Fraction as a Number: 3.5

• Indicator of Progress
• Teaching Strategies

Supporting materials

• Related Progression Point
• Developmental Overview of Proportional Reasoning and Multiplicative Thinking (PDF - 41Kb)
• Developmental Overview of Numbers and Operations (PDF - 2.8Mb)

Indicator of Progress

An important step in mathematical development is that students come to see a fraction ^a/_b as one number, even though it is written using two whole numbers, a and b.

The concept of fraction begins as a way of describing part-whole relationships. If I am getting two fifths of the pizza, then my part is obtained by dividing the pizza into 5 equal parts, and taking 2 of them. The fraction ²/₅ describes my part in relation to the whole pizza. In this first stage, the fraction describes the relationship between two quantities, and can be used as an operator on one quantity to get another. For example, students can observe that girls make up half of the class, they can describe two-fifths of the pizza and they can discover that if I take a half of a third of a pizza, then I get a sixth of a pizza.

Fractions can be used in these ways before students fully consider them as numbers in their own right. Using a linear model for fractions helps students to see a fraction as a number, with a place on the number line. Click here to find out more about models for fractions.

 

Illustration 1: Placing a fraction on a number line

A student who does not know that ²/₃ is one number will have trouble marking this point on a number line as a point between 0 and 1. Instead of marking the point ²/₃, they may mark ² /₃ of the given line (using the fraction as an operator), or one or both of the whole numbers 2 and 3.

Before they see a fraction as one number, students cannot place fractions on a number line and are unable to use the number line to model operations with fractions. They have not extended their number system from their original appreciation of the set of whole numbers.

 

Illustration 2: Number sense, especially with addition and subtraction of fractions

A student who does not know that ²/₃ is one number and ⁷/₈ is another number, will have trouble comparing their size. The student may be unable to estimate that the sum of these will be less than 2 because each number is less than 1.

 

Teaching Strategies

Many mathematical concepts undergo considerable refinement as students progress through the levels. Fraction concepts begin as a relation between a part and a whole, and as an operator connecting two quantities and as the outcome of division. The idea that a fraction is a number comes later as a culmination of these ideas.

The use of a linear model is recommended to develop the concept of a fraction as one number. Using a linear model is also an important step in the development of the concept of a number line. In order to assist students it is useful to know their current levels of understanding, so a diagnostic test is provided.

Activity 1: Comparing Fractions Diagnostic Test provides a diagnostic test to determine how students are conceptualising fractions.
Activity 2: Using area models better provides guidance on ensuring that fraction concepts are not lost when concrete models are used. A short diagnostic task to see if students understand that area models show the fraction as a part-whole relationship is included.
Activity 3: Number Between is a game that highlights the position of fractions on a number line, emphasises relative size, develops number sense and shows the property of number density for fractions.

 

Activity 1: Comparing Fractions Diagnostic Test

The Comparing Fractions Diagnostic Test (Word - 76Kb) will assist teachers to identify the misconceptions about fractions held by their students. The test asks students to compare 12 pairs of fractions. The instruction for the test is “For each pair of fractions, either circle the larger fraction or write = between them”. Look at a student’s test and try to match the pattern of choices with one of the samples on the Comparing Fractions Classification Sheet (Word - 235Kb). You will find that some students focus only on the denominator of the fractions (choosing either the smaller or the larger denominator throughout the test). Other students will focus only on the numerator, again choosing either the smaller or larger numerator throughout the test.

Some students know that they need to pay attention to both numerator and denominator, but combine them incorrectly e.g.

• with multiplication (“³/₅ is somehow like 15 (3 × 5)”)
• with difference (“both ³/₅ and ⁵/₇ have a gap of 2, so they must be equal”)
• by choosing the numbers that look larger (e.g. by adding or averaging them).

 

If students have any of these misconceptions, they will get some questions correct; some correct but for the wrong reasons; and some incorrect. The table below shows that their scores might be quite different, ranging from 1 - 9, but they all have equally bad misconceptions. A few students will be at a lower stage, and may simply circle the larger individual number (e.g. 8) instead of either fraction.

Choice based on:	Number correct (out of 12)	Choice based on:	Number correct (out of 12)
Larger numerator	7-9	Larger sum/product	6
Small numerator	1-3	Smaller sum/product	4
Larger denominator	3-5	Larger gap	7
Smaller denominator	5-7	Smaller gap	1

 

Activity 2: Using area models better

This activity focuses on ensuring that students appreciate that area models rely on considering relative sizes of area. Many fraction activities are left at the stage where students are only using whole number knowledge. For example, some worksheets may give students a series of equal-sized rectangles all pre-divided into 8 equal parts (by area) and ask students to colour ¹/₈ (i.e. one part), ³/₈ (three parts) etc.

This can easily be just a colouring-in and whole number counting exercise for students (count how many equal parts, count how many shaded, write them down), unless they are asked to look at the coloured diagrams and make comments on the proportion of the rectangle that has been shaded.

To emphasise the fraction concept (i.e. relative amount), ask questions such as:

(i) "Has more than a half of the rectangle been shaded here?"

(ii) "Show me two rectangles of different sizes with about the same fraction shaded."

(iii) "If I shaded this circle inside the rectangle, about what fraction would be shaded?" (Accept answers such as "less than a half", "about a third", etc.)

Developing the idea of the relative amount of the area that has been shaded is not possible unless the size of the basic rectangle is sometimes varied within one exercise.

Diagnostic task

Students who understand that a fraction indicates the relative amount and the part-whole relationship, not just the number of pieces shaded, will be able to explain why the thinking of the student who drew the pictures below is correct.

Toula drew this picture to show that a quarter is bigger than a sixth, and that three quarters is less than five sixths. She said: " In the first two circles, the orange quarter is shading more of the circle than the green sixth. A quarter is more of the whole shape than a sixth is. [In the two circles on the right], more of the yellow circle is showing, than of the blue circle. This means that five sixths of a circle is more than three quarters of a circle. It does not matter how big the circle is ."

 

Activity 3: Play 'Number Between'

This activity focuses on the 'density' of the real number line: we can always place a real number in between two numbers.

Draw a number line on the board. It may be from 0 to 1, or from 0 to 3, or some other range. Students in turn have to place a fraction reasonably accurately on the number line in a specified region. Be sure to label the fraction. Continue, placing the new fractions within progressively smaller intervals.

For example, start with number line [0,1]. The first student places a fraction (e.g. ¹/₅) on the line. The next student has to place a fraction between ¹/₅ and 1 (e.g. ²/₃ ). The next student has to place a fraction between ²/₃ and 1 (e.g. ⁷/₁₀). The next student has to place a fraction between ⁷/₁₀ and 1, etc.

Variations include making the fractions smaller each time (i.e., getting closer to 0, or the smaller end of the interval), or placing them in the central regions (e.g., between ¹/₅ and ²/₃), etc.

There are many opportunities for class discussion arising from this game, e.g. to check suggestions.

Of course, this game is much easier to play with decimals, than with common fractions. The ease of comparing decimals gives a great advantage over fractions. (Click here for Number Between as a decimal game)