B2.9 Use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to solve problems.
Skill: Using Ratios to Solve Problems Involving Proportional Reasoning
The ability to use proportional reasoning is developed throughout the mathematics learning process.
The analysis of proportional relationships is done by applying proportional reasoning. This reasoning comes into play when comparing 2 ratios and recognizing a multiplicative relationship.
In grade 3 , the study of proportional relationships focuses on recognizing and describing the multiplicative relationship in a variety of problem-solving situations. Students intuitively use proportional reasoning to solve problems involving 2 quantities that are in a ratio of 1 to many (for example, 1 person for 10 fingers), many to one (for example, 5 people per table) or many to many (for example, 4 litres of juice for 8 people). They also use manipulatives and various models such as diagrams, fractions and number lines.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 50-51.
Example 1
In this set of 9 marbles below, there is a ratio of 1 to 2, \(\frac{1}{2} \) or 1:2 between the number of black marbles and the number of white marbles.
There are 3 black marbles and 6 white marbles. This means that the white marbles are twice the black marbles. Using a double number line, the top number shows a scale by one and the bottom number shows a scale by 2. In this scaling of 1:2, we see that the white marbles represent the double of the black marbles.
Example 2
Teachers ask primary students to determine the number of pieces in 3 chocolate bars if a bar contains 5 pieces (ratio of 1 to 5). This is a multiplicative relationship since the number of pieces is 5 times greater than the number of bars. However, to solve this kind of problem, students will first use repeated addition (\(5 + 5 + 5 \)). Later, when they have been exposed to the concept of multiplication, they can solve it by multiplying (\(3 \times 5 \)), which is a first step towards using proportional reasoning.
\(5 + 5 + 5\) or \(3 \times 5\)
The proportionality relationship can be represented using a table of values. In a table of values that represents a situation of proportionality, the ratios between the corresponding quantities are equivalent.
| Chocolate Bars | 1 | 2 | 3 |
|---|---|---|---|
| Ratio 1 : 5 | \(\times 5\) | \(\times 5\) | \(\times 5\) |
| Number of pieces | 5 | 10 | 15 |
So, in 3 chocolate bars, there are 15 pieces.
In the example above, we easily recognize the multiplicative relationship by 5 between the number of bars and the number of pieces. In addition, this table of values allows to establish proportions (for example, \(\frac{1}{5} = \frac{3}{15} \))
Students place ratios on the double number line equivalent to the one given in order to solve the problem. They can choose the ratios according to their needs and understanding of the problem.
Example 3
Abdala buys bread to make sandwiches for the school picnic. The cost of the bread is $2 to make 20 small sandwiches. How much bread will it cost to make 60 sandwiches?
Using a double number line, the top number shows a scaling by one, which is the cost of bread, and the bottom number shows a scaling by 10, which is the number of sandwiches. The student knows that the ratio here is 1 : 10 since the multiplicative relationship between 2 and 20 is 10.
The cost of bread for 60 sandwiches is $6.
Source : adapted from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 50-52.
The multiplicative relationship between 2 ratios is the basis of proportional reasoning. Ratios are present in everyday life and in several mathematical situations, notably in place values (for example, the ratio between units and tens is 1:10), in fractions (for example, \(\frac{ 1}{5} \) or 1:5), in similar figures (for example, a magnification of 1:2), in units of measurement of the metric system (for example, the ratio between metres and decimetres is of 1:10).
Students should use a variety of models in activities to develop their proportional reasoning skills. Moreover, these informal experiences will be used for further study of ratios, rates, percentages, and algebra in later grades.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 54.
Knowledge: Proportional Reasoning
Proportional reasoning is basically seeing numbers in terms of their relative rather than absolute values.
Source: Qu'est-ce que le raisonnement proportionnel ?, p. 3.
The analysis of proportional relationships is done by applying proportional reasoning. This reasoning is used when comparing two ratios and recognizing a multiplicative relationship. Note that multiplicative relationships include the operation of division, since any division can be transformed into multiplication (for example, dividing by 2 is equivalent to multiplying by \(\frac{1}{2}\)).
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 50.
Knowledge: Proportionality Relationship
There is a relationship of proportionality between 2 quantities when these quantities can increase or decrease simultaneously by the same factor.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 49.
Knowledge: Ratios
Relationship between 2 quantities expressed as the quotient of the numbers that characterize them.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 49.