Yitro- Basic division with large numbers

"And you shall see from among the entire people, men of means, G-d-fearing people, men of truth, people who despise money, and you shall appoint them leaders of thousands, leaders of hundreds, leaders of fifties, and leaders of tens." ~Shemot 18;21
"Moshe heeded the voice of his father-in-law, and did everything that he had said. Moshe chose men of accomplishment from among all Israel and appointed them heads of the people, leaders of thousands, leaders of hundreds, leaders of fifties, and leaders of tens." ~Shemot 18;24

Rashi on 18;21--
Leaders of Thousands- They were six hundred of these leaders for six-hundred-thousand adult male Israelites.
Leaders of Hundreds- Of these there were six thousand
Leaders of Fifty- twelve thousand
And Leaders of Tens- sixty thousand

From Parshat Bo (Shemot 12;37) we know "The Children of Israel journeyed from Ramses to Succoth, about six hundred thousand on foot, the men, aside from the children"

In this week's parsha, we learn about Moshe trying to arbitrate on issues for the Jewish people. We hear of how he sits from morning until evening just hearing cases for people. When Yitro sees all the time that Moshe dedicates to arbitration, Yitro makes a suggestion to appoint levels of leaders to help relieve some of Moshe's burden; a suggestion which Moshe heeds.

In Rashi's commentary on the suggested levels of leaders, he does some basic division for us to let us know how many leaders there were at each level. This division calculation is important for us to understand that the leaders were not separated from the whole group of Jewish people, but rather, when calculating how many leaders are needed, the leaders themselves are included in the number of Jewish people.

Definitions: Division: The process of separating a group or item into a number of smaller, equal-sized groups or pieces. With smaller numbers, it's easy to model the actual division with manipulatives (pieces that can be laid out and moved around to model a mathematical situation) or draw a diagram to aid in calculation until students become proficient with basic math facts.

Dividend: The whole group that is being divided

Divisor: The number of groups into which you are separating the dividend

Quotient: The answer to a division calculation

When we write basic division problems, we write dividend ÷ divisor. When we write division as a fraction, the dividend goes on top (numerator) and the divisor goes on the bottom (denominator). Many students find the algorithm set-up for long-division confusing, because when we set-up the division box, the dividend goes inside the box and the divisor goes on the outside. 

When we need to start dividing very large numbers- in the tens, hundreds, thousands, etc.- the numbers become too big to model, and students begin to rely more on algorithmic notations (those big long division bars that we all remember) to keep track of their calculations. The other option is to draw representative diagrams to help students feel more comfortable with their number sense and help them connect meaning to the division of large numbers. For example, students can draw cubes to represent numbers of thousands, squares to represent hundreds, lines to represent tens, and dots to represent ones. In this way, they can visualize the breakdown of the numbers into equal groups. For students who have difficulty switching to the algorithm, working through calculations side-by-side visually and with the algorithm can help them to connect the concept.

Division trick: When dividing with large numbers that end in zeros (are multiples of tens, hundreds, thousands, etc.), you can remove any number of zeros from the end of both dividend and divisor as long as you can remove the same number of zeros from both. You can then do your division and your quotient will be the same as it would be for the larger number. For example, 100 ÷ 20: we can take one zero away from each number, divide 10 ÷ 2 = 5, so, 100 ÷ 20 = 5. 

Connection to Parsha:
So, if we know how to divide, we can check on Rashi's calculations and make sure that all the numbers match up.

For 600,000 adult males, we have:
--Leaders of thousands- 600,000 ÷ 1,000 = 600 (same as 600 ÷ 1, if we remove the 3 zeros at the end of both numbers)
--Leaders of hundreds- 600,000 ÷ 100 = 6,000 (same as 6,000 ÷ 1)
--Leaders of fifties- 600,000 ÷ 50 = 12,000 (same as 60,000 ÷ 5)
--Leaders of tens- 600,000 ÷ 10 = 60,000 (same as 60,000 ÷ 1)

So, here we have a total of 78,600 leaders total for the whole group.

If the numbers of leaders had not been included in the number of the whole group, the numbers would have been much different. How different? Let's assume that the leaders from each previous tier are not included in the next lower breakdown.

600,000 - 600 (leaders of thousands) = 599,400 ÷ 100 = 5,994

600,000 - 6,594 (leaders of thousands + leaders of hundreds) = 593,406 ÷ 50 = 11,868.12 (let's assume 11,868)

600,000 - 18,462 (leaders of thousands + leaders of hundreds + leaders of fifties)= 581,538 ÷ 10 = 58,153.8 (let's assume 58,153)

So, all the number of leaders here would have been 76,615 (leaders of thousands + leaders of hundreds + leaders of fifties + leaders of tens) for the whole group.

All together, then, by having the leaders included in the total count of Jewish people, there were a minimum of about 2,000 additional leaders who were assigned to help Moshe with arbitration of matters.