Fractions and Decimals
Step 1: Identify Outcomes to Address
- What do I want my students to learn?
- What can my students currently understand and do?
- What do I want my students to understand and be able to do, based on the Big Ideas and specific outcomes in the program of studies?
See Sequence of Outcomes from the Program of Studies
Fractions form a number system that includes the whole numbers and also an infinite set of numbers between every whole number. Fractions are numbers that can be written in the form a/b, where a and b are whole numbers, b≠0. The density property of fractions states that between any two fractions there is another fraction. Both the whole numbers and the fractions are infinite but the set of fractions are dense and the whole numbers are not. Also, the set of fractions is closed under addition, multiplication and division, whereas the whole numbers are closed under addition and multiplication; e.g., the sum of any two whole numbers is another whole number.
Number sense with fractions and decimals requires that the students develop a conceptual understanding of fractions and decimals as numbers. To work effectively with fractions and decimals, the students should demonstrate the ability to:
- Represent numbers using words, models, diagrams and symbols and make connections among various representations.
- Give other names for numbers and justify the procedures used to generate the equivalent forms.
- Describe the relative magnitude of numbers by comparing them to common benchmarks, given simple estimates, ordering a set of number, and finding a number between two numbers.
Adapted with permission from James Vance, "Rational Number Sense: Development and Assessment," delta-K 28, 2 (1990), p. 24. delta-K is published by The Alberta Teachers' Association.
The students construct a firm foundation for fraction concepts by experiencing and discussing activities that promote the following understandings:
- Fractional parts are equal shares or equal-sized portions of a whole or unit. A unit can be an object or a collection of things. More abstractly, the unit is counted as 1. On the number line, the distance from 0 to 1 is the unit.
- Fractional parts have special names that tell how many parts of that size are needed to make the whole. For example, thirds require three parts to make a whole.
- The more fractional parts used to make a whole, the smaller the parts. For example, eighths are smaller than fifths.
- The denominator of a fraction indicates by what number the whole has been divided in order to produce the type of part under consideration. Thus, the denominator is a divisor. In practical terms, the denominator names the kind of fractional part that is under consideration. The numerator of a fraction counts or tells how many of the fraction parts (or the type indicated by the denominator) are under consideration. Therefore, the numerator is a multiplier—it indicates a multiple of the given fractional part.
Adapted from Van de Walle, John A., LouAnn H. Lovin, Teaching Student-Centered Mathematics, Grades K–3 (p. 251). Published by Allyn and Bacon, Boston, MA. Copyright © 2006 by Pearson Education. Adapted by permission of the publisher.
The iterative nature of a fraction describes the bottom number (denominator) as telling what is being counted and the top number (numerator) telling what the count is (Van de Walle and Lovin 2006, p. 259).
The three models for fractions include:
- the region or area model, such as two-thirds of a pizza. Note: The equal parts must be the same size (area) but not necessarily the same shape (congruence) (Van de Walle 2001, p. 211)
- the length or measurement model, such as two-thirds of a paper strip or two-thirds of the distance from 0 to 1 on a number line
- set model, such as two-thirds of a set of 3 kittens.
(Van de Walle and Lovin 2006)
Conceptual understanding of comparing fractions is developed when the students use a variety of ways to relate fractions including the following:
- more of the same size in which the denominators of the fractions are the same; e.g., five-eighths is greater than three-eighths
- same number of parts but parts of different sizes in which the numerators of the fractions are the same; e.g., three-quarters is greater than three-fifths
- more or less than one-half or one whole in which numerator of the fraction is compared to the denominator in deciding its relation to a given benchmark; e.g., three-eighths is less than one-half because three is less than half of eight
- distance from one-half or one whole in which the fraction is one fractional part away from a whole or one-half; e.g., four-fifths is greater than three-quarters because they are each one fractional part away from a whole. Fifths are smaller than quarters and therefore four-fifths is closer to a whole than three-quarters.
Adapted from Van de Walle, John A., LouAnn H. Lovin, Teaching Student-Centered Mathematics, Grades K–3 (p. 265). Published by Allyn and Bacon, Boston, MA. Copyright © 2006 by Pearson Education. Adapted with permission of the publisher.
Conceptual understanding of decimals requires that the students connect decimals to whole numbers and to fractions. Decimals are shown as an extension of the whole number system by introducing a new place value, the tenth's place, to the right of the one's place. The tenth's place follows the pattern of the base ten number system by iterating one-tenth ten times to make one whole or a unit (Wheatley and Abshire 2002, p. 152). Similarly, the hundredth's place to the right of the tenth's place iterates one-hundredth ten times to make one-tenth.
The connection between decimals and fractions is developed conceptually when the students read decimals as fractions and represent them using the same visuals. For example, 0.8 is read as eight-tenths and can be represented using fraction strips or decimal strips (Wheatley and Abshire 2002).