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Friday, November 1, 2013

Improper Fractions and Mixed Numbers

How can we help kids really understand the conversions between improper fractions and mixed numbers? Using manipulatives and/or pictures is essential for initial conceptualization.

Step 1: Representing Mixed Numbers - In my experience, students are not ready to convert until they can look at a picture and name the mixed number and improper fraction represented.

Step 2: Converting Improper Fractions to Mixed Numbers - This part is easy if you've already worked on fractions as division. Your students already know, for example, that 5/4 equals five divided by four. The only piece that's missing is representing the remainder as a fraction (in this case, 1/4, or one out of four left). 

Warning: Some students will insist that the denominator should be the number of possible pieces instead of the number of pieces in one whole. For example, in the problem above, some of my students wanted to write 5/8. To counter this, I repeat (and repeat and repeat), "The denominator is the number of pieces in ONE WHOLE!" Additionally, I tell them that the denominator is like the name of these pieces. (For example, if I bring apples to class and cut each into four pieces, giving each student one piece, each student is getting one fourth of an apple, not one twenty-eighth.)

After lots of practice and discussion together, they're ready to roll. For my class, I use worksheets found on the Fractions page of Common Core Sheets. Scroll the bottom of the page, and under Converting Fractions, you will find ten versions of Improper to Mixed Number worksheets. Awesome! Now we have sufficient versions for introduction, practice, tutoring, review before the test, and more review as the year goes on.

Step 3: Converting Mixed Numbers to Improper Fractions - More pictures, please! I show students how to cut the wholes into the same number of pieces as the denominator of the fractional part. We add the pieces in the wholes with the fractional piece for the numerator, and the denominator stays the same. 

After working with a number of pictures, students can see that the number of pieces in each whole (denominator) times the number of wholes plus the numerator equals the numerator of the improper fraction. And the denominator stays the same (because the denominator is always the number that each whole is cut into).

Once again, I turn to the Fractions page of Common Core Sheets, and once again I find ten lovely versions of worksheets for practice. To locate them, just click on Mixed Number to Improper under Converting Fractions (near the bottom of the page).

I hope these hints help you and your students with converting mixed to improper and vice-versa!