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?

Re'eh- Activity investigations related to yearly tithings

In this week's parsha, we have two mentions of the tithings that were to be given at different times over the course of the shmita cycle. Additionally, tithings are referred to again in 3 weeks, in Parshat Ki Tavo. These three instances are looked at collectively in commentaries regarding the shmita cycle. 

• "You may not eat in your cities: the tithe of your grain, and your wine, and your oil; and the firstborn of your cattle and your flocks; all your vow offerings that you vow and your free-will offerings; and what is raised of your hand. Rather you shall eat them before Hashem, your G-d, in the place that Hashem, your G-d, will choose- you, your son, your daughter, your slave, your maidservant, and the Levite who is in your cities- and you shall rejoice before Hashem, your G-d, in your every undertaking. Beware for yourself lest you forsake the Levite, all your days in your land." ~Devarim 12:17-19
• Rashi on 12:18- regarding eating the tithings with the Levite in your city:
• Eat your first tithing together with him (1/10 portion of the crop given to Levites in each of the first 6 yrs of shemita)
• If you have no first tithe, then give from tithe for poor (1/10 of portion given to poor in third and sixth years (additional reference see 14:28-29, 26:12)
• If you have no poor tithe, then share your peace offering
• "At the end of three years you shall take out every tithe of your crop in that year and set it down within your cities. Then the Levite shall come- for he has no portion or inheritance with you- and the convert, and the orphan, and the widow who are in your cities, and they shall eat and be satisfied, in order that Hashem, your G-d, will bless you in all your handiwork that you may undertake." ~Devarim 14:28-29
• "When you have finished tithing every tithe of your produce of the third year, the year of the tithe, and you will have given to the Levite, to the convert, to the orphan, and to the widow, and they will have eaten in your cities and will have been satisfied, then you shall say before Hashem, your G-d, "I have eliminated the holy things from the house, and I have also given it to the Levite, to the convert, to the orphan, and to the widow, according to the entire commandment that you commanded me; I have not transgressed any of Your commandments, and I have not forgotten..." ~Devarim 26:12-13 (from Parshat Ki Tavo)

What were the tithings and the cycles?

• 1st tithe- 1/10 to the Levites
• 2nd tithe- 1/10 to be brought to Jerusalem and eaten there
• poor tithe- 1/10 to be given to the poor

The cycle of tithings over the shmita cycle was as follows:

Year 1- 1st tithes and 2nd tithes given

Year 2- 1st tithes and 2nd tithes given

Year 3- 1st tithes and tithes for poor given

Year 4- 1st tithes and 2nd tithes given

Year 5- 1st tithes and 2nd tithes given

Year 6- 1st tithes and tithes for poor given

Year 7- shmita year- no tithings

Activity suggestions:

• Younger children could investigate what it means to give 1/10. If you have a set-up for them to take a collection of items and divide them into ten groups, they would be donating one of those groups. The idea of 1 out of 10 might be conceptually easier for some than 1/10. In other words, for every 10 that's count out, 1 is going to be donated. 
• As children get to a point where they understand the concept of division by 10, they can practice some simple sample calculations of how much might have been donated based on a given monetary value. 
• Older children could be presented with sample scenarios of realistic incomes- some monetary based and some agriculturally based- and have them calculate projected tithings (per tithe and total donations) that families may have made over a 7 year shmita cycle. 
• Note that older students should also be able to make the connection that each 1/10 tithing would be equivalent 10% of their produce or earnings.
• It's also possible that, for some students, it would need to be clarified that all tithings were 1/10 of the total earnings; the 2nd tithing would not be calculated as 1/10 of what's left after the 1st tithing is given. This may be obvious to many, but for students who are making sense of the process, this may be a question that is unclear to them without explicit clarification.
• Extension thought- Based on these calculations, how could they think about how much the Levites had to live off of in a given year? This type of investigation could include researching information such as census information for each tribe listed in recent parshiot- if all the Levites were living off of tithings from those families, how much did they really have to live off of, per Levite family?

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.

Balak- Mathematical Insights Using Percentages and Other Basic Math

"These are the counted ones that Moses, Aaron, and the princes of Israel counted- twelve men, one man for his father's house, were they- these were all the counted ones of the Children of Israel, according to their fathers' house, from twenty years of age and up, everyone who goes out to the army in Israel: All their counted ones were six hundred and three thousand, five hundred and fifty. The Levites according to their fathers' tribe were not counted among them." ~Bamidbar 1;44-47 (Parshat Bamidbar)

In this week's parsha, following the plague as punishment for Israel becoming attached to Baal-peor:
"Hashem said to Moses, 'Take all the leaders of the people, and hang [the idol worshipers] before Hashem  opposite the sun- and the flaring wrath of Hashem will withdraw from Israel.' Moses said to the judges of Israel, 'Let each man kill his men who were attached to Baal-peor.'" ~Bamidbar 25;4-5
"Those who died in the plague were twenty-four thousand." ~Bamidbar 25;9

Rashi on Bamidbar 25;5 says:

Let each man kill his men: Each and every one of the judges of Israel would kill two and the judges of Israel were seven myriads (seventy thousand), and eight thousand as stated in Sanhedrin.

At first, Rashi's commentary confused me. At the end of the parsha, we are explicitly told that 24,000 died as the result of the plague. Rashi here is stating that 78,000 judges each killed 2 people. The numbers alone don't make sense- how did 78,000 judges kill only 24,000 people (comparing numbers- good lower elementary skill)? If we do the basic multiplication (middle elementary skill), to calculate Rashi's understanding that each judge killed 2 sinners, then 

78,000 Judges x 2 sinners killed = 156,000 men killed.

How does this make sense? I did a little more research, and it seems that Gemara Sanhedrin 82a talks about the Israelite man and the Midianite woman being Zimri son of Salu (from the tribe of Shimon) and Cozbi daughter of Zur (Midianite princess). In the gemara, it says that Zimri brought with him 24,000 Israelites from his tribe, which is what led him to sin (that's some peer pressure!). While it's not stated explicitly, my understanding from this explanation would then be that the 24,000 that we are told of in Bamidbar 25;9 are just the 24,000 Israelites from the tribe of Shimon, and that the additional sinners who were killed by the judges actually did number 156,000.

Now that we have some numbers to go on, let's work with some calculations:

*Let's start by figuring out how many people actually died all together as a result of this situation: (Good practice of long addition algorithm for younger students- remember to line up place values)

24,000 + 156,000 = 180,000 total Israelites who died 

*If we go based on these two accountings, how many Jews were left in the general population (non-Levite) after the plague? (Good practice of long subtraction algorithm for younger students- remember to line up place values)

603,550 - 180,000 = 423,550 total non-Levite Israelites who were left

*What percentage of the general population of Jews were killed in the plague? (Good calculation practice for Middle School)

Following our methods from Parshat Vayigash:
180,000/603,550 = 0.29823544031, or approximately 30%
This means that approximately 70% of the general population was left after the plague was finished. Let's check and see if those numbers are consistent:
423,550/603550 = 0.70176455968, or approximately 70%

An interesting follow-up activity:
If we are assuming that the 24,000 mentioned in Bamidbar 25;9 were all from Shimon, how would our percentage calculations look if we just took the census information for Shimon from back in Bamidbar and compared what percentage of Shimon died as a result of the plague. When making these calculations, we could also consider that Zimri was from Shimon and was killed, but not accounted for in the 24,000. Also, the Judges from Shimon each killed 2 people, so the Israelites from Shimon who were killed by the Shimonite judges would also need to be added into the Shimonite death toll for an accurate calculation.

Tzav- Statistics part 2

Once again, this week we have a significant portion of the parsha dedicated to specifics of korbanot, or sacrificial offerings. Again, rather than quoting from the parsha, I've organized some of the information related to the offerings in order to analyze.

Pie Chart/Circle Graph:
Whereas the bar graphs we looked at last week compared a measured amount between different categories, a circle graph compares portion sizes of categories that together make up 100% of something. For example, out of your total budget allotment, what percentage of the budget goes to different categories (food, medical, housing costs, etc.)? Out of all after school enrollment, what percentage of students go to each of the different activities? Out of the total hours in a day, what percentage of time is spent sleeping, eating, working, etc.? The key to circle graphs is that all categories together add to 100% and create a complete circle. 

When analyzing data to convert into a circle graph, we'll need our knowledge of fractions (Parshat Vayishlach), proportions (Parshat Ki Tisa), and percentages (Parshat Vayigash) to convert the raw data into a format that will fit the circle graph.

Connection to Parsha:
Let's take a look at a sample data set from this week's parsha to walk us through the process. Let's look at what percentage of sacrifices were to be fully burned and what percentage of sacrifices were allowed to be partially eaten in some form.

We can start by making a chart to organize the information so that we see which category each sacrifice falls into.

Next, we use the chart to count up how many sacrifices are in each category out of the total number of sacrifices. According to our chart, out of the 6 total sacrifices, we have 3 sacrifices that are completely burned and 3 sacrifices that are partially burned and partially eaten.

Let's restate this as fractions- 
3/6 = burned 
3/6 = burned then eaten

If we want, we can reduce these fractions (each reduces to 1/2), but we don't have to. In this case, we will reduce them, since it will make our next calculation easier to work with. 

So, we have-
1/2 of the sacrifices are burned
1/2 of the sacrifices are burned then eaten

Now, remember that the most important part of a circle graph is that it shows a percentage out of 100%, where 100% (or all) of what you're analyzing makes up the full circle. So, if 100% of the sacrifices is all of them, we know that 1/2 of the sacrifices were burned: 100% x 1/2 = 50% and 1/2 of the sacrifices were burned then eaten: 100% x 1/2 = 50%. So now we have calculated that 50% of the sacrifices were totally burned and 50% were burned then eaten. Once we have this calculation, we can create our graph. 

While I made the graph electronically, you can also construct a circle and measure out the angle measures for each section using pencil, paper, drawing compass, and protractor. Such an activity offers a good opportunity for art/geometry integrations in math. To calculate the sections by hand, we need to first know a full circle is 360°. Since we need 1/2 of the circle for each category, we multiply 360° x 1/2 = 180°. This means that the angle measure for each section of the circle needs to be 180°. Measuring out this 180°, you'll find that you've perfectly divided the circle into two 50% sections, one for each category.

It happened that the numbers for this activity fell perfectly into two equal groups. While it may seem simplistic because of the easy numbers and category divisions, it would make a very simple way to practice going through the steps of the process and practicing drawing circles and interior angles. Once practicing with this easy example, another problem with more complex category divisions could be introduced.

Vayigash- Percentages

"And it will be at the ingatherings that you will give a fifth to Pharaoh; the [other] four parts shall be yours, as seed for the field, and for feeding yourselves and for those who are in your household, and to feed your young ones." ~Bereishit 47;24

Back in Parshat Vayishlach, we looked at how to create ratios and fractions with a set of numbers that represent portions of a whole group. In this week's parsha, we actually have fractions given to us straight out. Working as an agent of Pharoah, throughout the course of the years of famine, Joseph purchased land from all of the landowners across Egypt in exchange for giving them food from his food storage. We are then told that the Priests were given food, and the landowners- who now all lived on land belonging to Pharoah and Joseph- were given seeds to plant for crops. They were instructed to work the fields, grow crops from the seeds that they received, and when they harvested their crops, 1/5 would be given to Pharoah, and the other 4/5 was to be kept for feeding themselves, their families, and their servants.

Percentage:
A rate or proportion per one hundred (dictionary.com)

While fractions are great for comparing parts of a group, there are times when you want to be able to compare across different groups, which can become messy and difficult if you're trying to compare fractions with different denominators. One way to create a standardization for comparison of information is to use percentages. Percentages readjust all the different groups so that for each group, the whole amount is represented by 100. Any portion of the group is then a percentage from 0-100%. All the total percentages within a group, when added together, should equal 100%.

How do we convert our information from fractions to percentages?
There are two basic ways to convert a fraction into a percentage.
1) Just as we could simplify fractions by dividing both numerator and denominator by the same divisor, we can make equivalent larger fractions by multiplying both numerator and denominator by the same factor. Since a percentage is a fraction out of 100 (per cent), then if you can easily identify how to multiply or divide your original denominator to make it 100, then you do the same to your numerator to find your percent.
Two basic examples:
--To find the percentage of 1/2, we know that 2x50=100, and then we can calculate 1x50=50. So, the percentage for 1/2 is 50/100 or 50%.
--To find the percentage of 350/500, we know that 500÷5=100, and then we can calculate 350÷5=70, so the percentage for 350/500 is 70/100 or 70%.
2) Some numbers just don't calculate nicely using the first method- try 1/8 or 1/3, for example. For these cases, you can divide the numerator divided by the denominator (eg. 1÷8 or 1÷3). This calculation will result in a decimal number. (1/8=.125 and 1/3=.33333...). These numbers multiplied times 100 (so the decimal moves 2 places to the right) are your percentages (1/8=12.5% and 1/3=33.3% -rounded off).
Intuitive connection- the first two decimal places, from left to right are tenths and hundredths; the fractions over 100 are read as "fifty hundredths" or "seventy hundredths"; when we multiply the decimal times 100, we are just changing it into the number that we would use in the fraction over 100

Using our method for calculating percentages, we can now look back to see what percentage of crops Joseph was collecting as taxes on Pharoah's behalf, and what percentage the previous landowners were allowed to keep for themselves and their families.

We are told that 1/5 was to be for Pharoah and the other 4/5 would be for the families. We lucked out here, since fifths are a fairly easy denominator to work with- 5x20=100. So, the percentages were:
Taxes for Pharoah: 1x20=20; 20/100 or 20%
Remaining crops for families: 4x20=80; 80/100 or 80%

To check our work, let's test our 100%. 20% taxes for Pharoah + 80% remaining for families should equal 100%. 20+80=100, so our math works and our calculated numbers are correct.