Errata for A Problem Solving Approach to Mathematics for Elementary School Teachers, 9th edition, by Billstein et al.

Thanks for my students and colleagues for pointing out some of these errors.

As a warm-up, we'll first list the errors in the NCTM's Principles and Standards for School Mathematics:

• page xii, lines 3-4 of third full paragraph: The quotation marks make no sense, since the word "and" is not a word they are talking about. Probably no quotation marks should be used at all, and instead the words should, will, can, and must should be italicized.

• page 11 of the on-line version is mislabeled as page 16.

• page 33, line -9: This is nonsense. It does not follow from the fact that each addend is less than or equal to 1/2 that the sum is strictly less than 1.

• page 38, line 20: This is not true. Using recursion to COMPUTE the Fibonacci sequence is a horrible way to do it, terribly inefficient. For computational purposes, one should use iteration or the explicit algebraic formula involving the square root of 5.

• page 59, Figure 3.6: Stephanie's "proof" is neither elegant nor correct. She certainly has some nice ideas, but holding this up as an example of excellence in reasoning and proof is ludicrous. Her claim that the answer is twice the number of towers of height 2 is correct, but her drawing in no way gives any justification for that claim. She has given what she claims is the entire set of towers of height three (and she has done so), but her "reasons" for why her list is complete are nonsense, and, despite her repeated assertion at the end, it has nothing to do with multiplying by 2.

• in the on-line version, algebra standards for each grade band: The comma after "Understand patterns" should not be on a separate line.

• in the on-line version, e-example 4.1.3: A space needs to be inserted between the words "so" and "can" (second line of the paragraph beginning "The examples above. . .").

• in the on-lin version, e-examples for Section 4.5: There are two examples, but they are both numbered 4.5.2 (rather than one of them being 4.5.1) in their title bars.

• page 99, line 6 of first paragraph of new section: "programs, with" should be "programs with". (This has been corrected in the electronic version.)

• page 99, last paragraph: The definition of pentomino is wrong. Actually the definition doesn't even say what was intended, since the "at least" makes no sense (there is no way an edge could coincide with MORE THAN ONE other edge). But even besides that, it doesn't work, since one needs to say that EVERY edge that touches any other square does so by sharing a side in common, and that the figure is connected. Think about a row of two squares above a row of three squares (touching), offset a bit. It satisfies the definition but is not legal. And a row of two squares entirely separate from and not touching a row of three squares also satisfies the definition and is not legal.

• page 106, yellow box, line 2: The tense is wrong: "will" should be "would".

• page 183, line -12: "work" should be "to work" (same error appears in on-line version).

• in the on-lin version, e-example 5.5.1, Collecting, Representing, and Interpreting Data: In the last sentence of the second paragraph under Additional Tasks, the words "and communities" don't make sense alone; they need an adjective describing the communities, such as "inland".

• page 238, figure 6.21: Why is there no white square at one of the intersections?

• page 238, line 5: Why the reference to Hamiltonian circuits? These are Eulerian circuits.

• page 244, exercise 42: All the minus signs are supposed to be in superscript as in the other problems (this typo occurs five times in this example). This was correct in the 8th edition.

• page 252, lines 14-15: This conclusion is really unwarranted. There is so much overlap of the distributions, that there seems to be no clear difference.

• page 264, line -2 to -1: This is backwards! An even number is, BY DEFINITION, one that is divisible by 2. The TEST for divisibility is that the numeral in Base Ten ends in an even digit. This same error is made on page 265, line 14.

• page 285, lines 15 and 17: "e.g." should be "i.e." in both cases.

• page 285, figure 6.44: The equal sign should be a hyphen (with no surrounding space).

• page 291, exercise 11a: "module" should be "modulo".

• page 293, figure 7.1: It is not clear whether d or e is closest to ab. There is too much in this figure.

• page 297, line -2: This function is out of date as of January 2001. Should have a date ("in 2000").

• page 298, line -6: This is not true. In the polar regions, the period of sunlight stays at 24 hours for a finite period of time (six months at the poles, less as one moves toward the circles).

• page 299, line -1: This is not true. As is stated in the next sentence, the movement is not along a line.

• page 302, second displayed equation: Remove parentheses.

• page 304, line 15: "31" should be "30", since we are told we are starting at 440.

• page 304, line 17: The subscript n is too big (print version only).

• page 305, fifth paragraph: The example in Chapter 6 did not have these two phone rates. One of them was different.

• *page 2 and elsewhere: George Polya's last name needs an accent over the o.

• page 11, sample book page: This is a poor choice of book page, because the problem is confusing, in that the "total" pay is doubled each day. (People don't get paid that way.) Also, saying that the answer is reasonable just because it is even seems like a stretch.

• *page 18: The statement in the last sentence in "Solution" isn't very satisfying. WHY is the answer no? Pretty fuzzy reasoning. The point is that each game has only one winner, so if there are six winners here, there must be at least six games.

• *page 54, last bullet item: "all have" should be "both have" or just "have" -- it doesn't make sense to use the word "all" when there are only two items being discussed.

• *page 62, line 2: This statement is clearly NOT true -- lots of healthy people have never even heard of Super-Bran cereal, let alone eat it. The reader is being asked to ASSUME that it is true.

• *page 62, Example 1-21: This makes no sense. Delete "natural" twice, and make it "the numbers" in the second instance. Even better would be to make all the implications singular, as is done in the solution.

• page 63, exercise 22: The labeling is inconsistent. Since the universe is spies, either "Women" is sufficient for that circle, or else "Poor Spies" and "Tall Spies" are needed for those circles. Also, the answer in the back of the book is silly, in that "not poor" does not mean "wealthy", and "not tall" does not mean "short".

• *page 64, questions from the classroom #1: This is not true for n = 1, so the student's arithmetic is faulty. That clearly is not meant to be the point of the problem. Also, the student's conjecture should explicitly refer only to whole numbers (bigger than 1).

• *page 71, line 2: "relationship" should be "relationships".

• *page 72, first solution: This notation is poor. What is the status of n? In fact, it is being existentially quantified here, but nowhere does it say that. Change to "x = 2n for some n [element of] N" and similarly in part b.

• *page 78, line 1 of Example 2-5: Replace ", and" by "for some". Notice also that the text is inconsistent in these bad usages, in that sometimes a comma is used and sometimes it is not (see page 83, exercise 10).

• page 83, exercise 4a: This duplicates part (c) of the "Now Try This" on page 74,

• *page 83, exercise 6: The descriptions of D, I, and J need the word "and" in the conditions, as is used in Example 2-1 of this section. A comma as used here is not a logical connective and it's therefore imprecise what is meant.

• *page 83, exercise 7d: Replace first comma by "for some".

• *page 83, exercise 11e and f: replace "and" in each case by "for some".

• *page 83, exercise 12a, e, and f: These cannot be made true with either the "is a subset of" or the "is not a subset of" inserted, since both of these symbols are used only in the context of sets, and 0, 1024, and 3002 are not sets. (See definition of "is not a subset of" on page 79.) The problem needs to be changed or reworded somehow.

• *page 88: Under the solution for "d." in the middle of the page, there is an extra closing parenthesis. ((A U C) - B)) should be ((A U C) - B).

• *page 92, exercises 1 and 2: Fix the misuses in set-builder notation (e.g., in #1, we need the word "or" before "9". This is a pervasive problem throughout this section, in the exposition and the exercises. Students should be shown clear mathematical usage if we want them to speak and write clearly.

• *page 92, exercises 6 and 11: There is gross inconsistency here -- in one case only one diagram is provided, and in the other, there is a separate diagram for each part.

• *page 95, exercise 46: The definitions of B and C are incorrect, in that the variable n seems to have some sort of existence, when in fact it needs to be existentially quantified. Both should have "for some" in place of "and".

• page 126, exercise 40: Parts b, c, and d make no sense as written. They must read "f feet, in yards", and so on.

• *page 132, figure 2-38, and page 142, figure for exercise 18, and page 147, figure for exercise 20: The label inside the function machine makes no sense. What's written there has to stand alone and certainly can't depend on what is written above the input tube. Notice that the function machines on page 131, 138, 141, and 144 are correct -- what's written inside is generic. But in figure 2-38, the inside should read "output is less than input"; on page 142, the inside should read "f(l,w) = 2l + 2w" (or use P for the function name instead of f, since it hints at the meaning of this function); on page 142, the inside should read "f(x,y) = x^2 + y^2".

• *page 138, fourth line after figure: The little open circle between 2 and 7 should be a multiplication dot symbol instead of the function composition symbol.

• *page 139, exercise 5c: There needs to be a statement as to what "x" is, as in the other five parts; otherwise, it is unclear what this is supposed to mean. So add something like "for all x [an element of] W". Also, the preamble doesn't really need to say "or a subset of W", and it needs to be made clear that the "x" in part b also has to be in W.

• page 141, exercise 13: The example is misquoted from Principles and Standards, in that the cost per minute in PSSM is $0.45, not $.30.

• *page 142, exercise 23b; page 271, exercise 43a; page 327, exercise 24a: Inconsistent style as to subparts (color and punctuation).

• *page 143, exercise 21: This is an unrealistic problem, since only pitchers in baseball can throw that fast (87 MPH), and they'd be hard pressed to do it upward.

• *page 147, exercise 1a: font of "M" is wrong (twice).

• *page 153 and elsewhere to a large extent throughout the book: The spelling of "pre-K" is inconsistent (here is it "preK"; on page 96 it is "PreK"; on page 571 it is "Pre-K") and also not consistent with the NCTM's Principles and Standards, which it claims to be quoting. In fact, the alleged word-for-word quotes on this page from pages 392 of that book do NOT appear there as quoted, aside from the misspelling of "pre-K". Worse than that, the quote is OBVIOUSLY wrong, in that it says that kids in grades pre-K to 2 should "develop fluency in ... dividing whole numbers", whereas PSSM talks about fluency only for adding and subtracting.

• page 161, line 4 after Table 3-7: There should be two bars above the V, not one. Also, two lines above that, the bar above the V is too high -- it runs into the tail of the "p" in the line above it. Finally, the third example purporting to be about "more bars", namely CXI with a bar above the CX part, is NOT about "more bars" -- there is just one bar. The issue there seems to be using the bar to cover just a portion of the numeral (is that really done with Roman numerals?). The sentences should be reworked to reflect this.

• page 178, exercise 2: inconsistent use of punctuation and color for parts a and b, compared to rest of text.

• *page 197, exercise 40b: "so" should be "such". Furthermore, making this part (b) makes no sense -- it has nothing to do with part (a), and it has nothing to do with this section (it belongs in Section 3-1).

• *page 198, lab activity #2: This is poorly explained. It refers to "the number game in Figure 3-27" but it never says what the game is! It asks "how this works" -- how WHAT works?

• page 228, exercise 10: There seems to be no exercise asking student to do subtraction with a charged field or chip model; there needs to be.

• page 242, exercise 5b and 5d: These make no sense. The "situation" 4 or n years ago depends on the situation (i.e., how many students are eating in the cafeteria) now. The "situation" 4 years ago was not 80.

• page 242, exercise 11, at least parts b, c, and d: The tenses make no sense. You can't say "was" for 8:00 and "is" for an earlier time. It's best to rethink the wording in all parts.

• *page 234, line 3 of historical note: The name of the University is Erlangen, not Erlanger.

• *page 246, third line of second paragraph, and throughout: The symbol for "does not divide" must be the symbol for "divides" with a slash through it. In this text it is not done that way -- the "not divides" symbol is much shorter.

• page 253, second line above last shaded box: wrong font for "c|b".

• *page 253, paragraph above last shaded box: The logic is incomplete. One needs the converse of this statement as well in order to assert the "if and only if" statement in the box. (It's true, but it would require a separate, admittedly easy, proof.)

• page 256, exercise 16: $1.03 rather than $1.00 is clearly a typo. The extra three cents play no role in the solution.

• page 270, exercise 13: This is not satisfying. The obvious question to ask after part (a) is whether one can conclude that ab divides n, not whether it divides n-squared. Perhaps there should be three parts to this exercise, with my suggestion as part (b), and the current part (b) as part (c).

• page 270, exercise 27, line 3: "; for example" should be ", namely,". These are not examples but the entire list.

• page 271, exercise 40: In line 1, 120 should be 121; compare with the last line of part (c).

• page 271, exercise 43b: It is unrealistic to expect students to find the next perfect number -- it is too big to find by trial and error, and they have no reason to suspect that Mersenne primes play a role.

• page 273, definition: The case of a=b=0 needs to be excluded, since every number is a common divisor. You don't want to exclude a=0 OR b=0; see Example 4-27b.

• page 282, exercise 5, line 2: The n-dash between the 4 and the 8 should be a hyphen.

• page 283, exercise 17: The layout of this exercise is wrong. The statement that a and b are to be natural numbers needs to go in the preamble (as in exercise 16), and it should then be removed from parts (d) and (e). As it stands, the answer given for (c) is wrong (if a=b=0, GCD(a,b) is undefined), as is the answer for (f) in the solutions manual and teacher's edition of the text, because a could be 0.

• page 283, exercise 20: Inconsistency of style in "twelfth", "13th", and "twentieth".

• *page 302, last box: Commas can't be used to splice sentences. Change to "... number. Then ..." or "If ..., then ...".

• *page 304, first line of Example 5-1: "so" should be "such". This is a misuse of English: "so" modifies verbs; "such" modifies nouns.

• *page 310, exericse 4a: The 0 and 1 should not be blue.

• *page 316, Example 5-4a: 1/15 should be 2/15 (that's the problem that is solved).

• page 326, exercise 10: "meutally" should be "mentally".

• *page 328, brain teaser: This makes no sense. How do we count those who study French? We aren't told how many study both French (or languages) and physics. This error has persisted for the last three editions, at least. Doesn't anyone READ the book?

• page 335, cartoon: The Peanuts character who is confused is Peppermint Patty, not Sally.

• page 344, exercise 35c: The equals sign should be a plus sign.

• *page 345, exercise 45: The placement of the points is wrong. C is about 1/2, so C times D is about halfway between 0 and D, and there is no point there. This error has persisted for at least three editions, despite its having been pointed out to the authors years ago.

• *page 347, paragraph following Example: This definition of "proportional" is incorrect. Ratios aren't said to be proportional; ratios are said to be equal. Varying quantities are said to be proportional. For example, the number of representatives in Congress is (approximately) proportional to the population of a state. See also Example 5-21 for another correct usage of this terminology.

• *page 351, middle of page: This discussion is poorly expressed. The equation is perfectly correct if x represents the number of inches. Since we are not told what x represents, we can't say that this equation is "incorrect". Students need to be taught not to be sloppy, and this is an excellent example of what NOT to do.

• page 351, line 6 of quote from PSSM: missing dollar sign in front of 15.00 (this error appeared in 8th edition, but was not on this list).

• *page 354, Figure 5-22: There are no doors to get into this house.

• page 373, exercise 2: Whenever there is a negative number in the exponent, the minus sign (-) should be raised (this was correct in the 8th edition); see page 368 for comparison. This typo occurs eight times in this problem.

• page 379, long division display at bottom of page: The arrow showing the movement of the decimal point in the divided is too long -- the point should be moved two places, not three.

• page 385, exercise 5: Electric rate are no longer anywhere close to this cheap.

• page 393, line -3: the bar should not extend over the 3, just over the 45.

• page 393, line -1: the bar should extend over the 235, not just over the 35.

• *page 396, exercise 2: These are too easy. There needs to be one like the example in the second half of page 393.

• page 405, exercise 26, line 4: "Using these methods" is silly. The procedure in the figures cannot be used to place the square root of 237 -- it would take 237 steps, and the inaccuracy of the pictures would be overwhelming. Instead, one can simply note that 237 is between the square of 15 (225) and the square of 16 (256).

• page 417, exercise 6b: The first "a" should be in italics just like the second one (this was correct in the previous edition).

• *page 493, Table 7-6: The label on the left is inappropriate. The left margin labels are simply counting the possibilities. It's the a, b, and c that are the letters, and these are in the body of the table.

• *page 498, chapter review exercise 4: This question is invalid; there is not enough information given to answer the question. In 1960 there were other candidates on the ballot, such as Orval E. Faubus (National States' Rights Party), and in fact about 0.7% of the people (200,000 or so) voted for someone else. Therefore the intended answers are not correct.

• *pages 535-536: This example is very badly chosen. One should not display this kind of data -- a time series of paired data -- in this manner (side by side box plots). Such displays are appropriate for independent samples, like test scores of students in two classes taking the same test. For paired time series, an appropriate display would be two line graphs on the same coordinate system, with time as the horizontal axis. Computing medians and such for a time series is nonsense.

• *page 543, line 3 of Example 8-10: The term "mound-shaped" has not been defined. The discussion of the normal distribution called the curve "bell-shaped". This usage occurs more than once, I believe, so a check for consistency throughout is needed.

• page 574, remark after table 9-2: "Betweeness" should be "Betweenness".

• page 574, Table 9-3: The definition of skew lines is poorly stated grammatically. It should read "... are lines that do not intersect and that no plane contains." Or even better, "... are lines that are not coplanar and do not intersect," since "coplanar lines" is defined in the line above that.

• page 574, Table 9-3: The definition of intersecting lines should not contain the word "coplanar" because that is a theorem (Property 7 on page 575), not part of the definition.

• page 578, last line: The "interior" of an angle has not been defined.

• *page 584, exercise 3, line 3: The minus signs should be hyphens (it's done correctly in parts (a) and (b) of this exercise).

• *page 587, exercise 20: Parts (b) and (c) make no sense. Knowing that no three points are collinear does not tell you how many planes are formed. What you need to know is that no four of the points are coplanar. For example, the answer to (b) could be 1 or 4, depending on whether or not the four given points are coplanar. The needed wording for (b) is "...four points, not all coplanar"; and the wording for (c) needs to be "...point, no four of which are coplanar."

• *page 592: The definition of regular polygon is incomplete -- it needs to require the polygon to be convex. Otherwise one can have a regular dodecagon with twelve right angles (a thick cross). In fact, the definition of interior angle on page 591 makes no sense if the polygon is not convex, because the interior angle is not an angle (you want it to be 360 degrees minus the actual angle). One solution is to allow reflex angles (whose measures are between 180 and 360).

• page 594, definition of isosceles trapezoid: The two definitions given are not equivalent. By the first, a rectangle is not an isosceles trapezoid, but by the second it is. In fact, the first definition is wrong, since the congruent sides could be adjacent.

• *page 600, second line above box: The second "3" should be "4".

• *page 616, Figures 9-37 and 9-38: Since the instructions are to use dashed segments for those that cannot be seen, many of the sides of the shades bases need to be dashed. Figure 9-39 has it right.

• page 622, Figure 9-44: It makes no sense to call (b) an oblique cone; "oblique" and "right" make sense only when the base is a circle.

• page 626, exercise 25: This is silly. One can only complete the table by knowing Euler's formula. First of all, Euler's formula is not stated in the text -- it is to be discovered by the student in the "Now Try This" on page 540. So unless the student has done this activity -- correctly! -- this exercise cannot be done. Second, the instruction "state the relationship suggested by the table" makes no sense, because that relationship was used TO MAKE the table!

• page 626, exercise 26d: Euler's formula is not stated in the text -- it is to be discovered by the student in the "Now Try This" on page 540. So unless the student has done this activity -- correctly! -- this exercise cannot be done.

• page 626, exercise 30: This problem seems too hard. No answers appear in the book (the instructor's edition says "Answers vary" -- very helpful!) or the instructor's manual. For some good answers to this, click here.

• *pages 629 and following, discussion of Königsberg bridges. The city does not have two islands. D is not an island but a land mass. The river splits as it move to the right but never comes back together. See, for example, this website. The fixes needed include the text in the first paragraph of 9-5, figure 9-46, line 3 of page 629, figure 9-47.

• page 629, line after Figure 9-47: The standard, preferred terminology, both among mathematicians and among K-12 mathematics educators, is to call the curves in a graph (at least when they don't have a direction or orientation to them) "edges" rather than "arcs". This usage should be followed throughout the book (with perhaps an initial mention of alternative terminology, which also includes "node" as a synonym for "vertex").

• *pages 629 and following: The definition of Euler circuit is not correct. The whole network is not called an Euler circuit if it has a tour that covers all the edges once and returns to its starting point; rather, that tour itself is called an Euler circuit. This is very standard in graph theory, and the definition given here makes no sense. The correct wording is that such a network "has an Euler circuit", not "is an Euler circuit". The correction needs to be made in several places.

• page 636, figure 9-54: The figure does not make sense. The tape on the bottom loop is shown taping it to the upper loop, and the cut marks don't go through the tape (so that we aren't shown cutting "completely around the middle of each strip" as the exercise states). Presumably what is intended is that the two loops are not taped together, but that each goes through the hole created by the other, and the cuts go completely around.

• page 659, exercise 12: The phrase "the respective triangles" makes no sense in this sentence -- there are no triangles to refer to with "the". Perhaps it should say simply "two triangles". But then what does this have to do with the first sentence? This problem is totally screwed up.

• page 660, exercise 19b: The first part of this sentence needs to be deleted; the angles aren't in (a) -- they're in the stem.

• page 661, exercise 34: The word "chord" is used here, but I don't think that word has ever been defined. (It should be! It's an important concept.) Furthermore, the words chord and diameter do not appear in the index. This can all be corrected by work on page 744 and the index.

• page 665, figures of parallelogram and rectangle: The vertex labels are not in alphabetical order; it would be better if they were (standard notation).

• page 665, definition of kite: "distinct pairs" is not enough. The point is that the two pairs have to involve four distinct _sides_. You could say "disjoint pairs"

• page 667, exercise 25: This cannot be proved with the tools available at this point. As the answer in the back of the book and the solutions manual suggest, you need to use hypotenuse-leg, or use the Pythagorean theorem to be able to get SSS (or we could use Theorem 10-3b, which is in the following section) -- none of which are known to the student at this point. So this exercise is totally out of place here.

• page 680, exercise 43: This is not necessarily correct unless it is specified that BC and AD are the parallel sides of the trapezoid, rather than AB and CD. Also, for the proof hint to work well, it should be specified that BC is shorter than AD.

• page 692, exercise 9: Are ALL unknown side lengths to be found, or just those marked or y? The instructions should be clear. Also, the y in part (d) is not well placed -- it should be centered on segment BC.

• page 747, exercise 3: The parenthetical comment should have no comma before it.

• page 748, exercise 24: This is out of place. The area formula is not introduced until section 11-2.

• *page 764, exercise 6a: The football field is 91 m long, to the nearest meter, not 100. The width of 49 m is correct (to the nearest meter).

• *page 766, exercise #27: This cannot be done without the Pythagorean theorem, which has not yet been discussed. Needs to be moved.

• page 769, Technology Corner: Two errors. In part (b), the segment symbol above AB should be a line symbol (we can't have C collinear with A and B, or else we won't be able to do step (c)). In part (k), the points D, C, B, A should be listed as D, E, B, A.

• page 772, Figure 11-39: The roles of a and b are not consistent between the parts of the figure (b is the longer side in part (a) but the shorter side in the other parts).

• *page 781, exercise 2f: It's not clear what the 12 is labeling -- the diagonal on the floor, or the width of the box?

• page 795, exercise 27: The figure is quite out of line with the given dimensions. In particular, the length labeled 40 is longer than the length labeled 60. It should be drawn more to scale.

• page 815, first line of section on temperature: There is no such thing as a "degree Kelvin". The unit is called a kelvin (lower case when spelled out, upper case "K" when used as a symbol). See this webpage from the United States government.

• page 817, exercise 5g: It's unclear whether the label 10 cm is for the radius or the diameter. Instead of showing the diameter as a dashed line segment, you should show only the radius as a dashed line segment.

• page 817, exercise 5j: The figure is quite out of line with the given dimensions. In particular, the side labeled 10 is about 2/3 the length of the side labeled 50. It should be drawn more to scale.

• page 818, exercise 39. The last part should read "y = 200 and x = 20"; by putting "cm" in the last sentence, the first sentence now reads "... 200 cm cm on a side", which makes no sense.

• page 819, exercise 40: A sphere cannot sit in a cone with its great circle along the base of the cone as shown. The intersection of the two will be a circle parallel to a great circle (a circle of latitude in the southern hemisphere). The circle is flatter than the cone near the equator.

• page 825, exercise 12a: The figure is not drawn correctly. The perspective is all wrong, and some edges need to be dashed.

• *page 840, exercise 13a: There is not enough space in the grid to show the answer.

• page 844, brainteaser: The statement is ambiguous -- there are three different ways one might imagine "rotating" the top quarter, and only one of these is intended. Say something like "without slipping" to make it clear that the corresponding points on the circumference are to be touching throughout the process.

• page 852, subsection on Congruence via Isometries: The authors miss the whole point here. There is absolutely no reason at all to introduce glide reflections if one is willing to live with compositions of the other three isometries, because after all a glide reflection is just the composition of a translation and a reflection. The point of introducing a glide reflection is so that one can say that if two figures are congruent, then there is ONE isometry that takes one to the other. This fix is needed elsewhere as well, such as in the definition of similar figures on page 862.

• page 853, fourth line above "Now Try This": "Principal" should be "Principle".

• *page 856, exercise 17: The question makes no sense, is poorly worded. The "original" what? Is the hexagon made up of part reality and part reflected image?

• *page 860, Theorem 12-1, part 1: This is not true. If the center is collinear with the segment, then the image segment is, too, so is not parallel to it.

• page 879: The answer to the brainteaser in the instructor's edition is incorrect: "of the body seen" should be "of the mirror".

• *page 885, exercise 3d: This duplicates part (a) and should be replaced.

• *index: There is no entry for "De Morgan's Laws" (exercise 1.4.21 and 2.2.17).

• *index: The entry for "division algorithms" in the index is wrong. There should be an entry for "division algorithm" (singular), with reference to pages 119-120 (and that reference should be removed from its current location under "division algorithms, overview"). The division algorithm is something quite different from algorithms for division (in fact it is a theorem, not an algorithm at all). The point that the division algorithm is NOT an algorithm should be made in the text around page 119.

• index: There is no entry for Möbius strip (which is described on page 635).

The following are errors in the answer section and/or student solutions manual. It is very annoying that numerous errors pointed out in the answers for the previous edition remain in the current edition.

• *In general, the student solutions manual is sloppy with punctuation. Complete sentences, such as answer to 1-4 #19a, line 5, and countless others, should end with periods.

• *student solutions manual answer to 1-1 #19a: There is an implication here that it is valid to assume always that the way to minimize the number of coins is to use as many of the larger coins as possible. This is not valid. For example, if the problem were the same except nickels were not used and the total was to be 80 cents (again, requiring at least one of each coin -- penny, dime, quarter), then this approach would have you use two quarters, two dimes, and ten pennies (14 coins), whereas you can do better using only one quarter, five dimes, and five pennies (11 coins). It's subtle but important.

• *book answer to 2-2 #10e: The second expression given for this region is wrong.

• *book answer to 2-2 #10a: The caption left out the bar over the B.

• *book answer to 2-4 #26: The correct placement of dots in sums is centered, as in the box on page 113, not on the line, as here. A similar mistake occurs at other places in the book. Apparently the copy-editors and typesetter don't care about such niceties as correct use of mathematical notation.

• *book answer to 2-4 #44: This is not a correct answer. It is true that multiplication is not distributive over multiplication, but this needs to be shown (using an example is perhaps the easiest way here, say let a = 1 and b = 2, then left side of Sue's equation = 6, but right side = 18). The first two sentences seem to imply that the fact that multiplication is not distributive over multiplication follows from some general fact that no operation is distributive over itself, and that is just not true; for example, union of sets distributes over itself, as does average of numbers. In addition to these two errors in the answer (not showing the counterexample, and implying that no operation distributes over itself), there is also a problem with the pedagogy. Sue should be shown the correct statement of the associative law for multiplication (which she may be intending her equation to be), as well as the correct statement of the distributive law of multiplication over addition, which is much more likely the thing she is confusing her statement with.

• *book answer to 2-5 #3 (teacher's edition only): The answer is 30, not 32, since the constant functions have 1-element ranges, not range {a,b}. The solutions manual has it wrong, too. This was pointed out in our SEVENTH edition errata -- why didn't they fix it?

• *book answer to 2-5 #12a: The answer is 11, not 7. The answer 7 is only for girls' scores, and that wasn't the question. This mistake was pointed out in our SEVENTH edition errata -- why didn't they fix it?

• *student solutions manual answers in Chapter 3: The font for stating the base in the manual is inconsistent with the text (it is Roman in the text, not italic). This font problem needs to be fixed in the entire chapter (and elsewhere?).

• *book answer to 3-1 #39: This answer is inadequate, since it doesn't respond to "why three" and "why commas".

• *book answer to 3-4 #3: The problem asked for the reasons for the first four steps, not all five steps.

• book answer to 4-3 #10c: This is sometimes true, for example, when the number is 33 or 17. (T iff T is true; F iff F is true.)

• book answer to 4-4 #6e: 10001 is not prime; it is 73 times 137. (Instructor solutions manual is similarly wrong.)

• *student solutions manual answer to 4-5 #3a: "2904" should be "2924" (two occurrences).

• book answer to 4-5 #3c: Comma and space between "122" and "368" need to be deleted.

• *book answer to 5-1 #27: The answer doesn't exactly answer the question as stated (rewrite a/b * c/d as ac/bd).

• *student solutions manual answer to 5-2 #1d: the second denominator is improperly rendered with a coefficient of 6 rather than 2.

• *book answer to 5-4 #34: There is no reason that price should be proportional to area. A lot goes into it besides amount of raw material -- labor, overhead, marketing strategies, etc.

• *student solutions manual answer to 6-2 #7: There is an inappropriate gap before "mi" in the next to last line.

• Exercises 24 and 25 in section 6-2 have blue numbers, but there are no anwers in the back of the book.

• *book answer to 6-2 #46: "7/8" should be "5/8".

• *book answer to 6-3 #22: This answer is incomplete. The idea that repeating 9s lead to finite decimals should also be discussed.

• *book answer to 8-2 #49: The last sentence is silly. Although any particular state would probably not have a fractional age for getting a license, reporting the mean age as 16.3 for all states is perfectly acceptable and conveys some information. The mode is certainly a useful statistic here, but the mean and median are, as well. Finally, it is illiterate to refer to non-whole numbers as "decimals"; that is how ignorant students talk, not how a writer of a mathematics textbook should talk. After all, 16.0 is a decimal.

• student solutions manual answer to 9-1 #3c: "if the four" should be "of the four".

• *book answer and student solutions manual answer to 9-2 #19: First of all, the intersection can also be a line, if the angle is a straight angle, which is a line. (Definition of angle allows this.) Second, the manual's statement that "the line starts at the vertex" makes no sense -- lines don't "start", and the statement makes no sense (you want to say that one of the rays of the angle lies on the line, but the other does not). Third, the manual's "One" and "Two" need to be "One point" and "Two points".

• *book answer to 9-4 #9a: There are a total of 16 faces glued together, not 10.

• book answer in instructor's edition to 9-4 #14b: Figure (i) CAN be used to form a square shadow. It's actually a little ambiguous as to whether the SIZE of the shadow as drawn in the book is relevant; presumably it is not, since otherwise the octahedron would not produce a square shadow that large.

• *book answer and student solutions manual answer to 9-4 #19: For one thing, the book and manual do not agree. Certainly the book is wrong for part (a), since it leaves out pentagon. But I don't think it's anywhere close to this simple. For example, you can get a figure with some curved edges and some straight edges for part (b). Furthermore, the answers are redundant; for example, a square is already included when stating rectangle and parallelogram. If there is little symmetry to the pentagon, it is not at all clear that one can get a square (I think you can't). This needs much further research on the part of the authors.

• book answer to 9-4 #35a: This is just wrong. As is well known but hard to see, the intersection of a cube and a plane can also be a hexagon (and once you see that, it's easy to see that it can also be a pentagon). Look at Exercise 30, for example! If the plane isn't required to actually slice THROUGH the cube, the answer can also be a line segment. This needs further research on the part of the authors.

• book answer to Now Try This 9-1c: There is a "First" dealing with the case of antipodal points, but there is no "Second" dealing with nonantipodal points. Also, in line 2, "which" should be "that".

• *book answer and student solutions manual answer to 10-1 #25: The height and base do not uniquely determine a pyramid up to congruence. You are assuming that it's a right pyramid (apex equally distant from each base vertex).

• book and solutions manuals answer to 10-2 #11: First of all, everything after the first sentence is irrelevant to answering the question. Also, "congruent" in the last line should be "perpendicular".

• *book and student solutions manual answer to 10-2 #15: This is an interesting problem, but the answers are problematic. First of all, there is one missing concave kite -- the one whose EXTERIOR angle is 75 degrees between the two 20cm sticks. One could perhaps argue that that doesn't fit the definition of "with a 75 degree angle", but it should be mentioned (look at your definition of angle). More seriously, two of the four kites look a lot like they might be congruent. It needs to be pointed out in the manual why they aren't. If there is a 75 degree angle between the 30 degree sides, then using the law of cosines we can find the angle between the noncongruent sides to be approximately 76.56 degrees, not 75 degrees.

• book and student solutions manual answer to 10-3 #9: There is a cheaper way. Locate points A and B (free). Draw circles centered at A and at B with radius AB (20 cents), intersecting at C and D. Draw circle centered at C with radius CA (10 cents), intersecting first two circles at E and F, respectively. Draw line CD (10 cents), intersecting third circle at G and H. Then EGFH is a square, which can be drawn with four more segments (40 cents). Total is 80 cents. (Solution due to Laszlo Liptak, Oakland University, October 25, 2007)

• book answer and student solutions manual answer to 10-3 #13: This is inadequate. "Symmetry properties" is not a valid method of proof here as to why these parallelograms are congruent. Furthermore, it's not clear what is meant to be given. Are the dashed segments meant to be extensions of the solid segments? If not, then no conclusions are possible. If so (and this is presumably what is intended), then the answer is incomplete, because if fact ABCD is a rhombus. One way to approach this is to note that angle E equals angle CBD, and angle E equals angle ADB, so BC and AD are parallel. This and the similar thing with the other angles shows that ABCD is a parallelogram, and hence a rhombus because of the given information that AD = DC. Experimentation with Geometer's Sketchpad will convince one that ABCD need not be a square.

• student solutions manual answer to 10-3 #25: "radius" is misspelled in line 3.

• book answers to 10-4 #41: This is a silly answer. The question is surely asking about the 3-d figure that one sees, not its 2-d projection onto the mirror (which doesn't exist anyway -- the mirror is just reflecting light rays). I think the answer is "yes". Furthermore, "arm's length" has nothing to do with it, so the question should be reworded.

• *book answers to 10-4 #42: The answer is #43 is actually the answer to #42, so replace #42 with it.

• *book answers to 10-4 #43: The answer is #44 is actually the answer to #43, so replace #43 with it.

• *book answers to 10-4 #44: The answer is #45 is actually the answer to #45, so replace #44 with it.

• *book answers to 10-4 #45: The actual answer to this is missing (see entry above).

• book answer to 11-1 #22b: It is not clear what "new figure" means here. The implication throughout this exercise seems to be that it is to be obtained by adding squares to the given figure. If so, then the minimum is 10, not 8. To justify the answer of 8, one needs to start from scratch and construct a figure with just one row in it.

• book answer to 11-1 #24: This is silly at best. In mathematics, upper case and lower case letters do NOT stand for the same thing, so the answer is incorrect in any case. Furthermore, just as sin when used to mean the sine of an angle is NOT the product of the three variables s, i, and n, so "Tart" when used as a word is NOT a product of single letters.

• *book answer to 11-2 #18c: The reason this is NECESSARILY false is that the area can never be more than 60 square centimeters, not that the area COULD BE 60 square centimeters. By comparison, the correct answer to (d) is that we don't know whether this is true or false -- it depends on the parallelogram (the answer given in the teacher's edition is correct). In fact, the question itself should be worded more clearly to get at this issue (always true, sometimes true, never true). Furthermore, the statement of the problem is wrong -- you can ask which "is" true, since more than one could be true.

• book answer and student solutions manual solution to 11-3 #11: Comment from Professor Liptak: "The problem asks about planes departing from the same point, one going south 1316 km, the other west 2268 km. Naturally, the solution in the back of the book assumes that we get a right triangle and uses the Pythagorean theorem to get the answer (2622 km). But it is clear that at such distances we can't ignore that Earth is not flat. According to a source on the Web, if the planes start from a point on the equator, they will be about 2608 km apart at the end, while if they start from a point on 45 degrees South latitude, they will be 2395 km apart. Of course, in the most extreme example, when the plane heading south ends up at the South Pole, the planes will be 1316 km apart."

• *student solutions manual solution to 11-5 #19: Typo in first line of displayed math: "92)" should be "(2)".

• *book answer to 12-1 #21a: The second A'B'C' should be A''B''C''. Furthermore, there should be no spaces between the letters that have these prime and double-prime superscripts.

• *student solutions manual solution to 12-3 #3c, line 2: "by by" should be "by".

• *book and student solutions manual answer to 12-5 #3b: This figure does tesselate; it is not hard to find the tesselation.

• *book and student solutions manual answer to 12-5 #3c: This drawing does not provide a tesselation. It is not at all clear how the pattern is to be extended. In particular, the piece on the far right is oddly placed. Instead, just repeat the "lower left to upper right" swatch in parallel upward sloping strips throughout the plane. The book answer also has a stray line segment.