Proportional Reasoning

The Dynamics of Proportional Reasoning – Uncovering Crucial Connections!

Proportional reasoning is an incredibly valuable math concept that is one of the four pillars of mathematics. It requires students to grasp relative size and ratio and proportional relationships between sets of numbers or objects. It also enables children to understand and make proportional judgments in various situations, which are integral to success in mathematics. 

When proportional reasoning is understood correctly, children can quickly break down complex mathematical problems into simpler parts, helping them form a strong foundation within mathematics and build a deeper comprehension essential for future math challenges. 

However, if a child is struggling in math, they may be missing the basic foundation, including proportional reasoning. As one of the leading math programs in Toronto, Dropkick Math Academy works with students to help repair any misunderstandings or gaps in learning. 

Does Your Child Enjoy Learning About Proportional Reasoning?

If your child enjoys spending time rationalizing and solving puzzles, chances are they enjoy learning about proportional reasoning. As one of the four pillars of math, proportional reasoning offers children the perfect combination of challenges, whether they want to test their problem-solving skills or just strengthen those mental muscles. 

With proportional reasoning paving the way toward understanding fundamental mathematical concepts, it’s no wonder why children must get a firm grasp on the subject. Children can power up their equation-solving abilities by fully understanding how deep relationships work in proportional reasoning.  

The Relationships

Relationships are a key component in mathematics, especially in proportional reasoning, as it relies heavily on comparing quantities and values. These relationships are evident through the study of ratios, rates and rational numbers. 

An example of how relationships work in proportional reasoning is shown when comparing values in a ratio. When children start to understand relationships, they will better understand the concept of the mathematical equation and how to solve it. 

Developing Proportional Reasoning

Students developing proportional reasoning can generally distinguish between additive and multiplicative relationships. They are able to solve problems involving proportional reasoning and can compare ratios. Students should learn proportional reasoning over several years, beginning in younger grades alongside the development of their rational number sense. 

For children who may be struggling with proportional reasoning, Dropkick Math Academy can help! Proportional reasoning is a fundamental component of mathematics and an invaluable asset for students of all ages. This skill can help students tackle any subject that requires arithmetic or mathematical understanding

Students of all academic levels must be equipped with proportional reasoning competencies to successfully get through school and have basic skills beyond their academic years. Employers in various fields may also appreciate workers who possess proportional reasoning abilities, so mastering this important concept should never be overlooked or belittled.

The Four Pillars Of Math

As experienced Ontario teachers, we have seen a lack of confidence and understanding from our students in mathematics over the years. Upon further review, we recognized four key areas that children struggled with when mastering math – Number Sense, Operational Sense, Proportional Reasoning and Algebraic Reasoning. 

We now understand that if students do not possess a strong base in these four pillars of math, they risk struggling to keep up in the classroom setting. Our goal is to improve our students’ foundational skills by incorporating the basic fundamentals in each of our programs to enrich math education for all.

Our Programs

Seeking relationships between mathematical concepts is a key component of learning, particularly when it comes to children. Understanding relationships between the different elements of mathematics can help children learn more efficiently and set them up for greater success beyond basic arithmetic. However, forcing children to uncover relationships before they understand the basics can be detrimental as it can disconnect them from math, causing them to shut down and be unwilling to engage in learning at all. 

Ultimately, introducing relationships into a child’s mathematics learning requires careful and thoughtful implementation; too little and their instruction could suffer, but too much will cause them to become disinterested. This is why we leave the stuffy textbooks behind and make learning fun

At Dropkick Math, our small-class math tutoring approach allows us to tailor lessons to the student’s needs and learning styles. So, whether your child is struggling with algebra or decimals and fractions, our team is here to provide the math learning techniques needed to support and guide your child every step of the way. Trust us for expert math help from caring professionals.

If your child is struggling with mathematics, it is essential to understand where they need extra help. An online math tutor like Dropkick Math Academy can help your child catch up and boost their confidence which will last for years to come. Get started with our program today by filling out our Free Early Indicators Check-In. 

Math Tutoring Proportional Reasoning

How Proportional Reasoning Can Simplify Division Problems

If your child struggles with division problems, don’t worry – they are not alone. But there is a solution: proportional reasoning. Proportional reasoning can simplify division problems and help better understand math concepts. In this blog post, we’ll discuss what proportional reasoning is, how it can help with division problems, and give some examples. We’ll also explain how we use it regularly in our math help services. So read on to learn more!

What Is Proportional Reasoning?

Proportional reasoning is perhaps best described as seeing relationships between numbers and understanding how those relationships can be applied in different situations. In many ways, proportional reasoning is the foundation of mathematics, providing a way to see the world in terms of numbers and to understand proportions and how those numbers work together. By understanding proportional reasoning, we can begin to see mathematics as a tool for solving problems and making predictions rather than simply a set of rules to be memorized. As we develop our proportional reasoning skills, we open up a world of possibilities for mathematical understanding.

Why Is Division So Challenging?

Division is one of the most fundamental operations in mathematics, and it can be one of the most challenging for children to learn. There are a number of reasons why division can be difficult for kids. First, division often requires children to think in terms of fractions and decimals, which can be confusing. Second, division problems can often be lengthy and complicated, making them difficult to solve. Finally, division often requires a high level of abstraction, which can be difficult for young minds to grasp.

Division can also be difficult for children as it is not always taught in a concrete way. For example, many children are first introduced to division by being asked to divide objects into groups. However, this can be difficult to visualize, so children may have difficulty understanding the concept. If children have difficulty with math in general, division (and other math operations), it may be even more challenging, and they may need a math tutor

However, most children can learn to divide effectively with patience and practice. By understanding division’s challenges, parents and teachers can help kids overcome these difficulties and develop a strong foundation in mathematics.

How To Use Proportional Reasoning In Division

As adults, we can talk about how relationships are a key component in mathematics, and proportional reasoning relies heavily on comparisons of quantities and values. However, when helping children with math, giving the formal definition of proportional reasoning may cause their eyes to roll to the back of their heads. So, always remember to try and leave the stuffy mathematical definitions to the textbooks. Here at Dropkick Academy, we think learning should be fun. So, let’s take a look at how to help children learn division through proportional reasoning in an entertaining way. 

In grade 7, students learn that a fraction can represent division. Just like we can simplify a fraction, we can simplify a division problem by using proportional reasoning to look at common factors. 

When the dividend and divisor have a common factor, the problem can be simplified. Let’s look at a division problem and see how proportional reasoning can help simplify the process. 

Dividend and Divisor

12 ÷ 4 = ?

If a child is given the question 12 ÷ 4 it may give them anxiety because it is a larger number. However, this can be simplified. Using math manipulatives, show them how to divide 12 pieces into 4 groups of 3 to get the answer.  

Next, you will want to show them how to simplify the problem by finding a common factor between the dividend and the divisor. In this case, the number 2 is a common factor. So, divide the problem by two, changing it to 6 ÷ 2. Is the answer still 3? Yes! Because both amounts were reduced at the same rate, it didn’t change the outcome. It just simplified the problem. 

Now, go one step further and show them how 2 is also a common factor of 6 ÷ 2. So, the problem can now become 3 ÷ 1. The answer is once again 3!

When children face a challenging division problem with larger numbers, they can now check to see if it can be simplified by thinking about proportional reasoning

Another way to look at this is by thinking how proportional reasoning is the ability to understand that two quantities are in proportion if they change at the same rate. For example, if we know that there are 2 apples for every 3 oranges, then we can also say that there are 6 apples for every 9 oranges. This is because both ratios are equivalent (2:3 = 6:9).

Math Manipulatives

By using math manipulatives to help children visualize each group’s proportions, they can better understand the concept at hand. Math manipulatives can also make learning more fun by creating a tactile way of learning. 

Manipulatives allow children to feel, touch and visualize what they can’t yet create on their own. They can enable children to receive immediate feedback about whether their idea makes sense. Using tools, a child can move hands-on objects to investigate and explore a math concept that may be challenging.

Understanding the fundamentals behind the mathematical foundation is critical for a child’s fluency and math development. Using both manipulatives and representations, children can build a deeper understanding of the four pillars of math (number sense, operational sense, algebraic reasoning, and proportional reasoning).

At Dropkick Math Academy, we believe using math manipulatives is an excellent way of learning. In fact, we even include them in some of our battle kits for our students to use in our programs!

Proportional Reasoning At Dropkick Math

As one of the leading math services in Ontario, our certified teachers support learning by focusing on the four pillars to give students an understanding of the root of mathematics. Our one-on-one math tutoring approach allows us to tailor each lesson to the student’s specific needs and learning style. 

So, whether your child is struggling with algebra or decimals and fractions, our team is here to provide personalized support and guidance every step of the way. Trust us for expert math help from caring professionals.

Learn more about our math tutoring today by visiting our website.

Proportional Reasoning

How Proportional Reasoning Is Used In Everyday Life

As a parent, you might wonder what mathematical concepts your child is learning in school and how they will be used in everyday life. Today, we’re going to take a look at proportional reasoning and how it’s used daily.

Proportional reasoning is the ability to understand that two quantities are in proportion if they change at the same rate. For example, if we know that there are 2 apples for every 3 oranges, then we can also say that there are 6 apples for every 9 oranges. This is because both ratios are equivalent (2:3 = 6:9). This math concept is just one of the four pillars (number sense, operational sense, proportional reasoning, algebraic reasoning) that we focus on at Dropkick Math tutoring company. 

Understanding The World Around Us

Too often, children see math as a dull and difficult subject that has no relevance to their lives. However, they are more inclined to want to learn when they understand why math is important in everyday life. After all, math is more than just a set of arbitrary rules. It helps us to understand the world around us and make sense of the patterns we see. 

In fact, math is a vital tool in many different fields, from medicine and engineering to finance and architecture. By showing children how math is used in the real world, we can help them to see its value and appreciate its importance. Only then will they be motivated to learn math and develop the skills they need for success in school and beyond.

What Are Some Examples Of Proportional Reasoning?

When we speak about proportional reasoning, it is essential to help children understand how it can be used in everyday life. If your child is struggling with math concepts, let them know that they have probably already been using proportional reasoning without even knowing it! If they hear that they already know how to do something, it can help boost their confidence and willingness to learn more. 

Some examples of everyday proportional reasoning include:

-When we’re trying to figure out how much shampoo to use, we usually start with a small amount and then add more if we need it. This is because we know that the amount of shampoo is directly proportional to the amount of hair we have.

-When we’re cooking, we often need to double or halve recipes depending on how many people we’re feeding. This requires us to understand that the ingredients are proportional to the number of people we’re serving.

-When we’re driving, we need to constantly adjust our speed based on the flow of traffic. We know that if the traffic is moving faster, we need to speed up; if it’s moving slower, we need to slow down. This is because our speed is directly proportional to the speed of traffic around us.

Why Is Proportional Reasoning Important?

As you can see, proportional reasoning is a useful skill that we use all the time without even realizing it! So next time your child does something where they need to make a quick calculation, let them know that they are using their proportional reasoning skills. 

Proportional reasoning is a skill that allows us to interpret graphs and charts accurately, make comparisons between quantities, and solve problems efficiently. In fact, many algebraic equations can even be solved using proportional reasoning skills. Without these abilities, it would be nearly impossible to navigate daily life tasks effectively. Proportional reasoning is a vital skill that should be developed and honed throughout our education and career.

Building Upon Math Concepts

Mathematics is often considered a series of isolated concepts, each to be memorized and applied to specific problems. However, this view overlooks the fact that mathematics constantly evolves and builds upon itself. One concept can lead to the discovery or development of another, leading to a deeper understanding of the underlying principles at play. 

For example, students who are developing proportional reasoning skills can also improve overall mathematical thinking and problem-solving abilities. Building upon previous concepts is central to the very nature of mathematics and helps explain its power as a problem-solving tool. Without this foundation, Mathematics would not be nearly as effective or versatile. So next time your child struggles with a complex math problem, remember that it may just be laying the groundwork for an even greater discovery in the future. Encourage them to work through the problems using proportional reasoning and demonstrate the equation in a real-life situation to make it more relatable to them. 

About The Four Pillars Of Math

At Dropkick Math Academy, we focus our curriculum on the four pillars of math. As one of the main pillars, we believe that the ability to reason and think proportionally is critical in developing an individual’s ability to understand mathematics. While students may have memorized how to solve a proportion problem, this does not mean they can think proportionally.

As one of the essential understandings of mathematics, all students must acquire good proportional reasoning skills to take them through school and later in life. If your child is struggling with mathematics, it is essential to understand where they need extra help. Our certified Ontario teachers can help your child with any gaps in education and help them catch up with their learning. 

Finding A Reputable Tutoring Company

When it comes to online math help, it can be difficult to find reliable and professional help. Many websites offer generic, automated solutions that can’t address individual student needs. But here at Dropkick Math, we have a team of dedicated math teachers who genuinely care about the success of our students. Our one-on-one approach allows us to tailor each lesson to the student’s specific needs and learning style. In addition, our teachers are all highly trained and passionate about math education. So, whether your child is struggling with algebra or decimals and fractions, our team is here to provide personalized support and guidance every step of the way. Trust us for expert math help from caring professionals.

Learn more about our math tutoring today by visiting our website.

Pillars of Math

What Are The Four Pillars Of Math?

The Four Pillars of Math are four concepts that are essential for students to understand in order to be successful in math. These pillars are: number sense, operational sense, proportional reasoning, and algebraic reasoning. By mastering these four fundamental concepts,  students will be able to solve any problem they encounter in math. In addition, the four pillars of math provide a strong foundation for students to build upon as they move on to more advanced concepts.

The Foundation Of Our Program

At Dropkick Math Academy, the foundation of our math tutoring program focuses on the four pillars of math. These four pillars are essential building blocks for understanding higher-level math concepts. We focus on one or more of these pillars throughout our program in each module. For example, in our first module, Operations, Whole Numbers, and Pirates, we focus heavily on developing a conceptual understanding of both number and operational sense. Without a strong understanding of these operations, it would be impossible to move on to more complex topics like algebra. However, we also touch on the other three pillars in this module. For instance, we use proportional reasoning to explore place value. By the end of the module, students have a well-rounded foundation in arithmetic and are prepared to move on to more advanced modules.

The Four Pillars Of Math

Let’s take a closer look at each of the four pillars:

Number Sense

Number sense is understanding the quantity represented by a number. It is not the same as numeracy, which is the ability to do calculations. Number sense develops gradually. For example, a baby starts by recognizing that they see one person or two people. By the time they start school, they can count up to 20 or more. They learn what numbers mean in terms of amounts and quantities. This knowledge forms the foundation for learning more complex math concepts later on. 

Fractions are an excellent example of a concept that builds on number sense. In order to understand fractions, students need to be able to visualize the quantity represented by a number. For example, if there are four pieces of candy and I eat two of them, then I have eaten half of the candy. Fractions are a way of representing numbers that are not whole numbers, and understanding fractions requires a strong foundation in number sense. 

Operational Sense

Operational sense is a critical math concept that refers to a child’s ability to make sense of addition, subtraction, multiplication, and division and use them in problem-solving situations. When a child is able to develop a strong understanding of the operations, they can recognize the relationships among them and develop systems for computing numbers. This understanding helps to form the basis for the four pillars of math.

Operational sense is not just about being able to do the computations but about understanding when and how to use them. For example, a child who knows that addition can be used to find out how many objects are in a group but does not understand that it can also be used to put together two groups of objects has not yet developed operational sense. Operational sense is a critical foundation for success in math, and children who develop a strong understanding of the operations are well on their way to becoming math problem-solvers.

Proportional Reasoning

Proportional reasoning is a unifying theme in mathematics and is often considered the foundation of abstract mathematical understanding. All four pillars connect to proportional reasoning. For example, in whole numbers, students learn about place value which is based on 10s. In decimals, place value is based on 10s with an added number after the decimal point. In fractions, there are different parts of a whole, and students use visual representations to help understand the relationships between those parts. Lastly, in percentages, there are hundredths and relationships are represented out of 100. 

By making connections to prior learning, students see that proportional reasoning is everywhere in mathematics! Proportional reasoning is not just a process but rather it’s a way of looking at the world and solving problems. In essence, proportional reasoning is the consideration of numbers in relative terms compared to absolute terms.

Algebraic Reasoning

Patterns are all around us. They can be found in the seasons, the way leaves grow on a tree, and the numbers we use every day. Understanding patterns is an essential mathematical skill that helps us to make sense of the world around us. 

Algebraic reasoning is a way of thinking that uses mathematics to solve problems and understand relationships. It is an essential skill that is introduced in the early years of learning and continues to play an increasingly important role through grades 4 to 9.

Algebraic reasoning can be used to solve problems in a variety of ways, including by using algebraic equations, graphing, and creating tables and charts. It is a powerful tool that can help us to understand complex ideas and make predictions about the future. By understanding algebraic reasoning, we can better understand the world around us and our place within it.

Learning Skills

The approach used in Dropkick Math Academy teaches children certain skills along with the four pillars to set them up for success in their future education. These include:

Concepts – Concepts are the building blocks of math. If students don’t understand the basic concepts, they’ll struggle to progress. That’s why we focus on teaching concepts in a way that is clear and easy to understand. We want students to really grasp the material so they can build on it later.

Fluency – Fluency is about being able to do math accurately. It’s important for students to be able to recall math facts and equations quickly so they can focus on solving problems. We use a variety of techniques to help students improve their fluency, including games and practice with real-world applications.

Problem-solving – Problem-solving is an essential skill for all students, not just those interested in math or science. Breaking down a problem and finding a solution is a valuable life skill that will serve students well no matter what path they choose. We encourage students to approach problem-solving with confidence and to persevere when they find themselves stuck.

Mastering The Four Pillars

The Dropkick Math team is committed to helping students master the Four Pillars of Math. We believe that this is the best way to set students up for success in their math education. We have created a variety of resources, such as articles, videos, and games, that all focus on helping students understand and master these four concepts and better understand math operations. Visit our website today to start exploring these resources and learn more about our math help services. By taking our FREE assessment, your child can begin to build a strong foundation in math!

Proportional Reasoning

Is Your Child A Proportional Thinker?

Is Your Child A Proportional Thinker?


Is Your Child A Proportional ThinkerWhen your child comes to you for math help, you may think they are speaking an entirely different language. The way that math operations are now taught have most likely changed since you were in school and appear completely different. Not only is the representation different, but the way your child speaks about math may also be different. 

We recognize these barriers between parent and child when discussing mathematics, which is why at Dropkick Math, we believe it is so important to educate the child and the parent. When you get involved and upgrade your education on the foundation of math, you can better help your child in their studies. 

The Four Pillars

We focus our programs on the four pillars of math (number sense, operational sense, algebraic reasoning, proportional reasoning). It is vital that every student fully understands each of these four pillars as the foundational concepts because one weak pillar can significantly impact future success. 

Proportional reasoning is often considered the foundation to abstract mathematical understanding as it is a unifying theme throughout mathematics. However, it is estimated that over half of the adult population are not proportional thinkers. So how can you tell if your child is a proportional thinker?

If your child can generally distinguish between additive relationships, (e.g., 8 is 6 more than 2) and multiplicative relationships, (e.g., 8 is 4 times as much as 2) they may be a proportional thinker. If they are able to compare ratios, they also may be a proportional thinker. Overall if your child can describe relationships between two or more things and see groups of items in various ways, they may be a proportional thinker.

Proportional Thinking Education

It is vital for all children to acquire good proportional reasoning skills to take them through school and later in life. The importance of proportional reasoning for students is often focused in junior and intermediate grades. Still, many educators believe it is vital to grasp these skills in kindergarten and primary grades when students begin grouping, unitizing, (e.g., one dime is a unit of 10 pennies) and sharing.

Students begin to use proportional reasoning early in math education. They often start by thinking of groups of numbers rather than whole numbers. For example, they may think of 10 as two groups of 5. Later on in education, proportional reasoning is used slightly differently. An example of this may be thinking how a speed of 50/km an hour is the same speed as 25/km per 30 minutes. Continuing in their education, students will use proportional reasoning when learning about percents, rates of change, and slopes of lines.

How Is Proportional Reasoning Used?

But proportional reasoning isn’t just used for mathematics. Many may be surprised to know that proportional reasoning can be helpful in other subject areas such as music, geography, and science. On a daily basis, people also use proportional reasoning to calculate taxes, find the best deals at a store, or adjust recipes.

From these examples, it is easy to see how thinking proportionally is critical in developing a child’s understanding of mathematics. But young children don’t think proportionately naturally. Proportional thinking must be developed as a foundational skill of our most basic math concepts, which is why many parents explore math tutoring for their children. 

As one of the four main pillars of mathematics, our programs help students learn proportional reasoning and fill in any underlying misconceptions they may have. Our math help services are built to suit each child, offering differentiated approaches, making it accessible for all learning needs.  

Learn more about our programs and how we can help ensure your child becomes a proportional thinker in mathematics and throughout life!