B2.8 Compare proportional situations and determine unknown values in proportional situations, and apply proportional reasoning to solve problems in various contexts.
Skill: Identifying Proportional and Non-Proportional Situations and Applying Proportional Reasoning
The ability to think and reason proportionally is one essential factor in the development of an individual’s ability to understand and apply mathematics. Susan Lamon estimates that over 90% of students who enter high school cannot reason well enough to learn mathematics and science with understanding and are unprepared for real applications in statistics, biology, geography or physics (Lamon, 2005, p. 10). While students may be able to solve a proportion problem with a memorized procedure, this does not mean that they can think proportionally.
Source: Paying Attention to Proportional Reasoning,p. 4.
Multiplicative Reasoning
This concept involves reasoning about several ideas or quantities simultaneously. It requires thinking about situations in relative rather than absolute terms. Consider the following problem. If one dog grows from 5 kilograms to 8 kilograms and another dog grows from 3 kg to 6 kg, which dog grew more? When a student is thinking in absolute terms or additively, they might answer that both dogs grew by the same amount.When a student is thinking in relative terms, they might argue that the second dog grew more since he doubled his previous weight, unlike the first dog who would have needed to be 10 kg to grow by the same relative amount. While both answers are viable, it is the relative (multiplicative thinking) that is necessary for proportional reasoning.
| Here is student thinking that is absolute: | Here is student thinking that is relative: |
|---|---|
| The first dog grew by 3 kg. The second dog grew by 3 kg. They grew the same amount. Dog 1:
Dog 2: 
|
The second dog grew more since he doubled his previous weight, unlike the first dog who
would have needed to be 10 kg to grow by the same relative amount. Dog 1: 
Less than doubled weight Dog 2: 
Doubled weight
|
Why is this important?
Helping students bridge from additive to multiplicative thinking is complex but starts early. It forms the backbone of the mathematics curriculum and includes important and interconnected ideas such as multiplication, division, fractions, decimals, ratios, percentages and linear functions. It requires time, a variety of situations and opportunities to construct their understanding in multiple ways.
Source: Paying Attention to Proportional Reasoning,p. 5-6.
Proportional Relationships
There is a proportional relationship between two quantities when these quantities can increase or decrease simultaneously by the same factor. For example, if one of the two quantities is tripled, the other is also tripled. The ratio between the two quantities is then constant (for example, \(\frac{1}{6}\; = \;\frac{3}{{18}}\)). Such equality between two ratios is called a proportion.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 39.
Example 1
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 Representation
Number of Litres According to Time
Source: Guide d'enseignement efficace des mathématiques de la 7e à la 10e année, p. 51.
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.
From the junior division onwards, students are introduced to the terms ratio and proportion, and the notations associated with them. When two ratios are equal, it is a proportion; for example: 2 : 3 = 10 : 15 or \(\frac{2}{3}\; = \;\frac{{10}}{{15}}\).
In the primary division, students intuitively use proportional reasoning to solve problems involving two quantities that are in a one-to-many ratio (for example, one cake for eight children), many-to-one (for example, three people per table), or many-to-many (for example, two litres of juice for five people).
Source: Guide d'enseignement efficace des mathématiques de la 7e à la 10e année, p. 49.
A relationship is non-proportional when two 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 therefore 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 |
Invoice Amount According to 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 \(\frac{2}{3}\) of a cup of sugar and 2 cups of flour. How much flour should Mrs. Larose add to the mixture if she has incorporated \(3\frac{1}{3}\) cups of sugar? Representing varying quantities using a diagram helps 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 table, the representation is similar to a familiar representation for the students, namely the table of values, which has the advantage of helping them to verify the presence of a multiplicative relationship between the quantities.
Example of a Problem With Variables
Martine works part-time at a water park. Her hourly rate is $15.25/hour. How can she calculate her salary?
Strategy A
She can use an equation with variables.
\(s\; = \;15.25\;\$ \; \times \;h\)
The variable s represents the salary earned and the variable h represents the number of hours worked.
Since for every hour she works she earns $15.25, then to find her salary, simply multiply $15.25 by the number of hours worked.
Strategy B
She can use equivalent ratios.
\(\frac{{15.25\;\$ }}{{1\;{\rm{time}}}}\; = \;\frac{s}{h}\)
This is proportional, so we are comparing two equivalent ratios. The first is the hourly rate $15.25/hour and the second is the salary earned based on the number of hours worked(s/h). The division of the first ratio will equal the division of the second ratio.
- Martine worked 12 hours this week, what is her salary?
To solve the problem with the 2 strategies listed above, I replace the variable h by 12.
Strategy A
\(\begin{align}s\; &= \;15.25\;\$ \; \times \;h\\ &= \;15.25\;\$ \; \times \;12\\ &= \;15.25\;\$ \; \times \;\left( {10\; + \;2} \right)\\ &= \;\left( {15.25\;\$ \ ; \times \;10} \right)\; + \;\left( {15.25\;\$ \; \times \;2} \right)\\ &= \;152.50\;\$ \; + \;30.50\\ &= \;183\;\$ \end{align}\)
Strategy B
Martine earns $183 for 12 hours of work.
Knowledge: Proportional Reasoning
Students use proportional reasoning in early math learning, for example, when they think of 8 as two fours or four twos rather than thinking of it as one more than seven. They use proportional reasoning later in learning when they think of how a speed of 50 km/h is the same as a speed of 25 km/30 min.
Students continue to use proportional reasoning when they think about slopesof lines and rates of change. The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms. Students are using proportional reasoning when they decide that a group of 3 children growing to 9 children is a more significant change than a group of 100 children growing to 150, since the number tripled in the first case; but only grew by 50%, not even doubling, in the second case.
Activities in the primary grades support the development of proportional reasoning. For example, if we ask students to compare the worth of a group of four nickels to the worth of a group of four pennies, we are helping them to develop proportional reasoning.
In the junior and intermediate grades, students work directly with fractional equivalence, ratio, rate and percent.
Proportional reasoning involves thinking about relationships and making comparisons of quantitiesor values. In the words of John Van de Walle, “Proportional reasoning is difficult to define. It is notsomething that you either can or cannot do but is developed over time through reasoning … It is thee ability to think about and compare multiplicative relationships between quantities” (2006, p. 154). Proportional reasoning is sometimes perceived as only being the study of ratios, rates and rational numbers such as fractions, decimals and percents, but it actually permeates all strands of mathematics. For example, proportionality is an important aspect of measurement, including unit conversions and understanding the multiplicative relationships of dimensions in area and volume.
An example of proportional reasoning in area measurement.
Which shape is more purple?
Giving students non-numerical representations which require qualitative reasoning can evoke rich discussions about proportionality. For examples: Small (2008), Van de Walle (2005) and Continuum & Connections: Big Ideas and Questioning: Proportional Reasoning K–12.
Source: Paying Attention to Proportional Reasoning,p. 3.
Knowledge: Ratios
Relationship between two quantities expressed as the quotient of the numbers that characterize them.
For example, in the set of 5 marbles above,
- there is a ratio of 2 to 3 ( \(\frac{2}{3}\) or 2:3) between the number of white marbles and the number of black marbles. (part: part ratio)
- there is a ratio of 2 to 5 (\(\frac{2}{5}\) or 2:5) between the number of white marbles and the total number of marbles. This can be interpreted as \(\frac{2}{5}\) marbles are white. (part:whole ratio)
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 49.
Knowledge: Equivalent Ratios
When 2 ratios can be increased or decreased simultaneously by the same factor.
Example
The ratio 3 : 5 or is equivalent to the ratio 6 : 10, obtained by multiplying the terms of the first ratio by the same factor of 2.
Note: Equivalent ratios are determined in the same way as equivalent fractions, that is, by multiplying or dividing each term (or part) of the ratio by the same factor. If the value of a term (or part) of a ratio is changed, there will be a direct effect on the ratio.
Source: En avant, les maths! Grade5 , CM, Nombres, p. 2.
Knowledge: Scale Factor
2 quantities are proportional if the value of one is obtained by multiplying or dividing the value of the other by the same number (scale factor).
Source: En avant, les maths! Grade5 , CM, Nombres, p. 2.
Knowledge: Ratio Table
Model that can be used to develop an understanding of multiplication, equivalent fractions, division, and proportional reasoning.
| Flour bags | 1 | 3 | 4 | 6 | ? |
| Water | 3 | 9 | ? | ? | 6 |
Source: En avant, les maths! Grade5 , CM, Nombres, p. 2.
Knowledge: Rate
A rate describes the relationship between 2 quantities expressed with different units (for example, objects with dollars or kilometres with hours).
Source: En avant, les maths! Grade5 , CM, Nombres, p. 2.
Knowledge: Percent
A percent is a particular way of presenting a fraction. It is often used in everyday life. A numeric expression such as 30% (which reads "thirty percent") is actually another notation for the number thirty hundredths, that is \(\frac{30}{100}\) or 0.30. To facilitate understanding of the concept of percent, students must first be introduced to the relationship between the percent and the fraction with a denominator of 100, using concrete or semi-concrete materials. The percentage also represents a ratio of 1:100.
Example
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 34