Programa analitica la matematica cls a 8-a Romania , comparata cu programele cls a 8-a , din UE , Ontario –Canada

decembrie 12, 2007 – 6:17 am

FEDERATIA NATIONALA A ASOCIATIILOR DE PARINTI – INVATAMANT PREUNIVERSITAR

STR. VIITORULUI NR. 60, SECTOR 2 BUCURESTI

TELEFON : 0722.252601, FAX : 021.4341175,

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www. federatieparinti.edu.ro

 

 

 

 

 

Algebra cls a8-a

Romania

Romania

 

Algebra cls a8-a

UE

UE

1.

Mulţimi de numere

 

1. Numere rationale,nr irationale,nr reale

2. N ÌZÌQÌR. Reprezentare pe axă.

3. Aproximarea nr .reale

4. Modulul unui nr real

5. Ecuatii cu module

6. Compararea nr. reale, min, max

7. Fractii

8. Partea intreaga si partea fractionara a unui nr real

9. Puteri,operatii cu puteri

10. Intervale

11. Reuniunea şi intersecţia intervalelor

12. Diferenţa intervalelor.

13. Operaţii între un interval şi o mulţime.

14. Inecuatii cu module

15. Sisteme de doua inecuatii cu module

 

 

 

 

 

 

1.

Calcul cu numere reale reprezentate prin litere

 

 

1. Adunarea, scăderea, înmulţirea, împărţirea şi ridicarea la putere a numerelorreale reprezentate prin litere.

2. Formule de calcul prescurtat.

3. Descompuneri în factori folosind formule de calcul prescurtat

 

2.

Operatii cu numere reale,radi-cali

16.Radicali

17.Calcule cu radicali

18.Adunarea, scăderea, înmulţirea, împărţirea şi ridicarea la putere numerelor de forma , unde a, b ÎR, b > 0.

19.Raţionalizare a numitorilor de forma, unde b ÎN şi , unde a, b ÎN.

2.

Operaţii cu rapoarte de numere reale reprezentate prin litere

 

4.Adunarea şi scăderea rapoartelor de numere reale reprezentate prin litere.

5.Înmulţirea, împărţirea şi ridicarea la putere a rapoartelor de numere reale reprezentate prin litere.

6.Aducerea la forma cea mai simplă a expresiilor care conţin rapoarte de numere reale reprezentate prin litere.

7.conditii pentru ca expresia sa aiba sens.

3

Calcul cu numere reale repre-zentate prin litere

 

 

20.Adunarea, scăderea, înmulţirea, împărţirea şi ridicarea laputere a numerelor reale reprezentate prin litere.

21.Suma algebrica

22.Formule de calcul prescurtat.

23.Descompuneri în factori folosind formule de calcul prescurtat şi alte metode.

 

3.

Mulţimi de numere

 

8.NÌZÌQÌR. Reprezentare pe axă. Valoare absolută.

9.Intervale (definiţie).

10.Reuniunea şi intersecţia intervalelor

11.diferenta

 

4

Operaţii cu rapoarte de numere reale reprezentate prin litere

 

 

24.Adunarea şi scăderea rapoartelor de numere reale reprezentate prin litere.

25.Înmulţirea, împărţirea şi ridicarea la putere a rapoartelor de numerereale reprezentate prin litere.

26.Aducerea le forma cea mai simplă a expresiilor care conţin rapoarte de numerereale reprezentate prin litere

27.Identificarea unor dependente functionale si a unor reguli de formare a sirurilor.

4.

Inecuatii de gradul I cu o singura necunoscuta

12.Inecuaţii de forma ax + b>0, (³,<,£) unde a şi b sunt numere reale.

 

13.Exerciţii aplicative la inecuaţii cu exemple din fizica ,economie,tehnologie, management.

5

Funcţii I

 

 

28.Funcţii definite pe mulţimi finite, exprimate cu ajutorul unor diagrame, tabele, formule,reprezentare grafică.

29.Funcţii de tipul f: R®R, f(x)=ax+b (a, b ÎR); reprezentareageometrică a graficului.

30.Funcţii de tipul f: A®R, f(x)=ax+b (a, b ÎR), unde A este un interval sau o mulţime finită; reprezentare grafică.

31. Punctele de intersecţie ale graficului unei funcţii cu axele de coordonate;punctul de intersecţie al graficelor a două funcţii; coliniaritateaa trei sau a mai multor puncte.

 

5

Funcţii I

 

 

14.Funcţii definite pe mulţimi finite, exprimate cu ajutorul unor diagrame, tabele,formule, reprezentare grafică.

15.Funcţii de tipul

16. f:R®R,f(x)=ax+b (a,bÎR); reprezentarea geometrică a graficului.

17.Funcţii de tipul

18.f: A®R, f(x)=ax+b (a, b ÎR), unde A este un interval sau o mulţime finită; reprezentare grafică.

19. Punctele de intersecţie ale graficului unei funcţii cu axele de coordonate;punctul de intersecţie al graficelor a două funcţii; coliniaritateaa trei sau a mai multor puncte.

20.Exemple si aplicatii in fizica,management, economie, etc

6

Funcţii II

 

32.Aplicarea teoriei specifice funcţiilor înprobleme de geometrie plană.

 

6

Sisteme de 2 ecuatii de gradul 1 cu 2 necunoscute

21.Sisteme de ecuaţii de forma:

, unde a1 ,

b1 , c1 , a2 , b2 , c2 sunt numere reale; rezolvare prin metodasubstituţiei şi prin metoda reducerii; interpretare geometrică

-aplicatii in fizica, economie etc.

7

 

 

 

Ecuaţii, inecuaţiişi sisteme I

 

33.Ecuaţii de forma ax+b=0, unde a şi b sunt numere reale.

34.Ecuaţii de forma ax+by+c=0, unde a, b, c sunt numere reale.

35.Sisteme de ecuaţii de forma:

, unde a1,

b1 , c1 , a2 , b2 , c2sunt numere reale; rezolvare prin metodasubstituţiei şi prin metoda reducerii; interpretaregeometrică.

 

 

 

 

8

Ecuaţii, inecuaţiişi sisteme II

 

35.Inecuaţii de forma ax + b>0, (³,<, £) unde a şi b sunt numere reale.

36.Sisteme de inecuaţii.

37.Exerciţii aplicative la inecuaţii.

38.Rezolvarea unor probleme cu ajutorul ecuaţiilor, inecuaţiilor şi a sistemelor de ecuaţii.

 

 

 

 

9

Ecuaţia de gradul al II-lea

 

 

39.Ecuaţii de forma ax2 + bx + c = 0, unde a ¹0; a, b, c ÎR (descriere şimulţimea soluţiilor). Rezolvarea ecuaţiei de forma ax2 + bx + c = 0, unde a ¹0; a, b, c ÎRîn cazurile b, c = 0 , b= 0 sau c = 0.

40.Rezolvarea ecuaţiei de forma ax2+ bx + c = 0, unde a ¹0; a, b, c ÎR prin descompuneri în factori sau în diferenţă depătrate.

41.Planul de rezolvare a ecuaţiei de forma ax2+ bx + c = 0, unde a ¹0; a, b, c ÎR (formule pentrudiscriminantul ecuaţiei şi soluţiile ecuaţiei).Ecuaţii echivalente

 

 

 

 

 

 

 

Geometrie cls a8-a

Romania

ROMANIA

Geometrie

Cls a 8-a ,UE

Geometrie cls a8-a

UE

1.

Introducere în geometria în spaţiu

 

1.Puncte,drepte,plane.

2.Determinarea planului.

3.Pozitia relativa a doua drepte in spatiu.

4.Pozitia relativa a doua plane.

5.Paralelism in spatiu

6.Alte teoreme de paralelism.

7.Masura unghiului a doua drepte.Drepte perpendiculare.

8.Dreapta perpendiculara pe un plan.

9.Calculul distantelor

a)distanta de la un punct la o dreapta.

b) distanta de la un punct la un plan

c) distanta dintre doua drepte paralele

d) distanta de la o dreapta la un plan

c) distanta dintre doua plane paralele

 

.

1.Geometrie

plana

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.Introducere în geometria în spaţiu

 

1.Metode pentru calcularea ariei si perimetrului

la:

-patrat

-dreptunghi

-triunghi

-trapez

2.suma unghiurilor intr-un poligon

3.calculul ariei unui poligon oarecare prin sumarea ariilor figurilor in care sedescompune poligonul

 

 

 

 

 

 

4.Puncte,drepte,plane.

5.Cubul,definitie, reprezentare în plan.

6.Sectiuni in cub

7.Proiectia unui cub intr-un plan

2.

Corpuri geometrice cunoscute

10.Prisma,definire, reprezentare în plan clasificarea prismelor

11.desfasurarea prismei. 12.înălţimea prismei; prismă dreaptă.

13.Tetraedrul şi piramida regulata.(definiţie, exemple).

14.Simetrie in spatiu

15.Sectiuni in corpurile studiate,prisma,piramida

16.Trunchiul de piramida

 

 

8.Piramida regulata, definiţie, elemente

9.Prisma

10.Calculul suprafetei diverselor prisme.

 

3.

Proiecţii ortogonale pe plan I

 

 

17.Proiecţii ortogonale de puncte, drepte şi segmente.

18.Teorema celor trei perpendiculare.

19.Reciprocele teoremei celor trei perpendiculare.

20.Calculul distanţei de la un punct la o dreaptă.

 

 

 

4

Proiecţii ortogonale pe plan II

 

21.Unghiul unei drepte cu un plan; lungimea proiecţiei unui segment.

22.Unghi diedru. Aria proiecţiei.

23.Plane perpendiculare. Aria proiecţiei.

24.Calculul distanţelor şi măsurilor de unghiuri pe feţele sau îninteriorul unor corpuri geometrice. 25.Probleme de perpendicularitate în spaţiu.

 

 

 

5

Calcul de arii şi volume I

 

 

26.Aria şi volumul unui corp geometric.

27.Aria laterală, aria totală şi volumul prismei drepte cu baza triunghi echilateral,pătrat sau hexagon regulat.

28.Aria laterală şi totală a paralelipipedului dreptunghic şi a cubului.

29.Probleme aplicative.

 

 

 

6

Calcul de arii şi volume II

 

 

30.Aria laterală, aria totală şi volumul piramidei triunghiulare regulate, piramideipatrulatere regulate şi piramidei hexagonale regulate.

31.Probleme aplicative.

32.Aria laterală, aria totală şi volumul trunchiului de piramidă triunghiularăregulată, al trunchiului de piramidă patrulateră regulatăşi al trunchiului de piramidă hexagonală regulată.

33.Probleme aplicative.

 

 

 

7

Calcul de arii şi volumeIII

 

 

34.Cilindrul circular drept: descriere, desfăşurare, secţiuni paralele cu baza şisecţiuni axiale; aria laterală, aria totală şi volumul.

35.Probleme aplicative.

36.Conul circular drept: descriere, desfăşurare, secţiuni paralele cu baza şisecţiuni axiale; aria laterală, aria totală şi volumul.

Probleme

 

 

8

Calcul de arii şi volume IV

 

 

37.Trunchiul de con circular drept: descriere, desfăşurare, secţiuniaxiale; aria laterală, aria totală şi volumul.

38.Probleme aplicative.

39.Sfera: aria şi volumul. Probleme aplicative.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Programa analitica si obiectivele urmarite la matematica cls a 8-a in Ontario,Canada

Dupa situl http://www.geocities.com/Athens/Oracle/8314/currgrid.htm

 

 

 

Grade 8: Patterning and Algebra

Planning: Term #

Tracking: Ach. Level

Overall Expectations

1

2

3

4

• represent linear growing patterns (where the terms are whole numbers) using graphs, algebraicexpressions, and equations;

 

 

 

 

• model linear relationships graphically and algebraically, and solve and verify algebraicequations, using a variety of strategies, including inspection, guess andcheck, and using a “balance” model.

 

 

 

 

Specific Expectations

 

 

 

 

Patterns and Relationships

 

 

 

 

– represent, through investigation with concrete materials, the general term of a linear pattern,using one or more algebraic expressions (e.g., “Using toothpicks, I noticedthat 1 square needs 4 toothpicks, 2 connected squares need 7 toothpicks, and3 connected squares need 10 toothpicks. I think that for n connected squaresI will need 4 + 3(n – 1) toothpicks, because the number of toothpickskeeps going up by 3 and I started with 4 toothpicks. Or, if I think ofstarting with 1 toothpick and adding 3 toothpicks at a time, the pattern canbe represented as 1 + 3n.”);

 

 

 

 

– represent linear patterns graphically (i.e., make a table of values that shows the term numberand the term, and plot the coordinates on a graph), using a variety of tools(e.g., graph paper, calculators, dynamic statistical software);

 

 

 

 

– determine a term, given its term number, in a linear pattern that is represented by a graph oran algebraic equation (Sample problem: Given the graph that represents thepattern 1, 3, 5, 7,…, find the 10th term. Given the algebraic equation thatrepresents the pattern, t = 2n – 1, find the 100th term.).

 

 

 

 

Variables, Expressions and Equations

 

 

 

 

– describe different ways in which algebra can be used in real-life situations (e.g., the value of$5 bills and toonies placed in a envelope for fund raising can be representedby the equation v = 5f + 2t);

 

 

 

 

– model linear relationships using tables of values, graphs, and equations (e.g., thesequence 2, 3, 4, 5, 6,… can be represented by the equation t = n + 1, wheren represents the term number and t represents the term), throughinvestigation using a variety of tools (e.g., algebra tiles, pattern blocks,connecting cubes, base ten materials) (Sample problem: Leah put $350 in abank certificate that pays 4% simple interest each year. Make a table ofvalues to show how much the bank certificate is worth after five years, usingbase ten materials to help you. Represent the relationship using anequation.);

 

 

 

 

– translate statements describing mathematical relationships into algebraic expressionsand equations (e.g., for a collection of triangles, the total number of sidesis equal to three times the number of triangles or s = 3n);

 

 

 

 

– evaluate algebraic expressions with up to three terms, by substituting fractions, decimals, andintegers for the variables (e.g., evaluate 3x + 4y = 2z, where x = 1/2, y =0.6, and z = –1);

 

 

 

 

– make connections between solving equations and determining the term number in a pattern, usingthe general term (e.g., for the pattern with the general term 2n + 1, solvingthe equation 2n + 1 = 17 tells you the term number when the term is 17);

 

 

 

 

– solve and verify linear equations involving a one-variable term and having solutions that areintegers, by using inspection, guess and check, and a “balance” model (Sampleproblem: What is the value of the variable in the equation 30x – 5 =10?).15 by using guess and check. First I tried 6 for x. Since I knew that 6plus 7 equals 13 and 13, is less than 15, then I knew that x must be greaterthan 6.”).

 

 

 

 

 

 

 

 

 

 

 

Grade 8: Geometry and Spatial Sense,Ontario , Canada

Planning: Term #

Tracking: Ach. Level

Overall Expectations

1

2

3

4

• demonstrate an understanding of the geometric properties of quadrilaterals and circles andthe applications of geometric properties in the real world;

 

 

 

 

• develop geometric relationships involving lines, triangles, and polyhedra, and solve problemsinvolving lines and triangles;

 

 

 

 

• represent transformations using the Cartesian coordinate plane, and make connections betweentransformations and the real world.

 

 

 

 

Specific Expectations

 

 

 

 

Geometric Properties

 

 

 

 

– sort and classify quadrilaterals by geometric properties, including those based on diagonals,through investigation using a variety of tools (e.g., concrete materials,dynamic geometry software) (Sample problem: Which quadrilaterals havediagonals that bisect each other perpendicularly?);

 

 

 

 

– construct a circle, given its centre and radius, or its centre and a point on the circle, orthree points on the circle;

 

 

 

 

– investigate and describe applications of geometric properties (e.g., properties of triangles,quadrilaterals, and circles) in the real world.

 

 

 

 

Geometric Relationships

 

 

 

 

– determine, through investigation using a variety of tools (e.g., dynamic geometry software,concrete materials, geoboard), relationships among area, perimeter,corresponding side lengths, and corresponding angles of similar shapes(Sample problem: Construct three similar rectangles, using grid paper or ageoboard, and compare the perimeters and areas of the rectangles.);

 

 

 

 

– determine, through investigation using a variety of tools (e.g., dynamic geometry software,concrete materials, protractor) and strategies (e.g., paper folding), theangle relationships for intersecting lines and for parallel lines andtransversals, and the sum of the angles of a triangle;

 

 

 

 

– solve angle-relationship problems involving triangles (e.g., finding interiorangles or complementary angles), intersecting lines (e.g., findingsupplementary angles or opposite angles), and parallel lines and transversals(e.g., finding alternate angles or corresponding angles);

 

 

 

 

– determine the Pythagorean relationship, through investigation using a variety of tools(e.g., dynamic geometry software; paper and scissors; geoboard) and strategies;

 

 

 

 

– solve problems involving right triangles geometrically, using the Pythagorean relationship;

 

 

 

 

– determine, through investigation using concrete materials, the relationship between the numbersof faces, edges, and vertices of a polyhedron (i.e., number of faces + numberof vertices = number of edges + 2) (Sample problem: Use Polydrons and/orpaper nets to construct the five Platonic solids [i.e., tetrahedron, cube,octahedron, dodecahedron, icosahedron], and compare the sum of the numbers offaces and vertices to the number of edges for each solid.).

 

 

 

 

Location and Movement

 

 

 

 

– graph the image of a point, or set of points, on the Cartesian coordinate plane after applying atransformation to the original point(s) (i.e., translation; reflection in thex-axis, the y-axis, or the angle bisector of the axes that passes through thefirst and third quadrants; rotation of 90°,

180°, or 270° about the origin);

 

 

 

 

– identify, through investigation, real-world movements that are translations, reflections, androtations.

 

 

 

 

 

 

Grade 8: Measurement,Mathematic,Canada

Planning: Term #

Tracking: Ach. Level

Overall Expectations

1

2

3

4

• research, describe, and report on applications of volume and capacity measurement;

 

 

 

 

• determine the relationships among units and measurable attributes, including the area of acircle and the volume of a cylinder.

 

 

 

 

Specific Expectations

 

 

 

 

Attributes, Units and Measurement Sense

 

 

 

 

– research, describe, and report on applications of volume and capacity measurement (e.g., cooking,closet space, aquarium size) (Sample problem: Describe situations wherevolume and capacity are used in your home.).

 

 

 

 

Measurement Relationships

 

 

 

 

– solve problems that require conversions involving metric units of area, volume, and capacity(i.e., square centimetres and square metres; cubic centimetres and cubicmetres; millilitres and cubic centimetres) (Sample problem: What is thecapacity of a cylindrical beaker with a

radius of 5 cm and a height of 15 cm?);

 

 

 

 

– measure the circumference, radius, and diameter of circular objects, using concretematerials (Sample Problem: Use string to measure the circumferences ofdifferent circular objects.);

 

 

 

 

– determine, through investigation using a variety of tools (e.g., cans and string, dynamicgeometry software) and strategies, the relationships for calculating thecircumference and the area of a circle, and generalize to develop theformulas (Sample problem: Use string to measure the circumferences and thediameters of a variety of cylindrical cans, and investigate the ratio of thecircumference to the diameter.);

 

 

 

 

– solve problems involving the estimation and calculation of the circumference and the area ofa circle;

 

 

 

 

– determine, through investigation using a variety of tools and strategies (e.g., generalizingfrom the volume relationship for right prisms, and verifying using thecapacity of thin-walled cylindrical containers), the relationship between thearea of the base and height and the volume of a cylinder, and generalize todevelop the formula (i.e., Volume = area of base x height);

 

 

 

 

– determine, through investigation using concrete materials, the surface area of a cylinder(Sample problem: Use the label and the plastic lid from a cylindrical containerto help determine its surface area.);

 

 

 

 

– solve problems involving the surface area and the volume of cylinders, using a variety ofstrategies (Sample problem: Compare the volumes of the two cylinders that canbe created by taping the top and bottom, or the other two sides, of astandard sheet of paper.).

 

 

 

 

 

 

 

Grade 8: Data Management and Probability

Mathematic

Planning: Term #

Tracking: Ach. Level

Overall Expectations

1

2

3

4

• collect and organize categorical, discrete, or continuous primary data and secondary data anddisplay the data using charts and graphs, including frequency tables withintervals, histograms, and scatter plots;

 

 

 

 

• apply a variety of data management tools and strategies to make convincing arguments about data;

 

 

 

 

• use probability models to make predictions about real-life events.

 

 

 

 

Specific Expectations

 

 

 

 

Collection and Organization of Data

 

 

 

 

– collect data by conducting a survey or an experiment to do with themselves, their environment,issues in their school or community, or content from another subject, andrecord observations or measurements;

 

 

 

 

– organize into intervals a set of data that is spread over a broad range (e.g., the age ofrespondents to a survey may range over 80 years and may be organized intoten-year intervals);

 

 

 

 

– collect and organize categorical, discrete, or continuous primary data and secondary data(e.g., electronic data from websites such as E-Stat or Census At Schools),and display the data in charts, tables, and graphs (including histograms andscatter plots) that have appropriate titles, labels (e.g., appropriate unitsmarked on the axes), and scales (e.g., with appropriate increments) that suitthe range and distribution of the data, using a variety of tools (e.g., graphpaper, spreadsheets, dynamic statistical software);

 

 

 

 

– select an appropriate type of graph to represent a set of data, graph the data usingtechnology, and justify the choice of graph (i.e., from types of graphs alreadystudied, including histograms and scatter plots);

 

 

 

 

– explain the relationship between a census, a representative sample, sample size, and apopulation (e.g., “I think that in most cases a larger sample size will bemore representative of the entire population.”).

 

 

 

 

Data Relationships

 

 

 

 

– read, interpret, and draw conclusions from primary data (e.g., survey results, measurements,observations) and from secondary data (e.g., election data or temperaturedata from the newspaper, data from the Internet about lifestyles), presentedin charts, tables, and graphs (including frequency tables with intervals,histograms, and scatter plots);

 

 

 

 

– determine, through investigation, the appropriate measure of central tendency (i.e., mean, median,or mode) needed to compare sets of data (e.g., in hockey, compare heights ormasses of players on defence with that of forwards);

 

 

 

 

– demonstrate an understanding of the appropriate uses of bar graphs and histograms bycomparing their characteristics (Sample problem: How is a histogram similarto and different from a bar graph? Use examples to support your answer.);

 

 

 

 

– compare two attributes or characteristics (e.g., height versus arm span), using a scatterplot, and determine whether or not the scatter plot suggests a relationship(Sample problem: Create a scatter plot to compare the lengths of the bases ofseveral similar triangles with their areas.);

 

 

 

 

– identify and describe trends, based on the rate of change of data from tables and graphs,using informal language (e.g., “The steep line going upward on this graphrepresents rapid growth. The steep line going downward on this other graphrepresents rapid decline.”);

 

 

 

 

– make inferences and convincing arguments that are based on the analysis of charts, tables, andgraphs (Sample problem: Use data to make a convincing argument that theenvironment is becoming increasingly polluted.);

 

 

 

 

– compare two attributes or characteristics, using a variety of data management tools andstrategies (i.e., pose a relevant question, then design an experiment orsurvey, collect and analyse the data, and draw conclusions) (Sample problem:Compare the length and width of different-sized leaves from a maple tree todetermine if maple leaves grow proportionally. What generalizations can youmake?).

 

 

 

 

Probability

 

 

 

 

– compare, through investigation, the theoretical probability of an event (i.e., the ratio ofthe number of ways a favourable outcome can occur compared to the totalnumber of possible outcomes) with experimental probability, and explain whythey might differ (Sample problem: Toss a fair coin 10 times, record theresults, and explain why you might not get the predicted result of 5 headsand 5 tails.);

 

 

 

 

– determine, through investigation, the tendency of experimental probability to approachtheoretical probability as the number of trials in an experiment increases,using class-generated data and technology-based simulation models (Sampleproblem: Compare the theoretical probability of getting a 6 when tossing anumber cube with the experimental probabilities obtained after tossing anumber cube once, 10 times, 100 times, and 1000 times.);

 

 

 

 

– identify the complementary event for a given event, and calculate the theoreticalprobability that a given event will not occur (Sample problem: Bingo uses thenumbers from 1 to 75. If the numbers are pulled at random, what is theprobability that the first number is a multiple of 5? Is not a multiple of5?).

 

 

 

 

 

Surse folosite in studiu de consilieri FNAP-IP

Manuale din Germania , editura Cornelsen Power Learning –Berlin ,gilt ab 01.01.2007 www.cornelsen.de,www.mentor.de

Matematica Oggi, Italia www.scuola.com,www.fabriscuola.it

http://www.geocities.com/Athens/Oracle/8314/currgrid.htm—Ontario -Canada

 

 

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