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The Progression of Fractions (Meaning, Equivalence, Comparison)

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Concrete Learning for Equivalent Fractions - Math Coach's Corner

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Concrete Learning for Equivalent Fractions - Math Coach's Corner

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Fractions are such an abstract concept, and children need lots of concrete and representational (pictorial) experiences to really understand the meaning of a fraction. Concrete learning also allows students to explore concepts and build understandings of their own, rather than having information delivered to them from a teacher.

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Multiplying Fractions by a Whole Number - Math Coach’s Corner

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Multiplying fractions isn’t as daunting as it may seem! Connect to what students already know and provide visuals. FREE DOWNLOAD!

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How to Teach Fraction Operations with Models

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I develop curriculum for middle school math. I focus on using critical-thinking skills in my content so the math makes sense to the students.

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Students should have flexible strategies for comparing fractions, and the strategies should be based on reasoning and fractions sense. In the first article of this series, we looked at comparing fractions with like numerators or denominators. You'll also find the vertical alignment of fraction comparison standards from 2nd through 4th grade in that post.

The next comparison strategy involves fractions that are one unit fraction from a whole. A unit fraction is a fraction with a numerator of 1, for example 1/8 or 1/12. Examples of fractions that are one unit fraction from a whole are 7/8 or 11/12. The key to this type of comparison is to think about, and visualize, the distance each fraction is from one whole. For example, 7/8 is 1/8 from a whole and 11/12 is 1/12 from a whole. Since 1/12 is smaller than 1/8, 11/12 is closer to one whole, so it is the greater fraction.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

To really understand this strategy, students need lots of hands-on practice and they need to understand the meaning of the denominator-the larger the denominator, the smaller the piece. Take a look at this graphic:

By using the visual, you can see that 11/12 is closer to the whole, because 1/12 is smaller than 1/8.

After we had practiced comparing fractions one unit fraction from a whole using fraction tiles, we played a good, old-fashioned game of War. I made two copies of this sheet for each pair of students on cardstock, laminated them, and cut them out. Each player turned over one card, and the player with the greater fraction had to explain why theirs was greater. Students had the fraction tiles handy to refer to, if they needed them. Check out this video of my students in action!

Click here for the next strategy! 🙂

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Comparing Fractions: One Unit Fraction from a Whole - Math Coach's Corner

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Students should have flexible strategies for comparing fractions, and the strategies should be based on reasoning and fractions sense. In the first article of this series, we looked at comparing fractions with like numerators or denominators. You'll also find the vertical alignment of fraction comparison standards from 2nd through 4th grade in that post.

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"Understanding why a fraction is close to 0, 1/2, or 1 is a good beginning for fraction number sense. It begins to focus on the size of fractions in an important yet simple manner."
(Van de Walle/ Lovin, 2006)

In this series of blog posts, I am exploring 5 different strategies for comparing fractions. The first post highlighted comparing fractions with like numerators or denominators, while the second post introduced a strategy for comparing fractions one unit fraction from a whole. Another useful strategy is to compare fractions to a benchmark of 1/2.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

Students should have many opportunities to explore fractions equivalent to 1/2 using hands-on materials. My favorite manipulative for this exploration is fraction tiles, because you can easily connect them to fractions on a number line. If you don't have access to fraction tiles, you can download printable ones here. When the students are using the tiles, I like them to draw a number line above the 1 whole tile (see picture) so they have a visual representation of the fact that proper fractions fall between 0 and 1 on the number line. Why is that important? Check out this post!

Challenge students to use their tiles to find fractions that are equivalent to 1/2. Be sure to have the students work in pairs to facilitate mathematical discourse. It's always interesting to watch this process-some kiddos are very methodical, trying each of the pieces in order (eg., thirds, fourths, fifths, etc.), while others employ a more 'helter skelter' approach. I like to highlight the students who use a more organized method and remind students that being organized often helps mathematicians do their work more efficiently.

After the students have found all of the fractions, write them on your white board or an anchor chart. I like to use the order shown below. By not putting the fractions in order (2/4, 3/6, 4/8, etc.), students are more likely to focus on the relationship between the numerator and denominator, which is what you want.

Now ask students a simple question: What do you notice? Be prepared for many different responses, but these are a few I usually hear:

The numerators are "in order" or count by ones (2, 3, 4, 5, 6)

The denominators skip count by twos (4, 6, 8, 10, 12)
Note: These are the relationships students usually see first. When students make them, I certainly record them, but I also challenge students to look for relationships between the numerator and denominator in each fraction

If you add the numerator again, you get the denominator (4 + 4 = 8)

Two times the numerator is the denominator (4 x 2 = 8)

Oddly enough, students rarely say that the numerator is half the denominator. Once they see it, though, the light bulbs go off! Next, I give them other denominators and ask what the numerator would need to be for the fraction to be equivalent to 1/2.

Now is a good time to go back and ask students which tiles they could not use to make a fraction equivalent to 1/2 (thirds and fifths). This is a GREAT discussion! Using the tiles, some students will totally get that half of 5 is 2 1/2, so 2/5 is less than 1/2 and 3/5 is greater than 1/2.
The next step is to have students determine if fractions are less than, equal to, or greater than 1/2. Even though students might see the pattern (numerator is half the denominator), they will still need lots of practice and hands-on learning. To practice, download these fraction cards, show students a card, have them build it with tiles, and then explain why it is less than, equal to, or greater than 1/2 (7/12 is greater than 1/2, because 6/12 is equivalent to 1/2, and 7/12 is more than 6/12). Verbalizing their thinking is really critical because that helps them internalize the relationship between the numerator and denominator in fractions equivalent to 1/2.

Because multiple representations are important, you can also use number lines for the same type of practice. Download a sheet with the number lines here, cut them into strips, and use them as described above.

One more strategy to go! Read about the last strategy here.

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Comparing Fractions: Using a Benchmark of 1/2 - Math Coach's Corner

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"Understanding why a fraction is close to 0, 1/2, or 1 is a good beginning for fraction number sense. It begins to focus on the size of fractions in an important yet simple manner." (Van de Walle/ Lovin, 2006) In this series of blog posts, I am exploring 5 different strategies for comparing fractions.

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"Learning to compare and order fractions is important to developing an understanding of fractions." Marilyn Burns ( About Teaching Mathematics, 2015, p 430)

"Helping students develop their 'fraction sense' is extremely important before they begin computing on and with fractions." Julie McNamara and Meghan M. Shaughnessy
( Beyond Pizzas and Pies, 2015, p 132)

In this series of blog posts, I'm discussing five different strategies for comparing fractions. As background to this last strategy, I suggest you go back to the first blog post and read the other three posts in the series, which describe the first four strategies. You'll also find activities you can download and use in your classroom.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

By now, your students should have four solid comparison strategies in their toolbox- compare fractions with like denominators, compare fractions with like numerators, compare fractions one unit fraction from a whole, and use a benchmark of one half. All of these strategies are based on a solid understanding of fractions. This type of understanding develops only through the extensive use of concrete materials, pictorial representations, and lots of great math conversations! These strategies will be sufficient for handling many, but not all, fraction comparisons. Before you go on, look at each pair of fractions below and see if you can determine which would be the most helpful strategy for making each comparison. Think about how you would explain your reasoning behind each comparison.

You probably came up with something like this:

1. In the first pair 5/6 is greater than 3/4. They are both missing one piece (one unit fraction from a whole), and 1/6 is a smaller piece than 1/4, so 5/6 is closer to one whole. That makes it bigger than 3/4.

2. In the second pair, 3/5 is greater than 3/8. I'm getting 3 pieces in both fractions, but fifths are bigger pieces than eighths, so I'd rather have 3/5.

3. I know 6/8 is greater than 2/6, because 2/6 is less than 1/2 and 6/8 is greater than 1/2.

4. In the pair 2/6 and 4/6, the pieces are the same size-sixths-so of course I want more pieces. That means 4/6 is greater than 2/6.

When deciding which strategy to use, you might have cycled through the strategies, thinking of the most simple strategies first: Are the denominators the same? Are the numerators the same? Are they one unit fraction from a whole? How are they related to 1/2?

That's really the process we want our students to go through as they compare fractions. Comparison of fractions should be based on an understanding of the fractions as numbers, not tricks.

But there will be some fractions that these first four strategies can't help with. Consider comparing 8/12 and 3/4. Hmmmm, unlike numerators, unlike denominators, both greater than 1/2, one is a unit fraction from the whole but the other isn't. Now what do I do? This is where an understanding of equivalency comes into play. We want students to notice a relationship between the denominators in this case and realize that fourths can be rewritten to twelfths. How do they develop that understanding? I think you know my answer to that-lots of experience with manipulatives, such as fraction tiles, exploring what fractions look like and represent. It's pretty hard to look at these tiles and not see that 3/4 is equivalent to 9/12. And since 9/12 is greater than 8/12, that means that 3/4 is greater than 8/12. You can download printable fraction tiles here.

There is also a multiplicative relationship between the denominators-4 and 12. Four times three is twelve, so it takes 3 times as many twelfths to have the same amount as fourths. In other words 1/4 is the same amount as 3/12. Pretty hard to miss that relationship looking at the tiles! After enough experience exploring these relationships with the manipulatives, it's easy for students to understand why we can multiply the numerator and denominator by the same number (a fraction equivalent to 1) to find an equivalent fraction. So, yes, they learn a procedure for finding equivalent fractions, but it's grounded in understanding.

Often overlooked is that sometimes it makes more sense to find a common numerator, rather than a common denominator. Take, for example, the fractions 3/8 and 15/60. I cycle through the first four strategies, which are no help in this circumstance. Then I look for a relationship between the denominators and find none. So instead, I look for a relationship between the numerators, and I find a multiplicative relationship between 3 and 15-I know that 3 x 5 is 15. I multiply 3/8 by 5/5 (a version of 1) and find out that 3/8 is an amount equivalent to 15/40. Now I'm right back to one of my first strategies-comparing fractions with like numerators, 15/40 and 15/60. I know that fortieths are bigger than sixtieths, so I'd much rather have 15/40, which is the same as 3/8.

Finally, what if none of the first four strategies work and there is no multiplicative relationship between either the numerator or denominator? Think, for example, of comparing 3/7 and 4/9. Unlike numerators, unlike denominators, neither is a unit fraction from the whole, both less than one half, no multiplicative relationship between the numerator or denominator. So do I have to find the least common denominator (LCD) in this case? Or, couldn't I change them both to have a common numerator? When you truly understand fractions, the options are limitless!
Now that your students know all the strategies and have had practice with each, they'll need practice deciding which strategy is most efficient for a given pair of fractions. I created a set of cards with pairs of fractions that lend themselves to each strategy, and they'd make a great workstation. You don't need a fancy recording sheet to use these cards. Have students choose a card and, in their math journal, record the correct comparison and explain which strategy they used to make the comparison. Remember, it's the explanation that you're really interested in to determine how well your students understand fractions. You can download the cards here.

One thing I feel the need to mention as I close this series. For your students to develop fraction sense, it's important that you have a conceptual understanding of fractions. Many of us were not taught to deeply understand fractions. We learned procedures and tricks that did not help develop a conceptual understanding. It is our responsibility to pass on something better than that to our students. If you are still using tricks in your classroom, set a personal goal to develop yourself in this area.

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Comparing Fractions: Find a Common Numerator or Denominator - Math Coach's Corner

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"Learning to compare and order fractions is important to developing an understanding of fractions." Marilyn Burns ( About Teaching Mathematics, 2015, p 430) "Helping students develop their 'fraction sense' is extremely important before they begin computing on and with fractions." Julie McNamara and Meghan M.

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The idea of fractions being equivalent is such a key understanding for helping students develop "fraction sense" that it absolutely must be done right. Students need to see a variety of representations of equivalent fractions to help them see the relationships between the numerators and denominators when fractions are equivalent. Beginning in 3rd grade, students should use concrete materials to experience equivalent fractions. The use of concrete materials helps students to develop images in their minds that they can draw on later when manipulatives or drawings aren't available. Another great visual representation for equivalent fractions is double number lines.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

A common practice for teaching equivalent fractions is to show students to either multiply both the numerator and denominator by the same number or divide both the numerator and denominator by the same number. Essentially, you are multiplying or dividing the fraction by 1, which doesn't change the amount of the fraction.

Without a visual representation, however, that is an incredibly abstract concept. But we teach the "trick" because it is quick and easy, even though it does nothing to further our students' understanding of fractions. Now don't get me wrong, I'm not knocking the mathematical way of finding equivalent fractions. I just think it should be the end result of LOTS of concrete and pictorial practice.

As students begin using fraction tiles to explore equivalent fractions, they notice that it takes, for example, four twelfths to make a third, because the twelfths are smaller pieces. There is a multiplicative relationship-for every third, four-twelfths are required.

Students can then transfer that knowledge to a pair of double number lines showing the equivalent fractions. The understanding of the multiplicative relationship actually even helped us construct the number lines as I recently worked with a group of 4th graders to compare numbers with different denominators. Consider the following picture as you read through the conversation we had while drawing the number lines you see pictured. You see my drawing, which was projected on a document camera, and the students were showing the same work in their own math journals.

Me: (writing 7/12 and 2/3 in my math journal) So, we are comparing 7/12 and 2/3. What relationship do you notice with the denominators here?
Student: Times 4.
Me: Explain.
Student: 3 times 4 equals 12.
Me: (drawing the arch between the denominators and labeling it) Oh! I saw it as dividing by 4, because 12 divided by 4 equals 3. I guess that's the same thing, right?
Students: Yes!
Me: (drawing two number lines and labeling them with 0 and 1) Okay, let's draw two number lines and see if that will help us decide on a way to compare these two fractions. Remember, you have to carefully line up your 0 and 1 on the number lines to make sure the wholes are the same size.
Me: (after both number lines are drawn) Let's show 2/3 on the top number line. I picked that one because I think it's the easiest one to draw, right?
Students: (laughing) Yes!
Me: (dividing the number line into thirds and labeling the parts) Okay, I know it won't be perfect, but let's try to make our three parts as equal as possible. By the way, how many lines will I have to draw to make three parts?
Student: Two!
Me: That's right. Okay, I think they look pretty good, don't you? Now, on our bottom number line, we need to show twelfths. That's a little trickier. Let's think about that times, or divided by, relationship. Right here at one third (drawing a line on the bottom number line), how many twelfths will I need?
Student: Hmmmm, four?
Me: (chuckling) You don't sound too sure. Why do you think four?
Student: Because it's times 4. It takes four twelfths to make one third.
Me: Sounds good to me. Let's label it 4/12. Now how many parts will we need between 0 and 4/12?
Students: Four.
Me: So how many lines will we need to draw?
Students: Three.
Me: (dividing the space between 0 and 4/12 into four equal parts) Let's do it.
Me: Okay. Looks pretty good. (drawing a line right below 2/3 on the bottom number line) Hmmm, what fraction would go right here?
Students: 8/12!
Me: Why?
Student: Because it will take another four twelfths.
Me: Makes sense to me. Can you guys draw and label the parts?
Students: Yep!
Me: Okay, can we finish out this number line to show all the twelfths?
Students: Yes.
Me: (after completing both number lines) Okay, so now we see the relationship between thirds and twelfths. Let's get back to our comparison. We were trying to compare 7/12 and 2/3. Can our number lines help with that?
Student: 2/3 is the same as 8/12, so it's greater than 7/12.
Me: Let's write that down and be sure it makes sense. (writing 7/12 under 7/12 and 8/12 under 2/3) So you're saying that 2/3 is equivalent to 8/12, and 8/12 is greater than 7/12?
Student: Yes.
Me: Give me a thumbs up if you agree with that.
Students: (all thumbs up)
Me: Think we can do another one?
Students: Yes!

Okay, I know that was really long, and I thank you for sticking with it. But I just thought it was important for you to "hear" the conversation we had while drawing the number lines.

Remember that finding common denominators is only one of the strategies for comparing fractions. Be sure to check out this series of blog posts for information on all of the other methods. There are several freebies sprinkled throughout the four posts.

I'd love to hear your comments on this lesson!

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Drawing Number Lines to Visualize Equivalent Fractions - Math Coach's Corner

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The idea of fractions being equivalent is such a key understanding for helping students develop "fraction sense" that it absolutely must be done right. Students need to see a variety of representations of equivalent fractions to help them see the relationships between the numerators and denominators when fractions are equivalent.

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Steve Wyborney\\\'s Blog: I\\\'m on a Learning Mission.

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Fraction Splats! have arrived! The original Splat! post rocketed up to over 35,000 hits in less than a month, which included over 19,000 lesson downloads! All 50 lessons in the Splat! post involved whole numbers. In this post, I've turned my attention to fractions, and have created 20 new lessons for you to download! This is The Fraction Splat! Series. Watch the video, download the lessons, and try them in your classroom!
Note: I've created the lessons in PowerPoint, so I recommend using PowerPoint to view them.
After reading this blog post, take a look at THE 12 MOST POPULAR MATH STRATEGIES AND DOWNLOADS ON THIS BLOG.
Note: These lessons use a single splat along with a mixture of unit fractions and whole numbers. In Level 11, you control the Splat! SPLAT 11.1 - Fraction Splats SPLAT 11.2 - Fraction Splats SPLAT 11.3 - Fraction Splats SPLAT 11.4 - Fraction Splats SPLAT 11.5 - Fraction Splats
Instant Fraction Splat!
Note: These lessons use a single Splat! which appears at the same moment as the whole numbers and fractions. Each question instantly presents missing information. SPLAT 12.1 - Instant Fraction Splats SPLAT 12.2 - Instant Fraction Splats SPLAT 12.3 - Instant Fraction Splats SPLAT 12.4 - Instant Fraction Splats SPLAT 12.5 - Instant Fraction Splats
Multiple Splats! with Fractions
Note: These lessons include multiple Splats! and you control the timing of those Splats! The entire screen can be seen, the total can be determined, and then Splat! Remember: Each Splat! that is the same color must be covering the same number. SPLAT 13.1 - Multiple Fraction Splats SPLAT 13.2 - Multiple Fraction Splats SPLAT 13.3 - Multiple Fraction Splats SPLAT 13.4 - Multiple Fraction Splats SPLAT 13.5 - Multiple Fraction Splats
Multiple Splats! with Fractions
Note: These lessons use multiple Splats! which appear at the same moment as the whole numbers and fractions. Each question instantly presents missing information. Remember: Each Splat! that is the same color must be covering the same number. SPLAT 14.1 - Instant Multiple Fraction Splats SPLAT 14.2 - Instant Multiple Fraction Splats SPLAT 14.3 - Instant Multiple Fraction Splats SPLAT 14.4 - Instant Multiple Fraction Splats SPLAT 14.5 - Instant Multiple Fraction Splats
Other Posts and Resources Which May Be of Interest

Splat! The original 50 downloadable Splat! lessons include 10 levels and variations.

Tiled Area Questions is a colorful, powerful, highly visual strategy for decomposing numbers - and offers many downloadable pages.

The Animated Multiplication Table remains one of the most popular downloaded resources from this blog, and this post includes several ideas about how to use it.

The Maze Hundreds Chart is another downloadable resource that can lead to some highly intriguing questions and rich discourse.

You may also be interested in Provide Massive Space to Notice.

Please take a moment to subscribe to the blog so you'll be notified each time new resources are posted!

I would treasure hearing about your experiences using The Fraction Splat! Series to promote number sense in your classroom!

All my best,
Steve

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The Fraction Splat! Series - Steve Wyborney's Blog: I'm on a Learning Mission.

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Fraction Splats! have arrived! The original Splat! post rocketed up to over 35,000 hits in less than a month, which included over 19,000 lesson downloads!

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Students often struggle with the concept of fractions. This interactive fraction strips tool helps students visualize fractions. With these online fraction strips, students can model fractions, solve fraction problems, explore equivalent fractions, and observe the differences between denominators.

To use the interactive fraction strips tool, click and drag the strips onto the canvas. You can drag as many of each fraction as you'd like. To remove the strips, drag them off the canvas. Change the chart to accommodate two whole units by pressing the "Unit Double" button. Add annotations using the pencil tool and erase using the eraser. To clear the canvas, press the trash can button.

A few activities you can do with students while using the online fraction strip tool include:

Have students model a given fraction, mixed or not. For example 1 1/4 or 3/6.

Ask students to model addition, subtraction, multiplication, or division problems using the fraction strips.

Ask students to find equivalent fractions for a certain fraction. For example, how many sixths are equivalent to 1/3?

Ask students to use the pencil tool to divide one of the wholes on the chart into fourths. Then, students can check their work by dragging fourths onto the chart afterward.

Looking for more fraction manipulatives? Try our interactive fraction circles.

Visit classplayground.com for more resources and printables about fraction strips.

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Fraction Strips | Free Virtual Manipulatives | Toy Theater

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Use the interactive fraction strips tool to help students visualize, model, and explore fractions. Students can also solve problems using fraction strips.

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This applet allows you to create equivalent fractions by dividing and shading squares or circles, and match each fraction to its location on the number line.

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Dividing Fractions by Fractions using Models - 3 examples

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Fraction Four (equivalent fractions) is a fun simple game to practice equivalent fractions with visual help. The players roll dice or spin a spinner to get a fraction. Then they find an equivalent of their fraction on the board. The first player to make four in a row wins.

The game is available in print and google slides version.
Print version
Material

Print out of the game board (Two game boards are given one the basic and one for the challenge game. The challenge game has a bigger table of fractions)

Two dice 6 with sides. or the spinner.

A color pencil for each player (different colors)

The cheat sheet to help the players verify their answers.

You can print and use the spinner below instead if dice. Paste the spinner or card stock and use a paper clip or a plastic spinner. Here is a video of how you can make a spinner with a wire , or a paper clip. You can also use the digital one from the google slides version. Download the spinner with the game boards at the end of the post.

The players take turns rolling the two dice. Every time they use the two numbers from the roll to form a fraction. They use the smaller number as the numerator and the greater number as the denominator. (so that the fractions will not be improper) They look for the equivalent to their fraction on the table/board. They circle the fraction on the board or mark it with an X using their color (or place a cover/ counters). If all the equivalents of that fraction are already taken then the player waits for their next turn to roll.

The player that gets four fractions in a row first (horizontally, vertically or diagonally) is the winner.

The players try to block each other from getting four like in a typical four in a row game.

This game offers good practice and helps students see patterns, find relationships, and understand equivalent fractions better.

It is a good idea to laminate this game board so that you can use it multiple times. Use different color marker for each player.

The players can use the cheat sheets to verify their answers. The cheat sheets are a useful resource for teaching equivalent fractions and can be used in many ways.

Here are the files to print.
Google slides Version
Here is the link to the free google slides version.
Check out our fractions and decimals activity cards Fractions Activity Cards - equivalent fractions, complete one (+google slides version)

Decimals, Decimal fractions, and Percentages - Visual Activity cards (google slides version included)

Representing Decimal Numbers using base 10 blocks. (Printable and Digital Activity Cards)

and our fraction card games Fractions Card Games

more posts and games for fractions.
Fraction Four (make one) Learning fractions Fraction Escape (equivalent fractions) Fraction Escape (make a whole)
Explore the rest of our free print to play games, task cards, worksheets, puzzles and more.
Check out our best selling games, now back in stock and available at amazon.com and amazon.ca

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Fraction Four (equivalent fractions) is a fun simple game to practice equivalent fractions with visual help. The players roll dice or spin a spinner to get a fraction. Then they find an equivalent of their fraction on the board. The first player to make four in a row wins.

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The division of fractions is often a challenging concept for students. Providing visual explanations and activities that include visuals helps them grasp the meaning of the division of fractions and gain a deep understanding. Keep in mind that before starting with the division of fractions students must have a concrete understanding of fractions as part of a whole, improper fractions and mixed numbers, equivalent fractions, like and unlike fractions, addition, and subtraction of like and unlike fractions.

When dividing fractions as well as when dividing whole numbers we use two techniques/interpetations. The measurement and the partitive technique.

In the measurement technique, we try to see how many times the divisor fits into the dividend. For example for the question 18÷6 we will think "How many times 6 fits into 18, or how many groups of 6 fit into 18. We can add or take away a group of 6 at a time.

In the partitive technique, we solve for the one. For example, for the question 18÷6 we will think "If 18 is divided into 6 groups, how much will be in one group.

Similarly, when dividing fractions we can use these two techniques.

Here is an example of the measurement technique of dividing fractions. With this method both fractions are shown as part of the same whole.

Problem: I have 4/8 of a pizza left. If my friends and I take 1/8 each, how many friends can have pizza?

Different denominators
Students learn that to add or subtract fractions with different denominators they need to find the LCM and turn them into like fractions (with the same denominators) so that they show part of the same whole. Following the same method when dividing fractions with the measurement method helps them see both fractions as part of the same whole and figure out how many times the divisor fits in the dividend. Example below

Partitive technique
Below is an example of the partitive technique of dividing fractions. With this method, I only show the first fraction and divide it based on the divisor.

Problem: 1/4 of a pound of blueberries costs 2/3 of a dollar. How many blueberries can I buy with 1$?

1/4 pound of blueberries costs 2/3 of a dollar. The shape shows the 1/4 pound of blueberries divided by 2 since it costs 2/3 of a dollar. This means that each of the two parts of the 1/4 pound of blueberries costs 1/3 a dollar. Therefore, with 1$ I can buy 3 parts. 3/8

The partitive technique is responsible for the algorithm that we all use. To divide a fraction by another fraction, multiply the first one by the reciprocal of the second one.

The partitive technique can explain why the algorithm works. 1/4 was multiplied by 1/2 (when we divide it into two pieces) and it became 1/8. Then multiplied by 3 it became 3/8
1/4÷2/3 = 1/4÷3/2 = 3/8
An algorithm is a series of steps that we follow to get to the answer. However, when these steps are followed blindly and don't have any meaning for the students they tend to get confused and make mistakes. Mistakes such as "flipping" the first fraction or forget to "flip" the second fraction. Explaining visually why this algorithm works will help students understand and apply the algorithm correctly.
Dividing whole numbers by fractions and fractions by whole numbers visually also demonstrates how or why the algorithm works.
Below are some examples.
Dividing whole numbers by fractions.

One whole has 6 pieces of 1/6, therefore, 4 wholes will have 4×6=24 pieces of 1/6. 4 is multiplied by the reciprocal of 1/6 which is 6. 4 can be shown as an improper fraction 4/1. 4/1×6/1=24 We multiply the first fraction (in this case 4) by the denominator of the second and divide by the numerator. We are actually multiplying with the reciprocal of 1/6.

Here is another example

3 wholes have 3×3 1/3 so we have a total of 9/3. I am dividing by 2/3 which means that I need to put the thirds in groups of 2. 9/2=4 1/2 It is the same as multiplying 3×3/2 which is the reciprocal of 2/3.
Dividing fractions by whole numbers.

Dividing 1/4 by 3 is the same as finding 1/3 of 1/4 or multiplying 1/4×1/3 1/3 is the reciprocal of 3.

Once students solve many different problems, they will understand why and how the algorithm works and they will not make mistakes while applying it. Most importantly they will gain a deep understanding of the division of fractions.

I have created some activity cards (print and digital) that you can use to practice division of fractions. The activities are organized in the following order to help students gradually gain understanding of the algorithm.

Dividing whole numbers by fractions

Dividing fractions by whole numbe

Dividing unlike fractions

Reciprocals and the algorithm

I focused on providing visuals and word problems for deeper understanding. I hope you find them useful.
As always here is a free print and google slides version.
f ree google slides version

Find the premium version below

The premium version includes 52 print activity cards and 58 google slides to help practice division of fractions.

Dividing whole numbers by fractions

Dividing fractions by whole numbers

Dividing fractions with like denominators

Dividing fractions with different denominators

Reciprocals and the algorithm

Solving word problems

Check out more of our fractions activities
Fractions Activity Cards - equivalent fractions, complete one (+google slides version)

Fractions Card Games

Decimals, Decimal fractions, and Percentages - Visual Activity cards (google slides version included)

Check out these free fun games to practice fractions (complete 1 and equivalent fractions)
Fraction Four (equivalent Fractions) Fraction Four (Make one) Fraction Escape game -complete one whole, equivalent fractions (+google slides version) Fractions Escape (equivalent fractions)
Explore the rest of our free print to play games, task cards, worksheets, puzzles and more.
Check out our best selling games available at amazon.com and amazon.ca

Thanks for visiting.

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Dividing Fractions -Understanding the algorithm (Print and Digital Activities)

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The division of fractions is often a challenging concept for students. Providing visual explanations and activities that include visuals helps them grasp the meaning of the division of fractions and gain a deep understanding. Keep in mind that before starting with the division of fractions students

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Compare Fractions Gameboard (page 5)

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Both educators and parents agree that fractions and their operations are one of the most difficult concepts for students to grasp. What makes fractions so challenging for students?

The introduction of the concept of fractions comes with a new vocabulary, new rules, new symbols, and the new concept of values between zero and 1. All this new information is hard to handle. However, when students are presented with visual simulations, activities, and examples from their everyday life they realize that fractions are all around them and they had been using them before learning about them. The vocabulary, the new symbols, and the rules become meaningful through hands-on activities and discussions.

There are a lot of rules concerning fractions and their operations and when students learn fractions as a series of rules they are much more likely to lose interest due to a lack of understanding. This is why is very important to help students gain a concrete understanding of the concept of fractions early on and avoid misconceptions. If misconceptions are not corrected in the beginning they make it difficult for students to understand more advanced concepts such as comparing fractions and operations of fractions.

Below are some common mistakes/misconceptions that students make. Being aware of these possible misconceptions helps us create activities that will help students gain a concrete and clear understanding of fractions.

Introducing Fractions-Parts and Wholes

It is important to help students construct the idea of fractional parts of the whole (for the area model of a fraction) and realize that a fraction is a part of a whole divided into equal parts. Students need to practice partitioning a variety of shapes into an equal number of parts. There is often a misconception that counting the number of parts of the shape instantly gives them the denominator.

For example, the shapes below are not all equally partitioned so the colored part of all them cannot be called 1/5. Only the first shape shows 1/5.

When it comes to number lines, the same misconception appears as many students often do not correctly partition the linear space relative to the fraction being shown.

Showing many right and wrong examples as well as encouraging the students to partition the shapes themselves will help clear this misconception.

Often students do not recognize a fraction when fractional parts are separated by space. Presenting examples and providing different kind of manilulatives to create different types of groups is recommended.

Moreover, students often need guidance to understand that the fractional part always depends on the whole. When students practice fractions with circles/discs they get used to the third blue shape below representing 1/4 and in their minds that particular shape becomes 1/4. However if the first red shape becomes the whole then the same shape that represented 1/4 now represents 1/2.

When working with tools such as cuisenaire rods try using different sizes of rods as the whole. Fraction circles that don't have the fraction marked on them can also be used to clear this misconception.

Students often do not recognize a fraction when the fractional parts are not all the same shape. For example the colored part of the shape below is not recognised as 1/4. Showing more examples of area models of fractions where the parts are not all the same shape will help students understand.

Students might misidentify fractions of a set due to confusion with the area model as they expect all objects in a group to be the same. For example, in the group below students find it difficult to accept that the jelly bean is 1/4 of the candy because it is different.

Here are some ides of activities you can do when introducing fractions.

Ask students to show fractions using lego bricks.

Use coins to show parts of 1 dollar.

Cut fruits for a fruit salad.

study a simple recipe and measure the ingredients.

use play dough to create objects that you can "cut" into different parts.

Use cuisenaire rods to show parts of different wholes.

Make paper plate fractions.

We have created some print and digital activity cards that will help students practice fractions (level1). There is a small sample below.

This resource includes 77 print and 80 digital activity cards to practice

recognizing fractions as part of a shape

recognizing fractions as part of a set

partitioning into equal parts, groups

Comparing parts to the whole

Complete 1

word form

Introducing addition and subtraction of proper like fractions

Introducing equivalence

As always here is the free print and digital version.
Free google-slides version The premium version includes 77 print activity cards and 80 google slides activity cards. How to use the activity cards.
You can use the activity cards as exploration and discussion, warm-ups, practice in groups, pairs or individual practice, homework, or assessment. The print cards can be used with manipulatives like rods, tiles, or discs. Provide objects for the students to use when working with fractions of groups.

Check out more of our fractions activity cards.
Fractions Activity Cards - equivalent fractions, complete one (+google slides version)

Fractions Card Games

Dividing Fractions -Understanding the algorithm (Print and Digital Activities)

Check out these free fraction games. (recognizing fractions, make 1, equivalent fractions)
Fraction Four (equivalent Fractions)+google slides version Fraction Four (Make one) Fraction Escape game -complete one whole, equivalent fractions (+google slides version) Fractions Escape (equivalent fractions)
Explore the rest of our free print to play games, task cards, worksheets, puzzles, and more.
Check out our best selling games available at amazon.com and amazon.ca

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Introducing Fractions- misconceptions, print, and digital activity cards.

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Both educators and parents agree that fractions and their operations are one of the most difficult concepts for students to grasp. What makes fractions so challenging for students? The introduction of the concept of fractions comes with a new vocabulary, new rules, new symbols, and the new concep

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You can learn about:

Fractions

Fractions on a pizza and the number line

The three different types of fractions

How to simplify a fraction, and how the "Greatest Common Factor" can help
♫ "Multiplying fractions no big problem,
Top times top over bottom times bottom, "Dividing fractions, easy as pie,
Flip the second fraction, then multiply, "If adding or subtracting is your aim,
The bottom numbers must be the same! "Change the bottom using multiply or divide,
But the same to the top must be applied, "And don't forget to simplify,
Before it's time to say goodbye"

How to compare fractions, and how the "Least Common Multiple" can help

How to add, subtract, multiply and divide fractions

How to convert fractions to decimals and vice versa

How to convert fractions to percentages and vice versa

Reciprocal fractions

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♫ "Multiplying fractions no big problem, Top times top over bottom times bottom, "Dividing fractions, easy as pie, Flip the second fraction, then multiply, "If adding or subtracting is your aim, The bottom numbers must be the same!

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A couple of days ago, I blogged about equivalent fractions, and I promised a follow-up post about fractions on the number line.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

We've been using lots of different representations for fractions, including pattern blocks, Cuisenaire rods, and fraction tiles to explore fractions. To introduce fractions on a number line, I wanted to use a familiar manipulative, Cuisenaire rods, to help the students make a connection between fractions as parts of a whole and fractions on a number line. I wanted to introduce an element of problem-solving, so I didn't want to directly teach the concept. Instead, I wanted to see if the students could use what they already knew about fractions to develop their own understanding of fractions on a number line.

Students were given the reproducible shown below and a set of rods to work with. First, I asked them to talk with another mathematician about the first number line and to make some observations.

Some students noticed that the number line showed 0, 1, and 2, but that A was between 0 and 1. Still others noted that the A was in the middle between 0 and 1. One student even commented that the A split the space between 0 and 1 into equal parts [smile!]. I referred the class to the number line on the wall and explained that the number line we were looking at was just a small part of the big number line-just the space between 0 and 2. I then gave them a few minutes to work with a partner, using the Cuisenaire rods, to see if they could identify the number represented by the letter A. It was really fun to see them work and hear the great math conversations! It actually wasn't too long before a couple of partnerships came up with one half. That's all you need!! I selected one pair of students to come up to the interactive whiteboard and use the Notebook file, which has the same number lines as the reproducible, and the infinite clone Cuisenaire rods to show how they arrived at their solution. I love it when the kids can teach a lesson instead of me.

We went through the same process with each of the other number lines. First students worked in pairs to try to solve the problem and then a pair of students were asked to present. The kids definitely had a much more difficult time with the second number line than the first, but that's called problem-solving, right?

This is a screenshot of the Notebook file and how we showed our work for the second number line.

Grab your copy of the Notebook file here and the reproducible here. Comments?

Do your students need more practice? Check out my Fractions on a Number Line Scoot game.

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Fractions on a Number Line with Cuisenaire Rods - Math Coach's Corner

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A couple of days ago, I blogged about equivalent fractions, and I promised a follow-up post about fractions on the number line. This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission.

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Generating Equivalent Fractions: Understanding the Process - Math Coach's Corner

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Generating Equivalent Fractions: Understanding the Process - Math Coach's Corner

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Help students understand the process for generating equivalent fractions before introducing the algorithm. FREE download!

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Composing and Decomposing Fractions - Math Coach's Corner

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Composing and Decomposing Fractions - Math Coach's Corner

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Part/whole thinking applies to fractions as well as whole numbers. Grab a FREE resource for composing and decomposing fractions!

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Visualizing Fractions {EQUIVALENT FRACTIONS FREEBIE} - Math Coach's Corner

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Visualizing Fractions {EQUIVALENT FRACTIONS FREEBIE} - Math Coach's Corner

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Visualizing fractions is easy when students have lots of hands-on and pictorial experiences early on. Grab a FREE resource!

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How to Teach Multiplying Fractions with Pattern Blocks

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How to Teach Multiplying Fractions with Pattern Blocks

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Planning a Hands-On Fractions as Division Lesson | 5.NF.3

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Planning a Hands-On Fractions as Division Lesson | 5.NF.3

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Comparing Fractions: Like Numerators or Denominators - Math Coach's Corner

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Comparing Fractions: Like Numerators or Denominators - Math Coach's Corner

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Students should have multiple strategies for comparing fractions based on number sense. A series of articles with FREE downloads!

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As elementary teachers, we rarely have the opportunity to explore division of a fraction by a fraction. When we do, it's normally accompanied with Keep-Change-Flip or the saying "Yours is not the reason why, just invert and multiply."

Both are conceptual cripplers.

I've been drafting the 4th installment of the Making Sense Series involving fractions and I'm sharing this post as more of a personal reference should K-C-F make its way round these parts again...and I'm sure it will.
Side note: A while back Fawn and Christopher each shared a post about division of fractions using common denominators. Both posts left lots of math residue and are well worth your time.
Let's start with a model for 3/5 ÷ 1/4.

At some point along the way it becomes inefficient for students to draw models once the conceptual understanding is established. As students represent measurement division of fractions they should be formally recording their thinking.

From here students generate their own algorithm (shortcut). They begin to recognize that they will always get a denominator of "1 whole" so they begin to purposefully leave it out. In doing so, they become more efficient in the procedure of dividing fractions.

Some students begin to eliminate the green and red steps from the above equation because they're seen as repetitive. We've even had one student that "invented" and generalized cross multiplication for division of fractions as they searched for ways to record fewer numbers and symbols.

It looked something like this...

I keep reminding myself that if fractions are the gatekeeper to algebraic reasoning then I need to slow the process down and conceptually understand what's happening. This includes K-C-F.

As students understand the power of creating a whole number denominator they begin to search for more efficient ways to get 1 whole. It's here that they being to explore with equivalent fraction and the idea of using the reciprocal to create a whole.

What I've realized is that aside from complex fractions, the underpinnings of this equation are developed in the elementary grades.

5.NF.3 Interpret a fraction as division of the numerator by the denominator (a÷b = a/b).

4.NF.1-Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b).

3.NF.3.c- Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Just like before, students will look to generalize the equation above and find shortcuts. They'll do so by eliminating repetitive steps which would leave them with...

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As elementary teachers, we rarely have the opportunity to explore division of a fraction by a fraction. When we do, it's normally accompanied with Keep-Change-Flip or the saying "Yours is not the reason why, just invert and multiply." Both are conceptual cripplers.

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Developing Fraction Sense - Math Coach's Corner

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Developing Fraction Sense - Math Coach's Corner

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Students often lack "fraction sense", or a deep understanding of fractions. Hands-on experiences with a variety of materials can help.

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Unit Overview On the first day of instruction, the teacher will show two videos. The Khan Academy (2013) video will introduce the students to the concept of fractions in general and will allow the teacher to introduce Khan Academy (2018) which will be integrated throughout the unit, via a

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Fractions with Unequal Partitions - Math Coach's Corner

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Fractions with Unequal Partitions - Math Coach's Corner

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Do your students understand the basic idea that fractions must have EQUAL parts? Check out this lesson for building understanding.

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On a Roll: Adding Fractions - Math Coach’s Corner

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Help students understand adding fractions by using hands-on and pictorial experiences. Download a free game for extra practice!

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Teaching Math Without Worksheets - Math Coach's Corner

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Worksheets have long been a staple in mathematics classrooms. But should they be? Read on to explore how teaching math without worksheets can lead to more engaging and meaningful lessons and actually reduce your prep time.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

For as long as I can remember, worksheets have been used for math practice. I think that stems from the fact that textbooks used to the be primary resource for teaching math. Worksheets are a logical transition from practice pages in a textbook. But recently, I've seen worksheets being used during both whole-class and small-group instruction. Observing a recent small-group lesson, the teacher was spending more time explaining the mechanics of the worksheet to students than the math she was trying to teach. And something else that worried me is that there were no manipulatives being used. The concrete, representational, abstract sequence of instruction is a framework proven to increase student understanding. A reliance on worksheets usually means a lack of manipulatives, resulting in instruction that is less impactful.

A word I've been thinking about a lot lately is intentional. With so many standards to teach, and more students displaying serious gaps due to missed learning, we have to be intentional about every lesson and every resource. Is that worksheet really the best way to teach students a concept? Is it the best way to provide meaningful practice? Read on to see how to use one resource for whole-class, small-group, and practice.

To demonstrate how we can teach without worksheets, I'm going to use a set of cards showing fractions represented in three ways-pictorial, word form, and fraction notation.

The 48 cards are designed for use with 3rd-grade students. Third grade is a foundational year for fractions, which is an incredibly abstract concept, and it's important that nearly all instruction includes manipulatives or pictures. That's actually even written into the standards! The 48 cards can be printed, which is ideal for small-group instruction and workstation games, but they are also provided as Google Slides, which works well for whole-class instruction.

SMALL-GROUP OR WHOLE-CLASS INSTRUCTION

The following mini-lessons can be used for either whole-class or small-group instruction. Consider having students work with a partner for both whole-class and small-group instruction. This fosters mathematical conversations and adds a social component to learning. It makes learning more student-centered, rather than teacher-centered. From a practical standpoint, it also means you only need half the manipulatives and materials.

If using for whole-class instruction, you probably want to use the Google Slides images so they are big enough for the whole group to see. At your small group table, you'd likely use the printed cards. You have a little more flexibility using the cards at the small group table, because you can show more than one card at a time, so you can use them, for example, to compare fractions.

Identify the fraction shown. Show a picture card/image. Students write the fraction notation for the fraction shown on their individual whiteboards.

Show using a different representation. There are two different versions of this lesson.

1. Representation is the teacher's choice. This activity is designed to focus on a single representation as they are learning about each different model (bar, set, number line). Show a word or fraction notation card and students draw a model of your choosing on their individual whiteboards.

2. Representation is the student's choice. This activity is used after students have been introduced to all the models to show a fraction using multiple models. Show a card/image. At this point, you could use any of the cards-picture, word form, or fraction notation. Using individual student whiteboards, students show the fraction using a different representation. For example, if a picture card/image of 3/4 is shown, students can represent the fraction in one of these three ways.

Generate equivalent fractions. Show a card/image. Students work in pairs with manipulatives to build an equivalent fraction. Fraction tiles would be a great manipulative for this. First students build the fraction shown on the card, then they use their tiles to build an equivalent fraction. Note here that students are likely to come up with different equivalent fractions. For example, there could be many fractions equivalent to 2/4. Students share out the equivalent fractions they built. Have students share what strategy they used to determine an equivalent fraction and what patterns they see among the equivalent fractions.

Generate a fraction greater than or less than the fraction displayed. Show a card/image. Students work in pairs with manipulatives to build a fraction that is either less than or greater than (whichever you tell them) the fraction displayed. Again, they build the fraction displayed first and then the one that is greater/less. Like the first activity, this one is very open-ended and many different fractions could be generated. Have students share their strategies and reasoning.

Compare two fractions. This works best at your small group table because you need to show two cards. Start with the pictorial cards. Show students two cards and ask them to talk with a partner about which fraction is greater and why. This sets the expectation for the conversations you expect them to have when playing War in a workstation. It also provides you with valuable assessment data. Have students write the inequality on their student whiteboards in two ways. For example 2/3 > 3/6 and 3/6 < 2/3. As students' understanding of fractions develops, you can repeat this activity using a mix of the three types of cards-pictorial, word form, and symbolic form.

WORKSTATION SUGGESTIONS

Note that the best way to introduce these activities is to use them at your small group table. It's much more effective than just explaining or modeling a game before putting them in your workstations. At the small group table, you can closely monitor students to make sure they understand the game/task, you can set your expectations for using manipulatives and for student conversations. After you've used the game/activity at your small group table, move it into your workstations to provide students with practice for each concept.

Fraction Representation Memory-From the full set of 48 cards, create decks of 12 cards made up of 6 pictorial cards and 6 matching word or symbolic cards. Here are two examples of matching pairs of cards.

You would need 6 pairs of cards like this for a total of 12 cards.

Lay the cards face down in a 3-by-4 array. Players take turns turning over two cards. If the cards show/name the same fraction, the player takes the two cards. If not, the player turns them back face down in their original place. The player with the most cards at the end of the game wins.

Equivalent Fraction Memory-From the full set of 48 cards, create decks of 12 cards made up of equivalent fraction pairs (e.g., 3/4 and 6/8 ). Initially, you will want to use the pictorial cards. As students become more confident with their skills, you can mix in the word and symbolic cards. Play as outlined in the previous activity.

What Was Eaten?-Students draw a picture card, which shows the part of the pizza that's left on the pan. In their math journal, they write sentences describing the part of the pizza that is left and the part eaten.

_____ of the pizza is left. _____ of the pizza was eaten. _____ equals one whole.

Example: 1/4 of the pizza is left. 3/4 of the pizza was eaten. 4/4 equals one whole.

Create Another Representation-Put the cards in a workstation. Students choose a card, sketch the fraction using a different representation in their math journal, and write the fraction name. Examples of different representations include a bar model, set model, or number line.

See the example for in the whole class/small group notes.

Fraction War-This game can be played by 2, 3, or 4 players. Initially, use just the picture cards. Over time, as students' understanding of fractions develops, mix in the word and symbolic cards. Copy the cards on cardstock, laminate (optional), and cut apart into a deck of cards. If you are only using the picture cards, you might want to make two copies of the two pictorial sheets, so students have a deck of 36 cards. Players deal out the cards equally. Each player turns over a card. All players name their fractions. The player with the greatest fraction takes the cards.

If players have equivalent fractions, each player turns over another card and the player with the greatest fraction takes the cards.

As I hope you can see, one set of cards with multiple uses can replace a stack of worksheets and provide more meaningful learning experiences.

For a similar discussion using a 1st Grade example, check out this post.

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Teaching Math Without Worksheets - Math Coach's Corner

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Worksheets have long been a staple in mathematics classrooms. But should they be? Read on to explore how teaching math without worksheets can lead to more engaging and meaningful lessons and actually reduce your prep time. This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission.

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Discover the world's best virtual manipulatives and digital tools for learning mathematics! Be creative, explore new ideas, discover patterns and collaborate with others! It is impossible to balance these weights. This is not a valid Polyhedron net. Sorry, this Polypad could not be loaded.

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Developing Fraction Number Sense Through Part/Whole Thinking - Math Coach's Corner

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Developing Fraction Number Sense Through Part/Whole Thinking - Math Coach's Corner

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Students need lots of concrete and pictorial experiences to create a visual understanding of fractions and build fraction number sense.

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Common misconception 1: not understanding the denominator

A common misconception is that learners believe the numerator and denominator are the same. Let's start with the fractions basics to help address this misconception.

'Denominator' means 'that which names' in Latin. This translation identifies the denominator as a name the same way 'one', 'two' and 'three' are names.

The denominator can also tell you the size of the parts. Children learn that:

shows us that each part is called a half and two of them will make one whole.
shows us that each part is called a third and three of them will make one whole.
shows us that each part is called a quarter and four of them will make one whole.

And so on.

Children need to use physical materials to discover this for themselves, like in the following example from Maths - No Problem! Textbook 1B, Chapter 14, Lesson 1.
How do we know that Charles' part is the same as Holly's part?
We could put Charles' piece on Holly's piece to see if they are the same size.
How many parts are needed to make one whole piece of paper?
It would take two parts to make one whole piece of paper.

Children need to discover this using physical materials first and develop the language associated with the idea.

For example take Maths - No Problem! Textbook 1B, Chapter 14, Lesson 2.

If we concentrate on the size of the pieces and understand that each size has a name, in this case a 'quarter', the notation becomes easier to understand.

Once learners understand this, we can find the name of the fraction.
What fraction of the shape is green?
Using the ideas that were first introduced in Year 1, this should be quite straightforward to answer.

It takes 6 green pieces to cover the yellow shape. There is one green piece.
of the shape is green. We can develop the idea further and ask, "How many green pieces would it take to cover half of the shape?".

If we concentrate on the denominator first, the understanding of the numerator becomes easier to understand. The numerator is simply showing us the number of parts.

shows us that there are three parts and it takes four parts to make one whole.

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4 common fractions misconceptions and how to address them | MNP Blog

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A common misconception is that learners believe the numerator and denominator are the same. Let's start with the fractions basics to help address this misconception. 'Denominator' means 'that which names' in Latin. This translation identifies the denominator as a name the same way 'one', 'two' and 'three' are names.

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practice, activities, and projects for middle school math

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I came across this problem a few years ago and love using it as a Warm Up. Hoo boy, does it elicit major misconceptions my middle school kids have about fractions. (Edit: These are called number searches. You can find more here!)

My 7th graders really struggled with it at first. I told them the dark blue rectangle was one whole and they just couldn't wrap their head around it. Thinking of the lime green as a whole was more accessible so we ran with that. I love that it is open-ended and kids can take it to different depths.

It's the only one I've found like it, so I've been itching to create my own. I came up with these:

I also created some more traditional fraction talk images, inspired by Nat Banting's site, fractiontalks.com. He has great tips for implementing and extending fraction talks.

I've put them all into a google slideshow for easy use. After getting some feedback, I modified them to improve their accessibility for folks with colorblindness.

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I came across this problem a few years ago and love using it as a Warm Up. Hoo boy, does it elicit major misconceptions my middle school kids have about fractions. (Edit: These are called number searches. You can find more here!) My 7th graders really struggled with it at first.

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Why do our students struggle to understand fractions? A big part of the problem is that fractions are incredibly abstract. If I were to ask you, Which is greater, 1/8 or 1/4? you would likely have an image in your head corresponding to those fractions. Unfortunately, many students see only symbols in their heads. Without the ability to visualize these unit fractions, there's a good chance they'll say 1/8 is greater than 1/4 because 8 is greater than 4. Not understanding what the denominators 8 and 4 actually represent, they are applying whole-number thinking to fractions.

So how do we put those images in their heads? The answer is actually amazingly simple-we provide them with lots of concrete and pictorial experiences so they can connect a visual to the abstract symbols. Too often we bypass the manipulatives, blaming lack of time or too much to "cover." But when we don't use manipulatives, we end up spending more time in the long run trying to fill the gaps that our students end up with.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

The foundations for fraction understanding begin in 3rd Grade, and there's a LOT packed into the 3rd Grade fraction standards. Included is an introduction to unit fractions, which are the building blocks of all fractions. A unit fraction is any fraction with a numerator of 1. It's basically one part of a whole divided into n parts. In 3rd grade, students learn that a fraction is the sum of unit fractions. For example, 3/4 = 1/4 + 1/4 + 1/4. In 4th grade, students extend that thinking to an understanding that fractions can be composed and decomposed in different ways. For example, 5/6 = 2/6 + 3/6 but also 5/6 = 4/6 + 1/6.

What better way to engage students in learning than with math games? Introduce this simple game called Cover Up at your small group table. You'll need a set of fraction tiles for each player (yourself included), or students can make fraction strips. Students will love this game, and, with well-placed probing questions, you can generate some terrific mathematical discussions. Once students are familiar with the game from playing in a small group, move it to your workstations. If students play this game enough, I'd be hard-pressed to see how they could ever say that 1/8 is greater than 1/4! Or add together the denominators when adding fractions. Or NOT see that 2/4 is equivalent to 1/2. I think you get the idea. 🙂

Check out the video for how to play and questioning techniques you can use while playing. Then scroll down for the link to download two spinners for working with different sets of fractions.

You can grab the spinners for playing Cover Up here . Enjoy!

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Unit Fraction Game: Cover Up - Math Coach's Corner

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Why do our students struggle to understand fractions? A big part of the problem is that fractions are incredibly abstract. If I were to ask you, Which is greater, 1/8 or 1/4? you would likely have an image in your head corresponding to those fractions. Unfortunately, many students see only symbols in their heads.

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Cover Up (Directions Video for Teachers)

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Cover Up (Directions Video for Teachers)

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Unit Fraction Game: Cover Up - Math Coach's Corner

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Why do our students struggle to understand fractions? A big part of the problem is that fractions are incredibly abstract. If I were to ask you, Which is greater, 1/8 or 1/4? you would likely have an image in your head corresponding to those fractions. Unfortunately, many students see only symbols in their heads. Without the ability to visualize these unit fractions, there's a good chance they'll say 1/8 is greater than 1/4 because 8 is greater than 4. Not understanding what the denominators 8 and 4 actually represent, they are applying whole-number thinking to fractions.

So how do we put those images in their heads? The answer is actually amazingly simple-we provide them with lots of concrete and pictorial experiences so they can connect a visual to the abstract symbols. Too often we bypass the manipulatives, blaming lack of time or too much to "cover." But when we don't use manipulatives, we end up spending more time in the long run trying to fill the gaps that our students end up with.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

The foundations for fraction understanding begin in 3rd Grade, and there's a LOT packed into the 3rd Grade fraction standards. Included is an introduction to unit fractions, which are the building blocks of all fractions. A unit fraction is any fraction with a numerator of 1. It's basically one part of a whole divided into n parts. In 3rd grade, students learn that a fraction is the sum of unit fractions. For example, 3/4 = 1/4 + 1/4 + 1/4. In 4th grade, students extend that thinking to an understanding that fractions can be composed and decomposed in different ways. For example, 5/6 = 2/6 + 3/6 but also 5/6 = 4/6 + 1/6.

What better way to engage students in learning than with math games? Introduce this simple game called Cover Up at your small group table. You'll need a set of fraction tiles for each player (yourself included), or students can make fraction strips. Students will love this game, and, with well-placed probing questions, you can generate some terrific mathematical discussions. Once students are familiar with the game from playing in a small group, move it to your workstations. If students play this game enough, I'd be hard-pressed to see how they could ever say that 1/8 is greater than 1/4! Or add together the denominators when adding fractions. Or NOT see that 2/4 is equivalent to 1/2. I think you get the idea. 🙂

Check out the video for how to play and questioning techniques you can use while playing. Then scroll down for the link to download two spinners for working with different sets of fractions.

You can grab the spinners for playing Cover Up here . Enjoy!

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Unit Fraction Game: Cover Up - Math Coach's Corner

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Why do our students struggle to understand fractions? A big part of the problem is that fractions are incredibly abstract. If I were to ask you, Which is greater, 1/8 or 1/4? you would likely have an image in your head corresponding to those fractions. Unfortunately, many students see only symbols in their heads.

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How to Multiply Fractions with Whole Numbers in 3 Steps — Mashup Math

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Ready to learn how to multiply fractions with whole numbers? This free step-by-step guide will teach you everything you need to know about multiplying fractions with whole numbers. We will share an easy 3-step method for multiplying fractions with whole numbers, and we work through four different pr