Would You Rather…Fractions

Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

A common misconception students have about fractions is that a larger denominator means a larger fraction. Ask a handful of third graders (or 4th graders or 5th graders…) which fraction is greater, 1/8 or 1/4, and most are likely to quickly tell you 1/8.  With big, proud smiles on their faces.  You’re nodding your heads out there–I see you!  You’ve been there.

There’s a reason this misconception is so widespread.  Up to this point in their educational career, bigger numbers always meant bigger values.  Eight is greater than four.  When students begin to learn about fractions, they erroneously apply whole number reasoning to fractions.  One-eighth must be greater than 1/4, because 8 is greater than 4.

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So how do we address this misconception?  First and foremost, students must have tons of experience with a variety of concrete and pictorial models of fractions.  Use fraction tiles, fraction circles, Cuisenaire rods, number lines, and cut paper strips.  It’s pretty hard to look at models of 1/8 and 1/4 and not see that 1/4 is greater.  The idea that fractions should be explored using manipulatives and models is very apparent from the wording of the 3rd grade Texas TEKS, but not so much in the CCSS.  I cannot overemphasize, however, do NOT rush to abstract symbols.

The other way we can overcome this faulty reasoning is to help students truly understand the meaning of the denominator.  The more parts an object is divided into, whether that object is a pizza or a number line, the smaller the parts.

Both the CCSS and new Texas TEKS address the issue of comparing fractions in a way that will help students deeply understand the denominator. The 3rd grade standards are really well written, once you get past understanding all the 1/b and a/b references, and will definitely result in better fraction number sense for our students.

Here are the standards for comparing fractions for both the CCSS and the TEKS:

Notice a few key points…

  • Students are only comparing two fractions, not ordering more than two
  • The denominators are limited to 2, 3, 4, 6, and 8
  • Students only compare fractions with either the same numerator or the same denominator
  • The TEKS specifically mention words, objects, and pictorial models along with symbols
  • Both include verbiage about reasoning about their size and justifying the conclusion
  • The CCSS states that students must understand that the reasoning only works when referring to the same whole
Let’s first consider comparing fractions with the same numerator.  When two fractions have the same numerator, it emphasizes that a larger denominator means smaller parts.  Look, for example, at the representations below of 1/8 and 1/4.  When you look at one piece of each pizza, the idea that eighths are smaller than fourths is pretty clear. It works anytime the numerators are the same.  For example 2/6 and 2/4, or 3/8 and 3/6.
Third-grade students are also required to compare fractions with the same denominator.  This emphasizes that the denominator describes how many parts the whole has been partitioned into (thereby influencing the size of the parts) and the numerator describes the number of those equal parts you have.
 

That’s it!  Those are the only two types of comparison 3rd-grade students need to do. And if they truly understand and can explain these comparisons, the next generation of students we send up won’t leave us scratching our heads when they say 1/8 is greater than 1/4.

Do you like the cards pictured above?  Well, guess what?  They are my gift to you tonight.  This freebie includes 16 cards–8 comparing equal numerators and 8 comparing equal denominators–along with a recording sheet students can use to document their thinking in both words and symbols. Click here to grab your freebie!

37 Comments

  1. Anonymous

    I’m a fifth grade teacher, and the hardest job is reteaching fractions and overcoming math fear in general.

    Reply
  2. Ms. Morse

    I still struggle with teaching the application of fractions after years of teaching to a test.

    Reply
  3. Kelly Pulham

    Very simply presented. Thanks for helping us to understand our students’ misconceptions.

    Reply
  4. J Wilson

    Knee deep in fractions right now with my 3rd graders and I so appreciate this free resource! Thank you!

    Reply
  5. Emily Levine

    My students have a hard time grasping equivalent fractions and also comparing fractions.
    Emily

    Reply
  6. Angela

    I believe that the more students are exposed to the concepts of fractions the more understanding they will have. It’s important for them to understand that what the denominator is and what it means before we start asking them to compare, add, subtract etc fractions. There appears to be so much pressure in all education systems to move kiddo’s too fast long the learning and not enough time is able to be given to concrete learning.

    Reply
  7. pendidikan

    Taught mathematics in children is indeed the appropriate tools needed. Because their logic is often not good.
    The number of mathematics learning method, sometimes quite confuse children.
    Sometimes going back to the basic method is precisely to help.
    Thanks, your article adds insight to me.

    Reply
  8. kmanzano

    I’m in Texas too! I agree that getting the students to understand the bigger the denominator the smaller the piece is tricky!

    Reply
  9. Suzy Q

    I have many language learners and in their language you say the denominator first, so naturally, that’s how they write it…on top!

    Reply
  10. Toni Nicholson

    My biggest challenge is exactly what you have stated in your post. When numerators are the same, how can the bigger denominator be the smaller piece? Thank you so much for this post.
    [email protected]

    Reply
  11. TheElementary MathManiac

    Excellent post! It is so important for students to have hands on experience with models and model drawing to help them develop these big ideas. I have found that by not introducing fraction symbols to early and instead using words like one-fourth and one-half, my students have more time and energy to focus on the models and visual representations without getting confused by the symbols.

    Tara
    The Math Maniac

    Reply
  12. Christie - Fine Lines

    This is PERFECT for what I am teaching right now!! Thanks!

    Reply
  13. The Math Spot

    I think that one of the biggest misconceptions in math teaching is that you can speed past the manipulatives! Your explanation and visuals show how powerful an understanding of the concept is towards future accuracy. Thank you for the fraction comparison cards as well!

    Reply
    • dibarth1

      I agree. Sometimes we get in such a hurry. I have found that working with the manipulatives helps overcome many misconceptions. The students gain confidence and success.

      Reply
  14. Daisy Fryer

    This is always something my 3rd graders are confused about at first. I love seeing the light bulb come on when they get it. Thank you for your freebies. I will use them with my students next week.
    Daisy Fryer at Not Your Mother’s Math Class

    Reply
  15. Jenny G.

    Most everything about comparing fractions is confusing for students. Starting with the problem you just wrote about–the fractional size of the piece vs. how “large” the denominator is. Also, comparing the missing piece is confusing, too… 5/6 or 6/7? The smaller missing piece actually belongs to the greater fraction. Everything is backwards in fractions land!! Thanks for the freebie!

    Reply
  16. Sarah M

    Thank you for this wonderful post, Donna! This is a misconception I see early on my second graders, and I agree that using models is so important. It is so natural to see this as students have developed such a strong understanding of whole numbers. Even though comparing fractions is not a second grade standard, it always comes up in our explorations. I try to give students many opportunities to compare models and hope this helps to squash the misconception early. Although, having taught third and fifth grade as well, I know how important it is to consistently reinforce this idea. Thanks so much for sharing your thoughts and freebie!

    Smiles,
    Sarah

    Reply
  17. Denise Strawn

    My biggest challenge is to get kids to visualize what fractions should look like and to not rely solely on “tricks”. Obviously, we spend a good deal of time on models/fractions tiles, etc. However, I’ve been trying lately to also focus on decomposing/composing and comparing to benchmarks. I’ve seen a lot more growth in fraction number sense when using those methods for my third graders.

    Reply
  18. Karen H

    I struggle with using models to multiply fractions. My students can’t seem to wrap their brains around it.

    Reply
  19. Sharon

    I always use my plastic fraction strips because the students can move and manipulate the fractions to see which is bigger. I then go to a single sheet (paper) fraction strips. The students really need to see the exact fractions and put them side to side or even on top of each other.

    Reply
  20. Sherry

    would love to have this to teach fractions

    Reply
  21. sfmuellr

    I see students struggling the most with having the dexterity to draw models, either when comparing fractions in 3rd grade, or when multiplying in 5th. Thanks for all your support Donna!

    Reply
  22. Tricia

    Thanks for sharing this. I think these are great examples. I still see students (and some teachers) struggling with the circle model, so I have been encouraging my students (preservice teachers) to use squares.

    Reply
    • Donna Boucher

      I agree, Tricia! So hard to divide circles equally, so I always have my kiddos draw rectangles. I tell them to draw “candy bars”, not ‘”pizzas”!

      Reply
  23. Greg Coleman

    Excellent post Donna! I am in full agreement about students needing multiple encounters with math manipulatives and fraction models. This helps them to “see” the magnitude of the fractions because they have concrete examples in front of them.

    Greg
    Mr Elementary Math

    Reply
  24. Νηπιαγωγός Πυθαγορείου

    I find it challenging to teach fractions to kindergarten but because also in Greece kids love eating pizza, it is a great example to use! Thank you for the opportunity but also for all your ideas and suggestions. Especially those with the 100chart. I found them very very useful! Maria Hatzipanayiotou

    Reply
  25. Jennifer O.

    My students have been struggling with the concept that 1/2 is not always the same size. It depends on how large your whole is. Half of a large salad is not going to be the same amount as 1/2 of a small salad.

    Reply
  26. Cynthia Burrell

    Equivalent fractions are difficult for my students. If they are using manipulatives, they can see the difference. When they draw the fractions and compare, they don’t always have them the same size and that’s where the problems lies. Great post!

    Reply
  27. Tchur8

    We are currently working on equivalent fractions and need LOTS of practice!!! I’ve been following you for a while now…and love all your materials! Very kid friendly. Thank you!!!

    Reply
  28. Ashley Waters

    It is so ironic that you posted this because this is a standard in the unit I am currently covering! I am a 2nd grade math teacher in Texas. One of the second grade TEKS is: (2.3) Number and operations. The student applies mathematical process standards to recognize and represent fractional units and communicates how they are used to name parts of a whole. (B) explain that the more fractional parts used to make a whole, the smaller the part; and the fewer the fractional parts, the larger the part.

    Initially, students assumed that the larger the denominator, the larger the fractional parts. They have been continuously learning number sense up to this point so it is natural for them to think that every time they saw a larger number it means more. However, like you mentioned, once I started using manipulatives and visuals they started to understand why the more you divide something, the smaller it will be. I use word stories that are very relatable to them. For instance, I will ask them if they would rather split a cake with their table of four (fourths) or two tables (eighths). When they picture it or draw it out they understand that the more pieces there are, the smaller the pieces will be or the fewer the parts, the smaller they are. It is imperative that students understand this at a young age so that they will understand more advanced fractional concepts in the future. Thank you for your post!

    Reply
    • Donna Boucher

      Thank YOU for using so many great strategies with your 2nd graders, Ashley! I know your third grade teachers appreciate it!

      Reply
  29. epigami

    Pizza will always work to get kids attention! Getting them to really understand the concepts is the hardest, but when they do it’s really fullfilling, it makes it feel so much more worth it. Teaching them how to mentally square a number makes them feel like superheroes, and I’ve seen a few change there mind about maths thanks to it.

    -E

    Reply
  30. Carrie

    a Texas teacher here…. Thanks… We have to cover this in 2nd…the bigger the fraction the smaller the piece or part… Love the visual

    Reply
  31. Bridgett W.

    Love your fraction series. I have been using so many of your resources in the classroom this past week. It’s really helped my students’ fraction sense. I’m excited to use this Would You Rather resources as well, but I wanted to point out a small typo on the recording sheet. It says “sybols” rather than “symbols.”

    Thanks again for all your work and shared knowledge.

    Reply
    • Donna Boucher

      Thanks for letting me know, Bridgett!

      Reply
  32. Kalli Nelson

    Busy with 3rd grade fractions as we speak! Thanks so much for the free resource! I’ve found that having the students smash and then cut play-doh is helpful by teaching them to see what is happening to the pieces each time we “cut” and what is happening to the denominator. Of course, ideally, they would cut it into equal pieces, but even if they don’t, it seems to help them get the idea that the more cuts (or more denominator), the smaller the pieces. Happy teaching!!

    Reply
    • Donna Boucher

      That’s a really neat idea!

      Reply

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