Teacher2Teacher 
Q&A #522 
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From: Ray M <raypublk@san.rr.com> To: Teacher2Teacher Public Discussion Date: 1999020501:40:48 Subject: Re: teaching substraction(sic) & in refrence(sic) to barrowing(sic) You don't have to know a thing about 10's to learn about borrowing. It is rather easy to teach most four year olds how to count in binary: put 1 dot on your r. thumb, 2 dots on r. index, 4 dots on right middle, etc. Now you can uniquely represent 0 to 31 on the right hand. Using both hands, you can uniquely represent 0 to 1023. Transfering the finger notation to paper is easy: finger up =1, finger down = 0. Once counting in binary is second nature, you can add in binary. Dots can be used to visualize numbers and carrying. There are close parallels to all the rules in base ten. Once you know binary, octal and hex come almost for free and their rules parallel those of base ten too. So there are three useful bases other than ten in which to learn about borrowing. Most spreadsheets, symbolic math programs, and scientific calculators can help you check your work. It is simply not true that "all the numbers on the top represent 10" and to say so will conflict with trying to teach place value. In the example given: 458900 235774 the statement is made that you have to start with borrowing from the 9. In some sense that is true, but it seems like bad advice to give to a second grader. I would always give the advice to start in the ones place, note that I need to borrow, look at the next place value (10s), note the need to borrow, look in the next place(100s), etc. (I kind of like Judy's story!) But that is certainly not the only algorithm. I can subtract first and then clean up the answer in a weird base if I want: 4523=22 8957=32 0074=74 Now, it would bother most people if I wrote my answer as (22,32,74) but that's a nice ordered triplet and there's no rule against them. And there is a fairly simple rule to clean it up so that it looks like (22,31,26) Which is parallel to what I'd get if I did my math base 100. And base 100 is easy to convert to base 10. So 223126 is the answer. In hex the problem looks like 070094H 0398FEH =036796H In octal it looks like 01600224 00714376 =00663626 and in binary it looks like 111 0000 0000 1001 0100 011 1001 1000 1111 1110 =011 0110 0111 1001 0110 While gifted 7 year olds can certainly do that, I don't recommend trying it in an average 2nd grade classroom. But if you, as the teacher, will take the time to learn how to do it, you will understand what borrowing is really about.
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