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 Thank you for visiting our site! You landed on this page because you entered a search term similar to this: examples of quadratic equations with fractional exponential.We have an extensive database of resources on examples of quadratic equations with fractional exponential. Below is one of them. If you need further help, please take a look at our software "Algebrator", a software program that can solve any algebra problem you enter!
Publisher:  McDougal-Littell Title: Algebra 1
Reviewers Name: Bishop, Friedman, Kerckhoff, Poon


With regard to mathematics content only, this program does not sufficientlyaddress the content standards and applicable evaluation criteria to berecommended for adoption.

This submission nominally does cover the standards of algebra 1 soasserting that it does not requires explanation, but it is an easy explanation. So much of the important material is in last third of the book, even thelast chapter, with so much review in the first two thirds, that reasonableuse would be almost a duplication of the material of Course 1 and 2 inthe same submitted series, not a course that is fundamentally algebra.  A quick summary of the situation is given by a glance at the first pagesof the same Section 9.1 of Course 2 (P. 449) and of Algebra 1 (P. 499). For all intents and purposes, they are interchangeable.  If it wereonly that page, and the algebra book went deeper faster, there would beno grounds for complaint but that is not the case.  There is far toomuch duplication, redundancy, and review that it should not be consideredadequate; the first quarter of the book is exclusively reviewing the standards of Grade 7 and below.

Although the basis for understanding linear equations and functionsis a Grade 7 standard, the repetition and more thorough treatment of thetopic is universally understood as appropriate in algebra I.  Howwell does this submission do?  At this level, students should be beingforced to hone their skills in writing appropriate equations from wordproblems, cookbook though they may be.  In 70-80 problems per section,there are at most a couple - often none at all - where the student is requiredto independently assign a variable and write an appropriate equation thatmathematically describes the verbal situation.  For example, considerthe 69 problems in Section 3.5, a section entitled "More on Linear Equations".  Of the seven word problems, none requires the student to write his or herown equation that describes the situation mathematically.  Item 40pretends to but, in fact, it is all but given.  In 41, students choosebetween two given choices, and in 43, a choice among four.  Thatsit.  Sections 3.6, has exactly one, Item 43.  Sections 3.7 and3.8 have none.   This is representative of real word problemsthroughout the book.  There are so few that they can be avoided entirelywhile students get "algebra"credit for meeting 7th, 6th, or even 4th or3rd grade standards.  Looking again at Section 3.5, Items 64-69 areall subtracting decimals; e.g., Item 64 is 11.9 - 1.2 .  That is,the last six items of the exercise set address only the Grade 3 standard,NS 2.1.  Jumping far ahead to Section 10.4 (literally randomly chosen),P. 593, Items 75-86 are twelve problems factoring natural numbers of atrivial nature, 12, 20, 18, up to 112.  Thats Grade 4, NS 4.2.

This submission fails in another way, the mathematical coherence impliedin Criteria 11 and 12, that mathematical discussions are to be broughtto closure in the sense that a discussion of a mathematical concept, onceinitiated, should be completed and that formulas and theorems should beproved as appropriate for the grade level, and that reasons should be givenwhen an important proof is not proved.  This submission has a seriousorganizational problem in that concepts are introduced and used, but theneeded development is then presented much later, so it becomes effectivelymoot as to whether or not the standard has, in a real sense, been met. Consider, for example, the quadratic formula.  It is simply givenin a highlighted box in Section 9.6 with a note that it will be derivedlater.  In fact, completing the square and a derivation of  thequadratic formula are delayed beyond the end of the school year for allbut the most advanced students.  Because the explanation, throughthe algebraic process of completing the square, will not be seen by moststudents, this text does not adequately cover this standard (19.0). Understanding the derivation of the quadratic formula is grade-appropriateand an important part of the standard. In contrast, the level of coveragehere is well below that of an algebra text; it has been reduced simplyto the mechanical process of plugging values into a formula which magicallygives the right answers.

A similar situation arises with exponential functions in Chapter 8. Only exponential expressions with integer exponents have any meaning sofar in the book.  No mention is made as to what meaning, if any, mightbe assigned to 2^1.5 (Example 2b of Section 8.3) because fractional exponentsare not dealt with until Chapter 12, beyond the course for most students. Even after rational exponents are introduced, the connection with whathad been done previously in Chapter 8 is never made.  Instead, exponentialfunctions are treated simply as a button on the calculator.  Studentsare told that certain situations are modeled by exponential functions butno reason is given; no reasonable explanation could be given at this stagesince the function itself has been given no meaning.  Students arefurther asked to identify the domain and range simply by looking at thepicture.  Not only is this a below-standard method of dealing withthis standard (17.0), but it is likely to lead to incorrect responses sincethe graph quickly disappears from view.

The presentation of both the quadratic formula and exponential functionssimply as "black boxes" are representative of the texts inadequate attentionto Mathematical Reasoning and logical discussion.  In particular,it fails to meet the Content Criterion 11, which is of critical importance.

Mathematics Content/Alignment with StandardsMathematics Content/Alignmentwith Standards

A systematic review of determinations regarding the criteria in thissection.  Citations of standards not adequately addressed (if any)are of particular importance with regard to Content Criterion 1.

Content Criterion 1.  The content supports teaching the mathematicsstandards at each grade level (as detailed, discussed, and prioritizedin Chapters 2 and 3 of the framework).


As explained in the general criticism above, the critical words hereare "supports teaching the mathematics standards at each grade level [subjectmatter content for algebra and beyond]" not "can somehow be referenced". Under the latter interpretation, the submission would pass, under the former,it does not.

Content Criterion 2.  A checklist of evidence accompanies the submissionand includes page numbers or other references and demonstrates alignmentwith the mathematics content standards and, to the extent possible, theframework.


Content Criterion 3.  Mathematical terms are defined and used appropriately,precisely, and accurately.


There are a few problems with the definition of mathematical terms.For example on page 710 it is not made clear that a^(1/n) is the principalnth root of a if more than one root exists. In fact, it is stated that``because 8^2=64, you know that 64^(1/2)=8''. But it is also true that(-8)^2=64.

On page 588, the text defines what it means for a polynomial to be infactored form, but gives several examples of polynomial equations wherea factored polynomial is equal to zero. This could lead students to confusepolynomials, with polynomial equations.

The algebra text is an improvement over the previous two texts in theseries in that, when concepts or properties are defined and examples given,the examples are clearly labeled as such.  For example, the clearlabeling of the examples of the distributive property on page 101 of thisbook compare favorably with the confusion of example and general statementof the distributive property on page 139 of Course I.

Content Criterion 4.  Concepts and procedures are explained andare accompanied by examples to reinforce the lessons.


Content Criterion 5.  Opportunities for both mental and writtencalculations are provided.



Content Criterion 6.  Many types of problems are provided: thosethat help develop a concept, those that provide practice in learning askill, those that apply previously learned concepts and skills to new situations,those that are mathematically interesting and challenging, and those thatrequire proofs.


As describe in the introduction, the problems are misleadingly trivialor are impossible except as concept-limited calculator exercises.

Content Criterion 7.  Ample practice is provided with both routinecalculations and more involved multi-step procedures in order to fosterthe automatic use of these procedures and to foster the development ofmathematical understanding, which is described in Chapters 1 and 4.


The earlier described problem with algebraically based word problemsis beyond rectifiable.  It is clear that the authors perceive thatthey are hard for students so instead of building on an acceptable basefrom the preceding Courses 1 and 2, they have avoided the problem entirely. That alone would be fatal but it is exacerbated with too many other "problem"situations being delayed too long to be developing anything close to automaticity. Factoring, and its various multi-step uses, and completing the square comequickly to mind.

Content Criterion 8.  Applications of mathematics are given whenappropriate, both within mathematics and to problems arising from dailylife. Applications must not dictate the scope and sequence of the mathematicsprogram and the use of brand names and logos should be avoided. When themathematics is understood, one can teach students how to apply it.



Content Criterion 9.  Selected solved examples and strategies forsolving various classes of problems are provided.


Content Criterion 10.  Materials must be written for individualstudy as well as for classroom instruction and for practice outside theclassroom.


In some places there is not enough explanatory text. For example onpage 568, a monomial in one variable is defined, and examples are given,sqrt(x) and 1/x, of expressions that are not monomials. But it is not explainedwhy these are not monomials, this is something which a student may notunderstand at first reading.

Content Criterion 11.  Mathematical discussions are brought toclosure. Discussion of a mathematical concept, once initiated, should becompleted.


Exponential functions and quadratic equations were discussed in theinitial summary.

Content Criterion 12.  All formulas and theorems appropriate forthe grade level should be proved, and reasons should be given when an importantproof is not proved.


  An example is the treatment of radical expressions in Section5.3.  Sqrt(ab) = sqrt(a) sqrt(b) credited to the "Product Propertyof Radicals", with no justification nor explanation, let alone proof. It would be easy enough to lead students through a proof, just square theright side and use properties of multiplication and the definition of squareroot given at the beginning of the chapter, but the idea of proof is underplayedquite consistently throughout the book.   One could assume thatthe authors avoided the proof since fractional exponents are not yet takenup and this is only a special case of that more general property of exponentsbut it is the wrong approach. It is reasonable to "do" square roots moredeliberately first, in part as motivation for the more general setting. Showing how the proof of new situations can be based on known properties,as in this case, is an important part of that developmental process.

Another example is Section 10.3 pgs 581-587 regarding special products.The proofs of these are well within the capacity of students at this level,since they only depend on basic properties of numbers. This opportunityto provide proofs is missed except for the use of an area model to justifyone of the special products on page 583.

Content Criterion 13.  Topics cover broad levels of difficulty.Materials must address mathematical content from the standards well beyonda minimal level of competence.


As commented at the beginning, the student involvement in appropriateuse of the ideas developed is below adequate, much less beyond.

Content Criterion 14.  Attention and emphasis differ across thestandards in accordance with (1) the emphasis given to standards in Chapter3; and (2) the inherent complexity and difficulty of a given standard.


The guidance of Mathematics Framework Chapter 3 is not followed, particularlyin regard to quadratic functions.  Factoring is not taken up untilChapter 10 (and the word "factor" is not explained, although "term" was,much earlier) so its importance in solving quadratic equations of one variableis delayed and its importance in identifying the zeros of a quadratic functionare postponed.  For many students, its importance will be lost entirely. Just plug numbers into the quadratic formula or let the graphing calculatordo it all.  The fact that this produces an adequate approximationis irrelevant when judged against the first paragraph of the quadraticfunction portion of the Algebra 1 section of Chapter 3.

Content Criterion 15.  Optional activities, advanced problems,discretionary activities, enrichment activities, and supplemental activitiesor examples are clearly identified and are easily accessible to teachersand students alike.


Content Criterion 16.  A substantial majority of the material relatesdirectly to the mathematics standards for each grade level, although standardsfrom earlier grades may be reinforced. The foundation for the mastery oflater standards should be built at each grade level.


Too much is below grade level.

Content Criterion 17.  An overwhelming majority of the submissionis devoted directly to mathematics. Extraneous topics that are not tiedto meeting or exceeding the standards, or to the goals of the framework,are kept to a minimum; and extraneous material is not in conflict withthe standards. Any non-mathematical content must be clearly relevant tomathematics. Mathematical content can include applications, worked problems,problem sets, and line drawings that represent and clarify the processof abstraction.


But as with the other books of the series, the books are too busy. Too many colors, photos and other visual distractions clutter them, distractingfrom the mathematics.
 The emphasis on multiple choice test practice is also misplaced.This is a mathematics text not a test-taking text. Test tips should notbe part of this book. Many of these tips are very general in nature andhave nothing at all to do with mathematics such as ``Start to work as soonas the testing time begins. Keep working and stay focused on the test''on pg 194.  Other examples are on pages 60, 264, 463, 752.


Content Criterion 18.  Factually accurate material is provided.


Content Criterion 19.  Principles of instruction are reflectiveof current and confirmed research.

The CRP members generally agreed that they would not comment on thiscriterion.

Content Criterion 20.  Materials drawn from other subject-matterareas are scholarly and accurate in relation to that other subject-matterarea. For example, if a mathematics program includes an example relatedto science, the scientific references must be scholarly and accurate.

Content Criterion 21.  Regular opportunities are provided for studentsto demonstrate mathematical reasoning. Such demonstrations may take a varietyof forms, but they should always focus on logical reasoning, such as showingsteps in calculations or giving oral and written explanations of how tosolve a particular problem.


It pretends to offer this but it falls far short.  The text offersproblems throughout which are labeled logical reasoning. Some of theseare good, but too often they do not require sufficient thought. Some ofthe logical reasoning problems only ask the students to make a conjecturebased on a number of examples without requiring the student to explainhis/her reasoning. See page 92, for example, problems #9, 10, 11. Other ``logical reasoning'' problems only ask the student to fill in ablank, again without explaining their reasoning, see for example #50, 51,52, -53 on page 334, or #28, 29l, 30, -31 on page 358.
Other problems do ask the student to explain his/her reasoning or justifysteps in a computation for example #61 on page 137, #43 on page 148, #137on page 161, #24 on page 176, #5, 6, 7 on page 415. More of this type ofproblem would strengthen the text.

The last section of the book which is about proof, has a very few problemsrequiring a student to "use properties of numbers to construct simple,valid arguments for ... claimed assertions", only #10, 11, 20, 21 on pages744-745 and #43 on page 750. Given the difficulty and importance of proofin mathematics more practice should be provided. Also all the problemsin this section tell the student what needs to be done, i.e., find a counter-example,construct an indirect proof etc. There should be some problems of the proveor disprove type.

Content Criterion 22.  Homework assignments are provided beyondgrade three (they are optional prior to grade three).


Additional Comments and Citations.

Corrections and Edits.