Toldot- Base-10 and Number Sense related activity ideas

"Yitzchak sowed in that land, and in that year he reaped a hundredfold; thus had Hashem Blessed him." ~Bereishit 26;12

Rashi on 26;12-
"A Hundredfold- For they assessed [the field] to determine how much it is fit to produce and it produced for every unit they estimated it could produce a hundred units. Our Rabbis said, This assessment was done for tithes."

In this section of the parsha, we learn of Yitzchak's time living in Gerar, under the rule of Avimelech. Bereishit Rabbah (64;6) asks: if we're not supposed to find a blessing in a measurable quantity, how could this crop be considered a blessing? You are not supposed to count on a crop before it is harvested (like the old adage- "Don't count your chickens before they hatch"). How does this make sense? The Mizrachi explains that the crop was traditionally estimated in order to make an advanced calculation for tithings to be easily set aside. The midrashim (Tosafot HaShalem citing Rivah) teach us that this crop that is referred to was sowed during a time of famine in the land. Yitzchak had estimated what his tithe should be in order to be able to set it aside for the poor as soon as possible, and he had estimated much less of a crop due to the famine in the land.  It is with this understanding that we see how this was such a blessing for Yitzchak. 

Activity Connections:
There are two primary mathematical concepts that connect to this crop:
1) the concept of calculating and setting aside tithings
2) the concept of one hundredfold 

Interestingly enough, both of these concepts connect to powers of 10. 
*Tithings are classically 1/10 of a crop (the root of the hebrew word- מעשרות/מעשר- is actually related to עשר, the number 10)
*one hundredfold means times 100, which is a power of 10 (10²;or X10 and then X10 again)

Activity ideas in increasing conceptual difficulty:
*What does a tithing or 1/10 of a group really look like? For the youngest students, they could begin by just taking a group of items and separating the items into 10 equal groups. Having a set-up of 10 cups, bowls, or plates to sort the pieces into will help students keep track of their sorting. This is doable even for preschool children. After they divided their items into 10 piles, they can count each pile to make sure that they all have the same number of items in them. From there, they can act out separating 1 of the piles to give away and keeping 9 of the piles for themselves. For beginners, you want to simplify the process for them by making sure that the number of items they start with is a multiple of 10 (so that it divides equally into 10 piles).

*As a step up from the previous activity, you can have students divide larger groups of items, and incorporate estimation or calculation of what amount the tithing will actually be when it's separated out. Students who have had exposure to the concept of fractions and decimals could also estimate and calculate the value of tithings for amounts that are not strictly multiples of 10.

*One hundredfold- what does x100 really look like? What if Yitzchak had anticipated having 5 bundles of grain? How much did he actually have that year? What if he had anticipated 10 bundles of grain? Younger students can physically count or draw out the difference of a group of 5 and a group of 500. Small or medium sized graph paper can be good for drawing out a model of these differences (with each box modeling 1 unit). If students have the dexterity, small manipulatives like paper clips can also be good. For younger students, these can be modeled with blocks or Legos, but you need to have a good supply in order to model the hundreds. (Tip: Creating stacks or grouping of 10 to count everything out will help make counting easier and will help children develop their number sense.)

*Older students can graph the difference between the anticipated crops and the actual crops. Is there a pattern to the difference? How would the difference be expressed as an algebraic expression?

*Another activity for older students would be a comparison of the tithings between the anticipated crops and the actual crops. What would a graph of these differences look like? How could the differences be expressed algebraically? How does the pattern of the difference in tithings (anticipated vs. actual) compare to the pattern of the difference in crops (anticipated vs. actual)? How do the algebraic expressions of each comparison compare to each other?

Toldot- A Timeline Through Deductive Reasoning

Deductive reasoning: a logical process in which a conclusion drawn from a set of premises contains no more information than the premises taken collectively (ref. dictionary.com)

When trying to problem solve, deductive reasoning helps us logically move from one step to the next. Deductive reasoning allows us to look at the information that we are provided with and state, with certainty, that the next logical step can also be relied on, because the original information proves the next step. Each step that is proven through deductive reasoning can then be relied on as a fact to support our logical thinking towards a final solution to the problem at hand.

As we've been reading through the parsha each week, we have followed storylines which offer some concrete sense of time. However, taken together, how much time is actually passing between each phase of each story? If we take the individual pieces of information that we are provided with from the text and Rashi's commentaries, we can use deductive reasoning to calculate some of the missing pieces of information in order to put together a fuller picture for ourselves. Is the story between the parshiot a straight timeline, or is it broken into parts dealing with each "main character"? With some deduction, we can find out. (My completed storyline for the primary Toldot "characters" with some added information appears at the bottom of this post. Check your calculations and timeline against mine and see how they match-up. Can you fill in the extra information that I calculated?)

Whenever sorting information for problem solving, the first step is to map out the information that you have.

• From Lech Lecha 17;17 and Vayeira 21;2 we know that when Isaac was born, Sarah was 90 years old and Abraham was 100 years old
• From Chayei Sarah 23;1 we know that Sarah lived to be 127 years old
• From Rashi's commentary on Sarah's death, we know that she died upon hearing the news that Isaac was to be sacrificed at the akedah
• From Rashi on Toldot 25;20, we know that Rebecca was born in the year of the akedah
• From Chayei Sarah 25;7, we know that Abraham lived to be 175 years old
• From Toldot 25;20, we know that Isaac was 40 years old when he married Rebecca
• From Toldot 25;26, we know that Isaac was 60 years old when Esau and Jacob were born
• From Rashi's commentary on Toldot 25;27, we know that Jacob bought Esau's birthright when they were 13 years old
• From Rashi's commentary on Toldot 27;2, we know that Jacob took Esau's blessing from Isaac when Isaac was 123 years old
• From Toldot 26;34, we know that Esau was 40 years old when he married Judith and Basemath

The second step to problem solving is to identify the questions that you are trying to answer.
--How old was Isaac at the akedah (and Sarah's death)?
--How old was Isaac at Abraham's death?
--How old was Rebecca when she married Isaac?
--How old was Rebecca when Esau and Jacob were born?
--How old were Isaac and Rebecca when Esau sold his birthright?
--How old were Jacob, Esau, and Rebecca when Isaac gave Jacob Esau's blessing?
--How old were Rebecca and Isaac when Esau was married?
--How old is Jacob (and Esau) at the end of Toldot?

Now let's see what information we can use to answer these questions:
--How old was Isaac at the akedah (and Sarah's death)?

• Sarah was 90 when Isaac was born, and she was 127 when she heard of the akedah. 
• 127-90=37, so Isaac was 37 at the akedah and when Sarah died

--How old was Isaac at Abraham's death?

• Abraham was 100 when Isaac was born, and Abraham lived to be 175.
• 175-100=75, so Isaac was 75 when Abraham died

--How old was Rebecca when she married Isaac?

• Isaac was 37 at the akedah, which was the same year that Rebecca was born.
• Isaac was 40 when he married Rebecca.
• 40-37=3, so Rebecca was 3 when they were married (keeping strictly to math here, no societal norms commentaries)

--How old was Rebecca when Esau and Jacob were born?

• Isaac was 60 when they were born
• Isaac was 37 years older than Rebecca
• 60-37=23, so Rebecca was 23 when they were born

--How old were Isaac and Rebecca when Esau sold his birthright?

• Isaac was 60 when they were born; Rebecca was 23 when they were born
• Esau and Jacob were 13 when the birthright was sold
• 60+13=73, so Isaac was 73; 23+13=36, so Rebecca was 36

--How old were Jacob, Esau, and Rebecca when Isaac gave Jacob Esau's blessing?

• Isaac was 60 when they were born; Rebecca was 23 when they were born
• Isaac was 123 when he gave Jacob Esau's blessing
• 123-60=63, so Jacob and Esau were 63; 23+63=86, so Rebecca was 86

--How old were Rebecca and Isaac when Esau was married?

• Isaac was 60 when they were born; Rebecca was 23 when they were born
• Esau was 40 when he was married
• 60+40=100, so Isaac was 100; 23+40=63, so Rebecca was 63

(click on the picture for a larger view of the timeline)

From this timeline, we can see that, although most of the storylines in these few parshiot are in order, there are some pieces of information that are placed out of order, seemingly to keep them together with the sections that discuss the particular "character" in more depth.