Fraction Arithmetic

Posted on by hpicciotto

[I posted an edited version of this post, including an approach to the division of fractions, here.]

If you are familiar with my Web site, you probably know my interest in visual representations of algebraic ideas. The seed for this was probably planted during my decade as an elementary school teacher, where I learned about visual representations of ideas in arithmetic. (This was in part, but not exclusively, through my exposure to manipulatives.)

Recently, I was thinking about visually representations of fraction arithmetic. I remembered that when teaching 5th grade, I had used a method on grid paper. Here is how it went.
<!– /* Font Definitions */ @font-face {font-family:Times; panose-1:2 0 5 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:Times; panose-1:2 0 5 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 0 0 0 1 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin-top:12.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in; margin-bottom:.0001pt; line-height:14.0pt; mso-pagination:widow-orphan; font-size:12.0pt; mso-bidi-font-size:10.0pt; font-family:Times; mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Times New Roman";} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"ＭＳ 明朝"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} @page WordSection1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.WordSection1 {page:WordSection1;}  

Let’s say we want to add 2/3 and 1/5. We will use grid paper to make it easier to visualize what is going on. If one unit was defined as a 3 by 5 rectangle, we could easily represent either fraction in it:
But now we see that the sum has to be 13/15. The same figure helps us see that the difference has to be 7/15. Students can decide on the appropriate rectangle for a given sum or difference. For a problem like 2/3 + 1/6, using a 3 by 6 rectangle would work, but it is more elegant (or saves paper, for the conservation-minded) to use a 1 by 6 rectangle, which will work for both fractions.
For multiplication, we can use a similar approach.  <!– /* Font Definitions */ @font-face {font-family:Times; panose-1:2 0 5 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:-536870145 1107305727 0 0 415 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin-top:12.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in; margin-bottom:.0001pt; line-height:14.0pt; mso-pagination:widow-orphan; font-size:12.0pt; mso-bidi-font-size:10.0pt; font-family:Times; mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Times New Roman";} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"ＭＳ 明朝"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} @page WordSection1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.WordSection1 {page:WordSection1;}

Let’s say we’d like to multiply 2/3 by 1/5, but this time, we will represent them one-dimensionally using line segments on the sides of the rectangle. Again, a 3 by 5 rectangle will come in handy:

The product is 2/15. (Or equivalently, we can represent these fractions as we did above, and look at their overlap for the product.)

I do not recall what I did about division back then, but a couple of weeks ago, I came up with an idea for that, though it turns out it is a bit trickier. Check it out here.

Are there better approaches I’m not familiar with? Let me know!

–Henri

5 thoughts on “Fraction Arithmetic”

  5. Good question, Hugues! Unfortunately, I don't know the answer. My guess is that if there are techniques to do that, they may not illuminate the underlying mathematics.

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