# 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

Program: Ämneslärarprogrammet

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Abstract

Master thesis: 15 hp

Course: LGMA1G

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.

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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

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

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

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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

4.3.8 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 .......................................................

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