"You stayed in Kadesh many days, [as many] as the days that you dwelt." ~Devarim 1;46
"We turned and journeyed to the wilderness by way of the Sea of Reeds, as Hashem spoke to me, and we went around Mount Seir for many days." ~Devarim 2;1
"The Emim dwelled there previously, a great and populous people, and tall as the giants." ~Devarim 2;10
"Sihon went out toward us- he and his entire people- for battle, to Jahaz. Hashem, our G-d, gave him before us, and we smote him and his sons and his entire people. We captured all his cities at that time, and we destroyed every city- the men, the women, and the small children; we did not leave a survivor." ~Devarim 2;32-34
"All these were fortified cities, with a high wall, doors and a bar, aside from open cities, very many." ~Devarim 3;5
"For only Og king of the Bashan was left of the remaining Rephaim. Behold! his bed was an iron bed, in Rabbah of the Children of Ammon; nine cubits its length and four cubits its width, by the cubit of a man." ~Devarim 3;11
Math Connection:
There are mathematically related words that we use on a regular basis, words that have a general meaning. If the people using the words don't have the same understanding of the meaning, then it can create great confusion. From a recent, personal example, my daughter was supposed to do something 5 or 6 times a day. I was checking-in with her nightly to find out how many times she had taken care of this responsibility, and she would consistently answer, "Several." Then, one night I asked her how many she thought several was, and she told me that it meant 2 or 3. What a difference!
Words such as "all" or "none" are clear even when we don't know how many are in a group, since we know that the situation refers to either the entire group or none of the group. Other words have commonly understood meanings (although even my ranges can be subjective):
a couple = 2
a few = 3-4
several= 5-6
many = 7+
Parsha Connection:
In this week's parsha, we have many examples of non-numerical vocabulary from which we should still be able to understand a certain quantity. Some of the words are concrete and give clear meaning, even without giving specified quantities, while other words leave us with the need for common understanding in order to have a full understanding of the meaning.
Examples of non-numerical concrete vocabulary:
*Devarim 2;32-34 Sihon went out with his entire people, his entire people were smote, and all his cities were captured and destroyed
Examples of non-numerical subjective vocabulary:
*Devarim 1;46 Children of Israel stayed in Kadesh many days.
*Devarim 2;1 Children of Israel traveled around Mount Seir for many days.
*Devarim 2;10 The Emim were a great and populous people, and tall as the giants.
*Devarim 2;32-34 The Children of Israel conquered Sihon and his people, including the men, the women, and the small children
*Devarim 3;5 The cities captured from Og "were fortified cities, with a high wall, doors and a bar, aside from open cities, very many."
Sometimes, even when measurements are given, clarification is needed. When we are told in Devarim 3;11 of the measurements of Og's iron bed (Og being a giant), we are given the measurements in Amot, which is a standard biblical measurement from the elbow to the tip of the middle finger of an average sized man. However here, Rashi specifies for us that Og's bed was actually measured in Amot based on Og's proportions, not that of an average man. This would make quite the difference in the size of his bed!
Everyday Connection:
Have you ever had a situation when you were given non-numerical quantitative information and you felt that you needed to research or discuss further to fully understand the information? Are you always clear when using such vocabulary yourself? How often do we think that we are being clear, when really we end up with a misunderstanding? I know that my daughter has helped me try to be more conscious of my own clarity when using such terms.
Thursday, July 31, 2014
Thursday, July 24, 2014
Masei- Lines and Angles
"Hashem spoke to Moshe, saying: 'Command the Children of Israel and say to them: When you come to the land of Canaan, this is the land that shall fall to you as an inheritance, the land of Canaan according to its borders. The southern side shall be for you from the Wilderness of Zin at the side of Edom, and the southern border shall be for you from the end of the Salt Sea to the east. The border shall go around for you south of Maaleh-akrabbim, and shall pass toward Zin; and its edges shall be south of Kadesh-barnea; then it shall go out Hazar-addar and pass to Azmon. The border shall go around from Azmon to the stream of Egypt, and its edges shall be to the west. And the western border: It shall be for you the Great Sea and the bounded area; this shall be for you the western border. This shall be for you the northern border: from the Great Sea you shall draw a slanting line for yourselves to Mount Hor. From Mount Hor you shall draw a slanting line to the approach to Hamath, and the edges of the border shall be toward Zedad. The border shall go forth toward Zifron and its edges shall be Hazar-enan; this shall be for you the northern border. You shall draw a slanting line for yourselves as the eastern border from Hazar-enan to Shefam. The border shall descend from Shefam to Riblah, east of Ain; the border shall descend and reach the bank of the Kinnereth Sea to the east. The border shall descend to the Jordan, and its edges shall be the Salt Sea; this shall be the Land for you, according to its borders all around.'" ~Bamidbar 34;1-12
Rashi on Bamidbar 34;7: (one explanation that Rashi gives for this passage)
You shall draw a slanting line: This means, you shall move on a slant to veer from west to north to Mount Hor.
Math Connection:
Lines and angles are amongst the earlier Geometry concepts that are introduced to students towards the middle of elementary school. The idea of angles being measured in degrees is a very abstract concept, but there are basic concepts that relate to angles that are taught early on.
*The point where 2 lines connect is called a vertex.
*When 2 lines connect at a vertex and continue to make a longer, straight line, it's called a straight angle, and it measures 180° (read: "180 degrees").
*When 2 lines connect at a vertex and turn to make an angle that looks like a proper corner, it's called a right angle, and it measures 90°.
*An angle that is greater than 90° is called an obtuse angle.
*An angle that is less than 90° is called an acute angle.
Parsha Connection:
For reference, the red borders on this map of Canaan shows you the borders listed in Bamidbar 34;1-12.
In this week's parsha, we are given the border demarcations for the land of Canaan, which is to be an inheritance to the Children of Israel. The beginning landmarks for the borders are given in such a way that the border is simply drawn by connecting straight lines between listed points, or following a described path between listed points.
However, when we get to certain border descriptions, we are told that "you shall draw a slanting line" between two locations. One of Rashi's explanations on these words is that it is a slanted line between the two directions. In other words, when we are told to travel north, south, east, or west, we automatically picture making a 90° turn, or following a path at 180° degrees. By being told that the line is on a slant in 34;7, for example, we learn that it's not a path straight up from the western point and then turning a 90° corner when you reach the proper location to the north; rather, the border line is an acute angle from the western location of the Great Sea to the northern location of Mount Hor.
An interesting follow-up activity:
Rashi on Bamidbar 34;7: (one explanation that Rashi gives for this passage)
You shall draw a slanting line: This means, you shall move on a slant to veer from west to north to Mount Hor.
Math Connection:
Lines and angles are amongst the earlier Geometry concepts that are introduced to students towards the middle of elementary school. The idea of angles being measured in degrees is a very abstract concept, but there are basic concepts that relate to angles that are taught early on.
*The point where 2 lines connect is called a vertex.
*When 2 lines connect at a vertex and continue to make a longer, straight line, it's called a straight angle, and it measures 180° (read: "180 degrees").
*When 2 lines connect at a vertex and turn to make an angle that looks like a proper corner, it's called a right angle, and it measures 90°.
*An angle that is greater than 90° is called an obtuse angle.
*An angle that is less than 90° is called an acute angle.
Parsha Connection:
For reference, the red borders on this map of Canaan shows you the borders listed in Bamidbar 34;1-12.
In this week's parsha, we are given the border demarcations for the land of Canaan, which is to be an inheritance to the Children of Israel. The beginning landmarks for the borders are given in such a way that the border is simply drawn by connecting straight lines between listed points, or following a described path between listed points.
However, when we get to certain border descriptions, we are told that "you shall draw a slanting line" between two locations. One of Rashi's explanations on these words is that it is a slanted line between the two directions. In other words, when we are told to travel north, south, east, or west, we automatically picture making a 90° turn, or following a path at 180° degrees. By being told that the line is on a slant in 34;7, for example, we learn that it's not a path straight up from the western point and then turning a 90° corner when you reach the proper location to the north; rather, the border line is an acute angle from the western location of the Great Sea to the northern location of Mount Hor.
An interesting follow-up activity:
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.
Thursday, July 10, 2014
Pinchas- Multiplying Fractions by Whole Numbers
"And on the Sabbath day: two male lambs in their first year, unblemished, two tenth-ephahs of fine flour for a meal-offering, mixed with oil, and its libation" ~Bamidbar 28;9
"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
"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.