Tetzaveh- Statistical Investigations

"You shall take the two shoham stones and engrave upon them the names of the sons of Israel; six of their names on one stone, and the names of the six remaining ones on the second stone, according to their birth." ~Shemot 28:9-10

Rashi 28:10- 
according to their birth- Rashi explains that this means that their names were to be written in the order in which they were born. In his explanation, Rashi elaborates to tell us that based on this ordering of the names, they were not just divided 6 names on one stone and 6 names on the second, but they were also divided evenly by the number of letters that were engraved on each stone. 

The brothers' birth order is discussed, among other places, in Parshat Miketz.

Listed in order, as the names were divided on the stones, the brothers were:

1st stone In Hebrew Reuven ראובן Shimon שמעון Levi לוי Yehudah יהודה Dan דן Naftali נפתלי

2nd stone In Hebrew Gad גד Asher אשר Isachar יששכר Zevulun זבולן Yoseph יוסף Binyamin בנימין

I've included the Hebrew spellings in the charts above, since this is the basis for the calculation of number of letters.

Investigation for Younger Students:
With the pasuk and Rashi's explanation in mind, younger students could have a discussion, first putting the brothers' names into birth order, followed by an activity wherein they separate the names into the two groups of six. After writing out the names in the two separate sections, students can count the letters on each "stone" to discover the 25 letters on each. A modification for early literacy students could be to have the names pre-written for students to identify, cut, and paste. For higher ability levels, students can count the number of letter per name, record their numbers next to each name, and add them up at the end, rather than counting out all the letters on the stones. 

Some statistics investigation for Older Students:
After discussing Rashi's insight into the pasuk, older students could graph the number of letters in each brother's name. They could calculate the range, mean, median, and mode of the number of letters in the brother's names. Knowing that there are 25 letters on each stone, how do the range, mean, median, and mode of each stone compare to each other and compare to the range, mean, median, and mode of the total group of 12 names?

Elementary level statistics for this investigation could include making a pictograph or tally marks representing the number of letters in each brother's name. Elementary students of varying ages can graph the information on a line plot (or statistical dot plot)- a simple number line with a dot or x above each number to represent the number of letters in each name. For example, if one name has 6 letters, you put a dot over the number 6 on the number line; if another name has 3 letters, you put another dot over the number 3 on the number line; if two names have 5 letters, you put two dots over the number 5 on the number line. Older elementary students can begin to incorporate aspects of the more complex statistical calculations, possibly looking at one statistical average across the different groups (whole group and each of the two stones), rather than calculating and comparing all of the statistical averages. They can also analyze the similarities and differences between the shapes of the 3 different line plots.

Other analytical possibilities:
What fraction of the total letters is each brother's name? What percentage?
What fraction of each stone is each brother's name? What percentage?
How do the fraction/percentages compare from whole group to individual stone for each name?
Older students could use their calculated percentages to display the information in a circle graph.

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.

Mishpatim- Understanding and Applying a concept

Case #1: 

"When men will fight...if there will be a fatality, then you shall give a life for a life; an eye in place of an eye, a tooth in place of a tooth, a hand in place of a hand, a foot in place of a foot; a burn in place of a burn, a laceration in place of a laceration, a bruise in place of a bruise." ~Shemot 21:22-25

Case #2:

"When a man's ox will push his fellow's ox which dies, they shall sell the living ox and divide its money, and the carcass, too, shall they divide. Or if it was known that it was a goring ox from yesterday and the day before yesterday, but its owner did not guard it, he shall surely pay an ox in place of the ox, and the dead body shall be his." ~Shemot 21:35-36

Case #3:

"When a man will steal an ox, or a sheep or goat, and slaughter it or sell it, he shall pay five cattle in place of the ox, and four sheep in place of the sheep." ~Shemot 21:37

Understanding a Concept:
When learning a new concept, it is often difficult to have a full understanding just by reading an explanation or watching someone work out a single example. There are nuances, specifics of concepts that vary slightly based on given parameters. When learning a concept, there are levels to understanding what is meant. The base level is usually a straightforward understanding with an example that falls very neatly into the basic parameters. As you investigate and test out the way small changes effect the way the concept works, you are developing the complexity of your understanding of the concept. 

Let's look at a simple example first: a student learning the algorithm for subtraction (as they grow through elementary school)-

• First, the student will learn how to simply subtract a small single digit number from a slightly bigger single digit number (7 - 4 = ?)
• Next, they might start working with subtracting a smaller multi-digit number from a relatively slightly multi-digit number (395 - 273 = ?)
• Next, they would learn how to handle a situation where one of the place values doesn't have a large enough number, so they need to regroup (407 - 263 = ?)
• Next they would learn how to handle subtracting numbers with different place values (3,654 - 475 = ?)
• Finally, they start to transform their understanding of number sense by introducing subtracting a larger number from a smaller number (5 - 7 = ?). From here, they can also incorporate all their previously understood nuances of subtraction into this new framework of number sense based on the introduction of being allowed to take away more than you actually have (ie negative numbers).

The above example is a basic framework. Even here, there are other nuances that are incorporated which I have left out (how to handle numbers different types of numbers with decimal points, how to handle numbers with zeros in different place values, etc.)

Parsha Connection:
This week's parsha has many lists of various specific laws. Above we have 3 different specific cases where the laws related to valuation of damages are explained for us. Rashi also goes into lengthy, specific details in each of these cases to explain more clearly the nuances of each of these cases. 

Note that my summary of Rashi for each case below does not delve into specific details. Rashi, in these cases, actually goes through and explains methods for valuation of certain nuanced cases, which help give us a clearer picture. In order to really work with the problems presented in this post, it is important to look at Rashi's explanations in full to help develop the full understanding of valuation for each case.

For Case #1, Rashi on 21:24 and on 21:25 (note all Rashi links are in Hebrew) goes into detail about how the textual meaning is not literal. Rather, in essence, the restitution to be made should be based on valuation of loss of work time and cost of medical recovery. 

For Case #2, Rashi on 21:35 and on 21:36 goes into detail to explain what the division of money, animals, and profit from sale of animals should look like based on different potential specifics.

For Case #3, Rashi on 21:37 explains that the verse is specifically referring only to the animals that are stated in this case. This law is not generalizable to a larger group of other animals.

For a basic understanding of each case, we would look at cases that most closely match the exact situations that are presented. As we test out different changes or adjustments to the possible situations, we are learning the different nuances of each concept.

For students, after learning the specifics of each case, along with Rashi's nuanced explanations, appropriately leveled sample cases could be presented for them to work through and try to determine whether or not one of the case examples applies to the presented sample case, and, if so, what the valuation of damages would be for the case. More straightforward examples could be used with elementary level students, while more complex examples that involve more thought, dissection of information, or research, would be appropriate for middle or high school students.

Some example cases: (Do they apply to our sample cases? If so, what valuation can be made to settle the damages?)
For Case #1 
(Note: These cases would all be better for higher level students, as they all need research regarding valuation of medical damages and potential lost wages.)

1. Two men have an altercation and one of them has a tooth knocked out and a deep cut on his hand.
2. Two teenagers have an altercation and one of them has a baby tooth knocked out, while the other's eye is injured causing permanent blindness.
3. Two men are fighting. One is 40 years old and one is 60 years old. The 60 year old is killed. 

For Case #2

1. One man's $700 ox kills his friend's ox.
2. One man's $700 ox kills his friend's $300 ox.
3. One man's $400 ox kills his friend's $500 ox.
4. One man's ox is known for goring. The goring ox kills an ox worth $500.
5. One man's ox is known for goring. The goring ox is worth $300 and kills another ox.
6. One man's ox is known for goring. The goring ox, worth $600, kills an ox worth $300.

For Case #3

1. A man stole a $600 ox and killed it.
2. A man stole a goat and killed it.
3. A man stole a $300 sheep and it was found alive.
4. A man stole a $700 horse and killed it.
5. A man stole a $600 horse and it was found alive.

These are just a few possible examples. Working together with Rashi's complete explanations, examples can be created for multiple grade levels based on students concrete understanding and their level of inferential understanding of situations as they develop their understanding.

Yitro- Sequencing over time

Rather than dealing with a specific section of the parsha, this week I will look at the parsha as a whole. Parshat Yitro (Shemot 18-20)

Sequencing, or putting items in proper order, is a skill that children begin working on at a preschool level and work their way up to more complex sequencing. Younger students can begin by taking pictures or statements of given related tasks and putting them into the proper order. Older students can be asked to identify the tasks for a given project or series of events in the order in which they happen. They can also be asked to break down a task into individual, minute steps, which may ordinarily be overlooked or assumed.

A basic, younger level example is having pictures of (pictured not necessarily in proper order)

1. a child waking up in the morning
2. a child being dropped of at school
3. a child learning or eating in school with other students
4. a child coming back home from school
5. a child getting into pajamas 

Students can be asked to cut out these 5 pictures and paste them onto a chart numbered 1-5 in order. 

Another, similar level example would be pictures of 

1. a child taking out ingredients to make a sandwich
2. a child setting up the food to make a sandwich
3. a child assembling the sandwich
4. a child eating the sandwich

There are many examples of such picture sequencing projects. As students become more proficient, they can handle a series of pictures that involve more pictures or pictures that require more careful scrutiny in order to identify the proper order. A more complex example for elementary students might be 3 pictures of a child eating, but cues on the table, a clock in the background, or sun and lighting positioning through a window could indicate how to identify breakfast, lunch, and dinner. 

Older students (grades 5-8 and older) can be given a task and asked to break it down into steps themselves. How can you give someone step by step instructions for making a sandwich? For getting ready to go to school? For your daily schedule? Remembering that the minute details are all important is key to the complete picture of the task. Often the clues to proper sequence are found in the minute details.

In this week's parsha, we are given a series of events that happened over a period of time. According to the order in which they are presented in the parsha:

• Yitro brings Tzipporah, Gershom, and Eliezer into the desert to rejoin Moshe
• Yitro brings olah and peace offerings and eats with Aaron and the elders of Israel
• Moshe judges the Israelites in disputes and matters that require judgement & Yitro advises him to appoint elders as judges over segments of the population
• In the 3rd month after the exodus from Egypt, the Israelites arrived in the wilderness & Moshe speaks with Hashem
• Hashem directs Moshe to give the people 3 days to prepare for receiving the Torah
• Moshe directs the Israelites to prepare for receiving the Torah
• Hashem speaks to Moshe
• Moshe relays 10 commandments to the people

Understood at face value, it would appear that if we wanted to put these events on a timeline, the events with Yitro, Moshe and family would be placed within the first 3 months after the exodus from Egypt. They appear to have happened after the exodus, but before the Israelites arrive in the wilderness to prepare for receiving the Torah. 

Rashi, in response to Shemot 18:13, explains that it's not possible that Moshe sat in judgement for the Israelites before they received the Torah- how could he judge for them before they received laws and statutes by which to live? This sequence doesn't make sense, logically. (The details are key!) This means that, minimally, the order of events of the parsha should actually be:

• Yitro brings Tzipporah, Gershom, and Eliezer into the desert to rejoin Moshe
• Yitro brings olah and peace offerings and eats with Aaron and the elders of Israel
• In the 3rd month after the exodus from Egypt, the Israelites arrived in the wilderness & Moshe speaks with Hashem
• Hashem directs Moshe to give the people 3 days to prepare for receiving the Torah
• Moshe directs the Israelites to prepare for receiving the Torah
• Hashem speaks to Moshe
• Moshe relays 10 commandments to the people
• Moshe judges the Israelites in disputes and matters that require judgement & Yitro advises him to appoint elders as judges over segments of the population (note that this event had originally been 3rd in the list)

Note: For a timeline consideration, Rashi indicates that this judgement actually began the day after Yom Kippur in their 2nd year in the desert. 

Rashi also makes reference to the gemara in Avodah Zarah 24a and Zevachim 116a where there is a dispute as to whether the entire section with the events involving Yitro actually took place after the Israelites received the Torah. If that were the case, than rather than the interpretation above, the actual sequencing of events would have been:

• In the 3rd month after the exodus from Egypt, the Israelites arrived in the wilderness & Moshe speaks with Hashem
• Hashem directs Moshe to give the people 3 days to prepare for receiving the Torah
• Moshe directs the Israelites to prepare for receiving the Torah
• Hashem speaks to Moshe
• Moshe relays 10 commandments to the people
• Yitro brings Tzipporah, Gershom, and Eliezer into the desert to rejoin Moshe
• Yitro brings olah and peace offerings and eats with Aaron and the elders of Israel
• Moshe judges the Israelites in disputes and matters that require judgement & Yitro advises him to appoint elders as judges over segments of the population (note that these 3 events had originally been the first 3 events in the list)

In actuality, these events would not even just be moved to the end of the parsha to be put in proper sequence. What follows in the next parshas are itemizations of laws that were given from Hashem to Moshe- the laws that were given before Moshe was able to sit in judgement for the people. So these events would, according to sequence, be moved much farther back, possibly to the end of the book of Shemot, or even to a different book, if we were to look at proper timeline sequencing alone.