Concrete, Symbolic and Pictorial Representations of Fractions

Fractions are definitely hard to grasp.  Just when the children demonstrated understanding, they asked a question that showed me that there was more work to do.  Last class we struggled with connecting fractions and division. After a conversation with Jean, a mathematician extraordinaire, I came armed with geometric shapes to explore the connection between division and fractions.  The yellow hexagon can be represented as halves (red), thirds (blue) and sixths (green).  The girls were able to show 1/2, 1/3 and 1/6 quite easily with the blocks.   They were also able to show that 1 divided by 3 equals the one of the blue blocks.

Hexagon shapesNext, we worked on placing fractions on a number line.  I followed Heitschmidt’s (2008) Fractional Clotheslines lesson.   I tied string across the room and placed a ‘0’ on one end and a ‘4’ on the other.  The girls were challenged to place 1, 2, and 3 on the line.  At first the girls placed the numbers in the correct order, but unevenly spaced on the line.  I asked, “is what a number line looks like?” After a conversation amongst themselves, they spread the numbers equidistance apart.

Then, I gave them a card with the fraction 3/4 to put on the line.  Finding where 3/4 fit took a bit of conversation. The placed the card at approximately 1/4. The girls demonstrated understandings of unit fractions, but hesitated with non-unit fractions.  We pulled out the fraction strips from last class and counted three of four quarters.  That seemed to help and they moved the fraction card.

Placing 1⅓  on the number line took a long time.  A fraction that was greater than 1 threw them a bit of a loop.  I resorted to discussing dividing cookies with siblings.  They could understand that two cookies could be divided into fractions. The connection to cookies, siblings and division helped the light bulbs come on and M & J placed 1⅓ accurately.
There are two Grade 4 and one Grade 5 specific outcomes in the Program of Studies about fractions.

Grade 4: Demonstrate an understanding of fractions less than or equal to one by using concrete, pictorial and symbolic representations to:

  1. Name and record fractions for the parts of a 
whole or a set
  2. Compare and order fractions
  3. Model and explain that for different wholes, two 
identical fractions may not represent the same 
quantity
  4. Provide examples of where fractions are used.

Grade 5: Demonstrate an understanding of fractions by using concrete, pictorial and symbolic representations to:

  1. Create sets of equivalent fractions
  2. Compare fractions with like and unlike denominators. (Alberta Education, 2007, p. 18)

Connecting fractions to quantity and the number line are challenging.  If there was a ranking system for the specific outcomes in the Program of Studies, they would get a 5 star alert for difficulty.  These outcomes are likely the most difficult concepts to grasp in both grades.  Teaching fractions is known internationally to be difficult (Fazio & Siegler, 2011).

Each different representation of fractions caused confusion for the girls. With the hexagon blocks, the girls showed concrete representation of quantity.  The number line was symbolic and turned things topsy turvy for the girls. Placing three different fraction cards on the string number line was enough for the day.

Next, on the agenda was Colour Iraq.  This problem is based on the Four Colour Theorem that states any map can be coloured with just four colours with no borders sharing the same colour.  Once the girls successfully coloured their map, I asked them to represent each colour as a fraction of the total provinces in Iraq.

The girls had no problem representing the number of colours as a fraction of the total.  Parts 1 and 4 of the Grade 4 specific outcome above were addressed. However, being able to “name and record fractions for the parts of the whole” demonstrates a procedural understanding.  I am not convinced they understand what fractions really mean, yet.   In other words, their understanding is still superficial.  We have more exploring to do.

Krista

Alberta Education. (2007). Mathematics: Kindergarten to grade 9. Alberta Government. Retrieved from http://www.education.gov.ab.ca/k%5F12/curriculum/bySubject/math/Kto9Math.pdf
Fazio, L., & Siegler, R. (2011). Teaching fractions (Vol. 22). Geneva: International Bureau of Education; International Academy of Education. Retrieved from http://www.unesco.org/ulis/cgi-bin/ulis.pl?catno=212781&set=4F8D848F_3_273&gp=1&lin=1&ll=1
Heitschmidt, C. (2008). Fractional Clothesline. NCTM Illuminations. Retrieved from http://illuminations.nctm.org/LessonDetail.aspx?id=L784