B1.1 Read, represent, compose, and decompose whole numbers up to and including 100 000, using appropriate tools and strategies, and describe various ways they are used in everyday life.
Skill: Reading Whole Numbers Up to 100 000
Reading numbers involves interpreting them as a quantity when they are expressed in words, in standard notation, in expanded notation, or on a number line.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
The Base Ten System
The base ten number system commonly used today in many countries uses 10 different symbols, namely the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. It is a positional system since the symbols are given different values depending on their position or place within a number. For example, the digit 2 has a value of 2 ones in the number 56 742, but it has a value of 20 000 ones in the number 23 487. Understanding the relationships between the value of digits and their position in a number is essential to the development of number sense. In the primary grades, students develop an understanding of the relationships between the place values of ones, tens and hundreds. However, in the Junior and Intermediate Divisions, students do not automatically transfer this understanding to larger numbers. Therefore, teachers must ensure that they support students to understand that the value of any digit in a number is always 10 times larger than its value would be if it occurred in the position immediately to its right, and 10 times smaller than if it occurred in the position immediately to its left. It is also important to examine the 100- or 1000-times greater than or less than relationships between place values in order to support students to develop number sense for large numbers.
Students should also recognize, for example, that 10 000 can be represented as 10 one thousands, 100 hundreds, 1000 tens, or even 10 000 ones. These equivalent representations help to build student understanding of numbers (for example, 2534 represents 25 hundreds and 34 ones).
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 44-45.
It is important that students understand that a zero in a column or number indicates that there is no group or quantity at that place value. It serves as a placeholder and holds the other digits in their correct “place”.
Source: Ontario Curriculum. Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
A number is an abstract representation of a very complex concept. The relationship between the way a number is named and the quantity it represents may not be obvious to students. Many adults mistakenly believe that if students can count, they understand the meaning of each number. However, a student may be able to read and name a number, for example, 58 000, without really having an understanding of the quantity it represents.
Teachers should help them make connections between the base ten number system and how to name and write numbers.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 65.
One strategy that encourages the association of the number with the quantity it represents is to emphasize the place value of each of the digits that make up a number when naming it (for example, instead of reading the number 62 098 as sixty-two thousand ninety-eight, students can say 6 groups of 10 000, 2 groups of 1000, 9 groups of 10, and 8 groups of one) or certain combinations (for example, 62 groups of 1000 and 98 groups of one).
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 70.
Mental Representation
The mental representation of a quantity is the image, developed through thinking, of a number. When students hear and read a number, they need to "see" the quantity that the number represents and understand the "how much" of it.
It is therefore important that students have mental representations of different numbers in different contexts. For example, the number "20 000" can be mentally represented by 2 grids of 10 000 or 20 grids of 1000, by the number of seats in a lecture hall, by 400 school buses carrying 50 students, etc.
It is important for students to develop a variety of mental representations for numbers. These visualizations may simply represent quantities. For example, the "how many" of 100 000 may generate a mental representation of 10 grids of 10 000. However, the presence of units in a situation may promote a different, more precise mental representation that would be related to a given situation. For example, students may visualize that 50 000 can be represented by 100 cars parked in 500 parking lots or 100 schools with 500 students.
In addition, if the context suggests it, students can look critically at the quantity (for example, determine whether it is "a lot" or "a little"). The mental representation will then be affected by the context. For example, the mental representation of 100 people at a family party is not the same as 100 people in a crowd at a field hockey game. The various mental representations are all valid; they depend primarily on the context of the situation and the students' sense of number. The mental representation remains personal, but the ease with which an individual can visualize numbers is an indicator of their number sense.
In order to develop mental representations, students use different strategies that respond to various situations and needs. With very small numbers, it is possible for them to subitize, which involves quantifying the elements of a small set of objects. To recognize larger quantities, students will use other strategies. For example, counting each pea in a package of peas can be laborious, so grouping can be used.
After counting 10 three times, they can see three equal sets for a total of 30 peas. In this case, the counting to 10 strategy is combined with times 3 to see what 30 looks like. As students progress, they will increasingly use grouping to create mental representations.
The mental representation of large numbers cannot easily be done by identifying the elements individually. However, it can be created by using benchmarks that will be associated with a large quantity (for example, identifying that 60 000 people is about 3 times the number of spectators in an amphitheatre with 20 000 seats).
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 29-31.
Benchmarks
In general, a benchmark is a reference. As part of the development of number sense, the use of benchmarks promotes mental representations, which facilitates the understanding of number and the notion of “how much”. Benchmarks, without which it is difficult to understand quantity, are numbers or quantities that can be easily represented mentally since they have already been seen and manipulated. Students will have difficulty understanding quantity if they do not use benchmarks. For example, when reading a book of world records, students see impressive numbers, but often do not have a true understanding of the size of these numbers unless they establish links between these numbers and meaningful benchmarks. Such is the case of a student who reads the following excerpt:
The Heaviest Newborn
On January 19, 1879, the Canadian Anna Bates [...] gave birth at home [...] to a baby boy who weighed 10.8 kg and measured 76 cm.
(Guinness World Records, 2005, p. 22)
A student may respond to this fact and finds it extraordinary, without relating the quantity 10.8 kg to a benchmark. This fact is extraordinary because it is a "world record", but does the student grasp that this baby is probably more than double their own birth weight?
With practice (for example, counting in intervals of 10 000, 20 000, 25 000, and 50 000) and internalizing relationships such as \(4\; \times \;25\;000\; = \;100\;000\), \(100\;000\; = \;2\; \times \;50\;000\;\), \(2\; \times \;2\; \times \;25\;000\; = \;100\;000\), numbers such as 10 000, 20 000, 25 000, and 50 000 can become benchmarks. Thus, 32 000 can be understood as 25 000 plus 7 000, or 25 000 plus 5 000 plus 2000, and the number 80 000 can be quickly associated with 4 groups of 20 000.
Although the majority of students entering Grade 5 can read and write numbers to 10 000 symbolically, they do not necessarily understand the quantities represented by these large numbers. By creating benchmarks and by visualizing groupings, students will develop a better understanding of the concept of quantity as represented by large numbers.
Benchmarks are particularly useful for understanding large numbers since it is usually impossible to recognize these quantities globally or grasp them by counting. Students need to get a sense of them by comparing them with a benchmark. For example, the school just received 50 000 sheets of paper for photocopying. Should the custodian ask for help carrying them? In order for students to truly understand the situation and the quantity involved, they need to create a mental image of what 50 000 sheets of paper might represent. Using a package of 500 sheets as a guide, students can imagine this quantity and think proportionally to deduce that 2 packages contain 1000 sheets. Therefore, 50 000 sheets would be 100 packages. Students can even recognize that 100 packages are equivalent to 10 boxes of paper. The mental representation of the space occupied by these 50 000 sheets then becomes possible. Then, the answer to the original question of whether the custodian should ask for help in carrying the sheets can be debated with full knowledge of the situation.
Students should develop their own personalized benchmarks, paying attention to the context and numbers being dealt with. There is no list of benchmarks. They are personal and come from the experiences of each individual. However, real-life situations provide teachers with ample opportunities to draw their students' attention to the quantity and creation of benchmarks.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 33-35.
Approximation (Estimating and Rounding)
Numbers can represent quantities with a high degree of accuracy. They provide a precision that terms like "more", "some", "many" and "few" do not. However, they can also be used to indicate, in a more general way, the relative size of a quantity. When a number is used to approximate a quantity, as in "about 200 people were at the party", it is not meant to provide an exact representation of that quantity. In general, the approximation is a number that is close enough to a known quantity (rounding) or an unknown quantity (estimation).
The terms "estimating" and "rounding" are often, and incorrectly, used interchangeably. The fundamental difference between these two concepts lies how the number is determined. Estimating uses the relationship between an unknown quantity and prior knowledge, usually in the form of benchmarks. Rounding, on the other hand, uses the relationship between a known number (precise or approximate) and its relative proximity to other numbers. Generally, estimating and rounding are used to visualize the quantity in question and to get a sense of the size of the quantity. The following table demonstrates this distinction.
| Rounding a Number | Estimating a Quantity | |
|---|---|---|
| Definition | Replace a number with a value appropriate to the situation, following some predefined or personal criteria. | To evaluate a quantity approximately. |
| Example | If the list price of a new car is $18 753, we can say that it costs around $19 000. | Walking through a parking lot, you notice a car and estimate its price to be $20 000. |
| Explanation | The actual price (a known number) has been rounded to the nearest thousand. | The price is not based on any specific information received, but on prior knowledge. |
Skill: Representing Whole Numbers Up to 100 000
Students need to learn to represent numbers in a variety of ways and to recognize them in their multiple representations. These skills help them make connections between a number, its representation, and the quantity it represents. It is therefore essential that students be exposed to different representations of numbers. It is also important that they be exposed to a variety of contexts that lead them to represent a number in each of the representations shown in the following diagram, as well as to move from one representation to another.
Teachers need to be aware of the order in which they should introduce these four modes of representation with students. Baroody and Coslick (1998, pp. 3-8 to 3-16) suggest that a new concept should be presented in a real and meaningful context so that students can first create representations using words, followed by concrete and visual representations. Only when students have developed some understanding of the concept can they move on to its symbolic representation. Students should be able to make connections between representations and move easily from one representation to another.
Representations Using Words
In Grade 5, students learn to read and write numbers up to 100 000 in words. The challenges of representing numbers with words should not be underestimated. To help students overcome these challenges, teachers should include numbers on the word wall and co-create reference materials (anchor charts) with them.
Concrete Representations
Using manipulatives (for example, counters, base ten materials) to represent numbers helps students develop number sense.
A word of caution is in order when it comes to using manipulatives. It is important to recognize that the materials represent a mathematical concept, not the concept itself; for example, the flat is not a hundred, but it represents a hundred small cubes.
The danger is that students will use the manipulative mechanically without making the connections to the underlying mathematical concepts. For this reason, teachers need to ensure that there is real learning taking place, with conscious and deliberate use of the manipulative.
For example, it is easy for students to fill in the blanks in the following sentence by looking at the place value mat.
| Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|
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The representation is _______ ten thousands, _______ thousand(s), ________ hundred(s), ______ ten(s) and _____ one(s).
Do students understand that there are also thirty-one thousand seven hundred twenty-four ones (units) on the mat and that this is one of the realities that is represented by the number 31 724? It is important to ask students to explain in different ways what is being represented. For example, 31 724 is also 31 thousands and 724 ones or 3172 tens and 4 ones.
Another way to check for understanding is to ask students, for example, how many hundreds there are in 31 724. Many students will say that there are 7. It is important to clarify that the number in the hundreds position is a "7", but that the number 31 724 actually has 317 hundreds (317 groups of 100 per group). We can recognize these hundreds by regrouping the ten thousands and the thousands into hundreds. A counter that represents ten thousand is equivalent to 10 counters of one thousand and 100 counters of one hundred. So, 3 ten thousands represents 300 hundreds, and 1 thousand represents 10 hundreds. We must then add the 7 hundreds to the 310 hundreds obtained in the grouping, for a total of 317 hundreds.
Using a place value mat or board, we notice that even though numbers are written from left to right, they are formed from right to left: ones grouped together form tens, tens grouped together form hundreds, and so on. However, once the regroupings are completed, the number is written starting on the left.
The choice of materials available to students can also make a difference in their development of conceptual understanding. There are a variety of commercial materials available to represent numbers: marbles, interlocking cubes, or any other object that can be used for counting. Some of these materials clearly and concretely represent the relationship between ones, tens, hundreds… (for example, base ten blocks). However, with other materials, this relationship is represented in a more abstract way. For example, on a Rekenrek, groups are represented according to the position of the digit from left to right, as in the symbolic writing of numbers.
By exposing students to a variety of manipulatives, teachers can help them develop a better understanding of numbers.
Visual (Semi-Concrete) Representations
Students can also represent numbers with visuals (for example, illustration, hundreds chart, number line).
Illustration: A number can be represented with drawings that show certain groupings or parts that compose it. For example, the number 376 can be illustrated by groupings as follows:
The drawings can also be related to manipulatives (hence the term "semi-concrete").
Hundreds Chart: The hundreds chart is widely used in the primary grades. Although more difficult to manipulate, a grid of 1000 or even 10 000 can help students in the junior grades to better understand numbers by providing them with opportunities to compare and contrast.
Hundreds chart
Thousands chart
Ten thousands chart
Number Lines: In the primary grades, students used and constructed number lines to skip count or to identify the number of tens in a number. In the junior grades, students will use and construct a variety of number lines to represent larger numbers and to recognize the relationships between them. Examples of number lines showing the number 457 include the following:
- number line scaled in intervals of 250;
- number line that does not start at 0, with a scale of 20;
- an open number line (not partitioned or divided) on which numbers are placed in relation to each other;
- a vertical number line with numbers in ascending order upward and has connections to other strands or subjects including Data (for example, y-axis) and Science (for example, thermometer).
Ten Frames: In the junior grades, the use of ten frames is another way to represent the value of the digits that make up large numbers.
Example
1256 represented using place value mat, ten frames and base ten blocks:
Symbolic Representations
Numbers are represented symbolically by the digits that make them up. They are written from left to right in 3-digit increments, called periods, that make up the thousands and ones. Each 3-digit group shows the hundreds (h), the tens (t) and the ones (o) for that group.
Note: Numbers are written by adding a space between the 3-digit increments (for example, 567 232).
The writing of large numbers requires a good mastery of place value, otherwise the student who is asked to write symbolically "one thousand two hundred thirteen" could write 1000 200 13 or 1000 213 or 1200 13. It also requires an understanding of the role of 0 as a placeholder to indicate the absence of a quantity in one of the positions.
The following are examples of reasoning that students might use to symbolically represent a large number such as twenty-six thousand four hundred:
- twenty-six thousand is represented by 2 and 6 in the thousands period, but it is necessary to insert a 0 in the position of hundred thousands, because in the word "hundred" is not heard in the thousands group of the number;
- four hundred is represented by a 4 in the hundreds position of the ones period. Since there are no tens and ones, a 0 must be used in each of these positions.
Twenty-six thousand four hundred is written symbolically as 26 400.
A number can be represented in different ways using mathematical symbols, either by respecting the value of the position of each digit (1 236 is equal to \(1\;000 + \;200 + \;30 + \;6\)), by using variations of place values (1236 equals 12 hundreds and 36 ones) or by performing different operations (1236 equals \(1\;000\; + \;236\) or \(1\;240\; - \;4\)) or \(1\;200\; + \;36\) or \(1\;000\; + \;100\; + \;100\; + \;15\; + \;15\; + \;6\)). In fact, there are an infinite number of ways of representing a number, each of which allows students to develop a different way of interpreting it and understanding its meaning.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p.48, 64-72.
In the Junior Division, the use of a table is another way to represent the value of the digits that make up a large number.
| Hundreds thousands | Ten thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|
| 429 156 | |||||
| 42 915 | 6 | ||||
| 4 291 | 5 | 6 | |||
| 429 | 1 | 5 | 6 | ||
| 42 | 9 | 1 | 5 | 6 | |
| 4 | 2 | 9 | 1 | 5 | 6 |
Skill: Composing and Decomposing Whole Numbers Up to 100 000
- Numbers can be composed and decomposed in various ways, including by place value.
- Numbers are composed when two or more numbers are combined to create a larger number. For example, the numbers 100 and 2 can be composed to make the sum 102 or the product 200.
- Numbers can be decomposed as a sum of numbers. For example, 53 125 can be decomposed into 50 000 and 3000 and 100 and 25.
- Numbers can be decomposed into their factors. For example, 81 can be decomposed into the factors 1, 3, 9, 27, and 81.
- Composing and decomposing numbers in a variety of ways can support students in becoming flexible with their mental math strategies.
Source: Ontario Curriculum. Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
To support the development of their number sense, junior students are expected to play with numbers: manipulating, decomposing, and recomposing numbers to uncover the key characteristics of numbers and the relationships among them. These activities also allow students to uncover several relationships between mathematical operations.
It is not a question of trying to ensure that students recognize all the relationships between numbers in a given situation. Rather, the focus should be on their ability to identify the most relevant relationships that will allow them to effectively deal with these numbers in context.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 43.
Equality Relationships
A number is represented symbolically using digits. For example, the number one thousand two hundred fifty-six written symbolically gives 1256. It can also be represented using various numerical expressions. For example, the representation \(1000 + \;200 + \;50 + 6\) allows you to recognize 1256 according to the place value of the digits that make it up. There are many other ways to decompose or represent this number:
A number can be represented in expanded form as 34 187 = 30 000 + 4000 + 100 + 80 + 7, or as 3 × 10 000 + 4 × 1000 + 1 × 100 + 3 × 10 + 7, to show place value relationships.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Exploring multiple representations of a number supports students in developing a better understanding of that number. In problem solving situations, students must learn to choose the representation that is the most appropriate for the context and intention.
Examples
To Calculate
\(\displaylines{\begin{align} 25 \times 9 &= 25 \times (10 - 1) \\ 25 \times 9 &= (25 \times 10) - (25 \times 1) \\ 25 \times 9 &= 250 - 25 \end{align}}\)
So
\(25 \times 9 = 225\)
To Do Mental Math
\(\begin{array}{l}325\; + \;527\; = \;325\; + \;525\; + \;2\\325\; + \;527\; = \;850 \; + \;2\\325\; + \;527\; = \;852\end{array}\)
To Estimate
\(\begin{array}{l}24\; \times \;26\;{\rm{is \ near \ }} 25\; \times \;25\\25\; \times \;25 \; = \;25\; \times \;\left( {20\; + \;5} \right)\\25\; \times \;25\; = \;\left( {25\; \ times \;20} \right)\; + \;\left( {25\; \times \;5} \right)\\25\; \times \;25\; = \;500\; + \; 125\\25\;\times\;25\;=\;625\end{array}\)
Thus, we can therefore say that \(24\; \times \;26\) is approximately 625.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p.48.
Skill: Describing the Ways in Which Numbers are Used in Everyday Life
The context of a problem-solving situation includes all the information surrounding that situation. This information helps us to understand the situation in which the quantities are used and facilitates looking critically at the numbers in question. In addition, context facilitates making links between numbers, mathematical concepts and the mathematical world. For all these reasons, it is important for students to explore and learn mathematics through problem solving and contextual situations.
What exactly does 22 014 mean? We are talking about 22 014 "what"? In short, a number without context has little meaning. Therefore, it must be accompanied by units (22 014 animals, people, marbles, metres…) if it is to be understood. Students in the primary grades engaged in number activities in a variety of contexts using a variety of units. In the junior grades, this contextualization continues to further develop a sense of quantity with numbers in the thousands (up to 100 000).
An important step is to support students in understanding that the same number represents the same quantity even though the contexts are different. The number is the symbolic representation of the quantity. If one has 1000 apples, 1000 heartbeats, or 1000 buildings, the quantity that is one thousand does not change. Yet, if students are asked if they believe there are more apples than buildings, many are likely to respond that there are more buildings than apples. They have focused on the space occupied by the objects rather than the quantity of objects (100 000).
Students should also recognize that depending on the context of the given situation, different interpretations can be made of the same quantity. For example, for young people, $100 may represent a lot of money. However, in context, the meaning of the number invites nuance: it is a lot for the price of a sweater, but not much for the price of a new bicycle. The context changes, but the quantity remains the same. Similarly, 100 000 blocks of wood is a lot of wood, while 100 000 hairs on the head is the equivalent of an average head of hair. Similarly, students may consider that 13 is not a large quantity, but if we add that it is the number of brothers and sisters we have, it takes on a whole new meaning. These simple, concrete examples encourage students to think about and critically analyze quantities.
In the Junior Division, understanding numbers in context becomes increasingly important. Students need to begin to make critical judgments about quantities and to exercise discernment about numbers. Learning activities should therefore help students develop other skills, such as recognizing the reasonableness of a given number, recognizing that it is an exact value, or recognizing that it is an approximate number resulting from estimation or even rounding. The development of these skills can be initiated by having class discussions about the meaning of numbers from newspapers and/or scientific journals and discussing their actual meaning and relevance.
The context also allows us to recognize that the same number does not always have the same meaning. The common interpretation associates them with a quantity, which are cardinal numbers. However, some numbers do not represent quantities, but rather a position, as is the case of ordinal numbers. For example, in a marathon, the runner who is in 11 582nd place is the 11 582nd person to cross the finish line. Numbers can also be used to label or identify as is the case with social insurance numbers, bar codes, license plates, and addresses.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 31-32.
Knowledge: Whole Numbers
Numbers that belong to the set W = {0, 1, 2, 3, 4, 5, 6, 7, 8,…}.
Examples
0, 2, 17, 36, 134, etc.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.