At a Glance...

B1.1 Represent and compare whole numbers up to and including one billion, including in expanded form using powers of ten, and describe various ways they are used in everyday life.

Skill: Representing Whole Numbers Up to and Including One Billion

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The base 10 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 in a number. For example, the digit 2 has a value of 2 ones in the number 1 356 742, while it has a value of 20 000 ones in the number 5 623 487. 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 division, students develop an understanding of the relationships between the place values of ones, tens and hundreds. However, in the junior and intermediate divisions, they do not automatically transfer this understanding to larger numbers. This is why teachers must ensure that they understand that the value of any position in a number is always 10 times greater than the value of the position immediately to the right, and 10 times less than the value of the position immediately to the left. It is also important to examine the relationships of 100 times or 1000 times greater or lesser between place values in order to develop in students a deep number sense, especially a sense of large numbers.

Students should also recognize, for example, that 10 000 represents a grouping of 10 one thousands, 100 hundreds, 1 000 tens, or even 10 000 ones. These regroupings allow students to recognize equivalent representations of numbers (for example, 2 534 represents 25 hundreds and 34 ones).

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 44-45.

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. Therefore, it is essential to expose students to different representations of numbers. It is also important to expose them 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 tap into these 4 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, then concrete and semi-concrete representations. Only when they 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.

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 64-65.

Representations Using Words

Number is an abstract representation of a very complex concept. For this reason, the relationship between the way a number is named and the quantity it represents is not 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, such as 3 000 047, without really having a sense of the quantity it represents.

In the junior division, students learn to read and write numbers up to one million and in the intermediate division, up to one billion. The challenges of writing numbers in words should not be underestimated. To help students overcome these challenges, teachers should include numbers on the word wall and build repositories with them to summarize the number agreement rules.

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 65-66.

Symbolic Representations

Numbers are represented symbolically by the digits that make them up. They are written from left to right in 3-digit periods that make up the trillions, billions, millions, thousands and ones. Each of the periods includes hundreds (h), tens (t) and ones (o).

Notes:

• In French, the term "billion" does not have the same meaning as the English term billion. In French, a trillion corresponds to 1 000 billion (\({10^{12}})) whereas in English it corresponds to 1 000 million (\({10^9}})). In other words, the English term billion corresponds to the French term "milliard".
• Note: In English as well as in French, numbers are written by adding a space between the 3-digit periods (for example, 13 567 232). Although writing 4-digit numbers without using a space is accepted (for example, 3543), writing them with a space (for example, 3 543) is preferred.

The writing of large numbers requires a good mastery of the concept 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 the 0 to indicate the absence of a quantity in one of the positions.

Here is an example of reasoning that students might use to symbolically represent a large number such as "thirteen million twenty-six thousand four hundred":

• thirteen million is represented by 1 and 3 in the millions period, which implies that there are 6 positions to the right of it to fill;

• 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 hundreds of thousand, because the word "hundred" is not heard in this period;

• four hundred is represented by a 4 in the hundreds position of the ones period. Since there is no indication for the tens and ones position, a 0 must be added to fill each of these positions.

• thirteen million twenty-six thousand four hundred written symbolically gives 13 026 400.

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 \(1;000; + \;200; + \;30; + \;6)), or from some place values (1 236 equals 12 hundreds and 36 ones) or by performing different operations (1 236 equals 1;000\; + \;236\) 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 to represent a number, each of which allows students to develop a different way of interpreting it and understanding its meaning.

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 70-71.

Equality Relationships

A number is represented symbolically using digits. For example, the number one thousand two hundred fifty-six written symbolically gives 1 256. It can also be represented using various numerical expressions. For example, the representation \(1\;000\; + \;200\; + \;50\; + \;6\) allows you to recognize 1 256 according to the place value of the digits that make it up. There are many other ways to decompose or represent this number.

Equality relationships make it possible to establish equivalence between various representations of the same quantity. Exploring multiple representations of a number helps students develop a better understanding of the meaning of that number. In problem-solving situations, students must learn to choose the representation that is the most appropriate for the context and intention.

Here are some examples:

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 48-49.

Large numbers can be expressed as decimal numbers by expressing the place value in words. For example, 36.2 million is 36 200 000 = (36.2; times; 10^6).

Expanded notation with powers of ten shows a number as an expression by using addition, multiplication, and exponents. The number “three hundred seven million, twenty thousand, and fifty”, can be expressed as 307 020 050 and \(3\; \times \;{10^8}\; + \;7\; \times \;{10^6} \; + \;2\; \times \;{10^4} + \;5\; \times \;10\).

Source: The Ontario Curriculum. Mathematics, Grades 1-8, Ontario Ministry of Education, 2020.

Skill: Comparing Whole Numbers Up To and Including One Billion

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Numbers can be compared by their place value or they can be compared using proportional reasoning. For example, 1 billion is 1000 times greater than 1 million.

Source : The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020..

Order Relationships

The ability to recognize order relationships is acquired by comparing numbers, decomposing them, placing them in ascending and descending order, counting backwards and analyzing the relative proximity of 2 numbers.

In the intermediate division, students are expected to recognize the order relationship between numbers by comparing them. They can describe the relationship by stating, for example, that 350 198 052 is smaller than 432 840 590. The following are examples of strategies they can use to compare large numbers.

Comparison of the numbers 23 942 and 12 273

Students can recognize that 23 942 is larger than 12 273 :

• by targeting a significant portion of each number (note that 23 000 is larger than 12 000);
• by visualizing large quantities (visualize 23 groupings of 1 000 and 12 groupings of 1 000);

• by locating the numbers on a number line (the number 23 942 is located to the right of the number 12 273);

• by comparing the numbers in the various positions from the left (2 represents 20 000 while 1 represents 10 000);

• by comparing the position values expressed using the power of 10.

Tokyo and Delhi have a higher value at the tens of millions position (¹⁰⁷) (3 tens of millions compared to 2 tens of millions)

Source: En avant, les maths! 7e année, CM, Nombres, p. 5.

To help students develop the skill of recognizing order relationships among large numbers, teachers can, starting with a given number, ask them to count up by 1 (for example, 12 998, 12 999, 13 000, 13 001…) or by intervals (for example, 32 200, 32 400, 32 600…) and count backword by 1 (for example, 26 271, 26 270, 26 269…) or by intervals (for example, 45 650, 45 600, 45 550…).

These activities help students recognize that when counting by 1s or intervals, any named number is greater than those before it and less than those after it, while when counting backwards by 1s or intervals, any named number is less than those before it and greater than those after it. While these relationships may seem obvious to adults, students often get them wrong because they do not consider the concept of grouping. When asked, for example, what number precedes 300, many students tend to spontaneously answer 399 because their attention is focused on the two 0's; they know that a number ending in two 0's is always preceded by a number ending in two 9's, and they forget to consider grouping by hundreds. In contrast, when the same problem is posed in context, students give more thoughtful responses; for example, in a situation where a person has 300 field hockey cards and loses one, students will readily respond that they have 299 cards left.

Once students have mastered the relationships "greater than" and "less than," they need to learn to make these relationships more specific by drawing on their understanding of quantity. They then use expressions such as close to, about, the same as, much more than , and a little less than. For example, students may say that the population of a village of 15 239 is about 15 000; that 304 is a little more than 300; that 12 894 is a little less than 13 000; that 32 523 is about a hundred more than 32 432; and that 620 and 618 are closer together than 630 and 680.

This ability to specify order relationships between numbers becomes important when students use numbers in problem solving, rounding, estimating and comparing. The following activity allows students to demonstrate this skill.

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 45-48.

Skill: Describing the Ways in Which Numbers are Used in Everyday Life

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Context is all the information surrounding a given situation. This information helps to identify the situation in which the quantities are used and facilitates the exercise of a critical eye on the numbers in question. In addition, the context facilitates the establishment of connections between numbers, mathematical concepts and the mathematical world. For all these reasons, the exploration of mathematics in problem-solving situations is advocated.

What does 21 560 814 mean exactly? We are talking about 21 560 814 "what"? In short, a number without context has little meaning. Therefore, it must be accompanied by units if it is to be understood. Students in the primary division have already engaged in number activities in a variety of contexts using a variety of units. In the junior and intermediate divisions, this contextualization must be maintained in order to develop a sense of quantity, and this must be done with numbers in the thousands and millions.

An important step to take is to get students to understand that the same number represents the same quantity even though the contexts are different. The number is only a symbolic representation of the quantity. If one has 100 000 apples, 100 000 heartbeats, or 100 000 buildings, the quantity that is the grouping of 100 000 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 focus 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, 200 000 blocks of wood is a lot of blocks, while 200 000 hairs on the head is the equivalent of an average head of hair. Or, students may consider that 13 is not a large quantity, but if we add that it is the number of our brothers and sisters, it takes on a whole new value. These simple, concrete examples encourage students to think about and critically analyze quantities.

In the junior and intermediate divisions, understanding numbers in context becomes increasingly important. Students need to begin to make critical judgments about quantities and to be discerning 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 discussing their actual meaning and relevance.

The context also allows us to recognize that the same number does not always have the same meaning. Indeed, when it comes to numbers, the common interpretation associates them with quantity or cardinal numbers. However, if we look at the context, we notice that some numbers do not represent quantities, but a position. This is the case of ordinal numbers. For example, in a marathon, the runner who is at position 11 582 does not represent 11 582 objects. The number indicates the position of a single person in the given series. Numbers are also used as an identification code. Social insurance numbers, bar codes, license plates and addresses fall into this category.

Benchmarks

In general, a benchmark is a mark, a milestone, a reference element. As part of the development of a sense of number, the use of benchmarks promotes mental representation and thus facilitates the understanding of number and the notion of "how much". The 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.

Benchmark numbers are particularly useful for understanding large numbers, since it is usually impossible to recognize these quantities globally or to grasp them by counting. Students need to get a sense of them by comparing them with a benchmark number. For example, the school just received 10 000 sheets 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 10 000 sheets might represent. Using a package of 500 sheets as a guide, students can imagine this quantity and apply the proportionality relationship to deduce that 2 packages contain 1000 sheets. Therefore, 10 000 sheets would be 10 times as many packages, or 20 packages of sheets. We can even recognize that 20 packages are equivalent to 2 boxes of paper. The mental representation of the space occupied by these 10 000 sheets then becomes possible. And the answer to the original question, whether the janitor should ask for help in carrying the sheets, can then be debated with full knowledge.

Students need to develop benchmarks to make it easier for them to use them depending on the context and the 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: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 31-35.

Knowledge: Expanded Form

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The expanded form shows a number as an expression using addition, multiplication and exponents.

Example

The number three hundred seven million twenty thousand fifty can be expressed as 307 020 050 or \(3\; \times {10^8}\; + \;7\; \times {10^6}\; + \;2\; \times {10^4}\; + \;5\; \times {10^1}\).

Knowledge: Power of Ten

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Number 10 multiplied by itself the number of times indicated by the exponent.

Example

\({10^2} = \;(10 \times 10 \;) = 100\)

Source: En avant, les maths! 7e année, CM, Nombres, p. 3.