## Table of contents

The *rule of three* is a mathematical concept that we tackle in different elementary school grades. The rule of three algorithm has become a very recurrent tool for solving problems related to proportionality. This being said, however, is it really necessary to know the algorithm to understand proportionality? Could we present the rule of three from different points of view? In this article, we show you how learning that may seem mechanical can in fact be competency-based.

## What Is the Rule of Three?

In our daily lives, we often encounter **situations that we can analyze mathematically thanks to proportionality**: buying different amounts of the same product in bulk; adjusting a recipe for a different number of guests; sharing out a task in a group to later find that the number of members has changed; working out discounts in a sale; converting units of measurement, etc.

These are situations that are traditionally solved with the rule of three. Below are some examples solved using this procedure.

*4 bottles of water cost €2.30. How much will 10 bottles cost?*

*6 operators do a job in 10 hours. How long will it take 15 operators to do the same job?*

*A shop on the High Street is having a sale. On the door, there is a sign that reads: “20% off on pants”. What is the new price of a pair of pants that used to cost €45?*

*In the US, height is measured in feet (abbreviated as “ft”). We know that 3 m is approximately 10 ft. How tall, in meters, is a 6.5 ft tree?*

This type of algorithmic solving requires **memorization skills**. **The rule of three is an algorithm that allows us to figure out the hidden piece of information by arranging the known facts in a specific way to perform arithmetic operations in a particular order**. As in any closed algorithm, neither the position occupied by the numbers, nor the operations, nor the order in which they are performed can be altered, since we’d end up with the wrong answer. We know that prematurely introducing algorithms to our students can lead to a certain degree of passivity in their thinking processes. They end up not being able to reason, getting stuck when the steps of the algorithm fail. So, if we introduce the algorithm too early, we cause them to struggle to develop tools to persevere when faced with a challenge.

Picture the following example:

*If 10 musicians can play Beethoven’s Ninth Symphony in 65 minutes, how long would it take 5 musicians?*

Students who are used to applying the rule of three without thinking may fall into the trap and make a mistake. With traditional algorithms, since they are not very transparent, it isn’t easy to make connections with mathematical content that helps us to understand the reason for the operations performed.

Next, we’ll be taking a look at different situations involving both direct and inverse proportionality, as well as others that have to do with calculating percentages or changing units. Traditionally, percentages and unit changes are solved using their own specific algorithms, but, actually, they share the idea of proportionality. As a result, the rule of three can provide a more or less efficient answer to the situation. Let’s take a look at the connections!

## Rule of Three vs. Proportionality Tables

At Innovamat, we strongly believe that algorithms should only be introduced when we’re sure that the student, while they might forget the steps, will have enough resources to solve the situation. In our lesson plans, we build on algorithms gradually instead of simply providing them.

We don’t believe that the rule of three should disappear from the math classroom, but we are convinced that **learning should be competency-based**; in other words, it should arise from understanding. Instead of saying “I’m going to solve this situation with the rule of three”, we should say “**I’m going to solve this situation using the concept of proportionality**”.

Using an algorithm for the sole purpose of gaining efficiency is insufficient, and it can’t be the only reason for giving the students a resource. Such practices come with the risk of projecting a distorted view of what it means to do math. Mathematical activity is not about doing long calculations or just following algorithms; rather, it involves thinking and looking for patterns and relationships between pieces of information.

On the other hand, the main difficulty students run into with the rule of three is that they’re unable to remember the order of operations or the position of the numbers on paper. When students are introduced to the rule of three without the right guidance and other more transparent algorithms, they are often unaware of what they’re doing. As a result, the rule of three ends up being a meaningless strategy and a purely mechanical exercise.

Not only this, introducing it too early in elementary school makes it harder to work on the multiplicative relationship between numbers and also obscures certain non-additive relationships between them. At the same time, in secondary school it can prevent us from going deeper into the ideas of *ratio and proportion*.

As an alternative to the rule of three, **we believe in solving these situations based on a deep understanding of proportionality** through logical reasoning, reducing to the unit, and mental math. As for proportionality tables, these are an effective resource to build on with students.

Next, we’ll be going back to the previous examples and focusing on the advantages of using proportionality tables to solve them.

*4 bottles of water cost €2.30. How much will 10 bottles cost?*

A reasoning method that students can easily apply is reducing the independent value to the unit. By bridging through the unit, they realize that they can find the price of any number of bottles of this type. Here, we can work on the concept of **proportionality** ratios. For each bottle, we will pay $0.575 (the result of dividing €2.30 by 4).

On the other hand, if we observe that 10 bottles are “two and a half times” the pack of 4, the same goes for the price we pay.

*6 operators do a job in 10 hours. How long will it take 15 operators to do the same job?*

This way of solving problems helps us to avoid making the common mistake of not knowing what the multiplication and division operations in the inverse rule of three are and in what order they are performed.

In this situation, students naturally understand and apply the fact that, the more operators working, the less time they will need to finish the job.

*A shop on the High Street is having a sale. On the door, there is a sign that reads: “20% off on pants”. What is the new price of a pair of pants that used to cost €45?*

These types of tables help us to grasp the nature of percentages understood as a fraction with 100 as the denominator (20% is the same as ⅕ of a unit).

In this case, we also place great value on the student seeing that a 20% discount is the same as paying 80% of the price. In turn, this saves us from having to do the subtraction at the end and instead get the answer directly.

*In the US, height is measured in feet (abbreviated as “ft”). We know that 3 m is approximately 10 ft. How tall, in meters, is a 6.5 ft tree?*

Here, students don’t need to worry about which piece of information they need to know. They know the equivalence (10 feet is the same as 3 meters), so they can measure in feet and give the answer in meters, like in the formulation. They could also measure in meters and give the answer in feet, just like we suggest below.

## Let’s Reflect on Math Learning

To conclude, we’d like to highlight the idea that the rule of three can end up turning into a crutch of sorts that gives students a false sense of security, since it doesn’t promote the development of mathematical competencies. By relying exclusively on the mechanical application of the formula, their conceptual understanding and ability to solve problems autonomously is limited. There is a need to rethink how we teach the rule of three and look for more holistic approaches that promote understanding, reasoning, and autonomy in math. This is why talking about understanding proportionality could be useful to avoid promoting a distorted view of math and instead see it for what it really is: a way of experiencing, perceiving, and understanding the world around us.