At a Glance...

B1.6 describe relationships and show equivalences among fractions and decimal numbers up to thousandths, using appropriate tools and drawings, in various contexts

Skill: Describing Relationships and Representing Equivalences Among Fractions, Decimal Numbers to Thousandths, and Percents, Using Appropriate Tools and Diagrams, in a Variety of Contexts

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Students should understand that since decimal notation is just another way of representing a decimal fraction, then it is possible to establish an equality relationship between the two notations (for example, \(0.3\; = \frac{3}{{10}}\)). By recognizing this equality, they are able to associate a place value with each of the decimals that make up a decimal number, that is, tenths, hundredths, thousandths and so on.

Example:

Students who do not understand this association are sometimes inclined to represent a fraction such as \(\frac{2}{5}\) by 0.2 or 0.25. The following description describes observable behaviours of students who have developed a conceptual understanding of numbers and the relationship between their different representations.

Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 46.

Conceptual Understanding of Numbers

Number: 0.3

Observable behaviours:

• The student can read it (three tenths).
• The student can write the number as a fraction that is, \(\frac{3}{{10}}\).
• The student can represent the number using concrete or semi-concrete materials.

Example

Number : \(\frac{16}{100}\)

Observable behaviours:

• The student can read it (sixteen hundredths).
• The student can write the number as a decimal, that is 0.16.
• The student can represent the number using concrete or semi-concrete materials.

Example

Number : \(\frac{2}{7}\)

Observable behaviours:

• The student can read it (two sevenths).
• The student knows that the fraction \(\frac{2}{7}\) is not represented by 0.2 since they know that 0.2 = \(\frac{2}{10}\).
  

To establish the equality relationship between a fraction whose denominator is not a power of ten (for example,\(\frac{1}{4}\)) and the corresponding decimal number, it is necessary to use the concept of equivalent fractions. For example, students can use strips of equal length as shown below to see that \(\frac{1}{4}\) is between \(\frac{2}{10}\) and \(\frac{3}{10}\).

They can then subdivide the tenths into ten equal parts, thus creating one hundred equal parts, or hundredths of the whole, and recognize that \(\frac{25}{100}\) is an equivalent fraction to \( \frac{1}{4}\).

Since \(\frac{{25}}{{100}}\; = \;0.25\), they can conclude that the fraction \(\frac{1}{4}\) can also be represented in decimal notation by 0.25 (\(\frac{1}{4}\; = \;\frac{{25}}{{100}}\; = \;0.25\)). This kind of example allows students to recognize that all fractions that can be expressed by an equivalent decimal fraction can be represented by a decimal number. This is particularly the case for fractions expressed in halves, fourths, fifths and twentieths, as shown in the following table.

Note: Some fractions (for example, \(\frac{2}{3}\), \(\frac{3}{7}\), \(\frac{5}{{11}}\)) cannot can be represented by an equivalent decimal fraction. They can, however, be expressed by decimal numbers with a repeating decimal part (for example, \(\frac{2}{3}\; = \;0,\overline{6}\); \(\frac{3 }{7}\; = \;0,\overline{428 \ 571}\); \(\frac{5}{{11}}\; = \;0,\overline{45}\)), the decimal representation being obtained by dividing the numerator by the denominator.

Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p 47-48.