B2.10 Identify proportional and non-proportional situations and apply proportional reasoning to solve problems.
Skill: Identifying Proportional and Non-Proportional Situations and Applying Proportional Reasoning
The ability to master proportional reasoning is a critical factor in the understanding and application of mathematics. Lamon estimates that more than 90 percent of students entering high school lack the reasoning skills necessary to fully understand mathematics and science, and are ill-prepared for real-world applications in statistics, biology, geography or physics (Lamon, 2005, p. 10). The student may be able to solve a problem based on proportions using a method that has been memorized, but this does not mean that the student is able to apply proportional reasoning in the course of solving the problem.
Source : Qu'est-ce que le raisonnement proportionnel?, 2012, Ministère de l’Éducation de l’Ontario, p. 4.
Multiplicative Reasoning
It is a concept that requires the ability to deal with several ideas or quantities at once. The idea is to see problems in relative rather than absolute values. Consider the following problem, "If one dog's weight goes from 5 kg to 8 kg and another dog's weight goes from 3 kg to 6 kg, which dog gained more weight?" If the student is approaching the problem from the perspective of absolute values or addition, they may be inclined to answer that the 2 dogs have gained equal weight. However, based on relative values, the student may argue that the2nd dog gained more weight since it doubled its starting weight as opposed to the1stdog, which would have had to reach 10 kg for its relative weight gain to be equivalent. The following table visually illustrates the 2 answers to this problem. If it is true that the 2 answers can be defended, it is on the relative value (multiplicative reasoning) that one must base oneself to apply a proportional reasoning.
| Representation of reasoning based on absolute values. | Representation of reasoning based on relative values. |
|---|---|
The first dog gained 3 kg and so did the second. They gained as much weight as each other.   | The second dog gained more weight as it doubled its weight, while the first one should have reached 10 kg for a relatively equivalent weight gain.  Less than doubled its weight  Doubled its weight |
Why is this important?
It is difficult to move students from additive to multiplicative reasoning, which is why it is important to start at a young age. This is the basis of the Ontario Mathematics Curriculum, Grades1 to8 , which includes important interrelated ideas such as multiplication, division, fractions, decimals, ratios, percentages and linear functions. It takes time, a variety of learning situations, and a variety of opportunities to build knowledge in different ways.
Source : Qu'est-ce que le raisonnement proportionnel?, 2012, Ministère de l’Éducation de l’Ontario, p. 5-6..
Proportional Relationships
There is a proportional relationship between 2 quantities when these quantities can increase or decrease simultaneously by the same factor. For example, if one of the 2 quantities is tripled, the other is also tripled. The ratio between the 2 quantities is then constant (for example, \(\frac{1}{6}\; = \;\frac{3}{{18}}\)). Such equality between 2 ratios is called a proportion.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 39.
Example
Example 2
An aquarium, empty at the beginning, fills at a rate of 2 litres of water per minute. Represent, in several ways, the number of litres of water in the aquarium according to time, in minutes.
Visual Representation (Table of Values)
Graphical Representations
Number of Litres According to Time
In this situation, the line starts at (0,0) and the student can see from the graph that if the time doubles, the number of litres also doubles.
Situations involving a proportional relationship can be solved intuitively with proportional reasoning.
In the junior grades, students are introduced to the terms ratio and proportion and their notation. When 2 ratios are equal, it is a proportion, for example, 2:3 = 10:15 or \(\frac{2}{3}\; = \;\frac{{10}}{{15}}\).
In the primary grades, students intuitively use proportional reasoning to solve problems involving 2 quantities that are in a ratio of 1 to many (for example, 1 cake for 8 children), many to 1 (for example, 3 people per table), or many to many (for example, 2 litres of juice for 5 people).
Source: Guide d'enseignement efficace des mathématiques de la 7e à la 10e année, p. 49-51.
A relationship is non-proportional when 2 variables do not change at the same rate. For example, a deposit of $5 one month and $2 the next is not proportional because the growth is not constant. Its graphical representation would be irregular, not a straight line.
However, some straight lines are considered to be non-proportional relationships.
Example
A plumber charges a customer for a service call and the hours required to repair a problem. She charges $40 for the service call and $30/hour afterwards. The table of values and the graph below represent this situation.
| Number of Hours | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Invoice Amount, $ | 40 | 70 | 100 | 130 | 160 | 190 |
Graph of the invoice amount according to the number of hours worked
In this situation, the line starts at (0, 40) and the student cannot apply proportional reasoning. The student will address this concept in C1.1.
Outside of math class, proportional reasoning is applied in subjects such as science, music and geography, as well as in everyday activities. People use proportional reasoning to make calculations when shopping, filing taxes, or investing, to draw plans and maps, to take measurements or convert foreign currencies, to follow recipes and adapt them to their needs, or to determine different concentrations for mixtures and solutions.
Source : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
Example of a problem in which proportional reasoning is used in the junior division
In the cake recipe, there is 1 cup of sugar and 2 cups of flour. How much flour should Mrs. Larose add to the mixture if she has added 3 cups of sugar? Represent the varying amounts in a diagram that will help students analyze proportional relationships.
This type of representation shows that the quantity of ingredients in the recipe is increased fivefold if \(3\frac{1}{3}\) cups of sugar are used. Transposed into a ratio table, the representation is similar to a familiar representation for the students, namely the ratio table, which has the advantage of helping them to verify the presence of a multiplicative relationship between the quantities.
Knowledge: Proportional Reasoning
The student uses proportional reasoning early in learning mathematics, for example, seeing that 8 equals \(2\; \times \;4\;{\rm{or}}\;{\rm {4}}\; \times \;{\rm{2}}\) and not just "1 more than 7". Proportional reasoning is used at later stages of learning when the student understands that a speed of 50 km/h is the equivalent of a distance of 25 km traveled in 30 minutes.
Students continue to use proportional reasoning when studying the slope of a curve or derivatives. Proportional reasoning is basically viewing numbers in terms of their relative values rather than their absolute values. The student applies proportional reasoning to determine that a group that grows from 3 to 9 children experiences a greater increase than one that grows from 100 to 150 children because in thefirst case the number has tripled, while in thesecond case it has only increased by 50% and has not even doubled.
Activities in the primary and junior grades contribute to the development of proportional reasoning. For example, by comparing the value of a set of 4 nickels to the value of a set of 4 pennies, students develop proportional reasoning.
In the junior and intermediate divisions, students work directly with fractions and their use in the context of ratios, rates, and percentages.
In proportional reasoning, we focus on relationships and compare quantities or values. As Van De Walle said, "Proportional reasoning is difficult to define. It is not a type of reasoning that one is able to do or not do: it is acquired gradually over time. In particular, it can be described as the ability to think about multiplicative relationships between quantities and to compare such relationships, represented symbolically as ratios." (2008, p. 163) There is sometimes a tendency to believe that proportional reasoning is limited to the study of ratios, rates, and rational numbers such as fractions, decimals, and percents when, in fact, it touches all areas of mathematics. For example, proportionality is an important aspect of measurement, including unit conversion and the dimensional multiplication relationships that exist between area and volume.
Source: Guide d'enseignement efficace des mathématiques de la 7e à la 10e année, p. 49-51.
An example of proportional reasoning in measurement
Which figure contains the most colour?
Source : Qu'est-ce que le raisonnement proportionnel?, 2012, Ministère de l’Éducation de l’Ontario, p. 3..