"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.
"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?
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?
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.
"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.
