Below is adapted from the video.  If you want the full context PLUS all the visuals I shared, watch the video.

Welcome fellow Recovering Traditionalists!!

Have you ever had a student look at 3/4 + 3/4 and confidently tell you it’s 6/8?  They add the numerators, add the denominators, and move right along. We usually label that a procedure problem. They just don’t know the steps yet, right? But it’s not a procedure problem. It’s a WHOLE problem.

That student has lost track of what the whole is. And one very common fraction visual makes that mistake feel completely logical.

Even if you teach early elementary, stay with me on this one. It all starts with the visuals kids see for fractions, and those first visuals show up way before third grade.

Every Fraction Is a Fraction of Something

Three-fourths only has meaning if we know what the whole is. The trouble is, kids naturally don’t pay attention to the whole. They pay attention to the pieces.

And the part of a set visual makes that even harder.

Picture it. You show three red counters out of four total counters. Kids see it as “three out of four.” Great. Now add another set: three red out of four. Put them together and what do you have? Six red out of eight counters. 6/8.

You just created a brand new whole of “eight.” The ‘whole’ changed and nobody noticed, especially your students.

So we need a visual that keeps the whole the same.  One of the best is the number line.  On a number line from 0 to 1, three-fourths gets you most of the way to 1.  Add three-fourths more and you fill that whole and go right past it.  You land at 6/4, or one whole and two-fourths more.  The whole does not change.

The Whole Is the Whole Point

In the video, I flash three fraction visuals on the screen and ask one question: is this showing three-fourths?

The first two?  Most people say, “Yep, that’s three-fourths.”  Then the third one shows up, with pieces spread across more than one whole, and suddenly you hesitate.  Hmm.  Does that show three-fourths?

It all depends on what the whole is.

Think of them as pans of brownies.  If I ask how much of a pan of brownies is left, one pan is the whole, and yes, that’s three-fourths.  But if I ask how much is left out of all the pans we started with, then no, that’s not three-fourths.

Same picture.  Different whole.  Different answer.  That’s why we have to help kids pay attention to the whole we’re starting with.

Why Part of a Whole Comes First (Not Part of a Set)

Number lines are great for keeping the whole consistent, but they can be tricky for our youngest learners.  That’s why we usually start with parts of a whole.  It’s also why a lot of textbooks throw in parts of a set early on.

Watch out for that.  With a set, the whole is a group of separate things, and it’s really hard to see that the whole is changing when you combine sets.

If you want backup, look at your standards.  Each state is a little different, but you’ll typically see fractions show up in first and second grade:

• First grade: usually in the geometry standards, where kids partition shapes. That’s the first whole kids work with.
• Second grade: number lines start showing up.
• Third grade: fractions really take hold, and the standards typically frame them as part of a whole.

Now, sets are not bad.  They’re a correct visual for fractions.  They just need to come later, once kids really own the idea of what a fraction is.

My Favorite Visuals for Building Fraction Sense

Start with part of a whole: a number line or one whole shape.  Then move to visuals with multiple wholes, where there are parts from each one.  That’s where you get to dig into the conversation that matters most: What is the whole we’re talking about?  Fractions only have value in relation to the whole.

Fraction Subitizing

One of my favorite activities is to flash a fraction visual and ask:

• How much do you see?
• How do you know?
• What is the whole in this scenario?

Kids need to see fraction quantities in relation to the whole, and recognize that the whole can change. When they can look at a visual and tell you how much is there without counting every piece, they’re subitizing. That’s one of the eight number sense concepts, and it’s one of the best places to start with any kind of number. Grab the Fraction Subitizing Cards HERE: https://buildmathminds.com/fraction-subitizing 

Iterating, Not Just Partitioning

The most common way we have kids model fractions is partitioning: start with a whole, cut it into four pieces, color three, and call it three-fourths.  Kids do need that.  But it’s almost too common.

The other side is iteration.  Instead of starting with the whole, you start with the part.  Hand a student a piece and say, “If this is one-fourth, show me three-fourths.”  They copy that piece over and over until they build three-fourths.  (I love construction paper fraction strips and blank fraction bars for this.)

While they iterate, they’re counting with fractions: one-fourth, two-fourths, three-fourths.  So when they see 1/4 + 1/4 + 1/4, it makes sense that only the numerators get added.  The denominator stays the same because the size of the piece never changed.

Back to 3/4 + 3/4

Let’s go back to the students who want to say 6/8.

The more visuals, modeling, and counting kids do with fractions, the more they understand that three-fourths is three one-fourth pieces.  So 3/4 + 3/4 is three one-fourth pieces plus three more one-fourth pieces.

Some kids will count on: “I have three-fourths.  One more fourth fills the whole.  Then I have two more fourths.”  That comes from lots of time with number lines and subitizing cards where we talk about filling the whole.

Other kids will lean on the true meaning of numerator and denominator:

• The numerator tells us how many pieces we have.
• The denominator tells us the size of those pieces, like a denomination.

When we add fractions, the size of the pieces doesn’t change. How many pieces we have does.

Where to Start

The best place to start is simple: get kids seeing fraction visuals and talking about them.  Lean on whole shapes and number lines, and save sets for later.

My Fraction Subitizing Cards are free to download, and they’re built around part-of-a-whole visuals, including number lines.  Grab them at buildmathminds.com/fraction-subitizing

If you want to go deeper on building number sense for whole numbers and fractions, that’s what we do inside my course, The Flexibility Formula, the version for 3rd-5th grade teachers is where you will find out how to build fraction sense with your students. https://buildmathminds.com/flexibility-formula-3rd-5th 

I hope this helped build your math mind so you can build the math minds of your students.