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Räkna med bråk i skolan - Göteborgs universitet · PDF file Räkna med bråk i skolan En litteraturstudie om elevers inlärning av bråkräkning i skolan Namn: Pontus Andersson

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  • Räkna med bråk i skolan

    En litteraturstudie om elevers inlärning av bråkräkning i skolan

    Namn: Pontus Andersson &

    Simon Ingvarsson

    Program: Ämneslärarprogrammet

  • 2

    Abstract

    Master thesis: 15 hp

    Course: LGMA1G

    Grade: Elementary

    Semester: HT16

    Supervisor: Johan Wästlund

    Examiner: Laura Fainsilber

    Code: HT16-3001-006-LGMA1G

    Keywords: fraction calculations, mathematics, mathematic didactics, fractions

    misconceptions, rational numbers and literature study.

    Summary Fractions and fraction calculation, is something that students seem to have problems with,

    even though it is a subject that is following them from the early ages of preschool, all the way

    up to the upper secondary school, which is something that both PISA and TIMSS studies can

    agree on. This literature study has its focus in the mathematics of upper secondary school,

    where problems and misconceptions regarding fractions become apparent, compared with the

    elementary school. The Swedish curriculum LGY11 does not even mention the word fraction

    once, even though it is a huge part of the mathematics of the upper secondary school. Almost

    all series of literature have chapters specifically involving fractions, even though it is not

    required form the curriculum. Studies made by didactics have proven that calculation with

    fractions is much more complicated than it seems at first glance and that it requires a solid

    foundation of mathematical understanding, which is something that not all students or

    teachers possess. Our conclusion, is that the problems regarding fraction calculation, has its

    roots during the elementary school, where the foundation of mathematical understanding is

    created. We highlight some of the aspects regarding the understanding of the mathematics of

    fractions, and compare them with teaching materials of the upper secondary school. Acknowledgments We would like to start off with thanking Johan Wästlund for being our supervisor and for his

    great ideas throughout the work phase. We would also like to thank Thomas Lingefjärd for his

    teachings of Geogebra, since we had great use of it during the creation of demonstration

    images. Laura Fainsilber will also receive our thanks for all the advice regarding possible

    literature during the creation of this thesis. We would also like to thank Johanna Pejlare for

    taking her time to read through the didactic aspects of this thesis. A big thanks goes to Kurt

    and Helena Andersson for reading our paper and helping us with grammar, it is greatly

    appreciated. At last we want to thank all the participants for taking part of our interview.

  • 3

    Innehållsförteckning

    1 Bakgrund ................................................................................................................................ 8

    1.1 Syfte och Frågeställningar .............................................................................................. 8

    1.2 Metod .............................................................................................................................. 8

    2 Vad är ett bråk? ..................................................................................................................... 9

    2.1 Varför skall vi kunna bråkräkning? .............................................................................. 10

    2.2 Räknelagar .................................................................................................................... 10

    2.2.1 Kommutativa lagen ..................................................................................... 11

    2.2.2 Associativa lagen ........................................................................................ 11

    2.2.3 Distributiva lagen ........................................................................................ 11

    2.3 Räkna med bråk ............................................................................................................ 12

    2.3.1 Det rationella talet ....................................................................................... 12

    2.3.2 Addition och subtraktion med samma nämnare .......................................... 13

    2.3.3 Multiplikation av bråk .................................................................................. 15

    2.3.4 Förlänga och förkorta .................................................................................. 17

    2.3.4.1 Addition och subtraktion med olika nämnare .................................. 18

    2.3.5 Minsta Gemensamma Nämnare (MGN) ...................................................... 18

    2.3.6 Division av bråk .......................................................................................... 19

    2.3.6.1 Omvandling till decimaltal ................................................................ 20

    2.3.6.2 Omvandling från decimaltal ............................................................. 20

    2.3.6.3 Distributiva lagen ............................................................................. 21

    2.3.6.4 Division utan multiplikation .............................................................. 21

    2.3.7 Jämförelse av bråk. .................................................................................... 22

    2.4 Bråkets historia ............................................................................................................. 23

    2.4.1 Egyptisk division ......................................................................................... 24

    2.4.2 Sylvesters algoritm för division .................................................................... 26

    3 Skolverkets styrdokument och bråk ................................................................................... 27

    4 Analys av gymnasiala matematikböcker ........................................................................... 28

    4.1 Origo 1b & 1c ............................................................................................................... 28

    4.1.1 Förkorta och förlänga bråk och minsta gemensamma nämnare .................. 28

    4.1.2 Addition och subtraktion av bråk ................................................................. 28

    4.1.3 Multiplikation ............................................................................................... 29

    4.1.4 Inverterade tal ............................................................................................. 29

    4.1.5 Division ....................................................................................................... 29

    4.2 Exponent 1a .................................................................................................................. 29

  • 4

    4.2.1 Addition och subtraktion med bråk .............................................................. 30

    4.2.2 Förkortning och Förlängning ....................................................................... 30

    4.2.3 Bråkform och blandad form ......................................................................... 31

    4.2.4 Från bråktal till decimaltal ........................................................................... 31

    4.2.5 Minsta gemensamma nämnare ................................................................... 31

    4.2.6 Multiplikation med bråk ............................................................................... 32

    4.3 Exponent 1b .................................................................................................................. 32

    4.3.1 Rationella tal ............................................................................................... 32

    4.3.2 Förlänga och förkorta ett bråk ..................................................................... 32

    4.3.3 Jämföra två bråk ......................................................................................... 33

    4.3.4 Minsta gemensamma nämnare ................................................................... 33

    4.3.5 Addition och subtraktion .............................................................................. 33

    4.3.6 Multiplikation ............................................................................................... 33

    4.3.7 Inverterade tal ............................................................................................. 33

    4.3.8 Division ....................................................................................................... 33

    4.4 Exponent 1c .................................................................................................................. 34

    4.4.1 Stam- och kedjebråk ................................................................................... 34

    4.4.2 Bråkform och decimalform .......................................................................... 34

    4.4.3 Problemlösning ........................................................................................... 34

    4.4.4 Avslutning ................................................................................................... 34

    4.5 Matematik 5000 1a .......................................................