Teaching fractions are very important, yet so poorly understood even by many adults. If many adults don’t understand them, that means our students are in poorer shape. Fractions have been given a bad rap and are expected to be confusing and hard. For me, I love fractions and honestly, could do them all day!
The disconnect actually has started among us, the teachers. I know that is not what you wanted to hear immediately upon coming to my website, but it is true and if you continue on you’ll see why. After extensive research into the matter for my math workshop units, I discovered many best practices that even I was “breaking” that added to student confusion.
Unfortunately, students are often applying whole number practices to fractions and few teachers are correcting this (more on this in a minute). Further, most teachers are teaching more about the procedures rather than the actual meaning of fractions, or “fraction sense.” To help “correct” these and break these areas of confusion, I bring to you the best practices or the 10 ways for students to master fractions.
The 10 Ways for Students to Master Fractions
1.) Watch the language we use in the classroom. Sometimes the language we use in the classroom can really confuse students. For instance, if we say an improper fraction (though I know some districts require it), it can be confusing and feel like if they write a fraction this way, it’s wrong. Instead maybe call it a fraction “greater than one,” which can also be used when you use the term mixed number. (I know it seems crazy, but it’s true!) Another example is when we use the term “reducing fractions.” To students this sounds like the fraction is getting smaller when in reality, we are not making smaller fractions – remember, they are equivalent with larger-sized pieces. Do you see how quickly a student can start thinking that 2/5 is smaller than 4/10 because we said we reduced it? It doesn’t always seem confusing to us because we know the material well.
2.) Help students see that the numerator and the denominator of a fraction are one single value– a single number. Often students view them as two separate values because we refer to them as “the top number” and “the bottom number.” (see, there’s that language again). Sometimes we will also call it, “three out of four” or “three over four.” Instead, we should call it the number it is, three-fourths. If you want to emphasize the relationship to division, then remind students to see the denominator as the divisor and the numerator as the multiplier. This means, 3 times what you get when you divide a whole into 4 parts, or 3 ÷ 4. Additionally, you can help students see that fractions are numbers, by continuously using a number line.
3.) Students must understand that the parts have to be equal sized parts. I have met many students over the years that believe that 2/3 means any 2 parts, not equal sized parts. In this example below, if you were to ask your students how much is shaded, would they say 3/4 or 1/2?
What furthers this misunderstanding is sometimes our student drawn area models. Since students in elementary have a lack of precision, they can easily create unequal partitions and believe that the parts don’t need to be equal. One way to combat this to provide outline models, but another is to show counterexamples of inaccurately drawn models. The continuous reminder that parts need to be equal is important too.
4.) We need to build “fraction sense.” This means we need to make sure that we are emphasizing the meaning of fractions a lot more. In fact, it is strongly recommended that teachers wait to teach the “algorithms” or “procedures” of any fractions until students have fully explored the concrete methods of any fraction concepts. For instance, often students are rushed to the cross-multiplication method when it comes to comparing fractions, the multiplication for equivalent fractions or improper/mixed numbers, or the algorithm for multiplying and dividing without even understanding why. In my post, Teaching Math so Students Get It, I explain how students learn math best.
5.) Help students really understand the size of fractions. Often students get confused because they think in whole number terms. With whole numbers, 5 is smaller than 10. With fractions, 1/5 is actually much larger than 1/10. This is confusing to children–unless they have LOTS of practice looking at a TON of visuals. They need to practice until they can instantly tell you that 1/10 is smaller because it has more pieces. Just telling a child that the bigger the denominator the smaller the number isn’t going to help at all. It especially won’t help when they get to problems like 7/10 vs 1/5. Have students practice this and look at it over and over until they can visualize it! It’s SOOO critical!
6.) Use a variety of fraction models and connect those models to real-world contexts. The repeated use of these physical tools can lead to the use of mental models and understanding. It’s sometimes useful to do the same activity with two different representations to help students really understand. These fraction models would be:
- Area Models – Typically sharing tasks, cut into smaller parts. This is the most commonly used. Examples would be “pie” pieces, rectangular regions, geoboard, pattern blocks, paper folding, drawings on grid paper, or dot paper.
- Length or Measurement Models – These show continuous lengths or measurements are compared. They are number lines or fraction strips. Lines are typically subdivided, though a measuring tool with a scale can be used too (ruler, measuring cup, thermometer). Examples would be fraction strips, Cuisenaire rods, folded paper strips, ruler, number line. **The number on a line designates the distance of the identified point from zero, not the point itself.**
- Set Models – The whole is understood to be a set of individual (discrete) objects and subsets of the whole, making up fractional parts. An example would be a set of 12 objects is the whole, while 3 objects are circled with yarn, so 3/12 or 1/4. Can be used with counters of two colors.
7.) Encourage the use of estimation and benchmarks. Estimation helps students know “about” how big a particular fraction is and students should be able to use that for comparing fractions and then again later with operations. Since students are typically less confident with estimating, help them by using benchmarks on a number line. I use the benchmarks (reference points) 0, 1/2, and 1. If the number is greater than one, I still use those same benchmarks, just with the numbers that the mixed number falls between. Like above, practice until students can really visualize this.
8.) Spend an extensive amount of time on teaching equivalent fractions. Equivalent fractions are an important concept that is foundational for everything in fractions – from the operations to relationships, to magnitude. Provide students with a lot of practice with equivalent fractions in a variety of models. Make sure they really grasp it and understand why fractions are equal. Don’t even teach the multiplication method until students can illustrate for you why fractions are equivalent. Then have them start noticing the pattern and work their way to the multiplication method.
9.) Watch out for the use of operations rules with whole numbers being used with fractions. I’m sure we have all seen students add, even unit fractions, before as if they were adding whole numbers. For instance, 1/2 + 1/2 = 2/4. What they are doing is adding the fractions as if they were whole numbers instead of thinking about the meaning of fractions. This demonstrates that they don’t have fraction sense and aren’t visualizing the fraction. We can prevent this by providing lots of practice with fractional pieces so they can begin visualizing it– even if it’s just the unit fractions.
10.) Incorporate fractions as often as possible. For instance, if you have a minute during class just quickly ask, “what fraction of the class is wearing sweaters today?” Get creative. Finding ways to bring fractions into your daily routine will help give students regular daily practice, keep it fresh on their mind, and help them see the relevance of it.
With these 10 best practices, you are definitely sure to help your students master fractions – a difficult concept in elementary math classrooms everywhere!