GEEM - Skills and Knowledge

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

Skill: Rounding Whole Numbers

The skill of rounding up requires analysis and reflection. The operation is not limited to rounding a given number to a certain place value; 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 real 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 position value (for example, should we round to the nearest hundred or to the nearest ten?). The next step is to remember benchmark numbers based on the chosen position (for example, to round 362 to the nearest 10, some students will recognize the progression of tens - 300, 310, 320, 330, 340, 350, 360, 370 - mentally or on a number line, while others will see immediately that 362 is between 360 and 370). Thus, by recognizing the closest number to the number in question (362), 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 precisely from number sense; in other words, it is the use of our knowledge that supports these decisions.

In the classroom, many decisions have already been made for students. This is when students are involved, for example, in rounding a number like 865 to the nearest ten and the nearest hundred, or simply in 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 instructional leads.

There are usually between 700 and 800 spectators at the "La Scène" theater. After the show, the box office clerk counts down the attendance and determines that there were 736 spectators. He ran into the manager and decided to give him an overview of the evening's sales. What will he tell the manager?

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 the clerk can round the number. However, if the clerk were to enter the numbers in the accounting ledger, then they would have to use the correct number.
Determine at which place value the rounding will take place There is no rule 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 force students to round to a certain place value, but to discuss with them giving examples and counter-examples so that they understand how to make a wise choice.	If we were typically selling 800-900 tickets, the rounding would be done to the nearest 100 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 usually between 700 and 800 tickets. If he wanted the rounding to give a higher degree of accuracy, he would do it to the nearest ten.
Determine benchmark numbers based on choice of place value Locating numbers on a number line helps students see the number in relation to other numbers. The use of number lines allows for a better grasp of the meaning of the number and to visualize the relative magnitude of the number. Identifying benchmarks according to the combination gives importance to the chosen combination and allows one to find the limits of an appropriate interval to complete the rounding. This involves recognizing that a number such as 736 is between the number of hundreds of the number in question (7) and the next hundred (8). Identifying benchmark numbers can also be done by recognizing a set of values related to the chosen unit (for example, 3000, 3100, ...). Note: It is very important that students use the full range of place value when rounding. For example, in trying to round to the nearest ten, some would only retain the number of the place value position in question and say, for example, that 736 is between 3 tens (30) and 4 tens (40) and therefore the rounding is to 40 (instead of 740).	In the example, the clerk determines that 736 is between 700 and 800.
Round up the number Students should grasp that rounding is done to a value close to the number. For example, 736 rounds to the nearest hundred to 700, because it is closer to 700 than to 800. Note: By convention, if the number is equidistant from 2 values, rounding is usually to the higher of the 2 values. For example, 750 is rounded to the nearest hundred to 800.	In the example, the clerk needs to understand that 736 is between 700 and 800, but it is closer to 700. Therefore, the rounding will be done at 700.
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 700 spectators, a little over 700 spectators or even close to 700 spectators. Note: It would be wrong to say that there were 700 spectators.

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 the clerk can round the number.

However, if the clerk were to enter the numbers in the accounting ledger, then they would have to use the correct number.

Determine at which place value the rounding will take place

There is no rule 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 force students to round to a certain place value, but to discuss with them giving examples and counter-examples so that they understand how to make a wise choice.

If we were typically selling 800-900 tickets, the rounding would be done to the nearest 100 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 usually between 700 and 800 tickets.

If he wanted the rounding to give a higher degree of accuracy, he would do it to the nearest ten.

Determine benchmark numbers based on choice of place value

Locating numbers on a number line helps students see the number in relation to other numbers.

The use of number lines allows for a better grasp of the meaning of the number and to visualize the relative magnitude of the number.

Identifying benchmarks according to the combination gives importance to the chosen combination and allows one to find the limits of an appropriate interval to complete the rounding. This involves recognizing that a number such as 736 is between the number of hundreds of the number in question (7) and the next hundred (8).

Identifying benchmark numbers can also be done by recognizing a set of values related to the chosen unit (for example, 3000, 3100, ...).

Note: It is very important that students use the full range of place value when rounding. For example, in trying to round to the nearest ten, some would only retain the number of the place value position in question and say, for example, that 736 is between 3 tens (30) and 4 tens (40) and therefore the rounding is to 40 (instead of 740).

In the example, the clerk determines that 736 is between 700 and 800.

Round up the number

Students should grasp that rounding is done to a value close to the number. For example, 736 rounds to the nearest hundred to 700, because it is closer to 700 than to 800.

Note: By convention, if the number is equidistant from 2 values, rounding is usually to the higher of the 2 values. For example, 750 is rounded to the nearest hundred to 800.

In the example, the clerk needs to understand that 736 is between 700 and 800, but it is closer to 700. Therefore, the rounding will be done at 700.

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 700 spectators, a little over 700 spectators or even close to 700 spectators.

Note: It would be wrong to say that there were 700 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 place value. However, the action of "rounding up" on a daily basis can be done in a broader sense. For example, a fundraiser raised $484, and the newspaper article will headline, "What a success, event raises nearly $500!" 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.

Strategies	Examples
Define an interval	If there are 736 people at a show, we can round up and say that there are between 700 and 800 people.
Round up to a benchmark	If each apple costs 44¢ and we bought a dozen, we can round up the price to 50¢.
Rounding by thinking about the effect of rounding on the quantity	If you are preparing small gifts for each participant in a contest, it is best to buy a little more than the number of participants. For example, if there are currently 63 participants, you may choose to round up to 70 and purchase 70 gifts to ensure that you have enough for all participants.

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e 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.

	Rounding a Number	Estimating a Quantity
Definitions	Replace a number with a value appropriate to the situation, following some predefined or personal criteria.	To estimate a quantity.
Examples	If the list price of a bike is $353, it costs about $400.	While walking through a park, you notice a bike and estimate its price at $300.
Explanations	The actual price (known number) has been rounded to the nearest hundred.	The price is not based on any specific information received, but on prior knowledge.

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