At Mentorhood, we employ the Singapore Math method in order to emphasize a gradual progression from concrete to pictorial to abstract teaching models. This is rewarded with a strong, stair-stepped understanding in learners. As we’ve seen, bar models are a fantastic visual tool that helps to bring students along this journey with arithmetic. Other math topics can also benefit from a concrete and visual approach first, and as such, we use this step-down approach to teach fractions, as well.
Fractions are often taught very visually and with concrete examples, especially at first. We aim to preserve the ease that visual representation provides to the understanding of fractions for as long as possible, until students are comfortable working with the symbolic, “numerator over denominator” representations of fractions that most of us are familiar with.
Easy as Pie
It’s no secret that fractions are often first taught with the help of concrete thought experiments, such as dividing up a pie or a pizza, and Mentorhood’s methods are no exception! Visualizing a “whole” object being divided into “parts” with different assignments (eaten and not eaten, for instance) is a fantastic way to impart fractional understanding. We simultaneously introduce the written form of fractions and their various pronunciations so that students develop fluidity between the visuals and their symbols.
Rudimentary lessons rely heavily on visuals, and this gives students the chance to learn basic concepts such as ordering fractions by size in tandem with knowing how to express these relationships symbolically. They can visually see that fractions with the same denominators (“like fractions”) can be compared based on their numerators. They can also make the leap to understanding that between fractions with the same numerator, the fraction with the smaller denominator is superior in size. This is a lot easier to grasp visually or with a thought experiment than just through the use of symbols: any child who knows what’s good for him or her would prefer one third of a pizza to one one-millionth.
Before the student knows it, they have mastered simple comparative relationships and can answer basic questions about the properties of fractions without the use of a visual aid at all. They begin to treat the obelus (the division symbol) with as much comfort and familiarity as a fruit-flavoured circle being split into several equal slices.
Visuals for Higher Concepts
Once the abstract tools are given and students understand the purpose of the numerator and denominator, many textbooks abandon the visual slicing and sectioning all together. There is still more room, however, for visualization to play a role as new learners of fractions advance to higher, arithmetic concepts.
When it comes to adding and subtracting fractions, the most foundational skill is the ability to convert a fraction into an equivalent fraction. (This is often necessary in order to produce two fractions that have the same denominator.) While the algebraic rules of converting fractions are simple enough to give students as a formula, it is much more effective to show them exactly what they are doing visually.
We prefer to showcase a fraction as a visual part of a whole and demonstrate that converting it into an equivalent fraction is simply a matter of dividing or combining slices, without changing the amount you had to begin with. We also like to show the difficulty of combining fractions of different denominators. While it is possible to do this visually, the trouble comes when you don’t know how to numerically express the answer you arrive at. We therefore need to make use of equivalent fractions so that we have a way to communicate our calculated result.
Having now understood how to get two fractions to share a denominator, the actual addition and subtraction becomes as easy as their first fractional lessons.
A solid understanding of finding equivalent fractions is mutually reinforcing to concepts such as converting between improper and mixed fractions or converting fractions into their simplest forms. The more a student can feel the visual implications behind these number-line-number combinations, the easier they can manipulate them into all kinds of formats.
Multiplication and division of fractions can be trickier to represent visually when you are working with two fractions, but a visualizable intermediate step can be taken in the form of multiplying and dividing by whole numbers. Multiplication becomes repeated addition with already-like fractions. Division turns into an equivalency calculation, determining what the equivalent fraction would be if the proverbial pie were divided into several more equal slices.
By the time the students have moved on to working purely with two fractions for more advanced arithmetic, they have blended confidence and a well-rounded understanding with the inherent rules that fractions abide by. It is then not so far a leap to teach the more abstract functions of fractions, such as getting fractions of fractions through multiplication, or reverse engineering totals from parts.
The goal is not to sever the link between fraction arithmetic and a cheesy New York pie, but to seamlessly join the two so that they are practically interchangeable in the students’ minds. The more they are able to see in their minds’ eyes what they are doing to a fraction, either consciously or subconsciously, the more they will be able to think outside the box to solve ambiguous word problems or ask their own questions. A deep, three-dimensional understanding is crucial to being able to turn numbers over in their minds to find properties that may have seemed hidden at first.
Three-dimensional understanding is rooted in concreteness as solid as the three-dimensional world around us. The more we can coach our students into creating that world in their minds, governed by the language of math, the more durable their understanding will be, and the more they will set themselves up on a foundation of success as they continue in their future studies.
Welcome to the team, parents—the Singapore Math team! We’re excited to have you on board and we look forward to helping you coach your little ones.
We’re happy to come alongside your learner in their learning journey. And if you want to do more from home, you can check out our ever-growing library of games, now on YouTube!
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