# A Balanced Approach to Fractions: Wholes, Sets, and Measurement

Welcome back!  If you have been following this series of blog posts all about fractions and improving our fraction instruction and learning, I welcome you to day 3!  If you are new to the series, feel free to snoop around at POST 1 and POST 2 when you have time for more foundation information about teaching fractions.

Today I want to focus on something I briefly touched upon in that first post–the fact that fractions can be so challenging because they appear in so many different formats and contexts.  Whether you teach in a Common Core state or any other state/country with rigorous standards, we need to strive for a balanced approach to teaching fractions.

When we learn whole numbers, three means three.  You can have 3 dogs.  Or you can eat 3 chocolates.  You can read 3 books.  We can count 3 on our fingers.  We can pound a drum three times.  This makes sense.  We can see it and hear it.

When we talk about fractional amounts, the game changes a little bit, especially for students who may struggle with math concepts. As I mentioned in the first post, consider the concept of “1/2”.  I love to start a math class by sitting back and patting my stomach and telling my students that I ate half a pizza for dinner last night.  Inevitably, they laugh and make comments about how much I love pizza, and so on.  I play it up a little bit and hope (usually it happens!) that someone finally says…

“Wait…how big WAS that pizza?”

And then we begin.

### Fractions of Wholes

When students are younger, we place a lot of emphasis on fractions of wholes.  We talk about whole pizzas.  Whole candy bars.  Whole pattern blocks.  We then work to divide these “wholes” into evenly partitioned shapes that we call “fractions”.  Typically we then divide them into reasonable numbers of parts–often 2, 3, 4, 6, 8, and 12.  We occasionally throw in fifths or other “odd” numbers but often steer clear as they are much harder to represent.  (Probably not the best reason, right?)
We typically ask students to identify a fraction divided into equal parts with a certain amount shaded…like this–which they quite quickly can learn is “1/4”.

Even something as simple as moving the shaded area is important…this is ALSO 1/4.  What about the unshaded area?  We need to ask questions about that too. For example, if I always ask, “What fraction of this cake is frosted green?”, we train our students to automatically label the shaded area.  What if the question was, “How many more fourths need to be shaded to make 3/4 of the cake green?”…haven’t we gotten our students thinking more deeply?

Or what about this.  “Here is 1/4 of a cake.  What would the entire cake look like?”

Here is what we might expect to see (with the original fourth shaded)…

But what about these? Wouldn’t these “atypical” examples work as well?

So, a little food for thought!
1.  Consider presenting things in atypical ways.
2.  Use non-standard shapes at times.
3.  Ask students to represent “wholes” in different ways–not just identify them.
4.  Make sure students see fractions as real-world situations…not just pictures in a textbook.

### Fractions of Sets

Many math series address “fractions of a set” in perhaps one or two lessons–and they often skip right to the computational part of the math.  Asking students to find “1/4 of 12” or “2/3 of 24” is not nearly as challenging if they have had ample opportunity to make the connection between division and fractions.

Knowing that 1/4 of 12 means that there are 12 objects (or even more abstract concept like minutes or correct answers!) and that we need to count ONE out of FOUR equal groups is much more manageable if they have first used 12 counters, made the fourths, and then physically SEE the groups.  We can then even count the groups…1/4 of the twelve would be 3.  2/4 of the twelve would be 6…and so on.  So often we teach algorithms that aren’t rooted in understanding; tell students to divide 12 by 4 and multiply by one is far more meaningful when they have built it and seen it for themselves!

Providing these constructivist experiences builds so much more understanding than merely teaching students the algorithm.  Does the algorithm work?  You bet–but it will make so much more sense after they have seen WHY it works.  Eventually, we can tie those counters to more abstract things…1/4 of 12 counters could REPRESENT 1/4 of 12 math problems on a test.

I love the Skittles activity I do with students that really gets them thinking about how fractions can be parts of sets.  It’s one of the activities in my fraction unit if you are interested in checking that out.

As I hinted at above–we can certainly grab math counters and model this type of math.  I also feel it’s critical to help students see how it’s true in the real world.  When WOULD we need to find fractions of a set?  Can your students brainstorm examples?  It’s essential that we do everything we can to help our students understand that we teach fractions for a reason–not because it’s the next unit in a textbook!  If 1/3 of the oranges are moldy…or 3/4 of the class joined the band…you get the picture!
One more point.  We can certainly find fractions of sets of OBJECTS, but in the “real world”, we also find fractions of sets of less “visible” things.  We can talk about counting 2/3 of a crowd…about getting 9/10 of the problems right on a test…or “feeling 9/10 better”.  All of these are more abstract concepts that we can tie to fractions of sets.

### Fractions with Units of Measure

Another area of fraction instruction that is often overlooked–or taught briefly during measurement units–is the idea that we use fractions with measurement concepts as well.  Again, this is a much more abstract concept for some…less tangible than half of a pizza, for sure.
Because measurement is such an important real-life skill, it is important that we DO address fractions with respect to measurement.  There are many ways to tackle this, but let’s just consider a length model.  What does a ruler look like?  A number line!  It’s really a perfect tool to explore “wholes” and “parts”.  Helping students recognize that there are numbers between the whole inches (or cm) is a perfect tie to fraction studies.  After all, there is no way everything in the world could be measured in whole inches, right?  We need those fractional parts.  Many students struggle to located fractions on a ruler, so it’s a great time to tie in some fraction work.
Consider putting a ruler under the document camera and studying it.  Do some paper strip folding where you measure inches, fold halves, and so on.  It’s also a great way to do some equivalent fraction work…1/2 inch is the same as 2/4 which is the same length as 4/8.  It isn’t just length, however.
What is a 1/2 hour?  1/4 hour?  How about the 3 1/4 cups of flour you need for a recipe?  Making these connections is a logical extension of fractional reasoning–and one that is often overlooked.  Quality fraction word problems can help make these real-world connections!

### Students Need to “DO” Fractions!

So…my final thought for today is that we need to have students DOING fractions…they need to be looking at rulers and folding paper and making connections to the real world.  If we invest in that type of work, the computation will follow.I hope I’ve given you a little to think about as you do your fraction planning.  We so often rely on what our math series provide us–but sometimes we need to think past what is given to us and make our students’ math experience more diverse and rich.

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