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, area, 670–71, 675–76, 689–93 asymmetrical distribution, 497–98 blocks/pieces, 71, 287 223 abstract, 689–93 Aubel’s theorem, 788 addition, 172–73 integer computation, 352 circles, 693 avoirdupois ounces and pounds, 672 Dienes blocks, 71 long division, 302–3 English system, 670–71 axiomatics – level 4, 585 long division, 178–79, 181 notation, standard to scientific, formulas (See inside back cover) axis manipulation and scaling, multiplication, 177 364–65 height, 692, 710, 718, 720–26 465–67 subtraction, 175–76 percents, 322, 326–27 bindex.qxd 11/8/07 11:57 AM Page I3

Index I3

proportions, 326–27 central tendency, measures of, rational numbers, 386 real numbers, 402 repeated-addition approach, 135 484–88 real numbers, 402 whole numbers, 125–26, simplifying fractions, 257 mean, 486–88 whole numbers, 111 127, 128, 158–59, 266 sums and differences of rational median, 485–86, 488 of multiplication comparison approach to numbers, 388 mode, 485, 486, 488 fractions, 269 subtraction, 117–18 calculators, 163–66 centroid of a triangle, 779, 787, integers, 360–61 compass, 608, 781, 865 algebraic logic, 163–64 835–37 rational numbers, 389 compass constructions, 772, 780, arithmetic logic, 163 certain events, 519, 523 real numbers, 402 782 constant functions, 164–65 cevian line segment, 335 whole numbers, 125 compass properties, 766, 767 DEL key, 163 chain rule, 917 cluster estimation, 167 compatible fractions, 260, 275, distributivity and, 164, 166 chant, counting, 60, 129, 140 clusters and gaps, 443 322–23 exponent keys, 165 chaos theory, 756 codomain, 88–89 compatible numbers, 158 factorial key, 550 characteristic, 301 coefficients, 408 decimals, 291, 293 graphing, 420 charts. See bar graphs; graphs; solving simultaneous equations rounding to, 162 history of, 155 misleading graphs and and, 827 whole numbers, 158, 162 improper fractions, 268 statistics Cohen, Paul, 433 compensation, 158–59 logic, types of, 163–64 chess problem, 590 coin tossing probabilities addition, 261, 291 long division with, 163, 164–65 children combinations, 554 additive, 159 memory functions, 165 abstract thinking, 585, 588–90 conditional, 566–67 decimals, 291, 292, 293 Mϩ key, 165 holistic thinking, 581, 583–84, experiments, 516, 518–20 division, 167 M– key, 165 586 tree diagrams, 540–41 multiplication, 159, 292, 293 MR key, 165 “Child’s Thought and Geometry, Cole, Frank Nelson, 213 subtraction, 159, 261 ON key, 163 The” (van Hiele), 581 collinearity, reflections and, 879 complement order of operations, 163–64 Chinese abacus, 155 collinearity, slope and, 813–14, 836 odds, 563 parentheses, 163–64 Chinese numeration system, 69 collinearity test for distance, 810 of a set, 50 RCL key, 165 chip abacus, 72, 172–75 collinear medians of a triangle, 836 subtraction by adding the, 185 Reverse Polish notation, 163 Chudnovsky, David V., 411 collinear points, 615, 863, 879, complementary angles, 618, 620, scientific notation, 165, 301–2, Chudnovsky, Gregory V., 411 881, 893, 894 621 364–65, 369 circle(s), 608–9, 777–79, 829 combinations, 552–54 complementary triangles, 620 STO key, 165 analyzing shape, 585–86, 608 and Pascal’s triangle, 553–54 complex experiments, probability SUM key, 165 arc, 766, 767, 770 of r objects chosen from n and, 532–42 TI-34 II, 163 area, 693 objects, 553 complex fractions, 274 calendar array, 416 circumference, 689 commission, 333 composite numbers, 205–7, 206, Caley, Arthur, 368 circumscribed, 777–79, 838–39 common denominators, 247–48 214, 217 Caley-Purser algorithm, 368 diameter, 608 addition, 255–57 compound inequality, 145 cancellation property equations of, 829, 839 division, 270–72, 273, 391 compound interest, 421, 431 of addition Euclidean construction, 777–79 equality of rational numbers, compound statements, 913 integers, 347 hexafoils, 706 383, 387–88, 391 computational devices, 155 rational numbers, 387 inscribed, 777–78, 779 subtraction, 259–60, 387–88 computational estimation, 160–63 whole numbers, 123 noncollinear, 839 common difference of a sequence, cluster, 167 of multiplication radius, 608 86 decimals, 291–93 integers, 362 spheres (See spheres) common-positive-denominator fraction equivalents, 291, 322 whole numbers, 137 symmetry, 608 approach to ordering fractions, 250–51, 275 zero divisor, 362 tangent line to, 777 rational numbers, 392 front-end, 160–61, 261 Canterbury Puzzles, The circle graphs (pie charts), 448–49, common ratio of a sequence, 86–87 percents as fraction equivalents, (Dudeney), 215 451 commutative property, 111, 125–26 322–23 capacity, 668–69, 671 exploding, 472 of addition range, 160, 261, 275, 292 Cardano, 341 misleading, 472–73 clock arithmetic, 923, 924 rounding, 161–63, 182, 261, card deck probabilities, 517, multiple-circle, 448–49 decimals, 291 292–93 518–19, 555, 565 relative vs. absolute, 472 fractions, 244, 258, 260 of volume, 719 cardinal number, 59 when to use, 454 integers, 347 whole number operations, carrying, in addition, 172, 173 Circle Limit III (Escher), 849 rational numbers, 386–87 160–63 Cartesian coordinate system, circumcenter of a triangle, 777 real numbers, 402 See also mental math 417–19, 807, 819 circumference, 689. See also inside whole numbers, 111–15, computations, 155, 157 Cartesian product of sets, 51–52, back cover 158–59, 172–73 with calculators, 163–66 125, 526–27 circumscribed circles, 777–79, calculators and, 164 estimation, 157–63 cartographers, 419 838–39 of multiplication See also calculator activities; cashier’s algorithm, 184–85 Clinton, William Jefferson, 475 clock arithmetic, 924–25 computational estimation; Cavalieri’s principle, 724–26 clock arithmetic, 923–25 decimals, 291 mental math Celsius, degrees, 88, 678–79 clockwise orientation, 854–55, 860 fractions, 266, 269 computers, 155, 225, 305 center of a size transformation, 863 closure property integers, 360–61 concave (nonconvex) shapes, 607, center of a sphere, 646 of addition in mental math, 158–59, 617, 631 centimeter, 667, 675–76 fractions, 257 291 conclusion, 914 central angles, 607–8, 628–30 integers, 347 rational numbers, 389 concrete representation, 406, 408 bindex.qxd 11/8/07 11:57 AM Page I4

I4 Index

concurrent lines, 615 equations cross-products, 244 fraction approach to addition, concurrent medians of a triangle, of circles, 829, 839 cryptarithm, 8–9 297 836 of lines, 822–26, 838–39 cryptographic codes, 134 fraction method for ordering, conditional equations, 15, 405 simultaneous, 826–29, cubed numbers, 88 290, 300 conditional probability, 565–67 830–31 cubes (geometric), 640–42, 671, fractions as, 287–89 conditional statements, 914 mapping history, 815 718 front-end estimation, 292 cones, 646, 647, 711–12, 716, midpoint formula, 811–12 cubic hundreds square, 288, 289–90 722–23 midsegment proof, 910 centimeter, 675–76 Look for a Pattern strategy, congruence orthocenter, 837–38 decimeter, 675–77 303–4 angles, 588, 590, 606, 741–42, percent grade, 818, 821 foot, 671 mental math, 291–93 742–46 polar coordinates, 832 inch, 671 metric system and, 673, 676, line segments, 588–90, 741 problem solving, 834–39 meter, 675–76 677–78 medians of isosceles triangle, slope, 812–16, 826, 836 unit, 87, 717 monetary system, 249 835 slope-intercept equation of a yard, 671 multiplying by powers of 10, midsegment theory, 909 line, 823–25 cubic function, graphs of, 422–23 291–92, 301 parallelograms, 604 See also planes cup (unit of measure), 671 nondecimal systems vs., 75–78 parallelograms and rhombuses, coordinate plane curved shapes in three-dimensional nonrepeating, 400–402 744, 791 distance in, 809–12 space, 645–47 nonterminating, 304–5 polygons, 881, 883–84 graphing integers on, 815 curves, bell-shaped, 497–99 notation, 294 shapes, 880–81 correlation relationship, 453 cylinders, 645–47, 709, 720–21, number lines, 288, 289–90 transformations, 875–81 correspondence 723, 724 number system diagram, 400 triangles (See triangle angles, 618–19 operations with, 297–301 congruence) congruent triangles, 742 D ordering, 289–90, 291 congruence modulo m, 925–26 one-to-one, 46, 60 data, 484–500 percents and, 321–22 conjectures points, 645, 647 analyzing (See analyzing data) place-value Bertrand’s, 217 similar triangles, 752–56 cluster, 443 for decimal algorithm, 297 Goldbach’s, 203 counterclockwise orientation, comparison with multiple bar for ordering, 289–90, 291 Hamann’s, 217 854–55, 860, 877 graphs, 445 properties, 291 odd perfect number, 203 counting factors, 219–20 discrete values, 445, 446 repeating (See repeating twin prime, 203 counting numbers, 22, 45, 381 gap, 443 decimals) Ulam’s, 203 composite, 205–7, 214 grouping in intervals, 443–44 rounding, 292–93 conjunction of statements, 913 as distinct primes with histogram vs. bar graph, 445 squeeze method, 412 connectives, logical, 913–15 respective exponents, 220 inaccurate, 475–77 terminating, 289, 305 consecutive angles, 627 doubling function, 88–89 intervals, 445 decimeter, 667, 673–74, 675–76, consecutive sides, 608 factors of, 219–20 organizing, 442–44 (See also 676–77 consecutive vertices, 608 number systems diagram, 381, graphs; statistics) deduction—level 3, 585 constant function with calculator, 382, 392, 401 picturing information, 449–54 deductive arguments, 916–19. See 164–65 prime, 205, 212–13 utilizing, 441 also direct reasoning constructions, 781. See also relatively prime, 227 debugging, 305 deductive reasoning, 916 Euclidean constructions sequences of, 20–21 decay, 422 deficient numbers, 228 continuous set of numbers, 445 set of, 45 decimal point, 288, 297–300, 301 definitions of geometric shapes, continuum hypotheses, 433 Smith number, 231 decimals, 71–73, 287–93, 297–305 621, 647 contrapositive, 914 sum of, 12–13, 20–21 additive properties, 291 degrees Celsius, 88, 678–79 converse, 914 counting techniques, 14–15, algorithms, 297–301 degrees Fahrenheit, 88, 672, converse of the Pythagorean 549–55 addition, 297 678–79 theorem, 792–93 by 1 and 2, 113 division, 300–301 dekameter, 667, 673–74 conversions chant, 60, 129, 140 multiplication, 298, 300 Deming, W. Edwards, 441 between bases, 75–78 combinations, 552–54 subtraction, 298 Deming Award, 441 Celsius to Fahrenheit, 88, factors, 219–20 associative property, 291 DeMorgan’s laws, 54, 57 678–79 fundamental counting property, with calculators, 300, 302–3 denominator(s), 240 measurements, 680–81 534, 536, 540, 542, 549–52 commutative property, 291 common (See common convertibility of measurement permutations, 549–53, 555 compatible numbers, 291, 293 denominators) systems, 672–73 and probability, 555 compensation, 291, 292, 293 Egyptians, Mayans, and, 237 convex shapes, 607–8, 643–44 and tree diagrams, 532–34, concept of, 287–89 of equivalent fractions, 244 coordinate geometry, 809–16, 540–41 converting to and from of improper fractions, 246 822–29, 834–39 Cover Up strategy, 16 fractions, 289, 291–93, 298, least common, 256–57 Cartesian coordinate system, Coxeter, H. S. M., 845 300–301 and numerators approach to 417–19, 807, 819 cropping graphs, 468–70 distributive property, 291 division, 272–73 collinearity test, 810 cross-multiplication, 244–45 dividing by powers of 10, in rational numbers, 382–83 coordinates on a line, 616 of fraction inequality, 247–48, 291–92, 301 reducing fractions and, 244–45 coordinate system, 418–19 249 Draw a Picture strategy, 288 unlike denominators distance in coordinate plane, of rational number inequality, equal-additions, 291, 301 addition, 256–57 809–12 393 estimation, 291–93 division, 272–73 Earth’s surface, 821 of ratios, 312–13 expanded form, 288, 297, 300 subtraction, 260, 388 bindex.qxd 11/8/07 11:57 AM Page I5

Index I5

density property, 403 from perpendicular bisector, estimation for fractions, 275 whole numbers density property of fractions, 249 778 fractions, 270–75, 300–301 less than and addition, 141 denying the consequent (modus from sides of a triangle, 779 invert and multiply, 390, transitive property of “less tollens), 917–18 equidistant from endpoints, 616 409 than,” 140–41 Descartes, René, 51, 807, 816 isometries and, 853, 876–77, integers, 357, 362–63 Draw a Picture strategy, 9–11 description of geometric shapes, 895 invert and multiply, 272–74 addition, 112–13, 114 589 on a number line, 616, 686–87 long (See long division) algebraic equations, 406, 408 diagonals, 605, 606, 629 parsec, 685 measurement, 129–30, 270–73 angle sum of a pentagon, 629 of a kite, 746, 768 from point P to point Q, PQ, mental math, 158, 275, 292 centroid of a triangle, 836 of a quadrilateral, 894 686–87, 766, 770–71, missing factor approach, 132, circumcenter of a triangle, of a rectangle, 743, 744, 777–79, 809–10, 876–77 182–83, 924–25 838–39 790–91 ratio of, 881–84 missing factors (See quotients) combining with other strategies, of a rhombus, 768–69, 770, unit distance, 687 nonstandard algorithms, 186 27–30 791–92 distribution(s), 489–500, 496–500 numerators and denominators congruence properties, 746, slopes for analyzing, 816 bell-shaped, 497–99 approach, 272–73 790–91 of a square, 795 mean, 497–500 (See also mean) order of operations, 144–45, congruent medians of a diagrams medians, 497–99 (See also 163–64 triangle, 835 arrow, 83, 89 medians) partitive, 129–30 converse of the Pythagorean functions as, 89 mode, 496–99 (See also mode) by powers of 2, 78 theorem, 792–93 in sequences, 86 normal, 498–500 by powers of 10, 291–92, 301 cubic function, 422 graphs (See graphs) outliers, 489–92, 500 quotients (See quotients) decimals, 288 number systems, 381, 382, 392, relative frequency, 496 rational numbers, 390–92 division, 133 400, 401 standard deviation, 498–500 real numbers, 402–3 four-fact families, 116–17 tree (See tree diagrams) symmetrical and asymmetrical, remainders, 133, 164–65, fraction strips, 242, 247, 255, Venn (See Venn diagrams) 496–99 182–83 257 See also Draw a Diagram z-scores, 495–96, 498–500, repeated-subtraction approach, indirect measurement, 755 strategy 499–500 133–34 initial problem, 2, 38 diameter, 608, 646 distributive property, 127 standard algorithm, 182–83 orthocenter of a triangle, 837 dice throwing, 516, 518–19, decimals, 291 unlike denominators, 272–73 pizza-cutting, 9–10 520–21, 534, 564 division over addition, 158 whole numbers, 123, 129–34, repeated-subtraction, 133–34 Dienes blocks, 71 multiplication over addition 274 tetromino shapes, 10–11 difference fractions, 270 zero, property of, 132 See also number lines common, of a sequence, 86 integers, 360–61 See also tests for divisibility drawing with/without replacement, minuend minus subtrahend, 116 rational numbers, 390 “division makes smaller” 534–35, 536–39 order of operations, 144–45, real numbers, 402 misconception, 297 dual of a tessellation, 635 163–64 whole numbers, 135, 176 divisor(s), 132, 207 Dudeney, Henry, 215, 628 set, 110 multiplication over subtraction decimals into whole numbers, duplication algorithm, 186 of sets, 50–51 fractions, 270 300–301 dynamic spreadsheet activities on successive, 27–30 rational numbers, 390 division of fractions, 270–73 the Web site digits, 52, 67, 71, 74–75 whole numbers, 128 rounding up, 182 Base Converter, 79, 81 digits, random, 561, 568 distributivity, calculators and, 164, zero divisors property of Circle Graph Budget, 462 dihedral angles, 641–44 166 integers, 362 Coin Toss, 527 dilations/dilitations. See divide and average for square roots, Do a Simulation strategy, 514, 574 Cubic, 428 transformations 412 dodecahedron, 644–45 Euclidean, 231 dimensional analysis, 680–81 dividends, 132, 182 does not divide, 207 Function Machines and Tables, Diophantus, 341 and “division makes smaller” domain, 88–89 95 directed angles, 854–55 misconception, 297 dot plot, 442 Roll the Dice, 527 directed line segments, 853 divides, 207 double-bar graphs, 446 Scaffold Division, 186 direct reasoning, 108, 148, 916 divine proportion, 285 doubles, 114 Standard Deviation, 503, 505 Direct Reasoning strategy, 108, 148 divisibility. See tests for divisibility doubling function, 88–89 discounts, finding, 326–27 division, 123, 129–34, 270–75 dram, 672 E discrete values, 445, 446 algorithm, 133 Draw a Diagram strategy earned run average, 319 disjoint sets, 48, 109–10, 345 base five algorithm, 194–95 factor trees, 206 earth’s surface, 821 disjunction of statements, 913–14 compensation, 167 fractions, 240, 241–42, 243 edges of a cube, 642 dispersion, measures of decimals, 292, 300–301 measurement division, 129–30 edges of a polyhedron, 643, 644 standard deviation, 493–96 denominators (See common outcome probabilities, 533, Egyptian numeration system, variance, 493–94 denominators; denominators) 537, 539, 562 62–63, 66, 237, 411 distance(s) denominators and numerators partitive division, 129–30 Einstein, 198 circumcenter of an angle, 770 approach, 272–73 percent problems, 324 elementary logic. See logic/logical coordinate distance formula, dividend, 132 sample space, 524, 566 Elements, The (Euclid), 739, 784 810 divides, 207 sets, ratios, and probabilities, elements in a finite set, 59, 89, 110 in coordinate plane, 809–12 divisors, 132, 182, 207, 44, 99 elements of a set, 45–46, 59–60 equidistant 270–73, 300–301, 362 sets, ratios and probabilities, elimination method of solving from endpoints, 770, 778 does not divide, 207 99–100 simultaneous equations, 831 bindex.qxd 11/8/07 11:57 AM Page I6

I6 Index

Elliott wave theory, 425 embedded graphs, 473–75 marble drawing, 521 Eratosthenes, Sieve of, 206, 234 ellipse, 834 empty set, 46, 112 multiplicative property of Erdos, Paul, 281, 713 ellipsis, 22 endpoints of a line segment, 601, probability, 536–38 Escalante, Jaime, 902 eManipulatives on the Web site 608, 616 odds and, 563–65 Escher, M. C., 849 Balance Beam Algebra, 413, English system of measurement, Pascal’s triangle, 542 Escher-type tessellations, 849, 415 669–72 probability property for, 861–62 Base Blocks—Addition, 183, area, 670–71 517–18 estimation. See computational 188 avoirdupois ounces and pounds, sample space of, 524 estimation; front-end Base Blocks—Subtraction, 184, 672 unequally likely events vs., estimation 189 length, 669–70 520–21, 525, 565, 566 Euclid, 739, 784 Chip Abacus, 79, 81 metric system compared, 674, Venn diagram of conditional Euclidean algorithm, 223, 225 Chips Minus, 353, 356 678, 680–81 probability, 566–67 Euclidean constructions, 765–71, Chips Plus, 353, 355 notation, 669–72 equal sets, 46 777–80, 782 Circle 0, 355 ratios, 669–70 equals sign, 742 bisect an angle, 769–70 Circle 21, 18 temperature, 672, 678–79 equal to, mode and median, 486 circumscribed circles, 777–79 Circle 99, 20 volume, 671 equation(s), 15 compass properties, 766, 767 Coin Toss—Heads in a Row, weight, 672 abstract representation, 406, copy a line segment, 766–67 532, 574 English system units of 408 copy an angle, 767 Color Patterns, 34 measurement Cartesian coordinate system, equilateral triangle, 780, 782 Comparing Fractions, 250, 252, acre, 669, 670–71 417–19, 807, 819 inscribed circles, 777–78, 779 253 barrel, 671 of circles, 829, 838–39 line parallel to a given line Composition of board foot, 729 concrete representation, 406, through a point not on the Transformations, 888, 892 cup (unit of measure), 671 408 line, 771 Congruence, 749 degrees Fahrenheit, 672, conditional, 15, 405 parallel lines, 771 Coordinate Geoboard, 817, 818, 678–79 extremes and means, 311 perpendicular bisector, 768 820, 872–73 dram, 672 false, 15 perpendicular line through a Counterfeit Coin, 32 foot, 669, 670–71 identity, 15 point on the line, 770 Dividing Fractions, 276, 278, furlong, 669 linear, 830–31 perpendicular line to a given 279 gallon, 671 of lines, 822–26, 838–39 line through a point not on the Equivalent Fractions, 250, 253 grain, 672 percents, 325–28 line, 770–71 Factor Tree, 228, 230 inch, 669, 670–71 point-slope of a line, 825–26 regular polygons, 780, 782 Fill ‘n Pour, 229 mile, 669, 670–71 simultaneous, 826–29, 830–31, segment lengths, 783 Function Grapher, 426, 429, 430 ounce, 671, 672 831 straightedge properties, 766–67 Function Machine, 95 pint, 671 slope-intercept of a line, Euclid of Alexandria, 739 Geoboard, 595, 623, 626, 634, quart, 671 823–25 Euclid’s perfect number, 228 637, 697, 698, 700, 703, 865, rod, 669 solving (See solving equations) Eudoxus, 379 866, 868, 870, 871 tablespoon, 671 true, 15 Euler, Leonhard, 213, 411, 733 Geoboard—Triangular Lattice, teaspoon, 671 equiangular, 590 Euler diagram, 917 594, 598 volume, 671 equiangular polygons, 628–30 Euler’s formula Histogram, 454, 457, 459 weight, 672 equidistant polyhedra, 644, 733 Let’s Make a Deal, 548 yard, 669, 670–71 from endpoints, 770, 778 even Multibase Blocks, 79, 80–81, ENIAC, 155 from perpendicular bisector, counting numbers, 22 195, 196, 197 equal-additions 778 n th root and, 403–4 Multiplying Fractions, 276, 278 algorithm, 185 from sides of a triangle, 779 number of factors, 363 Number Bars, 119 compensation in subtraction equidistant from endpoints, 616 number of prime factors, Number Puzzles, 18, 20, 120, method, 159 equilateral, 588 399–400 123 decimals, 291, 301 polygons, 589, 628–30, 632–33 numbers, 88 Parts of a Whole, 254 subtraction, 261 squares as, 590 odd and, making groups of two, Pattern Blocks, 254 equality triangle, 588, 589, 601, 602, 209 Percent Gauge, 329, 330, 332, equal groups and equal shares, 632 rounding to nearest, 162 333 131 equilateral triangles event(s), 516 Perpendicular Lines, 818 equal sets, 46 Euclidean construction, 780, pairwise mutually exclusive Pythagorean Theorem, 701 of fractions, 244, 322–23 782 simpler, 536 Rectangle Multiplication, 191 of rational numbers, 383, Koch curve, 756 probability of, P(E), 517–24, Rectangular Division, 136 387–88, 391 equivalence relation, 85 529 Scatterplot, 456, 457, 461 of ratios, 311 equivalent See also equally-likely events; Sieve of Eratosthenes, 214, 215, equally-likely events directed line segments, 853 outcomes 217 additive property of probability, fractions, 242, 244–45, 256, exchanging Simulation, 568, 571, 572, 574 536–38 289, 291, 322–23 addition, 172–73 Slicing Solids, 655, 656 children as male or female, logically equivalent, 915 subtraction, 174–75 Tessellation’s, 635, 638 560–61 parts, 240 exclusive “or,” 913 Tower of Hanoi, 36 conditional probability, 566–67 ratios, 311 exercises, problems vs., 4 Venn Diagrams, 53–54, 56, 57 drawing with/without sets, 46 expanded form/notation, 72 Visualizing Fractions, 254 replacement, 535 Eratosthenes, 700–701 decimals, 288, 297, 300 bindex.qxd 11/8/07 11:57 AM Page I7

Index I7

exponents, 142–43 missing (See quotients) closure, 257 as numbers, 240–41, 242–46 place value and, 72, 76, 172–73 number of, 219–20 commutative, 244, 258, 260 number systems diagram, 381, whole numbers, 72, 76, 172–73 prime factors vs., 219–20 identity, 258 382, 392, 401 expected outcome, 560 primes and composites, 205–7 associative property of addition, as numerals, 240, 241–42, 245 expected value, 562–63 proper, 203, 228 260 numerator, 237, 240 experiment(s), 516 special, 159 commutative property of numerousness, 241 Cartesian product of sets, test for, 213 (See also tests for addition, 260 ordering, 247–49 526–27 divisibility) comparing two, 294, 393 part-to-whole, 241–42 complex, 532–42 factor trees, 206 complex, 274 percents and, 321, 322 simple, 515–25 facts concept of, 239–46 prime factors, 245 computing probabilities in, addition, 111–15, 174, 176 cross-multiplication, 244–45, as rational numbers, 382 517–24 for base five subtraction, 247–48, 249 reciprocals, 269, 382, 389, experimental probability, 519–20, 192–93 decimals, 287–89 390–91 532–42 four-fact families, 116–17 denominator (See “reducing,” 244 simulation, 560–62 multiplication, 128–29, 176 denominators) relative amount, 240–42, 244 exploding circle graphs, 472 base five, 194 density property, 249 repeating decimal exponent(s), 142–44 subtraction, 116–17 dimensional analysis, 680 representation, 304–5 additive property, 404 base five, 193 division, 270–75, 300–301 repeating decimals and, 304–5 base systems and, 77–78 Fahrenheit, degrees, 88, 672, with common simplified form, 242, 244, 245, with calculators, 165, 404 678–79 denominators, 270–72, 253 division, 143–44 Fahrenheit, Gabriel, 672 273 with calculator, 257 integers and, 357 false equations, 15 complex fractions, 274 LCD, 256 multiplication, 142–43 Fermat, Pierre de, 203, 213, 513 invert and multiply, 272–74, Solve an Equivalent Problem negative integer, 363–65 Fermat primes, 782 390, 409 strategy, 238, 281 and ones digit, 22–24 Fermat’s Last Theorem, 203 numerators and subtraction, 259–61 powers of 10, 62, 159, 289, Fermi problems, 201 denominators approach, terminating decimals, 289 291–92, 301 Feuerbach, Karl, 839 272–73 “to break,” 237 powers of three, 22, 88 Feuerbach circle, 839 unlike denominators, unitary, 263 prime factorization method Fibonacci sequence 272–73 See also decimals; rational greatest common factor, counting numbers and, 22 Draw a Diagram strategy, 240, numbers; ratios 221–23 golden ratio and, 285 241–42, 243 fraction method for ordering least common multiple, patterns and, 33, 36 Egyptian numeration, 237 decimals, 290, 300 224, 225 Pythagorean triples and, 705 equivalent, 242, 244–45, 256, fractions product of primes with, 219–20 sum of ten consecutive, 218 289, 291, 322–23 compatible, 322–23 rational, 403–5 finger multiplication, 186 equivalent parts, 240 fraction strips, 242, 247, 255, 257 real-number, 405 finite sets, 47–48, 59, 60, 89, 110 estimation, 250–51, 275 Franklin, Benjamin, 118 scientific notation of, 363–65 Fish (Escher), 849 greater than, 246, 247–48 frequency of a number, 442 unit fraction, 404 5-con triangle pairs, 746 history of, 237 histograms, 443–44 variable as, 421 Flannery, Sarah, 368 improper, 246, 258–59 stem and leaf plots, 442–43 whole number, 77–78, 142–44 flats, in base ten pieces, 71 calculator, 268 front-end estimation, 160 zero as, 144 foot, 669, 670–71 infinite number of, 245, 248, with adjustment, 160–61, 292 exponential decay, 422 formulas 383, 385 decimals, 292 exponential functions, graphs of, coordinate distance, 810 least common multiple, 223–26 one and two column, 160 420–22 coordinate midpoint, 811–12 less than, 247–48 one/two column, 160 exponential growth, 87, 421, 422 Euler’s formula lowest terms, 244, 245, 253 range, 160 compound interest and, 431 polyhedra, 644, 733 mental math, 260–61, 275 frustum of a right circular cone, exponential notation, 142–43, 225, functions as, 90–91 missing-addend subtraction, 716 363–65 Hero’s formula, 697 259–60 fuel gauge model, 324 exterior angles, 607–8, 617 population predictions, 428 mixed numbers, 246, 257, function(s), 82, 85–91, 417–25 measures, 630 as problem solving strategy, 258–59 arithmetic sequence, 86–87 exterior of the angle, 617 440, 506–7 multiplying, 267–68, 275 as arrow diagrams, 89 extreme outliers, 503 See also inside back cover monetary system and, 249 calculator memory, 165 extremes, 311 four-color problem, 713 multiplication, 266–70, 275 Cartesian coordinate system, four-fact families, 116–17 decimals and, 298, 300 417–19 F four-step process. See Pólya’s four- multiplicative properties codomain, 88–89 faces of a dihedral angle, 641–42 step process associative, 269–70, 275 common difference of a faces of a polyhedron, 643, 647 Fractal Geometry of Nature, The closure, 269 sequence, 86 factorial, 214 (Mandelbrot), 755 commutative, 266, 269 common ratio of a sequence, n factorial, 550–51 fractals and self-similarity, 755–56 distributive, over addition, 86–87 factorization. See prime fraction(s), 239–49, 255–61, 270 cubic, 422–23 factorization 266–75, 381 identity, 269–70 defined, 85 factors, 124 addition, 255–57 inverse, 269–70, 274 domain, 88–89 counting, 219–20 additive properties number lines, 247, 248, 255, doubling, 88–89 of integers, 352 associative, 244, 258, 260 266 exponential, 421 bindex.qxd 11/8/07 11:57 AM Page I8

I8 Index

function(s) (continued) compass, 608 geometric shape concepts, 583–90, tessellations (See tessellations) exponential growth, 87 coordinates (See coordinate 600–609, 615–22 tetrahedron, 644–45 as formulas, 90–91 geometry) adding zing to, 845 tetromino, 10–11, 611 geometric sequences, 86–87 diagonals, 605, 606, 629 analytic geometry, 816 three-dimensional (See three- as geometric transformations, diameter, 608, 646 congruent, 880–81 (See also dimensional shapes) 91 Escher’s art, 849, 860–61 congruence) trapezoid (See trapezoids) from graphs, 424–25 Euclidean constructions, definitions of shapes, 621, 647 triangles (See triangles) graphs of, 90, 417–24 765–71, 777–80, 782 describing shapes, 588–90 Germain, Sophie, 31, 39 greatest integer, 427 Euclid’s organization of, 739 models of shapes, 589 German low-stress algorithm, 188 initial term, 86 fractals and self-similarity, van Hieles’ stages of learning, Gleason, Andrew, 507 inputs into, 90, 93 755–56 581 glide reflection, 858–60, 875–77 linear, 419 great circle of a sphere, 651, See also base (geometric) collinearity and, 879 as machines, 90 712 geometric shape(s) congruent shapes, 880–81 notation, 87–88 height (See height) angles (See angles) distance and, 877 as ordered pairs, 90 length (See length) circles (See circles) glide axis of, 859, 879–80 quadratic, 420 mean proportional, 786 concave, 607, 617, 631 image, 859 range of, 88–89, 91 median, interpretation of, 497–99 cones, 646, 647, 711–12, 716, isometries, 858–60 representations of, 88–91 Mira®, 602, 608 723 notation, 875, 876 step, 423–24 models and abstractions, 589 congruent (See congruence; symmetry, 861 stock market and, 425 orthocenter, 779, 784, 837–38 triangle congruence) transformations, 858–60 as tables, 89 parallel lines and transversals, convex, 607–8, 643–44 Goldbach’s conjecture, 203, 215 fundamental counting property, 618–19 cubes, 640–42, 718 golden ratio, 285 534, 536, 540, 542, 549–52 patterns, 13–15 (See also cylinders, 645–47, 709, golden rectangle, 285, 786 Fundamental Theorem of patterns; tessellations) 720–21, 723, 724 grain (unit of measure), 672 Arithmetic, 206, 219, perpendicular bisector (See dodecahedron, 644–45 gram, 678 399–400 perpendicular bisector) ellipse, 834 Granville, Evelyn Boyd, 281 furlong, 669 polygonal regions, 630, 643–45 frustum, 716 graphing calculator, 420 probability of concept of hexafoils, 706 graphs, 417–25, 444–54, 464–77 G measure, 529 hexagons, 607, 632, 633, 781 back-to-back stem and leaf gallon, 671 problem solving hexahedron, 644 plot, 443 Gallup poll, 477 with algebra, 807 hyperbola, 834 bar (See bar graphs) gaps and clusters, 443 with coordinates, 834–39 icosahedron, 644–45 bell-shaped curves, 497–99 Gardner, Martin, 374 with transformations, kites (See kites) box and whisker plot, 489–92, Garfield, James, 705 893–95 n-gons (See polygons) 503 Gauss, Carl Friedrich, 36, 39, 665 with triangle congruence octahedron, 644–45 Cartesian coordinate system, Gauss’s theorem for constructible and similarity, 790–95 parabola, 834 417–19, 807, 819 regular n-gons, 782–83 protractor, 617–18 parallelograms (See circle, 448–49, 451, 454, 472–73 GCF. See greatest common factor Pythagorean theorem (See parallelograms) clusters and gaps, 443 Geometer’s Sketchpad on the Web Pythagorean theorem) pentagon, 605, 607–8 comparison of, 454 site sequence, 86–87 pentominoes, 342, 373, 613 coordinate system, 418–19 Centroid, 787 shape (See geometric shapes) planes (See planes) cropping, 468–70 Circumcenter, 787 skew lines, 642 polygons (See polygons) cubic functions, 422–23 Midquad, 797 slant height, 710, 711–12 polyhedra (See polyhedra) distortion (See misleading Name That Quadrilateral, 614 surface area, 707–12 prisms (See prisms) graphs and statistics) Orthocenter, 784 symmetry, 600–604 properties as distributions, 489–500 Parallelogram Area, 704 tangent, 777 angles, 617, 619–20 dot plot, 442 Rectangle Area, 698 teaching, 845 convex attribute, 607 ellipse, 834 Same Base, Same Height, Same terminology, 589, 606, 621, 647 diagonals of a kite, 605 exponential functions, 420–22 Area, 703 timeline for, 807 isosceles triangles and functions, 90 Size Transformation, 869 transformation (See trapezoids, 601–2 of functions, 417–24 Slope, 817 transformation geometry) parallelograms, 604 functions from, 424–25 Tree Height, 758 vector geometry, 909 points and lines, 616 gaps and clusters, 443 Triangle Inequality, 707 vertex arrangement, 632–33, pyramids (See pyramids) histograms, 443–44, 454 geometric concepts 638 quadrilaterals (See hyperbola, 834 analytic geometry, 816 vertices, 588, 602–3, 605, quadrilaterals) line (See line graphs) analyzing shapes, 600–609 643–44 recognizing, 583–90, 585–87 linear functions, 419–20 circles, 608–9 volume (See volume) rhombicuboctahedron, 645 line plot, 442 parallel line segments, See also coordinate geometry; rhombus (See rhombus) misleading (See misleading 605–7 number lines; transformation shape identification, 585–87 graphs and statistics) perpendicular line geometry simple closed curves, 607, multiple-line, 447–48 segments, 605–7 geometric fallacy, 285 645–46, 657 outliers, 453, 489–92, 503 regular polygons, 607–8 geometric mean, 786 sphere, 646, 648, 651, 712 parabola, 834 symmetry, 600–604 geometric representation of real square (See square) pictographs, 449–51, 454, angle bisector, 746, 769–70, 779 numbers, 401 table of, 589 471–72 bindex.qxd 11/8/07 11:57 AM Page I9

Index I9

pictorial embellishments, subtractive pairs, 63 Hopper, Grace Brewster Murray, Inferential Guess and Test strategy, 450–51 in tally numeration system, 62 198, 305 7–9 pie charts, 448–49, 451, 454, by tens, 62–63, 71–73, 74 horizontal axis. See x-axis infinite geometric series, 265 472–73 Guess and Test strategy, 5, 9, 38 horsepower, 685 infinite number of primes, 226, 402 plot, 442 for algebraic equations, 16 How to Solve It (Pólya), 1 infinite sets, 47–48 quadratic functions, 420 cryptarithm, 8–9 hundreds square, 288, 289–90 informal measurement. See regression lines, 453 Inferential, 6, 8 Hypatia, 800 nonstandard units of scatterplot, 452–54 Random, 6, 8 hyperbola, 834 measurement solving simultaneous equations, Systematic, 6, 8 hypotenuse of a right triangle, 401, information, 442–54 830–31 gumball probabilities, 525, 539–40 693–95 organizing, 442–44 stem and leaf plot, 442–43, hypothesis, 914, 916 picturing, 449–54 490–91, 503 H hypothetical syllogism, 917 See also data; graphs step functions, 423–24 Haken, Wolfgang, 713 initial side of a directed angle, 855 vertex-edge (See vertex-edge half-turn rotation, 893–94 I initial term, 86 graphs) Halmos, Paul, 149 icosahedron, 644–45 inscribed circles, 777–78, 779 vertical line test, 424–25 halving and doubling algorithm, identification numbers, 59 inscribed hexagon, constructing, See also diagrams 169, 185 Identify Subgoals strategy, 740, 781 great circle of a sphere, 651, 712 Hamann, Arthur, 217 799 Instituzioni Analitiche (Agnesi), 733 greater than Hamann’s conjecture, 217 identity element (one), 126 insurance company’s expected addition hamburger condiments probability, identity element (zero), 112–13 value, 562–63 inequalities, 410 534 identity equations, 15 integer(s), 343–52, 357–67, 381 fractions, 246, 247–48 Harding, Warren G., 513 identity property absolute value, 354 general set formulation of, 62 Hardy, George, 232 of addition addition, 345–48 inequalities, 405, 410 harmonic triangles, 265 fractions, 258 negative integers, 348 integers, 365–67 hectare, 675–76 integers, 347, 384 additive properties mode and medians, 486 hectometer, 667, 673–74 rational numbers, 384, associative, 347 multiplication by a negative height 386–87 cancellation, 347 inequalities, 410 cone, 711–12 real numbers, 402 closure, 347 multiplication by a positive cylinder, 709 whole numbers, 112–13 commutative, 347 inequalities, 410 parallelogram, 692 of multiplication identity, 347, 384 rational numbers, 392 prism, 718, 720 fractions, 269–70 inverse, 347, 352, 384–85 real numbers, 402–3 slant, of a pyramid, 710 integers, 360–61 black and red chips model, 344, transitive property of less than trapezoid, 692 rational numbers, 389 345, 348, 349–50, 352, 382 and, 141 triangle, 691 real numbers, 402 calculators, 352 whole numbers, 62 Hero’s formula, 697 whole numbers, 126–27 distributivity of multiplication greater than or equal to, 62, 140 hexafoils, 706 zero, 112–13 over addition, 360–61 fractions, 246, 247–48 hexagonal cloud, 895 if and only if connective, 915 division, 357, 362–63 properties for solving hexagons, 607, 632, 633, 781 if-then connective, 914–15 exponents and, 357 inequalities, 410 hexahedron, 644 image factor of, 352 greatest common factor (GCF), Hiele, Pierre van, 581, 583 half-turn, 893 graphing on coordinate plane, 220–23 Hiele-Geldorf, Dieke van, 581, of points, 852, 861–62, 875 815 calculator method, 222, 223 583 reflection and, 852, 859 greatest integer function, 427 Euclidean algorithm, 223, 225 Hilbert, David, 100, 433 translations and, 853–55 measurement model, 345, 346 least common multiple and, Hindu-Arabic numeration system, image of P, 852, 875 multiplication, 357–62 225–26 71–75, 71–78 implication of statements, 914 look for a pattern, 358–59 prime factorization method, base, 71–73, 77–78 impossible events, 519, 521, 523 zero divisors of, 362 221–23 chip abacus, 72, 172–75 improper fractions, 245, 258–59 multiplicative properties set intersection method, 220–21 concepts, 74–75 calculator and, 268 associative, 360–61 Venn diagrams, finding with, digits, 71 incenter of a triangle, 777–78, 779 cancellation, 362 228, 234 digit shapes, 52 inch, 669, 670–71 closure, 360–61 greatest integer function, 427 expanded notation/form, 72, 76 included angles, 743–44, 753–54 commutative, 360–61 Great Mental Calculators, The fractions, 237 included side of a triangle, 744 distributive, over addition, (Smith), 166 history of, 107 inclusive “or,” 913 360–61 Great Pyramid of Giza, 261 nondecimal systems, 75–78 index, 404 distributive, over Gregory, 411 numeral word names, 74–75 indirect measurement, 754–55 subtraction, 360–61 grid approach to percent problem place value, 72, 73 indirect reasoning, 156, 198 identity, 360–61 solving, 324 See also base ten problem solving with, 198, 226, negative (See negative integers) grouping Hippasus, 379 399–400 negative exponents, 363–65 data in intervals, 443–44 histograms, 443–44, 445 inductive reasoning, 22 number lines, 344, 365–67 equal groups and equal shares, holistic measurement, 667 inequalities number systems diagram, 381, 131 holistic thinking, 581, 583–84, compound, 145 382, 392, 401 by fives, 62, 75–76 585–86, 586 defined, 405 opposites of, 344, 352 nondecimal systems, 75–78 Honea, David, 726 solving, 409–11 ordering, less than properties, for ratios, 311 Hoppe, Oscar, 215 triangle, 695 365–67 bindex.qxd 11/8/07 11:57 AM Page I10

I10 Index

integer(s) (continued) distance and, 876–77 law of detachment (modus ponens), light-year, 683 positive, 343 glide reflection (See glide 916–17 Lilavat (Bhaskara), 276 as rational numbers, 382 reflection) least common denominator (LCD), line(s), 822–27 scientific notation, 363–65 properties of, 878 256–57 angle bisector, 746, 769–70, set model, 344, 345 reflection (See reflection) least common multiple (LCM), 779 sets of, 343 rotation (See rotation) 223–26 concurrent, 615 subtraction, 348–52 similitudes, 864 build-up method, 224–26 equations of, 822–27, 837–38, adding the opposite, size transformation prior to, greatest common factor and, 838–39 349–50, 352 864 225–26 graphs (See line graphs) missing addend, 350 translation (See translation) least common denominator, intersection of, 618, 621, pattern, 348 isosceles trapezoids, 589 256–57 826–29, 830–31 take-away, 349, 350, 352 reflection symmetry, 601–2 number lines, 224 number (See number lines) Use a Variable strategy, 350 isosceles triangle, 589, 644 prime factorization method, parallel (See parallel lines) zero as, 343, 362 acute, 630–31 224, 225 perpendicular, 587, 605–7, 618, See also rational numbers congruent medians and, 835 set intersection method, 224, 225 621, 642, 770–71, 814, 816 interior angles, 608, 619–20, 627, reflection symmetry, 601–2, Venn diagrams, finding with, in a plane, 615–16 769. See also vertex angles 604 228, 234 point-slope equation, 825–26 interior of the angle, 617 isosceles triangles LeBlanc, Antoine (Sophie properties of points and, 616 intermediate algorithms Pons Asinorum Theorem, 739 Germain), 31, 39 reflection in, 856, 858 addition, 173, 192 is sufficient for, 915 left-to-right methods, 159 regression, 453 long division, 180, 182 legs of a right triangle, 693–95 skew, 642 multiplication, 177, 194 J Lehmer, D. H., 217 slope-intercept equation, 823–25 interquartile range (IQR), 489–92 Japan, Deming and, 441 Leibniz, Gottfried, 155 slope of, 812–13, 826, 837–38 interrelatedness of measurement Jefferson, Thomas, 454, 696 Leibniz, Gottfried Wilhelm, 665 in three-dimensional space, 642 systems, 672, 678, 679 Johnson, C., 789 length, 669–70, 673–74, 686–87 See also parallel lines; intersecting arcs, 766–70 passim, joule, 685 abstract, 686–87 perpendicular bisector 780 circumference, 689 linear equations, 830–31 intersecting lines, 618, 621, K distance (See distance) linear functions, graphs of, 419–20 826–29, 830–31 Kemeny, John, 198 English system, 669–70 line graphs, 447–48, 468–70 intersecting planes, 641, 651 kilogram, 678 metric system, 673–74 and cropping, 468–70 intersection of sets, 49, 110 kiloliter, 676–77 perimeter, 688–89 distorting, 471 probability and, 522, 524–25 kilometer, 667 length of a line segment, 588, 616 multiple-line, 447–48 intervals, data, 443–44, 445 kites less than, 140–42, 365–67, 392 with pictorial embellishments, invalid argument, 916 attributes of, 589, 606 addition 451 inverse, 914 creating shape of, 590 inequalities, 409–11 plot and, 442 inverse property diagonal of a, 746, 768 integers, 365 scatterplot and, 453 of addition model and abstraction, 589 rational numbers, 393 three-dimensional effects, 471 integers, 347, 350, 352, one-to-one correspondence, real numbers, 403 when to use, 454 384–85 852 whole numbers, 141 line of symmetry, 600 rational numbers, 384–87 perimeter, 688–89 fractions, 247–48 line plot, 442 real numbers, 402 perpendicular diagonals, 605, inequalities, 405 line segment(s), 588 of multiplication 606 mode and median, 486 AA similarity property, 783 fractions, 269–70, 274 reflection symmetry, 852 multiplication bisector, 746, 768, 770–71, 778 rational numbers, 389 SSS congruence property, rational numbers, 393 cevian, 335 real numbers, 402 746 whole numbers, 141–42 congruent, 588–90, 741 invert and multiply, 272–74, 390, tessellations with, 631 multiplication by a negative copying, 766–67 409 Koch curve, 756 inequalities, 409–11 creating shapes with, 588–90 Ionian numeration system, 68 Koch curve (or snowflake), 34 integers, 366–67 definition, 588 irrational numbers Kovalevskaya, Sonya, 336 rational numbers, 393 diagonal, 605, 606 defined, 379 real numbers, 402–3 directed, 853 infinite number of, 402 L multiplication by a positive endpoints, 601, 608, 616 number systems diagram, 401 lateral faces, 643, 647. See also inequalities, 409–11 equivalent directed, 853 pi, 402, 411, 567, 689, 723–24 quadrilaterals integers, 366–67 Euclidean constructions, set of, 400–402 lateral surface area, 707 rational numbers, 393 766–67, 783 is an element of a set, 46 of a cone, 711–12 real numbers, 402–3 length of, 686–87 is divisible by, 207 of a cylinder, 709 transitive property length of a, 588, 616 is necessary and sufficient for, 915 of a prism, 708 inequalities, 409–11 midpoint, 616, 811–12 is necessary for, 915 of a pyramid, 710 integers, 366 midsegment, 704, 794 is not an element of a set, 46 latitude, 821 rational numbers, 393 model and abstraction, 589 isometries, 852–60, 864, 875–81 lattice, square, 588, 615, 695, 700 real numbers, 403 parallel, 606, 615–16, 642 congruence lattice, triangular, 588 whole numbers, 140–42 perpendicular, 589 polygons, 881, 883–84 lattice method algorithm less than or equal to, 140–42 perpendicular bisector, 746, shapes, 880–81 addition, 173, 192 properties for solving 768, 770–71, 778 transformations, 875–81 multiplication, 177, 194 inequalities, 410 slope of, 813 bindex.qxd 11/8/07 11:57 AM Page I11

Index I11

liquid measures of capacity, 671 Mandelbrot, Benoit, 755, 800 number lines (See number subtraction, 158–59 Literary Digest poll, 477 mantissa, 301 lines) of fractions, 260–61 liters, 676–77 mapping. See transformation perimeter, 687–89 whole numbers, 158–59 logical connectives, 913 geometry process, 667 See also computational logically equivalent, 915 mapping history on the coordinate with a protractor, 617–18 estimation; thinking strategies logic/logical plane, 815 with square units, 689 Mere, Chevalier de, 513 algebraic, 163–64 maps, geographical, 419, 594 standard deviation, 493–96, meridians, 821 arguments, 916–19 Marathe’s triangle, 123 498–500 Mersenne number 267–1, 213, arithmetic, 163 marbles probabilities, 521–24, surface area, 707–12 217 connectives, 913–15 532–34, 535–38 system of units, ideal features meter, 667, 673–74, 675–76 statements, 912–13 matching sets, 46 of, 672 meter prototype, 673 long division, 178–83 mathematical art, 849 temperature, 672, 678–79 metric system, 662–79, 672–79 base five, 77 mathematical games, 373 variance, 493–94 area, 675–76 base five algorithm, 194–95 Mathematical Sciences Research volume, 671, 676–77, 717–26 converter diagram, 673–74 base ten algorithm, 178–79, Institute (MSRI), 845 measurement process, 667 convertibility of, 672–73, 181 mathematics of chance. See measures of central tendency, 676–78 calculator activities, 302, 303 probability 484–88 decimals, 673, 676, 677–78 with calculators, 163, 164–65 Mayan numeration system, 43, mean, 486–88 English system compared to, decimals, 300–301 65–66, 70 median, 485–86, 488 674, 678, 680–81 intermediate algorithm, 180, Maya people, 43 mode, 485, 486, 488 ideal features of, 672 182 mean (arithmetic average), measures of dispersion interrelatedness of, 672, 678, missing factor approach, 486–88 standard deviation, 493–96 679 178–80, 182, 195 central tendency, 486–88 variance, 493–94 length, 673–74 missing factors approach, 301, distribution, 497–500 median(s), 485–86 mass, 678 303 median and, 485 box and whisker plot, 489–92 notation, 673, 676, 677, 678 remainders, 178–80, 182 standard deviation, 493–96, central tendency, 485–86 powers of 10, 62, 159, 289, repeating decimals, 300–304 498–500 centroid, 779, 787 291–92, 301 scaffold method, 180, 195 variance, 493 distribution, 497–99 prefixes, 673–74 standard algorithm, 182–83, z-score, 495–96 finding, 488 temperature, 678–79 302–3 mean proportional, 786 geometric interpretation of, volume, 676–77 thinking strategy, 180, 182 means, extremes and, 311 497–99 world acceptance of, 696 longitude, 821 measurement, 667–81, 686–95, mean and, 485 metric system units of longs, in base ten pieces, 71 707–12, 717–26 of a triangle, 773, 779 measurement Look for a Formula strategy, 440, of angles, 617–18 collinear, 836 are (metric area of a square), 506–7 dihedral, 641–42 concurrent, 836 675 Look for a Pattern strategy, 20–23, directed, 854 congruent, 835 centimeter, 667, 675–76 27–28 in polygons, 628–31 ratio of, 837 decimeter, 667, 673–74, combining with other strategies, angle sum in a triangle, members of a set, 45–46 675–76, 675–77, 676–77 27–30, 440 619–20 memory function of calculators, degrees Celsius, 678–79 counting factors, 219–20 area (See area) 165 dekameter, 667, 673–74 decimals, 303–4 capacity vs. volume, 671 mental math, 157–59, 163 dram, 672 distance involving translations, central tendency (See measures addition, 158–59 gram, 678 895 of central tendency) of fractions, 260–61 hectare, 675–76 downward paths in grid, 21–22 circumference, 689 compatible numbers, 158 hectometer, 667, 673–74 find ones digits in 399 problem, dispersion, 492–96 (See also compensation, 158–59, 167 kilogram, 678 22–23 measures of dispersion) decimals, 291–93 kiloliter, 676–77 integer multiplication, 358–59 distance (See distance) developing child’s ability, liters, 676–77 negative exponents, 363–64 division, 129–30, 270–73 163 meter, 667, 673–74, 675–76 repeating decimals, 303–4 English (See English system of division, 158 metric ton, 678 Loomis, Elisha, 784 measurement) of fractions, 275 milliliter, 676–77 lower quartile, 489–92 holistic, 667 fraction equivalents, 322–23 millimeter, 667, 673–74 lowest terms indirect, 754–55 fractions, 260–61, 275 temperature, 678–79 fractions (simplified), 242, 244, informal (See nonstandard units halving and doubling, 169 metric ton, 678 245, 253, 256, 257 of measurement) left-to-right method, 159 midpoint formula, 811–12 rational numbers, 383 length, 669–70, 673–74, multiplication, 158–59 midpoint of a line, 616 686–87 of fractions, 275 midquad theorem, 794–95 M liquid measures of capacity, order of operations, 158–59 midsegment, 704, 794, 909–10 machines, functions as, 90 671 percents, 322–28 mild outliers, 503 magic squares, additive, 19, 217, metric (See metric system) percents as fraction equivalents, mile, 669, 670–71 295, 296 models 322–23 milliliter, 676–77 magnifications. See transformations addition, 110–11 powers of 10 multiplication, millimeter, 667, 673–74 Make a List strategy, 23–25 fractions, 255 159 million, 74, 104 combining with other strategies, integers, 345, 346 properties, 158 minuend, 116 26–30 take-away, 115–16 scaling up/down, 314 Mira®, 602, 608 bindex.qxd 11/8/07 11:57 AM Page I12

I12 Index

misleading graphs and statistics, Moebius, A. F., 622 identity property, 126, 269–70, multiplicative properties. See under 439, 464–77 Moore, R. L., 658 360–61, 389 properties bar graph distortion Morawetz, Cathleen Synge, 658 integers, 357–62, 366–67 musical scale, 416 compressing y-axis, MSRI (Mathematical Sciences inverse property, 269–70, 274, mutually exclusive events, 524–25, 444–45, 447, 465–67 Research Institute), 845 389 536 pictorial embellishments, multibase pieces, 71, 72, 76 lattice method algorithm, 177, 474 multi-digit addition, 114–15 194 N three-dimensional effects, multiple, 207 less than (See under less than) naming numbers, 74–75 470–72 common, 224, 225 mental math, 158–59, 275 Napier, John, 155, 294 circle graph distortion, 472–73 least common, 223–26 mixed numbers, 267–68, 275 Napier’s “decimal point” notation, line graph distortion, 468–70, multiple-bar graphs, 445 multi-digit by multi-digit 294 471 multiple-circle graphs, 448–49 numbers, 176–77 NASA, 895 pictorial embellishments, multiple-line graphs, 447–48 by a negative, less than National Council of Teachers of 473–75 multiplication, 123–29, 159, properties, 366–67, 393, Mathematics (NCTM), 3 samples and bias, 475–77 266–70 409–11 necessary, 915 scaling and axis manipulation, by 0, 1, 2, 5, or 9, 128–29 negative integers, 357–62 Needle Problem, Buffon’s, 567 465–67 associative property nonstandard algorithms, 185, negation of a statement, 912–13 three-dimensional effects, decimals, 291 186, 188 negative integers, 343–45, 352 470–72 fractions, 269–70, 275 obtaining ranges for estimation, addition, 346, 348 missing-addend subtraction integers, 360–61 160 additive inverse vs., 352 fractions, 259–60 rational numbers, 389 order of operations, 144–45, division, 362–63 integers, 350 real numbers, 402 163–64 exponents, 363–65 whole numbers, 116–17, 128, whole numbers, 126, 129, powers of 10, 159 multiplication, 357–62 192–93 158–59 by powers of 10, 159, 291–92, number lines, 344, 345, 346, missing factor approach base five, 194 301 358 division, 132, 182–83, 924 cancellation property, 137, 362 probability property, 536–38 opposite vs., 352 missing factors, 132, 300, 326. See Cartesian product approach, 51, products (See products) negative numbers, 341, 343 also quotients 125 properties (See under rational, 385 mixed numbers, 246, 257, 258–59 Cartesian product of sets, properties: multiplication) real, 400–401 Killie’s Way solution, 394 526–27 rational numbers, 388–90 See also negative integers multiplying, 267–68, 275 closure property, 125, 269, real numbers, 402 Newton, Sir Isaac, 16, 665 rational numbers, 382 360–61, 389 rectangular array, 124–25, 267 n factorial, 550–51 mode coefficients, 408 repeated addition approach, n- gons. See polygons central tendency, 485, 486 commutative property 123–24 Noether, Emmy, 100, 575 distribution, 496–99 decimals, 291 with calculator, 135 nonadjacent angles, 617, 621 finding, 488 fractions, 266, 269 clock arithmetic, 924–25 nonadjacent vertices, 605 models integers, 360–61 with fractions, 266 nondecimal numeration systems, black and red chips, 382 rational numbers, 389 negative integers, 357 75–78 blocks/pieces (See under base real numbers, 402 properties, 123–24 nonillion, 79 ten: blocks/pieces) whole numbers, 125–26, Russian peasant algorithm, nonintersecting lines, 642 bundles of sticks, 71, 75 127, 128, 158–59, 266 185 nonnegative (principal) square root, chip abacus, 172–75 decimals, 291–92, 298–300 single-digit by multiple-digit 402 fractions as part-to-whole, distributive property, over numbers, 129 nonrepeating decimals, 400–402 241–42 addition, 135 by special factors, 159 nonstandard algorithms fraction strips, 242, 247, 255, fractions, 270 standard algorithms, 176–77, addition, 183, 184 257 integers, 360–61 194 division, 186 fuel gauge, 324 rational numbers, 390 thinking strategies, 128–29 multiplication, 185, 186, 188 geometric shapes, 589, 616 real numbers, 402 tree diagram approach, 125 subtraction, 184–85 for integers, 344, 345, 348, whole numbers, 127, 129, whole numbers, 123–29 nonstandard units of measurement, 349–50, 352 176 algorithms, 169, 176–77, 667–69, 681, 685 measurement, 110–11, 115–16, distributive property, over 185, 186, 188, 194 board foot, 729 255, 345, 346 subtraction, 128 as product of primes with horsepower, 685 number lines (See number duplication algorithm, 186 their respective joule, 685 lines) estimation for fractions, 275 exponents, 219–20 light-year, 683 place value (See place value) of exponents, 142–43 set models, 123–24 parsec, 685 region, 255 facts, 128–29 zero, property of, 128 Smoots, 681 set (See set models) base five, 194 zero divisor property, 362 nonterminating decimals, 305, 400 solving problems with, 582, fingers, using, 186 “multiplication makes bigger” nonzero numbers, 382, 924 657 fraction of a fraction, 267 misconception, 297 normal distribution, 497–99 See also number lines fractions, 266–70, 275 multiplicative compensation, 159 notation modus ponens (law of detachment), German low-stress algorithm, multiplicative numeration system, absolute value, 354 916–17 188 72 angles, 616–17 modus tollens (denying the halving and doubling algorithm, multiplicative numeration systems, conversions with calculators, consequent), 917–18 169, 185 63–66 364–65 bindex.qxd 11/8/07 11:57 AM Page I13

Index I13

decimals, 294, 300 ordering (See ordering) numbers and, 59 O’Neal, Shaquille, 353 directed line segments, 853 ordinal, 59 word names for, 74–75 one-column front-end estimation, elements in a finite set, 60 perfect, 228 numeration systems, 61–66, 381 160 English system of positive, 343, 346, 357–62, additive, 62–66, 72 ones digit, 22–24 measurement, 669–72 363, 385, 400–401 Babylonian, 64–65, 66 one-stage tree diagrams, 542–43 equals sign, 742 powers of, 22–24 base, 71–73 one-to-one correspondence, 46, 60, expanded (See expanded prime (See prime numbers) binary, 77, 78 852, 880–81, 883 form/notation) properties as problem solving Braille, 69 operations expected value, 562–63 strategies, 204, 231–32 Chinese, 69 in base five, 192–95 exponents, 142–44, 225, rational (See rational numbers) comparison of, 66 binary, 78, 110, 923, 924 363–65 real (See real numbers) converting between bases, compatibility with respect to, factorial, 214 rectangular, 35, 87 75–78 158 fractions, 241 relatively prime, 227 decimal (See decimals) order of, 144–45, 158–59, function, 87–88 Smith, 231 digit, 71 163–64 intersection of sets, 522, square, 22 Egyptian, 62–63, 66 real numbers, 402, 403 524–25 systems (See numeration expanded notation (See on sets, 48–52 is congruent to, 742 systems) expanded form/notation) whole number, four basic, 134 isometries, 875–76 theory (See number theory) grouping, 62, 75–78 See also addition; division; metric system, 673, 676, 677, whole (See whole numbers) Hindu-Arabic, 52 multiplication; subtraction 678 zero (See zero) Ionian, 68 opinion polls, 475–77 negative number, 341, 352 number line(s) Mayan, 43, 65–66, 70 opposite of the opposite for rational negative signs and subtraction, base five, 192–94 multiplicative, 63–66, 72 numbers, 387 351 decimals, 288, 289–90 nondecimal, 75–78 opposites pi, 402 distance on, 616, 686–87 pictographic, 62–63 of integers, 344, 352 points and lines, 601, 615 fractions, 247, 248, 255, 266 placeholder, 64, 66 of rational numbers, 387 scientific, 165, 301–2, 363–65, integers, 344, 358, 365–67 positional, 43, 64–66 (See also or connective, 913–14 369 least common multiple, 224 place value) ordered arrangement of objects set, 241, 405 metric converter diagram, Roman, 63–64, 66 (permutation), 549–52 set-builder, 45–47 673–74 subtractive, 63–64, 66 ordered pairs, 51 subtraction, 350, 352 plotting statistics on, 489–90 tally, 62, 66 fractions as, 247 sum of a ϩ b, 111 rational numbers, 381, 385, 392 with zero, 65–66 functions as, 90 union of sets, 522, 524–25 real numbers, 401, 403 See also Hindu-Arabic graph coordinates, 417–18 See also inside back cover transitive property of less than, numeration system relations and, 82–83 not equally likely events, 520–21, 140–41 numerators, 237, 240 ordering 525, 565, 566 whole number, 110–11, comparing, 247–48 counting chant, 60, 129 nth percentile, 492 140–41, 248 and denominators, 249, 271–73 decimals, 289–90, 291 nth root, defined, 403–4 ordering with, 61 in rational numbers, 382, fractions, 247–49 null set, 46 number of a set, 59–60 388–89, 391 greater than (See greater than) number(s), 59–61 number sense, developing, 163 numerousness, 241 inequalities, 409–11 abundant, 228 number sequence, 20–22. See also integers, 365–67 amicable, 228 sequences O less than (See less than) base, 71 number theory objects one-to-one correspondence, 46, betrothed, 228 composite numbers, 205–7 no particular order of 60 cardinal, 59 counting factors, 219–20 (combinations), 552–54 PEMDAS, 145 compatible, 158, 291, 293 Euclidean algorithm, 223, 225 ordered arrangement of rational numbers, 392–93 counting (See counting greatest common factor, (permutation), 549–52 real numbers, 403 numbers) 220–23, 225–26 oblique circular cone, 646, 647 whole numbers, 60–61, 140–42 decimal (See decimals) prime factorization method, oblique cylinder, 645, 647 See also number lines deficient, 228 221–23 oblique prism, 643, 647 order of operations, 144–45, diagrams of relationship set intersection method, oblique pyramid, 644, 647 158–59, 163–64 between, 381, 382, 401 220–21 obtuse angle, 617, 620, 621 ordinal numbers, 59 rational numbers, fractions, least common multiple, obtuse triangle, 620, 621, 630–31 Oregon rain data, 501 and integers, 392 223–26 octahedron, 644–45 organizing information, 442–44. real numbers, 400 build-up method, 224–26 octillion, 79 See also graphs even, 22, 88 prime factorization method, odd origin, 417 fraction (See fractions) 224, 225 counting numbers, 22 orthocenter, 779, 784, 837–38 frequency of, 442–44 set intersection method, even and, making groups of ounce, 671, 672 identification, 59 224, 225 two, 209 outcome(s), 516 integer (See integers) prime numbers, 213, 226 nth Root and, 403–4 card decks, 517, 518–19, 555, natural, 45 infinite number of, 226, 402 number of factors, 363 565 negative, 341, 343, 348, tests for divisibility, 207–13 number of prime factors, certain, 519 357–62, 363–65, 400–401 numerals, 59 400–402 certain or impossible, 523 numerals vs., 59 Braille, 69 perfect number conjecture, 203 coin tossing (See coin tossing odd, 22, 203, 363, 400–402 fractions as, 240, 241–42, 245 odds, 563–65 probabilities) bindex.qxd 11/8/07 11:57 AM Page I14

I14 Index

outcome(s) (continued) model and abstraction, 589 permutations, 549–53, 555 patterns, 73 dice throwing, 516, 518–19, perimeter, 687–89 counting techniques, 549–53, positional numeration systems, 520–21, 534, 564 rectangles as, 619 555 43, 64–66 drawing gumballs, 525, 539–40 rhombuses as, 606 n factorial, 550–51 subtraction algorithm, 174–75 drawing marbles, 521–24, symmetry, 601, 604 of r objects chosen from n plane(s), 615–16, 640–42 532–34, 535–38 tessellations of a plane with, objects, 551 coordinate equally-likely (See equally- 860–61 perpendicular bisector, 768 distance in, 809–12 likely events) tessellations with, 631 angles, 769–70 graphing integers on, 815 expected, 560 translations, distance, and, 876 construction of circumscribed parallel, 645, 647 experiments with two trapezoids as, 698 circles, 777–79 points in, 615–16 outcomes, 540–41 parallel planes, 645, 647 Euclidean construction, 768 circle, 608 fundamental counting property, parallels of latitude, 821 of line segments, 746, 768, points on, 807 534 parentheses on calculators, 163–64 770–71, 778 similar shapes via similitudes, impossible, 519, 521 parsec, 685 in reflection transformations, 883 mutually exclusive, 524–25, 536 Parthenon (Athens), 285 856, 879 slope (See slope) odds, 563–65 partition of a set, 85 rhombus properties and, of symmetry, 652 pairwise mutually exclusive partitive division, 129–30 768–71 passim tessellations and, 630–31, 632, simpler, 536 part-to-part ratios, 311 sides of the triangle, 778, 779 860–61 permutations, 549–53, 555 part-to-whole model, 241–42 through a point not on a line, three-dimensional, 640–47 Pick Six wager, 566 part-to-whole ratios, 311 770–71 Platonic solids, 644 possible, 516 Pascal, Blaise, 155, 513 through a point on a line, 770 point(s) probability of, 517–24, 529 Pascal’s triangle triangle, 856, 879 arrays of, 818, 820 spinning spinners, 516–17, combinations, 553–54 perpendicular diagonals, 605, 606 collinear, 615, 863, 879, 881, 518–19, 525, 538–39 and different downward paths, perpendicular lines, 618, 621 893, 894 unequally likely, 520–21, 525, 22 parallel lines and, 587, 605–7 corresponding, 645, 647 565, 566 Fibonacci sequence and, 285 slopes of, 814, 816 distance See also events; probability predicting sums of diagonals, in three-dimensional space, 642 equidistant from endpoints, outliers, 453, 489–92 33 through a point 770, 778 box and whisker plot, 489–92, probability and, 541–42 on the line, 770 equidistant from 503 patterns not on the line, 770–71 perpendicular bisector, explaining occurrence of, counting dots using, 14–15 perpendicular line segments, 589 778 489–90 Escher-type, 861–62 Peter, Rozsa, 433 equidistant from sides of a extreme and mild, 503 and Fibonacci sequence, 33, 36 pi triangle, 779 integer subtraction, 348 area of a circle, 693 from point P to point Q, P See also Look for a Pattern Buffon’s Needle Problem, 567 PQ, 766, 770–71, 777–79 P(E), 517–24, 529 strategy; tessellations circumference of a circle, 689 distance from point P to point pairwise mutually exclusive PEMDAS (mnemonic for mnemonic, 411 Q, PQ, 686–87, 809–10, simpler events, 536 ordering), 145 notation, 402 876–77 palindromes, 120, 308 pentagon, 604, 607–8, 629 perimeter, 689 endpoints of a line, 601, 608, pan balance, 25–26, 108, 148, 407 pentagonal numbers, 35 volume of a sphere, 723–24 616 parabola, 834 pentagonal prism, 645 Pick Six wager, 566 equidistant parallel lines, 615–16, 618–22 pentagonal pyramid, 644 Pick’s theorem, 700 from endpoints, 616 angles associated with, 618–22 pentominoes, 342, 373, 613 pictogram numeration systems, image of, 852, 861–62, 875 Euclidean construction, 771 percent grade, 818, 821 62–63 image of P, 852, 875 perpendicular lines and, 587, percentile, 492 pictographs, 449–51, 454, 471–72 midpoint formula, 811–12 605–7 percents, 320–28 pictorial embellishments of graphs, midpoint of a line, 616 slopes of, 814, 816 with calculators, 322, 326–27 450–51 noncollinear, 839 in three-dimensional space, 642 converting, 320–22 deceptive, 473–75 perpendicular line through, and transversals, 618–19 decimals and, 321–22 picturing information, 449–54. See 770–71 parallel line segments, 606, Draw a Diagram strategy, 324 also graphs in a plane, 608, 615–16 615–16, 642 fractions and, 321, 322–23 pie charts, 448–49, 451, 454, on a plane, 807 directed, 853 percent grade, 818, 821 472–73 properties of lines and, 616 parallelogram as ratios, 324–25 pint, 671 point-slope equation of a line, base, 692 solving problems, 322–23 pizza order probability, 534 825–26 parallelograms equation approach, placeholder, 64, 65 polar coordinates, 832 area, 692 325–28 place value, 72, 73 Polk, James K., 513 attributes of, 606 grid approach, 324 addition algorithm, 172–73 Pólya, George, 1, 145 congruent, 604, 744, 791 proportion approach, decimals and, 289–90, 291 Pólya’s four-step problem-solving creating shape of, 589, 590 324–25, 326–27 expanded notation, 72, 76, process, 1, 4–6, 30 defined, 744, 791 perfect numbers, 203, 228 172–73 utilizing with strategies, 7–13, Escher-type patterns, 861–62 perfect square, 146 Hindu-Arabic system, 72, 73 21–28 height, 692 perimeter(s), 687–89. See also multi-digit numbers, 115 polygon(s), 419 lateral faces of polyhedra, 643, inside back cover multiplication algorithm, 177 angles of, 607–8 647 period of a decimal, 303 nondecimal systems, 75–78 arbitrary, 630 bindex.qxd 11/8/07 11:57 AM Page I15

Index I15

concave and convex, 607 Egyptian numeration system, counting techniques, 549–55 problem-solving process, Pólya’s, congruent, 880–81, 883–84 62–63 dice throwing, 516, 518–19, 1, 4, 6, 30 equilateral, 589, 628–30, Hindu-Arabic numeration 520–21, 534, 564 utilizing with strategies, 7–13, 632–33 system, 71–73 drawing with/without 21–28 equilateral triangles, 780, 782 multiplying by, 159, 291–92, replacement, 534–35, 536–39 problem-solving strategies, 4 Fermat prime, 782 301 events (See events) combining, 1–2, 26–30 as fundamental units of area, Precious Mirror of the Four expected value, 562–63 Cover Up, 16 670 Elements (Chu), 341 experimental, 519–20 Do a Simulation, 514, 574 one-to-one correspondence, prime factorization simulation, 560–62 Draw a Diagram (See Draw a 880–81, 883 of composite number, 206, 214 experiments with two Diagram strategy) perimeter of, 687–89 counting factors and, 219–20 outcomes, 540–41 Draw a Picture (See Draw a regular, 607–8, 628–30, divisibility tests for, 212–13 fundamental counting property, Picture strategy) 632–33, 780, 782–83 fundamental theorem of 534, 536, 540, 542, 549–52 Guess and Test, 7–9 angle measures, 628–30 arithmetic and, 206 intersection of sets, 522 Identify Subgoals, 740, 799 as equiangular and greatest common factor (GCF), multiplicative probability, Look for a Formula, 440, 506–7 equilateral, 628 221–23 536–38 Look for a Pattern (See Look Euclidean constructions, least common multiple (LCM), multiplicative property, for a Pattern strategy) 780, 782 224, 225 536–38 Solve an Equation, 380, 432, Gauss’s theorem for prime factors odds, 563–65 838, 839 constructible n-gons, counting factors vs., 219 outcomes (See outcomes) Solve an Equivalent Problem, 782–83 fractions, 245 Pascal’s triangle, 541–42, 238, 281 tessellations with, 632–33 odd number of, 399–400 553–54 Solve a Simpler Problem, 6 similar, 883 rational numbers, 383 properties Use a Model, 582, 657, 695, 707 size transformations, 863 test for, 213 additive probability, 536–38 Use a Variable, 11–13, 350, 695 tessellations, 632–33 prime meridian, 821 fundamental counting Use a Variable with algebra, polygonal regions, 630, 643–45. prime numbers, 205–13 property, 534, 536, 540, 13–16, 27–28 See also polyhedra 2, nature of, 399–400 542, 549–52 Use Cases, 342, 373 polyhedra, 642–45 composites and, 205–7 multiplicative probability, Use Coordinates, 808, 845 cubes, 640–42 factorization (See prime 536–38 Use Dimensional Analysis, 666, dodecahedron, 644–45 factorization) as ratios, 525 733 Euler’s formula, 644, 733 factor test, 213 relative frequency, 496, 516, Use Direct Reasoning, 108, 148 hexahedron, 644 Fermat prime, 782 517, 566 Use Indirect Reasoning, 156, icosahedron, 644–45 finding, 212–13 simple experiments, 515–25 198, 226, 399–400 octahedron, 644–45 infinite number of, 226 computing probabilities in, Use Properties of Numbers, prism, 643, 645, 650 largest known, 227 517–24 204, 231–32 pyramid, 643–44, 645, 654 Mersenne number, 213, 217 simulation, 560–62 Use Symmetry, 850, 901–2 regular, 644 relatively, 227 theoretical, 519–20, 560, 574 Work Backward, 286, 335 rhombicuboctahedron, 645 Sieve of Eratosthenes, 206, 234 tree diagrams and, 532–42 See also Look for a Pattern semiregular, 645 square root, 213 unequally likely outcomes, strategy tetrahedron, 644–45 with their respective exponents, 520–21, 525, 565, 566 products, 124 three-dimensional aspect, 219–20 Venn diagrams and, 566 counting factors and, 219–20 642–45 triples, 217, 413, 705 probability of an event, P(E), cross-multiplication of vertices, 643–44 twins, 215 517–24, 529 fractions, 244–45, 247–48, Pons Asinorum Theorem, 739 primitive Pythagorean triple, 413 problem of the points, 513 249 pool shot paths, 893, 894 principal square root, 402 problems, exercises vs., 4 from decimal to fraction population, in statistics, 475–77 prisms, 643, 645, 650, 718 problem solvers, suggestions from, conversions, 292 population predictions, 428 Cavalieri’s principle, 724 6, 30–31 exponents of the, 142–43 portability of measurement height, 718, 720 problem solving and “multiplication makes systems, 672, 679 oblique, 643 algebraic, 13–16 bigger” misconception, 297 positional numeration systems, 43, right, 708, 718, 724 geometric order of operations, 144–45, 64–66. See also place value right rectangular volume, 718 with algebra, 807 163–64 position function, 420 right regular, 709 with coordinates, 834–39 in probability, 536–38 positive numbers right square, 643 with transformations, See also multiplication integers, 343, 346, 357–62, 363 right triangular, 720 893–95 proper factors, 203, 228 rational, 385 surface area, 708 with triangle congruence proper subsets, 47, 60–61 real, 400–401 volume, 718 and similarity, 790–95 properties possible events, 516. See also probability, 515–25, 532–42, importance of, 3 AA similarity, 754–55, 783 outcomes 549–55, 560–67 Pólya (See Pólya’s four-step additive cancellation, 123, 347, pound (weight), 672 additive probability, 536–38 process) 387 power of. See exponents additive property, 536–38 strategies (See problem-solving additive probability, 536–38 powers of three, 22, 88 coin tossing (See coin tossing strategies) aesthetics property of golden powers of 10 probabilities) suggestions for, 30–31 ratio, 285 decimals, 289, 291–92, 301 complex experiments, 532–42 See also algorithms; properties; angle sum in a triangle, 620, dividing by, 291–92, 301 conditional probability, 565–67 solving equations 629–30 bindex.qxd 11/8/07 11:57 AM Page I16

I16 Index

properties (continued) for triangles, 743–44, 746, Q rates, ratio, 310 ASA congruence, 744, 746, 778 quadrants, 418 ratio(s), 310–11 791, 794 SAS similarity, 754, 793–94, quadratic functions, 420 avoirdupois measures of associative (See associative 881 quadrilateral(s), 588 weight, 672 property) similarity of triangles, 752–56 arbitrary, 631 cross-multiplication property, closure (See closure property) of similitudes, 882 attributes of, 606 312–13 commutative (See commutative of size transformations, 882 congruence, 749, 790, 793 dimensional analysis, 680 property) SSS congruence creating shape of, 589, 590 of distances, 881–84 compass, 766, 767 angles, 767 diagonals of, 894 English system of cross-multiplication, of ratios, converse of Pythagorean half-turn rotations, 894 measurement, 669–70 312–13 theorem, 792–93 midquad, 794–95 equality of, 311 directed angles, 854 kites, 746 parallelograms (See extremes and means, 311 distance on a line, 687 parallelograms, 791 parallelograms) golden ratio, 285 distributive (See distributive triangles, 745–46 perimeter, 687–89 liquid measures of capacity, property) SSS similarity, 754, 793–94, rectangles (See rectangles) 671 of exponents, 142–44, 405 881 rhombus (See rhombus) medians of a triangle, 837 of fractions straightedge, 766–67 SASAS congruence property, metric convertibility of, 673, additive, 244, 257, 258, 260 symmetric, 84 749 674, 676 density, 249 transitive (See transitive SAS congruence property, odds and, 563–65 multiplicative, 266, 269–70, property) 790–92, 793 part-to-part comparison, 311 274, 275 trapezoids (See trapezoids) size transformation of, 863 part-to-whole comparison, 311 fundamental counting property, triangle square (See square) percents as, 324–25 534, 536, 540, 542, 549–52 similarity of, 752–56 tessellations from, 631 probabilities as, 525 geometric shapes SSS congruence, 745–46 translation and, 853, 895 rates, 310 angles, 617, 619–20 of triangles, 600–601, 619–20, quadrillion, 74, 79 scaling up/down, 314 convex attribute, 607 834–39 quality control, 441 whole-to-part comparison, 311 diagonals of a kite, 605 whole number (See under quart, 671 See also proportions isosceles triangles and whole numbers) quartile, upper/lower, 489–92 rational exponents, 403–5 trapezoids, 601–2 of zero, 112–13, 128, 132, 362 quotients (missing factors) properties, 405 parallelograms, 604 zero divisor of integers, 362 calculators and, 163–65 rational number(s), 382–93 points and lines, 616 proportion, 312–15 decimals, long division, and, addition, 384–87 greater than, 402–3, 410 with calculators, 326–27 301, 303 additive properties, 384, identity (See identity property) percent problem-solving, from decimal to fraction 386–87 of integers, 347, 360–61 324–25 conversions, 292 identity, 383, 387–88 inverse scaling up/scaling down, 314 defined, 132 inverse, 384–87 additive, 347, 350, 352, See also ratios denominators and numerators comparing two, 393 384–87 protractor, 617–18 approach to division, 271–73 cross-multiplication of multiplicative, 389 public opinion polls, 475–77 division and, 182–83 inequality, 393 of isometries, 878 Purser, Michael, 368 and “division makes smaller” division, 390–92 less than, 366–67, 393, 402–3, pyramids, 654, 709–10, 721–22 misconception, 297 invert and multiply, 390, 409–11 apex, 643–44, 654 long division and, 178–80, 182, 409 for mental math, 158 Cavalieri’s principle, 726 195 equality of, 383, 387–88, 391 midquad, 794–95 slant height, 710 ratio and, 310 as exponents, 403–5 multiplication property of zero, square, 721–22 simplified form, 391 fractions as, 382 128 surface area, 710 infinite number of fractions, multiplicative cancellation, 137, volume, 721–22 R 383, 385 362 Pythagoras, 315, 379 r objects chosen from n objects, integers (nonzero) as, 382 multiplicative probability, Pythagorean theorem, 379, 693–95 551, 553 multiplication, 388–90 536–38 application to nontriangular radical, 404 multiplicative properties, multiplicative rectangular array, shapes, 757 radicand, 404 389–90 124–25 area of the square radius nonzero integers as, 382 of points and lines, 616 interpretation, 757, 797 arc of radius, 767, 769, 780 number lines, 381, 385, 392 of probability, 524, 534, Bhaskara’s proof, 699 of a circle, 608 products with fraction 536–38, 540, 542, 549–52 converse of, 792–93 of a sphere, 646 calculator, 389 as problem solving strategy, Euclid’s proof, 739, 784, 797, Ramanujan, Srinivasa, 18, 231, set models, 382 204, 231–32 900–901 232 set of, 382 of rational exponents, 405 Garfield’s proof, 705 random digits, 561, 568 simplifying, 383, 388, 389, 390, of rational numbers, 384–87, and irrational numbers, 379 Random Guess and Test strategy, 7, 391 389, 393 length of PQ, 809 8 subtraction, 387–88 of real numbers, 402, 403 length of the hypotenuse, 401 random-number table, 561, 568 sums and differences with reflexive, 83–84 origin of, 795 range fraction calculator, 388 SAS congruence proof, 586–87, 694 estimation, 160, 261, 275, 292 ratio of distances, 881–84 for quadrilaterals, 790–92, transformational proof, 897 functions, 88–90, 91 ray, 616, 855 793 Pythagorean triples, 398, 413, 705 interquartile, 489–92 reading numbers, 74–75 bindex.qxd 11/8/07 11:57 AM Page I17

Index I17

real number(s), 399–405 Reid, Constance Bowman, 232 Riese, Adam, 294 Cartesian product of sets, additive properties, 402 relations, 82–91 right angles, 589, 617 526–27 diagram of decimals and, 400 arrow diagrams, 83 right circular cone, 646, 711, 716, conditional probability, 565–67 exponents, 405 defined, 72–73 724 counting techniques in place of, geometric representation, 379, equivalence, 85 right circular cylinder, 645–46, 549–55 401 functions, 85–91 709, 724 equally-likely outcomes, multiplicative properties, 402 number systems diagrams, 381, right distributivity of division over 524–25 n th root, 403–4 382, 392, 401 addition, 158 large, dealing with, 524, 549, number line, 401, 403 partition, 85 right prism, 708, 718, 724 551 number systems diagram, 401 reflexive, 83–84 right rectangular prism volume, not equally-likely probability, ordering properties, 403 symmetric, 84 718 525 roots of, with calculators, 404 transitive, 84–85 right regular prism, 709 properties of probability, 524 set of, 400–402 relationships – level 2, 585 right regular pyramid, 710, 711, See also tree diagrams reasoning relative amount, 240–42, 244, 311 724 SAS. See side-angle-side direct, 108, 148 relative complement of sets, 50–51 right square pyramid, 710 SAT scores, 446 problem solving with, 108, relative frequency, 496, 516, 517, right triangle, 401, 589, 620, 621, scaffold method, long division, 180, 148 566. See also probability 693–95. See also 195 indirect, 156, 198 relatively prime numbers, 227 Pythagorean theorem scale factor, 863, 883 problem solving with, 198, relative vs. absolute in circle right triangular prism, 720 scalene triangle, 589, 630 226, 399–400 graphs, 472 rigid motion. See isometries scaling and axis manipulation, inductive, 22 remainders, 133, 164–65, 178–80, rise over the run, 812 465–67 reciprocals, 269, 382, 389, 390–91 182–83 road signs, 609 scaling up/scaling down, 314 fractions, 269 repeated addition approach to Robinson, Julia Bowman, 149, 232 scatterplot, 452–54 rational numbers, 389 multiplication, 123–24 rod, 669 School Mathematics Study Group, recognition—level 0, 583–84 with calculator, 135 rolling dice probabilities, 516, 902 rectangle clock arithmetic, 924 518–19, 520–21, 534, 564 scientific notation area, 689–91 with fractions, 266 Roman numeration system, 63–64, calculators and, 165, 301–2, attributes of, 606 negative integers, 357 66 364–65, 369 creating shape of, 589, 590 properties, 123–24 rotation, 853–55, 857 characteristic, 301 diagonals of a, 743, 744, repeated subtraction, division by, axis of rotational symmetry, mantissa, 301 790–91 133–34 652 negative integer exponents, golden, 285, 786 repeating decimals, 300–304 distance and, 876 363–65 horizontal vs. vertical, 586 calculator, 302–3 Escher-type drawings, 862 standard notation vs., 364 model and abstraction, 589 fraction representation, 304–5 half-turn, 893–94 scratch addition, 184 as parallelograms, 619 long division algorithm, 302–3 image, 893 segment construction, 588, 783. perimeter, 687–89 nonterminating, 304–5 notation, 875–76 See also line segments squared, 355 real numbers, 399 symmetry, 602–4, 606, 608, self-similarity, 425 symmetry, 601, 604 repetend, 303, 304–5, 400 860–61 fractals and, 755–56 rectangular array repetend, 303, 304–5, 400 of triangles and quadrilaterals, Koch curve, 756 description of divides, 208, 210 replacement, probability and, 631 semiregular polyhedra, 645 fraction multiplication, 267 534–35, 536–39 rotation symmetry, 860–61 semiregular tessellations, 633, 635, multiplication, 124–25 Reuleaux triangle, 900–901 rounding, 161–63 638 rectangular numbers, 35, 87, 88 reverse Polish notation, 163 to compatible numbers, 162 sequences, 22 rectangular regions, 641 Rhind Papyrus, 237 computational estimation, arithmetic, 86–87 “reducing” fractions, 244 rhombicuboctahedron, 645 292–93, 323 common difference, 86–87 Rees, Mina, 507 rhombus, 768, 769 decimals, 292–93 common ratio of, 86–87 reflection, 856–60 attributes of, 606 division, 182 Fibonacci (See Fibonacci distance and, 877 children’s recognition of, 585 down, 161–62, 182 sequence) image, 852, 859 coordinate geometry, 834 a five up, 161–62 geometric, 86–87 notation, 875–76 creating shape of, 589, 590 to nearest even, 162 identifying (See Look for a symmetry, 600, 602, 604, 606, diagonals of a, 768–69, 770, percents, 327 Pattern strategy) 652, 852, 860–61 791–92 truncate, 161–62 initial term, 86 See also glide reflection model and abstraction, 589 up, 161–62, 182 number, 20–22 reflex angles, 617, 621 as parallelograms, 606, 744, whole numbers, 161–63 set(s), 45–52 reflexive property, 83–84 791 RSA algorithm, 368 as basis for whole numbers, region model for fraction addition, perimeter, 687–89 Rudin, Mary Ellen, 658 45–52 255 perpendicular bisector Russian peasant algorithm, 185 binary operations using, 110 regression lines, 453 construction and, 768–71 Cartesian product, 51–52, 125 regrouping passim S Cartesian product of, 526–27 in addition, 172, 183 sides of a, 768 sample, defined, 475 combinations and, 552–54 in subtraction, 174, 184 slopes for analyzing diagonals, samples and bias, 475–77 complement of, 50 regular inscribed hexagon, 781 816 sample space, 516–21 continuous set of numbers, 445 regular pentagon, 604, 607–8 symmetry, 601, 604 additive property of probability, counting numbers, 45 regular polyhedra, 644 Rick, Killie, 394 536 DeMorgan’s laws, 54, 56, 57 bindex.qxd 11/8/07 11:57 AM Page I18

I18 Index

set(s) (continued) Shiing-Shen Chern, 845 simultaneous equations, 827 spreadsheet activities difference, 110 Shi-Ku Chu, 341 solutions, 828 Base Converter, 79, 81 difference of, 50–51 side-angle-side (SAS) elimination method, 831 Consecutive Integer Sum, 17 disjoint, 48, 109–10, 345 congruence property, 743–44 graphical method, 830–31 Function Machines and Tables, elements/members of, 45–46 for kites, 746 substitution method, 831 97 empty, 46, 112 for quadrilaterals, 790–92, simultaneous equations, solutions spreadsheet activities on the Web equal, 46 793 of, 826–29 site equivalent, 46 for rotation transformation, size transformations, 862–64, Consecutive Integer Sum, 17 finite, 47–48, 59, 60, 89, 110 876–77 881–83 Standard Deviation, 502 fractions for triangles, 743–44, 746, scale factor, 863, 883 square (geometric) concept, 246 778 skew lines, 642 8-by-8, 118 ordering, 247–49 similarity property, 753, 754, slant height, 710 acre, 670–71 functions (See functions) 793–94, 881 slide, 852–53, 859–60 additive magic, 19, 217, 295, infinite, 47–48 side-side-side (SSS) slope, 812–16 296 inherent rules regarding, 46 congruence property, 745–46, collinearity, 813–14, 836 area integers, 343–45 767, 791, 792–93 in coordinate plane, 812–16 English system, 689–91 intersection method, 220–21, similarity property, 754, lines, 826, 837–38 metric system, 675–76 224 793–94, 881 of lines, 812–13 area of a, 757, 797 intersection of, 49, 110 sides of a parallelogram, 790, 792 parallel lines, 814, 816 attributes of, 606 probability and, 522, sides of a rhombus, 768 percent grade, 818, 821 as base of pyramids, 654 524–25 sides of the triangle, 588 perpendicular lines, 814, 816 centimeter, 675–76 matching, 46 congruent, 745 point-slope equation of a line, creating shape of, 589, 590 null, 46 copying an angle, 767 825–26 decimeter, 675–76 number of a, 59–60 included, 744 Smith, Steven B., 166 dekameter, 675–76 operations on, 48–52 midpoint of, 793–95 Smith number, 231 diagonal of a, 795 ordered pair, 51, 82–83 perpendicular bisector, 778, 779 Smoot, Oliver, 681 as equilateral and equiangular, partition of a, 85 Sierpinski triangle (or gasket), 37 solution of an equation, 15 590 probability and, 521–25 Sieve of Eratosthenes, 206, 234 solution set, 405 foot, 670–71 proper subset, 47, 60–61 similarity, 881–84 solutions of simultaneous hectometer, 675–76 rational numbers, 382 AA similarity property, 754–55, equations, 828 inch, 670–71 real numbers, 400–402 783 Solve an Equation strategy, 380, kilometer, 675–76 relations (See relations) fractals and self-similarity, 432, 838, 839 meter, 675–76 relative complement, 50–51 755–56 Solve an Equivalent Problem mile, 670–71 sample space, 516–21, 549–51 geometric problem-solving strategy, 238, 281 millimeter, 675–76 set-builder notation, 45–47 with, 793–95 Solve a Simpler Problem strategy, model and abstraction, 589 solution, 405 SAS similarity property, 754, 6, 25–30, 38 oblique prism, 643 subset of, 46 793–94, 881 solving equations, 15 perimeter, 687–89 theory, 79–80 self-similarity, 425, 755–56 algebraically, 13–16, 405–6, 408 right square prism, 643 union of, 48–49, 110, 522, of shapes, 883 Balance Beam Algebra, 405–6, squared square, 357 524–25 SSS similarity property, 754, 408 symmetry, 601, 604 universal, 47 793–94, 881 balancing method, 405–6, 408 tessellations of, 632 Venn diagrams of, 47–52 of triangles, 752–56 Cover Up method, 16 units, 689 set-builder notation, 45–47 and similitudes, 882, 883 inequalities, 409–11 yard, 670–71 set intersection method, 220–21, similar shapes, 883 percents, 325–28 square (of a number) 224 similitudes, 862–64, 882 simultaneous, 826–29 of 2, 399–400 greatest common factor, properties, 882 solutions, 831 counting numbers, 22 220–21 similar shapes in the plane, 883 strategies (See problem-solving perfect, 146 least common multiple, 224, size, 862–63 strategies) squared rectangle, 355 225 size transformations, 862–64 transposing, 409 squared square, 357 set models center, 863 Work Backward method, 16 square lattice, 588, 615, 695, 700 integers, 344, 345 scale factor, 863, 883 See also problem solving; square pyramid, 721–22 rational numbers, 382 simple closed curves, 607, 645–46, problem-solving strategies square root take-away, 115–16 657 special factors, 159 with calculators, 213, 402 whole numbers simple experiments, 515–25 sphere(s), 646, 648, 723–24 defined, 402 addition, 109–10 computing probabilities in, Archimedean method for divide and average method, 412 division, 130 517–24 deriving volume, 665 irrational, 412 multiplication, 123–24 simplified method for integer Cavalieri’s principle, 725 primes and, 213 subtraction, 115–16 subtraction, 350 diameter, 646 principal, 402 set notation, 241, 405 simplifying great circle of the, 651 squeeze method for decimals, 412 sextillion, 79 fractions, 242, 244, 245, 253, surface area, 712 SSS. See side-side-side shape identification, 585–87. See 256, 257 volume, 723–24 standard algorithms also geometric shape rational numbers, 383, 388, spinning spinners probabilities, addition, 172–73, 192 concepts 389, 390, 391 516–17, 518–19, 525, division, 182–83 Shells and Starfish (Escher), 849 simulation, 514, 560–62, 574 538–39 long division, 195, 302–3 bindex.qxd 11/8/07 11:57 AM Page I19

Index I19

multiplication, 176–77, 194 subtraction, 115–18, 259–61, summands, 110 images of points under subtraction, 174–75, 192–93 348–52 supplementary angles, 618, 621 transformations, 875 standard deviation, 493–96, by adding the complement, 185 surface area, 707–12 linear functions, 419 498–500 adding the opposite approach, cones, 711–12 measure of a vertex angle in bell-shaped curves, 497–99 349–50, 352, 387 cylinders, 709 regular n-gon, 629, 632 dispersion, 493–96 algorithms, 174–75, 184–85, formulas (See inside back metric prefixes, 674 distribution, 498–500 192–93 cover) metric system units, 673, 676, mean, 493–96, 498–500 base five algorithm, 192–93 prisms, 708 677, 678 unbiased, 506 cashier’s algorithm, 184–85 pyramids, 710 number system summary, 66 z-scores, 495–96, 498–500 clock arithmetic, 923–24 spheres, 712 operations on sets, 52 standard meter prototypes, 673 comparison approach, 118 survey of college freshmen outcome frequency, 520, 521, standard units of measurement. See compensation, 159, 261 Draw a Diagram strategy, 44, 524 English system of decimal algorithm, 298 99 percents and fractions, 322 measurement difference (See difference) symbols, list of. See inside back Platonic solids, 644 statements, logical, 912–13 eliminating extra information, 5 cover quadrilateral attributes, 606 statistics, 439, 441–54, 464–77, equal-additions method, 159, symmetrical distribution, 496–97, random-number, 561, 568 484–500 185, 261 498–500 sequences, arithmetic and analyzing data (See analyzing estimation for fractions, 260–61 symmetry, 600–604, 860–61 geometric, 86 data) fractions, 259–61 analyzing shapes, 600–604 stem and leaf data, 442–43, central tendency, 484–88 integers, 348–52 of circles, 608 490–91 data (See data) left-to-right method, 159 in Escher’s art, 849, 861–62 summary of grafts and charts, distribution, 489–500, 496–500 mental computation, 158–59 glide reflection, 861 454 graphs (See graphs) missing-addend approach, line of, 600 surface area, 724 interquartile range, 489–92 192–93 Mira®, 602, 608 truth table, 913 lower quartile, 489–92 fractions, 259–60 plane of, 652 unit cubes, 87 mean (See mean) integers, 350 property, 84 z-scores, 495, 500 median, 485–86, 488, 489–92, whole numbers, 116–17, 128 reflection, 600, 602, 604, 606, tablespoon, 671 497–99 nonstandard algorithms, 652, 852, 860, 861, 862 take-away approach, 115–16 misleading (See misleading 184–85 rotation, 602–4, 606, 608, 652, clock arithmetic, 923–24 graphs and statistics) notation, 351, 352 860, 862 fractions, 259–60 mode, 485, 486, 488, 496–99 order of operations, 144–45, translation, 860–61, 862 integers, 349, 350, 352, 358–59 normal distribution, 497–99 163–64 triangle congruence and, 588, set models, 115–16 plotting on a number line, pattern, 348 601–2 whole number subtraction, 489–90 rational numbers, 387–88 Use Symmetry strategy, 850, 115–16 samples and bias, 475–77 real numbers, 402–3 901–2 tally numeration system, 62, 66 standard deviation (See regrouping, 174, 184 symmetry transformations, 860–61 tangent line to a circle, 777 standard deviation) simplified method, 350 Systematic Guess and Test strategy, tangram puzzles, 586–87 trends, 439, 447–49, 454 standard algorithm, 174–75 7, 8 Taylor, Richard, 203 upper quartile, 489–92 take-away approach system of equations. See Tchebyshev, 217 visual comparison (See graphs) clock arithmetic, 923–24 simultaneous equations teachers and teaching z-score, 495–96 fractions, 259–60 system of units, ideal features of, children’s “reversals stage ,” 67 stem and leaf plot, 442–43, 490–91 integers, 349, 350, 352, 673 of geometry, 845 back-to-back, 443 358–59 Killie’s [teacher’s] Way, 394 outliers, 503 whole numbers, 115–16 T “Ten Commandments for step functions, graphs of, 423–24 unlike denominators, 260, 388 table(s) Teachers” (Pólya), 1 Stevin, Simon, 294 whole number set models, addition, 113–15 thinking aloud, 6 Stiller, Lewis, 590 115–16 for base five operations, teaspoon, 671 stock market, 425 subtractive principle of Roman 192–93, 194 temperature straight angles, 617, 620, 621, 631 numeration system, 63–64, bias survey sources, 477 English system, 672 straightedge constructions, 772, 66 box and whiskers data, 492 metric system, 678–79 780, 781, 782, 865 subtrahend, 116 box volumes, 423 “Ten Commandments for Teachers” straightedge properties, 766–67 successive differences, 27–30 decimals and fractions, 291 (Pólya), 1 strategy, defined, 4. See also sufficient, 915 earthquakes of the 1960s, 452 term, 86 problem-solving strategies sum educational level vs. income, terminal side of a directed angle, subset consecutive whole numbers, 453 855 combinations, 552–54 11–12 English System units, 669, 670, terminating decimals, 289, 305, events, 516 first n counting numbers, 671, 672 399, 400 of populations, 475 12–13, 20–21 exponential functions, 420 terms of a sequence, 22 proper, 47, 60–61 infinite geometric series vs., exponential growth, 87 tessellations, 630–33 subset of a set, 46 265 functions as, 89 of arbitrary quadrilaterals, 631 substitution method of solving order of operations, 144–45, function values of t, 420 of arbitrary triangles, 630 simultaneous equations, 831 163–64 geometric shape dual of, 635 subtract-from-the-base algorithm, of a plus b, 110 definitions, 621, 647 Escher-type patterns, 849, 175–76, 193 of probabilities, 563 descriptions, 589 861–62 bindex.qxd 11/8/07 11:57 AM Page I20

I20 Index

tessellations (continued) image of P, 852 one- and two-stage, 542–43 right triangular prism, 643, 720 plane and, 630–31, 632 isometries (See isometries) probability, 532–42 rotation transformation, 862 plane with parallelograms, midsegment proof, 910 trends, statistics and, 439, 447–49, scalene, 589, 630 860–61 notation, 875–76 454 sides (See sides of the triangle) plane with squares, 670 similarity, 881–84 triangle(s), 588, 752–56, 778–82 Sierpinski, 37 with regular polygons, 632–33 similitudes, 862–64, 881–84, AA similarity property, 754–55 similarity, 752–56, 793–95 rotation transformation of a 882 acute, 620, 621, 630 similarity and similitudes, 882, triangle, 862 properties, 882 altitude, 773, 779, 837–38, 839 883 semiregular, 633, 635, 638 similar shapes in the plane, angle sum property, 620 square lattice and, 588, 695, tests for divisibility, 207–13 883 angle sums in n-gons and, 698 by 2, 207–8, 210 size, 862–63 629–30 tessellations with, 630, 632 by 3, 210–11 size transformations, arbitrary, 630 triangle congruence, 741–46, by 4, 210 862–64, 863, 883 area, 691 790–93 by 5, 207–8, 210 slides, 852–53, 859–60 base, 691 AA-side property, 746 by 6, 211–12 symmetry, 860–61 centroid, 779, 787 analysis with isometries, by 8, 210 in Escher’s art, 849, 861–62 centroid of a, 835–37 878–80 by 9, 210–11 isometries and, 852, 860, cevian line segments, 335 ASA property, 744, 746, 791, by 10, 207–8, 210 861, 862 circumcenter, 777 794 by 11, 211 Use Symmetry strategy, circumscribed circle and, correspondence, 742 prime factorization, 212–13 850, 901–2 777–79, 838–39 geometric problem solving, tetrahedron, 644–45 transformations, 852–64, 875–84, compass properties, 766, 767 790–93 tetromino shapes, 10–11, 611 893–95 complementary, 620 reflections, distance, and, 877 Theon of Alexandria, 800 isometries, 852–60 congruence (See triangle reflections and, 875, 877, 880 theorem, defined, 142 congruence and, 875–81 congruence) SAS property, 743–44, 746, theoretical probability, 519–20, 574 making Escher-type patterns, coordinate system, 834–39 778 experimental probability vs., 861–62 creating shape of, 589, 590 SSS property, 745–46 560 notation, 875–76 equilateral, 588, 589, 601, 602, symmetry, 588, 601–2 thermal imaging, 895 similitudes, 862–64 632, 780, 782 triangular lattice, 588 thinking strategies similarity and, 881–84 5-con pairs, 746 triangular numbers, 32–33 addition facts, 113–15 size transformations, 881–83 fractals and self-similarity, trillion, 74, 79, 104 facts for base five, 192–93, 194 solving problems with, 755–56 triples, prime, 217, 413, 705 long division, 180, 182 893–95 geometric problem solving, troy ounce, 672 multiplication facts, 128–29 symmetry, 860–61 793–95, 834–39, 893–95 true equations, 15 See also mental math transitive property harmonic, 265 truncate, 161–62 thousand, 74 less than height, 691, 692 truncated cube, 645 three-dimensional effects, 470–72 inequalities, 409–11 Hero’s formula, 697 truth table, 913 three-dimensional shapes, 640–47 integers, 366 hypotenuse, 401, 693–95 twin prime conjecture, 203 curved, 645–47 rational numbers, 393 incenter, 777–78, 779 twin primes, 215 dihedral angles, 641–44 real numbers, 403 inequality theorem, 695 two-column front-end estimation, planes, 640–47 whole numbers, 140–41 inscribed circles, 777–78, 779 160 polyhedra, 642–45 relations, 84–85 isosceles, 589, 644, 739 two-stage tree diagram, 542–43 skew lines, 642 translation, 852–53, 857 acute, 630–31 surface area, 707–12 distance and, 853, 876, 895 reflection symmetry, 601–2, U volume, 671, 676–77, 717–26 notation, 875–76 604 Ulam, Stanislaw, 575 three-dimensional space quadrilaterals, 853, 895 legs, 693–95 Ulam’s conjecture, 203 Cartesian coordinates in, 819 symmetry, 860–61, 862 Marathe’s, 123 unbiased standard deviation, 506 Todd, Olga Taussky, 575 transposing, 409 median, 773, 779 unequally likely events, 520–21, ton, 672 transversals, 618–19, 627 medians of a 525, 565, 566 transformation geometry, 851–64, trapezoid collinear, 836 union of sets, 48–49, 110 875–84, 893–95 area, 692 congruent, 835 probability and, 522, 524–25 applied problems, 894–95 base angles of, 614 ratio of, 837 unitary fractions, 263 clockwise/counterclockwise creating shape of, 589, 590 multi-digit midsegment, 794 unit cubes, 87, 717 orientation, 854–55, 860, 877 height, 692 obtuse, 620, 621, 630–31 unit distance, 687 congruence isosceles, 589, 601–2 orthocenter, 779, 784, 837–38 unit fraction exponents, 404 isometries, 875–81 midsegment, 704 Pascal’s, 285, 541–42, 553–54 unit fractions, 237 polygons, 881, 883–84 model and abstraction, 589 perpendicular bisector, 856, units, in base ten pieces, 71 shapes, 880–81 parallelograms as, 698 879 universal set, 47 Escher-type patterns, 849, symmetry, lack of, 601, 604 perpendicular bisector of sides, universe, 47 861–62 tessellations with, 631 778, 779 unknowns. See variables functions as, 91 tree diagrams, 532–42 properties of, 600–601, 619–20, unlike denominators geometric problem-solving as approach to multiplication, 834–39 addition, 256–57 using, 893–95 125 Reuleaux, 900–901 division, 272–73 glide axis, 859, 879–80 counting techniques, 532–34 right, 401, 589, 620, 621, subtraction, 260, 388 half-turn, 893–94 expected value, 562–63 693–95 upper quartile, 489–92 bindex.qxd 11/8/07 11:57 AM Page I21

Index I21

Use a Model, 582, 657, 695, 707 vertex arrangement, 632–33, 638 division, 178–83, 194–95, number systems diagram, 381, Use a Variable strategy, 11–13, 350, vertex figures, 635 302–3 382, 392, 401 695 vertex of the angle, 616–17 multiplication, 169, 176–77, numerals and, 59 Use a Variable with algebra, 13–16, vertical angles, 618, 619, 621 185, 186, 188, 194 ordering, 60–61, 140–42 27–28 vertical axis. See y-axis subtraction, 174–75, ordinal, 59 Use Cases, 342, 373 vertical line test, 424–25 192–93 pentagonal, 35 Use Coordinates, 808, 845 vertices, 588 amicable, 228 primes with their respective Use Dimensional Analysis, 666, congruence and, 602–3 cardinal, 59 exponents, 219–20 733 nonadjacent, 605 computation, 157–66 reading, 74–75 Use Direct Reasoning, 108, 148 of a polyhedron, 643–44 with calculators, 163–66 rectangular, 35 Use Indirect Reasoning, 156, 198, Vieta, Francois, 294 estimation, 160–63 rounding, 161–63 226, 399–400 volume, 676–77, 717–26 decimal point and, 287–88 set models Use Properties of Numbers, 204, abstract, 717–26 decimals and, 287–88 addition, 109–10 231–32 capacity vs., 671 division, 123, 129–34, 274 division, 130 Use Symmetry strategy, 850, 901–2 Cavalieri’s principle, 724–26 decimals, 300–301 multiplication, 123–24 cone, 723 fractions, 274 subtraction, 115–16 V cylinder, 720–21 division algorithms, 178–83, sets as basis for, 45–52 valid argument, 916 English system, 671 194–95 square roots, 213 value estimation, 719 elements in a set, 59–60 subtraction, 115–18 absolute, 354 formulas (See inside back cover) estimation, 160–63 algorithms, 174–75, 192–93 discrete, 445, 446 metric system, 676–77 expanded notation/form, 72, 76, missing-addend approach, expected, 562–63 prism, 717–18, 720 172–73 116–17, 128 place (See place value) pyramid, 721–22 exponents, 77–78, 142–44 take-away approach, van Hiele Theory, 581, 583–85 sphere, 723–24 factors, 205–7 115–16 level 0—recognition, 583–84 von Neumann, John, 145, 149 a divides b, 207 triangular, 32–33 level 1—analysis, 584–85 voter opinion polls, 475–77 four basic operations, 134 in Use a Variable strategy, level 2—relationships, 585 greater than, 61 11–13 level 3—deduction, 585 W greatest common factor, 220–23 writing, 74–75 level 4—axiomatics, 585 Wallis, 411 identification, 59 zero, property of, 128 variable(s), 15 Web site activities intermediate algorithms whole-to-part ratios, 311 coefficients, 408 prime number competition addition, 173, 192 Who Wants to Be a Millionaire?, concept of variables in algebra, updates, 227 long division, 180, 182 726 15 spreadsheets multiplication, 177, 194 Wiles, Andrew, 203 exponent as, 421 Consecutive Integer Sum, least common multiple (LCM), Work Backward strategy, 16, 286, greatest integer function, 427 17 223–26 335 problem solving with, 11–13, Standard Deviation, 502 less than, 61 World Cup Soccer Championships, 350, 695 See also dynamic spreadsheet measurement division, 129–30 633 problem solving with algebra, activities on the Web site; mental math, 158–59, 163 13–16, 27–28 eManipulatives on the Web missing-addend subtraction, X Use a Variable strategy, 11–13, site; Geometer’s Sketchpad on 192–93 x-axis, 418 350, 695 the Web site multiplication, 123–29 of bar graphs, 444–45 Use a Variable with algebra, Weierstrass, Karl, 336 algorithms, 169, 176–77, of histograms, 445 13–16, 27–28 weight 185, 186, 188, 194 polar coordinates, 832 variance, 493–94 English system, 672 as product of primes with x-coordinate, 417–18 vector geometry, 909 metric system, 678 their respective equations of lines, 822–23, 826 Venn diagrams, 47–52 Wells, H. G., 439 exponents, 219–20 simultaneous equations, conditional probability, 566 whole number(s), 45–52, 59–61, properties 826–29, 831 DeMorgan’s laws, 54, 57 125–27, 134, 381 associative, 126, 129, difference of sets, 50–51 addition, 109–15, 158–59, 158–59 Y freshmen survey problem, 99 172–73 cancellation, 137 yard, 669, 670–71 GCF and LCM, finding, 228, algorithms, 172–73, 184, closure, 125 y-axis, 418 234 185, 192 commutative, 125–26, compressing, 444–45, 447, intersection of sets, 49 commutative property, 257 127, 128, 158–59, 266 465–67 problem-solving with, 52 additive properties distributive, over histograms vs. bar graphs, 445 union of sets, 49 associative, 112–15, addition, 127, 129, manipulation of, 444–45, 447, vertex (interior) angles, 607–8 158–59, 172–73 135, 176 465–67 adjacent, 617 cancellation, 123 distributive, over y-coordinate, 417–18 alternate, 619–20 closure, 111 subtraction, 128 equations of lines, 822–23 angle sum in a triangle, 620 commutative, 111–15, identity, 126 midpoint formula, 811 measure of, 617 158–59, 172–73 rectangular array, 124–25 orthocenter of a triangle, in regular polygons, 632–33 identity, 112–13 multiplying decimals as, 299 837–38 on the same side of the algorithms, 171–83 naming, 74–75 simultaneous equations, 826–29 transversal, 627 addition, 172–73, 184, 185, number lines, 61, 110–11, slope-intercept equation, See also vertices 192 141–42, 248 823–25 bindex.qxd 11/8/07 11:57 AM Page I22

I22 Index

y-intercept, 823 Z factorial, 550 zero pair, 344 point-slope equation of a line, zero as identity property, 112–13 z-scores, 503 825–26 as additive identity, 112–13 as integer, 343, 352 dispersion and, 495–96, slope-intercept equation of a division property of, 132 in Mayan numeration system, 499–500 line, 823–25 divisor property of integers, 65–66 mean, 495–96 Young, Grace Chisholm, 373 362 multiplication property of, 128 Young, William, 373 as exponent, 144 repetend not, 400 bindex.qxd 11/8/07 3:39 PM Page I23 bindex.qxd 11/8/07 3:39 PM Page I24