V'Zot HaBracha- Estimating based on Non-Standard units

This Shabbat, being Chol HaMoed Sukkot, we will be reading a portion from parshat Ki Tisa. However, since next week (on Monday in Israel and Tuesday outside of Israel) we read V'Zot HaBracha and Bereishit in the same week, I felt it would be appropriate to appropriate V'Zot HaBracha for this week.

"And of Naphtali he said: Naphtali, fulfilled of desire, and filled with the blessing of Hashem; take possession of the sea and south." ~Devarim 33:23

Rashi on 33:23:
Take possession- Rashi explains that The Sea of Kinnereth (or Sea of Galilee) were Naphtali's portion, and at the Southern section of the sea, he took an additional section that was the width of a length of fishing net rope so that they would be able to fish by spreading out their nets and snares.

Baba Kamma 81b further explains that while the land of Naphtali was actually to the north and west of the Sea, the tribe was also given a strip at the south of the Sea that was wide enough to allow for their finishing nets to be pulled.

Parsha/Math Investigation:
Based on the usage of non-standard units to indicate the southern section of the Sea that was portioned to Naphtali, how can we estimate a section that could have realistically been given to them?

Using current information on the Sea of Galilee (source: Wikipedia.org), we know that the Galilee is 13 km (or 8.1 mi) at it's widest section. It's narrower sections lead from the Upper Jordan River and lead down into the Lower Jordan River. In our case of Naphtali's land portion, we are looking at the southern section of the Sea, so we are looking at the section that leads down into the Lower Jordan River. Using even just basic visual estimation, the narrowest southern section is approximately 1/6 of the widest section of the Sea. That would mean that, even at its narrowest, it's still about 2 km wide, or a little over a mile wide. If their section of inheritance from the Sea was only wide enough for fishing ropes, then we have to understand that they didn't receive a full section from one shore to the opposite shore. Rather, it had to be a section with a border somewhere between the two shores. Without knowing too much about standard lengths of fishing nets, based on some quick research, my best guess would be that the nets themselves wouldn't be more than a couple hundred yards at absolute most, which would seem to necessitate only needing to grant access or propriety to about 1/10 of a mile or approximately 1/10 of the distance out from the western shoreline on the southern section of the Sea.

More extensive researching of specific maps of the Sea of Galilee and historical fishing nets and techniques could offer information for a closer estimate to the exact Sea area that was given to Naphtali.

Related Math Investigations:

• Younger students could use different items as base units for non-standard measurement around the classroom-
• How many paper-clips long is the table?
• How many blocks tall (using identical blocks) is the bookshelf?
• Older students could measure things around the room using non-standard units, but then also compare these measurements to standard measurements.
• For older students, an idea for the classroom that is similar to the concept above from the parsha might be: 
• How many papers long is the table?
• How many inches/feet/meters long is the table?
• If I need to work on one side of the table and have space for my paper, how many inches/feet/meters of space do I need to allow myself on my end of the table?
• A step above for even older students might be to consider how much space would need to be allotted to individuals in order to have space to work at the table, and then consider, based on their estimates, how many students could effectively share the table and be able to get their work done without bumping into each other or invading each other's work space.

Haazinu- Looking for patterns

Parshat Haazinu is made up entirely of a single Chapter- Perek 32- towards the end of the book of Devarim. The majority of the parsha is a song written by Moshe. If you look at the parsha directly in the Torah scroll, the section of the song is written completely in two divided columns. Below is a picture that I took of this section from a tikun, or book that people use to practice the cantillation and pronunciation of the words. It is written exactly as it appears in the Torah scroll, but I took pictures of the practice side rather than the actual Torah side. (Please note that because these pictures contain G-d's name written out in them, please do not print them out or print them out and discard them.) 

The reading is in the two larger print righthand columns on each page. The pictures only include the sections with the song, not    the entire parsha.

If you've ever studied poetry in school, you'll be familiar with patterns of rhyme, beat, measure, line fragments, word placements, etc. that writers use to enhance the aesthetics of their poetry. Reading through the parsha as it is written in the Torah, do you find any patterns? Lyrical patterns- beat or measure to the pattern of the words? Visual patterns- patterns in the placement of the words or repetitions of words in the scroll? Is there a pattern in the number of words in the lines, or the way the words are broken up into columns or lines throughout the song?

Younger students can listen to the parsha being read and listen for beat or measure. Older students can investigate this on their own, or in partners. Students of all ages can look at patterns in numbers of words per line, or looking for any alliteration of letters that may appear. Younger students could handle smaller sections and maybe look at the whole as a class, while older students can look at the entire text as in partners. 

Please note that I have not investigated these questions myself, yet, but rather this is a proposed investigation. 

Vayeilech- Communicating Clearly

"Take this book of the Torah and place it at the side of the Ark of the Covenant of Hashem, your G-d, and it shall be there for you as a witness." ~Devarim 31:26

Rashi on 31:26 
At the side of the Covenant of Hashem- The Sages of Israel argued about this in Tractate Bava Basra. There are those among them who say that there was a board protruding from the Ark on the outside and there [the Torah scroll] was placed. And there are those who say that it was placed beside the Tablets inside the Ark.

Discussion Point:
Clarity of descriptions and directions are a key aspect of communication. If I say something that I think is clear, it's only really as clear as the way the other person hears and understands it. Sometimes there are language or cultural barriers that create a need for extra-specific clarity or explanation, but, in general, even in an everyday setting with people who speak the same language and use the same cultural cues, it is still important to include details and descriptions that add to the clarity of the information that you are trying to communicate. In the statement above from this week's parsha, there is enough ambiguity that it's unclear exactly where the Torah should be stored in relation to the Tablets in the Ark. 

Consider the following- I could tell you to:

1. get the dish from the kitchen
2. get the dish from the cabinet in the kitchen
3. get the dish from the bottom left cabinet under the window in the kitchen

All three of those directions request the same item and sound fairly clear, but the third one will leave my messenger the most sure of their task and help them formulate their plan for following my directions.

Suggested Activity:
Students of any age who have reached a level of interactive play and communication with peers can practice formulating a direction and sharing it with the class or in partners. Partners should test their friend's direction to see how clear it really is when they try to follow through. How much do you need to amend your directions or help your friend in order for them to do what you asked? The fewer changes you need to make, and the less help you need to give, the clearer your original directions are. Younger students might want to do this with directions for a physical task- moving around the room to complete a task. Older students can give directions for a physical or written task- moving around the room, following a specific path through a section of the school, creating a drawing, a pattern, or a sequence.
Students can adapt this activity to whatever their developmental level. Moving from verbal directions to clear, specific written directions also increases the difficulty level for students.

Nitzavim- Thinking about Mathematical Language

"...your small children, your women, and your convert who is in the midst of your camp, from the hewer of your wood to the drawer of your water..." ~Devarim 29:10
"Hashem, your G-d, will bring you to the land which your forefathers took possession and you shall take possession of it; He will do good to you and make you more numerous than your forefathers." ~Devarim 30:5
"...that which I command you today, to love Hashem, your G-d, to walk in His ways, and to keep His commandments, His statutes, and His ordinances; and you will live and you will multiply, and Hashem, your G-d, will bless you in the land to which you come, to take possession of it." ~Devarim 30:16
"...I tell you today that you will surely be lost; you will not lengthen your days upon the land that you cross the Jordan to come there, to take possession of it." ~Devarim 30:18

Math Connection:
I first wrote about mathematical language back in my 1st posting on Parshat Shemot. When speaking about mathematically related words and descriptions, students (and sometimes adults) will mix up or interchange terms without realizing the implications of the words that they choose to use. Careful word choice is important in any use of language, and that applies to mathematical language, as well. 

Some specific mathematically related descriptions to consider from this week's parsha:

• "small children"- what information does that tell you? why is that an important detail? how might you define small?
• First it says "make you more numerous", and later it says "you will multiply"- (note that my quotes are from my Artscroll edition; in the Sefaria.org translation, it actually translates both locations as "multiply") How does the second description further define the first description? Does it make it clearer? In what way will you become more numerous? How much greater will you be? In what way will the numbers increase?
• Later it says "lengthen your days"- what's the difference between multiplying and lengthening? Is one description more specific than the other? Or do they indicate different meanings?

Ki Tavo- Leveled activities for division

"Moshe commanded the people on that day saying, 'These shall stand to bless the people on Mount Gerizim, when you have crossed the Jordan: Shimon and Levi and Judah and Issachar and Joseph and Benjamin. And these shall stand for the curse on Mount Ebal: Reuven, Gad and Asher, and Zevulun, Dan and Naphtali...'"~Devarim 27:11-13

Some activity suggestions:

• Younger students can talk about dividing the 12 tribes evenly between the two mountains. How many should be on each mountain? Connect the separation to division of 12 by 2-- will have 6 on each mountain. 
• Pattern finding: If you cross-reference with charts of children from each mother (Rachel, Leah, Bilha, and Zilpah) and the birth order of the 12 sons, can you find any connection between the siblings who were grouped together on each mountain?
• For Older students- if we take the most recent census information- from Bamidbar 26:1-51, we have a listing of the numbers of men in each tribe. Taking these numbers, was the division of people equal between the two mountains? Did one mountain have more people and one have fewer people? Could there have been a different way to divide the tribes that would have made the division more equatable between the two mountains?

Ki Teitzei- Activity suggestions related to weights & measures and fractions

A couple of suggested activities related to this week's parsha:
In last year's post, I wrote about the algebraic connection to the mitzvah (commandment) of being honest in weights and measures. Here are some suggested pre-algebra activities-

• using a balance scale, students can have different sized blocks which they need to place on the scale to figure out which is greater (heavier) and which is lesser (lighter). From there, they can investigate how many of the lesser weights it takes to balance the greater weight.
• a more difficult variation- using multiple unlabeled blocks/weights, students can work to:
•  order them from lightest to heaviest
• determine if multiples of the same block are equal to (balance) any of the other blocks
• determine if any combinations of blocks are equal to (balance) any of the other single blocks
• determine if any combinations of blocks are equal to (balance) any combinations of other blocks
• using labeled weights, students can confirm different combinations of smaller weights that should equal (or balance) with a single larger weight- for example, they should find that 2+3 is equal to (balances) a 5.
• students can then "mix and match" to find different combinations of weights that are equivalent to each other- for example, they should find that a 2+5 is equal to (balances) a 3+4.

There are many commandments (mitzvot) enumerated in this week's parsha. It has been counted that, in fact, 74 out of the 613 commandments have a basis in this parsha. Students can consider-

• What fraction of mitzvot are based in the parsha?
• Can this fraction be reduced?
• a related concept- what is the prime factorization of these two numbers? Did you know that you can quickly reduce large fractions by finding the prime factorization of both the numerator and denominator and crossing out "pairs" of common numbers that appear in both.
• a simple example- to reduce 10/15, we could say that 10 = 2x5 and 15 = 3x5; since there is a 5 in the prime factorization of both the numerator and the denominator, we can cross out both 5's, and we are left with 2/3. In this way, we have reduced 10/15 to 2/3 using prime factorization.
• another related concept- investigating common factors and identifying prime numbers (numbers that have only 2 factors- 1 and the number itself)
• What fraction of mitzvot are not based in the parsha? In other words, what's the fraction of other mitzvot that are not listed?
• Can this fraction be reduced?
• What percentage of mitzvot are listed in this parsha? What percentage of mitzvot are not listed in this parsha?
• What is the ratio of mitzvot in this parsha to the mitzvot not in this parsha?

Shoftim- Activity Suggestions related to Understanding Fractions

"This shall be the due of the Kohanim from the people, from those who perform a slaughter, whether of an ox or of a lamb/kid: he shall give the Kohen the foreleg, and the jaw, and the stomach. The first of your grain, wine, and oil, and the first of the shearing of your flock you shall give to him." ~Devarim 18:3-4

Rashi on 18:4 "The first of your grain":
"This is terumah. [Scripture] did not specify an amount regarding it, but our Rabbis established an amount regarding it: One who has a nice eye, i.e., one who is generous, give one part out of forty. One who has a bad eye, i.e., one who is miserly, gives one part out of sixty. One who is in the middle, i.e., who tends toward neither extreme, gives one part out of fifty. And they drew support from Scripture not to give less than one out of sixty, for it says, "[This is the terumah which you shall separate:] a sixth of an ephah from a chomer of wheat." A sixth of an ephah is half of a se'ah. When you give a half se'ah per kor, see now, there is one part out of sixty, for the kor is thirty se'ahs.

Activity Suggestions:
One of the most difficult conceptual aspects of fractions is that, when comparing sizes of fractional pieces, the larger the denominator (bottom number), the smaller the piece size. Once students understand fractions, they realize that this incongruence is because the denominator explains how many equal size pieces you have cut the whole into, and the more pieces you make, the smaller each piece will be. 

• With this in mind, younger students could test this by having fractional templates to cut or count and compare. Starting with the same size whole, which fractional pieces are larger and which are smaller? 1/2, 1/3, 1/4, etc. 
• Students who can make sense of this concept can extend the idea to larger number without comparing visually- compare 1/27 and 1/30, for example. And then extend to connect back to our Parsha. There, we are comparing donations of 1/40, 1/50, and 1/60. Understanding this idea, why does it make sense that 1/40 is considered generous, 1/60 is considered miserly, and 1/50 is considered average?
• Older students can look more carefully at the end section of Rashi's commentary and investigate the fractional equivalences for ephah, chomer, se'ah, and kor. Rashi's explanation indicates a comparable equivalency that relates directly to the fractions that he lists in the first part of his explanation. How do these equivalences support his explanation of the appropriate amounts to be given in donation from the first crops?

Re'eh- Activity investigations related to yearly tithings

In this week's parsha, we have two mentions of the tithings that were to be given at different times over the course of the shmita cycle. Additionally, tithings are referred to again in 3 weeks, in Parshat Ki Tavo. These three instances are looked at collectively in commentaries regarding the shmita cycle. 

• "You may not eat in your cities: the tithe of your grain, and your wine, and your oil; and the firstborn of your cattle and your flocks; all your vow offerings that you vow and your free-will offerings; and what is raised of your hand. Rather you shall eat them before Hashem, your G-d, in the place that Hashem, your G-d, will choose- you, your son, your daughter, your slave, your maidservant, and the Levite who is in your cities- and you shall rejoice before Hashem, your G-d, in your every undertaking. Beware for yourself lest you forsake the Levite, all your days in your land." ~Devarim 12:17-19
• Rashi on 12:18- regarding eating the tithings with the Levite in your city:
• Eat your first tithing together with him (1/10 portion of the crop given to Levites in each of the first 6 yrs of shemita)
• If you have no first tithe, then give from tithe for poor (1/10 of portion given to poor in third and sixth years (additional reference see 14:28-29, 26:12)
• If you have no poor tithe, then share your peace offering
• "At the end of three years you shall take out every tithe of your crop in that year and set it down within your cities. Then the Levite shall come- for he has no portion or inheritance with you- and the convert, and the orphan, and the widow who are in your cities, and they shall eat and be satisfied, in order that Hashem, your G-d, will bless you in all your handiwork that you may undertake." ~Devarim 14:28-29
• "When you have finished tithing every tithe of your produce of the third year, the year of the tithe, and you will have given to the Levite, to the convert, to the orphan, and to the widow, and they will have eaten in your cities and will have been satisfied, then you shall say before Hashem, your G-d, "I have eliminated the holy things from the house, and I have also given it to the Levite, to the convert, to the orphan, and to the widow, according to the entire commandment that you commanded me; I have not transgressed any of Your commandments, and I have not forgotten..." ~Devarim 26:12-13 (from Parshat Ki Tavo)

What were the tithings and the cycles?

• 1st tithe- 1/10 to the Levites
• 2nd tithe- 1/10 to be brought to Jerusalem and eaten there
• poor tithe- 1/10 to be given to the poor

The cycle of tithings over the shmita cycle was as follows:

Year 1- 1st tithes and 2nd tithes given

Year 2- 1st tithes and 2nd tithes given

Year 3- 1st tithes and tithes for poor given

Year 4- 1st tithes and 2nd tithes given

Year 5- 1st tithes and 2nd tithes given

Year 6- 1st tithes and tithes for poor given

Year 7- shmita year- no tithings

Activity suggestions:

• Younger children could investigate what it means to give 1/10. If you have a set-up for them to take a collection of items and divide them into ten groups, they would be donating one of those groups. The idea of 1 out of 10 might be conceptually easier for some than 1/10. In other words, for every 10 that's count out, 1 is going to be donated. 
• As children get to a point where they understand the concept of division by 10, they can practice some simple sample calculations of how much might have been donated based on a given monetary value. 
• Older children could be presented with sample scenarios of realistic incomes- some monetary based and some agriculturally based- and have them calculate projected tithings (per tithe and total donations) that families may have made over a 7 year shmita cycle. 
• Note that older students should also be able to make the connection that each 1/10 tithing would be equivalent 10% of their produce or earnings.
• It's also possible that, for some students, it would need to be clarified that all tithings were 1/10 of the total earnings; the 2nd tithing would not be calculated as 1/10 of what's left after the 1st tithing is given. This may be obvious to many, but for students who are making sense of the process, this may be a question that is unclear to them without explicit clarification.
• Extension thought- Based on these calculations, how could they think about how much the Levites had to live off of in a given year? This type of investigation could include researching information such as census information for each tribe listed in recent parshiot- if all the Levites were living off of tithings from those families, how much did they really have to live off of, per Levite family?

Eikev- An activity in comparisons and estimated measures

Last year for Parshat Eikev, I wrote about Rashi and Tanchuma's commentaries regarding the size and weight of the stone tablets that Moshe brought down with the 10 commandments inscribed on them. Their commentaries explained that despite there being differences between the two tablets, the size and weights of the two tablets were actually identical.

Thoughts on a follow-up activity:

• Last week in Parshat Va'etchanan (Devarim 5:6-18) we read the 10 commandments. These commandments were divided into the first 5, which are between man and Hashem and inscribed on the first tablet, and the second 5, which are between man and man and inscribed on the second tablet.
• If you compare, side by side, the wording, number of letters, number of characters, etc. between the two subsets of commandments, what are the actual differences between them? Which has more words, letters, characters, etc. and which has fewer? 
• Think about how you might compare the inscriptions proportionally to each other? 
• Is there a way to calculate- if I start with two 25 lb. tablets, and then inscribe in each one, is there a way to estimate how much each one should have weighed at the end? 
• Could you estimate surface area if you take a standard font and printed onto a sheet of paper- what's the ratio of print to total surface area for each tablet? What might the proportional volume differences be? What about weight differences? 
• There is a discussion in Gemara Bava Batra 14a wherein it is determined that the size of each tablet was 6 tefachim (handbreadths) high, 6 tefachim wide, and 3 tefachim deep. 
• Based on these measurements, what might the original volume of each tablet have been?
• Based on these measurements, what might the original weight of each tablet have been?
• Using estimated proportionally carved out sections from each tablet, what would one have assumed the new volumes/weights of each tablet to have been after the inscription of the commandments?
• Older students can also work through the measurement calculations from the source in Bava Batra related to the ark (where the tablets were kept) and the tablets, to see how the measurements were determined.

Va'etchanan- Internalizing External Directionality

"But Hashem became angry with me because of you, and He did not listen to me; Hashem said to me, 'It is much for you! Do not continue to speak to Me further about this matter. Ascend to the top of the cliff and raise your eyes westward, northward, southward, and eastward, and see with your eyes, for you shall not cross this Jordan. And command Joshua, and strengthen him and give him resolve, for he shall cross before this people and he shall cause them to inherit the land that you will see.' And we stayed in the valley, opposite Beth-peor." ~Devarim 3:26-28

In this section of this week's parsha, we have Moshe retelling the Children of Israel about how he begged Hashem to let him join them when they entered the land of Israel, and Hashem's response to his request. He confirms that he is not allowed to enter the land, but he is told to go up to the top of the cliff and look around in all directions- West, North, South, East- to see the land that will be given.

Last year, in Parshat Behaalotcha, I wrote about the difficulties that children have in making sense of right and left and how the use of right and left is specifically confusing when trying to assign it to different people or objects- my left, your right, to the left of the desk, etc. An overlay to that understanding is the concept of cardinal directions, which is used here (and in many other descriptive sections of the Torah).

Cardinal directions are north, south, east, and west. While these are classically a geographically linked concept, they tie into students understanding of direction, and therefore expand beyond just the basic map-reading skills and fall into spatial understanding and reasoning, which promotes their understanding of geometry and spatial awareness. 

The biggest aspect for students to explore is based on the idea that the descriptions of left and right change based on a person's orientation. For example, when a stand here, the window is on my left, but when I turn, now the door is on my left. However, with cardinal directions, they maintain their location and description no matter how you move around. If the window is on the north side of the room, and the door on the east side, they will remain on the north side and the east side, respectively, no matter which way I turn around. 

Students can also rely on the fact that once they remember the relationship of the directions to each other, then as long as they can definitively identify the location of one direction, they can figure out the remaining directions. In other words, if I know that my window is north, and I need to figure out the direction of my door, if, when I face my window the door is 90 degrees on the right, then I know that my door is to the east.

As students advance, they can extend their knowledge throughout a building. So, if I give my students a mock blueprint of a school, and I tell them that the chalkboard in the 2nd grade classroom is on the northeast side of the building, they should be able to use that information to determine the cardinal location of any other item on the floor, or even in the entire building, that I ask them to locate. Going another step beyond, rather than just lining up blueprints or maps in front of them, can they use identifying markers to physically move through the building, or school campus, or beyond and identify the cardinal directions of specific requested items?

While knowing left from right is a critical skill for life, knowing cardinal directions is an even greater skill, since the cardinal directions allow for greater extrapolation of the relationship of the space that you're moving in. 

Extension Thought:
Visually, students can make a connection between the directions on the compass and a coordinate plane. Imagine:

• The N-S line = y-axis
• The W-E line = x-axis
• Center point of the compass = the origin or (0, 0)
• NE = the 45 degree split of the (+, +) Quadrant I
• NW = the 45 degree split of the (-, +) Quadrant II
• SW = the 45 degree split of the (-, -) Quadrant III
• SE = the 45 degree split of the (+, -) Quadrant IV

Conceptually speaking, these may not have a strong connection to each other, but connecting information in different ways can make a big difference in the way that students make sense of, understand, and internalize disparate information to create meaningful and useful practical applications of the information for themselves. This is why we teach, use, and create mnemonic devices for ourselves when trying to remember information. 

Devarim- Standard or Non-Standard Measurements?

"For only Og king of the Bashan was left of the remaining Rephaim. Behold! his bed was an iron bed, in Rabbah of the Children of Ammon; nine cubits its length and four cubits its width, by the cubit of a man." ~Devarim 3:11 

In this week's parsha, we have Moshe retelling about the journey of the forty years through the desert. This is actually the second retelling, since last week in Parshat Masei we also had a retelling of the journey. Last week we read a more factual retelling (traveling from point A to point B to point C...), while this week we read a more emotionally connected retelling by Moshe to the Children of Israel. Within this week's parsha, we learn more details about when The Children of Israel were fighting various groups of people towards the end of their journey, as they approached the land of Israel. In the specific section quoted above, we learn that Og the king of Bashan was the only remaining giant from the Rephaim following their battles with the Rephaim.

Math Connection:
In this section, we are told the dimensions of Og's bed, and Rashi specifically explains that the measurements given were proportional to Og's size, rather than an average man. I first explained about non-standard units of measure back in Parshat Vayeira. Here, we have an interesting measurement- Og's bed is described by measure of cubits. A cubit is commonly defined as the distance from a man's elbow to the tip of his middle finger. Would this be considered a standard or non-standard measurement? While the definition is based upon using a standard measure using reference points on a person's arm, relatively speaking, each person's cubit will be measured differently. It is for this reason that it was meaningful for the parsha to specify "by the cubit of a man" and for Rashi to clarify that this specifically meant that it was a cubit measured by Og's arm. The argument could be made that general measurements based on cubits measured using measurements from an average man would result in fairly standardized sizes- maybe give or take a foot or so if you have final measurements of the same item measured with cubits from a shorter man compared to final measurements measured with cubits from a taller man.

Activity Suggestions:

• How long is a cubit by your arm?
• Take measurements for the entire class. 
• How does the data compare for all the students in the class? 
• What would the average cubit size for your class be? 
• Does that include or exclude your teacher's cubit?
• Which measure of statistical average is most appropriate in this case- mean, median, or mode?
• How big would Og's bed be if it were built according to your cubit?
• Compare your measurements with measurements for your classmates. 
• How do the measurements compare for all the students in the class? 
• What would the average bed size for your class be? 
• Does that include or exclude your teacher?
• What would the average bed size be for your class? 
• Would everyone in the class fit on that bed? 
• Which measurement would you choose to use if you needed to build a bed to any student in your class? 
• Would your teacher fit?
• Using sizes for current-day standard bed sizes- twin, double, queen, king-
• How would your calculated bed(s) compare to a modern-day standard bed? Which bed is the closest in size to your calculated bed?

For Further Thought:

The following question is one that would make an interesting follow-up and "self-to-text" connection for students, but I was unable to find enough clear information regarding Og's height to confirm that this is an answerable question. I am including it here as a follow-up and would love to hear from anyone who might know of sources that offer information regarding Og's height.

• How big was Og? How can we estimate what size the bed might have been based on Og's cubit?
• How would Og's calculated bed compare to a modern-day standard bed? Which modern day bed is the closest in size to your calculated bed for Og? 

Mattot/Masei- Activity Suggestions for Creating Charts with Guided Calculations to Understand Information

Calculate and Divide the Plunder: 

"Hashem said to Moshe, saying: 'Calculate the total of the plunder of the captives, of people and animals, you, and Elazar the Kohen, and the heads of the fathers of the assembly. Divide the plunder in half, between those who undertook the battle, who go out to the army, and the entire assembly. You shall raise up a tribute to Hashem from the men of war who go out to the army, one living being of five hundred, from the people, from the cattle, from the donkeys, and from the flock. You shall take it from their half and give it to Elazar the Kohen, as a portion of Hashem. And from the half of the Children of Israel you shall take one drawn from fifty, from the people, from the cattle, from the donkeys, from the flock- from all the animals- and you shall give them to the Levites, the guardians for the charge of the Tabernacle of Hashem.'" ~Bamidbar 31:25-30

Itemization of Division of the Plunder: 

"Moshe and Elazar the Kohen did as Hashem commanded Moshe. The plunder, beyond the spoils that the people of the army took as spoils was: the flock, six hundred seventy-five thousand; and cattle, seventy-two thousand; and donkeys, sixty-one thousand; and human beings, and of the women who had not known lying with a male, all the souls, thirty-two thousand. The half which was the share of those who went out to the army, was: the count of the flock, three hundred and thirty-seven thousand, five hundred- the tribute of Hashem from the flock was six hundred and seventy-five; and the cattle, thirty-six thousand- and their tribute to Hashem, seventy-two; and the donkeys, thirty thousand, five hundred- and their tribute to Hashem, sixty-one; and the human beings, sixteen thousand- and their tribute to Hashem, thirty-two people. Moshe gave the tribute that was raised up for Hashem to Elazar the Kohen, as Hashem had commanded Moshe."

From the half of the Children of Israel that Moshe had divided from the men who went to the army, and the half of the assembly was: of the flock, three hundred and thirty-seven thousand, five hundred- and the cattle, thirty-six thousand; and the donkeys, thirty thousand, five hundred; and the human beings, sixteen thousand. Moshe took from the half of the Children of Israel the one drawn from the fifty, from the people and the animals, and gave them to the Levites, the guardians of the charge of Hashem's Tabernacle, as Hashem had commanded Moshe." ~Bamidbar 31:31-47

This week's parsha is a double-parsha (we read two parshiot). For this post, I will focus on a portion from parshat Mattot. In this section, we have Hashem directing Moshe as to how the plunder of the Israelites' war with Moav was to be divided and donated. Following this, we have a section that itemizes what plunder was collected and exactly how they were divided. This set-up offers an opportunity for students to first organize the information related to how to divide the plunder and then test out their calculations on their own to see if their calculated divisions and donations match what is listed for the itemization of divisions and donations.

Some guiding questions for organizing and calculating:

• To begin, how many groups was the plunder to be divided into? 
• Was this an even division? 
• What/who were the groups?
• What were the categories of plunder that was being divided?
• How many were there?
• What exactly were they?
• Where was the donation from each group to be given?
• What was the calculation for donations from each group? Were there different calculations for the subgroups from each group?
• How did the calculations for donations compare within groups (looking at subgroups) and between groups?
• Can you use these questions to organize the basic set-up into a chart that will be usable for charting the actual division and donation of the plunder?

Using the chart created above, if you organize the information about the actual plunder that was collected, how do your calculations for division and donations match with the calculations given in the second quoted section above?

Problem Solving For Younger Students:
While younger students might not be able to organize a chart or perform the calculations necessary, some possible activities could be:

• As a group, with teacher guidance, identifying where information would fit into a pre-structured chart.
• Testing out donation samples for "1 out of 500" and "1 out of 50" using manipulatives.
• Note that counting out such large groupings would likely be too drawn out and tedious for most young students, but if they have pre-prepared sample groups already counted out into sub-groups of 500 and/or 50, then they can use these groups to model donating 1 from each group. Students could then use skip counting to show that, for example, 3 groups of 500 means that there were 1,500 paperclips and 3 of them would be donated.

Extension thoughts for Older Students:

• This type of information can be expressed using fractions, ratios, and percentages. How could you express the information using each of these forms?
• How might this information be displayed using graphs?

Pinchas- Comparing and Collating Large Information Sets

This week, I'd like to focus on two sections from the parsha that offer opportunities for larger investigative approaches to help students make sense of the information given. I find, myself, that when I read through the sections of parsha that are more of a list format, I tend to lose focus rather than making meaning out of the information. So, if there are a list of census numbers, I'll make a mental note that the parsha talks about a census, but I don't usually fully process the numbers.

This week, we have at least two sections like this. The first is in Bamidbar 26:1-51, where we have current census from each tribe and then the tally of the entire group. The second is in Bamidbar 29:12-38, where we have a list of the sacrifices that are to be offered on each day of the holiday of Sukkot. Each of these is part of an even larger section that could also be looked at- census from Levi is also in the parsha, as are the lists of sacrifices for every holiday. I'm choosing these two larger sub-sections to focus on presenting my thoughts, but many of these ideas can be adapted or extended to other sections of the parsha, as well. Adaptations to meet differing developmental abilities are also possible.

The Census:
We can compare the current census for our parsha, found in Bamidbar 26:1-51, with the original census at the beginning of Bamidbar, found in Bamidbar 1:1-46.
Some ideas for comparisons:

• What are the similarities and differences between the text sections for each tribe? Do you notice any major differences in the texts? (Hint for one major difference- look carefully at the section for Yoseph's children in each of the two passages.)
• What are the similarities and differences between the individual census numbers for each tribe?
• Did the numbers for each increase or decrease?
• How do the census numbers for each tribe compare to each other? How did the original numbers compare to each other? If you order the tribes by according to size, does the order change from the first to the second census? If you create a graph for each group of census numbers, how do the graphs compare? 
• How do the total sum numbers compare between the two different census counts? Did the total sum increase or decrease?
• Some statistical thoughts for grades 5 or 6 and above: (Note that while the calculations may be straightforward or rote, the full understanding of the significance of these evaluations of the data may be hard for younger students to make sense of and truly understand and explain.)
• What was the average population per tribe after the first census? after the second census?
• In the first census, was each tribe above or below the average? What about in the second census?
• How far above or below the average was each tribe's population in each census? How do these statistics compare for each tribe between the two sets of census numbers?
• What fraction of the population was each tribe in the first census? in the second census? How do these compare?
• What percentage of the population was each tribe in the first census? in the second census? How do these compare?
• How would different types of graphs of the two sets of census data look and compare to each other? (Separate bar graphs, a double bar graph, line plots, box-and-whisker plot, pie charts.) Which graph would you personally choose to best represent the data, and why?
• Extra thought- when evaluating the data, does the 24,000 who were killed in the plague on the tribe of Shimon effect statistical changes from the first census to the second?
• Adapted but similar evaluations can be made between the original and new census for the tribe of Levi, as well.

Korbanot/Sacrifices:
As I mentioned above, there is a large section in this parsha that is dedicated to the description of the sacrifices that were to be brought on each holiday. I will focus, here, specifically on the sacrifices for Sukkot, but these questions can be extended and adapted to compare information related to other holidays or even to compare between different holidays.

Sacrifices for Sukkot:
The description for these sacrifices is found in Bamidbar 29:12-38.
Some ideas for comparisons:

• What are the similarities and differences in the text for the sacrifices for each individual day of the holiday?
• What are the specific lists of sacrifices that were to be brought on each day?
• How do the sacrifices for each different day compare to each other? How are they similar? How are they different? Are there any patterns that you notice? 

Some ideas for calculations with fractions and measurements:

• Think about making a collated list of the items needed for each sacrifice or each group of sacrifices-
• How much of each "ingredient" did they need for each sacrifice?
• How much of each "ingredient" did they need for all of the sacrifices on each individual day?
• How do the totals for each individual day compare to each other?
• How much of each "ingredient" did they need for the entire holiday?
• Do you see any significant numbers in your tally lists? Rashi on 29:35 refers to a significant connection to the tallies.

Balak- Quantity vs. Quality

"The officers of Moav arose and came to Balak and said, 'Balaam has refused to go with us.'

Balak kept on sending officers- more, and higher ranking than these. They came to Balaam and said to him, 'So said Balak son of Zippor, "Do not refrain from going to me, for I will honor you very much, and everything that you say to me I shall do; so go now and imprecate this people for me."'

Balaam answered and said to the servants of Balak, 'If Balak were to give me his houseful of silver and gold, I am unable to transgress the word of Hashem, my G-d, to do anything small or great. And now, you, too, please stay here for the night, and I will know what Hashem will add in speaking with me.'" ~Bamidbar 22:14-19

Quantitative vs. Qualitative:
Quantitative data is information that can be counted or measured.
Qualitative data is information that describes something. 

Parsha Connection:
In this week's parsha, Balak is trying to convince Balaam to invoke a curse upon the Children of Israel. When the first set of elders were sent from Moav, they were unsuccessful in convincing Balaam to curse the Israelites. Without giving up, we learn that Balak sent more, higher ranking officers. We aren't told how many officers were sent each time, nor are we told their specific rankings. Rather, we are given a qualitative description of the officers. We know that the second group was larger and of higher standing than the first group.

In an effort to convince Balaam, Balak sends a message that he will honor Balaam "very much". Again, Balak's offer was a qualitative one- he will honor Balaam "very much"; there is no indication of how he will honor Balaam, or what he will offer him, we just know that it will be a lot of honor.

Balaam's response, however, is a quantitative one. He responds that even if Balak gave over his entire houseful of silver and gold, Balaam is not willing to go against Hashem. Although we don't know how much silver and gold was contained in Balak's house, it is a quantifiable, calculable amount. 

Academic Connection:
Sometimes the difference between quantity and quality is important, and other times it's not critical. For example, if I have 2 sizes of containers to put away toys, the specific sizes probably don't matter- I can look and see that one size is too small, but the other size will work well. If I have half a gallon of water left and want to transfer it to a smaller container, it would be important to know the exact sizes of the two containers so that I will know if the water will fit in one or the other of the containers.

Suggested Activity:

• Younger students could have scenarios set up for them- either as part of a whole class discussion or individual stations for students to go through in partners or individually. Students could have 2 sample situations presented to them where they need to determine "Quantity or Quality?" Sample situations might include:
• Case 1: Your friend's bag of cookies is bigger than your bag of cookies. Case 2: We need 7 more cookies in order to have enough for everyone in the class.
• Case 1: The cookies have black and white frosting on top. Case 2: There are 5 black and white cookies in each package.
• Case 1: Our class collected $50 in tzedaka money. Case 2: Our class collected more money for tzedaka than we did last year.
• Case 1: My house has 3 bedrooms. Case 2: My house is blue.
• Case 1: Your friend has fewer siblings than you. Case 2: Together, you and your friend have 5 siblings.
• After working through some examples, older students could create their own examples and test each other. Another activity might be for students to create 2 short writing samples, one with only qualitative descriptions and the other with only quantitative descriptions. As a class, sharing writing samples, students can think about the differences they notice between the two types of writing samples.
• Note that qualitative information can sometimes be turned into quantitative information, and this might be confusing to students. For example, if I have more cookies than you, that's qualitative. However, we could each count our cookies and then describe the difference using actual numbers, which would be quantitative. Other qualitative information cannot be explained quantitatively- for example, my house being blue.

Chukat- Two Approaches to Distance Traveled as Related to Rate of Speed

This week I would like to follow-up on my post from Parshat Chukat last year. In last year's post, I looked at the distance that the Children of Israel traveled in Bamidbar Chapters 20-21. Using an estimated route, I calculated that they traveled approximately 312.5 miles in this final section of their wandering in the desert.

"The Children of Israel, the entire assembly, arrived at the Wilderness of Zin in the first month and the people settled in Kadesh. Miriam died there and she was buried there." ~Numbers 20:1

From Midrash, we learn that these two chapters take place in the final year of the wandering. We learn in this first statement that we are beginning this section from the first month of this final year of wandering. Throughout the course of these 2 chapters, we have Miriam's death with 30 days of mourning and Aaron's death with 30 days of mourning. This means that out of the final full year of wandering, the Children of Israel had at most 10 months of travel time. While we know that there were periods of traveling and periods of encampment, we can still calculate an average rate of travel in terms of miles traveled per month. We could break it down further to calculate average miles traveled per day or even per hour. Let's remember that when you look at an average rate, it doesn't mean that you are traveling consistently over the whole time period, but if you broke up your entire travel so that you were traveling equal distances each hour, day, week, or month, this is how much you would have traveled in that space of time.

Back in Parshat Vayetzei, I first looked at Rate of Speed, where you express how fast or slow something or someones covers a distance by comparing the distance covered to the amount of time that it takes to cover the distance. This comparison is expressed as a ratio. So, if I travel 30 miles over the course of 1 hour, then my rate of speed would be 30 miles per hour (30mi/1hr or 30mph). If I travel 100 miles in 2 hours, then my rate could be expressed as 100mi/2hr or reduced as 50mph (since 100/2 reduces to 50/1).

Using this understanding of rate of speed, how could we look at the travel speed of the Children of Israel during the end of their wandering in the desert?

Rate of travel per month: 

Based on the estimates that I calculated last year, the distance traveled was 312.5 miles, and using my timeline logic above, they had 10 months of potential travel time.

Therefore, the rate per month would have been: 

312.5mi/10mo

This reduces to an average of 31.25mi/1mo or 31.25 miles per month

Follow-up:

Following this thinking-

*you can compare how many days would have been in the 10 months of travel time, and then calculate the estimated rate of miles per day

*how many hours would have been in those 10 months? what would the average rate of miles per hour have been?

*would the miles per hour calculation be a realistic estimate? Think about what this would mean for travel time and rest time during each day or week. How might you adjust this calculation to account for resting time each day (don't forget stopping for Shabbat)? How might you adjust the original calculations to allow for estimated periods of encampment based on the timeline that we're given in the parsha?

An alternate approach:
Looking at it from a different angle, if we estimate a reasonable rate of speed for their travel, how many hours/days/weeks/months would they have needed to be traveling and how much time would they have had for their periods of encampment?

*What does this mean? What would be a reasonable estimate for the amount of distance they were able to travel in an hour? a day? a week? a month? By quantifying some of these rates, how might their travel have lined up with the information that is provided in the parsha? How long would it have taken them to get between their different stops where they requested and were denied permission to cross through other lands? How might you create an estimated timeline of their travel using all of this information?

*Personal connection: Think about it similar to planning a road trip- how much mileage do you have to cover? What stops or activity locations might you be planning, and what are the distances between the stops? What is a reasonable rate of speed for your travel? How long will it take you between each stop? Can you fit your trip within a given timeframe or do you need more time to reasonably make your full trip?

Note: It is important to remember that this information is all hypothetical, since we don't have day by day or month by month itemizations of travel time or encampment periods. Additionally, my calculations from last year were merely estimates as well. That being said, thinking about the situation within this context can help students create a semblance of understanding around what the Children of Israel might have actually been going through in their travels- offering a realistic link to this "story".

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?

Beha'alotcha- Mixed Bag of activity suggestions

In this week's parsha, there were a number of smaller activity ideas that caught my attention. Different activities apply to different learning levels, and some are more easily modified up or down for varying age learners. Below I've listed each passage pertaining to the activities, with the specific activity ideas summarized directly below the applicable passage.

"Hashem spoke to Moshe, in the Wilderness of Sinai, in the second year from their exodus from the land of Egypt, in the first month, saying: 'The Children of Israel shall make the pesach-offering in its appointed time. On the fourteenth day of this month in the afternoon shall you make it, in its appointed time; according to all its decrees and according to all its laws shall you make it.' Moshe spoke to the Children of Israel to make the pesach-offering. They made the pesach-offering in the first [month], on the fourteenth day of the month, in the afternoon, in the Wilderness of Sinai; according to everything that Hashem had commanded Moshe, so the Children of Israel did." ~Bamidbar 9:1-5

"In the second month, on the fourteenth day, in the afternoon, shall they make it; with matzot and bitter herbs shall they eat it." ~Bamidbar 9:11 

• Calendar activity- In the above passage we are given dates for when the Israelites first offered the pesach offering in the desert, and then when they offer the secondary pesach offering for anyone who was impure at the time of the first offering. Using a calendar, students can map out the dates given in this passage, and they can also compare them to the dates listed on modern-day Jewish calendars to make the connection between the dates listed in the passage and the dates on which we now celebrate Pesach and Pesach Sheni. 

*****

"Hashem spoke to Moshe, saying, 'Make for yourself two silver trumpets- make them beaten out, and they shall be yours for the summoning of the assembly and to cause the camps to journey. When they sound a long blast with them, the entire assembly shall assemble to you, to the entrance of the Tent of Meeting. If they sound a long blast with one, the princes shall assemble to you, the heads of Israel's thousands. When you sound short blasts, the camps resting to the east shall journey. When you sound short blasts a second time, the camps resting to the south shall journey; short blasts shall they sound for their journeys. When you gather together the congregation, you shall sound a long blast, but not short blasts. The sons of Aaron, the Kohanim shall sound the trumpets, and it shall be for you an eternal decree for your generations.'" ~Bamidbar 10:1-8

• Codes- The above passage gives directions for Moshe to make two silver trumpets and use different combinations of blasts to indicate different messages to the groups of Israelites. Students could first map out the different blasts and what they indicate- long blasts? short blasts? combinations of long and short? number of each type of blast? Younger students could role play having the teacher sound a trumpet or call out a sound in a certain way and following different directions (move to the front of the room, sit down in a circle, sit down at the project tables,...) based on the sounds made. Older students could use this as a launching point into a discussion of other coding that is used- morse code, for example- and create a code of their own to then share with the group.

*****

"Moshe said, 'Six hundred thousand foot soldiers are the people in whose midst I am, yet You say I shall give them meat, and they shall eat for a month of days!...'" ~Bamidbar 11:21 

• Rounding numbers & Comparing numbers- Rashi on this passage wrote about how when Moshe is complaining about having to provide meat for the Israelites after they complained about the manna, he lists the foot soldiers at an even 600,000 men, yet when we read about the census in Bamidbar 1:46, there were 603,550 soldiers counted. What happened? 
• Rashi offers 2 explanations: 
1. Moshe was not focused on the detail of the specific numbers. Thinking about it, this makes sense- when you are upset or excited about something, you are focused on the big picture or larger situation, not the very specific details.
• Students can talk about how we round numbers mathematically. Find the greatest place value that we want to round to, and then look at the next smaller place value. If the number in that place is 5 or greater, then we round our desired place value up by one number; if the number in that place is 4 or less, then we round down by keeping our desired place value at its current number. In Moshe's excitement, did he round accurately according to the rules of mathematics?
2. Moshe was only referring to the men who left Egypt, who were the ones who complained, since only those who left Egypt had something to compare their experience to and say that this was worse.
• Students can refer back to Shemot 12:37 to compare the numbers. Students can calculate that, based on this comparison, there were therefore 3,550 men who were born and raised in the desert.

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"Moshe left and spoke the words of Hashem to the people; and he gathered seventy men from among the elders of the people and had them stand around the tent. Hashem descended in a cloud and spoke to him, and He set aside some of the spirit that was upon him and gave it to the seventy men, the elders; when the spirit rested upon them, they prophesied, but did not continue to do so. Two men remained behind in the camp, the name of one was Eldad and the name of the second was Medad, and the spirit rested upon them; they were of those who had been written, but they had not gone out of the Tent, and they prophesied in the camp." ~Bamidbar 11:24-26

• Probability- In the passage above, Hashem instructed Moshe to gather 70 elders to help in the spiritual guidance of the Israelites. Rashi explains that there were supposed to be 6 elders from each tribe (12 x 6 = 72) who were to gather and draw lots. The lots would indicate 70 spiritual leaders and 2 who were not able to attain prophetic status. Eldad and Medad, however, assumed that they would not achieve prophetic status, so they did not join the group. When the lots were drawn, 2 of the present elders drew blank lots, and the lots for Eldad and Medad were left undrawn. A few questions to begin thinking about this statistically:
• What was the probability of each elder drawing a prophetic lot? What was their probability of drawing a blank lot? 
• Was their probability changed in any way by Eldad and Medad abstaining from drawing lots? 
• What percentage/fraction of the elders achieved prophetic status? What percentage/fraction of the elders did not achieve prophetic status?