# Comparing Fractions: Find a Common Numerator or Denominator

“Learning to compare and order fractions is important to developing an understanding of fractions.”ย 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.”ย 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 goย back to the first blog post and read theย other three posts in theย series, which describe the first four strategies. You’ll also find activities you can download and use in your classroom.

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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 whichย 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 greaterย 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.ย 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?ย 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 Iย have to find the least common denominator (LCD) in this case? Or, couldn’t I change themย bothย 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 thatย you have a conceptual understanding of fractions. Many of us were not taught to deeply understand fractions. We learned proceduresย 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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1. Sara Franza says:

Finally, with this last post, my comparing fractions strategies are made whole!! ha ha ๐ Thank you Donna ,for this great post. I am one of those students who was taught tricks and procedures without conceptual understanding. I want better for myself and the precious kiddos that I teach! Thank you for the freebie and encouragement to do better with our students!

1. Donna Boucher says:

Thanks so much, Sara! I learned exactly the same way, and I love that I now have a deeper understanding. It makes math so much more interesting!

2. Thanks for a great post! What is exciting is that if early on students understand the identity property of multiplication, they understand WHY we can multiply a fraction times 4/4 (for example) and why the quantity doesn’t change! “Properties, Place Value, and the Relationship”! ….key in conceptual understanding and efficient strategies!

1. Donna Boucher says:

I couldn’t agree more, Becky! It’s so important that students understand that multiplying by 4/4 is the same as multiplying by 1.

3. Julie Woods says:

Donna,
I was unable to download your cards and fraction tiles when pressing “here”. Has this been a problem for others? Would you know of how to fix this? I am on a mac.

1. Donna Boucher says:

I just checked the link, Julie, and it works for me. It takes me right to the Google doc. I’m not a Mac user, so I don’t know if it has something to do with that. Sorry I can’t help!

I love this series of posts, and they are just in time for me and my students. Your blog has made me fall in love with teaching kids to use number sense rather than tricks. Thank you so much for your thoughts and generous freebies!! ?

5. Amy Kuc says:

Iโve thoroughly enjoyed this series on comparing fractions. Iโve always relied on tricks and realize my kids know nothing about fractions! I am a Texas teacher, teaching 3.-5 math. How far would you go with third graders? Thank you!

1. Donna Boucher says:

So glad you like it! In Texas, 3rd graders are only required to compare fractions with either like numerators or like denominators. It’s designed that way to really make sure they understand the meaning of the numerator and the denominator.

6. Brenda says:

I need advice on teaching fractions to above high students who only want to use computation. Their parents have pretaught and I find they are unable to use strategies you have mentioned.

1. Donna Boucher says:

That is a tough road! Once kids have learned by memorization, they are usually very reluctant to entertain other strategies. It can be done, though by educating parents about the need for developing conceptual understanding as well as setting an expectation for mathematical communication and understanding in the classroom. It truly is changing the culture of how we teach math, and that can be like turning the Titanic. Keep fighting the good fight. ๐

7. Agustina says:

And everything makes sense now! After learning this years ago based on tricks and no understanding at all, I have re-learnt and I am now able to provide my students with the manipulatives and material necessary for them to build conceptual understanding. I used all your lessons on comparison of fractions and now that we’ve come to the end I am SO proud of my 4th graders! And they are proud of their achievements as well, which is what matters the most.
Thank you SO, SO much!!!! I am an Argentinian teacher working at a school in NC, this is my first year here and you have truly helped me.
LET’S KEEP DOING MATH!

1. Donna Boucher says:

MOST of us learned without understanding! I’m so glad that we are teaching using better methods than how we learned. Welcome to the US!

8. I find this article insightful. The math coach in me wonders if teachers allow students to discover the concepts. I am curious about how students acquire their math. knowledge. I also wonder what type of questions to ask students about what it means to be an equivalent fraction. What do you wonder about the model? What do you notice between a circle and a bar model (fraction sets) How do they compare? How large how small are the pieces? How does that affect the denominator? What relationships or patterns do you see in the denominators? (This would require comparing more than two fractions. Allowing students to internalize the magnitude of fractional size. Discovering the meaning of the denominator.

Moving from whole models to set/group models. Looking at sets of fractions, 3 sets of the 5 =3/5 compared to 6 sets of the 10 =6/10. How can those be equivalent? You told me with models they were! Again asking what do you wonder, what do you notice, what do you think?

I wonder what type of questions I ask to prompt students to see that equivalent fractions can be found on the same point of the number line. The magnitude of numbers comes to mind.
Evoking SMP’s 1, 2, 6, 7 and 8.

The progressions in which we teach fractions MAKES ALL THE DIFFERENCE.
Enjoy this group Donna! Keep posting.