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?
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