Beast 4A, Binary chapter. Is anybody else as lost as I am?

[sweetpea3829]

I never learned binary math and am thoroughly confused on how to explain it to DS who is just as confused as I am, lol.

 

Any easier explanations out there? Is it something we even need to worry about?

[kateingr]

Binary's apparently very useful for understanding programming (according to my techie husband--I don't know much about programming myself). But it's certainly not an essential 4th grade math topic, so don't sweat it if it's stressing out you and your ds . :) 

 

If you want to give it one more go, I think the most helpful tip is to start on the "left" of the binary number when you're converting base-ten to binary. But, start on the "right" of the binary number when you're converting binary to base-ten.

 

So, looking at p. 94, to convert 101 to binary, you start on the right side of the number (the lower place-value): "Okay, I have 1 one, no 2s, and 1 4. So, that's 5." This is a lot like adding up the expanded form of a number (such as thinking of 364 as 300+60+4), but each digit stands for one of the powers of 2 rather than our usual multiples of a power of 10. 

 

But, when you convert base-ten to binary, you start on the left side of the number (with the higher place-value). So, on page 95, to convert 50 to binary: "No 64s fit into 50. I'll put a 0 in the 64s spot. One 32 fits, so I'll put a 1 in the 32s spot. Now I have 18 left to account for. So, I can fit one more 16 in, and one 2 to use up the rest of the number." So, the answer is 110010 (since we drop the initial 0 for the 64s place). It's kind of like the repeated subtraction we do in the long-division algorithm, but you're always subtracting powers of 2.

 

Clear as mud? ;) Truly, not a big deal if you skip this one.  

[deerforest]

I personally think the real joy of that section is understanding that there are other ways of basing number systems than our standard base 10 system.

 

If you're interested in playing with alternative number systems that might be easier to swallow before binary, I highly recommend "Can You Count in Greek?" by Prufrock Press. They cover binary in there too, and we had done that before we did the section in BA.

[vonfirmath]

AS a child I LOVED learning binary, hexadecimal, etc. becuase they were like secret codes. (And honestly, also helps understand the base 10 system and what it really means better too)

 

[Miss Tick]

Remember that old Place Value Town game for teaching traditional place value? Make yourselves a little house that only one (whatever manipulative you want, let's say) single-bop Lego fits into. The house next door can fit 2, then 4, 8, etc. Spend some time counting legos with base-10 and base-2 systems.

 

I find those math problems bring out my meditative side. I have to disconnect from all my mathematical instincts. Ommmm...

[Carol in Cal.]

I think base 2 is the hardest of the alternative bases to learn.  I'm glad I learned base 8 before that, so my head did not explode as much.  

 

Having said that, base 2 (binary) is very useful for programming because there are a lot of electronics that have exactly two positions, so you can encode things using combos of 'off' and 'on' states by expressing them in binary.  So it's a fairly important skill down the road.

 

When I taught it, I went back to the fundamentals of base 10, and taught by analogy.  

[Black-eyed Suzan]

I agree with the PP who said the larger bases are easier to understand. The School House Rock video Little Twelve Toes helped introduce us to the concept. 

 

 

The base ends up being written as 10.

The base is the number of different digits available for you to use. In the fingers and toes analogy, it is the number of fingers the person has.

 

In base 10, we count 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

So if it is base 5, you count 0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20.

In base 4, you count 0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22.

In base 3, count 0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101.

In base 2, count 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010.

 

I need to leave and don't have time to triple check these, so if someone sees an error, let me know and I will correct it. :)

 

ETA the zeros at the beginning. Thanks, Carol in Cal!

Edited May 24, 2016 by Black-eyed Suzan

[UCF612]

Welp now I'm scared of BA4! I've never done anything like this so I'm hoping the book will explain it well enough for my kid to understand somehow!

Edited May 24, 2016 by UCF612

[epi]

https://en.wikipedia.org/wiki/Mathematical_joke
"There are only 10 types of people in the world: those who understand binary, and those who don't."
 

Edited May 24, 2016 by epi

[sweetpea3829]

https://en.wikipedia.org/wiki/Mathematical_joke

"There are only 10 types of people in the world: those who understand binary, and those who don't."

 

 

LOLOL!!!!

 

 

Thanks for the advice everybody.  I'm exhausted and not braining too well right now...so I'll look it over well in the morning when I'm "fresh".  

[redsquirrel]

I lucked out. I barely understood it, but my son took to it like  duck to water. 

 

Honestly, if it is causing tears I would just skip it. I'm not even sure it comes up in 5th grade BA. And I'm pretty sure it wasn't part of AoPS pre-Algebra.  So it's a very interesting thing to know and understand, but not a hill to die on

[MrSmith]

I never learned binary math and am thoroughly confused on how to explain it to DS who is just as confused as I am, lol.

 

Any easier explanations out there? Is it something we even need to worry about?

 

 

I have not read the BA 4A explanation.  Binary number system is certainly interesting (Aliens with 2 fingers need to count, too!), but is generally not needed.

 

Binary is no different from decimal: It all comes down to place value.  The trouble is we spend 10 years using [0,9] but only a couple of weeks using [0,1].

 

Consider the following:

 

Base 10 (for reference)

42 = (4 * 10^1) + (2 * 10^0), where a^b is taken to mean 'raise a to the power b'.  It is further assumed that all powers of 10 larger than 0 have coefficient 0 (that is, 0 * 10^2 + 0 * 10^3 and so on).

 

Base 2

42 = 101010 = (1 * 2^5) + (0 * 2^4) + (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (0 * 2^0)

As with decimal, all powers of 2 larger than 6 are assumed to have coefficient 0.

 

Conversion from Binary to Decimal is a direct application of base-2 place value (as above).

 

Conversion from Decimal to Binary can be conceptualized like this:  Divide the number by 2 and keep track of the quotients and remainders separately.  The bit (binary digit) is 0 if the dividend is even, and the bit is 1 if the dividend is odd.

 

The thing to remember is the first division you perform results in the lowest valued bit (2^0), and advances to higher value ('to the left') for each subsequent division.  This is a departure from decimal division, where the initial operation results in the largest value in the quotient.

 

To convert 42 into binary:

 

42 / 2 = 21 remainder 0

21 / 2 = 10 remainder 1

10 / 2 =  5 remainder 0

 5 / 2 =  2 remainder 1

 2 / 2 =  1 remainder 0

 1 / 2 =  0 remainder 1

[stop since dividend is 0]

 

42 decimal = 101010 binary

 

There are other methods but I always liked this one to start.

 

[sweetpea3829]

I lucked out. I barely understood it, but my son took to it like duck to water.

 

Honestly, if it is causing tears I would just skip it. I'm not even sure it comes up in 5th grade BA. And I'm pretty sure it wasn't part of AoPS pre-Algebra. So it's a very interesting thing to know and understand, but not a hill to die on

DS told me yesterday afternoon that he thinks he gets it, ok. I might put him on Khan Academy just to work it a bit more while we move on in Beast.

 

This is what I love about Beast, though. It really does stretch these kids.

[Janeway]

My parents are STEM people so I could do binary before I could count to 100. I am not kidding. LOL. It was fun, like a puzzle! Still got made fun of in kindergarten when I could not count to 100, but hey, I knew binary! And 100 stood for something other than 99+1 when you are using binary. There were 100 kids in my family. My place number was 11. 

[Tumbatoo]

Ok, we're only on BA 3B, but now I'm excited to get to 4! That sounds fun :)

[daijobu]

It's also covered in AoPS' Introduction to Number Theory text, so if you don't get it now, no worries, you can always return to it later.  It isn't a standard part of an elementary curriculum, but a good thing to know long term, like some time before graduating high school.  

[Carol in Cal.]

I agree with the PP who said the larger bases are easier to understand. The School House Rock video Little Twelve Toes helped introduce us to the concept. 

 

 

The base ends up being written as 10.

The base is the number of different digits available for you to use. In the fingers and toes analogy, it is the number of fingers the person has.

 

In base 10, we count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

So if it is base 5, you count 1, 2, 3, 4, 10, 11, 12, 13, 14, 20.

In base 4, you count 1, 2, 3, 10, 11, 12, 13, 20, 21, 22.

In base 3, count 1, 2, 10, 11, 12, 20, 21, 22, 100, 101.

In base 2, count 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010.

 

I need to leave and don't have time to triple check these, so if someone sees an error, let me know and I will correct it. :)

Almost, but not quite, and the difference is crucial

 

In base 10 we count:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

 

In base 2 we count:

0, 1, 10, 11, 100, 101

 

Without the 0, base 2 is useless, because the choice between 0 and 1 at every digit is what enables you to use electronic switches in either the off or the on position to encode information.

[Tanaqui]

"There are only 10 types of people in the world: those who understand binary, and those who don't."

 

There are 10 types of people in the world: those who understand ternary, those who don't, and those who thought this was gonna be a binary joke.

 

Binary is very useful to learn, because with binary you can use your fingers to count to 1023.

 

And it's really not that hard. You understand place value using the tens system, right? The number just to the right of the decimal is the ones place, or 10⁰. You can fit 10 numbers in there (0 - 9) till you run out of numerals and have to roll over to the next place. The next place is 10¹, same deal - you can fit as many as 9 tens in there before you have to roll over. Then it's 10², or the hundreds place, and so on.

 

Well, all the place value systems use the same rule, they just have a different number of numerals. So the octal system (base 8) uses the numerals 0 - 7. The duodecimal system uses the numerals 0 - B, where A and B are eleven and twelve. The binary system only uses 0 and 1.

[mymommy1]

The joy of binary or other bases is realizing that not everyone used a base 10 system.  If you or your children are interested in different number systems, there are a couple of books I've used with my kids and math ed classes: How to Count Like A Martian by Glory St. John (old) and (newer) The History of Counting by Denise Schmandt-Besserat.  There are Egyptian, Mayan, Babylonian, and other systems. 

 

When I do this with my college elementary ed majors, they seem to do best when we revert to snap cubes and place value charts. 

[mymommy1]

Oh, another note.  At least one other professor, older than me, said alternate base systems (not just binary) where taught to all children when he went to school in the 70's.  Of course, they also learned to take square roots and logs without calculators.

[Kiara.I]

Oh, another note.  At least one other professor, older than me, said alternate base systems (not just binary) where taught to all children when he went to school in the 70's.  Of course, they also learned to take square roots and logs without calculators.

 

Yup.

My dad taught me to do square roots without a calculator.  We didn't do logs though.

[LEK]

I learned binary and bases other than 10 in school, 5th-6th grades. I loved it as a kid and am really looking forward to dd getting to beast 4 (it will not be long now). Hopefully she enjoys it as much as I did

[sweetpea3829]

Well....I finally get it...lol.  

 

Because it's a different way of thinking of numbers (number sense), I am nowhere near intuitive, but I finally get the concept underlying it and how it works.  

 

I think I have a better idea of how my dyscalculic daughter views the base-10 system now, too...lol.  

[luuknam]

Binary is also fun. I'm turning 100000 in a couple of months!

 

ETA: I think the importance of binary for computer programming is overrated. Yes, obviously computers use binary to do everything, but most people program use such high-level languages that what the computer exactly does stays hidden pretty well. Not saying it's ideal, but you probably could program quite a bit without understanding binary...

Edited May 27, 2016 by luuknam

[MrSmith]

Binary is also fun. I'm turning 100000 in a couple of months!

 

ETA: I think the importance of binary for computer programming is overrated. Yes, obviously computers use binary to do everything, but most people program use such high-level languages that what the computer exactly does stays hidden pretty well. Not saying it's ideal, but you probably could program quite a bit without understanding binary...

To a large extent this is true. Much in the same way that one can perform lots of arithmetic without understanding any of it.

[daijobu]

Coincidentally, I just came across this cute (wordless) video on binary.