Dedicated l'ilui nishmat Dov Nachman ben Aharon Yonah, my paternal grandfather, whose 86th birthday is today. May his memory be a blessing.
Isn't this how you count to 20?
We all count Sefirat Ha'Omer in two base systems - base ten and base seven. For example, on day 20, we count "הַיּוֹם עֶשְׂרִים יוֹם שֶׁהֵם שְׁנֵי שָׁבוּעוֹת וְשִׁשָּׁה יָמִים לָעֹֽמֶר", which basically tells us that 20 in base ten is 26 in base seven - two weeks and six days. (In fact, last night we counted 3 (or 11 in binary.) I chose the title of this post because of the time a few years ago when I saw a 6th grader of mine had been practicing her binary Sefirat Ha'Omer in her locker. This was a serious kvell moment for me.*
My interest in base system math was sparked by two things. The first was a class I took at Harvard called "Math For Teaching Arithmetic" (with my amazing professor Bret Benesh) where we invented and used our own number system to model how kids learn arithmetic. If you think long division is tough, you should try it in binary. That's rough! But most math teachers don't find long division tough, so these activities really helped us get into the heads of struggling students. The second impetus to begin the Binary Sefira project was a math team my strongest students were on. Though I was not coaching the math team, they were assigned base system problems during their competitions and they didn't know what base systems were.
For example, 10,100 in binary is 202 in what base? (Hence the title.) This is a challenging problem because most of us think in base 10 and would naturally turn 10,100 into 20 first before trying to figure out in which base 20 would be written as 202.* This is definitely a higher-level problem, so first, let's examine how this material would be taught to middle school students who would benefit from enrichment.
Base system work is an ideal extension - it's intellectually demanding, reinforces understanding of place value and exponents, and, most importantly, it is an entirely separate topic not in most curricula. Essentially, you're enriching the students without teaching the material someone else is planning to teach them next year. It fits the criteria for high-quality challenge materials. It's not for every kid, though most of our students are very comfortable answering the question "What time will it be in 12 hours?" Telling time, also known as "clock math" is math in a different base system.)
How to introduce the topic
With my initial groups of students, they knew what different base systems were from their math team, so we just jumped right in. For the first 3-5 days after Pesach, I would write the Sefira on the board in various different base systems, increasing the number of base systems every day. At first, I put the 'binary Sefira' (my term for Sefira in multiple base systems) up on the board. After a few days, student volunteers would put binary Sefira up on the board. There was a gradual build to comfort and mastery - over a period of 49 days, you can say to students, "You'll be more comfortable next week."
For example, 10,100 in binary is 202 in what base? (Hence the title.) This is a challenging problem because most of us think in base 10 and would naturally turn 10,100 into 20 first before trying to figure out in which base 20 would be written as 202.* This is definitely a higher-level problem, so first, let's examine how this material would be taught to middle school students who would benefit from enrichment.
Base system work is an ideal extension - it's intellectually demanding, reinforces understanding of place value and exponents, and, most importantly, it is an entirely separate topic not in most curricula. Essentially, you're enriching the students without teaching the material someone else is planning to teach them next year. It fits the criteria for high-quality challenge materials. It's not for every kid, though most of our students are very comfortable answering the question "What time will it be in 12 hours?" Telling time, also known as "clock math" is math in a different base system.)
How to introduce the topic
With my initial groups of students, they knew what different base systems were from their math team, so we just jumped right in. For the first 3-5 days after Pesach, I would write the Sefira on the board in various different base systems, increasing the number of base systems every day. At first, I put the 'binary Sefira' (my term for Sefira in multiple base systems) up on the board. After a few days, student volunteers would put binary Sefira up on the board. There was a gradual build to comfort and mastery - over a period of 49 days, you can say to students, "You'll be more comfortable next week."
Teaching Explicitly
After a few years, I started getting students who had not worked with other bases on the math team. I also always had a few weaker students in my class for whom it was not as obvious. I began to introduce the concept with explicit instruction.
So how does it work?
Differentiation
This exercise allows for a lot of differentiation. The strongest students can be most involved in the early days, as I mentioned above. You can also have different kids wrote different numbers on the board. Stronger students can do the smallest bases (binary, base 3) and the largest (base 14, base 15, hexadecimal - more about this later.) Students who are comparatively weaker can do the bases in the middle, such as 5, 6 and 7.
I also love that these worksheets are really easy to make!
Large bases
When you get past base 10, you need additional digits. If you are working in base 11, you need a single digit that represents 10 ones. If you are working in base 12, you need a digit for 10 and a digit for 11. With my students, I've gone as far as hexadecimal (base 16).
Here's the full list for 44.
For bases 11 through 14, the numbers work out nicely. However, since 44 is 2 groups of 15 and 14 ones, in base 15 we need to write it as 2E, with the E representing 14. Similarly, 44 is 2 groups of 16 and 12 ones, so we write it 2C, with the C representing 12.
We generally use A for 10, B for 11, C for 12, D for 13, E for 14 and F for fifteen.
Challenge materials:
Once students have been practicing for a while (for example, on day 44), you can ask them to work backwards in a few ways. The first is turning a number in another base back into base 10. Below, note that 44 is written as a number in bases 5 through 14. You can't have the number 44 in any lower base because none of those have enough room in the 'ones column' or the "tens column" for four groups of anything.
The students are instructed to turn 44 in various bases into numbers in base 10. For example, 44 in base 8 means (4)8^1 + (4)8^0 or 32+4 or 36 in base 10.
Challenge type #2 would be to ask kids to figure out what 44 in base 13 would be in base 7, much like the problem we did right at the beginning. In this case, two base changes are required.
Error analysis is a third way students can be asked to think differently. In the picture below, base 3 is right and base 4 is wrong. You can ask your students:
a) What day is it?
b) What is the correct base 4?
(Answers in *** below.)
The column you think of as the "ones" column could also be described as the 10^0 column. It tells you how many 10^0 (ones) you have. When that column fills up, we need to regroup into the 10^1 column. If the tens column fills up (i.e., you have ten groups of 10), you need a 10^2 (or hundreds) column.
As I mentioned before, we count Sefira in two bases - base 10 and base 7. Base 7 is a big enough base that we only get to the 7^1 column filling up on the last day. There is a beautiful mathematical and religious synergy to the fact that we count 7 weeks of 7 days - perhaps worthy of a Shmita or Yovel post another day.
When turning base 10 numbers into much smaller bases, we have to regroup really quickly. For all the big bases, day 44 is still in double digits. But for binary and base 3 (below), it's another matter.
Binary:
The procedure I teach kids is to think of the biggest power of two that can be subtracted from the target number (in this case, 44). The answer, 32, is in the 2^5 column. That tells us we're going to have to fill 6 columns. In my early years, I liked to write the exponential expression that was the size of the column below the blank line. It clarified which column was which but made them look too confusingly like fractions.
The number above is 44 in base 10: There is a group of 32, no groups of 16, one group of 8 and one group of 4, then placeholder zeroes in the twos and ones column.
The base 3 number is also quite rich. They have to figure out that the greatest power of 3 is 27, subtract that from 44 to get 17, figure out that 17 has only one 9, subtract that to get 8, and so on. 44 is what I call a "full count" kind of day in base 3, because the addition of one more day would fill the next two columns, leading day 45 in binary to be 1200.
This exercise allows for a lot of differentiation. The strongest students can be most involved in the early days, as I mentioned above. You can also have different kids wrote different numbers on the board. Stronger students can do the smallest bases (binary, base 3) and the largest (base 14, base 15, hexadecimal - more about this later.) Students who are comparatively weaker can do the bases in the middle, such as 5, 6 and 7.
Large bases
When you get past base 10, you need additional digits. If you are working in base 11, you need a single digit that represents 10 ones. If you are working in base 12, you need a digit for 10 and a digit for 11. With my students, I've gone as far as hexadecimal (base 16).
Here's the full list for 44.
We generally use A for 10, B for 11, C for 12, D for 13, E for 14 and F for fifteen.
Challenge materials:
Once students have been practicing for a while (for example, on day 44), you can ask them to work backwards in a few ways. The first is turning a number in another base back into base 10. Below, note that 44 is written as a number in bases 5 through 14. You can't have the number 44 in any lower base because none of those have enough room in the 'ones column' or the "tens column" for four groups of anything.
The students are instructed to turn 44 in various bases into numbers in base 10. For example, 44 in base 8 means (4)8^1 + (4)8^0 or 32+4 or 36 in base 10.
Challenge type #2 would be to ask kids to figure out what 44 in base 13 would be in base 7, much like the problem we did right at the beginning. In this case, two base changes are required.
Error analysis is a third way students can be asked to think differently. In the picture below, base 3 is right and base 4 is wrong. You can ask your students:
a) What day is it?
b) What is the correct base 4?
(Answers in *** below.)
Challenge type #4, the hardest by far, is "guess the base". You give the student the number (let's pick 47 in this case) in base 10 and then you say "47 in base 10 is 43 in what base?" In this case, the answer is base 11, so it is somewhat guessable since the two bases are close. However, make it "... is 133 in what base?"**** and it is a lot harder to guess. It has to be reasoned out. (Using, I'll admit, a series of educated guesses.)
Other methods:
There are other methods to calculate numbers in other bases (often referred to as "mod" or "modulo": 3 mod10 = 11 mod2) that don't involve using exponents explicitly, the way I have done in this post. These methods have their own distinct benefits and drawbacks. My colleagues John Watkins-Chow and Dr. Steven Steinsaltz shared two such methods with me, which I may yet summarize in another post. (After all, there are many more days of the Omer ahead of us.)
In conclusion:
I don't know why more strong students are not taught to convert between bases. It strengthens their understanding of place value, exponents and makes their thinking more flexible. Try it in your classroom and let me know how it works!
* Day 20 is also Yom Ha'Atzmaut, though this year Y"HA is pushed off to the 6th of Iyar - stay tuned for a future post for this important holiday.
*** (1)3^3 + (2)3^2 + (0)3^1 + (1)3^0 = 27+18+1=46. However, (2)4^2 + (3)4^1=32+12=44. To do 46 in base 4, you need to add 2 ones, so the correct answer would be 232.
****Base 5: 25+15+3
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