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. It is not limited to simply rounding a given number to a certain place value because 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 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. For example, they might be asked to round a number like 865 to the nearest ten and the nearest hundred, or to determine 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.

An activity that illustrates reasoning during number rounding is as follows:

There are usually between 700 and 800 in the audience at the "La Scène" theatre. After the show, the box office clerk counted the attendance and determined that there were 736 attendees. What can he tell the manager about the showing?

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 exact numbers should be used.
Determine at which place value the rounding will take place There is no rule for choosing the place value at which to round. 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 different scenarios with them and giving examples and counter-examples, so that they better understand how to make a wise choice.	As the typical sales are between 700 and 800 tickets, the clerk might round to the nearest ten to show that the sales were typical. The clerk could decide to round down to the nearest hundred if the goal is to show that sales were on the low side. 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 a number in relation to other numbers. The use of number lines supports visualizing the relative magnitude of the number. Identifying benchmarks that relate to a number or numbers allows one to determine an appropriate interval to complete the rounding. This involves recognizing that a number such as 736 is between 7 hundred and 8 hundred. Identifying benchmarks can also be supported by finding a set of values related to the chosen place value (for example, if rounding to the nearest ten, thinking about 70 tens, 71 tens, 72 tens, 73 tens, 74 tens...). Note: It is very important that students consider the full number when rounding. For example, in trying to round to the nearest ten, some might focus only on the tens and say, for example, that 736 is between 3 tens (30) and 4 tens (40) and therefore the number is closer 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 using values close to the number. For example, 736 rounds to 700, because it is closer to 700 than to 800. Note: If the number is an equal distance between two values, rounding is usually to the higher of the two 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 results in 700.
Communicating the rounding Since rounding is context-bound, communication must also include context.	The clerk might mention that there were about 700 in attendance, a little over 700 or even close to 700 attendees. 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 exact numbers should be used.

Determine at which place value the rounding will take place

There is no rule for choosing the place value at which to round. 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 different scenarios with them and giving examples and counter-examples, so that they better understand how to make a wise choice.

As the typical sales are between 700 and 800 tickets, the clerk might round to the nearest ten to show that the sales were typical.

The clerk could decide to round down to the nearest hundred if the goal is to show that sales were on the low side.

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 a number in relation to other numbers.

The use of number lines supports visualizing the relative magnitude of the number.

Identifying benchmarks that relate to a number or numbers allows one to determine an appropriate interval to complete the rounding. This involves recognizing that a number such as 736 is between 7 hundred and 8 hundred.

Identifying benchmarks can also be supported by finding a set of values related to the chosen place value (for example, if rounding to the nearest ten, thinking about 70 tens, 71 tens, 72 tens, 73 tens, 74 tens...).

Note: It is very important that students consider the full number when rounding. For example, in trying to round to the nearest ten, some might focus only on the tens and say, for example, that 736 is between 3 tens (30) and 4 tens (40) and therefore the number is closer 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 using values close to the number. For example, 736 rounds to 700, because it is closer to 700 than to 800.

Note: If the number is an equal distance between two values, rounding is usually to the higher of the two 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 results in 700.

Communicating the rounding

Since rounding is context-bound, communication must also include context.

The clerk might mention that there were about 700 in attendance, a little over 700 or even close to 700 attendees.

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 rounding 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 appropriate range	If there are 736 people at a show, we 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: 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. 38-41.

Knowledge: Approximation (Estimation and Rounding)

Numbers 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 show an order of magnitude for 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 there). 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, and incorrectly, used interchangeably. The fundamental difference between these two concepts lies in the how the number is generated. Estimating uses the relationship between an unknown quantity and prior knowledge, usually in the form of benchmark numbers. 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 magnitude of the quantity. The following table 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: 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. 35-36.