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.

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?

Shmini- Sharing portions

"Moshe spoke to Aaron and to Elazar and Ithamar, his remaining sons, 'Take the meal-offering that is left from the fire-offerings of Hashem, and eat it unleavened near the Mizbe'ach; for it is that which is holy of the highest degree. You shall eat it in a holy place, for it is your portion and the portion of your sons from the fire-offerings of Hashem, for so have I been commanded. And the breast of the waving and the thigh of the raising you are to eat in a pure place, you and your sons and your daughters with you; for they have been given as your portion and the portion of your sons from the sacrifices of the peace-offerings of the Children of Israel. They are to bring the thigh of the raising and the breast of the waving upon the fire-offering fats to wave as a wave service before Hashem; and it shall be for you and your sons with you for an eternal decree, as Hashem has commanded.'" ~Vayikra 10:12-15

In the above section of this week's parsha, Moshe explains to Aaron about how the meal-offering (mincha-מנחה), the wave offering (tenupha-תנופה), and the raise offering (terumah-תרומה) are divided amongst Aaron and his children. Rashi discusses the apportioning on 10:13 and 10:14, where there is differentiation between the mincha, which is the holiest, and the tenupha and terumah, which must be eaten in a pure place.

If we look at the comparison of these different sacrifices, we see that the mincha is to be apportioned between Aaron and his sons, with no mention of daughters at all. Regarding Tenupha and terumah, however, Moshe mentions both the sons and daughters eating along with Aaron, but he mentions portions for only Aaron and his sons. Rashi explains these differences in wording to mean that the mincha should only be portioned for and eaten by Aaron and his sons- his daughters may not partake. For tenupha and terumah, the portions are again only to be divided between Aaron and his sons, but for these, when they are eating their portions, they may share their portions with their daughters and wives. 

Practically speaking, what does this mean? For ease of calculating, let's think about the portions of sacrifices based on made up weights that would be divided between Aaron and his sons. Before we begin, we also need to know about the composition of Aaron's immediate family, to the best of our knowledge. 

*Aaron was married to Elisheva
*Their 2 oldest sons, Nadav & Avihu, were killed at the beginning of this chapter for bringing a fire onto the altar without checking with Moshe.
*Aaron and Elisheva's son Elazar was married
*Elazar and his wife had a son Pinchas
*Aaron and Elisheva's 4th son was Ithamar

So, in sum, there were 4 Kohanim (priests)- Aaron, Elazar, Ithamar, and Pinchas. Then, there were 2 women in the immediate family- Elisheva and Elazar's wife.

Now let's look at a sample situation to understand the calculations. Let's assume that there's 8 lbs. of each sacrifice to be apportioned. 

Mincha- This will be divided evenly between the 4 Kohanim, and each Kohen will have 2 lbs. (8 lbs. ÷ 4 Kohanim = 2 lbs. per Kohen). Each Kohen will also eat the full 2 lbs. on their own.

Tenupha & Terumah- This will also be divided evenly between the 4 Kohamin, and each Kohen will, again, have 2 lbs. (same calculation as above). However, they will not each have 2 lbs. of each sacrifice to eat.
*Aaron will share his 2 lbs. with Elisheva. So, if divided evenly, Aaron & Elisheva will each have 1 lb. of tenupha and 1 lb. of terumah.
*Elazar will share his 2 lbs. with his wife. So, as with Aaron, Elazar and his wife will each have 1 lb. of each of the 2 sacrifices.
*Pinchas and Ithamar having no family at this point, would each eat their entire portions of 2 lbs. of each sacrifice.

If we think about this situation related to fractions, we would say that each sacrifice was divided the same, and each Kohen received 1/4 of the sacrifice. When it came to eating the sacrifices, though, while each Kohen ate 1/4 of the mincha, this was not the case for the tenupha & terumah. Let's think about those fractions:
*Aaron had 1/4 of each, which he shared with Elisheva- What is half of 1/4? If we take our 1/4 pieces and break them each into half, we would have 8 pieces out of our original 4, so Aaron's 1/4 was the same as 2/8, and he and Elisheva each had 1/8 of each the tenupha & terumah.
*Comparably, Elazar and his wife also each had 1/8 of each of these 2 sacrifices.
*As with the portion weights sampled above, Pinchas and Ithamar each ate their entire 1/4 of each of these sacrifices.

To help students internalize these calculations (by weights and/or by fractions), we could have them personalize these calculations by applying them to their own family compositions. How much of each of the 3 sacrifices would the people in their own families receive? After they calculate for their own families, they can compare calculations with their peers, to extend their thinking again to a third sample size.

Tzav- Standard vs. Non-Standard Measurements

In last week's parsha and again in this week's parsha, we have reference to a non-standard unit of measurement, referred to as threefingersful (קמצו).

Regarding a person bringing a meal-offering:

"He shall bring it to the sons of Aaron, the Kohanim, and he shall scoop from there his full threefingersful, of its fine flour and of its oil, as well as all its frankincense; and the Kohen shall cause its memorial portion to go up in smoke upon the Mizbe'ach- a fire-offering, a satisfying aroma to Hashem." ~Vayikra (Parshat Vayikra) 2:2 

(Note the sefaria.org link uses the term "handful" in place of "threefingersful")

Rashi on 2:2 explains that "his full threefingersful" means the amount that fits in his palm and can be held onto within the space of his 3 fingers (not including the thumb or the little finger). This amount should fill up the entire space enclosed in between those 3 fingers and the palm, but it should not be overflowing.

Again regarding a meal-offering- this time brought by a Kohen:

"And he shall separate from it in his threefingersful from the fine flour of the meal-offering and from its oil, and all of the frankincense that is on the meal-offering; and he shall cause them to go up in smoke on the Mizbe'ach for a satisfying aroma- its memorial portion unto Hashem." ~Vayikra 6:8

Rashi on 6:8 explains that the Kohen should not make a measure, i.e., measuring cup, for the threefingersful. In other words, he needs to measure it directly in his hands, without using a specified measuring utensil.

For my first discussion of non-standard units of measure, you can look back to Parshat Vayeira where I discussed measuring the distance that Hagar maintained between herself and Ishmael. 

In addition to non-standard units, we also find reference to a standard Torah measure in this week's parsha.

"Hashem spoke to Moshe, saying: 'This is the offering of Aaron and his sons, which each shall offer to Hashem on the day he is inaugurated: a tenth of an ephah of fine flour as a meal-offering; continually, half of it in the morning and half of it in the afternoon. It should be made on a pan with oil, scalded shall you bring it; repeated bakings, a meal-offering of crumbs, you shall offer it as a satisfying aroma to Hashem..." ~Vayikra 6:12-14

 If you look back to Parshat Beshalach, the volume of 1/10 ephah was broken down into a chart and compared with equivalent measures.

Activities on Understanding the Difference between Standard & Non-Standard Measurements:
*Students can test out how big their own threefingersful are. With a big bucket of rice in the center of a table, students can test how many threefingersful it takes them to fill up (or fill to a marked spot) a small container or cup. They can each record their own results and then compare their results with their classmates. Having a teacher also model the activity will be helpful so that they have results from a larger hand to also compare to their own. Note that using something like rice, rather than flour or sand will be less stressful for younger fingers still developing their fine motor skills. At the same time, rice will offer more of the same feeling of filling up the space in their palm as opposed to larger items like beads or small blocks. 

*At another table, students can practice measuring with a container that is pre-measured to 1/10 ephah. They can use preset measurements to see how many of those measurements fill up a 1/10 ephah container. They could also see how many 1/10 ephah measurements it takes to fill various other containers. For each activity, students can record and track their findings to compare with the findings of their classmates.

*A follow-up discussion of the similarities and differences between measuring with standard & non-standard units will help students understand the bigger conceptual picture by sharing their thoughts and findings with each other. Some questions to consider: [Note that for younger students you may choose to limit the questions, and for older students you can push their thought process further. If students seem to be understanding the progression of concepts, you can push their understanding further by continuing the series of follow-up questions.]

• Did most people have similar or different amounts for using threefingersful to fill a cup? Did it take the teacher more or fewer "scoops" to fill their cup? Why? What does this mean about the hand sizes of people who took more "scoops" than others? people who took fewer "scoops" than others? Test out your thoughts by comparing hand sizes. 
• Did most people have similar or different amounts for the measurements using 1/10 ephah? Did the teacher have the same amounts as the students? Why do you think these results are different from the threefingersful investigation?
• Thinking about your findings, what is the biggest difference between using standard measurements vs. non-standard measurements? Can you think of examples when it would be better to use one type of measurement over the other? What times might it be ok to use non-standard measurements? Can you think of non-standard measurements that could be used and still have everyone get the same results? (think about using paperclips or blocks, for example)

Tzav- Statistics part 2

Once again, this week we have a significant portion of the parsha dedicated to specifics of korbanot, or sacrificial offerings. Again, rather than quoting from the parsha, I've organized some of the information related to the offerings in order to analyze.

Pie Chart/Circle Graph:
Whereas the bar graphs we looked at last week compared a measured amount between different categories, a circle graph compares portion sizes of categories that together make up 100% of something. For example, out of your total budget allotment, what percentage of the budget goes to different categories (food, medical, housing costs, etc.)? Out of all after school enrollment, what percentage of students go to each of the different activities? Out of the total hours in a day, what percentage of time is spent sleeping, eating, working, etc.? The key to circle graphs is that all categories together add to 100% and create a complete circle. 

When analyzing data to convert into a circle graph, we'll need our knowledge of fractions (Parshat Vayishlach), proportions (Parshat Ki Tisa), and percentages (Parshat Vayigash) to convert the raw data into a format that will fit the circle graph.

Connection to Parsha:
Let's take a look at a sample data set from this week's parsha to walk us through the process. Let's look at what percentage of sacrifices were to be fully burned and what percentage of sacrifices were allowed to be partially eaten in some form.

We can start by making a chart to organize the information so that we see which category each sacrifice falls into.

Next, we use the chart to count up how many sacrifices are in each category out of the total number of sacrifices. According to our chart, out of the 6 total sacrifices, we have 3 sacrifices that are completely burned and 3 sacrifices that are partially burned and partially eaten.

Let's restate this as fractions- 
3/6 = burned 
3/6 = burned then eaten

If we want, we can reduce these fractions (each reduces to 1/2), but we don't have to. In this case, we will reduce them, since it will make our next calculation easier to work with. 

So, we have-
1/2 of the sacrifices are burned
1/2 of the sacrifices are burned then eaten

Now, remember that the most important part of a circle graph is that it shows a percentage out of 100%, where 100% (or all) of what you're analyzing makes up the full circle. So, if 100% of the sacrifices is all of them, we know that 1/2 of the sacrifices were burned: 100% x 1/2 = 50% and 1/2 of the sacrifices were burned then eaten: 100% x 1/2 = 50%. So now we have calculated that 50% of the sacrifices were totally burned and 50% were burned then eaten. Once we have this calculation, we can create our graph. 

While I made the graph electronically, you can also construct a circle and measure out the angle measures for each section using pencil, paper, drawing compass, and protractor. Such an activity offers a good opportunity for art/geometry integrations in math. To calculate the sections by hand, we need to first know a full circle is 360°. Since we need 1/2 of the circle for each category, we multiply 360° x 1/2 = 180°. This means that the angle measure for each section of the circle needs to be 180°. Measuring out this 180°, you'll find that you've perfectly divided the circle into two 50% sections, one for each category.

It happened that the numbers for this activity fell perfectly into two equal groups. While it may seem simplistic because of the easy numbers and category divisions, it would make a very simple way to practice going through the steps of the process and practicing drawing circles and interior angles. Once practicing with this easy example, another problem with more complex category divisions could be introduced.

Vayikra- Statistics part 1

About the Parsha:
In Parshat Vayikra we have 5 chapters that just list different types of sacrificial offerings that can be brought. Rather than quoting sections of the parsha, I've reduced the information into the summary below:

olah-offering-- male cattle
olah-offering--male sheep or goat
olah-offering--turtledoves or young doves
meal-offering--fine flour, oil, frankincense

meal-offering--first fruits

peace-offering--male or female cattle

peace-offering-- male or female from flock (sheep or goat)

Kohen's sin-offering-- young male cattle

Community sin-offering-- young bull (male cattle)

Ruler sin-offering-- male goat

individual sin-offering-- female goat or sheep

guilt-offering--female sheep or goat

guilt-offering-- two turtledoves or two young doves (if can't afford sheep/goat)

guilt-offering-- flour (if can't afford doves)

guilt-offering-- ram

What is a graph?
A graph is a visual representation of information in order to easily compare information in a given situation. There are many types of graphs- line graphs, bar graphs, pie charts, histograms, venn diagrams- each type of graph is suited to a certain type of information and a certain format. A pie chart, for example, gives information about the division of categories within a total group, or sectional percentages out of 100%.

For the next few weeks, I'd like to look at different types of graphs and statistical analysis of information that's given in the parshiot.

For this week, we'll look at bar graphs. Bar graphs are graphs that are used to compare categories. A vertical bar graph has the names of the categories on the bottom of the graph, and the left side of the graph has the number scale to show how much is measured in each category. A horizontal bar graph can display the same exact information, but the graph is turned sideways- the names of categories are on the left of the graph, and the number scale is along the bottom of the graph.

It's interesting to look at different ways that the same information can be graphed- each graph bringing out different aspects of the information.

Let's first make a graph showing how many options there are for each type of offering. First we'll organize our data:

*olah-offering: (3 options)
--cattle
--sheep/goat
--doves
*meal-offering: (2 options)
--fine flour mixture
--first fruits mixture
*peace-offering: (2 options)
--cattle
--sheep/goat
*sin-offering: (4 options)
--Kohen (cattle)
--community (cattle)
--ruler (goat/sheep)
--individual (goat/sheep)
*guilt-offering: (4 options)
--sheep/goat
--doves
--flour
--ram

Now that we have our data organized the way we want it, we can graph it (chart on top, graph underneath):

Number of Options for Sacrifice Types:

Now, what if we make a graph of how many must be male animals, must be female animals, could be either male or female, or are grain/fruit?

Again, let's first organize our data:

*male animals: (8 options)

--olah-offering (x3)
--sin-offerings (x3)
--guilt-offering (x2)
*female animals: (2 options)
--sin-offering (x1)
--guilt-offering (x1)
*male or female: (2 options)
--peace-offering (x2)
*grain/fruit: (3 options)
--meal-offering (x2)
--guilt-offering (x1)

Now that we have our data organized the way we want it, we can graph it (chart on top, graph underneath):

Number of Sacrifices by Animal Type:

So, here we have two very different graphs, both based on the same exact data set, but focusing on different aspects of the information; one looks at which sacrifices have the most options for what can be offered for that type of sacrifice, and the other looks at how many sacrifice types each animal can be used for. 

Some good questions for testing for understanding of graphs might be: Which category has the most options? Which category has the least? Are there any that have the same amount? How much difference is there between two given categories? These questions can be tailored for each specific graph that you're reading.