"In the first month, on the fourteenth day of the month, shall be a pesach-offering to Hashem...And their meal-offering: fine flour mixed with oil; you shall make three tenth-ephahs for the bull and two tenth-ephahs for the ram. One tenth-ephah shall you make for the one lamb, for the seven lambs..." ~Bamidbar 28;16-21
Math Connection:
We are already familiar with the basic meaning of a fraction from Parshat Vayishlach. We know that the denominator tells us how many pieces the whole unit has been broken into, and the numerator tells us how many of those pieces we are including in our count of what we need- If I have 3/5 of a cookie, then the whole cookie was broken into 5 equal pieces, and I have 3 of those pieces for myself.
So what happens if we need to multiply a fraction by a whole number (positive number without fractions or decimals)? With integers (what we think of as "regular" numbers- positive or negative, but no fractions or decimals), multiplying is the same as repeated addition. 3 x 4, for example, is the same as 3 + 3 + 3 + 3. This is the same as what happens when we multiply fractions by a whole number. 1/4 x 3, for example, would be the same as 1/4 + 1/4 + 1/4. Now, how does that look as a single fractional number? We have 3 pieces that are each 1/4 of a whole unit. In other words, we have 3/4. Fractions can be confusing because there's a number on the top and the bottom, and children often get confused about which numbers they need to manipulate in different ways. If you remember that the bottom number is really a label for the section of the whole unit that you're working with and the top number tells how many pieces of that size you have, it becomes easier to remember when and how to manipulate (or not manipulate) each part of the fraction.
Parsha Connection:
In Chapter 28 of this week's parsha, we are given lists of all of the sacrifices that were to be offered on any day or for any special occasion. There are several instances where we are given multiple fractional measurements related to the sacrifices. In the quote above for the pesach-offering, for example, we are told of "three tenth-ephahs for the bull and two tenth-ephahs for the ram". What do these measurements mean, and what do their fractions look like?
Let's look at each individually:
"three tenth-ephahs for the bull"- ephah is the unit of measurement that's being used to measure the flour & oil mixture. 1/10 ephah is a standard fraction of the unit that is commonly used in sacrificial "recipes". Here, we are being told 3 x 1/10 ephah. So, the same as 1/10 + 1/10 + 1/10, or 3/10, when simplified. So, for the bull, they needed 3/10 ephah for the "recipe"
"two tenth-ephahs for the ram"- similar to above, we are being told 2 x 1/10 ephahs. Above, we used repeated addition, which we know is the same as multiplication. How do we manipulate the numbers if we just want to do the multiplication? Remember, we know that the denominator tells us how many equal pieces our unit is broken into, and the numerator tells us how many pieces we have. When we multiply the fraction by a whole number, we are not changing the size of the fractional piece in any way, so our denominator would stay the same. We are, however, changing how many 1/10 pieces we have- in fact, we now have 2 of those pieces- 1 x 2. So, we multiply the original numerator times the whole number to get our new numerator. So, for the ram, they needed 2/10 ephah for the "recipe". We could reduce this fraction (as we talked about in Parshat Vayishlach). Since 2/10 is the same as 1/5, we could also say that they needed 1/5 ephah for the ram.
These are just two examples of multiplied fractions in the parsha. If you continue through Chapter 28, you will find more examples of larger sacrifices, as they were changed for each holiday (or different days of different holidays), each with more examples of opportunities for multiplying fractions.
Everyday Connection:
Have you ever tried to double or triple a recipe? Particularly ones that call for fractional amounts of an ingredient? 1/3 C or 1/2 tsp? The same principles will apply. How many cups of sugar will you need, if you are making 4 times a recipe, and the original recipe calls for 3/4 C?
Our denominator of 4 stays the same. We multiply 3 x 4 for our numerator (=12). Now we have 12/4. Rather than taking our 1/4 C measure and counting out 12 of them, how can we simplify our new fraction? It takes 4 1/4 pieces to make 1 whole C.
4/4 = 1 C
8/4 = 2 C
12/4 = 3 C
What about 5 times the recipe?
Our denominator of 4 stays the same. We multiply 3 x 5 for our numerator (=15). Now we have 15/4. Again, rather than taking our 1/4 C measure and counting out 15 of them, how can we simplify our new fraction? It takes 4 1/4 pieces to make 1 whole C.
4/4 = 1 C
8/4 = 2 C
12/4 = 3 C
16/4 = 4 C, but we only have 15/4. So we'll have 3 full Cups, and then 3/4 C more that we need to add.
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