DOCUMENT RESUME

ED 093 704 SE 018 078

AUTHOR Crawford, GlendaTITLE Algebra 2s, Mathematics (Experimental); 5216.24.INSTITUTION Dade County Public Schools, Miami, Fla.PUB DATE 72NOTE 22p.; An Authorized Course of Instruction for the

Quinmester Program. Related documents are ED 084 161and 162 and SE 018 079

EDRS PRICE Mr-$0.75 HC-$1.50 PLUS POSTAGEDESCRIPTORS *Algebra; Behavioral Objectives; *Curriculum; Graphs;

Instruction; Mathematics; Mathematics Education;Matrices; *Objectives; *Secondary School Mathematics;*Teaching Guides; Tests

IDENTIFIERS Complex Numbers; *Quinmester Program

ABSTRACTThe fourth in a series of six guidebooks on minimum

course content for second-year algebra, this booklet covers linearand quadratic relations, absolute value, graphing complex numbers,determinants and matrices, graphing quadratic relations, and solvingsystems of linear and quadratic equations. Overall course goals arespecified, a course outline is provided, performance objectives arelisted, and text references keyed to the performance objectives areprovided. A sample posttest is included along with a 13-itembibliography. (JP)

U.S DEPARTMENT OF HEALTH,EDUCATION & WELFARENATIONAL INSTITUTE OF

EDUCATION[Ills DOCuMENI HAS KEEN REPROOUCED E :CAC It Y AS RECEIVE. I) ERON'tTHE PERSON OR ORDANIZA TION ORIGINA TING IT POINTS 01- VIEW OR OPINIONSSTATED DO NOT

NECESSARILY REPRESENT 041 ICIAL NATIONALINSTITUTE OFEDUCATION POSITION OR POLICY.

AUTHORIZED COURSE OF INSTRUCTION FOR THE

MATHEMATICS I Algebra 2S 5216.24

DIVISION OF INSTRUCTION1913

QUINMESTER MATHEMATICS

COURSE OF STUDY

FOR

ALGEBRA 2S

5216,24

(EXPERIMENTAL)

Written by

Glenda Crawford

for the

DIVISION OF INSTRUCTIONDade County Public Schools

Miami, Florida 331321971-72

DADE COUNTY SCHOOL BOARD

Mr. G. Holmes Braddock, ChairmanMr. William Turner, Vice Chairman

Mrs. Ethel BeckhamMrs. Phyllis MillerDoctor Ben SheppardMr. Alfredo Duran

Dr. E. L. Whigham, Superintendent of SchoolsDade County Public Schools

Miami, Florida 33132

Published by the Dade County School BoardMiami, Florida 33132

PREFACE

The following course of study has been designed toset a glipimum standard for studerit performance afterexposure to the material described and to specifysources which can be the basis for the planning ofdaily activities by the teacher. There has been noattempt to prescribe teaching strategies; thosestrategies listed are merely suggestions which haveproved successful at some time for some class.

The course sequence is suggested as a guide; anindividual teacher should feel free to rearrange thesequence whenever other alternatives seem more de-sirable. Since the course content represents aminimum, a teacher should feel free to add to thecontent specified.

Any comments and/or suggestions which will help toimprove the existing curriculum will be appreciated.Please direct your remarks to the Consultant forMathematics.

All courses of study have been edited by a subcommitteeof the Mathematics Advisory Committee.

ti

1

CATALOGUE DESCRIPTION

Further work with linear and quadratic relations. Includesabsolute value and graphing of complex numbers, determinantsand matrices, graphing quadratic relations, and solvingsystems of linear and quadratic systems,

Designed for the student who has mastered the skills and con-cepts of Algebra 2p,

TABLE OF CONTENTS

Page

Goals 3

Text Bibliography 3

Course Outline 4

Objectives-Strategies 61160 66 6

References 8

Sample Posttest Items . 0006 OOOOO 10

Bibliography 18

2

OVERALL GOALS

The student wills

1. Achieve competence in the basic arithmetic skills, gainunderstandings requisite for solving computational problems,and use the properties of mathematical structure.

2. Develop reading skills used in mathematics.

3. Develop the ability to define, categorize, analyze, evaluate,interpret, and communicate through symbolic mathematical ex-pressions in problem-solving situations.

4. Appreciate the significant role of mathematics in the develop-ment of civilization in the past, present, and future, andbecome more aware of the ever increasing dependence that manhas upon mathematics for his future development.

5. Develop both inductive and deductive reasoning in a mathema-tical context, with emphasis placed on their application tomathematical proofs and life situations.

Notes The above overall goals are from Florida Standards1971-72,

6. Develop those comprehensions and skills in the language ofmathematics which will allow for further study in mathematicsand science.

TEXT BIBLIOGRAPHY(*State-adopted)

D3- Dolciani, Mary P.; Berman, Simon L.; and Wooten, William.Podern Algebra and Trigonorvtri, Book 2. Boston: HoughtonMifflin Company, 1963.

*D8- Dolciani, Mary; Wooten, William; Beckenback, Edwin; Sharron,

Sidney, Modern School Mathematics Algebra II and Tri:o 0-metry. Boston: Houghton Mifflin Company, 19

N - Nichols, Eugene D.; Heimer, Ralph T.; Garland, Henry C.Modern Intermediate Algebra. New Yorks Holt, Rinehart andWinston, Inc., 1965.

*PL - Payne9 Joseph N.; Zamboni, Floyd F.; and Lankford, FrancisG., Jr, Alebra Two with TrigonometrI. New York: Harcourt,Brace and World, Inc., 1969.

*PA - Pearson, Helen R. and Allen, Frank B. Modern_102Logical Approach, Book Two. Boston; Ginn and Company, 19o::.

3

COURSE OUTLINE

RelatedObjectives I. Complex numbers

1,2 A. Graphing

B. Absolute value

3 II. Solution of systems of linear equations intwo variables

A. Substitution

B. Addition

C. Matrix

D. Determinant

4,5 III. Quadratic relations

A. Identify and graph

1. Circle

2. Ellipse

3. Parabola

4, Hyperbola

B. Determine the equation from a graph

1. Circle

2. Ellipse

3. Parabola

4. Hyperbola

4

Course Outline (continued)

RelatedObjectives

6,7 IV. Quadratic inequalities

A. In one variable

1. Solve

2, Graph

B, In two variables

1. Graph

8,9 V, Systems of equations involving quadratics

A, Solve by graphing

B. Solve algebraically

10 VI. Word Problems

5

REFERENCES

OBJECTIVE PL PA D8 D3 N

6101 172 615 585 398 251

432 176 610 585 45o 273

24-r3 299 652 205 97 298

240294 648 204 97 300

327 654 558 --655

341 657 ....... 566 307

4 418 442 300 322421 688 449 306 322425 337 445 227 322427 691 453 309 322

5 444 ........, - -- - -- ''. .".

437

444 - -- '' ''' ...

437

444 - -- Iwo we MN a ma. ow. ... my rA.

437

444 ....... -- ... '......".

437

6 276 374 363 297 240

7 277 374 444 236 326442 448 302

452 306458 309

316319

6

References (continued)

OBJECTIVES PL PA D8

D3 N

8 438 698 464 319 322703

9 438 699 465 321 314703 468 324 321

10 302 658 213 102 )02306 702 453 306 317308 467 316 323310 470 322433 325

7

.PERFORMANCE OBJECTIVES

The student wills

1. Graph complex numbers.

Assume students know how to graph (x,y) on coordinateaxis. Teach graphing of a + bi - Explain orderedpair (alb) can represent a + bi. Practice graphingpoints.

2. Define absolute value of complex numbers.Assume students can define the absolute value of realnumbers. Define the absolute value of complex numbers,Show this is consistent with definition of the abso-lute value of reals.

3. Solve a system of linear equations in two variables by:a, substitutionb. elimination by addition-subtractionc. matricesd. determinants

(a) substitution method of solving a system oflinear equations

(b) addition method of solving a system of linearequations--stress equivalent systems and familyof lines

(c) define a matrix--stress the three operations touse when solving a system of linear equations bymatrices. These operations are;(1) multiplication of all elements of a row by

the same number.(2) addition of the same multiple of the elements

of one row to corresponding elements of anotherrow.

(3) interchange rows. Notes Can easil beextended to a system of 3 and 4 linear equatioul;.

(d) have students solve for x and y.

alx + bly = cl

a2x + b2y = c2

4. Identify and graph quadratic relations including the circic.,ellipse, parabola, and hyperbola.Practice graphing as

(a) circle(b) ellipse(c) parabola(d) hyperbola

when the equation is in standard form.

8

Performance Objectives (continued)

5. Determine the equation of the relation given the graphof a circle, ellipse, parabola and hyperbola.Sketch a circle, ellipse, parabola and hyperbola on theboard and have students give the equation in standardform. After examples of this type, move from thegeneral case to specific eases,

(a) circle(b) ellipse(c) parabola(d) hyperbola

6. Solve quadratic inequalities in one variable and graphthe solution set.In explaining the method for solving quadratic in-equalities in one variable and the graph of the solutionset, Payne has the best example and stresses union andintersection.

7. Graph a. quadratic inequality in two variables.Graph examples letting students guess what area to shade.Show how to check the guess in order to determine whereshading should occur.

8. Solve a system of equations involving quadratics graphi-cally (linear and quadratic as well as quadratics).In solving quadratics graphically, it is best to chooseproblems which have small integral roots.

9. Solve a system of equations algebraically involvingquadratics (linear and quadratic), (2 quadratics).Sketch possibilities of line and curve for possiblenumber of points of intersection. Sketch possibility orcurve and curve for number of points of intersection.In working examples, be sure to choose one that gives ananswer which does,not satisfy the original equation.

10. Write and solve mathematical models for word problemswhich can be solved by the algebraic skills developedin this quin.Require many examples for all sections.

9

SAMPLE POSTTEST ITEMS

1. Graph each of the following on the complex plane.

a, (213) d. 4 + 6i

b. -6 e, 0

c. 4i

2, Find the absolute value of (-5 + 12i).

3. a. Solve by substitution

il5x +y = 7

3x + 2y = 0

b. Solve by addition

fx - 4y = 2

2x - 5y = 1

c. Solve by matrices

fx + y = 6

2x2 x - y = 3

d. Solve by using the ratio of determinants

+ y = 6

x - y = 4

4. Graph the following equations

a. x2

+ y2

= 25

b. x2 + 4y2

= 36

c, 2)2 +3

d,2

x2

- 116 9

10

5. Write the equations of the following relations.

( -6,-

( -3, -2)

(-21-3( -6)

6. Find the solution set and graph the solution on a number linc.

7. Graphs

x2 - x 6 >0

((x,y) I Y c (x-a)2 + 3.1

8. Find the solution set graphically.

2+ Y2 .--' 25

Ir.

a. x2 t y

2= 4

x = -211

2 e

9. Find the solution set al2ebraically.,

x2

+ y2 = 13 x2 y2 = 25a, b,

2x y = 4 x + y2

= 5

10, a. The square of the first number is equal to the secondnumber. The sum of the two numbers is 2, Find thenumber.

b, The difference of the squares of two positive numbersis 33. The sum of their squares is 65, Find thenumber.

12

1.

1

(-t+61)

\I (-5)2 + (12)2(o+4i)

2+ i V 25 + 144

(-6+01) (o+oi) 4 169

AmOta0n0 Mr% 7-%ne.mmnc,mitilhaVYKiPAD ruallzQl

2.

3. a. 5x +y =73x + 2y = 0

b.

13

x 4y = 22x - 5y =

Y =

3x + 2(7 -

3x + 14 -

7

5x)

10x

5x

=0

=0

2x - 8y =-2x + Liy =

-3Y =

41

3

-7x = -14

x = 2

y = -1

x - 4(-1) 2

x 4 = 2

5(2) y = x = -2

y = 7 - 10 -2 -4(-1) =

y = -3 -2 +4 = 2

5(2) + (-3) = 72(-2) - 5( -1.)

10 - 3 = -4 + ti

3(2) + 2(-3) = 0

6 - 6 = 0

13

3 0.

R 2+R1° 2

x2x

(12

y =y =

1- 1

63

6

d. 2xx

X

+ y- y

6 1

= 6= 4

4R2.-2, -20( -2 -12)

3 -3 - 9 2 1

R2°2+111 -2 -2 -12 1 -10

(1 3

)

x = _

R1.-_21 (-2 -6 -13

x10

03)

Y = 2 6

x = 3 1 4

= 3 -j

8 - 6-3

2-3

211-U3

6+3

=

14

20 2 .... 65 3

18 = 63

1:113 3

4, a, x2 + y2 = 25

b. x2 + 4y2 = 36x2 2

+ 91

d.

c. Y = x-2)2 + 3

15

x29

1

5. a. (x + 2)2 + (y - 3)2 = 25

2 2b. 25 9

6.

7.

C. y = x2 - 4

d. xy =6

3

8. a. C(-2,0))

16

b. (01-.5)

9. a, 3$2)

b. Q.50) (-413) (-4,-3)

10. a. -2 and 4

1 and 1

b. 7 and 4

17

BIBLIOGRAPHY

1. Courant, Richard and Robbins, Herbert. What is Mathematics?New York' Oxford University Press, 1941.

2. Dolciani, Mary P.; Berman, Simon L.; and Wooten, William.Modern Algebra and Trigonometry, Book 2. Boston: HoughtonMifflin Company, 1963.

3. Dolciani, Mary P.; Wooten, William; Beckenbach, Edwin F.;and Sharron, Sidney. Modern School Mathematics. Algebra2 and Trigonometry. Boston' Houghton Mifflin Company, 1968.

4. Fitzgerald, William M.; Dalton, Leroy C..; Brunner, Vincent F.;and Zetterberg, Jack P. Algebra 2 and Trigonometry_ Theorand Application. River Forest, Illinois: Laidlaw BrothersPublishers, 19 8.

5. Gibb, G.; Jones, P.; and Junge, C. The Twenty-Fourth Year-book of The National Council of Teachers of Mathematics.Washington, D.C., 1959.

6. Haag, Vincent H. Structure of Elementary Algebra. Studies inMathematics, Volumn III. New Haven, Conn.: Yale UniversityPress, 1961.

7. Hooper, Alfred. Makers of Mathematics. New York: Random house,Inc., 1948.

8. James, Glenn. The Tree of Mathematics. Pasadena, California.The Digest Press, 1957.

9. Johnson, Richard E.; Lendsey, Lona Lee; ;;Lesnick, William E,;and Bates, Grace E. Algebra and Trigonometry. Menlo Park,California: Addison-Wesley Publishing Company, 1967.

10. Kasner, Edward and Newman, James. Mathematics and theImagination. New York: Simon and Schuster, Inc., 1940.

11. Kemeny, J.G.; Snell, J.L.; and Thompson, G.L. Introduction toFinite Mathematics. Englewood Cliffs, N.J.: Prentice-Hall,Inc., 1957.

Kraitchik, Maurice. Mathematical Recreations. New York: W.W.Norton and Company, Inc., 1942.

13. Nichols, Eugene D.; Reimer, Ralph T.; and Garland, Henry E.Modern Intermediate Al ebra. New York: Holt, Rinehart andWinston, Inc., 19 56 .

18

1. Payne, Joseph N.; Zamboni, Floyd F.; and Lankford, FrancisG., Jr. Algebra Two with Trigonometry. New York; Harcourt,Brace and World, Inc., 1969.

15. Pearson, Helen R. and Allen, Frank B. Modern Al&rebru_Acal A.rach includin Trigonometry. Boston; Ginn

and CoMpany, 19

16, Polya, G. How to Solve It. 2nd ed. Garden City, N.Y.;Doubleday and Company, 1957.

19