Korach- Currency Equivalency Activities

"Every first issue of the womb of any flesh that they offer to Hashem, whether of man or of animal, shall be yours; but you shall surely redeem the firstborn of man, and the firstborn of an impure animal shall you redeem. And those that are to be redeemed- from one month shall you redeem according to the valuation, five silver shekalim by the sacred shekel; it is twenty geirah." ~Bamidbar 18:15-16

Towards the end of this week's parsha,  we read of what items are set aside to be given to Aaron and the Kohanim. Within this section, we learn that firstborn male infants and firstborn animals are part of what is to be given to the Kohanim. However, we also learn that firstborn children and impure firstborn animals are to be redeemed at the age of one month. The value for redeeming them is 5 silver shekalim, and we are told that one silver shekel is equivalent to 20 geirah.

Activity Suggestions related to Monetary Equivalence:
Learning equivalencies of monetary values can be confusing for young children. Opportunities for hands-on acting out with manipulatives, repetition, and practice help them become proficient in working with monetary conversions.

• Students can convert between geirahs and shekalim.                        
  Given that 1 shekel = 20 geirah:
• Representing the conversion-
• Younger students could calculate how many geirahs would be in 5 shekalim, by using manipulatives for representation. They could have cups to represent 1 shekel, and small chips or counters representing a geirah. Students could work together to create shekel value cups, each filled with 20 geirah. Once they have the cups set up, they can then go back and count out 5 shekalim. 5 cups would represent 5 shekalim. 
• Students who are ready could draw representative circles or boxes for shekalim and show 20 tally marks within each one to represent the geirahs in each. In this case, 5 circles or boxes would represent the 5 shekalim. 
• Making the calculations-
• Youngest students can work together (or as a class) to count out how many geirahs there are all together in the 5 shekalim. 
• Slightly older children could use their knowledge of "skip counting" or addition of simple large numbers to add the twenties together and make the calculation.
• Older students can use their knowledge of multiplication to problem solve and calculate how many geirahs would be in 5 shekalim.
• Follow-up calculations:
• Can students (young and old) figure out how many shekalim would be needed to redeem multiple firstborns? If 7 families redeem their firstborns, how many shekalim would be given to the Kohanim all together? How many geirahs would be given to the Kohanim all together?
• If you chart the redemption values in shekalim and geirahs for different numbers of firstborns, can you identify any patterns?
• For upper elementary or middle schoolers- can they convert the patterns that they find into functions? How would these functions look when graphed?
• Older students can also research conversion values to current day currencies. Based on the researched conversion rates, what is the value today of a geirah? of a shekel? How much would families need to pay to redeem firstborns in different countries today? Can they write a function to calculate redemption values for multiple firstborns using current currencies?

Tazria/Metzora- Investigating Patterns in Doubling

In Vayikra 12:2-5 we learn about the Biblical periods of impurity after a woman gives birth. We learn that after giving birth to a son, she is impure for one week (7 days) and she may not touch holy items or partake in eating the terumah for an additional 33 days. At the end of the 40 day period (7 + 33 = 40), she brings sacrifices. After giving birth to a daughter, she is impure for 2 weeks (14 days) and she may not touch holy items or partake in eating the terumah for an additional 66 days. At the end of the 80 day period (14 + 66 = 80), she brings sacrifices.

Doubling Numbers & The Sums of Numbers:
If we look at the timeline for a woman's post-delivery impurity and offering of sacrifices, we can see that the timeline after the delivery of a daughter is double the timeline after the delivery of a son. Let's take a minute to break it down piece by piece:

Post delivery impurity:
Son- 1 week (7 days)
Daughter- 2 weeks (14 days)

Additional time before bringing sacrifices:
Son- 33 days
Daughter- 66 days

Total time post-delivery until sacrifices are brought:
Son- 7 days + 33 days = 40 days
Daughter- 14 days + 66 days = 80 days

As a teacher, I have seen students many times find a pattern in the way numbers work together and automatically assume that the pattern will apply to all numbers. Sometimes their assumption is correct and they have found an accurate pattern in number theory; other times students stumble upon a neat trick that works with specific types of numbers but cannot be extrapolated to other numbers. Without thinking about and testing their pattern, students may inadvertently apply a neat trick as a greater application in number theory. 

With this in mind, I propose the following investigation for students:
If you double two numbers, will the sum of the doubled numbers always be double the sum of the original numbers?  (Note that this investigation is applicable for students of any age who aren't yet certain of the answer and explanation of why.)

Students can test this by picking a variety of different numbers and checking what happens when they double the numbers and compare the sums of different numbers- original and then doubled. Students can work in partners or individually and compare their findings with their classmates. It's important for students to consider that, while they can't test every number, if enough numbers follow their pattern, they can feel confident that their pattern holds true for most numbers. 

With enough testing, students should develop the basic premise of the theorem:
If a + b = c, then 2a + 2b = 2c

Through their testing, at least some students (at least by grade 5+) should develop enough of a thought process about their testing to begin to explain why this is true.

Some follow-up investigations:
Math related-
*Does this hold true when you add more addends to your original addition statement? i.e. if you add three numbers, will the sum of double those 3 numbers be double the original sum? 4? 5? etc. Why?
*Does this hold true for subtracting numbers?
*Does this hold true for multiplication? division?
*Does this hold true for negative numbers? fractions & decimals?

The more scenarios and number sets that they test, the stronger theory they can develop for use in their future work.

Math & Parsha related-
*Specifically related to this week's parsha, when you convert 33 days and 66 days into time measured in weeks & days, will they still appear to be doubled at first glance? Can you explain your findings? Based on your findings, do you think there's a reason that the Torah may have listed the time periods in the way that they are listed?

Mattot- Multiplying and Dividing powers of 10

"Moshe spoke to the people, saying, 'Arm men from among yourselves for the army that they may be against Midian to inflict Hashem's vengeance against Midian. A thousand from a tribe, a thousand from a tribe, for all the tribes of Israel shall you send the army.' 
So there were delivered from the thousands of the Children of Israel, a thousand from each tribe, twelve thousand armed for the army. Moshe sent them- a thousand from each tribe for the army- them and Pinchas son of Elazar the Kohen to the army, and the sacred vessels and the trumpets for sounding under his authority." ~Bamidbar 31;3-6

"Moshe, Elazar the Kohen, and all the leaders of the assembly went out to meet them outside the camp. Moshe was angry with the commanders of the legion, the officers of the thousands and the officers of the hundreds, who came from the army of the battle." ~Bamidbar 31;13-14

Math Connection:
For younger students, and even some older students, calculations with big numbers can be scary. The larger the number, the more room there is for errors. It is helpful for students to identify patterns in the way numbers work in order to help reduce anxiety and make calculations easier, and even faster. When dealing with multiplication and division of numbers that are powers of 10 (tens, hundreds, thousands,...), there is a simple trick that you can use to make the calculations easier. If you remove (or ignore) any extra zeros at the end of the number that aren't needed for the purpose of your calculation, you can multiply or divide as you normally would with your truncated number, and then add back your extra zeros onto the end of your product or quotient from your calculation. 

Some examples:

4,000 x 4: Set aside the 3 zeros at the end, leaving 4 x 4 = 16, and then add back the 3 zeros, making it 16,000.

650,000 x 2: Set aside the 4 zeros at the end, leaving 65 x 2 = 130, and then add back the 4 zeros, making it 1,300,000.

For multiplication, if both numbers are powers of 10, you can still set aside the zeros from the end of both numbers to simplify your calculation. After your calculation, you add all of the zeros from both numbers back to the end of your product. For example, 2,000 x 80 can be calculated by multiplying 2 x 8 = 16, and then add back on the 4 zeros (3 from 2,000 and 1 from 80), making it 160,000.

6,200 + 2: Set aside the 2 zeros at the end, leaving 62 + 2 = 31, and then add back the 2 zeros, making it 3,100. 

43,000 + 5: Here, 43 is not evenly divisible by 5, so we will take away just 2 zeros at the end, leaving 430 + 5 = 86, and then add back the 2 zeros, making it 8,600.

For division, if both numbers are powers of 10, you can permanently remove the same number of zeros from the end of both numbers to simplify your calculation. For example, 24,000 + 300 is the same as 2,400 + 30, which is the same as 240 + 3 (all of which = 80).

Parsha Connection:
In this week's parsha, we have both multiplication and division with powers of 10. First, when Moshe instructions the Children of Israel to create an army with 1,000 men from each of the twelve tribes, we can confirm the total of 12,000 that is given in the text. 

1,000 x 12: Remove the 3 zeros at the end, leaving 1 x 12 = 12, and then add back the 3 zeros, making it 12,000 men.

When they return from battle, Moshe is "angry with the commanders of the legion, the officers of the thousands and the officers of the hundreds, who came from the army of the battle." How many commanders were there?

Officers of the thousands:
12,000 + 1,000: Here, we have 3 zeros at the end of both numbers (dividend and divisor), so we can remove the 3 zeros from both, and are left with 12 + 1 = 12 commanders overseeing the thousands.

Officers of the hundreds:
12,000 + 100: Here, we have 2 zeros that we can take away from both numbers, leaving us with 120 + 1 = 120 commanders overseeing the hundreds.

So, there were 132 commanders (12 + 120) with whom Moshe was angry following the battle with Midian.

Everyday Connection:

Working with a committee on planning for a large event? What if you have 237 people attending your event and your tables each seat 8 people? How many tables will you need? Using estimation and our division trick, we can quickly calculate how many tables you'll need.

237 is just 3 short of 240. To calculate 240 + 8, let's use our trick- 24 + 8 = 3, then add the zero back, and we know that we need to set 30 tables for the event.

And what if you want to estimate a food budget for the event? Let's say $30 per person. Now we can use estimation and our multiplication trick to see if our food budget is realistic.

Again, we'll use 240. To calculate 240 x 30, we'll do 24 x 3 = 72, then add our 2 zeros back, and we have an estimate of $7,200, if you spend $30 per person.

Tetzaveh- Mixed Bag of Elementary Geometric Problem Solving

"You shall make the Choshen of Judgement the work of an artist, like the work of the Eiphod shall you make it, of gold, turquoise wool, and purple wool, and scarlet wool, and twisted linen shall you make it. Square shall it be, folded, a zeret its length and a zeret its width. You shall fill it with stone filling, four rows of stone: a row of odem, pit'dah, and barekes- the one row; the second row: nofech, sapir, and yahalom; the third row: leshem, shevo, and achlamah; and the fourth row: tarshish, shoham, and yashfeh; they shall be of golden settings with their fillings. The stones shall be according to the names of the sons of Israel, twelve according to their names, like the engraving of a signet ring, each man by his name shall they be, for the twelve tribes." ~Shemot 28; 15-21

This week's parsha gives us a description of the structure of the Choshen, or the breastplate, that the Cohen Gadol (High Priest) wore. The mathematical parameters that we are given are:

--1 zeret x 1 zeret square, when folded (this is following Rashi's interpretation of the text; also according to Rashi, 1 zeret = 1/2 amah or approx 9-12 inches)

--12 stones, representative of the 12 tribes

--the 12 stones should be organized as 4 rows of 3 stones each

I'll provide here some problem solving questions and answers, but the true way to learn through these questions is by making models or diagraming the problems to sketch out a way of calculating the answers.

Geometric problem solving:

If we know that it needs to make a 1 zeret x 1 zeret square when it's folded once, what are the dimensions of the original shape before it is folded? What is the name of the original shape?

*Answer- In order to be folded once and become a square, we know that, on the original shape, one side is 1 zeret long. The second side, in order to be 1 zeret when folded, would have to be twice as long originally, or 2 zerets (zratim? zratot?) long. Thus, the original shape, before folding, was 1 zeret x 2 zrat__, which would give us a rectangle.

Follow-up for higher levels could be to calculate the area of the Choshen (the original rectangle, and the folded square)- What's the area measured in zrat__, amot, and modern day measures? Do you see a pattern in the rectangular areas compared to the square areas? (hint: you should!)

Representing Multiplication Arrays as Rectangles:

We are told that the 12 tribes were represented in a 4 x 3 arrangement on the Choshen. What other possible organizations could have been used to lay out the 12 stones (assuming complete rows and columns)?

*Answer- When organizing a given number of items into evenly divided rows and columns, we are essentially looking for the possible combinations of factor pairs for the given number. Laying items out in rows and columns (or drawing them out) is a common technique for diagraming factor pairs when students are first processing multiplication facts. This both helps them come up with factor pairs for a given number and also helps set a basis for areas of rectangular shapes when they move into geometry. These rectangular diagrams of factor pairs are called arrays. 

Here we are looking for all possible factor pairs that will make 12. The Torah gives us 4 x 3 (4 rows, with 3 columns in each). What else can we find?

1 x 12

2 x 6

3 x 4

4 x 3 (Torah's description)

6 x 2

12 x 1

In total, there are 6 possible arrays for 12 items. Half of them are repetitions (1 x 12 can be the same as 12 x 1), but it's important for students to understand that swapping the length and width results in the same total number of items (or area)- an internalization of the Commutative Property: changing the order in multiplication or addition facts does not change the final product or sum.

What shapes could the stones be in order to fit into the Choshen as it is described?

*Answer- They could have had round or square stones, which would have left uneven spaces between rows and columns. Alternatively, if they were oval or rectangular, it would have made it possible to allow for equal spacing between rows and columns on the square plate. Try it and see.