fractions Archives - The Teacher Studio

Demystifying Fractions: Strategies to Help With Fraction Planning

M Anderson — Sun, 19 Jan 2025 18:35:24 +0000

Fractions often feel like a math mountain for upper elementary students—and let’s be honest, sometimes for teachers too! Students struggle to grasp concepts like numerators, denominators, and equivalence, leaving them frustrated and disengaged. Do you struggle when it comes to fraction planning for your students?

After teaching fractions for more than 30 years, I can tell you one truth. Fractions don’t have to be intimidating. With the right approach, we can make them accessible, relatable, and even enjoyable for students. Today, I’ll walk you through some of the most common fraction misconceptions and share some of my own classroom-tested strategies to help your students build confidence and mastery.

Understanding Common Fraction Misconceptions

For many students, fractions feel abstract and confusing. In fact, some don’t even think of fractions as “numbers” (see THIS POST for more about that!). Before jumping into teaching strategies, let’s address some of the most common misconceptions our students may have:

Numerator/Denominator Confusion: Students often think the numerator and denominator are separate numbers rather than part of a whole. Some students reverse them and need coaching with this. I like to say “out of” (3 out of 4 pieces) to help students understand this.
Improper Fractions Are “Wrong”: Many students view improper fractions as mistakes because they don’t fit their idea of a “whole.” Solid number line work can help with this.
Fractions Are Harder Than Whole Numbers: The shift from whole numbers to parts often feels like an entirely new language. If students don’t spend enough time with concrete models, this may present itself as a problem.

Try This! Use real-life examples, like slicing a pizza or dividing a bag of candy, to show how numerators and denominators work together to represent parts of a whole. Do as many “think alouds” as you can when talking about fractions!

Strategies to Build Confidence and Mastery

Visual Models and Manipulatives

Making fractions tangible is key to understanding.

Use fraction strips, pie charts, or number lines to visually represent fractions.
Encourage students to manipulate these tools to compare fractions, find equivalencies, and practice operations.

Example: Show students two fraction strips—one for ½ and one for ¼. Ask them to align the strips to see how many ¼ pieces fit into ½.

Try This! Provide students with printable fraction circles or paper strips that they can cut and manipulate during lessons. If you have plastic or even digital manipulatives, they work well too! I can’t stress enough how important it is for students to see, touch, and move “pieces” as they learn how to represent fractions.

Real-World Connections

Fractions become less intimidating when they’re tied to everyday experiences.

Talk about design and how people use fractions to create patterns (see the image above or CLICK HERE)
Use cooking recipes to teach measurement conversions (e.g., “How many ½ cups equal 2 cups?”).
Relate fractions to sharing, like dividing a class snack or splitting a group project workload.
My personal favorite involves talking about pizza or cakes or pans of brownies. These are simple for students to visualize and are VERY real to them!
Don’t forget about fractions of a set! Even looking at the colors of M&M’s in a snack size bag or using math counters to show “how many out of how many”! (6 red out of 15 total, and so on).

Example: Have students calculate how to double or halve a simple recipe. THIS PROJECT is a super fun way to let students explore that! Or make a class trail mix and create your own recipe!

Scaffolded Lessons

Break down concepts into manageable steps as you do your fraction planning. This sequence is one possible path you could take. I try to spiral my lessons so I continue to circle back to concepts that have already been taught.

Start with visualizing, drawing, and folding simple fractions (e.g., ½, ⅓, ¼). This can also involve identifying fractions from a picture, but don’t let that be your only early step!
Move to equivalencies (e.g., 2/4 = ½). This may involve more paper folding, more manipulatives, and more “real world” examples. For example, most students can visualize that 1/2 of a pizza is the exact same amount as 2/4 of a pizza. Be very careful to not introduce the algorithm of multiplying the numerator and denominator too early. Let students discover that on their own–with your coaching, of course!
Make sure you expose students to fractions on number lines as well. There are MANY misconceptions that happen with numbers lines, so you want to take your time with this. Here’s a blog post about fractions on number lines that can help if this is a challenge for your students.
Practice operations with fractions (e.g., adding/subtracting fractions with like denominators, then different denominators). Over the years, one thing I’ve noticed is that many students struggle with converting between mixed numbers and improper fractions, and vice versa. Don’t jump to that algorithm too soon! Play with manipulatives until students can see that 15/4 can be reorganized into 3 wholes and 3/4 of another one. I promise–if you take your time here, the lightbulbs will go off!

Interactive Resources and Games

Engaging activities make fractions fun while reinforcing concepts.

Use online tools like fraction apps or virtual manipulatives. Practice counting “unit fractions” until you get to a whole and beyond.
Incorporate board games or card games that involve fractions. Here’s one of our favorites!

Present fraction review experiences in a gamified “puzzle” format. Check out this amazing activity where students REALLY have to apply what they’ve learned! My students persevere on this so much–and some get frustrated. Their sense of satisfaction when they solve the puzzles is usually represented by shouts and cheers!

Don’t forget to do lots and lots of real-world problems and word problems related to fractions. Here are some great word problems if you need some!

Want help getting with your fraction planning?

Fractions don’t have to be scary—for students or teachers. With hands-on tools, real-world connections, and engaging activities, you can help your students move from frustration to confidence.

I have a TON of fraction resources available. You may not have noticed, but I have a bit of passion for this topic! Check these out if you want to learn more. Just click the images at the bottom of this post!

FAQ Section

Q1: What if my students still don’t grasp fractions after trying these methods?
A: Keep reinforcing concepts with consistent practice and try alternative representations, like virtual manipulatives or drawing fractions. It may take a lot of time with “hands on”, concrete learning before students are ready for the more representational and abstract work.

Q2: How can I differentiate for advanced and struggling learners?
A: Offer advanced students multi-step fraction problems or open ended math challenges while providing struggling learners with hands-on tools like fraction strips or visual aids.

Q3: What’s a quick way to check student understanding of fractions?
A: Use exit tickets with a simple fraction question or ask students to explain a fraction concept in their own words.

Want to learn more about fractions and fraction planning?

I have had such a passion for teaching fractions over the years that I have a TON of blog posts! I mentioned some above, but let me highlight just a few more for you if you want to dig in!

Try THIS POST that addresses even more about what makes fractions so tricky for students!

Or THIS ONE about how to use the gradual release model as you teach fractions.

Finally, CLICK HERE to read about how to use math sorts to find fraction misconceptions.

Interested in a few amazing fraction resources? Click the images below to learn more.

Want more fraction fun?

The post Demystifying Fractions: Strategies to Help With Fraction Planning appeared first on The Teacher Studio.

Fraction Unit Update

M Anderson — Fri, 07 Jan 2022 01:23:59 +0000

It’s getting to be “fraction season” for many of us, and having quality fraction lessons and activities is SO important as we try to teach this challenging topic! Teaching fractions is a passion of mine and led me to create an entire fraction unit that I’ve shared on my blog over the years. Over time, people were SO excited to try the lessons and wanted more and more detail about how to teach them. They wanted everything I used from writing prompts to assessments and more. Gradually, my fraction unit was “born”!

Over time, it has evolved and grown and gotten more components to help teachers teach fractions. It’s truly my “baby” and has helped LITERALLY thousands of teachers improve their instruction. It’s like lessons and professional development all in one. From “in the trenches” photos to teaching tips to helpful hints, it’s a great tool to use either on its own or as a supplement to a textbook or other resource.

How is this different from fraction lessons in a textbook?

I’m so glad you asked! I am on a quest to help teachers really learn WHY fractions are challenging to teach–and to help them develop strategies. I have a TON of blog posts, but I want to draw your attention to one short series in case you are interested. You can click the links below to learn more. Again, teaching fractions is a HUGE part of what we do in grades 3-5, but many of us haven’t been given the professional development needed to do so effectively. Let me help!

The goal of my fraction unit is to build deep understanding by having students USE and explore fractions, not merely learn how to compute with fractions. I want students to:

Play and model with fractions
Talk about fractions in increasingly sophisticated ways
Use deep reasoning to talk and write about fraction concepts
Explain their thinking using clear math language
Critique the thinking of others and defend their reasoning
Look for patterns and connections
Find and explore real-world examples of fractions
And so much more!

Fraction Unit Updates

Although the content hasn’t really changed over the years, I’ve continued to add features, explanations, and tools to help teachers. The latest? A more detailed table of contents complete with learning targets to help your planning.

In the unit, I include a NUMBER of different tools to help you get students talking about fractions and using quality math discourse. The unit weaves this into individual lessons with modeling and ideas. I also included additional tools to help you really get them talking!

This includes help making sure you tackle the Standards for Mathematical Practice which are completely woven into the lessons in this unit.

Fraction Lessons: The Progression

One thing I wanted to do is to help teachers see how these lessons unfold. Again, many teachers use this unit as a supplement to the fraction work they already do because they don’t feel it’s rigorous enough. This table of contents can help you tie these lessons to the other resources you have.

You can even print this document out, take notes on it, and use it as a great planning tool! You will notice a great deal of repetition as the lessons work to build on previous work to deepen understanding.

Want to learn more?

I’d love for you to check it out! Just click the image below or RIGHT HERE. Check out the preview I’ve included–it really gives a good feeling for exactly what you get! Want even more fraction resources? Make sure to check out my fraction BUNDLE which includes the unit plus a ton of other powerful fraction resources and tools!

Want some great fraction freebies? I’d love to send you some! Just CLICK HERE or the image below to get signed up!

The post Fraction Unit Update appeared first on The Teacher Studio.

Fractions are Numbers

M Anderson — Sun, 03 Jan 2021 21:15:35 +0000

Teaching fractions is a challenge for many teachers, and so much of it is because WE didn’t have good fraction instruction! I hope this blog post can give you a few key points to ground your thinking about fractions and the best approaches to take when teaching about these important numbers!

Fractions…more than pizza

So many textbooks focus on fractions of areas/regions. Students are asked to divide shapes and shade shapes…but when we focus only on this part of fractions, we do students a huge disservice. When students see “fractions” as the act of dividing a shape, they miss the big idea–that fractions are the numbers that we can use to show numbers between whole numbers.

Fractions are numbers.

That’s right–fractions are numbers. Fractions are not shading exercises. Fractions are not writing “a number on top of another number”. Students have to be exposed to the fact that fractions help us be precise with amounts. We don’t always have one full inch. We don’t always have one complete pizza. It is critical that we give students a way to represent these “parts” so that we can identify ANY quantity. Try something. Ask your students, “What are fractions?” See what you get.

Because fractions are, indeed, numbers–we can represent them in different ways. Sure, we can shade in pictures to show fractional amounts.l We can also have students draw their OWN representation of fractional amounts (FAR more powerful). We can also make sure students understand fractional notation so when they write 3/4 they understand that there is a REASON for the notation. Finally, we can help students build on their understanding of number lines with WHOLE numbers but modeling how we can represent fractions on number lines as well.

Fractions can be ordered and sequenced

Because fractions actually are “real numbers”, we can treat them as such. We can compare them. We can sequence them. With careful thinking and reasoning, we can help students visualize fractions and use their math reasoning. We do NOT need to teach students tricks and rote computation–at least not until we have built their conceptual understanding.

We can perform “operations” on fractions

So, if fractions are a part of this world of “real numbers”, then we can treat them as such. We can add them. We can subtract them. If students understand that 5 x 3 is five groups of three, then they can also learn that 5 x 2/3 is five groups of two-thirds. When we provide only computation rules, we take away the thinking and the sense-making.

I hope this gives you a little food for thought, and perhaps a centering idea. If we help students recognize that fractions are actually numbers that represent real amounts, we are helping build a foundation for understanding that will take them into their more advanced math work.

More Fraction Fun!

I have so many blog posts that focus on fraction understanding. Just search the sidebar for more! To get you started, here are a few!

THIS POST is all about balancing the types of fraction activities we do with students.
THIS POST is about using a gradual release approach while teaching fractions.
Want to learn more about math sorts? Check out THIS POST for ideas and inspiration!

There are literally dozens more–so snoop around!

Looking for some resources to help YOU teach fractions?

Click HERE or the image below for my fraction unit–16 lessons to build conceptual understanding.

Click HERE to see my huge fraction toolkit with 10 great resources to build fraction understanding.

Click HERE to see my fraction and decimal number line resource to help you guide students in their understanding of fractions on a number line.

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

M Anderson — Mon, 29 Jan 2018 04:12:00 +0000

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.

Interested in my fraction unit?

How about a bundle of 10 quality fraction resources at a discounted price? Here you go!

Rather pin this post for later? Here you go!

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Getting Students Ready to Learn Fractions: Gradual Release and More

M Anderson — Mon, 15 Jan 2018 19:10:00 +0000

In my earlier post, I talked about the many factors that make fractions challenging for students. If you missed it, you can read it by clicking RIGHT HERE. Before my next few posts where I tackle some “in the trenches” ideas about fractions, I want to talk about gradual release, an instructional strategy that is true in good math instruction across ALL topics.

What is “Gradual Release” of Instruction?

So although I want to continue to address some of these foundation fraction concepts that can be so difficult, I want to stress something that is true in math instruction overall–not just for fractions. When we want to utilize a gradual release of instruction model, we often think of the following:

I show the students.

We do it together.

They try it alone.

Now, I’m not going to lie. I feel this is a very over-simplified model of what a true gradual release of instruction plan should involve. A true gradual release is NOT linear; it is recursive and cycles around and around as we layer our instruction. That’s for another day! But what I see happening oftentimes is that we teach, we practice, and then we assess–and the results of the assessment don’t always make us so happy.

I’m going to propose a SECOND type of gradual release that is particularly pertinent in math instruction. It looks a little something like this.

Now, what this arrow represents is that when we introduce a concept like fractions, it is absolutely critical that we provide our students with the opportunity to spend time–lots of time–in the world of the concrete. Whether this means paper folding, using pre-made fraction manipulatives, base ten blocks, actual pans of brownies, paper pizzas–WHATEVER–we must immerse students in experiences where they see, feel, and move objects to make discoveries.

As we do this, it is important that we help students start to make connections between these “objects” and the real world. This is where stories, real-world examples, and meaningful problem solving come in to play.

An example…

A student with very little fractional understanding or experience can perhaps start to make connections when we present a problem like this.

“Sue walked into a bakery to buy some cookies for her family. She noticed that each tray held 24 cookies. If half of the cookies on one of the trays had sprinkles, how many would that be? What if half of the cookies with sprinkles also had chocolate chips–how many would that be?”

The problem we have shared is essentially this: “What is 1/4 of 24?”, a problem that surfaces in most fraction units later on…and usually with a page full of similar problems. By telling the story in context, helping students visualize (“Can you imagine what that tray of cookies might look like?”), perhaps even sketch or model it–this more advanced skill is very accessible to even the most beginning student.

Only after spending time building, folding, exploring, playing, and “seeing” how fractions make sense in the world should we move to these “bare number” tasks…tasks like:

3/4 + 4/4 = ?

5 x 3/8 = ?

Generate a list of 3 fractions equivalent to 4/5.

These are all the types of questions that we find rampant in our textbooks and teaching resources–and they DO have value. The question is WHEN. WHEN does it make sense to do these? In my view, the correct timing is when the understanding is there and students need simply to work on the fluency and accuracy of the tasks. Students do not learn best by doing tasks like this; they learn to get more fluent and adept at tasks like this if the foundations have been clearly set.

I really thought they were understanding…

One thing I have found–especially for the more beginning levels of fractions, is that students may seem to be understanding. They fill in the answers correctly on their practice sheets. In addition, they may be able to identify a given fraction. They may even be able to do the bare number tasks listed above…but do any of these show a TRUE understanding of fractions and their real-world application? That takes us, as teachers, getting in trenches with them watching, listening, and asking questions.

Provide students with open-ended tasks and challenges that give them time to explore, discuss, and for YOU to observe. If your math series is full of computation–that’s fine! Just WAIT to do that work until students have demonstrated readiness. Unfortunately, many series don’t give us enough of these experiences, which is what led to the unit I created to use with my own students and many of the other fraction resources in my store. To clarify, I simply didn’t feel as though we spent enough time on the left side of the arrow. If you want to see more, here are the links to a few resources. See you soon for more fraction thinking!

Want to check out the fraction unit I use?

How about a bundled set of TEN fraction resources that support best practices?

Do you want to pin this for later?

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What makes teaching fractions so challenging?

M Anderson — Wed, 10 Jan 2018 00:33:00 +0000

If you have followed me or seen any of my webinars, you know that I wholeheartedly believe that all students can learn math at a high level–and we, as teachers, need to constantly strive to refine our teaching strategies and methods so that we reach ALL students…no matter their starting point. This is especially true for fractions which can be one of the most challenging things we teach. This post begins a series about fraction instruction that I hope you find helpful and meaningful.

To begin, I don’t think it would be surprising if I told you that a majority of intermediate grade teachers declare that fractions are one of (or THE) most challenging concept for them to teach–and for their students to learn. So why is this? And more importantly–what can we do about it?! These questions will be the foundation for this series of blog posts. We CAN make a difference in how we reach our students and deepen their fraction knowledge.

What makes fractions so challenging to teach?

I have done some research (formal and informal) by looking at what the experts have said and by asking countless teachers in the trenches, and I think I have some answers as to what makes this topic so overwhelming for many. Why is this important? If we can identify the stumbling blocks, then we can start to chip away at them and begin to learn more about what WILL work and how we can overcome these obstacles. I am constantly on a quest to find ways to “make sense” of fractions for my students (which is what led to me creating a full fraction unit to use with my students!)

Fractions are not always “concrete”.

Although we may start our fraction instruction with objects–perhaps paper folding or plastic fraction pieces–we too often move very quickly past these real-world models to paper and pencil computation. I’m going to argue that even drawings of fractions fail to be concrete enough for some students with limited fraction sense. To many, this is just a circle with lines. The physical act of cutting, folding, and manipulating is so important for many.

To make matters more complicated–this is just the beginning! When we multiply whole numbers, we can make “groups of” objects, right? 3 times 6 can be represented by 3 piles of 6 objects. Do we take the time to really make our more advanced fraction concepts such as multiplying and dividing concrete? Do students really “see” what happens when we multiply fractions (HINT: We can still model 3 groups of 1/4 or 6 groups of 2/3…and it can make a huge difference!). When we move too quickly to “bare numbers”, we make a lot of assumptions about what our students truly understand.

Fractions are numbers–really!

Here’s another statement that seems a little obvious, right? Of course, fractions are numbers! But wait…try to get yourself into the mindset of a struggling student. We show them pictures of pizzas and candy bars cut into little pieces. We talk about the pictures in different ways…and we help students learn to “label” them as representing 1/5 or 3/8 or whatever the picture represents. Do we really help them understand that fractions allow us to represent real numbers–and even parts of numbers? That when we have a drawing of 3/8, it means that we have LESS than a whole object or amount? That fractions allow us to show amounts BETWEEN whole numbers? This is a critical part of fractional understanding; the drawings we use to show fractions are merely representations of numbers from a number line. More to come in upcoming posts.

(These number lines are a part of my fraction number line resource–also part of my 10 resource fraction toolkit)

The understanding of “unit fraction” is missing

This leads me directly to this concept. The term “unit fraction” refers to a fraction with a

“1” in the numerator…such as 1/4, 1/7, or even 1/500. Understanding that this is “unit” of measure or amount is so critical. I like to interchange the terms “unit fraction” and “counting fraction” with my students because I really want them to understand that, just like whole numbers, unit fractions can be counted, added, subtracted, composed and decomposed–and more.

In the primary years, a great deal of time is spent counting whole numbers in different ways–by ones. By 5’s. By 10’s. We even practice counting backward which helps develop subtraction understanding. The same is true for fractions–and we really cannot make assumptions that all our students can do this automatically.

When we count fractional parts (“unit fractions”) using manipulatives, on number lines, and even orally, it helps build that understanding that fractions are numbers. It helps students begin to develop their sense of the relative size of fractions. It helps lead to a greater understanding of what happens when we count and get the same numerator and denominator (a new “whole”) and then count past that to make improper fractions and mixed numbers. When we can count fractions we start to see the additive nature of them…I have 2/5 and I count up one more fifth and get 3/5–and this is something that can often be missing in student understanding if time isn’t spent working on this. It really forms the foundation for addition and subtraction of fractions–and is truly an essential skill.

Fraction interventions

In fact, this is one of the first interventions I do with students who are struggling. I get out a manipulative of some type (I linked to some of my favorites…note, these are affiliate links) and we practice counting. We notice what happens when we reach a “whole” and then beyond. We essentially “play” with these counting fractions and then begin to record our findings on paper and pencil. I will often pull these back out when reteaching is needed. Also, consider when you look for manipulatives that some have the fractional part written on them and others are blank–both are great, but make your decisions based on what you want to accomplish. Similarly, if you ONLY use circles or ONLY use fraction bars, students may struggle to generalize the learning. Even using tools like pattern blocks can be really helpful as you study basic fraction concepts.

Fraction notation and terminology can be challenging

So, when talking about playing with unit fractions and counting fractions, I mentioned that I work to connect the physical fractions (fraction circles, folded paper, etc) to the written symbols–and this leads me to yet another challenge associated with fractions–notation and terminology. We throw around words like “numerator” and “denominator” and sometimes forget that students may not have internalized those terms and have most likely never heard them. Words like “equivalence” and “improper” and “reduce” can also add to the confusion. If you see my other posts from the past (or other lessons from my fraction unit)To make it worse? We can write fractions in different ways!

Check out the image below–and then think through the lens of a struggling mathematician. Both of the green fractions represent the same thing–but they are written in two different ways. One, if written carelessly, could end up looking like 2,110–the other is easy to read but difficult to make using technology. How about the purple ones? Students need to understand that both of those are the same value–written in two different ways. Even understanding that the “1” is a “whole” and the 7/8 represents part of another whole can be confusing for many. We need to be constantly assessing for this type of misunderstanding. Finally, with a tie back to unit fractions, students need to understand that one “whole” is the same as when it is notated with the same numerator and denominator. We cannot make assumptions that this understanding is in place. Dig in and find out–and fix any misconceptions along the way.

Fractions appear in multiple contexts

I’m almost finished–I promise! I know so many of these things seem trivial–but if we don’t catch them in our strugglers, we just keep moving them forward with an unstable foundation–and we KNOW that really understanding fractions is the key to success in algebra and other upper math classes. We can’t fail our students! Another frustration for students is that fractions appear in so many contexts. We can divide a pizza in fourths. And we can divide SETS of things (like bags of jelly beans or baseballs) into groups.

Also, we need to use fractions of more abstract things like units of measure–like fractions of an inch or a pound. It’s no wonder students get confused–1/2 can mean so many different things! Even thinking about 1/2 of a pizza…is it a 7 inch pizza? A 16 inch pizza? Are those halves the same? What about 1/2 of the pizzas in the restaurant? Or half of a slice?

Because 1/2 can VARY, this is extremely confusing to many students. In fact, this challenge forms the foundation of a great deal of the fraction work I do in my class–to really help students grasp how fluid fractions are. You can see lots of examples of this in my complete fraction unit that has been such a game changer for me and literally thousands of teachers who have used it.

A growth mindset is missing!

This is perhaps the hardest to overcome. Students aren’t born thinking fractions are hard–somewhere along the way, they have gotten this message. Whether it be from their families (“I was terrible at fractions, too!”) or the way we, as teachers, present the information–somehow we are sending messages that fractions are our nemesis, something only to be understood by the lucky ones. We have to think about our own biases and make sure we are NOT sending those fixed mindset messages to our students. Fractions ARE accessible to everyone…but we do need to make sure that we build and nurture a strong foundation of understanding and a mindset of growth and perseverance.

I hope I gave you some things to think about today–and stay tuned for more fractions posts coming soon!

Want to check out the unit I was referring to?

Or maybe check out a bundle of 10 fraction resources I use to help build this understanding?

Want to pin this for later?

The post What makes teaching fractions so challenging? appeared first on The Teacher Studio.

Fraction Misconceptions and Math Language (Day 2 Fraction Concepts)

M Anderson — Sun, 19 Feb 2017 17:10:00 +0000

Fractions can be tricky—not because students aren’t capable, but because misconceptions about fractions are incredibly common. In this Day 2 lesson from my Fraction Concepts series, the goal is to intentionally surface those misconceptions while helping students develop clear, accurate fraction language. By creating space for discussion, questioning, and mathematical talk, students start to refine their thinking and build a stronger foundation for understanding fractions in meaningful ways.

Indeed, things got even more interesting as my work with fractions unfolded on day 2! For those of you who read my first fraction post, I discussed how we used paper folding to get our fraction concepts unit “launched”, and we ran out of time to finish the investigation before digging into some fraction misconceptions. I gave them another 25 minutes today to work on folding their different fractional amounts and then asked them to spend some time writing in their math notebook on any of the following:

What did you notice?

Was there anything challenging?

What was easy?

What patterns did you discover?

First I gave the students “alone” time to write and then let them go compare notes with their partner from the investigation…

After the students had their work time, we had a discussion about the things that they noticed…I jotted down their ideas on the Smartboard…

Math vocabulary and language

What really stood out to me was the very NEBULOUS (vocabulary word!) math language I was hearing. A phrase that kept coming up was “small fractions” as in “It was harder to fold the small fractions.”

So, being the pesky and annoying teacher that I am, I presented this question to the class:

I L-O-V-E to ask questions like this!

So, I didn’t let the students talk . . . I just said that I was hearing MANY students use this phrase, so I thought we had better make sure we all knew what it meant. I asked each student to write the So, answer to that question on a post it note and come slap it on the easel.

Being the obedient students that they are, they did.

And–seriously–I only wish I had taken some photos of the notes up close. SERIOUSLY! I had everything from . . .

1/100th

“When you make it smaller”

“Small fractions are like tenths”

At this point, any concerns I had about backing up this far and moving this slowly were GONE! I read off the notes to the students and then asked what we had learned about what a “small” fraction is and one of my delightful yet very shy students called out, “ABsolutely nothing!” and we all laughed hysterically.

What next?

So where do we go from here? I reminded the students about how we have been talking about the Common Core and how important it is to use precise mathematical language. I asked if “small fraction” was precise enough that anyone else would know what we meant. . . and we had complete agreement that it was not. Next, I asked them if, on our 4, 3, 2 , 1 rubric we were evaluate how clearly we communicated, we agreed that all our post its were 1’s with a few 2’s sprinkled in . . . until “T” asked, “If there really isn’t an answer, how could someone get a 4 on this?”

Have I mentioned how much I love fourth graders? He honestly asked–and honestly wanted to know! I told him I had to think about it. . . always a good strategy for the tricky questions! Don’t answer too soon!

Deep fractional thinking

So I set the students to work on a little activity to set them up for tomorrow’s lesson…I asked them to work completely alone to answer the following question:

If it is too small to read, the question asks “Is this shape divided into fourths? Explain your thinking.”

I let them work for a few minutes and then on their way out to recess made them commit to either a “yes” or a “no” response . . . and it will be our kickoff “debate” tomorrow! Remember, one key element of the Standards for Mathematical Practice is the ability to explain one’s own work and to critique the reasoning of others . . . so tune in tomorrow to see how it all shakes out!

After they worked for a few minutes we gathered back together to talk about T’s question . . . is it possible to get a 4 on a question with no answer? Maybe I should save that for another blog post . . . this one is already plenty long–but I told him that yes, I do believe you can get a 4 on a question with no answer. I reminded him that we are stressing explaining our thinking, and we could certainly do that with this question. I modeled something sort of like this (this was done aloud, not in writing . . . I’m just trying to give you the gist of it)

“I know that there are different ways to look at the idea of being small. Some people might think the numbers should be small–like in 1/2. Some people might look at the size of the pieces to determine what a “small” fraction is. Still others might think a small fraction is one where you only look at one piece of a whole–like 1/3 is small where 2/3 is big. So for the question “What is a small fraction?”, it is very important to understand that there might be more than one way to answer it.”

Reasonable and rigorous expectations

Would I expect a fourth grader to be able to answer like this? Maybe not today . . . but now that it has been modeled–perhaps. We must push their thinking, push their comfort level, and push them into the rigor that is required.

If we make it meaningful, they can do it! Thanks for sticking with me again–sorry these are getting so long! Stay tuned to see how our debate works out . . . I’ll give you a hint–the vote was 9 “yes” to 13 “no”!

***UPDATE***

This blog post is now a part of my comprehensive fraction unit available by clicking the image below. Hundreds of teachers have now used it to change the way they teach fractions!

Now available as a part of a ten-resource fraction toolbox that might help you teach to deep understanding.

Did you miss Day 1 of this fraction series? Go take a look!

Hey there! Looking for more?

Follow me on social media and by email!

You can find me on Facebook, Tiktok, Instagram, and Pinterest and

you can also sign up for my NEWSLETTER where I start you off with an awesome freebie–and then you will get even more teaching tips, freebies, and offers delivered right to your mailbox!

The post Fraction Misconceptions and Math Language (Day 2 Fraction Concepts) appeared first on The Teacher Studio.

Fraction Folding–discovery learning

M Anderson — Sat, 18 Feb 2017 23:47:00 +0000

Today we kicked off our fraction work, and every time I start this unit I’m reminded just how important it is for intermediate students. Fractions show up everywhere in our standards. The understanding students build here truly matters! I like to take the time to document the thinking, the misconceptions, and the little moments of clarity that happen along the way. Today’s focus? Fraction folding!

I always invite conversation around how we teach fractions because there are so many ways to help students make sense of them. As I walk through our work, I try to share photos, student examples, and the reasoning behind each step. I do this partly to keep myself honest about my choices, and partly because I love hearing what has worked well in other classrooms, too. If you have ideas or resources that have been successful for you, I’m always eager to learn from them.

Teaching fractions throughout the year

One thing our district math team emphasized when we built our pacing was the importance of giving fractions more than a single stretch of instructional time. We built in two units. We do one one now to develop foundational understanding. Then we built in a later one that connects fractions, decimals, and the more sophisticated ideas students will need. That structure has really shaped how I approach the beginning of this work.

Today I started by asking students to reflect on what they already know about fractions and to rate their overall confidence (using our 4 3 2 1 ) scale (Click here to revisit earlier blog post about this!) and the results were quite amazing! In a ten minute writing time, I got to witness a number of misconceptions, poorly explained reasoning, and a bunch of “3’s” and “4’s” in confidence!

Good thing I had planned on starting slowly! Today we started our new math notebooks, and I explained to the students that we are “raising the rigor” one level more. The poor things have heard this all year! I told them we are going to work hard to use our new notebooks to both record our thinking, our practicing, and our new learning.

I’ll blog more about this later as it unfolds! For today, we started the section we called “Fraction Concepts” and even talked about what the word “concept” means. Fascinating! I told the students that our job through this unit would be to determine some things we could determine to be “true” about fractions and that we would be working our way through a number of these “truths” during the unit.

Today’s fraction “truth”

Today’s “truth” involved ensuring that students understand that fractions represent equal parts of something. (I didn’t really want to use the term “whole” yet–I don’t really like to treat fractions of “objects” and “sets” differently until I have to!) I also told them that we would be spending some time creating equal parts.

Then, I put the students into pairs (I love popsicle stick picking!) and gave them each 3 minutes to find a classroom object that was either a square, rectangle, or circle. The item needed to be bigger than a deck of cards and smaller than a book. Each team was then assigned a color of paper and we got started.

The Task

With your partner, trace and cut out your shape. You will need many of these as the investigation unfolds. Your job is to find ways to divide your shape into equal parts. . . first in two equal parts, then three, and so on.

I made sure that we had plenty of objects accessible in a variety of sizes. You could also have cardboard tracers if that is more appropriate for your learners.

I then showed the students the following charts–each is labeled “halves”, “thirds”, “fourths”, and so on. I used pieces of bulletin board paper, but you could use construction paper or even just set your shapes on the floor by the headers. As they discovered a way to fold their shape, they were asked to put it on the correct chart. I was on the prowl for work that was not accurate and precise.

Voila! Posters are taped to the ground with deliberate “aisles” so they don’t get trampled on!

Each poster is labeled with the word for the fractional part–I will use the words and different symbols interchangeably throughout the unit. NOTE: The chart does NOT say “1/3” because we were not identifying 1 out of 3 parts.

FRACTION MISCONCEPTION ALERT! Make sure students are aware of the difference between drawing “thirds” and drawing 1/3. Watch for this and help students see the difference.

Digging In

Students dug into their work and did need some frequent reminders about using straight edges, working carefully, and so on. (I know–shocking!) That being said, they were very engaged and thoughtful. I heard some pretty nifty stuff like:

“Hey–as you try to get more pieces, the pieces get smaller!”

and

“Man–it’s a lot harder to fold the odd numbered pieces!”

and

“If you fold it in half and then in half again, each half gets cut in half!”

Tracing…

Using a straightedge for precision…

Circles are the toughest as this team found!

As the period unfolded (and it became clear that this was going to be a TWO DAY investigation!), our charts began to fill up and things moved a little faster. I was walking around and taking notes about who was struggling and about observations I wanted to share with the whole class.

The charts started to fill…first halves and thirds, then fourths and eighths…we’ll see how it “unfolds” tomorrow! Got to love a little math humor, right?

What’s next?

Tomorrow we will finish up our investigation and then ask ourselves if we have discovered any new “truths” to record in our notebooks. I am pretty confident that my students would have been able to answer correctly if given a sheet of “shaded fractions” to identify.

That being said, today’s activity showed me that many students are maybe missing some very critical understandings about equal parts and about patterns that arise when dividing shapes. I’m pretty sure that the work we will do over the next lessons is going to be very important as we build a foundation for the more advanced skills coming soon.

This is the difference between asking students to REPRESENT and MODEL fractions rather than just identifying fractions on a piece of paper. This activity deepens understanding, gets students noticing and talking, and helps you as the teacher to notice misconceptions and to better understand where things might be going wrong.

Finally, if you aren’t familiar with what your standards require, I would encourage you to dig in and follow along with us as we try to construct meaning over the next few weeks!

One more thing…

The terms “numerator” and “denominator” did come up in our lesson today. I made a poster to represent this and hung it on our math board. You may need to introduce the terms over the next days if your students do not bring them up!

UPDATE!

This blog post is now a part of my comprehensive fraction unit (18 complete lessons!) available by clicking HERE or the image below. Literally THOUSANDS of teachers have now used it to change the way they teach fractions!

This unit is also now available as a part of a huge 12 resource bundle which offers an amazing assortment of lessons and resources to help you teach fractions at a deep level.

Finally, are you interested in getting a short series of emails and freebies about teaching fractions? Just CLICK HERE to get signed up. Want to read about the misconception some students have that fractions aren’t numbers? Check out this blog post all about it!

Hey there! Looking for more?

Follow me on social media and by email!

Find me on Facebook, Tiktok, Instagram, and Pinterest and

you can also sign up for my NEWSLETTER where I start you off with an awesome freebie! You then will get even more teaching tips, freebies, and offers delivered right to your mailbox!

The post Fraction Folding–discovery learning appeared first on The Teacher Studio.

More Fraction Number Lines: More Critiquing Reasoning!

M Anderson — Sun, 29 Jan 2017 04:02:00 +0000

If you missed my post the other day about using number lines to improve fractional reasoning, you might want to take a peek at it before digging into this post! Just CLICK HERE if you missed it…because I want you to know that the lesson featured today was not the FIRST time my students had worked with this number lines or critiquing reasoning. This lesson is one that should come after they have had some other experiences using number lines and having conversations about them.

So let’s dig in!

Step one–present students with a number line task. As I mentioned the other day, even a number line task can be simplistic and obvious. I am partial to a more “open” number line–where partitions are not already drawn. When you include those partitions, you’ve done so much of the thinking FOR the students! Remember, I always ask students to THINK before they pick up their pencil so they have a starting point.

As students worked, I walked around and checked out their work and hunted for misconceptions. I would make note of how they were organizing their work or strategies they used that I thought might be worth talking about. In my last post, the next step involved pairing up and having discussions together and coming to consensus. Today? Not so much!

Whole class sharing

My next step was to ask students to come up and place a blue dot on this “class number line” to show where they had marked in in their math journal. I reminded them to NOT change their answer based on what they see–because they need to trust their gut!

By the time all the dots were up, it was time to have a discussion where we “critiqued the reasoning” and defended our thinking. We had to use math language and vocabulary with our sharing–so I heard things like…

“The dots on the far left can’t be accurate because they would really be past 0 and into negative numbers.”

and

“I think [pointing to a dot] this isn’t possible because there is no way that you can keep equal parts.”

and

“It was more clear to me when I put in all the whole numbers so I could find where they 1/3 really should go.”

Some students needed some convincing–and we had frequent “turn and talks” throughout to get EVERYONE involved in discussion the ideas people shared. It was great to see some light bulbs REALLY go off! After our discussion, I sent the students back to their notebooks to “revise” their thinking if necessary. SUPER powerful stuff!

This was a very nice and quick warm up for our lesson–and the process can be used not just with number lines but with ANY problem type! Stay tuned for another post coming soon with more fraction fun!

Want to see the resource that these number line problems are from? Lots of different problems and options…

Also available bundled with whole numbers to 1,000 and whole numbers to 1,000,000.

Want to pin this for later? Here you go!

The post More Fraction Number Lines: More Critiquing Reasoning! appeared first on The Teacher Studio.

Fraction Number Lines, Number Sense, and Math Discourse Part 1

M Anderson — Fri, 27 Jan 2017 04:48:00 +0000

It seems like I have picked a few topics lately that just don’t fit into one blog post! I wanted to share two different “warm ups” I did with fraction number lines–and two different types of “math talk” that resulted from them. Give these two lessons a try and see what you think! I went kind of “photo journalism” style because I thought the pictures helped tell the story! Watch not just for the important fraction understanding–but the immersion in the Standards for Mathematical Process as well!

Before we go on…if you want even MORE fraction help and freebies, just CLICK THIS BUTTON and watch your inbox for a series of free fraction tips and printables for you to use!

Fraction Number Lines to Build Number Sense and Math Talk

First of all…if you have followed me for long, you know I love number lines–especially number lines that think “outside the box”–even number lines can be rote and low level–so we want to watch for that!

The other day I gave my students this one…it had the 0 and the 2 marked–and asked students to identify what number they felt that “dot” was showing. My first step is always to ask them to THINK before they even pick up their pencil. While they think, I remind them to consider what they know and can tell from just looking at it. I really think slowing them down before they start writing can lead to deeper understanding and reduce careless errors. Plus–for those students who ARE slower processors…not having to watch 20 other students get to work feverishly while they sit is SO refreshing and validating for them. After some think time, they were off!

Some asked if they could use rulers…I simply said, “If you think it will help…”

Justifying Our Thinking

I noticed that some students seemed to be putting fractions on their number lines WITHOUT adding the whole numbers first. I asked, “Are you sure 1/4 goes there? How do you know?” Their answers told me a ton about their level of understanding. Some had already visualized where the “1” went–others were simply putting it “where it looked right”.

As with all my number line work, I want them to be able to explain their reasoning both in writing and orally. As students were working to finish, I had students begin to write their ideas down so they were ready to buddy up.

When we were all in a good place (in other words, essentially finished), I put students in pairs and trios and gave them the following direction:

You must all come to consensus about what fraction you will assign your dot. When you are all in agreement (and this took SOME debate in some groups!), you will mark it on the white copy and start explaining your thinking. When you present, I will call on whoever I want so make sure everyone is accountable for the information and explanation.

(or something to that effect)

I circulated and listened and coached and looked for misconceptions and mistakes. I also looked to see the variety of strategies shown so I could have a variety of ideas to share under the projector.

For this particular problem, we had some debate. About half the groups believed the dot to be at 3/4 while the other groups had a variety of answers. One by one, they presented their solution and defended their logic. Along the way, there were a few “a-ha’s”…and a few stubborn souls who stood their ground despite very good arguments from others!

At one point, we put two different solutions up and had students try to determine which one they felt was “more right”…and which explanations seemed most plausible.

We also showcased different strategies that seemed to help some groups. This group used different colors for each step along the way–and other groups agreed that this seemed to make their explanation easier to follow.

Finally, we ended up grabbing a spare number line and doing some folding to find those midway points and really “prove” what the 3/4 teams were proposing. We had a great discussion…and SO much learning happened–and the students did it ALL! Consider trying the strategy:

1. Think time
2. Independent time
3. Partner/consensus time
4. Whole group sharing and debating time

Watch how engaged your students will be…and let me know how it goes!

Looking for the fraction number line resource pictured?

I also have two other number line resources and a bundle of all three…see what you think! I use them ALL year long.

Want to pin this post for later? Here you go! And watch for several more fraction posts coming soon!

Looking for an entire fraction UNIT to get students thinking and talking? Check this one out!

(the number lines are not a part of this resource)

Or even my entire fraction toolkit of TEN great resources…such a great deal!

The post Fraction Number Lines, Number Sense, and Math Discourse Part 1 appeared first on The Teacher Studio.