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?
"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?
"Hashem spoke to Moshe, saying: Speak to the Children of Israel and say to them: When a man will express a vow to Hashem regarding the valuation of souls, the valuation of a male shall be: for someone twenty years of age to sixty years of age, the valuation shall be fifty silver shekels, of the sacred shekel. If she is female, the valuation shall be thirty shekels. And if from five years of age to twenty years of age, the valuation of a male shall be twenty shekels and of a female ten shekels. And if from one month of age to five years of age, the valuation of a male shall be five silver shekels; and for a female, the valuation shall be three silver shekels. And if from sixty years of age and up, if for a male, the valuation shall be fifteen shekels; and for a female, ten shekels. But if he is destitute for the valuation, then he should have him stand before the Kohen, and the Kohen should set his valuation; according to what the hand of the person who makes the vow can attain should the Kohen set his evaluation." ~Vayikra 27:1-8
In this section of this week's parsha, we learn about the standard valuation for donations that are vowed to the Beit HaMikdash on behalf of family members. We have the standard valuations, and we are also told that if someone makes a vow but doesn't have the funds to meet the standard valuations, the Kohen can make a determination of an appropriate valuation for that person. This week I would like to look at some activities for students of varying levels based on the standard valuations that are listed.
To begin, let's organize the information (older students could be asked to chart this information for themselves by identifying and organizing the information in the passage):
Note that this could also be organized into two charts- one for males and one for females. This breakdown might be clearer for some students when trying to separate information on males and females in follow-up activities.
Related Activity Suggestions:
- Graph It!- A good way to compare data of this nature is by graphing it. This information lends itself well to a double bar graph, with a category for each age grouping, and two bars within each category (one for males and one for females). The left side of the bar graph (y-axis) would indicate the number of shekels donated. As always, grade levels recommended are an approximation, and individualization for students is necessary to meet their actual ability levels. K-2 students could fill in a pre-made class-sized graphing chart as a class activity, with older students also copying the information onto personal graphs. Grades 3-5 students could create individual graphs on pre-printed graph templates with spacing indicated for different levels of labeling, and then graphing the information. Students in grades 6+ could create their own double-bar graph from start to finish.
- Is there a pattern?- Looking at the table above and/or the double-bar graph created in the first activity, students can look for comparisons between males and females at each age group. They can also look for comparisons between the different age groups for males and then between the different age groups for females.
- What trends do they see over a person's lifetime? Is this consistent for both men and women? Why do they think the valuations may have been set in the way that they were?
- After making note of comparisons, older students could look for specific numerical comparisons between males and females in each age group and then across the data for males and females at different age groups. Are any of the ratios between groups consistent in any way? Rashi's note on 27:7 is also relevant to these comparisons, as he specifically points out some differences in the ways that the valuations change between the different age groups.
- Problem Solving- Processing the information through sample family valuations- Using the valuations given, students could be provided with sample problems with different family groupings to calculate the valuation for each family. Students could also create their own sample families to calculate or switch with friends to calculate. What about students' own families? What would each student's personal family valuation be if they were donating to the Beit HaMikdash? Can older students come up with an algebraic formula for calculating the valuation for any given family scenario?
"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.
