How to Subtract and Regroup Mixed Numbers with Visual Models

Flashback to 4th grade when my math teacher asked me to subtract mixed numbers, my palms were sweaty, and I felt like I was looking at the demonstration cross eyed WITH brain fog. If you are a loyal follower of my blog, you might know about my past traumatic experience with fractions. It was not pretty, and the last thing I want is for any student of mine to feel this way.

I created the video below to try to show regrouping with visual models. I can’t stress enough how visual models have changed and revolutionized my teaching. It almost never fails that when students struggle, I’ve forgotten to include a visual to help them “see” the math. I am not alone, check out Dr. Jo Boaler, Berkeley Everett and many others who truly understand how we can deeply teach mathematics through visual representations.

The video below will speak for itself, but if you don’t have time to watch it (it’s less than five minutes), the idea is that you can regroup whole numbers into fractions just like students do with base ten models. It’s SO MUCH FUN when students get to learn this way with you. Please check it out and let me know what you think!

Fly on the Math Teachers Wall – Squashing Fraction Misconceptions

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When I was an elementary student, fractions were by FAR my most difficult subject.  I could not ever wrap my brain around them and embarrassingly enough, I still struggle to understand them. The other day I had trouble trying to figure what half of a 3/4 cup of butter would be! My cookies tasted a little more buttery than I would have liked. So today I am talking about misconceptions about fractions, because I’m really a pro!

This question was on a fourth grade test on fractions this week at one of my schools.

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Amazingly, many students answered the question by saying they agreed with Molly. Their explanations said things like:

  • “4/8 and 1/2 are the same number because they are equivalent.”
  • “The diagram below each shows a half, so they are always the same.”

So somehow, somewhere we have a misconception here. Students are missing the idea that fractions can be different amounts if the whole is a different size. After all, one 8 inch pizza is not the same as one 16 inch pizza, right? I’d MUCH rather eat 1/2 of the 16 inch pizza!

My favorite way to clear up misconceptions is to relate it to real life…especially food. Food lends itself beautifully to math in so many ways. Once I brought in the skittles, suddenly light bulbs turned on.  Equivalent fractions may be the same number, but they are not always the same amount.

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I think we miss this step very often when we work with students. I think that real life connection is what helps them figure out what the symbols stand for. When we leave that out, students are unable to make sense of a problem.

If you want to see more examples of misconceptions that us math nerds have uncovered, check out The Math Spot:

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Fractions with Food: Hands on Math

I am thrilled to guest post over at Teach Mama today for her Rockstar Sunday series!  Today I talk a little more about making connections between fractions and food. I hope you’ll check out how I used graham crackers to show representations of crackers, and to help students understand the fraction number line. Click on either of the pics below!

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Make those boxes make sense to students with a real world example.

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Show how you can count on a number line using fractions.

Discovering Numerator and Denominator with a Pan of Brownies

The way to a child’s stomach heart brain is most definitely with sweet treats. While I don’t like to sugar up my students, I do like when they can connect math to the real world.  That was exactly my mission when I brought in a pan of brownies.

Example of Real World Fractions

If all else fails, capture their interest with food!

So far at this point, we had examined the definition of a fraction, and thought about things that come in halves and quarters.  It was time to move into some more new vocabulary, the numerator and denominator of a fraction.

In came the pan of brownies.  I brought it over to a large rectangle table and had them all gather around me.  As they were salivating I asked them how I could split this pan into fractions so that we’d all get an equal amount.  I asked them to draw what that looked like in their math journals knowing that we had 25 students in the room.  This was easier said than done.

For some reason, a bunch of them abandoned the hard work we’ve done with arrays, and started drawing diagonals and squiggly lines all over their papers.  It was like they heard the word “fraction” and felt they needed to abandon everything they knew for this brand new concept.

*Sigh*

Then, I asked them to start sharing solutions, and we started to get somewhere. Arrays popped up on the chalkboard, 2 x 13 arrays (“I didn’t want to leave the teacher out!”), a 5×5 array and a 3 x 10 array.  I asked them which one would get them the best deal.

The settled on the 2 x 13 model so that I could get a brownie (how kind!).  That was when I began cutting.  I handed out the first one and asked them to think about what fraction of the brownie pan they were getting. That was when I introduced the fraction in number form and explained the difference between the numerator and denominator. The numerator was the number of pieces they were going to get to eat, and the denominator was the total pieces in the pan. For example (Hint: This is not the actual pan of brownies I used, since the cuts became VERY small and very messy…they were super gooey! So…I had to whip up another batch tonight for this picture, YUM!):

Real World Examples of Fractions

The numerator and denominator suddenly became clear!

They didn’t REALLY get it though, until the last person got their brownie. At that moment, I gave her my piece, telling her how proud I was that she was so patient to wait and be last. That was when the numerator part really sunk in, because I announced that she was getting 2/26 of the brownie, while everyone else only got 1/26. It was a lesson in patience as well as a lesson in math.

It was a pretty sweet mini lesson!

Things That Come in Halves and Quarters: The Real World Connection

Whenever I have connected math to the real world, I’ve seen a boost in achievement in my classroom. Fractions are REALLY important concepts that must be connected to student’s lives. When I first started teaching, I would just plod along in the book. I would hand out worksheets with rectangle boxes that students would just fill in.  They’d write the numbers without really connecting it to much of anything. It was kind of a disaster!

Now, I love to collect real world examples, and put them on an anchor chart.  Because fractions are so abstract, we put this anchor chart together after brainstorming with a partner first:

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On the student planner that day, I put an assignment to look for things that come in halves or quarters at home as well. We can always add more! Now, each time we talk about a fraction, we try to picture something from this list.

I am hoping that these concrete examples will really help them understand when I move them into representational symbols and numbers.

What Exactly is a Fraction Anyway?

When I was a student in elementary school, I dreaded learning about fractions. It was a very tough concept for me. All I remember is shading in boxes and finding common denominators. I never understood what I was doing.

I decided as a teacher that my mission was to help fractions make sense to my students.  So I introduce the concept very slowly and very carefully.  Because this is so abstract for students, it must be connected to the real world the whole way through the unit.

We started learning about fractions by trying to figure out what a fraction actually is.  I know that sounds obvious, but I need to find out what my students know. So I posted the question, What is a fraction?

I gave a large piece of paper to small groups and asked them to write everything they know. They all pretty much came back with something along these lines.  There were a lot of I don’t knows, and a lot of blank stares. The people who did write something just wrote symbols or numbers.

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Not one student could tell me what a fraction really was. So I tried to clarify it for them with a simple drawing.

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Now that the definition is out of the way, maybe we can move into the conceptual understanding part! I make it a point to say those words daily as we talk about what we learned the day before. The emphasis in the Common Core State Standards for fractions in third grade is on parts of a whole, so that is what we’ll focus on!

How do you help your students know the meaning of math vocabulary?