B1.4 read, represent, compare, and order decimal numbers up to thousandths, in various contexts
Skill: read decimal numbers to thousandths
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 3.72 or 12.13564.
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 can be expressed as decimal fractions, that is, fractions whose
denominator is a power of 10.
In the junior grades, the study of decimal numbers is related specifically to the use of decimal notation to express these numbers. A number expressed in decimal notation is composed of two parts, the integer part and the decimal part.
Example
The number \(8\frac{1}{4}\) is written as 8.25 when represented as a decimal number.
Source: A Guide
to Effective Instruction in Mathematics, Grades 4 to 6, p 28-29.
In teaching decimal numbers and related decimal notation, the focus is too often on learning procedures and rules,
rather than the concepts that support them. This prevents students from developing conceptual knowledge of decimal
numbers. Statements such as "you can add zeros 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 rule following.
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. Since the decimal part of a number is, in fact, just a different way of representing a decimal
fraction, 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.
Source: A Guide to
Effective Instruction in Mathematics, Grades 4 to 6, p. 30.
The use of decimal numbers to express a quantity responds to the need to express quantities with greater 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 say that it measures 1 m and "part of one metre".
A more accurate measurement can be given by expressing this "part of a metre" in decimal notation. Using a
ruler graduated in decimetres, we can determine that the diagonal is about 1.4 m. One could even obtain a measurement
of 1.41 m with a centimetre ruler or 1.414 m with a millimetre ruler.
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p 32-34.
Place Value Relationships
The beginning of the junior grades is the first time that students study the decimal part of a number. They need to deepen their understanding of the place values of the numbers and the relationship between the 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.
It is essential that students recognize the position of the units, since it is the unit that defines the whole, from which the tenths, hundredths and thousandths are formed on the one hand, and the tens, hundreds and thousands on the other. We can therefore say that the unit, 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 digit. Tens represent a quantity ten times greater than the unit, while tenths
represent a quantity ten times smaller than the unit. Similarly, one hundred is a hundred times larger than a unit,
while one hundredth is a hundred times smaller than a unit.
Note: Some students are under the impression that the decimal point is the center of the decimal system. As a result, they tend to call the first position to the right of the decimal point the position of ones rather than tens.
It is important that students also grasp the multiplicative relationship by 10 that exists between adjacent place
values. They 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 also true for decimal places.
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, 0.013 to represent thirteen hundredths. These students
attempt to place 13 in the hundredths column as follows.
0 , 0 13
However, the decimal system does not allow two digits to be written in one position. The students must recognize that
10 hundredths is the equivalent of 1 tenth, which makes 13 hundredths equal to 1 tenth plus 3 hundredths. Therefore,
we write 0.13.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 51-53.
Mental Representation
In order to develop number sense, it is important that students form images of the quantities represented by the numbers. In the case of decimal numbers, reading them correctly allows students to form better mental representations and to draw on their knowledge of fractions (for example, 0.175 reads "one hundred and seventy-five thousandths", not "zero point one hundred and seventy-five"). 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, as these same models were previously used to represent other concepts (for example, the rod represented ten ones). Students need to understand that the whole has changed. In the first of the two examples above, the whole object represents the whole; in the second example, the flat represents the whole.
Students must also form a mental representation of decimal numbers greater than one. When reading such a decimal number, they must mentally picture the quantity it represents by interpreting the two parts that make up the number: the integer part and the decimal part. For example, they must recognize that the number 8.245 represents 8 integers and part of another identical integer. They can then visualize a quantity between 8 and 9.
Over time, students are able to mentally picture the quantity in context. For example, in a situation involving an
item that costs $197.98, a student who recognizes that $197.98 is a little more than $197.00 can then visualize or
conceive that, in this context, the amount of $197.98 can be represented approximately by 10 $20 bills.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 36-37.
Skill: Representing Decimal Numbers to Thousandths
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 representations.
The following figure illustrates various ways to represent twenty-five hundredths concretely, semi-concretely,
symbolically, and with words, and suggests, through the phrases $0.25, 0.25 kg, 0.25 h, and 0.25 cm, different
contexts for its use. Students develop a better understanding of numbers if they are regularly asked in a variety of
situations to move from one representation to another.
Source: A Guide
to Effective Instruction in Mathematics, Grades 4-6, p 57-58.
Concrete Representations
In order to fully understand decimal numbers, it is important that students be able to represent them concretely using manipulatives (for example, tokens, base ten blocks). Unfortunately, students in the junior grades do not always have the opportunity to explore numbers using these materials. As a result, their knowledge of decimal numbers is often based on symbolic representation and does not reflect 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, \(\frac{3}{10}\)).
Example
Once students have become accustomed to recognizing tenths, hundredths, and thousandths, concrete materials can be
used to introduce decimal notation. Here is a concrete representation of the numbers 2.4 and 0.70.
The base ten material is an excellent tool to represent a decimal number, since it highlights the multiplicative relationship by 10 between the place value of the digits that compose it. However, it is very important to always clearly define the piece that constitutes the unit according to which the other pieces will be defined. When studying wholel numbers, the small cube was usually associated with the unit. However, if we choose the strip as the unit, then the small cube represents one tenth of a strip.
Example:
The small cube represents 0.1 of a rod.
If we choose the board as unit, then the small cube represents one hundredth of a board and the tab represents one tenth of a board.
Example:
The small cube represents 0.01 of a flat.
If we choose the large cube as the unit, then the small cube represents one thousandth of the large cube, the tab
represents one hundredth of the large cube and the board represents one tenth of the large cube.
The pictures below show the representation of different numbers in different units on a place value mat.
Change can also be used when exploring decimal numbers. However, it is important to use only the $1, 10¢, and 1¢
coins since their value respects the multiplicative relationship by 10. A representation using coins is more abstract
than one using base ten material 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 one-tenth of a unit (0.1 of a
dollar) and the 1¢ coin represents one-hundredth of a unit (0.01 of a dollar). From the multiplicative relation by 10,
the groupings are formed by respecting the place value: 10 coins of one-hundredth of a dollar are equivalent to 1 coin
of one-tenth of a dollar and 10 coins of one-tenth of a dollar are equivalent to 1 coin of $1.
Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p 59-62.
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, 100, or 1 000 equal parts can also be used to represent decimal numbers.
Example:
It is also important that students use set models to represent decimal tenths.
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 encouraged to represent 0.5 in a variety of ways on a grid of \(10 \times 10\).
Examples:
They can also be asked to use different semi-concrete representations to represent a given number.
Examples
Here, three 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 recognize what each interval corresponds to.
Number line to represent tenths and hundredths
Number line to represent hundredths and thousandths
Number line to position the numbers 5.2 and 5.32
Source: A Guide
to Effective Instruction in Mathematics, Grades 4 to 6, p 65-67.
Symbolic Representation
The conventional writing of decimal numbers is a symbolic representation of these concepts. To symbolically represent decimal numbers, a decimal point is embedded in the writing of the number to separate the integer part from the decimal part.
Example
Three and one hundred and twenty-five thousandths
Writing decimal numbers requires an understanding of the concept of regrouping. Thus the number "thirty-five
hundredths", after regrouping, corresponds to 3 tenths and 5 hundredths, hence the writing of the number 3 in the
tenths position and the number 5 in the hundredths position, or 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 "see" the groupings. For example, students can
conclude by counting that the following figures represent 9 thousandths, 11 thousandths, and 101 thousandths
respectively.
Their knowledge of fractions enables them to represent these quantities by \frac{9}{{1\;000}}\, \frac{11}}{{1\;000}}\ and \frac{101}{{1\;000}}\, and their understanding of the concept of position value enables them to recognize that the first fraction, expressed in decimal notation, is written 0.009. However, it's not obvious that the other two are written as 0.011 and 0.101. For example, to understand that 0.011 represents 11 thousandths, it is useful to demonstrate with a grouping that 11 thousandths is equivalent to "1 hundredth plus 1 thousandth".
It is important to distinguish between the number in the thousandths position and the amount of thousandths in the number. For example, in the number 0.011, the digit in the thousandths position is 1, but there are 11 thousandths in the number. In the number 0.101, the digit in the thousandths position is also a 1, but this number is 101 thousandths. If we want students to identify the digit in the tenths position in a given number (for example, 2.35), we need to 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?"
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 67-69.
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 "zero point
seventy-five," the meaning of the notation is left out. This way of reading a decimal number is merely a
sequential listing of the symbols that make up the number, just as it would be if one were to read the number 123 by
saying "one, two, three." However, if we insist on reading these numbers as "seven tenths" and
"seventy-five hundredths," we are emphasizing the meaning of the notation. 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 advantage of
reminding them of the correspondence between decimal numbers and the corresponding decimal fractions. It also allows
students to refer to a decimal number by its name.
It is important that teachers model this way of reading decimal numbers at all times. The numbers 12.34 and 1013.7 read "twelve and thirty-four hundredths" and "one thousand thirteen and seven tenths" respectively. Note that when reading a decimal number, the integer part is read as if it were a whole number, the word "and" (not "period") serves as a link between the two parts, and the decimal part is read according to the place value of the far right 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. 69.
Skill: Comparing and Ordering Decimal Numbers to Thousandths
The ordering relationship is based on the comparison of numbers. One of the great strengths of decimal notation 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 0.125 and 0.075 when expressed in decimal notation.
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, \(134 < 1\;275\)) 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, \(34 < 275\)).
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 ofsixth 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: Reading, Representing, Comparing and Ordering Decimal Numbers in Various Contexts
Context is all the information surrounding a given situation. To 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 the context helps clarify the quantity represented. For example, if 2.347 is mentioned, students understand that this number represents 2 times any unit and is the same unit. But, if it is stated that it is 2.347 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 his or her parents, while the child may find it a bargain for a collectible car. The more students are exposed to different mathematical situations in context, the more knowledge they gain and the more informed critical judgments they can make.
Newspaper headlines often provide topics for discussion that can help students understand and interpret decimal numbers.
Examples
- 13.7 million decrease in municipal budget
- New world record, an apple of 13.45 kg!
During a class discussion, teachers can ask questions that help students analyze the titles listed above.
Examples:
- Is it important to specify the 0.455 kg in the mass of the apple? Why or why not?
- At first glance, $13.7 million is a lot of money, but is it a significant amount in the budget?
- How can a number like 13.7 represent such a large amount of money?
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p 40-41.
Knowledge: Decimal Number
Set of decimal numbers (\(\mathbb{D}\))
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, \(\frac{2}{5}\; = \;0.4\), \(\frac{3}{{16}}\; = \;0.187\;5\), ( \(\frac{1}{3}\; = \;0.333\;3 \ldots \;{\rm{et}}\;\frac{7}{{11}}\; = \;0.636\; 363 \ldots \)). 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 can be expressed as a fraction whose denominator is a power of 10. (The powers of 10 are 1, 10, 100, 1,000… We include 1 as a power of 10 because, by definition, \(10^{0} \; = \;1\)
Examples
\(3,72 = 3\frac{72}{100} = 3\frac{372}{100}\)
\(5\; = \;5.0\; = \;\frac{5}{1}\)
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: 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.
Knowledge: Thousandths
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 unit is highlighted by the prefixes of the names given to the place value of the digits on either side of the unit. Thus, thousandths represent a quantity one thousand times smaller than the unit.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 51.