Early Fraction Ideas with Models: 2.5

Indicator of Progress

Success depends on students extending their understanding of fractions to include equivalent fractions and the ability to order fractions by reference to appropriate physical models. Students will appreciate that fractions can be renamed (as equivalent fractions) analogously to the way that whole numbers are renamed and that this renaming does not change the size of the number involved.

Before this, students often do not focus on the proportion of the model being indicated. Possibly they see a fraction just as a pair of whole numbers. This has many consequences, such as preventing them from appreciating equivalent fractions (e.g. they might think that ²/₄ is more than ¹/₂ ). While they are able to recognise certain familiar fractions, their inability to create or recognise equivalent fractions indicates that they do not appreciate the fundamental characteristic of fractions: that ‘more, smaller parts’ can be the same quantity as ‘fewer, larger parts’ of the same whole.

Illustration 1: Incorrect belief that portions must be adjacent

Students need to appreciate which features of fraction models have mathematical significance. For example, with area models (such as circles and rectangles), it is the total area of the portions that determines the size of the fraction modelled, rather than their spatial arrangement. In the following diagrams, the full circle is the whole and each of the diagrams represents ³/₈ of the circle. The arrangement of the portions is irrelevant, although some students will be distracted by this.

Illustration 2: Using the discrete model inappropriately

Some students believe that ⁴/₅ is the same size as ⁵/₆ as they interpret the symbols using a discrete model (such as in the following diagrams) and explain that both 'have one piece missing'. Discrete models can reinforce the idea that fractions are two numbers rather than one (‘4 of the 5 squares are black’) so that students may not appreciate that they are dealing with a new type of number which is, in fact, less than one. This is because discrete objects remind them of their counting experiences with whole numbers.

Students need to ‘stand back’ to see that proportionally more of the second set is coloured.

I have coloured 4 out of 5 squares black, so that's ⁴/₅

 

I have coloured 5 out of 6 squares black, so that's ⁵/₆

 

Illustration 3: Not partitioning the discrete model

Some students will correctly colour ⁴/₅ of a set of 5 objects yet they will incorrectly colour ⁴/₅ of 10 objects (see diagrams below). The incorrect response to Task 2 is because the student does not appreciate that the entire set is firstly to be divided or partitioned into 5 equal portions, of which 4 portions are then coloured. A correct response to Task 1 does NOT indicate fraction understanding; this task reminds some students of the counting and colouring tasks they completed when they started school and they do not see what is new until they encounter Task 2.

Task 1

Colour ⁴/₅ of the squares black

Correct

 

Task 2

Colour ⁴/₅ of the squares black

Incorrect

Illustration 4: Confusion with ordinal names

Some students confuse fraction names with ordinal names.

In this diagram, the student was asked to shade ‘a fifth’ and counted from the left “first, second, third, fourth, fifth” then shaded the fifth piece. This might be correct…

 

… but if the student adopts the same procedure to find ‘a fifth’ in this diagram, then they are definitely thinking only of ordinal names.

In this diagram, another student was asked to shade four fifths and counted along until they found the piece number 4.

Teaching Strategies and Further Resources

For Early Fraction Ideas with Models teaching strategies and further resources, see: Part 2