B2.9 multiply whole numbers by proper fractions, using appropriate tools and strategies
Multiplying Whole Numbers by Proper Fractions, Using Appropriate Tools and Strategies
In the junior grades, students already have a background of knowledge about multiplication. Since the primary grades, they have been exploring concepts related to multiplication using concrete materials, the calculator, illustrations and symbols. In Grade 5, fraction multiplication is limited to the multiplication of a whole number by a unit fraction. This type of multiplication can be understood by relating it to repeated addition. For example, students easily grasp that "3 times one-half" is a multiplication that can be represented by repeated addition, i.e., "3 times one-half. These should be explored to help students understand fraction multiplication.
However, it is more difficult to make sense of the multiplication of a fraction to a whole number (for example,
\(\frac{1}{2} \times \;3\)). These situations are explored beginning in Grade 5. There is a connection between the
concept of multiplication by a fraction and the concept of a fraction of a set. The fraction of a set is a concept
related to the concept of fraction. For example, in Grade 2, students learn the meaning of a group of 12 objects.
Later, they consolidate their understanding of the concept of a fraction of a set by applying it to other fractions.
In Grade 6, as they examine the concept of multiplying a whole number by a proper fraction, they will learn that the
fraction of a set \(\frac{2}{3}\) of 6) is related to multiplication and that this situation can be represented by
multiplication \(\frac{2}{3}\; \times \;6\). It takes a good degree of abstraction to accept that a situation like
\(\frac{1}{2}\) of 12 is considered a multiplication.
Mathematicians decided that this was a multiplication by doing something like this:
- we can consider \(4\; \times \;6\) as 4 groups of 6;
- we can consider \(2\; \times \;8\) as 2 groups of 8.
There is no difficulty in extending this observation to fractional numbers greater than 2 :
- we can consider \(4 \frac{1}{2}\; \times \;6\) as 4 and one half groups of 6;
- we can consider \(2 \frac{1}{3}\; \times \;12\) as 2 groups and one third of 12.
We generalize this situation (which implies an abstraction, since the group is not "multiplied" as such) by adding that :
- we want to consider \(1\frac{1}{2} \times \;6\) as a group and a half of 6;
- we want to consider \(\frac{1}{2} \times \;6\) as a half-group of 6.
Thus, it is as a result of an interpretation of the operation that \(\frac{1}{2}\) of 6 is considered as a multiplication of \(\frac{1}{2}\) and 6.
Example:
In a Grade 6 classroom, \(\frac{1}{2}\) students are wearing hats. If there are 24 students in the classroom, how
many students are wearing a hat?
Source: Adapted from Math: A Little, A Lot, A Lot of Fun, Teacher's Guide, Revised Edition, Numeration and Number Sense, Grade 6, Unit 2, Series 2, p. 281.
- Multiplication using a double number line
To determine \(\frac{1}{2}\) of 24, I divide 24 by 2, which gives me 12.
I represent the 24 students below the line and half of 24 on the top of the line.
There are 12 students wearing hats.
- Multiplication performed using an area model
I decompose 24 into \(20\; + \;4\). I determine half of 20 and half of 4, \(10\; + \;2\; = \;12\).
\(10\; + \;2\; = \;12\)
There are 12 students wearing hats.
In Grade 6, with a focus on concepts and contextualization, it is more relevant to deepen the meaning of a fraction
of a set by performing a calculation (for example, \(\frac{2}{3}\) of 9) than to move toward multiplying a whole number
by a fraction \(\frac{2}{3}; \times \;9\).
Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p 78-79.
Example
In a rectangular field with an area of 100 m2, Mr. Longpré has planted cucumbers on \(\frac{2}{5}\)of the
field . What area of the field, in m2, is devoted to cucumbers?
Source: Adapted from Math: A Little, A Lot, A Lot of Fun, Teacher's Guide, Revised Edition, Numeration and Number Sense, Grade6, Unit 2, Series 2, p. 309.
- Multiplication performed using an area model
I have decomposed 100 into \(25\; + \;25\; + \;25\; + \;25\). I also decomposed \(\frac{2}{5}\) into \(\frac{1}{5}\;
+ \;\frac{1}{5}\). I know that \(\frac{1}{5}\) of 25 is 5 since \(5\; \times \;5\; = \;25\).
I added up the partial products to arrive at 40.
The area of the field dedicated to the cultivation of cucumbers is 40 m2.
- Multiplication Using a Personal Algorithm
I decomposed \(\frac{2}{5}\) into \(2\; \times \;\frac{1}{5}\).
Using the associative property, I multiplied \(\frac{1}{5}; \times \;2\; \times \;100\).
I multiplied \(\frac{1}{5}\; \times \;200\).
Multiplying by \(\frac{1}{5}\) is the same as dividing by 5.
I get 40.
\(\begin{align}\frac{2}{5}\; \times \;100\; = \;2\; \times \;\frac{1}{5}\; \times \;100\; &= \;\frac{1}{5}\; \times \;2\; \times \;100\\ &= \;\frac{1}{5}\; \times \;200\\ &= \;200\; \div \;5\\ &= \;40\end{align}\)
The area of the field dedicated to the cultivation of cucumbers is 40 m2.
Knowledge: Proper Fraction
Fraction whose numerator is less than the denominator.
Examples
\(\frac{2}{3}\), \(\frac{{11}}{{15}}\), \(\frac{{53}}{{123}}\)