Math Matters

Fraction Focus

Visualizing Fractions

Fractions might not only be one of the hardest to learn concepts, but also the hardest to teach. Often students struggle with fractions once operations begin and perhaps it is because operating with partial amounts is a pretty abstract idea. I have found in my own experience, teachers lack resources, or knowledge in some cases, to make fraction operations visual for students so that conceptual understanding is developed. When teachers can bridge students through fractions and their operations using concrete and pictorial models, students are better equipped to understand the abstract presentation of fraction operation problems. The focus of this newsletter is to provide some ideas on how to make fractions, and their operations, more visually evident for students so they become less feared and dreaded by students and by teachers.

A Few Teaching Thoughts

  • Start with the concrete, no matter what area of fraction learning your students are in.
  • Build in time for pictorial modeling. This is an important step to getting students to form images in their mind when the models are gone.
  • Include number line connections. If we don’t show and talk about the connections, students may not make them.
  • Link the pictorial to the abstract mathematics going on behind the scenes.


Develop operations in the same way, building from concrete to pictorial to abstract (absence of models)

  • Add/Subtract with like denominators – begin with comparing, then composing/decomposing, then operating.
  • Use a number line and the concrete models as needed to show the concepts and then bridge to picture models.
  • Develop the idea of a common denominator through equivalence. We add and subtract like things so use equivalence to make them alike, then operate.
  • Continue picture modeling with unlike fractions by representing the common denominator that makes them alike.
  • Multiplication and division also can be done concretely and using relationships in pictorial models.
  • Multiplication is like addition - combining groups or shares together.
  • Division like subtraction - separating into groups or shares.
  • Remember that the remainder in a division problem represents the amount of a portion or share that is left out of the next group. For example 18/16 means how many groups of 16 can I get from 18. There is one full group of 16 in 18, and 2 parts of 16 leftover to make the next group. The answer is 1 and 2/16.

Click each image to see a modeled example.

Do you wonder why we invert & multiply? Check out this great video explanation.

Thank you!

I appreciate that you took the time to do some hands on math with me today. Be sure to check out other newsletters and my pre-made Math Cut Ups activities at www.mathcutups.com. And reach out anytime!


Kelli D. Mallory, Ed.D.

Math Cut Ups by Integral Mathematics

I offer ready-to-use, pre-printed math activities as well as Professional Learning. Contact me for details and questions.