"Joseph amassed grain like the sand of the sea, very much, until he ceased counting, for there is no number." ~Bereishit 41:49
For this week, I want to focus on the statement above, where Joseph was overseeing the collection of grains in Egypt during the 7 abundant years in preparation for the upcoming 7 years of famine. Using the commentaries of Rashi and Sforno we learn that Joseph was amassing as much grain as possible, until the grain counter stopped counting (Rashi) and the reason that he stopped counting was that he had run out of names of numbers to use for his counting (Sforno).
This statement struck me as one that would make for an interesting discussion with students on place-value and increasing numbers. For information on place value and thinking about how it works, functionally, see my post from Parshat Bereishit.
Concepts related to the progression of counting:
- Children first start by learning the base pattern in counting from 1-10. Once they have a solid ability to count 1-10, we begin working with them on recognizing the pattern of how we loop through 1-9 in each set of tens (teens, twenties, thirties,...) to make another set of 10. Interestingly, the tens places increase in the same numerical order as the ones.
- After tens, usually somewhere in the grades K-2, students solidify their counting from 1-100. Next, they practice understanding the next loop- counting from 100-200 will follow the same pattern as 1-100, but with "one hundred" before each number. Once they reach 199, they increase the hundreds place to 200. Again, interestingly enough, the numbers continue to loop in the same pattern with the hundreds place increasing in the same numerical order as the ones and tens did.
- As students move up in their understanding of place value through the hundreds, then we work with them on extending their understanding exponentially by reviewing with them the idea that this loop continues on and on, and as we reach our maximum capacity for each place value, we just add on the next place value and continue the pattern loop that we've created.
- Once students have mastered the concepts of the unlimited potential for counting by increasing numbers, we then extend the concept in 2 directions:
- 1) negative numbers and comparing how "increasing" negative numbers and place values relates to increasing positive numbers and place values
- 2) decimal numbers and comparing how "increasing" decimal numbers and place values relates to increasing positive/negative numbers and place values.
- Note that while positive and negative numbers are fairly comparable, conceptually, when we think about magnitude, or size, of the numbers, an added conceptual difficulty that decimals offer is the idea that the "larger" the number, the "smaller" the piece. The numbers we talk about as decimals, really means the number of pieces that 1 unit or whole is broken into- so more pieces, means cutting the same piece into smaller pieces.
The most basic construct that children are building on as they increase their understanding of counting through each level is that you can always make a bigger number by just adding one more; cycling through the understood loop of numbers and adding new place values as needed will always result in a new, bigger number.
Based on this construct, how can we understand the statement above in the parsha? An interesting Social Studies connection might be for the students to see if they can find information on how many of our place values the Egyptians actually used. Clearly, based on our understanding of counting, even if we didn't know how to read the number of counted grain, we would at least be able to write out the number representing how much grain was collected, and we could have kept on counting.
Some questions to consider for a conversation on the topic:
Obviously, the Egyptians had a limit- did they stop counting because they thought they couldn't have any more grain? No, they stopped counting because they reached a point where they could no longer track their collection. What implications might extend from such a situation?
"[Joseph's brothers] were seated before [Joseph], the firstborn according to his seniority and the youngest according to his youth..." ~Bereishit 43;33
"[The man in charge of Joseph's house] searched; he began with the oldest and ended with the youngest; and the goblet was found in Benjamin's saddlebag." ~Bereishit 44;12
In this week's parsha, we can see two instances where Joseph's brothers were organized according to their age. Putting items in order, or sequencing, is a critical math skill that is used by students of all ages (preschool to adulthood), both in school and in life.
Sequencing:
Sequencing means putting items into a specific order or arrangement.
The skill of putting any group of items or information into a designated order is an important one. Children can begin working on this skill at a very early age. By the time they reach Kindergarten and 1st grade, children should be able to put objects in a designated order, and they can work on designating attributes to objects by which to order them. For example, the same group of students could be placed together and first asked to put themselves in height order and then asked to rearrange themselves according to their birthdays.
Connection to Parsha:
In Bereishit 43;33, Rashi explains that Joseph arranged his brothers to sit first by mother, and then in birth order within their groups. So, they were seated in the following order:
Reuven, Shimon, Levi, Yehuda, Issachar, Zevulun [children of Leah], Dan, Naphtali [children of Bilhah], Gad, Asher [children of Zilpah], and Binyamin sat with Joseph [children of Rachel/"without a mother" to quote Rashi here].
In Bereishit 44;12, Rashi explains that they were searched by oldest to youngest so that they wouldn't sense that the messenger already knew where the goblet was packed. According to this understanding, at this time they were searched in the following order:
Reuven, Shimon, Levi, Yehuda, Dan, Naphtali, Gad, Asher, Issachar, Zevulun, and Binyamin.
Sequencing Activites for the young and old:
For younger students, sequencing activities could be:
--putting dates into a calendar (cut and paste)
--putting counting numbers in order
--setting out a pattern with items or shapes and having them continue the pattern
--asking students to put a group of people in order or put themselves, as a group, in order by height, birthday, day of the month that they were born, etc. A game of asking them to sequence a group of people based on a non-physical attribute would best be accomplished with the people holding cards so that the child has something concrete to look at while putting the people in order
--creating a pattern of shapes, numbers, or information and asking a child to identify the pattern is also a good thought-provoking exercise for students.
For older students, there are still very relevant sequencing activities. Such activities could be:
--writing a timeline of events
--expanding on a number line- ordering integers (positive and negative numbers), fractions, decimals, or any mixture of rational numbers
--looking to identify patterns in numbers or shapes (this is a skill that they will use in all subjects as they learn- patterns in poetry tempo or lines, patterns in a science experiment, patterns in social history, etc)
--asking students as a group to organize themselves in a particular sequence based on a physical or non-physical attribute (this is often used as an ice-breaker activity for new groups of people first meeting each other at conventions and seminars).
For an extra challenge: There is a variation where students have an index card with a piece of information on their forehead. The students do not know what their own cards say, only what they see on everyone else's cards. Using this information and speaking to others in the group, they need to work together to put themselves in proper sequence for the information on the cards (without explicitly stating what is written on other people's cards).
What sequences can you find in your lives?
