"Yitzchak sowed in that land, and in that year he reaped a hundredfold; thus had Hashem Blessed him." ~Bereishit 26;12
Rashi on 26;12-
"A Hundredfold- For they assessed [the field] to determine how much it is fit to produce and it produced for every unit they estimated it could produce a hundred units. Our Rabbis said, This assessment was done for tithes."
In this section of the parsha, we learn of Yitzchak's time living in Gerar, under the rule of Avimelech. Bereishit Rabbah (64;6) asks: if we're not supposed to find a blessing in a measurable quantity, how could this crop be considered a blessing? You are not supposed to count on a crop before it is harvested (like the old adage- "Don't count your chickens before they hatch"). How does this make sense? The Mizrachi explains that the crop was traditionally estimated in order to make an advanced calculation for tithings to be easily set aside. The midrashim (Tosafot HaShalem citing Rivah) teach us that this crop that is referred to was sowed during a time of famine in the land. Yitzchak had estimated what his tithe should be in order to be able to set it aside for the poor as soon as possible, and he had estimated much less of a crop due to the famine in the land. It is with this understanding that we see how this was such a blessing for Yitzchak.
Activity Connections:
There are two primary mathematical concepts that connect to this crop:
1) the concept of calculating and setting aside tithings
2) the concept of one hundredfold
Interestingly enough, both of these concepts connect to powers of 10.
*Tithings are classically 1/10 of a crop (the root of the hebrew word- מעשרות/מעשר- is actually related to עשר, the number 10)
*one hundredfold means times 100, which is a power of 10 (102;or X10 and then X10 again)
Activity ideas in increasing conceptual difficulty:
*What does a tithing or 1/10 of a group really look like? For the youngest students, they could begin by just taking a group of items and separating the items into 10 equal groups. Having a set-up of 10 cups, bowls, or plates to sort the pieces into will help students keep track of their sorting. This is doable even for preschool children. After they divided their items into 10 piles, they can count each pile to make sure that they all have the same number of items in them. From there, they can act out separating 1 of the piles to give away and keeping 9 of the piles for themselves. For beginners, you want to simplify the process for them by making sure that the number of items they start with is a multiple of 10 (so that it divides equally into 10 piles).
*As a step up from the previous activity, you can have students divide larger groups of items, and incorporate estimation or calculation of what amount the tithing will actually be when it's separated out. Students who have had exposure to the concept of fractions and decimals could also estimate and calculate the value of tithings for amounts that are not strictly multiples of 10.
*One hundredfold- what does x100 really look like? What if Yitzchak had anticipated having 5 bundles of grain? How much did he actually have that year? What if he had anticipated 10 bundles of grain? Younger students can physically count or draw out the difference of a group of 5 and a group of 500. Small or medium sized graph paper can be good for drawing out a model of these differences (with each box modeling 1 unit). If students have the dexterity, small manipulatives like paper clips can also be good. For younger students, these can be modeled with blocks or Legos, but you need to have a good supply in order to model the hundreds. (Tip: Creating stacks or grouping of 10 to count everything out will help make counting easier and will help children develop their number sense.)
*Older students can graph the difference between the anticipated crops and the actual crops. Is there a pattern to the difference? How would the difference be expressed as an algebraic expression?
*Another activity for older students would be a comparison of the tithings between the anticipated crops and the actual crops. What would a graph of these differences look like? How could the differences be expressed algebraically? How does the pattern of the difference in tithings (anticipated vs. actual) compare to the pattern of the difference in crops (anticipated vs. actual)? How do the algebraic expressions of each comparison compare to each other?
Showing posts with label powers of ten. Show all posts
Showing posts with label powers of ten. Show all posts
Thursday, November 20, 2014
Thursday, July 17, 2014
Mattot- Multiplying and Dividing powers of 10
"Moshe spoke to the people, saying, 'Arm men from among yourselves for the army that they may be against Midian to inflict Hashem's vengeance against Midian. A thousand from a tribe, a thousand from a tribe, for all the tribes of Israel shall you send the army.'
So there were delivered from the thousands of the Children of Israel, a thousand from each tribe, twelve thousand armed for the army. Moshe sent them- a thousand from each tribe for the army- them and Pinchas son of Elazar the Kohen to the army, and the sacred vessels and the trumpets for sounding under his authority." ~Bamidbar 31;3-6
"Moshe, Elazar the Kohen, and all the leaders of the assembly went out to meet them outside the camp. Moshe was angry with the commanders of the legion, the officers of the thousands and the officers of the hundreds, who came from the army of the battle." ~Bamidbar 31;13-14
Math Connection:
For younger students, and even some older students, calculations with big numbers can be scary. The larger the number, the more room there is for errors. It is helpful for students to identify patterns in the way numbers work in order to help reduce anxiety and make calculations easier, and even faster. When dealing with multiplication and division of numbers that are powers of 10 (tens, hundreds, thousands,...), there is a simple trick that you can use to make the calculations easier. If you remove (or ignore) any extra zeros at the end of the number that aren't needed for the purpose of your calculation, you can multiply or divide as you normally would with your truncated number, and then add back your extra zeros onto the end of your product or quotient from your calculation.
Some examples:
4,000 x 4: Set aside the 3 zeros at the end, leaving 4 x 4 = 16, and then add back the 3 zeros, making it 16,000.
650,000 x 2: Set aside the 4 zeros at the end, leaving 65 x 2 = 130, and then add back the 4 zeros, making it 1,300,000.
For multiplication, if both numbers are powers of 10, you can still set aside the zeros from the end of both numbers to simplify your calculation. After your calculation, you add all of the zeros from both numbers back to the end of your product. For example, 2,000 x 80 can be calculated by multiplying 2 x 8 = 16, and then add back on the 4 zeros (3 from 2,000 and 1 from 80), making it 160,000.
6,200 + 2: Set aside the 2 zeros at the end, leaving 62 + 2 = 31, and then add back the 2 zeros, making it 3,100.
43,000 + 5: Here, 43 is not evenly divisible by 5, so we will take away just 2 zeros at the end, leaving 430 + 5 = 86, and then add back the 2 zeros, making it 8,600.
For division, if both numbers are powers of 10, you can permanently remove the same number of zeros from the end of both numbers to simplify your calculation. For example, 24,000 + 300 is the same as 2,400 + 30, which is the same as 240 + 3 (all of which = 80).
Parsha Connection:
In this week's parsha, we have both multiplication and division with powers of 10. First, when Moshe instructions the Children of Israel to create an army with 1,000 men from each of the twelve tribes, we can confirm the total of 12,000 that is given in the text.
1,000 x 12: Remove the 3 zeros at the end, leaving 1 x 12 = 12, and then add back the 3 zeros, making it 12,000 men.
When they return from battle, Moshe is "angry with the commanders of the legion, the officers of the thousands and the officers of the hundreds, who came from the army of the battle." How many commanders were there?
Officers of the thousands:
12,000 + 1,000: Here, we have 3 zeros at the end of both numbers (dividend and divisor), so we can remove the 3 zeros from both, and are left with 12 + 1 = 12 commanders overseeing the thousands.
Officers of the hundreds:
12,000 + 100: Here, we have 2 zeros that we can take away from both numbers, leaving us with 120 + 1 = 120 commanders overseeing the hundreds.
So, there were 132 commanders (12 + 120) with whom Moshe was angry following the battle with Midian.
Everyday Connection:
Working with a committee on planning for a large event? What if you have 237 people attending your event and your tables each seat 8 people? How many tables will you need? Using estimation and our division trick, we can quickly calculate how many tables you'll need.
237 is just 3 short of 240. To calculate 240 + 8, let's use our trick- 24 + 8 = 3, then add the zero back, and we know that we need to set 30 tables for the event.
And what if you want to estimate a food budget for the event? Let's say $30 per person. Now we can use estimation and our multiplication trick to see if our food budget is realistic.
Again, we'll use 240. To calculate 240 x 30, we'll do 24 x 3 = 72, then add our 2 zeros back, and we have an estimate of $7,200, if you spend $30 per person.
So there were delivered from the thousands of the Children of Israel, a thousand from each tribe, twelve thousand armed for the army. Moshe sent them- a thousand from each tribe for the army- them and Pinchas son of Elazar the Kohen to the army, and the sacred vessels and the trumpets for sounding under his authority." ~Bamidbar 31;3-6
"Moshe, Elazar the Kohen, and all the leaders of the assembly went out to meet them outside the camp. Moshe was angry with the commanders of the legion, the officers of the thousands and the officers of the hundreds, who came from the army of the battle." ~Bamidbar 31;13-14
Math Connection:
For younger students, and even some older students, calculations with big numbers can be scary. The larger the number, the more room there is for errors. It is helpful for students to identify patterns in the way numbers work in order to help reduce anxiety and make calculations easier, and even faster. When dealing with multiplication and division of numbers that are powers of 10 (tens, hundreds, thousands,...), there is a simple trick that you can use to make the calculations easier. If you remove (or ignore) any extra zeros at the end of the number that aren't needed for the purpose of your calculation, you can multiply or divide as you normally would with your truncated number, and then add back your extra zeros onto the end of your product or quotient from your calculation.
Some examples:
4,000 x 4: Set aside the 3 zeros at the end, leaving 4 x 4 = 16, and then add back the 3 zeros, making it 16,000.
650,000 x 2: Set aside the 4 zeros at the end, leaving 65 x 2 = 130, and then add back the 4 zeros, making it 1,300,000.
For multiplication, if both numbers are powers of 10, you can still set aside the zeros from the end of both numbers to simplify your calculation. After your calculation, you add all of the zeros from both numbers back to the end of your product. For example, 2,000 x 80 can be calculated by multiplying 2 x 8 = 16, and then add back on the 4 zeros (3 from 2,000 and 1 from 80), making it 160,000.
6,200 + 2: Set aside the 2 zeros at the end, leaving 62 + 2 = 31, and then add back the 2 zeros, making it 3,100.
43,000 + 5: Here, 43 is not evenly divisible by 5, so we will take away just 2 zeros at the end, leaving 430 + 5 = 86, and then add back the 2 zeros, making it 8,600.
For division, if both numbers are powers of 10, you can permanently remove the same number of zeros from the end of both numbers to simplify your calculation. For example, 24,000 + 300 is the same as 2,400 + 30, which is the same as 240 + 3 (all of which = 80).
Parsha Connection:
In this week's parsha, we have both multiplication and division with powers of 10. First, when Moshe instructions the Children of Israel to create an army with 1,000 men from each of the twelve tribes, we can confirm the total of 12,000 that is given in the text.
1,000 x 12: Remove the 3 zeros at the end, leaving 1 x 12 = 12, and then add back the 3 zeros, making it 12,000 men.
When they return from battle, Moshe is "angry with the commanders of the legion, the officers of the thousands and the officers of the hundreds, who came from the army of the battle." How many commanders were there?
Officers of the thousands:
12,000 + 1,000: Here, we have 3 zeros at the end of both numbers (dividend and divisor), so we can remove the 3 zeros from both, and are left with 12 + 1 = 12 commanders overseeing the thousands.
Officers of the hundreds:
12,000 + 100: Here, we have 2 zeros that we can take away from both numbers, leaving us with 120 + 1 = 120 commanders overseeing the hundreds.
So, there were 132 commanders (12 + 120) with whom Moshe was angry following the battle with Midian.
Everyday Connection:
Working with a committee on planning for a large event? What if you have 237 people attending your event and your tables each seat 8 people? How many tables will you need? Using estimation and our division trick, we can quickly calculate how many tables you'll need.
237 is just 3 short of 240. To calculate 240 + 8, let's use our trick- 24 + 8 = 3, then add the zero back, and we know that we need to set 30 tables for the event.
And what if you want to estimate a food budget for the event? Let's say $30 per person. Now we can use estimation and our multiplication trick to see if our food budget is realistic.
Again, we'll use 240. To calculate 240 x 30, we'll do 24 x 3 = 72, then add our 2 zeros back, and we have an estimate of $7,200, if you spend $30 per person.