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)

Tzav- Statistics part 2

Once again, this week we have a significant portion of the parsha dedicated to specifics of korbanot, or sacrificial offerings. Again, rather than quoting from the parsha, I've organized some of the information related to the offerings in order to analyze.

Pie Chart/Circle Graph:
Whereas the bar graphs we looked at last week compared a measured amount between different categories, a circle graph compares portion sizes of categories that together make up 100% of something. For example, out of your total budget allotment, what percentage of the budget goes to different categories (food, medical, housing costs, etc.)? Out of all after school enrollment, what percentage of students go to each of the different activities? Out of the total hours in a day, what percentage of time is spent sleeping, eating, working, etc.? The key to circle graphs is that all categories together add to 100% and create a complete circle. 

When analyzing data to convert into a circle graph, we'll need our knowledge of fractions (Parshat Vayishlach), proportions (Parshat Ki Tisa), and percentages (Parshat Vayigash) to convert the raw data into a format that will fit the circle graph.

Connection to Parsha:
Let's take a look at a sample data set from this week's parsha to walk us through the process. Let's look at what percentage of sacrifices were to be fully burned and what percentage of sacrifices were allowed to be partially eaten in some form.

We can start by making a chart to organize the information so that we see which category each sacrifice falls into.

Next, we use the chart to count up how many sacrifices are in each category out of the total number of sacrifices. According to our chart, out of the 6 total sacrifices, we have 3 sacrifices that are completely burned and 3 sacrifices that are partially burned and partially eaten.

Let's restate this as fractions- 
3/6 = burned 
3/6 = burned then eaten

If we want, we can reduce these fractions (each reduces to 1/2), but we don't have to. In this case, we will reduce them, since it will make our next calculation easier to work with. 

So, we have-
1/2 of the sacrifices are burned
1/2 of the sacrifices are burned then eaten

Now, remember that the most important part of a circle graph is that it shows a percentage out of 100%, where 100% (or all) of what you're analyzing makes up the full circle. So, if 100% of the sacrifices is all of them, we know that 1/2 of the sacrifices were burned: 100% x 1/2 = 50% and 1/2 of the sacrifices were burned then eaten: 100% x 1/2 = 50%. So now we have calculated that 50% of the sacrifices were totally burned and 50% were burned then eaten. Once we have this calculation, we can create our graph. 

While I made the graph electronically, you can also construct a circle and measure out the angle measures for each section using pencil, paper, drawing compass, and protractor. Such an activity offers a good opportunity for art/geometry integrations in math. To calculate the sections by hand, we need to first know a full circle is 360°. Since we need 1/2 of the circle for each category, we multiply 360° x 1/2 = 180°. This means that the angle measure for each section of the circle needs to be 180°. Measuring out this 180°, you'll find that you've perfectly divided the circle into two 50% sections, one for each category.

It happened that the numbers for this activity fell perfectly into two equal groups. While it may seem simplistic because of the easy numbers and category divisions, it would make a very simple way to practice going through the steps of the process and practicing drawing circles and interior angles. Once practicing with this easy example, another problem with more complex category divisions could be introduced.