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".