B1.5 Read, represent, compare, and order decimal numbers up to hundredths, in various contexts.
Skill: Reading Decimal Numbers to Hundredths
Learning decimal numbers is closely related to understanding decimal notation. Decimal notation is commonly used in the International System of Units (SI) and in the monetary system, among other things. However, despite its frequent use in everyday life and in the classroom, decimal notation is far from being well understood and mastered.
In order to explore learning about decimal numbers, it is important to examine the terminology related to these numbers as well to decimal notation. A decimal number is a number that has a decimal point, such as 3.75. The part before the decimal point represents a whole number amount, and the part after the decimal point represents a value that is less than one. The set of decimal numbers includes all integers, because every integer can be expressed with as a number (for example, 3 = 3.0).
It is interesting to note that all decimal numbers that terminate can be expressed as decimal fractions, that is, fractions whose denominator is a power of 10. Be cautious if introducing this to students, however, since this property is NOT true for the infinite repeating decimal numbers and infinite non-repeating decimal numbers they will see later on in their math studies.
In the junior grades, the study of decimal numbers is linked more specifically to the use of decimal representations to express these numbers. A decimal number is composed of two parts, namely the integer part (whole number part in Grade 5) and the decimal part.
Example
The number \(8\frac{1}{4}\) is written as 8.25 when represented as a decimal number.
In teaching decimal numbers and related decimal representations, the focus is too often on learning procedures and rules rather than the concepts that support them. This prevents students from developing conceptual understanding of decimal numbers. Statements such as "you can add 0's after the last decimal place without changing the quantity, for example, 2.3 = 2.30" undermine understanding of the quantity represented by a decimal number, and their use reduces learning about decimal numbers to following rules that are, at best, poorly understood. According to the recommendations in Teaching and Learning Mathematics: The Report of the Expert Panel on Mathematics in Grades 4 to 6 (Ontario Ministry of Education, 2004a), effective mathematics instruction should focus on developing an understanding of the meaning of the concepts being taught. In Grade 5, the decimal part of a number is, in fact, just a different way of representing a decimal fraction, so it is essential that students have a solid understanding of fractions and place value before they are introduced to the concept of a decimal number. In terms of the quantity that a decimal number represents, connections must be made between the decimal part of the number and the concept of a fraction. Paradoxically, the writing and reading of decimal numbers are more similar to those of whole numbers than to fractions.
The use of decimal numbers to express a quantity responds to a need to express quantities with more precision.
Example
Consider a diagonal of a square whose sides measure 1 m. If we try to describe the length of this diagonal using whole numbers, we can only say that it measures 1 m and “part of another metre”.
It is possible to provide a more accurate measurements by expressing a part of one metre as a decimal number. Using a ruler graduated in decimetres, we can determine that the diagonal measures about 1.4 m. A measurement of 1.41 m could also be obtained with a ruler graduated in centimetres.
Source: Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 28-34.
Place Value
The beginning of the junior grades is the first time students study the decimal part of a number. Students need to deepen their understanding of the place value of numbers and the relationships between place values. Decimal numbers are part of everyday life and understanding the place values to the right and left of the decimal point is essential.
The decimal point plays a significant role when interpreting the value of a decimal number. It separates the integer part (in Grade 5, the whole number part) from the decimal part, and thus indicates the position of the ones.
However, it is essential that students recognize the position of the ones since it is the ones that defines the whole according to which the tenths, hundredths, tens, hundreds, and thousands are formed. It can therefore be said that the ones, identified by the decimal point, is at the heart of the decimal number.
This recognition of the role of the ones is evidenced by the name prefixes given to the place value of the digits on either side of the ones. Thus, tens represents a quantity 10 times greater than one, while tenths represents a quantity one tenth as large as one. Similarly, hundreds are 100 times larger than one, while hundredths are one hundredth as large as one.
Note: Some students are under the impression that the decimal point is the centre of the decimal number. As a result, they call the first position to the right of the decimal point the position of ones rather than tenths.
It is important that students also grasp the multiplicative relationship by 10 that exists between adjacent position values. Students have previously developed an understanding of this relationship in the study of whole numbers, that each position has a value 10 times greater than the one to its right and 10 times smaller than the one to its left.
This multiplicative relationship is true for all positions.
Students can develop an understanding of this by doing groupings using base ten materials. The idea is to demonstrate that just as 10 ones yield 1 ten, 10 tenths yield 1 one and 10 hundredths yield 1 tenth, and so on.
It is common for junior students to write, for example, 0.13 to represent 13 tenths. These students attempt to place 13 in the tenths column as follows.
0 , 13
However, a decimal number cannot have two digits in one position. Students must recognize that 10 tenths is the equivalent of 1 one, so 13 tenths is equivalent to 1 one plus 3 tenths. Therefore, we write 1.3.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 51-53.
Mental Representation
In order to develop a strong sense of number, it is important that students form mental representations of the quantities represented by the numbers. In the case of decimal numbers, reading them correctly will help them form a better image and to draw on their knowledge of fractions (for example, 0.75 is read as "75 hundredths", not "0 point 75"). Students should be encouraged to use multiple models to support the creation of various mental representations.
Example
When representing decimal numbers using models, there is an adjustment to be made since these same models were previously used differently (for example, the rod represented ten ones). Students must understand that the the whole has changed. In the first of the two previous examples, the rod (the whole) represents 10 tenths, so the unit is tenths; in the second, the large square (the whole) represents 100 hundredths, so the unit is hundredths.
Students must also form a mental representation of decimal numbers greater than 1. When reading such a decimal number, they must imagine the quantity it represents by interpreting each of its 2 parts: the whole number part and the decimal part. For example, they must recognize that the number 8.24 represents 8 ones and part of another one. They can then visualize a quantity between 8 and 9.
Over time, students are able to mentally picture the quantity taking into account the context. For example, in a situation involving an item that costs $197.98, the student who recognizes that $197.98 is a little less than $200 can then visualize or conceive that in In this context, the amount of $197.98 can be represented approximately by 10 $20 bills.
Skill: Representing Decimal Numbers to Hundredths
Students need to learn not only to represent decimal numbers in a variety of ways, but also to recognize them in their multiple representations. These skills help them make connections between numbers, their representations, and the quantities they represent. In some cases, the use of models facilitates the construction of other representations.
The following figure illustrates various ways of representing 25 cents concretely, semi-concretely, symbolically, and with words, and suggests, through the expressions $0.25, 0.25 kg, 0.25 hrs, and 0.25 cm, different contexts for its use. Students develop a better understanding of numbers if they are regularly asked to move from one representation to another in a variety of situations.
Concrete Representations
To fully understand decimals, it is important for students to be able to represent them concretely using manipulatives (for example, counters, base ten materials). Unfortunately, junior students do not always have the opportunity to explore numbers using manipulatives. Consequently, their knowledge of decimal numbers often relies on symbolic representation and does not demonstrate a solid understanding of these numbers. Before beginning to teach decimal numbers, it is important to have students use manipulatives to describe parts of a whole represented by a decimal fraction (for example, \(\frac{3}{ {10}}\), \(\frac{{12}}{{100}}\)).
Example
Once students have become accustomed to recognizing tenths and hundredths, concrete materials can be used to represent decimal numbers. Here is a concrete representation of the numbers 2.4 and 0.70.
Base 10 materials are excellent tools to represent decimal numbers since they highlight the multiplicative relationship between the position and the value of the digits that compose it. However, it is very important to always clearly define the part that constitutes the unit according to which the other parts will be defined. When studying whole numbers, the small cube was usually associated with ones. However, if we choose the rod as the unit, then the small cube represents 1 tenth.
Example
The small cube represents 0.1 of a rod.
If we choose the flat as the unit, then the small cube represents 1 hundredth of the flat and the rod represents 1 tenth of the flat.
Example
The small cube represents 0.01 of a flat.
The pictures below show the representation of different numbers in different units on a place value mat.
Money can also be used when exploring decimal numbers. However, it is important to use only $1, 10¢, and 1¢ coins since their value respects the multiplicative relationship by 10. A representation made with coins is more abstract than one using base 10 materials since the multiplicative relationship by 10 is based on the value of the coin and not its size. Students may identify the $1 coin as the unit. The 10¢ coin then represents 1 tenth of the unit (0.1 of a dollar) and the 1¢ value represents 1 hundredth of the unit (0.01 of a dollar). From the multiplicative relationship by 10, the groupings are formed respecting the place value: 10 times the value of the cent (1 hundredth of a dollar) is equivalent to 1 dime and 10 dimes are equivalent to 1 dollar.
Semi-Concrete Representations
Students can also develop a sense of decimal numbers and percents by representing them in semi-concrete ways. For example, a disk divided into 10 equal parts can represent tenths and a disk divided into 100 equal parts can represent hundredths.
Examples
Strips and grids separated into 10 or 100 equal parts can also be used to represent decimal numbers.
Example
It is also important that students use set models to represent decimal numbers.
Examples
To develop a good sense of number, it is important that students explore different representations of the same quantity. For example, they can be invited to represent 0.50 in several ways on a 10 × 10 grid.
Examples
They can also be asked to use different semi-concrete representations to represent a given number.
Examples
Here, 3 representations of 0.3.
Students have learned to use a number line to locate whole numbers and to count to large numbers in intervals. They can also use it to locate decimal numbers. To do this, students need to understand how to subdivide the intervals and to recognize what each interval corresponds to.
Number line to represent tenths and hundredths
Number line to position the numbers 5.2 and 5.32
Symbolic Representations
The conventional writing of decimal numbers is a symbolic representation of these numbers.
To symbolically represent decimal numbers, a decimal point is integrated into the writing of the number to separate the integer (or whole number) part from the decimal part.
Example
Writing decimal numbers requires an understanding of the concept of regrouping. After regrouping, the number "35 hundredths", corresponds to 3 tenths and 5 hundredths, so we write 3 in the tenths position and 5 in the hundredths position, i.e. 0.35.
This relationship between writing a decimal number and groupings can be understood by exploring decimal numbers using concrete and semi-concrete representations that allow one to experience the groupings.
It is important to distinguish between the digit in the hundredths position and the number of hundredths in the number. For example, in the number 0.11, the digit in the hundredths position is 1, but there are 11 hundredths in the number. In the number 1.01, the digit in the hundredths position is also a 1, but this number is 101 hundredths. Thus, if we want students to identify the digit in the tenths position in a given number (for example, 2.35), we should ask the question "What digit is in the tenths position in the number 2.35?" rather than "How many tenths are in the number 2.35?"
Representations Using Words
How students learn to read decimal numbers can affect their understanding. If they are taught to read the numbers 0.7 and 0.75 by saying "zero point seven" and "zero point seven five," or "0 point 75," the meaning of the number is left out. This way of reading a decimal number is merely a successive listing of the symbols that make up the number, just as it would be if one were to read the number 123 by saying "1, 2, 3." However, insisting on reading these numbers as "7 tenths" and "75 hundredths" emphasizes the meaning of the representation of the number. This way of reading decimal numbers gives students the opportunity to visualize the 7 parts of 10 and the 75 parts of 100 and has the added advantage of reminding them of the correspondence between decimal numbers that terminate and the corresponding decimal fractions.
It is important that teachers model this way of reading decimal numbers at all times. The numbers 12.34 and 1013.7 are read as "12 and 34 hundredths" and "1013 and 7 tenths" respectively. Note that when reading a decimal number, we read the whole number part; then the word "and" (not "point"), which serves as a link between the 2 parts; and the decimal part is read according to the place value of the rightmost digit in the number.
Note: Some students may confuse the terms ten and tenth or hundred and hundredth when reading because of their phonetic similarity.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 57-69.
Skill: Comparing and Ordering Decimal Numbers to Hundredths
The order relationship is based on the comparison of numbers. One of the great strengths of decimal numbers is the speed with which it is possible, through the concept of place values, to compare and order quantities. For example, it is much easier to compare the numbers 0.64 and 0.75 when expressed in decimal notation rather than as the fractions \(\frac{{16}}{{25}}\) and \(\frac{{15}}{{20}}\).
In general, students have little difficulty comparing decimal numbers with the same number of decimal places (for example, 0.34 < 0.46). They have more difficulty comparing numbers with a different number of decimal places (for example, 1.3 and 1.27). Some tend to compare these numbers without the decimal point (for example, 13 < 127) and conclude that 1.3 < 1.27. Others reach the same incorrect conclusion by comparing only the numbers to the right of the decimal point (for example, 3 < 27).
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 with good number sense can also compare 3.55 m and 3.7 m by first noticing that they each represent jumps greater than 3 m. Then, they can compare tenths to notice that the first number has 5 tenths (representing 5 decimetres), while the second has 7.
Samantha's jump is therefore longer than Remi's. 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-metre jump is therefore longer than the 3.55-metre jump.
Traditionally, a procedure was taught where you had to add a 0 to the end of 3.7 to give two numbers with the same number of decimal places. The decimal parts, 55 and 70, were then 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 method that the explanation and the concept were quickly lost. Not surprisingly, students 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 problems, which offer more than one answer and are thought-provoking, allow students to deepen their understanding of ordering relationships.
For example:
- Determine three decimal numbers between 0.55 and 0.62.
- Determine three numbers within 1 tenth of 2.8.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 54-55.
Skill: Reading, Representing, Comparing and OrderingDdecimal Numbers in Various Contexts
Context includes all the information surrounding a given situation. To deeply understand the meaning of the quantity represented by a decimal number, students must analyze it in its context. Like a fraction, a decimal number represents a part of a whole. The magnitude of the quantity represented by a decimal number is therefore entirely dependent on the magnitude of the whole.
Emphasize that context may help to make the image of the quantity more clear. For example, if 2.34 is mentioned, students understand that this number represents 2 times any unit and \(\frac{{34}}{{100}}\) of the same unit. But, if it is stated that it is 2.34 m, they can form a picture of the quantity in context.
However, in some situations, it is important to recognize that different decimal numbers are not referencing the same whole.
Example
In these examples, even though the number 0.2 is less than the number 0.4 (0.2 < 0.4), 0.2 kg is a larger mass than 0.4 g, just as 0.2 of the large square is a larger area than 0.4 of the small square.
Furthermore, without context, it is sometimes impossible to make a critical judgment about the size of the quantity represented by a decimal number. For example, a child's purchase of a toy car for $8.34 may be considered expensive by their parents, while the child may find it a bargain for a collectible car. The more students are exposed to different mathematical situations and context, the more they grow their knowledge and their ability to make informed critical judgments.
Newspaper headlines often provide topics for discussion that can help students understand and interpret decimal numbers.
Example
- A decrease of $13.7 thousand in the municipal budget.
- New world record, an apple of 13.45 kg!
During a class discussion, teachers can ask questions that help students analyze the headlines given above.
Example
- Is it important to specify the 0.45 kg in the mass of the apple? Why or why not?
- At first glance, $13.7 thousand seems like a lot of money, but is it a significant amount of the budget?
- How can a number like 13.7 represent such a large amount of money?
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 40-41.
Knowledge: Decimal Number
Set of decimal numbers (ⅅ)
The set of decimal numbers is made up of all numbers that have a decimal point, (for example, 3.72; 0; 12.13564). This set includes all integers, because they can be expressed with a decimal part (for example, 3 = 3.0). It also includes all fractions (for example, \(\frac{2}{5}\; = \ ;0.4\), \(\frac{3}{{16}}\; = \;0.187\;5\), ( \(\frac{1}{3}\; = \;0.333\;3 \ldots \;{\rm{and}}\;\frac{7}{{11}}\; = \;0.636\; 363 \ldots \)). Note: Students in Grade 5 only see terminating decimal numbers. They will not see repeating non-terminating decimal numbers or non-repeating non-terminating decimal numbers until later grades.
It is interesting to note that all decimal numbers that terminate can be expressed as decimal fractions, that is, fractions whose denominator is a power of 10. Be cautious if introducing this to students, however, since this property is NOT true for the infinite repeating decimal numbers and infinite non-repeating decimal numbers they will see later on in their math studies.
Note: The powers of 10 are 1, 10, 100, 1000… We include 1 as a power of 10 because, by definition, 10^0 = 1.
Examples
\(3.72\; = \;3\frac{{72}}{{100}}\; = \;\frac{{372}}{{100}}\)
\(5\; = \;5.0\; = \;\frac{5}{1}\)
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: A decimal number can be a terminating decimal number (for example, 0.45), a repeating non-terminating decimal number (for example, 0.333…) or a non-repeating non-terminating decimal number (for example, 3.14159…).
Source: Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 42.
Knowledge: Hundredths
The decimal point plays a significant role when interpreting the value of a decimal number. It separates the integer part (in Grade 5, the whole number part) from the decimal part, and thus indicates the position of the ones.
The position of the ones defines the whole according to which the tenths, hundredths, tens, hundreds, and thousands are formed. It can therefore be said that the ones, identified by the decimal point, is at the heart of the decimal number.
This recognition of the role of the ones is evidenced by the name prefixes given to the place value of the digits on either side of the ones. Thus, hundredths are one hundredth as large as one.
Source: Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 51.