At a Glance...

B2.11 represent and solve problems involving the division of decimal numbers up to thousandths by whole numbers up to 10, using appropriate tools and strategies

Representing and Solving Problems Involving the Division of Decimal Numbers up to Thousandths by Whole Numbers Up to 10

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During their learning, the pupils learned that a division like \(12\; \div \;3\) is carried out according to a context where one creates groups of 3 elements (direction of grouping) or according to the one where we create 3 equal groups (sense of sharing). These two contexts also exist in the presence of a division of a decimal number by a whole number.

Thus, having understood these two meanings of division with whole numbers, students can recognize that the situation of dividing $23.50 equally among 5 friends has a sharing meaning and can be represented by \(23.50\; \div \;5\), a division whose quotient represents the size of a group (number of dollars a friend receives). They may also recognize that the situation of pouring 23.50 litres of juice into 5-litre containers has a grouping meaning and can also be represented by \(23.50\; \div \;5\). However, in this case, the quotient represents the number of groups (number of filled containers).

Then, regardless of the given situation (sharing or grouping), students can estimate that the quotient is about 5 and they can use a strategy to determine that the quotient is equal to $4.70 (sharing) or 4.7 filled jugs (grouping).

The fractional nature of a decimal number makes the interpretation of quantities in a division more difficult than with whole numbers as shown in the following table.

In a sharing situation, understanding the quotient is usually quite simple since the decimal part of the quotient represents a part or fraction of an element. Thus, in the previous sharing problem, it is fairly clear that each group is composed of 3.2 km (3.2 elements), which is 3 whole kilometres and 0.2 \(\frac{2}{10}\) of another kilometre.

In a grouping situation, we count groups. The decimal part of the quotient is then more complex to interpret since this decimal part represents a part of a group. From the previous grouping problem, students can understand the meaning of the quotient by making an estimate using whole numbers. They can then say that it will take Peter just over 3 hours \(3 \times 4 = 12\) or between 3 and 4 hours to complete the hike. In order to determine a quotient by following the direction of grouping, we must create groups of a given size. In the example, we create groups of 4 km. We create 3 groups and we end up with another partially complete group. What does this part of a group represent? In a grouping situation, it is the whole group that becomes the unit to be counted. Thus, the 8 tenths of a tab should be considered a fraction of the group, the group being composed of 4 tabs or 40 tenths of a tab. So the 0.8 in this group then represents 0.2 of a group.

The sense of division reflects the most common sense of a division whose dividend is a decimal number. Moreover, as mentioned previously, the interpretation of the decimal quotient is also more accessible in a sharing context. It is still important that students are exposed to a few examples of divisions that take on the meaning of grouping.

As with multiplication, students can discover, using a series of related operations, that in divisions such as \(72\; \div \;4\); \(\;7,2\; \div \;4\); \(\;0.72\; \div \;4\) and \(0.072\; \div \;4\) the digits that make up the quotients are the same, in the same order, and that only the position of the comma is different.

Example:

Students have already seen that to perform a multiplication such as 4, they can estimate the answer (for example, the product will be close to 48, because 4, 12, = 48), perform the multiplication as if it were whole numbers (4, 12) to get the product 512, and then place the decimal point in the middle;48)), multiply as if they were whole numbers (\(4\; \times \;128\)) to get the product 512, and then place the decimal point in the appropriate place (51.2), using the estimate as an indicator of the order of magnitude. They can do the same thing with division.

Example 1

One seeks to carry out \(32,4\; \div \;3\).

We know that the answer is close to 10, because \(30\; \div \;3\; = \;10\).

Divide as if they were whole numbers \(324\; \div 3\) to get the quotient 108.

Then place the decimal point in the appropriate place, that is, 10.8, knowing that the answer is close to 10.

Note: Since 32.4 is 10 times smaller than 324, the quotient of \(32.4\; \div \;3\) (10.8) is 10 times smaller than the quotient of \(324\; \div \;3\) (108).

Example 2:

One seeks to carry out \(0,388; \div \;4\).

To estimate, we can think that we have about 400 mils to divide by 4, which gives about 100 mils (0.100).

We can also think that we have about 40 hundredths to divide by 4, which gives about 10 hundredths (0.10) or even that we have about 4 tenths to divide by 4, which gives about 1 tenth (0.1).

We divide \(388\; \div \;4\) to obtain 97. We place the comma in the appropriate place, i.e. 0.097, because we know that the answer is close to 0.100 or 0.10 or 0, 1.

It is best to stick to answers with a maximum of three decimal places. This can be done by working backwards to select appropriate numbers.

Example 3

The following situation comes to mind:

During her vacation, Valerie wants to travel 15.3 km in one week. She decides to travel the same distance every day for 7 days. How far will she travel each day?

Using a calculator, we determine that \(15.3 \div \;7 = \;2.185\;714\)… We decide that it is best to modify the data in the problem to give the answer as 2.2. To find the dividend, simply calculate \(7; \times \;2.2\). We get 15.4. The situation is then modified as follows:

During her vacation, Valerie wants to travel 15.4 km in one week. She decides to travel the same distance each day for 7 days. How far will she travel each day?

Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p 83-87.

When dividing a decimal by a whole number using various strategies, it is important that students interpret the result correctly. To do so, they must refer to the estimate made beforehand to ensure the order of magnitude of the quotient. They must also refer to the problem situation to ensure that the quotient is meaningful, or in other words, that the result represents the size of the groups or that it represents the number of groups. Students must also understand and be able to explain the strategy used to perform the operation.

Often, when division of a decimal number represents the grouping direction, the operation is not performed according to that direction since representing a grouping reasoning requires the creation and interpretation of a part of a group, which makes the process more difficult to grasp. Division is then most often performed following a sharing reasoning. Here are different strategies for dividing, depending on the direction of division, a decimal number by a whole number, for example:

In order to estimate the quotient, it is possible to reason as follows: 105 thousandths is almost 100 thousandths, and 100 thousandths \( \div \;3\), it is a little more than 33 thousandths (0.033) ; so 105 mils \( \div \;3\) is also a bit more than 33 mils (0.033).

• Division using base ten materials

Note: The large cube represents the unit.

• Division using student-generated algorithm

• Division using the standard algorithm

Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p 107-110.

Knowledge: Decimal Number

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A decimal number is a number that can be expressed in decimal notation with a finite decimal part.

Example

3.72 and 12.135

The set of decimal numbers includes all integers, because integers can be expressed with a decimal part.

Example:

\(3\; = \;3,0\)

It also includes some fractions, such as \(\frac{2}{5}\) and \(\frac{3}{16}\), since \(\frac{2}{5} = 0.4\) and \(\frac{3}{16} = \;0.187\;5\). We notice then that \(\frac{1}{2}\), \(\frac{5}{10}\) and 0,5 are symbolic representations of the same decimal number.

Source: A Guide to Effective Instruction in Mathematics Grades 4-6, p. 28.

Knowledge: Role of the Decimal Point

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The decimal point plays a significant role in decimal notation. It separates the whole part from the decimal part.

The position of the ones defines the whole according to which are formed on the one hand the tenths, the hundredths and the thousandths and on the other hand the tens, the hundreds and the thousands.It can therefoere be said that the ones, identified by the decimal point, is at the heart of the decimal system.

This recognition of the role of the unit is highlighted by the prefixes of the names given to the place value of the digits on either side of the unit. Thus, thousandths represent a quantity one thousand times smaller than the unit.

Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p. 51.

Knowledge: Algorithm

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Knowledge: Usual Algorithm

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Standardized method for performing an operation.

Example

Knowledge: Student-Generated Algorithm

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A strategy for solving a problem or performing a calculation, usually developed by the student.

Example:

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 76.