Chanukah Activities- Chag Sameach

When I first stopped posting weekly, I had intended to post holiday ideas a few times a year, as the chagim came up. Although life has gotten in the way, since I'm planning some activities for my classrooms, I thought I'd quickly share what I have. Chag Chanukah Sameach!

Activity 1:
For younger grades, I always love playing dreidel with the students and having them track and graph the results of their spins as bar graphs. If the students play for long enough, their results should approximate even results for all 4 letters. You can combine data from students in the class to have them see how the comparisons change and the results (usually) even out across the letters as they combine the data from more people. 
**For younger grades to visualize the graph, you can have the students cut out their bar graphs from their own papers and tape them together on the wall or board to quickly see the growth of adding together the data without having them tediously count and add the bar graphs. 
**For older students, they can calculate statistics for the results of each letter and compare how the statistics change when they combine data from classmates together.

Activity 2:
In my Algebra 1 class, we recently began graphing, distinguishing between discrete and continuous graphs, identifying restricted graphs, identifying domains, ranges, and graph intercepts. With these topics in mind, I created the following worksheet with practical application problem solving for my classes.

Chanukah Algebra

Question 1:

Thinking about Discrete vs Continuous-

Graphs of candles vs oil used over Chanukah.

1)    What would be the difference in a graph of the number of candles used over time or by number of people and the amount of oil used over time or by number of people?

2)    Assuming that 1 oz of oil burns for the minimum zman, make an argument for why the graphs of candles and oil should actually be similar.

a.     Can you make a counter-argument for why the graphs should still be different?

Question 2:

For your doughnut sale, you’re charging $1.50 for jelly doughnuts and $1.25 for glazed doughnuts. Assuming you sold $375 worth of doughnuts, draw the graph of your possible sales. Use the following questions to help guide you in creating your graph.

1)    The equation that represents your sales would be: __________________________

2)    What are the two intercepts of your graph? 

3)    What do they each represent? 

Question 3:

For your Chanukah carnival, each student is allowed 3 turns at a booth for each ticket that they have.

1)    Write a function for the number of turns that a student gets based on the number of tickets they have.

2)    Draw a graph of this function.

3)    What is the domain, if each student is allowed no more than 10 tickets? (use the table below to organize your answers)

4)    What is the range, given this limit on the number of tickets? (use the table below to organize your answers)

__________
__________

Question 4:

You’re lighting an oil menorah this year, which takes 1 oz of oil for each light. Assuming you use this menorah for all 8 nights, and you use a regular candle for the shamash-

1)    How many ounces of oil will you need for the full chag?

2)    You want to buy oil for a number of people to be able to light menorah for the whole week. Write a function for how much oil (in oz) you will need based on the number of people you give oil to.

3)    You bought the oil in 4 qt bottles (160 oz) and were able to buy 18 bottles, which gives you 2,880 oz of oil. What is the most number of people you can give oil to for them to have for the whole week?

4)    What are the domain and range for this situation? 

5)    Draw a graph to represent the possible number of people who you could provide oil to and how much oil you will have given out.

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?