Ki Teitzei- Activity suggestions related to weights & measures and fractions

A couple of suggested activities related to this week's parsha:
In last year's post, I wrote about the algebraic connection to the mitzvah (commandment) of being honest in weights and measures. Here are some suggested pre-algebra activities-

• using a balance scale, students can have different sized blocks which they need to place on the scale to figure out which is greater (heavier) and which is lesser (lighter). From there, they can investigate how many of the lesser weights it takes to balance the greater weight.
• a more difficult variation- using multiple unlabeled blocks/weights, students can work to:
•  order them from lightest to heaviest
• determine if multiples of the same block are equal to (balance) any of the other blocks
• determine if any combinations of blocks are equal to (balance) any of the other single blocks
• determine if any combinations of blocks are equal to (balance) any combinations of other blocks
• using labeled weights, students can confirm different combinations of smaller weights that should equal (or balance) with a single larger weight- for example, they should find that 2+3 is equal to (balances) a 5.
• students can then "mix and match" to find different combinations of weights that are equivalent to each other- for example, they should find that a 2+5 is equal to (balances) a 3+4.

There are many commandments (mitzvot) enumerated in this week's parsha. It has been counted that, in fact, 74 out of the 613 commandments have a basis in this parsha. Students can consider-

• What fraction of mitzvot are based in the parsha?
• Can this fraction be reduced?
• a related concept- what is the prime factorization of these two numbers? Did you know that you can quickly reduce large fractions by finding the prime factorization of both the numerator and denominator and crossing out "pairs" of common numbers that appear in both.
• a simple example- to reduce 10/15, we could say that 10 = 2x5 and 15 = 3x5; since there is a 5 in the prime factorization of both the numerator and the denominator, we can cross out both 5's, and we are left with 2/3. In this way, we have reduced 10/15 to 2/3 using prime factorization.
• another related concept- investigating common factors and identifying prime numbers (numbers that have only 2 factors- 1 and the number itself)
• What fraction of mitzvot are not based in the parsha? In other words, what's the fraction of other mitzvot that are not listed?
• Can this fraction be reduced?
• What percentage of mitzvot are listed in this parsha? What percentage of mitzvot are not listed in this parsha?
• What is the ratio of mitzvot in this parsha to the mitzvot not in this parsha?

Eikev- An activity in comparisons and estimated measures

Last year for Parshat Eikev, I wrote about Rashi and Tanchuma's commentaries regarding the size and weight of the stone tablets that Moshe brought down with the 10 commandments inscribed on them. Their commentaries explained that despite there being differences between the two tablets, the size and weights of the two tablets were actually identical.

Thoughts on a follow-up activity:

• Last week in Parshat Va'etchanan (Devarim 5:6-18) we read the 10 commandments. These commandments were divided into the first 5, which are between man and Hashem and inscribed on the first tablet, and the second 5, which are between man and man and inscribed on the second tablet.
• If you compare, side by side, the wording, number of letters, number of characters, etc. between the two subsets of commandments, what are the actual differences between them? Which has more words, letters, characters, etc. and which has fewer? 
• Think about how you might compare the inscriptions proportionally to each other? 
• Is there a way to calculate- if I start with two 25 lb. tablets, and then inscribe in each one, is there a way to estimate how much each one should have weighed at the end? 
• Could you estimate surface area if you take a standard font and printed onto a sheet of paper- what's the ratio of print to total surface area for each tablet? What might the proportional volume differences be? What about weight differences? 
• There is a discussion in Gemara Bava Batra 14a wherein it is determined that the size of each tablet was 6 tefachim (handbreadths) high, 6 tefachim wide, and 3 tefachim deep. 
• Based on these measurements, what might the original volume of each tablet have been?
• Based on these measurements, what might the original weight of each tablet have been?
• Using estimated proportionally carved out sections from each tablet, what would one have assumed the new volumes/weights of each tablet to have been after the inscription of the commandments?
• Older students can also work through the measurement calculations from the source in Bava Batra related to the ark (where the tablets were kept) and the tablets, to see how the measurements were determined.

Ki Teitzei- Weights & Measures, Balanced Scales

"You shall not have in your pouch a stone and a stone- a large one and a small one. You shall not have in your house a measure and a measure- a large one and a small one. A perfect and honest stone shall you have, a perfect and honest measure shall you have, so that your days shall be lengthened on the land that Hashem, your G-d, gives you. For an abomination of Hashem, your G-d, are all who do this, all who act fraudulently." ~Devarim 25;13-16

Rashi on these passages explains that these stones refer to weights and measures. He also clarifies that it's not saying that you're not allowed to use different size weights. Rather, it means that you may not use two weights that are different weights but look to be the same, which would enable you to trick someone else into thinking that you are using the heavier weight when you're really using the lighter one.

Weights & Measures- Balanced Scales:
Classic scales, ones that were used before analog and digital scales with internal weight mechanisms were developed, worked by balancing two sides with each other. With an item of weight on either side, if the right side dips lower, then the item(s) on the right side are heavier; if the left side dips lower, then the item(s) on the left side are heavier; if the sides are even with each other, then the items on the two sides weigh the same amount. 

The concept of balancing a scale is also one that is commonly used now when teaching pre-algebraic and algebraic concepts. In this format, the idea of balancing equivalent combinations of numbers and variables is compared to balancing weights on a scale. If you know that the two sides of an equation are balanced, then you can perform the same operation to both sides of the balance- similar to adding or subtracting the same amount of weights on both sides of a scale- in order to isolate a variable on one side while keeping the equation balanced so the other side tells you the value of the variable.

Some examples:

If you know that-

x + 3 = 15

the "=" tells us that x + 3 is the same as (balances with) 15. Imagine that the "x" is one weight with an unknown value and the "3" is another weight with a value of 3. The "15" is a single weight with a value of 15. For younger children who need to physically manipulate to help them work through the problem, it might be an unknown weight, 3 weights with a value of 1, and 15 weights with a value of one. This set-up means that if you take away 3 from both sides, the scale (so to speak) will remain balanced. This leaves us with just our unknown, "X" weight on one side and [15 - 3 =] 12 on the other side. So, now we know that the unknown weight has a value of 12.

A more complex example:

If you know that-

5 x Y = 20

again, the "=" tells us that the 5 x Y is the same as (balances with) 20. Here, we imagine that we have 5 weights which all have the same unknown value of "Y" on one side of the scale and a weight with value of 20 (or 20 weights with a value of 1) on the other side. This set-up means that if you divide both sides into 5 equal groups, you can match-up groups of equivalent values. When we divide the "5 x Y" side by 5, we will have the 5 weights separated into 5 groups of one weight in each. With the more simplistic set-up, we can divide the 20 weights into 5 groups, and we'll have 4 weights with a value of 1 in each group. This means that 1 weight "Y" is the same as 4 weights. So, the unknown weight has a value of 4.

Parsha Connection:
In this week's parsha, we are warned against not using two weights that have the same shape and size, but have different weights. If you think about labeled weights that we use nowadays, does that mean that we're not allowed to have more than one weight? I have to choose if I'm going to measure everything with a 1 lb weight or a 5 lb weight? Rashi explains that it means that I can have weights of different weight, but I can't have, for example, a 1 lb weight and a 5 lb weight that look to be the same shape and size. You need to have weights that are clearly distinguishable from one another so that when you are weighing out items of value, it will be clear that you are in no way cheating regarding the value of the items being weighed.

Everyday Connection:
Practicing with balancing scales gives children an opportunity to manipulate the concept of equivalence. Building an understanding of being able to manipulate both sides of a scale in the same way and still maintain equivalence is a critical skill for developing algebraic thinking.

Have you every tried playing on a see-saw? If you have two people who are closer in weight, then they will balance each other and can have fun bouncing each end up and down. Sometimes, if you have one heavier person and two lighter people, you can put the two lighter people together on one side and they will approximately balance the one heavier person. If the heavier person tried to balance with just one lighter person on the other side, the heavier person will be stuck down on the ground, while his friend is stuck up in the air- not heavy enough to weigh himself down against his friend.

Chayei Sarah - Unit Conversion

"And it was, when the camels had finished drinking, the man took a golden nose ring, its weight a beka, and two bracelets on her arms, ten gold shekels their weight." ~Bereishit 24;22

Unit Conversion:

Unit (or Unit of Measure): a standard amount of a physical quantity (ref. dictionary.com)

In Math and Science, units are a critical aspect of calculation. Units, in simple terms, are the base quantity that we choose to use to measure physical properties- a length, a weight, a volume, etc. You can measure with standard units- units that are already established by society and have specific value assigned to them (e.g inch, cm, pound, gram, gallon, liter, etc.); You can also measure with non-standard units- units that you choose for you convenience, but are not standardized to all places and situations (e.g. the paperclips that I have in a box on my desk, or the length of my right foot).

In order to compare measurements of different items, we need to measure them using the same units. This is where unit conversion comes in. If we have items measured in different units, then we need to convert them all to the same units of measure in order to be able to say how they compare to each other. We need to know how many cm are in an inch, and then make sure that all items we're comparing are measured in either cm or inches. In general, it doesn't matter which measure you choose to use, you just need to pick one and make sure that everything that you're comparing is measured with the same measure. If I want to compare my height over the past 5 years, I can measure myself consistently in ft, or I can have some measurements in ft and some in meters, but in order to compare, I need to convert all the measures to either ft or meters to know how my height has changed. Note that in order to calculate the conversion from ft to meters or vice versa, I need to know that 1 meter is approximately 3.28 ft.

So, how does this connect to the parsha?

In Chayei Sarah (Bereishit 24), Avraham's servant, Eliezer, is sent to find a wife for Yitzchak. When Rivkah meets him at the well in Nahor and she offers water to both him and his camels, he realizes that she is the right woman for Yitzchak. He then, in Bereishit 24:22, gives a gift to her-- and here is where we find units of measure. Eliezer gives her "...a golden nose ring, its weight a beka, and two bracelets on her arms, ten gold shekels their weight."

So, what if we want to know how much the weight of her jewelry was altogether? Or, how big was the nose ring compared to the bracelets? How can we compare the weight of a beka and the weight of ten shekels? Rashi helps us out here and tells us that a beka is the weight of a half-shekel (see Rashi on this Pasuk to see how he knows this). So, we now we know that the nose ring was the weight of a half-shekel and the two bracelets were the weight of 10 shekels. So, the weight of all of Rivkah's jewelry from Eliezer was 10.5 shekels. The weight of the two bracelets together was 20 times the weight of the nose ring. Or, if we assume that the bracelets were equal in weight, then each bracelet was 10 times the weight of the nose ring.