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From: Ray M <firstname.lastname@example.org> To: Teacher2Teacher Public Discussion Date: 1999020806:55:05 Subject: bits, bytes, bases, and borrowing The intent of my message is a little deeper. When a student is having trouble with borrowing, changing to binary can provide an analogy to draw on. The manipulatives are simpler to manipulate too. 4.-1. =>100b - 1b =11b only requires 4 unifix cubes on the desk. At the other extreme, I can find a base for any given problem that will not require borrowing. Notation may be a problem but.... WHEN I borrow is an artifact of base 10. And octal and hex provide generalizations of the borrowing. Hex and binary are used daily by thousands of EE's so they are "useful". The translation from hex to binary and binary to hex is nearly transparent (FFFh=1111 1111 1111b) but the borrowing behavior is very different. One other practical trick, since you mentioned fifth grade. Some of the various hand positions in binary finger counting are associated with certain obscene gestures (645. being particularly rude). Keep the back of your hand against a wall or flat on a desk or towards the ground until you can count very fast. I generally do this exposure to binary in kindergarten, so it's not such a problem. An interesting spin on it is to use the left hand either for larger numbers OR by crossing the arms a fractional notation: left thumb = 1/2 left index = 1/4 l. mid =1/8 l. ring =1/16 l. pinky = 1/32 just like on an inch ruler. And of course now you can demonstrate carrying from fractions to whole units and from one flavor of fraction to another. RE: the dots. I put the dots on the palm side. 3 and 4 year olds can count the exposed dots to convert binary to decimal. The dots are three dimensional. Demonstrating that in K-6 is a jaw dropper. I have three demos: a helium lamp with optical flats (very abstract), an electronic gage (less abstract), and finally least expensive and most concrete, a hardened steel pin and bushing. The pin goes in the hole and spins freely when clean but binds if you draw on it, Re: semi-concrete. Much of what Piaget did is bunk. Sometimes he didn't understand the physics (billiards), sometimes the methodology was flawed (water levels), and often his conclusions overreach the actual experiment.
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