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? 

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?

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)

Beshalach- Measurement related activities

"This is the thing that Hashem has commanded, 'Gather from it, for every man according to what he eats- an omer per person- according to the number of your souls, everyone according to whomever is in his tent shall you take." ~Shemot 16:16

"They measured in an omer and whoever took more had nothing extra and whoever took less was not lacking; everyone according to what he eats had they gathered." ~Shemot 16:18

"It happened on the sixth day that they gathered double bread, two omers for each; and all the princes of the assembly came and they told Moshe." ~Shemot 16:22 

Rashi on Shemot 16:22 explains that the Israelites actually collected the same amount when they went out on Fridays, but when they came home to measure how much they had, they found that they had 2 omers.

"The omer is a tenth of an ephah." ~Shemot 16:36

Rashi on Shemot 16:36 explains- 
1 ephah = 3 se'in
1 se'ah = 6 kabin
1 kav = 4 lugin
1 log = 6 beitzim
1 beitzah = the volume of 1 egg
Therefore, a tenth of an ephah is 43 beitzim & 1/5 of a beitzah

(Chart from Artscroll Series, The Sapirstein Edition; Student size edition; Vol 2- Shemos/Exodus pg 198)

Activity Connections:
Activity ideas in increasing conceptual difficulty:

• [Pre-activity 1] Younger students begin learning about measurement by playing with standard and non-standard measurements. Young students may begin by playing around with how many paper-clips long different items are, how many blocks tall each of their friends is (using standard size blocks), or how many classroom cups of water or rice it takes to fill a container. By playing around with these non-standard measurements, they get a sense of how to compare lengths of items without needing to worry about how to read rulers or measuring cups. Following-up on a non-standard activity, students can have a discussion about how you might be able to talk about your non-standard activities with students in other classes. What if their paperclips/blocks/cups are a different size? How could you express your measurements to those students? This helps illustrate for them where standard measurements are needed.
• [Pre-activity 2] Younger students can also have an opportunity to investigate different containers or items that have been pre-measured with standard units for them to compare quart/liter/gallon, inch/foot/yard- or metric equivalents. 
• Connecting to this week's parsha, a pre-measured container could also show the estimated volume of an omer (based on our understanding of the measurement- see below), so that students can see how much food each Israelite was apportioned for each day in the desert.
• Using a pre-measured "Omer Container" students could play with the measurement and see how it compares to our current day measurements. Using investigation- which is bigger- an omer or a quart? an omer or a gallon? How many of the smaller measure does it take to make up the larger measure? Can you express the smaller measure as a fractional amount of the larger measure? These are two different ways of expressing the same equivalency based on the measure you use for a frame of reference (ie 2 cups = 1 pint; 1/2 pint = 1 cup).
• Older students can work on calculating conversions based on the information provided by Rashi in 16:36. How do the different Biblical measures compare to each other? A modern day understanding of the volume of a beitzah is indicated here as 2.53 fl. oz. (or 75 ml). Using this understanding of beitzah, what would each of the other volume measurements be in Rashi's explanation? Ultimately, working through the calculations, what would the volume of an omer have actually been? What would that look like in current day measures? What was the volume of food that the Israelites had for each day's sustenance? What did the volume collected on Fridays (in preparation for Shabbat) look like?

Vayeira- Non-standard Units of Measure

"The water of the skin was finished, and she cast off the boy beneath one of the trees. She went and sat herself at a distance, some bowshots [away], for she said, 'Let me not see the death of the child.' And she sat at a distance, lifted her voice, and wept." ~Bereishit 21;15-16

Non-Standard Units of Measure:
We are all familiar with standard units that we use to measure on a regular basis- we measure length with inches, feet, miles, centimeters, meters, kilometers; we measure liquids with cups, pints, quarts, gallons; we measure weight with ounces, pounds, grams, kilograms. Sometimes, though, the standard units that we use either aren't available or aren't necessary for the purposes of what you're doing.

When children are first learning about measuring, a common activity is to choose a standard by which to measure different items, and then students can compare the items based on their measurements. For example, they may use stacking cubes (all of the same size) to measure items in their surroundings- a pencil, a chair, a water bottle, etc. They can record their findings and then compare which items are shorter and which items are longer (or taller). As long as the same standard is used throughout an activity, all the items measured can be compared to each other.

Anything that you choose can be used as a non-standard unit for measuring- a paper clip, a pencil, a shoe- as long as the repeated measurement is using an item (or items) that are the same exact length. So, you can't use a mixture of large and small paper clips to measure items, but you can choose one or the other. You can see how many water bottles it takes to fill up a pitcher, but you need to fill the water bottles full each time to establish a temporary standard for your measurements. 

If you want, or need, you can always convert your non-standard measurements into a standard unit afterwards by measuring your non-standard unit. If your paperclip is 3/4 of an inch long, you have enough information to convert your measurements into inches. If your water bottle holds 2/3 C of water, then you can calculate how many cups your pitcher holds.

Parsha Connection:
In this week's parsha, we learn of Hagar and Ishmael's wandering after they are sent away from Avraham and Sarah's home. We are told that when Hagar believes that Ishmael is dying, she sits "some bowshots" from him, so as not to have to watch him die. What does this mean practically, for our understanding of the distance between Hagar and Ishmael at this time?

Bereishit Rabbah 53;13 explains two aspects of this in order for us to understand the distance:
1) first it explains that since it is written "bowshots", plural, but with no specific quantity, it must mean that the minimum plural is implied. Since the smallest plural number is 2, therefore it's 2 bowshots away.
2) second it explains that a standard bowshot during this time was 2,000 cubits, or a little more than today's measure of 1/2 a mile.

Based on this explanation, we can use the parsha's non-standard measurement of a bowshot to calculate that, by current day measurements, Hagar was approximately a distance of a mile away (or just over a mile) from Ishmael during this time.

Everyday Connection:
Let's say you have a recipe for a sauce that calls for equal parts of each ingredient. You don't need to measure exactly with a 1 cup measure- if you have a cup without measurement markings, you can still use that cup to measure out your ingredients. All you need to do is make sure that you measure each ingredient to the same point on the cup, and you'll have equal parts of each ingredient.

What if you need to measure the length of an item, but you don't have a ruler nearby? You can use a pencil, a piece of paper, or even your shoe- count how many "shoes" long your item is, and then you can measure the length of your shoe later to convert your measurement to a standard unit.

Pikudei- Mixed Measures

"All the gold that was used for the work- for all the labor of the Sanctuary- the offered-up gold was twenty-nine kikar and seven hundred thirty shekel, in the shekel of the Sanctuary. The silver of the accountings of the assembly, a hundred kikar, one thousand seven hundred seventy-five shekel, in the shekel of the Sanctuary." ~Shemot 38;24-25

"The offered-up copper was seventy kikar and two thousand four hundred shekel." ~Shemot 38;29

Rashi on 38;24 explains that a kikar of the Sanctuary is three thousand shekalim, and, therefore, the individual shekalim that are listed are mentioned because together they don't add up to one kikar.

What is a "mixed measure"?:
Understanding how to measure with standard measurements is a basic skill that students begin practicing with from an early age. How many feet long is the room? How many cups of water are in the pitcher? But, what happens when you don't have enough to complete an exact measurement with the measure that you're using? What if the room is between 10 and 11 feet long? What if the pitcher has between 5 and 6 cups of water? Sometimes we use fractions to indicate these middle measures. However, we also have other, smaller measurements that can be used to measure these partial portions. We have a standard equivalency that 12 inches are equivalent to 1 foot, for example. So, if my room is between 10 and 11 feet long, I can measure the "extra" length using inches. Maybe my room is 10 feet and 3 inches. This type of measurement is known as a mixed measure. To simplify a measurement out of a mixed measure, we could convert the mixed measure into the larger measurement using a mixed number (a whole number with a fraction) or decimal, or we could convert the mixed measure into the smaller measurement. With our room example, 10 ft 3 in would be 10 1/4 ft or 10.25 ft when measured only in feet, or 123 inches when measured only in inches. For younger students, the calculation for converting the mixed measure into the smaller unit (from feet and inches to just inches) is an easier one and also helps to give them a better sense of the full meaning of the number, where mixed measures and mixed numbers are often harder for them to visualize and fully understand.

Let's calculate:
So how do we convert to the smaller measure unit? Let's use our 10 ft 3 in room example. We know that 1 foot is made up of 12 inches. So, we multiply 10 x 12 (10 feet, with 12 inches in each) = 120 inches. Then, we add the extra 3 inches that were left over, so 120 + 3 = 123 inches.

What does this have to do with the parsha?:
In this week's parsha, Rashi helps us understand that the weight measurements that we're given for the gold, silver, and copper that were donated to the mishkan are mixed measures using kikar and shekalim. Rashi explains that 1 kikar = 3,000 shekalim.

The amounts that we are given in the parsha are:
Gold: 29 kikar and 730 shekel
Silver: 100 kikar and 1,775 shekel
Copper: 70 kikar and 2,400 shekel

How can we convert these to all shekalim measures?
We know that 1 kikar has 3,000 shekalim, so:
Gold: 29 x 3,000 (29 kikar, with 3,000 shekalim in each) = 87,000 + 730 shekel = 87,730 shekel
Silver: 100 x 3,000 (100 kikar, with 3,000 shekalim in each) = 300,000 + 1,775 shekel = 301,775 shekel
Copper: 70 x 3,000 (70 kikar, with 3,000 shekalim in each) = 210,000 + 2,400 shekel = 212,400 shekel

So, our final calculations after converting from mixed measures to just shekalim are:
Gold: 87,730 shekel
Silver: 301,775 shekel
Copper: 212,400 shekel

A further thought:
The silver calculation is particularly meaningful, since we are told "a beka for every head, a half-shekel in the shekel of the Sanctuary for everyone who passed the counters, from twenty years of age and up, for the six hundred three thousand, five hundred fifty." (28;26) So, our silver calculation should be accurate for 603,550 men 20 years of age and older having each given a 1/2 shekel. Does this work? 
301,775 shekel of silver x 2 (two 1/2-shekel pieces for each shekel) = 603,550 1/2-shekel pieces.
So, it seems that there isn't even estimation here, but rather the calculations work exactly!

Vayakhel- A multi-level hands-on activity

"'Every wise-hearted person among you shall come and make everything that Hashem has commanded: the Tabernacle-spread, its Tent-spread, and its Cover, its hooks, its beams, its bars, its pillars, and its sockets; the Aron and its staves, the Lid, the Parochet screen; the Shulchan, and its staves, and all its implements, and the Bread of Surfaces; and the Menorah of illumination, and its implements, and its lamps, and the oil of illumination; and the Mizbe'ach for the Incense and its staves; and the anointing oil and the incense spices; and the entrance screen for the entrance of the Mishkan; the Mizbe'ach of the olah-offering and its copper lattice, its staves, and all its implements; the Kiyyor and its pedestal; the curtains of the Courtyard, its pillars, and its sockets, and the screen of the gate of the Courtyard; the pegs of the Mishkan, and the pegs of the Courtyard, and their cords; the mesh garments of the holy for Aaron the Kohen and the garments of his sons, to be Kohanim.'" ~Shemot 35;10-19

The above passage gives the list of items which the Jewish people were instructed to build as part of the Mishkan for worship in the desert. What follows afterwards in Shemot 36;8-38;20 are detailed descriptions for the building of the Mishkan and all its contents.

Note: Whereas most weeks I try to provide some added insight or understanding into the parsha through my mathematical investigation, this week's topic is an overview for an academic unit based on the information provided in the parsha. The students completing such a unit would certainly gain a deeper insight and understanding of the mishkan and it's contents. 

The major substantive part of this week's parsha is very specific blueprints for building the mishkan and it's contents- the aron, mizbeach, kiyyor, menorah, etc. 

An outline for a hands-on learning opportunity for multiple levels:
Older students can use the details to draw a scale blueprint. This would include:
--conversion from Torah measurements to modern day measurements
--adjusting measurements to a scale proportion
--accuracy of measurement and drawing in their blueprint
--attention to detailed descriptions provided in the text
Older students can create a list of required supplies and materials to build a scale model. This would include:
--accounting for amounts of each material needed based on length, width, height, area, etc. of where and how each material is used
--computation of conversions between the blueprint and model sizes
--being mindful of economic use of materials and trying to calculate to take only what's needed and use materials in such a way that minimal scrap is left over
Younger students can use the blueprint and/or text to help create materials lists. This would include:
--listing the different parts that are to be created
--listing the different materials that are used/needed
--counting out items, such as rings for attaching panels, to determine how many are needed all together throughout the project
Students of all ages can work together on building the model. This would include:
--measuring out and cutting materials to size
--following the blueprints of the text and that created by the student to accurately build the mishkan and it's contents.