Saturday, February 27, 2010
Fractions: Equal Parts
The mathematics education course focuses on curriculum structure, teaching strategies including ICT and students' responses (errors). For example in the Diploma programme, the first course (of three or four) includes Teaching of Whole Numbers, Teaching of Fractions, Decimals and Percent, Teaching of Ratio and Teaching of Rate and Speed.
Student teachers also learn the Singapore curriculum framework, learning theories, scheme of work and lesson planning, Ministry of Education initiativces including the role of calculators and ICT.
In a lesson on Teaching of Fractions, teachers were presented with this problem - is a rectangle cut into four equal parts by its two diagonals?
There were student teachers who said that the parts are not all equal because the triangles are not 'the same' meaning not congruent. Subsequent whole-group discussion with paper rectangles and a pair of scissors led to four responses that are shown at the top of this page.
There was a response that suggested cutting each the the four parts into two triangles which are equal parts as they are congruent triangles. Each of the four parts is then clearly 2 eighths. See Photograph 1.
A second response is based on using the formula to calculate the area of triangle. On Photograph 2, it can be seen that B = 2h and b = 2H. The area of the two non-conguent triangles can be shown to be the same.
Another response includes cutting each of the four parts into two congruent triangles and rearranging the pieces to form congruent triangles. Thus, the triangle shaded black and the one shaded red in Photograph 3 can be rearranged to form congruent triangles.
Finally there was another response that is based on using two congruent triangles to form a rhombus. This resulted in two congruent rhombii. If the two rhombii are equal, it follows that half of one rhombus (one of the triangles) is equal to half of the other (the other triangle).
The student teachers were told that in mathematics classrooms, teachers use different models to show fractions. In the example, we use the area model - where we use area to represent fractions. There is also the length model (e.g. the bar model or line model) where length is used to represent fractions. Other models include the volume model (e.g. using a cylinder of water to show fractions).
In terms of curriculum structure, we learn that equal parts is an important concept in understanding of the fraction notation. In terms of common misconceptions, we see that some students may have the wrong idea that equal parts refer to congruent parts rather than parts with equal area.
In Singapore curriculum, fraction is introduced formally in Primary 2.