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Copyrighted Material bindex.qxd 11/8/07 11:57 AM Page I1 Index A compensation, 158–59, 261, 291 aesthetics, golden ratio, 285 multiplication, 176–77, 194 AA (angle-angle) similarity decimals, 291, 297 Agnesi, Maria, 733 subtraction, 174–75, property, 754–55, 783 denominators Ahmes Papyrus, 237 192–93 AAS (angle-angle-side) triangle common, 255–57 a is congruent to b mod subtract-from-the-base, 175–76, congruence, 746 unlike, 256–57 multiplication, 925–26 193 abacus, 71, 107, 155, 172–75 exponent properties, 404 algebra, 13–16, 405–11 subtraction, 174–75, 298 absolute value, 354 facts, 111–15, 174, 176 balancing method, 405–6, 408 base five, 192–93 absolute vs. relative in circle for base five subtraction, coefficients, 408 whole-number operations, graphs, 472 192–93 concept of variables, 15 171–83 abstractions of geometric shapes, fractions, 255–59, 260–61 derivation of term, 830 al-Khowarizimi, 107, 830 588–90 identity property geometric problems, 807 alternate exterior angles, 624 abstract representation, 406, 408 fractions, 258 graphing integers on the alternate interior angles, 619–20, abstract thinking, 585, 588–90 integers, 347 coordinate plane, 815 743, 744, 771, 791–92 abundant numbers, 228 rational numbers, 384, Guess and Test strategy with, altitude of a triangle, 773, 779, acre, 669, 670–71 386–87 16 837–38, 839 acute angles, 617, 621 real numbers, 402 and one-to-one correspondence, amicable whole numbers, 228 acute triangle, 620, 621, 630 integer, 345–48, 352, 365–67 48 amoebas, exponential growth of, 87 addends, 110. See also missing- inverse property pan balance, 407 amounts, relative vs. absolute, 472 addend subtraction integers, 347, 352 solving equations, 405–6, 408, analysis—level 1, 584–85 adding the complement, subtraction rational numbers, 384–87 826–29 analytic geometry, 816 by, 185 lattice method algorithm, 173, solving inequalities, 409–11 analyzing data, 484–500 adding the opposite 192 transposing, 409 box and whisker plot, 489–92, integers, 349–50 left-to-right method, 159 Use a Variable strategy with, 503 rational numbers, 387 mental math, 157–59 13–16, 27–28 dispersion, 492–96 addition, 109–15, 255–59 multi-digit, 114–15 variables (See variables) distribution, 496–500 algorithms, 172–73, 184, 185, negative integer, 348 algebraic logic, 163–64 measures of central tendency, 192, 297 nonstandard algorithms, 183, algorithms, 107, 171–83, 297–301 484–88 associative property 184 addition, 172–73, 184, 185 Oregon rain, 501 clock arithmetic, 923, 924 numerators, 249 base five, 192 organizing, 442–44 (See also decimals, 291 obtaining ranges for estimation, lattice method, 173, 192 graphs) fractions, 244, 258, 260 160 base five, 192–95 standard deviation, 493–96, integers, 347 order of operations, 144–45, Caley-Purser, 368 498–500 rational numbers, 386–87 163–64 cashier’s, 184–85 variance, 493–94 real numbers, 402 with place-value pieces, 172–73 decimal, 297–301 and connective, 913 whole numbers, 112–15, probability property, 536–38 derivation of term, 830 angle(s), 588 158–59, 172–73 rational numbers, 384–87, 392 division, 133, 178–83 AA similarity property, 754–55, in base five, 192 real numbers, 402 base five, 194–95 783 as binary operation, 110 repeated addition for duplication, 186 acute, 617, 621 cancellation property, 123, 347, COPYRIGHTEDmultiplication, 123–24, 135, equal-additions, MATERIAL 185 adjacent, 617, 621 387 266, 357, 924 Euclidean, 223, 225 alternate exterior, 624 with chip abacus, 72, 172–73 sum of consecutive whole German low-stress, 188 alternate interior, 619–20, 743, clock arithmetic, 923, 924 numbers, 11–12 intermediate, 173, 176–77, 180, 744, 771, 791–92 closure property, 111, 257, 347, sum of counting numbers, 182, 192, 194 ASA congruence, 744, 746, 386 12–13, 20–21 lattice method, 173, 177, 192, 791, 794 common denominators, 255–57 unlike denominators, 256–57 194 base, 602, 604 commutative property whole numbers, 109–15, multiplication, 176–77 bisectors, 746, 769–70, 779 clock arithmetic, 923, 924 158–59, 172–73 base five, 194 central, 607–8, 628–30 decimals, 291 additive magic squares, 19, 217, RSA, 368 complementary, 618, 620, 621 fractions, 244, 258, 260 295, 296 Russian peasant, 185 congruent, 588, 590, 606, integers, 347 additive numeration systems, scratch addition, 184 741–42, 742–46, 791 rational numbers, 386–87 62–66, 72 Sieve of Eratosthenes, 206, 234 consecutive, 627 real numbers, 402 adjacent angles, 617, 621 standard copying, 767 whole numbers, 111–15, Adventures of a Mathematician addition, 172–73, 192 corresponding, 618–19 158–59, 172–73 (Ulam), 575 division, 182–83, 302–3 creating shapes with, 584, 587 I1 bindex.qxd 11/8/07 11:57 AM Page I2 I2 Index angle(s) (continued) Hero’s formula, 697 axis of rotational symmetry, 652 converting to, 76–77 dihedral, 641–44 lateral surface, 707 axis of symmetry, 600 converting to base five, 75–76 directed, 854–55 metric system, 675–76 grouping by tens, 71–73 Euclidean constructions, parallelogram, 692 B long division algorithm, 769–70 rectangle, 689–91 Babbage, Charles, 155 178–79, 181 exterior, 607–8, 617, 630 trapezoid, 692 Babylonian numeration system, powers of 10, 62, 159, 289, included, 743–44, 753–54 triangle, 691, 692 64–65, 66, 795 291–92, 301 interior, 608, 619–20, 627, 769 argument, 916 back-to-back stem and leaf plot, subtract-from-the-base interior of the, 617 arguments, logical, 916–19 443 algorithm, 175–76 measurement of, 617–18 Aristotle, 379 balancing method of solving base twelve, 78 dihedral, 641–42 Arithmetic, Fundamental Theorem equations, 405–6, 408 Begle, Edward G., 902 in polygons, 628–31 of, 206, 219, 399–400 bar graphs, 444–46 bell-shaped curves, 497–99 model and abstraction, 589 Arithmetica (Diophantus), 341 double-bar, 446 Bernoulli, 411 nonadjacent, 617, 621 arithmetic average. See mean histograms compared to, 445 Bertrand’s conjecture, 217 notation, 616–17 arithmetic logic, 163 misleading distortion of betrothed numbers, 228 obtuse, 617, 620, 621 arithmetic sequences, 86–87 compressing y-axis, Bhaskara, 276, 699 parallel lines and, 618–22 arithmogon, 17 444–45, 447, 465–67 bias and samples, 475–77 perpendicular bisector, 769–70 array pictorial embellishments, biased surveys, sources of, 475–77 polar coordinates, 832 8-by-8 square, 118 474 biconditional statement, 915 properties, 617, 619–20 calendar, 416 three-dimensional effects, Bidder, George Parker, 168, 195 reflex, 617, 621 coordinates and, 818 470–72 billion, 74, 79 right, 589, 617 of points, 818, 820 multiple-bar, 445 binary numeration system, 77, 78 SAS congruence property, rectangular with pictorial embellishments, binary operations, 78, 110 743–44, 746, 778, 790–92, division, 207, 208, 210 450–51 clock arithmetic, 923, 924 793 multiplication, 124–25, 267 SAT scores, 446 biorhythm, theory of, 231 straight, 617, 620, 621, 631 numbers, 87 when to use, 454 birthday probability problem, 513 sum in a triangle, 620, 629–30 Sieve of Eratosthenes, 206, 234 barrel, 671 bisector of an angle, 746, 769–70, supplementary, 618, 621 square lattice, 588, 615, 695, Barry, Rick, 353 779. See also perpendicular types of, 616–18 700 base, 71–73, 77–78 bisector vertex of the, 616–17 triangular lattice, 588 exponent of the, 142 black and red chips model, 382 vertical, 618, 619, 621 arrow diagrams, 83 subtract-from-the-base black and red chips model for See also vertex (interior) angles functions as, 89 algorithm, 175–76, 193 integers, 344, 345, 348, angle-angle-side (AAS) triangle in sequences, 86 base (geometric) 349–50, 352 congruence, 746 Ars Magna (Cardano), 341 angles of an isosceles triangle, Blackwell, David, 336 angle-angle (AA) similarity ASA (angle-side-angle) 602 blocks, in base ten pieces, 71 property, 754–55, 783 congruence property, 744, angles of a trapezoid, 614 board foot, 729 angle bisector, 746, 769–70, 779 746, 791, 794 of a cone, 646 borrowing angle-side-angle (ASA) associative property, 112, 126 of a cylinder, 645 subtraction, 174 congruence property, 744, of addition of a parallelogram, 692 box and whisker plot, 489–92 746, 791, 794 clock arithmetic, 923, 924 of a prism, 650 interquartile range, 489–92 angle sum in a triangle, 620, decimals, 291 of a pyramid, 647, 654 outliers, 489–92, 503 629–30 fractions, 244, 258, 260 three-dimensional figures, Brahmagupta, 237, 341 apex, 643–44, 646, 647 integers, 347 707–12 Braille numerals, 69 Appel, Kenneth, 713 rational numbers, 386–87 of a triangle, 691 Buffon, Georges, 567 applied problems in transformation, real numbers, 402 in volume, 718, 720–26 bundles-of-sticks model, 71, 75 894–95 whole numbers, 112–15, base design, 651–52 buoyancy, principle of, 665 Arabic numeration. See Hindu- 158–59, 172–73 base two, 78, 110, 923, 924 Arabic numeration system in mental math, 158–59, 291 base three, 77 C arc(s) of multiplication base five, 75–77 calculator activities of a circle, 766, 767, 770 clock arithmetic, 924–25 converting from base ten, comparing two fractions, 294 intersecting, 766–70 passim, decimals, 291 76–77 cross-multiplication, 248 780 fractions, 269–70, 275 converting to, 75–76 cross-multiplication of fraction of radius, 767, 769, 780 integers, 360–61 multiplication facts, 194 inequality, 248 Archimedean method for deriving rational numbers, 389 operations, 192–95 decimals, dividing, 302, 303 volume of a sphere, 665, real numbers, 402 subtract-from-the-base exponents, 404 712 whole numbers, 126, 129, algorithm, 193 fraction equality, 245 Archimedes, 411, 665, 712 158–59 subtraction facts, 193 fractions to percents, 322 are (metric area of a square), 675 astronomical unit (AU), 316 base ten greatest common factor, 222,
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