different items, we need to measure them using the same units. This is where unit conversion comes in. If we have items measured in different units, then we need to convert them all to the same units of measure in order to be able to say how they compare to each other. We need to know how many cm are in an inch, and then make sure that all items we're comparing are measured in either cm or inches. In general, it doesn't matter which measure you choose to use, you just need to pick one and make sure that everything that you're comparing is measured with the same measure. If I want to compare my height over the past 5 years, I can measure myself consistently in ft, or I can have some measurements in ft and some in meters, but in order to compare, I need to convert all the measures to either ft or meters to know how my height has changed. Note that in order to calculate the conversion from ft to meters or vice versa, I need to know that 1 meter is approximately 3.28 ft.

So, how does this connect to the parsha?

In Chayei Sarah (Bereishit 24), Avraham's servant, Eliezer, is sent to find a wife for Yitzchak. When Rivkah meets him at the well in Nahor and she offers water to both him and his camels, he realizes that she is the right woman for Yitzchak. He then, in Bereishit 24:22, gives a gift to her-- and here is where we find units of measure. Eliezer gives her "...a golden nose ring, its weight a beka, and two bracelets on her arms, ten gold shekels their weight."

So, what if we want to know how much the weight of her jewelry was altogether? Or, how big was the nose ring compared to the bracelets? How can we compare the weight of a beka and the weight of ten shekels? Rashi helps us out here and tells us that a beka is the weight of a half-shekel (see Rashi on this Pasuk to see how he knows this). So, we now we know that the nose ring was the weight of a half-shekel and the two bracelets were the weight of 10 shekels. So, the weight of all of Rivkah's jewelry from Eliezer was 10.5 shekels. The weight of the two bracelets together was 20 times the weight of the nose ring. Or, if we assume that the bracelets were equal in weight, then each bracelet was 10 times the weight of the nose ring.