At a Glance...

B1.3 Round whole numbers to the nearest ten, hundred, or thousand, in various contexts.

Skill: Rounding Whole Numbers to the Nearest Ten, Hundred or Thousand

Rounding a number makes it easier to use. This method is often used to make estimates, measurements and quick comparisons.

A rounded number remains close to the original number depending on the unit it is being rounded to. The larger the unit, the broader the approximation; the smaller the unit, the more precise.

Whether a number is rounded up or down depends on the context. For example, in a grocery store, people may round up to make sure they have enough money. Contrarily, if they are looking at a pile of coins, they may want to round down to again ensure they have enough money.

In the absence of a context, numbers are typically rounded on a midpoint, for example :

• Rounding 1237 to the nearest 10 becomes 1240, since 1237 is closer to 1240 than to 1230.
• Rounding 1237 to the nearest 100 becomes 1200, since 1237 is closer to 1200 than to 1300.
• Rounding 1237 to the nearest 1000 becomes 1000, since 1237 is closer to 1000 than to 2000.

If a number is exactly on the midpoint, convention rounds the number up (unless the context suggests differently). So, 1235 rounded to the nearest 10 becomes 1240. With the elimination of the penny, cash transactions are now rounded to the nearest five cents (or nickel).

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

Rounding

The skill of rounding up requires analysis and reflection. The operation is not limited to rounding a given number to a certain unit; it also requires the evaluation of the context in which the number is found. The activities that target the ability to round must then be done in context so that they reflect an authentic use of rounding, to give meaning to this learning, meaning that will be transferred to the cognitive foundation of the students.

In everyday life, the decision to round comes from the interpretation of a situation, not from a received instruction. Then, the situation and the number in question are evaluated to designate a rounding unit (for example, should we round to the nearest thousand or to the nearest hundred?). The next step is to remember benchmark numbers based on the chosen unit (for example, to round 3620 to the nearest hundred, some students will recognize mentally or on a number line the progression of hundreds - 3000, 3100, 3200, 3300, 3400, 3500, 3600, 3700, while others will see immediately that 3620 is between 3600 and 3700). Thus, by recognizing the number closest to the number in question (3620), students will be able to determine the rounded value of the number. The final step is to communicate the rounded number. Of course, some of the steps are done intuitively and almost automatically. This fluency comes specifically from number sense; in other words, it is the use of our knowledge that supports these decisions.

In the classroom, many decisions have often already been made for students. This is when students are involved, for example, in rounding a number like 2365 to the nearest ten, hundred, and thousand, or simply determining whether 365 is closer to 300 or 400. It would be beneficial to their learning if students had the opportunity to make the full range of decisions surrounding the rounding of a number. This helps develop their critical sense of number use and deepens their sense of rounding.

Here is a situation that illustrates reasoning during number rounding. The additional information offers some pedagogical leads.

There are usually between 3000 and 4000 people in attendance at "The Scene". After the show, the box office clerk counted the number of spectators and determined that there were 3736. He meets with management and decides to give them an overview of the evening's sales. What will he tell management?

Reasoning During Rounding	Reasoning of the Clerk
Decide to round up It makes sense to round up if the goal is to communicate a quantity using a number that can quickly become meaningful to the recipient.	In the example, if the clerk wants to communicate the magnitude of the sales and not the exact quantity, then he or she can round the number. However, if the clerk were to enter the numbers in the accounting ledger, then he or she would have to use the correct number.
Determine at which unit the rounding will take place There are no rules for choosing the place value. However, the choice is not arbitrary, as it depends on the interpretation of the situation and the individual's sense of number. It is important not to systematically impose on students to round to a certain unit, but to discuss with them by giving examples and counter-examples so that they understand how to make a judicious choice.	If sales were typically between 8000 and 9000 tickets, the rounding would be to the nearest thousand to show that there is a significant drop in sales. In the example, the clerk decides to round up to the nearest hundred since sales are typically between 3000 and 4000 tickets. If he wanted the rounding to give a higher degree of accuracy, he would round to the nearest 10.
Determine benchmark numbers based on the choice of unit Locating benchmark numbers helps students see the number in relation to other numbers. Using number lines helps students understand the meaning of the number and visualize the relative magnitude of the number. Locating numbers based on grouping gives importance to the grouping chosen and allows students to find the boundaries of an appropriate interval to complete the rounding. This involves recognizing that a number such as 3736 is located between the number of hundreds of the number in question (37) and the next hundred (38). Identifying benchmark numbers can also be done by recognizing a set of values related to the chosen unit (for example, 3000, 3100, ...). Note: It's very important that students use the full range of units when rounding. For example, when trying to round to the nearest hundred, some students would retain only the number of the unit in question and say, for example, that 3736 lies between 7 hundreds (700) and 8 hundreds (800) and that the rounding is therefore done to 700 (instead of 3700).	In the example, the clerk determines that 3736 is between 3700 and 3800.
Round the number Students should understand that rounding is performed to a unit close to the number. For example, 3736 is rounded to the nearest hundred to 3700, as it is closer to 3700 than to 3800. Note: If a number is exactly on the midpoint, convention rounds the number up. For example, 3750 is rounded to the nearest hundred to 3800.	In the example, the clerk needs to understand that 3736 is between 3700 and 3800, but that it is closer to 3700. Therefore, the rounding becomes 3700.
Communicating the rounding Since rounding comes from a context, communication must be done in context.	The clerk might mention to management that there were about 3700 spectators, a little over 3700 spectators or even close to 3700 spectators. Note: It would be wrong to say that there were 3700 spectators.

Classroom activities should allow students to develop the full range of rounding skills. It is important that teachers nuance their words during rounding activities so as not to direct students' thinking toward a particular strategy or answer. So far, rounding has been discussed to within one unit. However, the action of "rounding up" on a daily basis can be done in a broader sense. For example, a fundraiser raised $14 345, and the newspaper article will headline, "What a success, event raises nearly $15 000!" In this case, the roundig was used to "round up" an amount of money so that the information would be understood quickly. Here are some examples of possible rounding strategies.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 38-41.

Knowledge: Approximation (Estimation and Rounding)

Numbers were created to represent quantities with a high degree of accuracy. This is because they provide a precision that terms like "more", "some", "many" and "few" do not. However, they can also be used to show some order of magnitude of that quantity. In this case, the number is used to approximate the quantity (for example, about 200 people were at the party does not mean that there were exactly 200). In general, the approximation is a magnitude that is close enough to a known (rounding) or unknown (estimation) magnitude.

The terms "rounding" and "estimation" are often incorrectly used interchangeably. The fundamental difference between these 2 concepts lies in the origin of the number. The estimation comes from the relationship between an unknown quantity and prior knowledge, usually in the form of benchmark numbers. Rounding, on the other hand, comes from the relationship between a known number (precise or approximate) and its relative proximity to other numbers. Generally, estimations and rounding are used to paint a picture of the quantity in question and to convey a sense of the magnitude of the quantity. The following table, which discusses the example of the price of a car, demonstrates this distinction.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 35-36.