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?

Ki Tisa- Proportions

"Hashem spoke to Moshe, saying: 'Now you, take for yourself prime spices: five hundred shekel-weights of pure myrrh; and aromatic cinnamon, half of it, two hundred and fifty; and cane of aromatic spice, two hundred and fifty; and kiddah, five hundred- in the sacred shekel-weight, and a hin of olive oil. Of it you shall make oil of sacred anointment, a blend of mixture, the work of a spice-blender; it shall be oil of sacred anointment...'" ~Shemot 30;22-25

"To the Children of Israel You shall speak, saying: 'This shall be for Me oil of sacred anointment for your generations. He shall not smear on the flesh of man and in its measure you shall not make anything like it; it is holy, it shall be holy for you...'" ~Shemot 30;31-32

"Hashem said to Moshe: 'Take yourself spices- stacte, onycha and galbanum- spices and pure frankincense: they shall be equal one to another. You shall make it into incense, the work of a spice-blender, thoroughly mixed, pure and holy...'" ~Shemot 30;34-35

"'...The incense that you shall make- in its measure you shall not make for yourselves; it shall be holy to you for Hashem...'" ~Shemot 30;37

In this week's parsha, we are given 2 recipes, one for anointing oil and one for incense - both for use in the Mishkan, and given explicit instruction about whether or not they can be replicated in any form. There is an interesting commentary by the Nachalat Yaakov explaining Shemot 30;32 and 30;37. The commentary explains that while the recipe for the anointing oil can be recreated in proportion to the recipe in the Torah (but not in the exact recipe itself), the recipe for incense may not be recreated itself or in any proportional recipe, greater or lesser.

Definitions:
Proportion- comparative relation between things or magnitudes as to size, quantity, number, etc. (dictionary.com)
What does this mean? Back in Parshat Vayishlach we took a look at ratios and compared how quantities of items in a group compare to each other. The idea of proportion is maintaining these existing ratios within the whole group. Here we'll look at what that means.

Baking is always a fun, hands-on way of incorporating math skills into an activity. In lower grades, just measuring out ingredients and following the directions is an important skill. As students get older, calculating proportionally larger or smaller recipes gets them working with multiplication, division, ratios, and fractions. When working with recipes, the traditional form of a recipe (3 eggs, 1 C flour,...) is good for multiplying and dividing to make it larger or smaller, but a more basic, generic recipe that is solely based on proportions (1 part water, 2 parts flour,...) actually fits much more easily into lessons on proportionality, since the nature of the recipe is based on comparison from the start.

Let's take a look, now, at the recipes in this week's parsha.
Anointing Oil:
*500 shekel-weights pure myrrh
*half of it, 250 shekel-weights aromatic cinnamon (commentary explains that it's actually 500 total, but weighed out in half-size portions)
*250 shekel-weights cane of aromatic spice
*500 shekel-weights kiddah
*1 hin of olive oil

Incense:
*stacte
*onycha
*galbanum
*additional spices (7 other spices, according to Rashi citing the Mishna)
*pure frankincense
they shall be equal one to another.
Rashi's interpretation here is that the spices specifically mentioned (stacte, onycha, galbanum, and pure frankincense) were all equal amounts (1 to 1 ratio for all 4). Rashi explains that the amount for each was 70 maneh in weight. Rashi does not explain the possible ratio for the other additional spices.

It's interesting to note, that the recipe for the anointing oil, which can be replicated in larger or smaller quantities, is given in specific weight amounts, and the incense, which may not be replicated in any proportion, is given in a proportional recipe, without any measurements. Perhaps the missing proportions are missing to keep us from actually attempting to replicate the recipe?

So what if we wanted to make some nice tiny vials of anointing oil to use as a perfume? How could we make a smaller recipe? Let's divide the whole recipe by 5 to get our smaller proportionally similar recipe:

*500 ÷ 5 = 100 shekel-weights pure myrrh
*half of it, 250 ÷ 5 = 50 shekel-weights aromatic cinnamon (commentary explains that it's actually 500 ÷ 5 = 100 total, but weighed out in half-size portions)
*250 ÷ 5 = 50 shekel-weights cane of aromatic spice
*500 ÷ 5 = 100 shekel-weights kiddah
*1 ÷ 5 = 1/5 hin of olive oil

Therefore, our new, smaller recipe would be:
*100 shekel-weights pure myrrh
*two batches of 50 shekel-weights aromatic cinnamon (100 total weight)
*50 shekel-weights cane of aromatic spice
*100 shekel-weights kiddah
*1/5 hin of olive oil

Want to make it larger, rather than smaller? Just multiply each ingredient by the same number instead of dividing.

Parshat Vayishlach- Ratios and Fractions

"...then [Jacob] 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

While my other posts addressed more mid to upper level math skills (within a K-8 range), this week's topic deals with more simplistic concepts. A nice, engaging, educational method for younger students is to introduce a concept and then give them lots and lots of opportunities to practice that concept. With this chart below, once you get the hang of what's happening, there's lots of room for organizing the information in different ways to offer lots and lots of practice with the idea of fractions and ratios.

Let's start by organizing our information 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

A fraction is a number that tells you what part of a whole group you have. A fraction is written as one number over another with a bar between them (eg. 1/2). The top number, or "numerator", is the number that tells you how many pieces you have from the group; the bottom number, or "denominator", is the number that tells you how many pieces were in the group all together. So, the fraction 1/2 tells us that our group had 2 pieces all together, and we had 1 of those pieces.

A ratio is a comparison of numbers within a group. A ratio can be written as a fraction, with a colon separating the numbers that you're comparing, or writing "to" between the numbers. When written as a fraction, one number is written in the numerator spot and the other is written in the denominator spot; when written with a colon, the numbers are written next to each other with a colon separating them. So, if we have 15 marbles composed of 8 blue marbles and 7 red marbles, the ratio of blue to red is 8/7, 8:7, or 8 to 7. We can flip them around, too, to say that the ratio of red to blue is 7/8, 7:8, or 7 to 8.

Both fractions and ratios can be reduced to use smaller numbers to describe a situation. For example, if I have 8 cookies, and I ate 4 of them, I could say that I ate 4/8 of the cookies, or I could reduce that fraction to say that I ate 1/2 of the cookies. Another example- if I have 20 marbles composed of 5 purple marbles and 15 green marbles, I could say that the ratio of purple to green is 5:15, or I could reduce that to say that the ratio is 1:3. If both numbers in a fraction or ratio are divisible (can be divided) by the same number, then you can divide them to get to a reduced fraction or ratio.

Using our chart of the tribute gifts that Jacob set aside for Esav, we can use fractions and ratios to make different descriptions and comparisons of the types of animals. (Note: To avoid confusion, I'll use the colon for ratios and the bar for fractions.)

What fraction of the animals were female? 490/550, or reduced- 49/55. This means that if all the animals were divided into exactly the same groups, for every group of 55 animals, 49 of them would be female.

What was the ratio of female to male animals? 490:60, or reduced- 49:6. This means that if all the animals were divided into exactly the same groups, each group would have 49 female animals and 6 male animals. Notice that when you add the ratio numbers together, you get the total number of animals (490 females + 60 males= 550 animals) and this works for the reduced fractions and ratios, as well (49 females + 6 males= 55 animals in each group).

What do the comparisons look like for individual types of animals?

Goats: 
--Fraction of females: 200/220 or 20/22 or 10/11. So, for every 11 goats, 10 were female.
--Fraction of males: 20/220 or 2/22 or 1/11. So, for every 11 goats, 1 was male.
--Ratio of females to males: 200:20 or 20:2 or 10:1. So, for every 10 female goats, there was 1 male goat.
Do you see how nicely the fractions and ratios fit together? How the numbers are consistent, even when they're reduced, so that you can make comparisons between males and females and also compare sections of the whole group to each other? 

Let's try another animal. Ewes and Rams have the same ratio as Goats, and the Camels only had females and children- no males, so nothing to compare within the category.

Cows/Bulls:
--Fraction of females: 40/50 or 4/5. So, for every group of 5 cows & bulls, 4 were female.
--Fraction of males: 10/50 or 1/5. So, for every group of 5 cows & bulls, 1 was male.
--Ratio of females to males: 40:10 or 4:1. So, for every 4 cows, there was 1 bull.

Donkeys:
--Fraction of females: 20/30 or 2/3. So, for every 3 donkeys, 2 were female.
--Fraction of males: 10/30 or 1/3. So, for every 3 donkeys, 1 was male.
--Ratio of females to males: 20:10 or 2:1. So, for every 2 female donkeys, there was 1 male donkey.

There are lots of other fractions and ratios that we could look at here:
--What fraction of all animals were goats? What fraction were camels?...
--What was the ratio of goats to camels? goats to donkeys?...
--What fraction of all female animals were female goats? ewes? camels?...
--What was the ratio of female goats to female donkeys? male donkeys to bulls?...

If you're really getting into the comparisons, you can actually calculate how many different comparisons could be made between the animals, but that's getting into combinatorics- another topic for another time.

What interesting comparisons can you make? Do you see any interesting patterns in your fractions or ratios?