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)
"Hashem spoke to Moshe, saying: If a person will commit a misuse, and sins unintentionally against Hashem's holies, he shall bring his guilt-offering to Hashem, an unblemished ram from the flock, with a value of silver shekels, according to the sacred shekel, for a guilt-offering. For what he has deprived the Sanctuary he shall make restitution, and add a fifth to it, and he shall give it to the Kohen; then the Kohen shall provide him atonement with the ram of the guilt-offering and it shall be forgiven him." ~Vayikra 5:14-16
In this section of this week's parsha, we learn about a person who is required to bring a monetary restitution for an unintentional sin of misusing an item in the Tabernacle. His restitution, in addition to bringing a sacrifice of a certain value, is to also repay the value of the damages done, plus 1/5 of the value.
In the Sapirstein Edition of the Artscroll Vayikra (Torah with Rashi's Commentary), there is a footnote on Rashi 5:16 ("For what he has deprived the sanctuary he shall make restitution") that explains that when the Torah speaks of 1/5, it is really referring to 1/5 of the final total. In other words, the amount that was added is 1/5 of the final total after adding in that additional piece.
What does this mean when thinking of fractions as we know them? If we want to have a section be 1/5 of the total, it means that before that piece was added, there were only 4 of those identically sized pieces. In other words, the original whole was cut into 4 quarters. Then, another 1 piece of equal size to the other 4 pieces is added on. Now, the final added piece is 1/5 of the entire new whole.
Why might this be so confusing to understand?
Think of an optical illusion type of artwork that you may have seen- maybe this well known image where, from one angle, you see an old woman, and from a different angle you see a young woman. It is all a matter of perspective. If you look at one set of features in a certain way, then you can see the old woman; if you focus on a different set of features slightly differently, then you see the young woman.
The fraction situation that we have in our parsha this week can also be confusing to learners nowadays because of perspective. When we teach and think about fractions nowadays, in most cases, we think about our original value as one whole unit. If someone told us to add 1/5, we would assume that we needed to break our original unit into fifths (ie 5 even pieces) and then add on 1 additional piece of the same size. Alternatively, in the Torah and rabbinical writings through the Talmudic period, the entire unit was determined after the final additional amount was added on. Therefore, the 1/5 that was added on was 1/5 of the final unit, and in order to determine the size of that unit, the smaller segment actually needed to be broken into quarters.
Let's look at an example to see how this would differ, practically:
For ease of calculation, let's take a starting value of $100 restitution that's owed. I will break down the case, for the same starting value, first with our current day perspective, and then with the Torah perspective.
Current Day Perspective:
Starting value owed: $100
We need to add 1/5 of the starting value: 1/5 of $100 is $20 (100 ÷ 5 = 20).
The total value to be paid is the original value + 1/5, or $100 + $20 = $120
Torah Perspective:
Starting value owed: $100
We need to add an amount in order to have that amount be 1/5 of the final value. Remember, I noted above that if you have 4 pieces of equal size, and add one more that is the same size, now you will have 5 pieces of equal size and that last piece that you added will be 1/5 of the total final value.
So, we need to first figure out 1/4 of the starting value: 1/4 of $100 is $25 (100 ÷ 4 = 25).
When we add $100 + $25 = $125. We can check to confirm that $25 is 1/5 of $125 (125 ÷ 5 = 25).
Since this calculation does work out, we have confirmed that the total value to be paid is the $125.
If we compare these two sample calculations, we can see that, according to our calculations, the Torah Perspective actually turns out to be more costly for the sinner than the Current Day Perspective does, and the calculation process is slightly more involved (although maybe not if that's how you're used to calculating!).
Some questions to keep you thinking about fractions:
Does this perspective difference in the understanding of 1/5 always result in the Torah calculation being more costly than the current day calculation? Does it change if you use a smaller original value? a larger original value? What if you used different fractions and compared them in current day vs. Torah perspective?
[Note that with an understanding of what a fraction is and represents, these questions are developmentally appropriate for students in younger elementary grades and upwards. The approach to finding the answers would vary based on other skill levels and developmental understanding of the concepts needed to think about the questions and make sense of them.]
Note that I have a collection of various quotes this week from the double-parsha. The samples that I've chosen are just a couple examples of different types of information that are given and would be needed for the activity suggestions that I make below.
Samples of details from Parshat Vayakhel:
"He made fifty golden hooks and attached the panels one to the other with the hooks- so the Tabernacle-spread became one." ~Shemot 36:13
"He made fifty copper hooks to attach the Tent-spread so that it would be one." ~Shemot 36:18
Details of accountings of gold, silver, and copper that were collected:
"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." ~Shemot 38:24
"The silver of the accountings of the assembly, a hundred kikar, one thousand shekel of the Sanctuary;" ~Shemot 38:25
"The hundred kikar of silver were to pour the sockets of the Sanctity and the sockets of the Parochet; a hundred sockets for a hundred kikar, a kikar per socket. And from the one thousand seven hundred seventy-five he made hooks for the pillars, and covered their tops and belted them." ~Shemot 38:27-28
"The offered-up copper was seventy kikar and two thousand four hundred shekel. With it he made the sockets of the entrance to the Tent of Meeting, and the copper Mizbe'ach, the copper lattice that was on it, and all the implements of the Mizbe'ach; the sockets of the Courtyard all around, the sockets of the gate of the Courtyard, all the pegs of the Mishkan, and all the pegs of the Courtyard, all around." ~Shemot 38:29-31
Explanations of Rashi:
On Shemot 38:27
Rashi itemizes the 100 silver sockets that are referred to. He explains that there were 48 beams, each with 2 sockets [48 x 2 = 96]. Then, there were 4 sockets on the parochet. 96 + 4 =100. Aside from these 100 silver sockets, the blueprints indicate that all other sockets were copper.
On Shemot 38:28
Rashi explains that the remaining 1,775 shekalim of silver was used to cover the tops and bands of the pillars.
This week's double parsha begins with Parshat Vayakhel (link to last year's post) reviewing all the specifics of how the mishkan was built. This section is actually an exact review of the same blueprints that were given in Parshat Terumah. What's the difference between the two sections? In Terumah we are told that this is how the mishkan is to be built, and in Vayakhel we are told that this is how the mishkan was built. So, we are basically being told that Betzalel, Aholiav, and the other wise-hearted men who worked together built the mishkan according to the exact specifications that were given to Moshe.
Our double parsha continues with Parshat Pikudei, where we have an accounting of the gold, silver, and copper that was donated for the building of the mishkan. In last year's post, I first began by calculating the unit conversions to see how much gold, silver, and copper was collected when measuring with just the unit of shekel, rather than the mixed measure amounts that were given in Kikars and Shekels. At the end, I followed Rashi's example, and I finished off the post by working through the calculation to confirm that the amount of silver that was donated did indeed match up with the census population that was calculated.
The two parshas come together to give us a complete view of the materials used, and how they were divided up to build the mishkan, down to the smallest details.
Sometimes information is organized in one way for one purpose, but needs to be reorganized for another purpose. When the blueprints are given for the creation of the mishkan, the details are organized based on the way that the information would be most understandable when actually putting together each piece of each item that was being built. However, when figuring out how much of each material should be apportioned for each piece of the mishkan, the details from the blueprints should be reorganized according to the materials from which each item was to be made.
This activity of reorganizing the given information in order to have a better vision of the breakdown based on materials used for the project fits well into the activity I have mentioned before, that of organizing, mapping, and creating a model mishkan based on the blueprints.
This activity can be done on a larger scale, including the aspects of:
*a full start-to-finish organization of all the information related to building the mishkan
*creating blueprints from the information provided
*building a complete scale model mishkan
This activity could also be done on a smaller scale, by:
*choosing only a single item
*mapping out what materials would be needed for the chosen item
*creating blueprints for the chosen item
*possibly even building a scale model of the chosen item
Choosing as your activity to try to align the building information in Parshat Vayakhel directly with the detailed materials lists that are given in Parshat Pikudei would require organization of all the information given regarding building the mishkan, even if you're ultimately only interested in looking at the information for a constructing a single item. The reason for this is that the amount of gold, silver, and copper was to be divided based on what was donated to the construction. In order to divide up the materials to be used for the items, you need to account for all of the items and detailed pieces that needed to use each material. For example, from the pasuk and from Rashi above (38:27), we know that the 100 kikar of silver was used to make the 100 silver sockets, so there was 1 kikar of silver for each socket. For the remaining silver, however, there is a list of the remaining items amongst which it was to be divided, so additional calculation, thought, and possibly estimation would need to go into figuring out how many shekalim of silver went into each of the remaining items.
Real-Life Example:
Think about the sample situation of cooking for and hosting a dinner party. In order to host a dinner party, you will need to plan out your menu. When you plan a menu, you choose dishes to serve for each course. In order to plan the dishes that you are serving, you think about how the completed, cooked foods will go together, how they'll complement each other, and what order you will serve them in. At this point in the planning, you don't think about specific details of the ingredients, unless you are keeping in mind to specifically include or avoid certain individual ingredients. Even then, your focus is primarily on the final outcome of each cooked dish.
Once you have your menu planned, you need to start thinking about actually preparing and cooking the foods. In order to cook the foods, you need to go make sure that you have all of the ingredients. When you calculate to see what you have in stock and what you need to buy, it doesn't matter which 3 recipes call for orange juice, it only matters how much total orange juice you need in order to cook everything and whether or not you have that total amount of orange juice in stock. When you go to the store, it's not important how many carrots are in the soup and the side-dishes individually, or how many eggs you need first for the side-dishes and then for the desserts. When you walk into the store, you want to know how many carrots you need all together, and how many eggs you need all together.
The reorganization of information from individual recipes into a compilation of total ingredients needed in order to do all your cooking is exactly the same as the reorganization of information that I mention about in thinking about planning and dividing specific materials in order to build a model (or the real life) mishkan.
"Hashem spoke to Moshe, saying: 'When you will take a census of the Children of Israel according to their counts, every man shall give Hashem an atonement for his soul when counting them, and there will be no plague among them when counting them. This is what they shall give- everyone who passes among the counted- half of the shekel, by the holy shekel, the shekel is twenty geirah, half of the shekel as a portion to Hashem. Everyone who passes among the counted, from the age of twenty years and up, shall give the portion of Hashem. The wealthy shall not increase and the destitute shall not decrease from half of the shekel- to give the portion of Hashem, to atone for your souls. You shall take the silver of the atonements from the Children of Israel and give it for the work of the Tent of Meeting; and it shall be a remembrance before Hashem for the Children of Israel, to atone for your souls.'" ~Shemot 30:11-16
One-to-One Correspondence Activity:
[K-2 concept]
In this week's parsha, we learn of Hashem telling Moshe that at such time as he chooses to take a census- a counting of the number of people, either total Children of Israel or a portion of the population- he should not just count the people. Rather, he should have each person being counted give a half-shekel as a donation to the Holy Tabernacle. It explicitly states that a rich person should not give more, and a poor person should not give less; any person being counted must give exactly a half-shekel.
Socially, an important message about the value of human life can be embedded into this lesson- each person is equally as important; a rich person's life isn't worth more, and a poor person's life isn't worth less.
Mathematically, if each half-shekel is representative of a person in the given community, it is important for each person to give exactly a half-shekel (no more, no less). Children can discuss what would happen if someone put in an extra half-shekel, or if other people gave less than a half-shekel.
An activity of playing out taking a census in the classroom with a given token can help students understand the one-to-one correspondence of each person giving in one specific coin to represent their presence as part of the community. Together, they can look at the coins collected to see that the number of coins matches exactly to the number of people in the class.
A follow-up activity would be to have some students with extra coins and some students with fewer coins (or no coins). Children could be allowed to give as many coins as they want. Then go back, as a class, to look at how many coins were collected and see if they can figure out how many people are in the class based on the coins collected. How does the number of coins compare to the actual number of students in the class? Students will be able to see the confusion caused by the discrepancy- the missing aspect of the one-to-one correspondence.
Base-10 Connection:
[3-5 concept]
Students can also talk about the idea that, in Judaism, we don't count out individual people. If we need to know how many people are in a group, we use another method of tallying the number of people in the group. Rather than counting people, for example, we could count people by their shirts (17 shirts in a room means 17 people who are wearing those shirts). If you do need to "point and count" people in a group, there is a custom to use the passage:
"הושיע את עמך וברך את נחלתך ורעם ונשאם עד עולם"
"Save Your people, and bless Your inheritance; And tend them, and carry them forever"
~Tehillim 28:9
Mathematically, using this Hebrew phrase from Tehillim works out very nicely, since there are 10 words in the phrase. Each word represents one person, so every full phrase represents 10 people. Students can discuss different example situations of how many full passages and a number of extra words represents how many people in a group. For example, if I'm counting people in a crowd, and I go through the full passage 3 times and then begin the passage again, ending on the word "וברך", that would mean that there are 34 people in the crowd.
