B1.3 compare and order integers, decimal numbers, and fractions, separately and in combination, in various contexts
Skills: Reading, Representing, Comparing and Ordering Integers
The positive or negative sign indicates whether the number is greater or less than 0. Thus, if the number is positive, it will be placed to the right of 0 on a horizontal number line or over 0 on a vertical number line. If the number is negative, it will be placed to the left of 0 on a horizontal number line or below 0 on a vertical number line. The reference number for integers is therefore always 0.
A whole number is greater than another whole number if it is placed to the right of it on a horizontal number line or above it on a vertical number line.
For example, \(+ 8 > + 2\) and \(- 2 > - 6\).
Using the number line, it is easier to compare and order whole numbers since a number with multiple digits does not necessarily mean it is larger (for example, -584 is smaller than 8). By analyzing the line above, we can place the numbers in ascending order (-6, -2, 2, and 8) or descending order (8, 2, -2, and -6).
On the number line, the absolute value of a number is its distance from zero. For example, the absolute value of -3 and 3 is 3 since 3 units separate the two numbers from 0.
Source: Closing the Performance Gap , Grade 9, p. 11.
Skill: Comparing and Ordering Decimal Numbers
The order relation is based on the comparison of numbers. One of the great strengths of decimal notation is the speed with which it is possible, thanks to the concept of place value, to compare and order quantities. For example, it is much easier to compare the numbers \(\frac{1}{8}\)and \(\frac{3}{{40}}\)when expressed in decimal notation, i.e. 0.125 and 0.075.
In general, students have little difficulty comparing decimal numbers with the same number of decimal places (for example, 0.341 < 0.462). They have more difficulty comparing numbers with different numbers of decimal places (for example, 1.34 and 1.275). Some tend to compare these numbers without the decimal point (for example, 134 < 1275) and conclude that 1.34 < 1.275. Others reach the same erroneous conclusion by comparing only the numbers to the right of the decimal point (for example, \(34 < 275)). Thus, a number with more digits is not necessarily larger.
The order relationship should be addressed by comparing decimal numbers in contextualized situations. For example: "Remi did a jump of 3.55 m and Samantha did one of 3.7 m. Which of the two had the longer jump?" Students can respond and justify their choice if they understand place value. The number line is a powerful visual model for comparing decimal numbers. To place 3.7 on a number line, students can represent tenths from 3.0 to 4.0. To place 3.55, they must divide the interval between 3.5 and 3.6 into tenths, with each space representing one hundredth. They can then conclude that 3.55 < 3.7, so Samantha has taken a longer jump than Remi.
Students who have acquired good number sense can also compare 3.55 m and 3.7 m by first noticing that they represent two jumps greater than 3 m. Then they can compare the tenths to notice that the first number has 5 tenths, or 5 decimetres, while the second has 7.
The second jump is therefore longer than the first. Students can also, after comparing units, think of 3.7 as 3.70, or 3 metres and 70 centimetres. The number 3.55 represents 3 metres and 55 centimetres. The 3.7 m jump is therefore longer than the 3.55 m jump.
Traditionally, a procedure was taught where you had to add a zero to the end of 3.7 to give two numbers with the same number of decimal places. Then the decimal parts, 55 and 70, were compared to conclude that 3.70 was greater than 3.55. Although the teaching of the method was accompanied by an explanation, so much emphasis was placed on the procedure that the explanation and the concept were quickly lost. Not surprisingly, young people often answered such questions incorrectly. For example, in an international test of sixth graders, 87% reported that 6,987 is greater than 6,879, while only 52% concluded that 1.05 is greater than 1.015 (Brissiaud, 1998). Students who understand the concept of place value do not need to apply a procedure to compare decimal numbers.
Open-ended problems, which offer more than one answer and are thought-provoking, allow students to deepen their
understanding of order relationships. For example:
- Determine three decimal numbers between \(\frac{3}{{40}}\) and \(\frac{1}{8}\).
- Determine three decimal numbers between 0.555 and 0.623.
- Determine two numbers within one thousandth of 2.869.
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p 54-55.
Skill: Comparing and Ordering Unit Fractions
Relationships in Ordering and Comparing
It is important that students learn to order fractions by comparing them. The ability to order and the ability to compare should be based on students' constructed representations of fractions. They should arise from problem-solving situations that allow students to develop a sense of the value of a fraction. It is this sense of the fraction that allows students to compare 2 fractions and conclude that one is greater than the other. In contrast, teaching that is done out of context tends to promote learning algorithms that neglect the meaning of the fraction.
Junior students already have strong ideas about the properties of whole numbers. However, when they begin to work with fractions, students often experience a cognitive imbalance - a sense that what was true is no longer true. For example, they have learned that when two whole numbers are present, the larger number indicates a greater quantity. They try to transfer their knowledge and skills to fractions. For example, in trying to compare fractions and fractions, some students believe that 6 is larger than 3. They feel off balance if they are told that they are wrong. To avoid such situations, learning must be based on an analytical view of the fraction rather than on an automatic or algorithmic approach.
Activities that involve students in comparing and ordering fractions should promote the construction of fraction meaning. To compare fractions, it is rarely necessary to look for a common denominator. Thus, there are many strategies for comparing fractions. Generally, the first strategy learned is the use of a concrete or semi-concrete representation of the fractions involved. This strategy can be used on any occasion. One can also compare numerators if the denominators are the same, or compare denominators if the numerators are the same. Another effective strategy is the use of benchmark fractions.
It is important to work thoroughly on the comparison of fractions before the equivalence of fractions. This is very
complex and the comparison serves as a catalyst for understanding equivalent fractions.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 44-45.
Skill: Comparing Numbers Expressed in Different Forms
Junior students need to understand the relationships between various notations of a quantity, for example, the connections between equivalent decimals and equivalent fractions. Students should recognize that a decimal is a fraction or mixed number whose denominator is a multiple of 10 (10, 100, 1 000, 10 000, etc.). For example, the number 0.5 is equivalent to the fraction \(\frac{5}{{10}}\) and the number 2.63 is equivalent to the fractional number 2 \(\frac{{63}}{{ 100}}\). Models such as \(10\; \times \;10\) grids allow students to visualize the connections between the corresponding fraction and the corresponding decimal.
Example:
With a good grasp of this relationship, students are able to identify equivalent fractions and decimals and choose the most appropriate form for the context. For example, a merchant needs to order the exact amount of flour to meet the demands of his customers. The first customer requires 12.5 kg of flour, the second requires 11 kg, the third requires 10 kg, and the fourth requires 5 kg. The student who has mastered the relationship might decide to convert the fractions to decimal numbers to simplify the calculation. Therefore, 11 kg is the equivalent of 11.75 kg, 1.5 kg is the equivalent of 1.5 kg, and 5 kg is the equivalent of 5.5 kg. Again, it is important to emphasize the use of models to clearly see the relationships between fractions and decimal numbers rather than formally learning a procedure to convert from one form to another.
In the junior grades, instructional interventions should focus on developing students' versatility with numbers. To do this, teachers should present learning situations that involve a variety of numbers and should emphasize the different ways of representing these numbers, while reflecting on the advantages of one over another. This allows students to develop their number sense and expertise in using numbers. It is desirable that students have the opportunity to relate numbers in various forms, i.e., mixed numbers, decimal numbers, fractions, etc.
Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p 55-56.
Models, especially proportional ones such as the number line, can be used to order numbers, reveal relative sizes,
and show equivalences between fractions, decimal numbers, and whole numbers.
Source : The Ontario Curriculum, Mathematics, Grades 1-8, Ontario Ministry of Education, 2020..
Knowledge: Decimal Numbers
Set of decimal numbers (\(\mathbb{D}\))
The set of decimal numbers is made up of numbers that can be expressed in decimal form with a finite decimal part (for example, 3.72; -5.1; 0; -7.0; 12.135 64). This set includes all integers (\(\mathbb{Z}\)), because they can be expressed with a decimal part (for example, \(3\; = \;3,0\)). It also includes certain fractions, such as \(\frac{2}{5}\) and \(\frac{3}{{16}}\), since \(\frac{2}{5}\; = \ ;0.4\) and \(\frac{3}{{16}}\; = \;0.187\;5\). However, a large number of fractions are excluded, such as \(\frac{1}{3}\) and \(\frac{7}{{11}}\), because their decimal expansion requires an infinite number of decimal places ( \(\frac{1}{3}\; = 0.333\;33\))… ). and \(\frac{7}{{11}}\; = \;0.636\;363\)…Interestingly, all decimals can be expressed as a fraction whose denominator is a power of 10. (Powers of 10 are 1, 10, 100, 1 000…We include 1 as a power of 10, because, by definition, \(10^{0} \; = \;1\).)
Examples
\(3.72\; = \;3\;\frac{{72}}{{100}}\; = \;\frac{{372}}{{100}}\)
\( - 5,1\; = \; ^- 5\;\frac{1}{10}; = \;\frac{-51}{10}\)
\(5\; = \;5.0\; = \;\frac{5}{1}\)
Since all whole numbers are integers and all integers are decimal numbers, we can represent the relationship between
the sets of numbers by the Venn diagram below.
Note: There is no such thing as a set of decimal numbers. The name decimal number simply means that the expression of
the number contains a decimal point. Thus, a decimal number can be a decimal number (for example, 0.45), a periodic
number (for example, 0.333) or an irrational number (for example., 3.141 5).
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 42.
The decimal point plays a significant role in decimal notation. It separates the whole part from the decimal part.
The position of the ones defines the whole according to which are formed on the one hand the tenths, the hundredths and the thousandths and on the other hand the tens, the hundreds and the thousands.It can therefoere be said that the ones, identified by the decimal point, is at the heart of the decimal system.
This recognition of the role of the units digit is evidenced by the name prefixes given to the place value of the digits on either side of the unit. Thus, tenths represent a quantity ten times smaller than the unit.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 51.
Knowledge: Fractions
The word fraction comes from the Latin fractio which means "break". A part of a broken object can therefore represent a fraction, because it is a part of a whole. However, in order to determine a fraction of an object divided into parts, the parts must be equivalent. Note that when we talk about equivalent parts, we are not necessarily talking about identical shapes, although these are easier to use. The representations of 1 fourth \(\frac{1}{4}\) in the example below are based on the area of the whole. Since each whole has an area of 16 square units, each fourth has an area of 4 square units. Despite their different shapes, each of these fourths represents an equivalent part of the same whole.
Example
Six equivalent representations of a quarter \(\left( {\frac{1}{4}} \right)\) of
the same whole.
It is important for students to understand that the more the whole is divided, the smaller its parts.
Example
The fourths of a whole are larger than the tenths of the same whole.
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 33.