MODULE ONE. This module addresses the foundational concepts and skills that support all of the Elementary Algebra academic standards.

Transcription

1 Mdule Fundatinal Tpics MODULE ONE This mdule addresses the fundatinal cncepts and skills that supprt all f the Elementary Algebra academic standards. SC Academic Elementary Algebra Indicatrs included in this mdule are: EA-2. Exemplify elements f the real number system (including integers, ratinal numbers and irratinal numbers). EA-2.5 Carry ut a prcedure using the prperties f real numbers (including cmmutative, assciative, and distributive) t simplify expressins. EA-2.6 Carry ut a prcedure t evaluate an expressin by substituting a value fr the variable. EA-2.7 Carry ut a prcedure (including additin, subtractin, multiplicatin, and divisin by a mnmial) t simplify plynmial expressins. EA-2.9 Carry ut a prcedure t perfrm peratins with matrices (including additin, subtractin, and scalar multiplicatin). EA-2. Represent applied prblems by using matrices. The resurces prvided in this mdule are nt all inclusive. They are prvided t begin t build the cnceptual fundatin students need. Additinal resurces will be required t develp the cncepts. Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

2 Mdule Fundatinal Tpics The fllwing Frayer mdel strategy can be used during Lessn #2 as a strategy t help students develp a cnceptual understanding f the real number system and as a tl fr keeping track f their learning. This strategy can be adapted t use with ther Elementary Algebra Lessns. What is a Frayer Mdel? A Frayer Mdel is a cncept map that can help students develp an understanding f tpics. This strategy prvides students with an pprtunity t develp bth examples and nnexamples f a cncept, helping them further refine their understanding f that cncept. Example: Belw is a template that might be used in Elementary Algebra Lessns. Als, an example f hw this template might be used has been inserted. In place f cpying this template, students may als create their wn Frayer Mdels. See: Definitin in yur wn wrds Frayer Mdel Facts/characteristics Examples Cncept Nnexamples 2 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

3 Mdule Fundatinal Tpics Definitin in yur wn wrds Frayer Mdel Facts/characteristics A number that can be expressed as the qutient f tw integers x/y, y. Any number that can be written as a fractin Any decimal that is either repeating r terminates Examples -3 -¼ Ratinal Numbers Nnexamples NOTE: The respnses in the abve Frayer Mdel are nt all inclusive. They are prvided t give an example f pssible student respnses. 3 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

4 Mdule Fundatinal Tpics Lessn # Tpic: Writing and Evaluating Algebra Expressins Standard (s): EA 2.6 I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills related t this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 6 th grade students apply rder f peratins t simplify whle-number expressins (6-3.2). Students write variable expressins t represent quantities (6-3.3). In Elementary Algebra, students use substitutin t find a numerical value fr an expressin (evaluate). Student understanding shuld exceed rte peratinal prficiency. This essential skill is necessary in all subsequent study f mathematics. Taxnmy 3 -C Cgnitive Prcess Dimensin: Apply Knwledge Dimensin: Prcedural Knwledge Key Cncepts Evaluatin Simplificatin Substitutin II. Teaching the Lessn Students have been writing algebraic expressins since 6 th grade but may need review their cnceptual understanding f variables and uses. They have als applied t rder f peratins t simplify expressin and may need a review f this cncept. In this lessn, students becme fluent in evaluating algebraic expressins (EA-2.6) which an essential skill which is used t check slutins t linear equatins, t determine pints n a line and t write linear equatins. Essential Learning and Understanding It is essential fr students t d the fllwing fr the attainment f this indicatr: Understand hw variables are used t represent numerical quantities. 4 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

5 Mdule Fundatinal Tpics Substitute values fr ne r mre variables. Evaluate algebraic expressins fr specified real numerical values. Evaluate algebraic expressins that may invlve square rts and/r expnents Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that students shuld be able t successfully cmplete. Evaluate x + y fr x =.3, y = 2/5 Evaluate x 2 fr x = 3 Evaluate 2 xy z y 3 fr x = 5, y = 3, and z = -2 Nn-Essential Learning and Understanding It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra: Substitute and evaluate expressins that are nt related t the indicatrs f Elementary Algebra. Substitute and evaluate expressins invlving imaginary numbers. Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra. Evaluate sin x fr x = /2 Evaluate lg x fr x = Miscnceptins/Cmmn Errrs Students may nt recgnize xy, (x)(y), and x y, as x multiplied by y. Students may misapply the Order f Operatins. Technlgy Students may use technlgy fr cmplex cmputatin. Students may use the STO n TI graphing calculatrs t evaluate expressins. III. Assessing the Lessn Assessment Guidelines: The bjective f this indicatr is t carry ut a prcedure t evaluate an expressin by substituting values fr 5 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

6 Mdule Fundatinal Tpics variables. Therefre, the primary fcus f the assessment shuld be fr students t carry ut such prcedures. Assessment Item Examples: If a = 3 and b = 2, what is the value f the expressin A. B. C. D ? The area f a circle is given by the frmula A = r 2. Find A when r = 3. A. 3 B. 6 C. 9 D. 36 The frmula fr finding the perimeter f a square is P = 4s. Find P when s = 5. A. 9 B. 4 C. 2 D. 6 6 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

7 Mdule Fundatinal Tpics Lessn # 2 Tpic: The Real Number Line and Operatins with Integers Standard (s): EA 2. I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills related t this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 6 th grade, students applied rder f peratins (6-3.2). In 8 th grade students cmpare ratinal and irratinal numbers by using the symbls,, <, >, and = (8-2.4). Als, students apply an algrithm t add, subtract, multiply, and divide integers (8-2.). In Elementary Algebra students find examples f the fllwing subsets f the real numbers: integers, ratinal numbers, and irratinal numbers. In Intermediate Algebra, students build n their knwledge f the real number system by studying cmplex numbers. Intermediate Algebra students carry ut a prcedure t simplify expressins invlving pwers f i (IA-3.). In additin, Intermediate Algebra students carry ut a prcedure t perfrm peratins with cmplex numbers (including additin, subtractin, multiplicatin, and divisin) (IA-3.2). Taxnmy 2 -B Cgnitive Prcess Dimensin: Understand Knwledge Dimensin: Cnceptual Knwledge Key Cncepts Real Numbers Integers Ratinal Numbers Irratinal Numbers II. Teaching the Lessn In this lessn, students nt nly fcus n the fur arithmetic peratins but als transfer their understanding f the whle number system t integers, ratinal and irratinal numbers. The real number line is an example tl fr demnstrating the relatinship amng the types f numbers. Students perfrmed peratins with integers in 8 th grade but may require a shrt review f this cncept. Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2 7

8 Mdule Fundatinal Tpics Essential Learning and Understanding It is essential fr students t d the fllwing fr the attainment f this indicatr: Give examples f integers, ratinal numbers, and irratinal numbers. Understand hw integers, ratinal numbers, and irratinal numbers are interrelated. Classify numbers as integers, ratinal numbers, r irratinal numbers. Perfrm the fur arithmetic peratins with integers Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that student shuld able t successfully cmplete. Identifying 5 is an integer and a ratinal number. Prvide an example f a number that is an integer and a ratinal number, such as -7. Identify. 66 as a ratinal number. Give an example f a ratinal number such as - ⅜. Identify.25 as a ratinal number. Prvide an example f an irratinal number, such as (des nt repeat in a pattern r terminate) r. Identify 2 as an irratinal number. Explain why 7 is irratinal. (A frmal prf is nt required). 5 + (-5) Nn-Essential Learning and Understanding It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra: Knw additinal subsets f the real number system (whle numbers and cunting numbers). Study imaginary numbers. Study cmplex numbers. Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra. Identify 7 as a cunting number. 8 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

9 Mdule Fundatinal Tpics Knw that infinitely many irratinal numbers are between and n the number line. Give an example f a cmplex number, such as 5 + 3i. Miscnceptins/Cmmn Errrs Students may nt identify integers as a subset f the ratinal numbers. Therefre, students may fail t classify integers as ratinal numbers. Technlgy Students may use technlgy t identify the existence f a pattern when calculating a qutient. III. Assessing the Lessn Assessment Guidelines: The bjective f this indicatr is t exemplify elements f the real number system. Therefre, the primary fcus f the assessment shuld be fr students t give examples f integers, ratinal numbers, and irratinal numbers. Because this is cnceptual knwledge, assessments shuld test the student s ability t apply this cncept t any integer, ratinal number, r irratinal number, nt t be restricted t memrized examples. Students shuld understand the interrelatinships amng integers, ratinal numbers, and irratinal numbers. In additin t exemplify, students shuld be able t: Classify numbers as integers, ratinal numbers, r irratinal numbers. Assessment Item Examples (-5)(-3) (-3) -5 (-3) Which number is nt ratinal? A. 4 B..3 C. 3 D. 6.7 Which number is ratinal? A B C. 2 D. Nn-repeating, nn-terminating decimals are always 9 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

10 Mdule Fundatinal Tpics A. Irratinal B. Ratinal C. Natural D. Integers The number -4 is which f the fllwing? A. An irratinal number B. A natural number C. An integer D. A psitive number IV. Resurces See Frayer mdel example n pages and 2 f this mdule. Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

11 Mdule Fundatinal Tpics Lessn # 3 Tpic: Cmbining like terms Standards (s): EA 2.7 (additin and subtractin) I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills needed t meet this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 8 th grade students apply rder f peratins t simplify whle-number expressins (6-3.2). Apply an algrithm t add, subtract, multiply, and divide integers (8-2.). Understand equivalent symblic expressins as distinct symblic frms that represent the same relatinship (6-.4, 7-.4, and 8-.4). Use cmmutative, assciative, and distributive prperties t examine the equivalence f a variety f algebraic expressins (8-3.3). In Elementary Algebra students carry ut a prcedure (including additin, subtractin) t simplify plynmial expressins. NOTE: Plynmial multiplicatin will be addressed in Lessn #5 f Mdule and divisin by a mnmial will be addressed in Mdule Eight. In Intermediate Algebra, students build n their knwledge f the real number system by studying cmplex numbers. Intermediate Algebra students carry ut a prcedure t perfrm peratins (including multiplicatin, expnentiatin, and divisin) with plynmial expressins (IA-4.). Als, students carry ut prcedures t perfrm peratins n plynmial functins (including f(x) + g(x), f(x) g(x), f(x) g(x), and f(x)/g(x)) (IA-2.5). Taxnmy 3 -C Cgnitive Prcess Dimensin: Apply Knwledge Dimensin: Prcedural Knwledge Key Cncepts Expressins Terms Mnmial Plynmial Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

12 Mdule Fundatinal Tpics II. Teaching the Lessn In this lessn students becme fluent in prcedures t simplify plynmial expressin using additin, subtractin. NOTE: Multiplicatin and divisin by a mnmial will be addressed in Mdule Eight f Elementary Algebra. Students use their knwledge f peratins with integers, real number prperties and rder f peratins t simplify expressins. Essential Learning and Understanding It is essential fr students t understand the fllwing fr the attainment f this indicatr: Real number prperties Adding, subtracting plynmial expressins Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that students shuld able t successfully cmplete. (x + 5) + (2x + 3) (2x 3 + 5x 2 3) + (-4x 3 + 9x 2 ) (3x 4 8x 3 + 2x 2 3) + (6x 3 2x 2 + 5) (x + 5) (2x + 3) (2x 3 + 5x 2 3) (-4x 3 + 9x 2 ) (3x 4 8x 3 + 2x 2 3) (6x 3 2x 2 + 5) NOTE: Plynmial multiplicatin will be addressed in Lessn #5 f Mdule and divisin by a mnmial will be addressed in Mdule Eight. Nn-Essential Learning and Understanding It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra: Multiplicatin and divisin by a mnmial d nt need t be addressed in this mdule. Plynmial multiplicatin will be addressed in Lessn #5 f this Mdule and divisin by a mnmial will be addressed in Mdule Eight. Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra. x 2 (2x 2 3x 9) 3(x + 8) 2(x 2 4x 2) 2 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

13 Mdule Fundatinal Tpics 2x(x 3 2x - 3) + x 2 (-3x 2 + 5x ) 2x 2 4x x 3 + 2x 2 5x x Miscnceptins/Cmmn Errrs When subtracting tw plynmials students may frget t distribute the negative thus subtracting nly the first term f the subtrahend rather than the entire plynmial prvided that the plynmial has mre than ne term. (minuend subtrahend = difference) Technlgy Nte Students may use cmputer algebra system technlgy, which is capable f perfrming symblic manipulatins, t verify slutins. III. Assessing the Lessn Assessment Guidelines The bjective f this indicatr is t carry ut a prcedure (including additin and subtractin) t simplify plynmial expressins. Therefre, the primary fcus f the assessment shuld be fr students t carry ut such prcedures. Assessment Item Examples What is (x 2 + 3x + 4) (3x 2 +x )? A. 4x 2 + 4x +3 B. -2x 2 +2x +5 C. -2x 2 +4x +3 D. 4x 2 +2x +3 What is (x 4 3x 2 +7) + (2x 3 )? A. x 4 + 2x 3 3x B. 3x 7 3x C. x 4 +2x 3 3x D. 3x 7 3x Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

14 Mdule Fundatinal Tpics IV. Resurces Activity: Students may have difficulty determining which terms are alike. Using different clr markers t identify like terms r writing the terms n different clrs squares may help. Fr example, all the x 2 terms can be n red squares, the cnstants n blue squares and the x terms n white squares. Student srt them by clr and then discuss cmmn characteristics (ther than clr). Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2 4

15 Mdule Fundatinal Tpics Lessn # 4 Tpic: Real Number Prperties (Including Identity, Cmmutative and Assciative) Standards (s): EA 2.5 I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills related t this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 8 th grade students use cmmutative, assciative, and distributive prperties t examine the equivalence f a variety f algebraic expressins (8-3.3). In Elementary Algebra students carry ut a prcedure using the prperties f real numbers (including identity, cmmutative and assciative) t simplify expressins (EA-2.5). The distributive prperty will be addressed in Lessn #5 f this mdule. These essential skills are necessary in all subsequent study f mathematics. Taxnmy 3 -C Cgnitive Prcess Dimensin: Apply Knwledge Dimensin: Prcedural Knwledge Key Cncepts Prperties f real numbers Cmmutative Prperty Assciative Prperty Expressins Terms II. Teaching the Lessn In this lessn, students gain an understanding f cmmutative and assciative prperties by applying them t simplify algebraic expressins. Students als gain a deeper cnceptual understanding f the cncept f equivalency. Students used these prperties, in the previus lessn, t address the additin and subtractin prtins f indicatr EA 2.7 (carry ut additin and subtractin t simplify plynmial expressins). In Lessn #5, students will apply the distributive prperty t multiply plynmial Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2 5

16 Mdule Fundatinal Tpics expressins. In Mdule Eight, students will carry ut a prcedure t divide a plynmial by a mnmial (the divisin prtin f EA-2.7). Essential Learning and Understanding It is essential fr students t d the fllwing fr the attainment f this indicatr: Use the fllwing prperties f the real number system t simplify expressins: Cmmutative and Assciate Prperties. Identity Prperty f Additin a + = a and + a = a Identity Prperty f Multiplicatin a = a and a = a Cmmutative Prperty f Additin a + b = b + a Cmmutative Prperty f Multiplicatin ab = ba Assciative Prperty f Additin (a + b) + c = a + (b + c) Assciative Prperty f Multiplicatin (ab)c = a(bc) Simplify algebraic expressins, including numerical expressins. Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that students shuld be able t successfully cmplete. Write an equivalent expressin: x + (y + 2) = a(bc) = y + z = ts = a = + a = Nn-Essential Learning and Understanding It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra: Identify which prperty is used t justify equivalent expressins. Demnstrate knwledge f additinal prperties f the real number system beynd the Assciative, Cmmutative, and Identity Prperty. The Distributive Prperty will be addressed in Lessn #5 f Mdule One. 6 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

17 Mdule Fundatinal Tpics Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra. a(x + y) = ax + ay is true by what prperty? (m + n) + p = m + (n + p) accrding t what prperty? 7 = 7 accrding t what prperty? Create an example f the Cmmutative Prperty f real numbers using symbls r numbers Miscnceptins/Cmmn Errrs Nne nted. Technlgy Students may use technlgy t verify equivalent expressins by substituting values and/r using cmputer algebra systems (CASs). Students may use cmputer algebra system technlgy, which is capable f perfrming symblic manipulatins t verify slutins. III. Assessing the Lessn Assessment Guidelines: The bjective EA-2.5 is t carry ut a prcedure using the prperties f real numbers (including cmmutative and assciative) t simplify expressins. Therefre, the primary fcus f the assessment shuld be fr students t carry ut such prcedures. Assessment Item Examples Write an equivalent expressin fr each f the fllwing: (xy)z = (x + y) + z = x = + z = d + (c + b) = IV. Resurces Activity: Have students create mdels f the prperties using bjects r pictures instead f numbers. They can cut ut pictures frm magazines r draw them. Fr example, + = +. 7 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

18 Mdule Fundatinal Tpics Lessn # 5 Tpic: Distributive Prperty Standard (s): EA-2.5 and EA-2.7 I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills related t this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 8 th grade students apply rder f peratins t simplify whlenumber expressins (6-3.2); apply an algrithm t add, subtract, multiply, and divide integers (8-2.); understand equivalent symblic expressins as distinct symblic frms that represent the same relatinship (6-.4, 7-.4, and 8-.4); use cmmutative, assciative, and distributive prperties t examine the equivalence f a variety f algebraic expressins (8-3.3). In Elementary Algebra students carry ut a prcedure (the distributive prperty) t simplify plynmial expressins. In Mdule 8 f Elementary Algebra, students carry ut a prcedure t divide plynmials by a mnmial. In Intermediate Algebra, students build n their knwledge f the real number system by studying cmplex numbers. Intermediate Algebra students carry ut a prcedure t perfrm peratins (including multiplicatin, expnentiatin, and divisin) with plynmial expressins (IA-4.). Als, students carry ut prcedures t perfrm peratins n plynmial functins (including f(x) + g(x), f(x) g(x), f(x) g(x), and f(x)/g(x)) (IA-2.5). Taxnmy Indicatr EA-2.5: 3 -C Cgnitive Prcess Dimensin: Apply Knwledge Dimensin: Prcedural Knwledge Indicatr EA-2.7 (multiplicatin): 3 -C Cgnitive Prcess Dimensin: Apply Knwledge Dimensin: Prcedural Knwledge 8 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

19 Mdule Fundatinal Tpics Key Cncepts Distributive Prperty Expressins Terms Mnmial Binmial Trinmial Plynmial II. Teaching the Lessn In this lessn, students use the distributive prperty t multiply plynmial expressins. Essential Learning and Understanding It is essential fr students t d the fllwing fr the attainment f this indicatr: Add, subtract and/r multiply plynmial expressins. Simplify algebraic expressins, including numerical expressins. Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that students shuld be able t successfully cmplete. -2x 2 (x 2 ) 3x(x 2 2x + 9) 2(x 3 + 8x 2) (x 3)(2x + 4) (4x + 3)(-x 7) (x + 5)(x 5) (x )(-4x 2 + x + 2) Nn-Essential Learning and Understanding It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra: Multiply plynmial expressins that exceed the number f terms in a binmial times a trinmial. Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr 9 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

20 Mdule Fundatinal Tpics the attainment f ther indicatrs within Elementary Algebra. (5x 2 - x + 3) (-7x 2 + 8x + 2) Miscnceptins/Cmmn Errrs Students may nt prperly apply the distributive prperty stating that 3(x + 2) = 3x + 2 instead f the crrect answer, which is 3(x +2) = 3x + 6. Students may nt prperly distribute a term if the term is fund at the end f the expressin. Example: (x 3)x Technlgy Students may use technlgy t verify equivalent expressins by substituting values and/r using cmputer algebra systems (CASs). Students may use cmputer algebra system technlgy, which is capable f perfrming symblic manipulatins t verify slutins. III. Assessing the Lessn Assessment Guidelines: The bjective f EA-2.7 is t carry ut a prcedure t use multiplicatin t simplify plynmial expressins. Therefre, the primary fcus f the assessment shuld be fr students t carry ut such prcedures. Assessment Item Examples Simplify: x 2 (2x 2 3x 9) A. 2x 4 3x 2 9 B. 2x 2 3x 9x 2 C. 2x 4 3x 3 9x 2 D. 2x 2 3x 3 9 Simplify: 3(x + 8) 2(x 2 4x 2) A. -2x 2 5x B. -2x 2 + x + 48 C. -2x 2 x 4 D. -2x 2 + 7x + 2 Simplify: (x 9)(x + 8) A. X 2 x 72 B. X 2 + 7x + 72 C. X 2 x + 72 D. X 2 + 7x Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

21 Mdule Fundatinal Tpics Simplify: (2x + 3)(3x 8) A. 5x 2 + 2x + 5 B. 5x 2 5 C. 6x D. 6x 2 7x 24 Simplify: (x + 5)(x 2 5x ) A. X 2 26x 5 B. X 3 + x x + 5 C. X 3 26x 5 D. X 3 + x 2 26x Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

22 Mdule Fundatinal Tpics Lessn # 6 Tpic: Operatins with matrices Standard (s): EA 2.9 and EA 2. I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills needed t meet this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 8 th grade students rganize data in matrices r scatterplts as apprpriate (8-6.2). In Elementary Algebra students carry ut a prcedure t perfrm peratins with matrices (including additin, subtractin, and scalar multiplicatin) and represent applied prblems using matrices. In Gemetry, students apply transfrmatins (including translatin and dilatin) t figures in the crdinate plane by using matrices (G-6.4). Taxnmy Level Indicatr EA-2.9: 3-C Cgnitive Prcess Dimensin: Apply Knwledge Dimensin: Prcedural Knwledge Indicatr EA-2.: 2-C Cgnitive Prcess Dimensin: Understand Knwledge Dimensin: Cnceptual Knwledge Key Cncepts Matrix Rw Clumn Dimensin (size f a matrix) Matrix additin Matrix subtractin Scalar Scalar multiplicatin 22 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

23 Mdule Fundatinal Tpics II. Teaching the Lessn In this lessn, students apply their knwledge f peratins with integers t perfrm peratins with matrices. Althugh students have rganized data int matrices in the eighth grade, a review f this cncept will be useful in setting the cntext fr peratins with matrices. Essential Learning and Understanding It is essential fr students t d the fllwing fr the attainment f this indicatr: Indicatr EA-2.9: Add tw matrices f size n larger than a 3x3 matrix. Subtract tw matrices f size n larger than a 3x3 matrix. Multiply a matrix by a scalar fr matrices f size n larger than a 3x3. Indicatr EA-2.: Represent real-wrld data using matrices, n larger than a 3x3 matrix Distinguish relevant frm irrelevant data. Represent data using matrices, understanding the meaning f clumns and rws in the applied situatin. Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that students shuld be able t successfully cmplete. Indicatr EA-2.9: Perfrm the peratin r explain why it is nt pssible Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

24 Mdule Fundatinal Tpics Suth Carlina S 3 Mathematics Curriculum Cpyright July, (Explain why it is nt pssible t perfrm the peratin.)

25 Mdule Fundatinal Tpics Indicatr E-2. Example : The Table A gives the number f peple (in thusands) wh visited Australia and Suth Africa in 998. Figures are runded t the nearest,. Surce: The New Yrk Times, January 4, 2. Table A T Frm Australia Suth Africa Nrth America 44 9 Eurpe Asia,79 2 Represent the turism data in Table A using a 3x2 matrix. Example 2: The fllwing is sales data frm a department stre sales persn wh wrks part time. Sales Data: Mnday: shirts, 8 pants, shrts, 2 hats Tuesday: 9 shirts, 3 pants, 7 shrts, hats Wednesday: 5 shirts, 4 pants, 6 shrts, 7 hats Represent the sales data using a matrix. Example 3: In a discunt department stre, similar items sale fr the same price. Shirts are $8, pants are $25, shrts are $3, and hats are $. Represent the price f the items using a matrix. Example 4: The number f grams f prtein, carbhydrates, and fats are given fr three samples f fd. Each unce f Fd I cntains 6 grams f prtein, 2 grams f carbhydrates, and 37 grams f fat. Each unce f Fd II cntains grams f prtein, 5 grams f carbhydrates, and 32 grams f fat. Each unce f Fd III cntains 2 grams f prtein, 8 grams f carbhydrates, and 77 grams f fat. Represent the number f graphs f prtein, carbhydrates, and fats fr each unce f the three fd samples using a matrix. Nn-Essential Learning and Understanding 25 Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2

26 Mdule Fundatinal Tpics It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra: Perfrm matrix multiplicatin. Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr the attainment f ther indicatrs within Elementary Algebra x 5 3 Miscnceptins/Cmmn Errrs Students may frget t distribute the negative t each term in the subtrahend when subtracting matrices. [minuend subtrahend = difference] Technlgy Nte Students may use technlgy t verify slutins. III. Assessing the Lessn Assessment Guidelines: The bjective f this indicatr is t carry ut a prcedure t perfrm peratins with matrices (including additin, subtractin, and scalar multiplicatin). Therefre, the primary fcus f the assessment shuld be fr students t carry ut such prcedures. Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2 26

27 Mdule Fundatinal Tpics Assessment Item Examples Explain why each example is r is nt pssible. If pssible, perfrm the peratin: Suth Carlina S 3 Mathematics Curriculum Cpyright July, 2