Solutions+NPMaBVT05 Eng

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Skolverket: NpMaB vt 2005 Version 1 NV-College - Sjdalsgymnasiet

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Concerning test material in general, the Swedish Board of Education refers to the Official SecretsAct, the regulation about secrecy, 4th chapter 3rd paragraph. For this material, the secrecy is valid

until 10th June 2005.

NATIONAL TEST IN MATHEMATICS COURSE B

SPRING 2005DirectionsTest time 240 minutes for Part I and Part II together. We recommend that you spend no more than 60

minutes on Part I.

Resources Part I: Formulas for the National Test in Mathematics Course BPlease note that calculators are not allowed in this part.

Part II: Calculators and Formulas for the National Test in Mathematics Course B.Test material The test material should be handed in together with your solutions.

Write your name, the name of your education programme / adult education on allsheets of paper you hand in.

Solutions to Part I should be handed in before you retrieve your calculator. You should

therefore present your work on Part I on a separate sheet of paper.Please note that you may start your work on Part II without a calculator.

The test The test consists of a total of 17 problems. Part I consists of 8 problems and Part IIconsists of 9 problems.For some problems (where it says Only answer is required) it is enough to give short

answers. For the other problems short answers are not enough. They require that youwrite down what you do, that you explain your train of thought, that you, when necessary,draw figures. When you solve problems graphically/numerically please indicate

how you have used your resources.

Problem 17 is a larger problem which may take up to an hour to solve completely. It

is important that you try to solve this problem. A description of what your teacherwill consider when evaluating your work is attached to the problem.Try all of the problems. It can be relatively easy, even towards the end of

Try all of the problems. It can be relatively easy, even towards the end of the test, to

receive some points for partial solutions. A positive evaluation can be given even forunfinished solutions.

Score and The maximum score is 47 points.mark levels

The maximum number of points you can receive for each solution is indicated after

each problem. If a problem can give 2 Pass-points and 1 Pass with distinction-point this is written (2/1). Some problems are marked with, which means that theymore than other problems offer opportunities to show knowledge that can be relatedto the criteria for Pass with Special Distinction.

Lower limit for the mark on the testPass: 14 points

Pass with distinction: 27 points of which at least 6 Pass with distinction-points.Pass with special distinction: In addition to the requirements for Pass with distinction you

have to show most of the Pass with special distinction qualities that the-problems give theopportunity to show. You must also have at least 12 Passwith distinction-points.

Name:: School:

Education programme/adult education:


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Part I

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This part consists of 8 problems that should be solved without the aid of a calculator. Your

solutions to the problems in this part should be presented on separate sheets of paper that must

be handed in before you retrieve your calculator. Please note that you may begin working on

Part II without the aid of a calculator.

1. Solve the equations

a) (2/0)0822 =+ xx

b) 010402 =+ xx

Suggested solutions:

a) 08 22

=+ xxFind two numbers such that their products is 8 and their sum is .

The larger of these numbers must be positive, and the smaller nega-tive. Two such numbers are: and

2

4 2 :( ) 824 = ( ) 22424 ==+

Therefore: ( )( ) 024822 =+=+ xxxx

=

=+

02

04

x

x [2/0] Answer:

=

=

2

4

2

1

x

x

+x

b) 02 =x 1040

Factorize 10 [2/0] Answer:x ( ) 0410 =+xx

=+=

040

xx

==

40

2

1

xx


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2. The diagrams below show the distribution of employee-age at three differentplaces of work.

Which place of work has the largest range of distribution and how large is it?Only answer is required (1/0)

Suggested solutions:The IT-Company has the largest range of distribution: 5020 x

3.

a) Determine the anglex (1/0)

b) Which of the following geometric relationships did you use when youdetermined the anglex? Only answer is required (1/0)

A Pythagoras theorem

B The sum of all angles in a triangle is 180

C The sum of supplementary angles is 180

D Exterior angle theorem

E Triangle proportionality theoremF Inscribed angle theorem

Suggested solutions:+= 104144 x

== 40104144x [1/0] Answer: = 40xAnswer: I used D: Exterior angle theorem to determine the angle.

[1/0]I could as well use B and C.



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4.

a) Find the equation to the line drawn in the coordinate system.Only answer is required (1/0)

b) Decide whether the point (-4, 11) is on that line. (1/0)

c) Draw a coordinate system and then draw a line that has a gradient4

3=k .

Only answer is required (1/0)

Suggested solutionsa) Answer: The equation to the line drawn in the coordinate system

is: 32 + x [1/0]=y

To find the equation of the line, we may recognize that the y-intercept is and the slope of the line is:3=m

21

2

01

31

12

12 =

=

=

=

xx

yyk

mkxy += 32 += xy

Two clear points on the line are used to find the slope of the line3,0 11 == yx , and 1,1 22 == yx .

b) If the coordinates of the point (-4, 11) satisfy the equation32 + x it is on that line. Therefore, we may substitute=y 4=x

if 11=y it is on the line:

( ) 1138342 =+=+=y [1/0]

0

1

2

3

4

5

6

8

10

x

y

xy =4

37

c) The line xy =3

is plotted

in the figure to the right.

4

[1/0]0 2 4 6 8



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5. Solve the simultaneous equations: (2/0)

=

=+

xy

xy

22

1622

Suggested solutions

=

=+

xy

xy

22

1622

Rewrite the second equation

=

=+

22

1622

xy

xy We may eliminate x by adding these two equations together:

216222 +=++ xyxy 183 =y 3

18=y Answer: y 6= [1/0]

Substitute 6=y in one of the original equations, for example into findxy 22 = x :

xy 22 = x226 = x24 = 2

4=x Answer: 2=x [1/0]

We may check the validity of the answers by substituting 2=x and6=y in :1622 =+ xy

164122262 =+=+ : Checks!

6. a) Solve the inequality 3 713+


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7. You are participating in a game show on TV and you can win SEK 1000 on a

game of dice.

The rules are that the host casts two dice that you cannot see. You are then going

to guess the number of dots the two dice show together.

If your guess is correct you win SEK 1000.

How many dots should you guess at to have the highest probability of winning?

Justify your answer. (0/2)

Suggested solutions Answer: [0/1]7

As illustrated in the figure below, choosing seven dots gives the highestprobability. There are six choices for 7 dots: ( )6,1 , ( )5,2 , ( ) , ,

, and ( . The total number of possibilities are 36 . Therefore:4,3 ( )3,4

( 2,5 ) )1,6

( )6

1

36

67 ==P [0/1]

0

1

2

3

4

5

6

0 1 2 3 4 5 6

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8. Write down the equation to a straight line that neverintersects the graph of the func-tion: . Only answer is required (0/1)xxy 4

2 =

Suggested solutions


Answer: [0/1]6=y

Note that all parallel lines with x-axis, i.e lines with equations as

ay = 4>awhere will never inter-

sect the function . [0/1]xxy 42 =


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Part II


This part consists of 9 problems and you may use a calculator when solving them.Please note that you may begin working on Part II without your calculator.

9. Simplify the following expressions as far as possible:

a) Endast svar fordras (1/0)16)4(2 x

b) Endast svar fordras (1/0))3(2)52( xxx ++

Suggested solutions

a) xx [1/0]xxx 81616816)4( 222 =+=

b) 68 [1/0]226102)3(2)52( 22 +=+=++ xxxxxxxx

10. In October 2003 the Swedish national team in womens football enjoyed a degreeof success by winning a silver medal in the Womens World Cup. Out of the 20

players in the team, 6 came from Ume IK, the same number of players came from

Malm FF and the rest of the players were from four other football clubs.

At one time during the Womens World Cup, two players were to be picked out at

random to undergo a doping test.

a) What was the probability that the first player picked for a doping test camefrom Ume IK? Only answer is required (1/0)

b) What was the probability that both players picked for a doping test camefrom Ume IK? (1/1)

Club:NumberofPlayers

ProbabilityFirst Playerto be picked upcame from

ProbabilitySecond Playerto be picked upcame from

Probability thatfirst and second Playerto be picked up came from

Ume IK 6( )

20

6=UmeP

( )10

3=UmeP

( )19

5=UmeP ( )

19

5

10

3, =UmeUmeP

( )38

3, =UmeUmeP

Malm FF 6( )

20

6=MalmP

( )10

3=MalmP

( )19

5=MalmP ( )

19

5

10

3, =MalmMalmP

( )38

3, =MalmMalmP

Others 8( )

20

8=OthersP

( ) 52

=OthersP

( )19

7=OthersP

( )19

7

5

2, =OthersOthersP

( ) 9514

, =OthersOthersP

Total 20


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11. Patrik is going to buy Picknmix sweets for his mum Ellen. She tells Patrik thatshe wants 5 hg of sweets and gives him SEK 30 to spend.

At the shop, there are two different prices of Picknmix sweets. The price of the

more expensive sweets is 7.90 SEK/ hg and the cheaper sweets is 4.90 SEK/hg.

Patrik asks himself: Is it possible to buy exactly 5 hg of sweets for SEK 30?

After some thought, he comes up with a way of calculating it and writes down thesimultaneous equations:

=+

=+

3090.790.4

5

yx

yx

a) Explain whatx andy represent in the simultaneous equations.Only answer is required (1/0)

b) Choose one of the equations above and explain what the equation describes.Only answer is required (1/0)

c) Solve the simultaneous equations and then answer Patriks question above.(2/0)

Suggested solutionsa) x represents the weight of sweets that costs 4.90 SEK/hg , and y the

weight of sweets that costs 7.90 SEK/hg [1/0]

b) 5=+y describes that the total weight of the sweets is 5 hg.x

3090.790.4 =+ yx describes that x hg of sweets costing 4.90 SEK/hg

together with y hg of sweets that costs 7.90 SEK/hg cost 30 SEK.[1/0]

c) We may solve the simultaneous equations= 30y

by means

of substantiation. We may first

+

=+

90.790.4

5

x

yx

x in terms ofy from 5=+y . Then

substitute it in the second formula and find y :

x

5=+yx yx = 5

Substitut yx = in5 3090.790.4 =+ yx :

( ) 3090.7590.4 =+ yy

3090.790.45.24 =+ yy 5.24303 =y

5.53 =y

ghgy 1836

11

30

55

3

5.5=== [1/0]

ghgx 3176

19

6

1130

6

11

6

30

6

11

6

65

6

115 ==

==== [1/0]

Answer: Patrik buys gx 317= of sweet costing 4.90 SEK/hg and

of sweet costing 7.90 SEK/hggy 183



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12. In the 2005 Alpine World Ski Championships Anja Prson won the competition in

Super G on a course that can be described by the simplified figure below. The

course starts at a height of 2335 metres above sea level (ASL) and has a drop of

590 metres.

Pontus is standing at a ski lift station along the course, watching the competition. His

altitude indicator shows that he is at an altitude of 2000 metres above sea level. A

sign at the ski lift station tells him that the lift goes 1132 metres up to the start area,

see figure.

How far do the competitors still have to go to get to the finish when they have passed

Pontus? (0/2)Suggested solutionsLets name the start point , ASL 1745 mS A , ASL 2000 m , finish

point , and the present position of pontos

B

F P :Due to the fact that BP is parallel with the base of the triangle AF, thetransversal rule states that:

PF

SP

BA

SB=

x

1132

17452000

20002335=

[1/0]

x1132

255335 =

Cross product:2551132335 =x

mmx 867.861335

2551132=

= Answer: [1/0]mx 860



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13. In April 2003 the citizens of Hungary voted on membership in the EU. At thecounting of the votes it turned out that 84% voted Yes to membership in the EUandthat 45% of those having the right to vote in the referendum did so.

Investigate between which percentages the share of Yes-votes could be if everyonewith the right to vote had participated in the referendum. (0/2)

Suggested solutionsIf all of those who have the write to vote had voted If all of them were for EU membership then the percentage of

those who are for EU-membership would be:%9393.0928.055.045.084.0 ==+=yes [0/1]

If all of them were against EU membership then the percentageof those who are for EU-membership would be:

%3838.0378.0045.084.0 ==+=yes

Answer: If everyone with the right to vote had participated in the refer-endum, the percentage of those who are for EU-membership would bebetween 38 and 93 .i.e. 38% % %93%


yes . [0/1]


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14. Each one of the situations I, II and III applies to one graph in the figure below.

I For many goods, the VAT corresponds to 20% of the price of the item. Theamount of VAT is a function of the price of the item.

II You are going to build a rectangular kennel from 40 m of fencing. The areaof the kennel is a function of the length of the kennel.

III In the beginning there are 50 bacteria in a culture. Every hour the number ofbacteria increases by 20%.

The number of bacteria is a function of time.

a) Combine the situations I, II and III with the functionsf, g and h.

Only answer is required (2/0)

b) What is they-value at point P? Only answer is required (1/0)

c) What is thex-value at point Q? Only answer is required (0/1)

d) Writey as a function ofx for situation II. (0/1/)

Suggested solutionsa. I is described by x20.0= , II is described by( )xh ( ) ( )x 20 , and

III is illustrated in the figure by

xxf =

( ) x20.1 . [2/0]xg 50=

b. The y-value at P is the initial number of bacteria in the culturewhich was 50. [1/0]

c. The x-value of Q is 20. This corresponds to one of the two solu-tions of ( ) ( ) 0=xxf 20 =x . [0/1]

d. For the situation II: ( ) x20.1 . [0/1/]xgy 50=

MVG- quality 14d

M1 Formulates and develops the problem, uses general methods with problem solving.

M2 Analyses and interprets the results, concludes and evaluates if they are reasonable.



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15. The two most common picture formats for a TV are the standard picture formatand the widescreen format. The length of the diagonal of the screen, measured in

inches, is used to describe the size of a TV. One inch is approximately 2.54

centimetres.

Example: A common size of TV is 28

(28 inches).

A standard picture formatTV has a

screen where the width is3

4of the

height.

A widescreen TV has a screen where the

width is

9

16of the height..

Consider two TVs that are the same size, which means that the diagonals of the

screens are the same length, but where one of them is of the standard picture format

and the other one is of the widescreen format.

Determine which format gives the screen with the largest area. (0/3/)

Suggested solutionsLets assume the diameter of the screen is D . If the width of the TV is

and its height is , we may calculate the area of each TV-format

using Pythagoras Theorem,

w h

STST hw 3

4

= and WSWS hw 9

16

= where WS repre-

sent the widescreen TV-format and ST represents standard- TV format.

STST hw3

4= 22222

2

222

25

9

9

25

3

4DhhhhhwD STSTSTSTSTST ==+

=+=

222

25

12

25

9

3

4

3

4DDhhwA STSTSTST ==== [0/1]

2

25

12DAST =

WSWS hw9

16=

22222

22

2

222

337

81

81

337

81

8116

9

16DhhhhhhwD STSTSTSTSTSTST ==

+

=+

=+=

222

337

144

337

81

9

16

9

16DDhhwA WSWSWSWS ====

2

337

144DAWS = WSAD

144

3372 = [0/1]

WSWSST AADA300

337

144

337

25

12

25

12 2 ===

Answer: The area of the standard TV is larger than that of widescreen

TV: WSST AA300

337= [0/1]

MVG - quality15M1 Formulates and develops the problem, uses general methods with problem solving.


M5 The presentation is structured, and mathematical language is correct.



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16. The figure shows the letter M standing on a horizontal surface. The two vertical

supporting legs are of equal length.

Show that xv 2= (0/2/)

Suggested solutions:

x90

AB

C

x90Lets connect the top ver-texes ofM, with a hori-zontal line ofAB to eachother. In the newly formedisosceles triangle

. Therefore,

the third angle of thetriangle is:

ABC

xBA == 90

( ) xx 2902180 == QED. [0/2/]

MVG- quality 16


M3 Carries out mathematical proof, or analyses mathematical reasoning.



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Vid bedmningen av ditt arbete med fljande uppgift kommer lraren att tahnsyn till: Hur vl du genomfr dina berkningar

Hur vl du redovisar och kommenterar ditt arbete Hur vl du motiverar dina slutsatser Vilka matematiska kunskaper du visar Hur vl du anvnder det matematiska sprket Hur generell din lsning r

17. This problem is about forming figures by using matches. The object is to connecta number of simple regular polygons after each other into a row. The examples

below show how it is done with regular triangles and rectangles.

From 3 matches 1

triangle can be

formed.

From 5 matches 2 triangles

can be formed.

From 7 matches 3 triangles in a row

can be formed.

The relation between the number of matches and the number of connected triangles

can be found if they are connected in a row as shown in the picture.

In the table below,x is the number of matches andy the number of connected

triangles.

x y

3 1

5 27 3

.. ..

Plot the points in a coordinate system. The points are on a straight line. Findthe equation of the line of the form y kx m= + .

How many triangles can be formed from 20 matches if you connect the trianglesas in the pictures above? Comment on your answer and draw a conclusion about

the number of matches needed to form a row of triangles in this way.



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What will happen if you instead form a row of squares in the same way as shownin the pictures below? Write down and describe a relation between the number of

matches and the number of connected squares.

One square. Two squares. Three squares placed in a row.

A polygon is sometimes called an n-gon, where n is a positive integer thatstates the number of corners. Imagine that you form a line of a certain kind of

n-gons that are connected in the same way as before.

Try to find the relation between the number of matches and the number of connected

n-gons. Describe this relation in words and with a formula. Explain why your

relation is true to all n-gons. (3/4/)

Suggested solutions:Using the data tabulated inthe table to the west wemay find the equation ofthe linear function:

x: number ofmatches

Y: number of trianglesbuild by x matches

3 1

5 27 3

.. ..

mkxy +=

2

1

35

12

12

12 =

=

=

xx

yyk

2

1

2

31

2

313

2

11

2

1==+=+=+=+= mmmmmxymkxy

Answer:


2

1

2

1= xy [1/0]

( ) ( ) ( ) 9205.92

110

2

120

2

120 ==== yy

From 20 matches 9 triangles in a row can be formed. [1/0]

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

3 5 7 9 11 13 15 17 19 21 23 25 27 29

x: Number of matches

y:Numberoftriangles 2

1

2

1= xy

Note that two extramatches are neededto form one more tri-angle in a row. On theother hand due to the

fact that2

1

2

1= xy to

make a complete set

of triangles in a rownumber of matchesmust be odd. [1/0]


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Using the data tabulated in the table below we may find equation of thelinear function:

x: number ofmatches

Y: number of squaresbuild by x matches

4 1

7 210 3

.. ..

mkxy +=

3

1

47

12

12

12 =

=

=

xx

yyk


3

1

3

41

3

414

3

11

3


Answer: From x matches3

1

3

1= xy squares in a row can be formed.

[0/1]Note that threeextra matches are needed to form one more square in a

row. On the other hand due to the fact that31

31 = xy to make a com-

plete set of triangles in a row number of matches must be...,3,2,1,043 =+ ii .

Using the data tabulated in the table below we may find equation of thelinear function:

x: number ofmatches

Y: number of pentagonsbuild by x matches

5 1

9 2

13 3.. ..

mkxy +=

4

1

59

12

12

12 =

=

=

xx

yyk

4

1

4

51

4

515

4

11

4



1

4

1= xy pentagons in a row can be formed.

[0/1]Note that fourextra matches are needed to form one more pentagon in

a row. On the other hand due to the fact that4

1

4

1= xy to make a com-

plete set of triangles in a row number of matches must be...,3,2,1,054 =+ ii .

We may generalize the method and conclude that


1

1

1

=

nx

ny n-gons in a row can be formed.

[0/1/M1]Note that extra matches are needed to form one more n-gonin a1n

row. On the other hand due to the fact that1

1

1

1

=

nx

ny to make a

complete set of n-gonin a row number of matches must be( ) ...,3,2,1,01 =+ inin . This is due to the fact that already one sideof the n-gon is already formed. [0/1/M2M3M5]


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MVG- quality 17



M3 Carries out mathematical proof, or analyses mathematical reasoning.

M5 The presentation is structured, and mathematical language is correct.

MVG - kvalitet MVG-QualityFormulerar och utvecklar problem, anvn-der generella metoder/ modeller vid pro-blemlsning

Formulates and develops the problem, usesgeneral methods with problem solving.

M1

Analyserar och tolkar resultat, drar slutsat-ser samt bedmer rimlighet

Analyses and interprets the results, con-cludes and evaluates if they are reasonable.

M2

Genomfr bevis och/eller analyserar mate-matiska resonemang

Carries out mathematical proof, or analysesmathematical reasoning.

M3

Vrderar och jmfr metoder och/eller mo-deller

Evaluates and compares different methodsand mathematical models.

M4

Redovisar vlstrukturerat med korrekt ma-

tematiskt

The presentation is structured, and mathe-

matical language is correct.

M5

Uppgiften ska bedmas med s.k. aspektbedmning. Bedmningsanvisningarna innehller

tv delar:

Frst beskrivs i en tabell olika kvalitativa niver fr tre olika aspekter p kunskapsom lraren ska ta hnsyn till vid bedmningen av elevens arbete.

Drefter ges exempel p bedmda elevlsningar med kommentarer och pongstt-ning.



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20/28

NpMaB vt 2005 Version 1

20

Skolverket hnvisar generellt betrffande provmaterial till bestmmelsen om sekretess i

4 kap. 3 sekretesslagen. Fr detta material gller sekretessen fram till och med den 10

juni 2005.

Bedmningsanvisningar (MaB vt 2005)

Exempel p ett godtagbart svar anges inom parentes. Bedmningen godtagbar ska

tolkas utifrn den undervisning som fregtt provet. Till en del uppgifter r bedmda

elevlsningar bifogade fr att ange nivn p bedmningen.

Uppg. Bedmningsanvisningar Pong

Del I

1. Max 4/0

a) Redovisad godtagbar bestmning av en rot +1 g

Redovisad godtagbar bestmning av ytterligare en rot ( ) +1 g2,4 21 == xx

b) Redovisad godtagbar bestmning av en rot +1 g

Redovisad godtagbar bestmning av ytterligare en rot ( ) +1 g0,4 21 == xx

2. Max 1/0

Korrekt svar (IT-fretaget, 30) +1 g

3. Max 2/0

a) Redovisad godtagbar lsning (40) +1 g

b) Korrekt svar utifrn lsningen i a) (t.ex. B och C) +1 g

4. Max 3/0

a) Korrekt svar ( 32 += xy ) +1 g

b) Redovisad godtagbar lsning (Ja) +1 g

c) Godtagbart ritad linje, t.ex. en linje genom punkterna (0, 0) och (4, 3) +1 g

5. Max 2/0

Redovisad godtagbar metod +1 g

med korrekt svar ( 6,2 == yx ) +1 g


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21


6. Max 2/0

Redovisad godtagbar lsning ( 2


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22


12. Max 0/2

Redovisad godtagbar ansats, t.ex. korrekt uppstlld ekvation +1 vg

med i vrigt redovisad godtagbar lsning (860 meter) +1 vg

13. Max 0/2

Visat insikt i att alla i bortfallsgruppen kan rsta antingenJa ellerNej +1 vg

Godtagbar berkning av grnserna (38 % och 93 %) +1 vg

14. Max 3/2/

a) Minst en korrekt kombination +1 g

Alla kombinationerna korrekta ( I h, II f, III g) +1 g

b) Korrekt svar (50) +1 g

c) Korrekt svar (20) +1 vg

d) Redovisad godtagbar lsning ( )20( xxy = ) +1 vg

MVG-kvalitet visar eleven i denna uppgift genom att:

Formulerar och utvecklar problem, an-

vnder generella metoder/modeller vid

problemlsning

formulera problemet, det vill sga stlla upp en

korrekt funktion som modell fr hundgrdens

area.*

Analyserar och tolkar resultat, drar slut-

satser samt bedmer rimlighet

analysera och tolka situationen med tillhrande

graf, t.ex. genom att ange en korrekt definitions-

mngd fr funktionen.

Genomfr bevis och analyserar matema-

tiska resonemang

Vrderar och jmfr metoder/modeller

Redovisar vlstrukturerat med korrekt

matematiskt sprk

* Eftersom denna uppgift krver MVG-kvalitet fr sin lsning s kommer godtagbara

elevlsningar att ge vg-pong och visa p MVG-kvaliteter p samma gng.


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15. Max 0/3/

Redovisad godtagbar ansats, t.ex. korrekt uppstlld ekvation fr berkning

av hjd eller bredd fr en av tv-apparaterna +1 vg

med i vrigt redovisad godtagbar lsning(Tv:n i standardformat har strst bildskrmsyta) +1-2 vg

MVG-kvalitet visar eleven i denna uppgift genom att:

Formulerar och utvecklar problem, an-

vnder generella metoder/modeller vid

problemlsning

formulera och utveckla problemet, t.ex. genom att

fra in lmpliga variabler och stlla upp relevanta

matematiska samband fr att lsa problemet.*

Analyserar och tolkar resultat, drar slut-

satser samt bedmer rimlighet

korrekt analysera och tolka sina resultat och dra

slutsatsen att standardformat ger strsta bild-

skrmsytan.Genomfr bevis och analyserar matema-

tiska resonemang

Vrderar och jmfr metoder/modeller

Redovisar vlstrukturerat med korrekt

matematiskt sprk

redovisa vlstrukturerat med ett lmpligt och i hu-

vudsak korrekt matematiskt sprk.

* Eftersom denna uppgift krver MVG-kvalitet fr sin lsning s kommer godtagbara

elevlsningar att ge vg-pong och visa p MVG-kvaliteter p samma gng.

16


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17. Max 3/4/

Uppgiften ska bedmas med s.k. aspektbedmning. Bedmningsanvisningarna innehller

tv delar:

Frst beskrivs i en tabell olika kvalitativa niver fr tre olika aspekter p kunskap somlraren ska ta hnsyn till vid bedmningen av elevens arbete. Drefter ges exempel p bedmda elevlsningar med kommentarer och pongsttning.

KravgrnserDetta prov kan ge maximalt 47 pong, varav 28 g-pong.

Undre grns fr provbetyget

Godknd: 14 pong.Vl godknd: 27 pong varav minst 6 vg-pong.Mycket vl godknd: Fr provbetyget Mycket vl godknd gller utver kraven frVl godknd att eleven ska ha visat prov p minst tre av de fyra MVG kvaliteter som de

-mrkta uppgifterna ger mjlighet att visa (se tabellen nedan). Eleven ska dessutom haminst 12 vg-pong.

MVG- quality 14d 15 16 17 Other

Problems

M1 Formulates and develops the problem, uses generalmethods with problem solving.

M2 Analyses and interprets the results, concludes andevaluates if they are reasonable.

M3 Carries out mathematical proof, or analyses mathe-matical reasoning.

M4 Evaluates and compares different methods andmathematical models.

M5 The presentation is structured, and mathematicallanguage is correct.


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MVG - kvalitet MVG-QualityFormulerar och utvecklar problem, an-vnder generella metoder/ modeller vidproblemlsning

Formulates and develops the problem,uses general methods with problem solv-ing.

M1

Analyserar och tolkar resultat, drar slut-satser samt bedmer rimlighet

Analyses and interprets the results, con-cludes and evaluates if they are reason-able.

M2

Genomfr bevis och/eller analyserar ma-

tematiska resonemang

Carries out mathematical proof, or analy-

ses mathematical reasoning.

M3

Vrderar och jmfr metoder och/ellermodeller

Evaluates and compares different methodsand mathematical models.

M4

Redovisar vlstrukturerat med korrektmatematiskt

The presentation is structured, andmathematical language is correct.

M5


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Goals

Goals that pupils should have attained on completion of the course

Pupils should:

be able to formulate, analyse and solve mathematical problems of importance for appli-

cations and their selected study orientations with an in-depth knowledge of concepts

and methods learned in earlier courses

be able to explain, prove and when solving problems, use some important propositionsfrom classical geometry

be able to calculate probabilities for simple random trials and multi-stage random trials

as well as be able to estimate probabilities by studying relative frequencies

use with judgement different types of status indicators for statistical material, and be

able to explain the difference between them, as well as be familiar with and interpret

some measures of dispersion

be able to plan, carry out and report a statistical study, and in this context be able to

discuss different types of errors, as well as evaluate the results

be able to interpret, simplify and reformulate expressions of the second degree, as well

as solve quadratic equations and apply this knowledge in solving problems

be able to work with linear equations in different forms, as well as solve linear differ-

ences and equation systems with graphic and algebraic methods

be able to explain the properties of a function, as well as be able to set up, interpret and

use some non-linear functions as models for real processes, and in connection with this

be able to work both with and without computers and graphic drawing aids.

Grading criteriaCriteria for Pass

Pupils use appropriate mathematical concepts, methods, models and procedures to for-

mulate and solve problems in one step.

Pupils carry out mathematical reasoning, both orally and in writing.

Pupils use mathematical terms, symbols and conventions, as well as carry out calcula-

tions in such a way that it is possible to follow, understand and examine the thinking

expressed.

Pupils differentiate between guesses and assumptions from given facts, as well as de-ductions and proof.

Criteria for Pass with distinction

Pupils use appropriate mathematical concepts, methods, models and procedures to for-

mulate and solve different types of problems.

Pupils participate in and carry out mathematical reasoning, both orally and in writing.

Pupils provide mathematical interpretations of situations and events, as well as carry out

and present their work with logical reasoning, both orally and in writing.

Pupils use mathematical terms, symbols and conventions, as well as carry out calcula-tions in such a way that it is easy to follow, understand and examine the thinking they

express, both orally and in writing.


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27

Pupils demonstrate accuracy concerning calculations and solutions to different kinds of

problems, and use their knowledge from different fields of mathematics.

Pupils give examples of how mathematics has developed and been used throughout his-

tory, and the importance it has in our time in a number of different areas.

Criteria for Pass with special distinction

Pupils formulate and develop problems, choose general methods and models for prob-

lem solving, as well as demonstrate clear thinking in correct mathematical language.

Pupils analyse and interpret the results from different kinds of mathematical reasoning

and problem solving.

Pupils participate in mathematical discussions and provide mathematical proof, both

orally and in writing.

Pupils evaluate and compare different methods, draw conclusions from different types

of mathematical problems and solutions, as well as assess the reasonableness and valid-

ity of their conclusions.

Pupils describe some of the influences of mathematics in the past and present on the

development of our working and societal life, as well as on our culture.

Ml fr matematik kurs B Kursplan 2000Geometri (G)G3. kunna frklara, bevisa och vid problemlsning anvnda ngra viktiga satser frnklassisk geometri,

Statistik (S)S2. kunna berkna sannolikheter vid enkla slumpfrsk och slumpfrsk i flera stegsamt kunna uppskatta sannolikheter genom att studera relativa frekvenser,

S3. med omdme anvnda olika lgesmtt fr statistiska material och kunna frklaraskillnaden mellan dem samt knna till och tolka ngra spridningsmtt,

S4. kunna planera genomfra och rapportera en statistisk underskning och i dettasammanhang kunna diskutera olika typer av fel samt vrdera resultatet,

Algebra (A)A3. kunna tolka frenkla och omforma uttryck av andra graden samt lsaandragradsekvationer och tillmpa kunskaperna vid problemlsning,

A4. kunna arbeta med rta linjens ekvation i olika former

A5. lsa linjra olikheter och ekvationssystem med grafiska och algebraiska

metoder,

Funktionslra (F)F2. kunna frklara vad som knnetecknar en funktion samt kunna stlla upp, tolka ochanvnda ngra icke-linjra funktioner som modeller fr verkliga frlopp och i samband

drmed kunna arbeta bde med

och utan dator och grafritande hjlpmedel,

vrigt()1. kunna formulera, analysera och lsa matematiska problem av betydelse frtillmpningar och vald studieinriktning

4. med frdjupad kunskap om sdana begrepp och metoder som ingr i tidigarekurser,


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Betygskriterier 2000Kriterier fr betyget GodkndG1: Eleven anvnder lmpliga matematiska begrepp, metoder och tillvgagngsstt fratt formulera och lsa problem i ett steg.

G2: Eleven genomfr matematiska resonemang svl muntligt som skriftligt.G3: Eleven anvnder matematiska termer, symboler och konventioner samt utfr berk-ningar p ett sdant stt att det r mjligt att flja, frst och prva de tankar som kom-

mer till uttryck.G4: Eleven skiljer gissningar och antaganden frn givna fakta och hrledningar eller

bevis.

Kriterier fr betyget Vl godkndV1: Eleven anvnder lmpliga matematiska begrepp, metoder, modeller och tillvga-gngsstt fr att formulera och lsa olika typer av problem.

V2: Eleven deltar i och genomfr matematiska resonemang svl muntligt som skrift-ligt.

V3: Eleven gr matematiska tolkningar av situationer eller hndelser samt genomfroch redovisar sitt arbete med logiska resonemang svl muntligt som skriftligt.

V4: Eleven anvnder matematiska termer, symboler och konventioner p sdant stt attdet r ltt att flja, frst och prva de tankar som kommer till uttryck svl muntligtsom skriftligt.

V5: Eleven visar skerhet betrffande berkningar och lsning av olika typer av pro-blem och anvnder sina kunskaper frn olika delomrden av matematiken.

V6: Eleven ger exempel p hur matematiken utvecklats och anvnts genom historienoch vilken betydelse den har i vr tid inom ngra olika omrden.

Kriterier fr betyget Mycket vl godkndM1: Eleven formulerar och utvecklar problem, vljer generella metoder och modellervid problemlsning samt redovisar en klar tankegng med korrekt matematiskt sprk.

M2: Eleven analyserar och tolkar resultat frn olika typer av matematisk problemls-

ning och matematiska resonemang.M3: Eleven deltar i matematiska samtal och genomfr svl muntligt som skriftligt ma-tematiska bevis.

M4: Eleven vrderar och jmfr olika metoder, drar slutsatser frn olika typer av mate-matiska problem och lsningar samt bedmer slutsatsernas rimlighet och giltighet.

M5: Eleven redogr fr ngot av det inflytande matematiken har och har haft fr ut-vecklingen av vrt arbets- och samhllsliv samt fr vr kultur.

Kravgrnser

Detta prov kan ge maximalt 47 pong, varav 28 g-pong.

Undre grns fr provbetyget

Godknd: 14 pong.Vl godknd: 27 pong varav minst 6 vg-pong.

Mycket vl godknd: Fr provbetyget Mycket vl godknd gller utver kraven fr Vl

godknd att eleven ska ha visat prov p minst tre av de fyra MVGkvaliteter

som de -mrkta uppgifterna ger mjlighet att visa (se tabellen

nedan). Eleven ska dessutom ha minst 12 vg-pong.

Documents

Solutions+NPMaBVT05 Eng