Vayigash- Family Tree and Information Organization related activity ideas

"So Jacob arose from Beer-Sheva; the sons of Israel transported Jacob their father, and their young children and their wives, in the wagons which Pharoah had sent to transport him. They took their livestock and their possessions which they had amassed in the land of Canaan and they came to Egypt- Jacob and all his offspring with him. His sons and the sons of his sons with him, his daughters and the daughters of his sons and all his offspring he brought with him to Egypt" ~Bereishit 46:5-7

What follows in Bereishit 46:8-27 is an enumeration of all the family members who traveled to Egypt with Jacob when Joseph sent for them to move to Egypt. We learn that, in total, there were 70 people who moved down to Egypt.

While it's not directly related to calculation, the ability to organize of information is a key skill for success in mathematics. For successful problem solving, students need to be able to identify important information, classify how the different pieces of important information relate to each other, identify the question or problem that they are tasked with answering or solving, identify the steps necessary to find the solution to their question/problem, follow through with their planned steps for finding the answer, and finally critically look at their answer to determine if it makes sense and is a sensible solution to the problem that are trying to solve. 

Activity Connections:
Activity ideas in increasing conceptual difficulty:

While the Torah offers us a certain amount of organization in the way in which information is given to us, it is often helpful to map out information to gain a broader picture or greater understanding of the information. Mapping out a family tree of Jacob's descendants based on the information provided to us in the parsha is an organizational task that students of many varying levels can benefit from. Note that the grade ranges that I've listed are guideline approximations. There will always be students for whom activities come more easily and other students for whom activities are more difficult. I recommend shifting up or down within activity suggestions to properly accommodate each student's level.

• At the lower levels (grades K-2), teachers can provide students with a cut and paste chart of a family tree and the names that need to be placed on the tree and the family tree map with all the appropriate slots organized where they should be. For younger students, 70 names will be particularly overwhelming. To make it more manageable, students can work in partners or groups and just work with a single branch- one group can cut and paste Reuven and his children onto the tree, another can work on Levi and his children, a 3rd can work on Shimon and his children, etc. When all the groups have put their trees together, the class can come together to see how all the brothers' families fit together on the tree as children of Jacob. 
• For slightly older students (grades 2-6), teachers can provide a blank family tree with the proper organization and slots provided. The students could find the information from the passages and write in the names. Again, to alleviate some of the pressure from the large number of names, students could be assigned a section of the tree and come together with all their pieces of the tree to see how they fit together. 
• For students attempting to fill in the full chart by themselves (grades 4+), a helpful accommodation could be to have some of the names pre-filled in order to help the students begin to envision the organization on the tree and to give them some guide-points to work with. In it's entirety, such a project would need to be broken up into smaller time chunks so that students can work on sections at a time without feeling overwhelmed or frustrated.
• As students get older (grades 6+), they can fill in more and more of the chart independently. Ultimately, the goal would be for the students to be able to read the passages, create an organizational system on a blank piece of paper, and place the names of the 70 family members in their organized family tree without assistance. This task takes a good deal of vision and organizational skill, and while some students are capable of this task by middle school, others may take well into high school to be able to master these organization skills.

Some organizational points to keep in mind when planning out the family tree- how do you want to organize the sons on the tree? are you going to follow the organization in the passages? are you going to organize the sons according to birth order? are you going to group the sons according to their birth mothers?

Some follow-up analysis of the family tree:

* Based on the passages, do you have 70 people on the family tree?

Note the following which will affect the actual counting of the people:

• Er and Onan are listed as sons of Judah, but it states that they died in Canaan (therefore, they aren't counted in the 70 people).
• Yocheved is missing from the list of children of Levi, since her mother was pregnant with her when they traveled to Egypt. Since she was born on their arrival in Egypt, she is counted in the 70 people even though she is missing from the list.

Miketz- How high can you count?

"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?

Vayeishev- Timeline and basic arithmetic related activity ideas

"Then Jacob rent his garments and placed sackcloth on his loins; he mourned for his son many days." ~Bereishit 37;34

This week we learn of when Joseph's brothers sold him and implied to Jacob, their father, that he was attacked and killed by a wild animal. Upon seeing Joseph's bloody tunic and surmising his death, we are told that Jacob mourned for "many days". Rashi  here explains that "many days" actually means that he mourned for Joseph the entire time that Joseph was missing to him- which actually calculated to be 22 years.

Activity Connections:
Activity ideas in increasing conceptual difficulty:

• The most basic calculation connection would be to begin with the statement 1) in Bereishit 37;2 (at the start of this week's parsha) that Joseph was 17 years old at the time of the story of his sale and 2) in Bereishit 41;46 (in next week's parsha- Miketz) that Joseph was 30 years old at the time that he first was called to come before Pharoah. Younger students can just calculate the passage of time between Joseph's sale and his appearance before Pharoah. Before learning subtraction, this could be drawn on a time line for students where they can physically count the years. Children who are a little older can count up from 17 to 30 or subtract 30-17.
• The next step would be to look at the calculation that is offered by Rashi regarding the 22 years that pass before Jacob sees Joseph again. Children can take the calculation from the activity above and pull out (from Parshat Miketz) the additional years that passed- 7 plentiful years followed by 2 years of famine before Jacob travels to Egypt. Students would then add the 3 numbers together to confirm that the time that passed was truly 22 years.
• A larger activity, which could be leveled in many ways depending on the age and abilities of students, would be to pull out the information from larger segments of Sefer Bereishit and place the information onto a more comprehensive timeline. Students could either create timelines for individual sections, or a more comprehensive timeline that incorporates multiple people's storylines for students to see the crossover of time between the different stories that we are given throughout Sefer Bereishit. Along with Rashi's interpretations and explanations throughout the sefer, students can use their own calculations for comparison of information and calculating where everything that we learn would actually fall on a grand-scale timeline.

We can use the information given in this week's parsha and next week's parsha to analyze the passage of time that Rashi refers to on Bereishit 37;24. Creating a timeline helps students formulate an understanding of the chronology of events, which provides them a framework for the both the logical progression of events and the greater picture of how multiple events fit together with each other. In addition to the sequential understanding that is gained, students also practice basic addition and subtraction by calculating the time lapses between different events. Students also work on identifying which information needs to be added or subtracted in order to find missing information, which is a critical problem solving skill. Finally, when the completed timeline has been pieced together, students can look to compare and contrast information to gain a deeper insight that they may not have picked up on in their original, single-story understanding of the events.

Vayishlach- Leveled activity ideas related to organizing and analyzing information

"...then [Yaakov] took, from that which had come into his hand, a tribute to Esav his brother: She-goats, two hundred, and he-goats, twenty; ewes, two hundred, and rams, twenty; nursing camels and their young, thirty; cows, forty, and bulls, ten; she-donkeys, twenty, and he-donkeys, ten." ~Bereishit 32;14-16

In this week's parsha, we learn about Yaakov's gift to Esav to try to appease him and keep him happy.

Activity Connections:
When analyzing information, there are multiple levels of understanding that are needed. We can break down these levels for students in order for them to develop an understanding of each level of processing information.

Activity ideas in increasing conceptual difficulty:
*Last year, for Parshat Vayishlach, I used a chart to organize the information about Yaakov's gift to Esav. Figuring out a way to organize the information that you are given in the first step to analyzing the information. Younger students will need help to work together to put the information into a pre-written chart, while slightly older students can work on organizing and charting the information independently.

Here is the gift information organized into a chart:

Type of Animal	# of Females (& children)	# of Males	Total #
Goats	200	20	220
Ewes/Rams	200	20	220
Camels	30	0	30
Cows/Bulls	40	10	50
Donkeys	20	10	30
Total # of Animals	490	60	550

Children can also use this information to learn about representing the information in different types of charts- how would this look displayed in a pictograph? bar graph? double-bar graph (comparing Males & Females of each)?

Additionally, students can look at the chart or graphs and talk about quantitative comparisons between different groups.

*After children are comfortable organizing the information, they can use the information to talk about how pieces of the whole compare to each other as part of the whole or sub-parts of the whole. Ratios of males to females by animal or within the whole group; what fraction of each animal is male or female? what fraction of the whole? what fraction of the whole is each animal? What are the basic fractions? Can the fractions or ratios be reduced to simpler numbers? What is the significance of these reduced fractions/ratios? See last year's post for more on this.

*The next level of calculation would be converting the fractions into percentages of the whole. See here for my post explaining the concept of percentages. Students can calculate the percentages of the total gift that are made up for varying subcategories- how do the percentages breakdown comparing males to females? comparing different animals?

*An additional level of complexity to this organization of information could be to have students draw circle graphs with accurately calculated segments. See my post here to read about calculating the angle measures of the segments of a circle graph.

Vayetzei- Sorting, Sets, Subsets, and Unions related activity ideas

"[Lavan] said, 'What shall I give you?' And Yaakov said, 'Do not give me anything; if you will do this thing for me, I will resume pasturing and guarding your flocks: Let me pass through your whole flock today. Remove from there every speckled or dappled lamb, every brownish lamb among the sheep and the dappled or speckled among the goats- that will be my wage. Let my integrity testify for me in the future when it will come regarding my wage before you; any that is not speckled or dappled among the goats, or brownish among the sheep, is stolen in my possession.'
And Lavan said, 'Yes! If only it will be as you say.'
So he removed on that day the ringed and dappled he-goats and all the speckled and dappled goats- every one that had white on it, as well as all the brownish ones among the sheep- and he put them in the charge of his sons. And he put a distance of three days between himself and Yaakov; and Yaakov tended Lavan's flock that remained." ~Bereishit 30;27-36

In this week's parsha, we learn of how Lavan paid Yaakov for his years of work by separating his flocks into two groups. Lavan kept for himself all of the sheep and goats that were completely white or completely brown, and he gave to Yaakov all of the sheep and goats that had any kind of mixture of white and brown, regardless of the patterning- speckled, dappled, ringed, etc.

Activity Connections:
The concept of separating is a basic one that we begin working on with students at the preschool level. It is often thought of just as a primary skill, but the understanding of sorting, separating, and comparing groups is one that is necessary through higher levels of mathematics, as students work with logical thinking, and sets and subsets of information.

Activity ideas in increasing conceptual difficulty:
*The most basic level of sorting items is creating two or more simple categories and separating the items into the different categories. If I have a box of buttons, I might choose to sort them by color (green, blue, red, etc.), by size (large, medium, small), by shape (round, square, other), by number of holes (1, 2, 4, etc), or any other distinguishing feature that I notice. To make it even simpler, you can create just two categories- green & not green, large & small, round & not round, 1 or 2 holes & 3 or more holes. Sometimes it's easiest to just create two categories, and sometimes it's helpful to have the additional categories. This decision will be based on the items that you're using, if there's a purpose for the sorting beyond just sorting (do you need specific items for different projects?), and the way in which the students see the groupings of items that they're trying to sort. Even at an early age, you can see aspects of how each child processes information based on sorting decisions that they make when categorizing items in a group.

*The next step in sorting complexity is learning how to deal with items that might fall in two categories at the same time. The ultimate model for this situation is a Venn diagram. Students can practice sorting out actual items on an oversized mat that has a Venn diagram drawn onto it, or you can use string or hula-hoops to create overlapping circles for them the place objects into. This works for children as young as older preschool students. As students get up into early elementary classes, they should be able to record their work on a Venn diagram map, ultimately not needing to actually physically move items around. 
--When using Venn diagrams for sorting, you begin crossing categorizations. It could be straightforward- maybe some of your buttons are solid colors and others have multiple colors. Blue buttons go in one circle, red buttons in a second circle, and the buttons that have both red and blue will go in the overlapping section between the two circles. Or, you can begin crossing different attributes- you could sort buttons into green, 3 holes, and round. Different buttons will fit into different sections of the diagram- where will the green star with 2 holes go? what about the round, blue button with 3 holes? the round, green one with 3 holes?

*The next level of categorization is to identify sorted groups and then think about what the groups will look like if you subcategorize them or mix them in certain ways. This is where students begin thinking about subsets (a smaller group within a category), unions (combining of two categories), and intersections (what two different categories have in common). Really, this is an articulation of what they are drawing out on the Venn diagram, but as the concepts move from categories of items to categories of geometric attributes or categories of number types, the complexities increase. A Venn diagram showing categories and intersections of polygons, prisms, angle measures, etc. could have value into high school. Sorting numbers into sets, subsets, unions, and intersections- integers, prime numbers, factors of 26, multiples of 75- has applications in number theory into high school and beyond, depending on your field of study.

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?

Chayei Sarah- Currency related activity ideas

A note to readers- Having written a full cycle of parsha posts, I've decided to try a slightly different approach. For this next cycle, I'm beginning with identifying a concept and then posing concept specific questions and activities for investigating the concepts at varying academic levels- still K-8 level.  I'd love feedback from readers as I keep the process going. Are you just an interested reader? Am I still keeping you interested? Are you a teacher/parent looking for ideas? Which ideas do you find most helpful? Have you tested out any ideas with your students/children? Looking forward to your feedback!
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"...And Ephron replied to Abraham, saying to him: 'My lord, hear me! Land worth four hundred silver shekalim; between me and you- what is it? And bury your dead.' Avraham listened to Ephron, and Avraham weighed out to Ephron the money that he had mentioned in the hearing of the children of Heth, four hundred silver shekalim in negotiable currency. And Ephron's field, that was in Machpelah, that was facing Mamre stood- the field and the cave within it and all the trees in the field, within its entire boundary all around- as Avraham's as a purchase in the view of the children of Heth, with all who came to the gate of his city. And after that, Avraham buried Sarah his wife in the cave of the field of Machpelah facing Mamre, which is Hevron, in the land of Canaan." ~Bereishit 23;14-19

Activity Connections:
The two ideas that stand out to me from this passage are:
1) Avraham's weighing out of the money
2) The fact that the Torah specifically says that he was using "negotiable currency"

For some previous thoughts about scales and balance, see my post on Parshat Ki Teitzei where I talk about weights and measures.

Rashi on 23;16 explains that "negotiable currency" is stated because the shekel that the Torah usually refers to is a smaller shekel (equal to 1 sela). In our case, though, Avraham chose to use shekels which are acceptable as currency even in areas that use larger shekels ("kanterin", which are equal to 100 selas). 

Activity ideas in increasing conceptual difficulty:
*Discussion and exposure to local current-day currency; discussion around equivalent currencies
eg. for US currency, exposure to pennies, nickels, dimes, quarters, half-dollars, dollar coins, dollar bills, etc and discussion/practice with the value of each and equivalencies (5 pennies = 1 nickel; 25 pennies = 1 quarter = 5 nickels = 2 dimes + 1 nickel)

*Combine currency with the idea that Avraham weighed out his money. Does this still work with current day currency? Will 5 pennies balance with 1 nickel on a balance scale? Our current money is more of a representation of value than true equivalence between coin sizes, weights, and values.

*Look at the difference between what 400 shekalim would have been if they were each 1 sela instead of being 1 kanterin. This can be done at varying levels for different ages and abilities. What does it physically look like using manipulatives to represent 1 sela? How could you calculate how many selas Avraham actually paid Ephron? 

*If you look at the comments section in last year's post on Chayei Sarah, you'll find a link to some calculations on current day equivalence for a shekel (one that's the value of a sela). How can you use this information to calculate the current day value for what Avraham actually paid for Maarat HaMachpelah? How different is that from what he would have paid in current day currency if he used selas instead of kanterin?

*How does the amount of money paid toward Maarat HaMachpelah compare to the value of Rivkah's dowry (the bracelets and nosering discussed in last year's post)? How does a beka compare to a sela? to kanterin?

*If you compare the difference between values of selas and kanterin, is the difference linear growth or exponential growth? (Hint: 1 kanterin = 100 selas; 2 kanterin = 200 selas; and so on)

Vayeira- Non-standard Units of Measure

"The water of the skin was finished, and she cast off the boy beneath one of the trees. She went and sat herself at a distance, some bowshots [away], for she said, 'Let me not see the death of the child.' And she sat at a distance, lifted her voice, and wept." ~Bereishit 21;15-16

Non-Standard Units of Measure:
We are all familiar with standard units that we use to measure on a regular basis- we measure length with inches, feet, miles, centimeters, meters, kilometers; we measure liquids with cups, pints, quarts, gallons; we measure weight with ounces, pounds, grams, kilograms. Sometimes, though, the standard units that we use either aren't available or aren't necessary for the purposes of what you're doing.

When children are first learning about measuring, a common activity is to choose a standard by which to measure different items, and then students can compare the items based on their measurements. For example, they may use stacking cubes (all of the same size) to measure items in their surroundings- a pencil, a chair, a water bottle, etc. They can record their findings and then compare which items are shorter and which items are longer (or taller). As long as the same standard is used throughout an activity, all the items measured can be compared to each other.

Anything that you choose can be used as a non-standard unit for measuring- a paper clip, a pencil, a shoe- as long as the repeated measurement is using an item (or items) that are the same exact length. So, you can't use a mixture of large and small paper clips to measure items, but you can choose one or the other. You can see how many water bottles it takes to fill up a pitcher, but you need to fill the water bottles full each time to establish a temporary standard for your measurements. 

If you want, or need, you can always convert your non-standard measurements into a standard unit afterwards by measuring your non-standard unit. If your paperclip is 3/4 of an inch long, you have enough information to convert your measurements into inches. If your water bottle holds 2/3 C of water, then you can calculate how many cups your pitcher holds.

Parsha Connection:
In this week's parsha, we learn of Hagar and Ishmael's wandering after they are sent away from Avraham and Sarah's home. We are told that when Hagar believes that Ishmael is dying, she sits "some bowshots" from him, so as not to have to watch him die. What does this mean practically, for our understanding of the distance between Hagar and Ishmael at this time?

Bereishit Rabbah 53;13 explains two aspects of this in order for us to understand the distance:
1) first it explains that since it is written "bowshots", plural, but with no specific quantity, it must mean that the minimum plural is implied. Since the smallest plural number is 2, therefore it's 2 bowshots away.
2) second it explains that a standard bowshot during this time was 2,000 cubits, or a little more than today's measure of 1/2 a mile.

Based on this explanation, we can use the parsha's non-standard measurement of a bowshot to calculate that, by current day measurements, Hagar was approximately a distance of a mile away (or just over a mile) from Ishmael during this time.

Everyday Connection:
Let's say you have a recipe for a sauce that calls for equal parts of each ingredient. You don't need to measure exactly with a 1 cup measure- if you have a cup without measurement markings, you can still use that cup to measure out your ingredients. All you need to do is make sure that you measure each ingredient to the same point on the cup, and you'll have equal parts of each ingredient.

What if you need to measure the length of an item, but you don't have a ruler nearby? You can use a pencil, a piece of paper, or even your shoe- count how many "shoes" long your item is, and then you can measure the length of your shoe later to convert your measurement to a standard unit.

Lech Lecha- Vectors and Velocity

"Hashem appeared to Avram and said, 'To your offspring I will give this land.' So he built an altar [in Canaan] to Hashem Who appeared to him. From there he relocated to the mountain east of Beth-el and pitched his tent, with Beth-el on the west and Ai on the east; and he built there an altar to Hashem and invoked Hashem by Name. Then Avram journeyed on, going and traveling toward the south. There was a famine in the land, and Avram descended to Egypt to sojourn there, for the famine was severe in the land." ~Bereishit 12;7-10
"So Avram went up from Egypt, he with his wife and all that was his- and Lot with him- to the south. Now Avram was very heavy with livestock, with silver, and with gold. He proceeded on his journeys from the south to Beth-el, to the place where his tent had been at first, between Beth-el and Ai, to the site of the altar which he had made there at first; and there Avram invoked Hashem by Name." ~Bereishit 13;1-2

Vectors:
In their introductory to Geometry, younger students are introduced to the concepts of points, lines, line segments, and rays.
*A point is a single point or location.
*A line is a straight connection between two points that continues straight and extends indefinitely in both directions
*A line segment is the straight connection between two points with endpoints at the two points
*A ray is the straight connection between two points with an endpoint at one of the points and extending indefinitely through the other point

 A vector is a concept that is not classically introduced to younger students. Essentially, a vector is the same idea as a segment, but it also includes information about distance and tells which direction you're moving in. The two key aspects to a vector are distance (referred to as "magnitude") and direction. Velocity explains the speed at which something is moving (again, it's magnitude) and the direction in which it is moving (23 miles per hour NorthEast). With this understanding of vector and velocity, we can see that velocity is a type of vector measurement, and we can use this understanding to measure how far something travels and the direction in which it is traveling. One simple example would be calculating how far something is traveling either towards (+) or away from (-) a given point. To think about vectors in their simplest form, every straight movement in one direction can be shown as one vector. This means that if I'm traveling in one direction, and then I change my direction, I would need to use 2 different vectors- 1 to explain the first part of my trip, and 1 to explain my trip after I change my direction.

The vector label is based on directions relative to a specific reference point. It's important to realize that this information only tells you have far and in what direction you've moved from your starting point; by itself it doesn't tell you how close or far you are from your reference point. In order to know where you are related to your reference point, you need to add or subtract different vectors to calculate the distance. Using the calculations of multiple vectors, this type of information can be charted on graphs showing distance compared to a location over time on a coordinate plane, and adding the aspect of direction adds an additional quality of information to the graph. For today, we'll just look at simple vectors.

Parsha Connection:
In this week's parsha, we learn of Avram's travels over the course of 25 years. If we look specifically at his travel from the beginning of the 12th chapter through the beginning of the 13th chapter of Bereishit, we see him set-up his tent in Canaan between Beth-el and Ai, travel down to Egypt for the duration of a famine, and then travel back up to return to the exact same location between Beth-el and Ai. How can we use vectors or velocity to explain his trip?

Based on a rough estimate found here, let's assume that Avram's travel distance for his trip from Beth-el down to Egypt was approximately 225 miles. If we designate Beth-el as his starting point (the location where he wants to be) and we use the standard of North and East representing positive directions and South and West representing negative, then his trip down to Egypt is mileage expressed as a negative velocity. So, for example, he starts at 0 miles (when he's standing in Beth-el), and after traveling 100 miles towards Egypt, he had traveled 100 miles SouthWest, or -100 miles, because he was moving away from Beth-el in a negative direction. When he reached his final destination in Egypt, he had traveled 225 miles SouthWest, or -225 miles. On his return trip, he is now heading NorthEast, which is a positive direction. As he traveled back toward Beth-el, let's say he traveled 60 miles per day (velocity = 60 mph NE)- assuming he was traveling by camel.
*After 1 day, he was 60 miles closer to Beth-el, so he had traveled 60 miles NorthEast, or +60 miles
*After 2 days, he was another 60 miles closer, so he had traveled 120 miles NorthEast, or +120 miles
*After 3 days, he was another 60 miles closer, so he had traveled 180 miles NorthEast, or +180 miles
*On his 4th day, he covered the final 45 mi to reach his destination, and he stopped once he reached his 225 miles NorthEast, or +225 miles, to get to Beth-el.

Note that you can see here that by adding the individual vectors from each day, we get an accumulated larger vector, only because Avram was continuing his travel in the exact same direction. 

Extension Thoughts:
If we add the two vectors from each trip together, we get zero - he's back to his initial starting point. (-225 miles) + (+225 miles) = 0 miles. If we add the vectors after the first day of Avram's return trip, his total travels still end up as a negative vector. He travelled 225 miles SouthWest, and then 60 miles NorthEast. (-225 miles) + (+60 miles) = (-165 miles), or 165 miles SouthWest of the vector's starting point.

Everyday Activity:
The concept of vectors can be quite easily integrated into younger students learning about measurement and distance. Simple classroom activities can include students measuring distance from a reference point and then having them walk certain distances in directions related to your reference point and expressing their walks using vector language.

A simple everyday example of vectors: Have you ever paid attention to the highway distance markers as you drive along? Using those markers, you can say that you've driven 23 miles North toward your destination.

Noah- Geometry & Pythagorean Theorem

"G-d said to Noah, 'The end of all flesh has come before Me, for the earth is filled with robbery through them; and behold, I am about to destroy them from the earth. Make yourself an ark of gopher wood; make the ark with compartments, and tar it inside and out with pitch. This is how you should make it- three hundred cubits the length of the ark; fifty cubits its width; and thirty cubits its height. A light shall you make for the ark, and to a cubit finish it from above. The entrance of the ark you shall put in its side; make it with bottom, second, and third decks...'" ~Bereishit 6;13-16

Rashi on 6;16 explains that "And to a cubit finish it from above" refers to a sloped roof which narrowed at the top and was 1 cubit high at the topmost point, allowing the water to flow downward off the roof on both sides.

The idea for this post developed when I was working with a student on a different math project related to the ark. This was a question that arose, but since she was focused on a different mathematical concept, she chose not to pursue the question below. Since our conversation last year, I've been looking forward to investigating this question for this parsha.

Pythagorean Theorem:
The Pythagorean Theorem is a theorem that we use when calculating with right triangles (triangles that have a right or 90° angle).

In the diagram above, we have a right triangle with legs a and b and hypotenuse c. In a right triangle, the hypotenuse is always the longest side and it is always the side across from the right angle. The legs are the two sides that join together to make the right angle. In our diagram, around the sides of the triangle are squares. Each square is made from sides that are the lengths of the three sides of the triangle. The Pythagorean Theorem tells us that if you take the squares of a and b, the areas of those two squares can be used to make up the exact same area as the square of c. Mathematically, for older students, this concept is easily written out as:

a² + b² = c²

For younger students, if they are able to calculate a number multiplied by itself, then they can make the calculations to figure out the areas of these squares.

For practical calculation purposes, if you know any two sides of a right triangle, you can use this theorem to calculate the missing third side.

Parsha Connection:

We are given specific dimensions for the construction of the ark: 

*length- three hundred cubits 

*width- fifty cubits

*height- thirty cubits

However, we are also told that the roof should be 1 cubit high and, according to Rashi, it slopes for the flow of water off the roof. Essentially, the roof is created in the form of an isosceles triangle (a triangle with at least 2 sides of equal length), with the vertex of the triangle up at the top of the ark. 

Using what we know about the dimensions of the ark and the Pythagorean Theorem, how can we figure out the width of the wood needed to form the two sloping sides of the roof?

Let's visualize a view of the roof from the front of the ark.

If we're looking head-on, this could be a sketch of the roof of the ark:

*The 2 sides, x, would be each of the two sloping sides of the roof. 

*The height, h, at the center, would be the 1 cubit height that is described in the parsha.

*The bottom, b, would be the width from one side of the ark to the other- 50 cubits, as described in the parsha.

But how does this help us? This is an isosceles triangle. We know that the Pythagorean Theorem only applies to right triangles. If you look carefully at the triangle above, you can see that we can actually break the isosceles triangle into two congruent (exactly the same) triangles- if we cut the isosceles triangle along the height line, h, then we have two congruent right triangles. The legs of the triangle will be h and half of the bottom b; the hypotenuse of each triangle will be the slopes- x. 

Let's put this information back into the format for the Pythagorean Theorem:

a² + b² = c<sup>2

The a will be replaced with our h- the height line.
*Remember we said that h was the height given in the parsha- 1 cubit.
The b will be replaced with our new b number- 1/2 of the width distance b.
*Remember we said that this was the width given in the parsha- the full width of the ark was 50 cubits, so half of that width (to calculate for the right triangle) is 25 cubits.
The c will be replaced with our x- the slopes of the roof.
*This is the missing information that we're looking for.</sup>

So, now we can put our own numbers into the set-up for calculation:
a² + b² = c²
h² + b² = x²
1² + 25² = x<sup>2

Now we calculate:</sup>
1² = 1
25² = 625

So, 1 + 625 = x²
626 = x<sup>2

Now we need to know what number multiplied by itself will give us 626. For older students, they would either use a square root function, or, by hand, they could use prime factorization to possibly calculate the number. Younger students might use guess, check, and correct to narrow down the number to an approximate length.</sup> 

If we just look at the numbers, we know that 25² = 625, and we're looking at 626- a number very close. Just by looking at it, we can see that the length of the sloped side will be very close to 25 cubits, but a little bit longer (since 626 is larger than 625).

If we actually calculate the square root using a calculator, we get 25.02 cubits (if we round the number).
If we calculate using prime factorization, we get that 626 = 2 x 313. Since these are both prime numbers, this doesn't help us to come to a number.

So, with our calculation, we know that the pieces for the sloped roof of the ark needed to be approximately 25.02 cubits wide. The complete measurements of these sloped pieces needed to be 300 cubits long (to reach from front to back of the ark) and 25.02 cubits wide to reach from the tip of the roof down to the top of the sides of the ark.

Everyday Connection:
The world is a spacial environment. We have squares and rectangles all around us. Do you know that if you break squares and rectangles in half by making a diagonal cut from one upper corner to the opposite bottom corner you are actually creating two congruent right triangles? That length of the horizontal cut is actually the hypotenuse of each of those right triangles. Did you know that TV and computer monitor sizes are actually measured by that hypotenuse length? 

An activity for younger students is a geometry scavenger hunts, where they need to list different shapes that they see in the world around them and connect them to what they've learned in class. This can actually be done at any age and really helps students open their eyes to how the material they are learning is part of their environment. What triangles do you see around you every day?

Bereishit- Place Value and Regrouping

Addition and Place Values:
When younger children begin learning to add (and subtract) multi-digit numbers, the algorithms alone for these calculations can be confusing. As a starting point, children first need to understand the meaning behind what each digit in a number represents. As they expand their understanding, it is helpful for children to recognize the pattern of every three place values from the decimal point moving towards the left. Larger decimal place values come later, but, again, understanding the pattern of place value will help students pick up the larger picture of what's happening with the numbers.

Basic place values to the millions:

The biggest hurdle for students to overcome is that each place value can't have more than 9 of it's kind, and every time you reach a group of 10, you have reached the next place value and need to "regroup" your numbers.

*10 ones = 1 ten and 0 ones [10]
*10 tens = 1 hundred and 0 tens and 0 ones [100]
*10 hundreds = 1 thousand and 0 hundreds and 0 tens and 0 ones [1,000]
*10 thousands = 1 ten-thousand and 0 thousands and 0 hundreds and 0 tens and 0 ones [10,000]
*10 ten-thousands = 1 hundred-thousand and 0 ten-thousands and 0 thousands and 0 hundreds and 0 tens and 0 ones [100,000]
*10 hundred-thousands = 1 million and 0 hundred-thousands and 0 ten-thousands and 0 thousands and 0 hundreds and 0 tens and 0 ones [1,000,000]

It helps for children to visualize this concept using manipulatives such as base-10 blocks, which have models for ones, tens, hundreds, and thousands. Paper clips are a good make-shift manipulative, since students can clip and unclip into and out of groups of 10, 100, etc. as they make sense of the numbers.

One key that is critical for students to remember is that they need to line up the place values- you can only add ones with ones, tens with tens, hundreds with hundreds, etc. (and then regroup as needed).

Parsha Connection:
Chapter 5 of this week's parsha gives details of the direct lineage from Adam to Noah. We are given information on names and ages for this direct line of descendants. We are given enough information about each person to provide 9 sample addition problems, where we can check the numbers that we are given in the parsha.

As a first step, let's organize the individual pieces of information that we are given. As we organize information for each person, we can check the math.

1st generation:
"This is the account of the descendants of Adam- on the day of G-d's creating of Man, He made him in the likeness of G-d. He created them male and female. He blessed them and called their name Man on the day they were created- when Adam had lived one hundred and thirty years, he begot in his likeness and his image, and he named him Seth. And the days of Adam after begetting Seth were eight hundred years, and he begot sons and daughters. All the days that Adam lived were nine hundred and thirty years; and he died." ~Bereishit 5;1-5

Adam was 130 years old when Seth was born.
Adam lived 800 years after Seth was born.
Adam died at 930 years old

  130
+800 
  930    so this works

2nd generation:
"Seth lived one hundred and five years and begot Enosh. And Seth lived eight hundred and seven years after begetting Enosh, and he begot sons and daughters. All the days of Seth were nine hundred and twelve years; and he died." ~Bereishit 5;6-8

Seth was 105 years old when Enosh was born.
Seth lived 807 years after Enosh was born.
Seth died at 912 years old

  105
+807
  912    so this works

The above example offers regrouping in the ones (to tens) digits.

3rd generation:
"Enosh lived ninety years, and begot Kenan. And Enosh lived eight hundred and fifteen years after begetting Kenan, and he begot sons and daughters. All the days of Enosh were nine hundred and five years; and he died." ~Bereishit 5;9-11

Enosh was 90 years old when Kenan was born.
Enosh lived 815 years after Kenan was born.
Enosh died at 905 years old

   90
+815
  905    so this works

The above example offers a sample with different sized numbers (one tens and one hundreds), which offers a good check on understanding lining up the place values. Also, we have regrouping in the tens (to hundreds) digits here.

4th generation:
"Kenan lived seventy years, and begot Mahalalel. And Kenan lived eight hundred and forty years after begetting Mahalalel, and he begot sons and daughters. All the days of Kenan were nine hundred and ten years; and he died." ~Bereishit 5;11-14

Kenan was 70 years old when Mahalalel was born.
Kenan lived 840 years after Mahalalel was born.
Kenan died at 910 years old

   70
+840
  910    so this works

Again, here we have different sized numbers and regrouping in the tens (to hundreds) digits.

5th generation:
"Mahalalel lived sixty-five years, and begot Jared. And Mahalalel lived eight hundred and thirty years after begetting Jared, and he begot sons and daughters. All the days of Mahalalal were eight hundred and ninety-five years; and he died."  ~Bereishit 5;15-17

Mahalalel was 65 years old when Jared was born.
Mahalalel lived 830 years after Jared was born.
Mahalalel died at 895 years old

   65
+830
  895    so this works

Again, here we have different sized numbers.

6th generation:

"Jared lived one hundred and sixty-two years, and begot Enoch. And Jared lived eight hundred years after begetting Enoch and he begot sons and daughters. All the days of Jared came to nine hundred and sixty-two years; and he died." ~Bereishit 5;18-20

Jared was 162 years old when Enoch was born.
Jared lived 800 years after Enoch was born.
Jared died at 962 years old

  162
+800
  962    so this works

7th generation:
"Enoch lived sixty-five years, and begot Methuselah. Enoch walked with G-d for three hundred years after begetting Mehuselah; and he begot sons and daughters. All the days of Enoch were three hundred and sixty-five years. And Enoch walked with G-d; then he was no more, for G-d had taken him." ~Bereishit 5;21-24

Enoch was 65 years old when Methuselah was born.
Enoch lived 300 years after Methuselah was born.
Enoch died at 365 years old

   65
+300
  365    so this works

8th generation:

"Methuselah lived one hundred and eighty-seven years and begot Lamech. And Methuselah lived seven hundred and eighty-two years after begetting Lamech, and he begot sons and daughters. All the days of Methuselah were nine hundred and sixty-nine years; and he died." ~Bereishit 5;25-27

Methuselah was 187 years old when Lamech was born.
Methuselah lived 782 years after Lamech was born.
Methuselah died at 969 years old

  187
+782
  969    so this works

Again, here we have regrouping in the tens (to hundreds) digits.

9th generation:
"Lamech lived one hundred and eighty-two years, and he begot a son. And he called his name Noah, saying, 'This one will bring us ease from our work and from the toil of our hands, from the ground which Hashem had cursed.' Lamech lived five hundred and ninety-five years after begetting Noah, and he begot sons and daughters. All the days of Lamech were seven hundred and seventy-seven years; and he died." ~Bereishit 5;28-31

Lamech was 182 years old when Noah was born.
Lamech lived 595 years after Noah was born.
Lamech died at 777 years old

  182
+595
  777    so this works

Again, here we have regrouping in the tens (to hundreds) digits.

10th generation:
"When Noah was five hundred years old, Noah begot Shem, Ham, and Japheth." ~Bereishit 5;32

Noah was 500 years old when Shem, Ham, and Japheth were born.

Follow-up Activities:
If you take this base information and use it to try to place the births and deaths in a timeline, students can calculate how old each person was at the births of successive descendants, as well as how old each of the descendants was at the time of death of their ancestors. These calculations will offer students more, similar calculation practice.

For an added exercise, having students identify which numbers to use for these different comparative calculations offers practice with problem solving and identifying important information.