Fast Fractions – Easy System

We were playing around trying to figure out an easy way to memorize the different techniques for operations on fractions last Friday in my intermediate algebra class.  Suddenly this simple pattern emerged.

ADD/SUBTRACT FRACTIONS – CRISSCROSS APPLESAUCE

MULTIPLY FRACTIONS – APPLESAUCE APPLESAUCE

DIVIDE FRACTIONS – CRISSCROSS NO APPLESAUCE

This diagram shows how the system works in action. Please note:  This diagram has been improved to show which direction for the crisscrosses.  Thanks to the commenters  for this suggestion.

Operations on Fractions

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About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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18 Responses to Fast Fractions – Easy System

  1. Cara Ziegel says:

    very cool – except of course the crisscross no applesauce – how do you remember which one ends up on the bottom – the first crisscross, both products end up on top.

  2. Amber Dittert says:

    I really think this confuses students. So many students try to multiply fractions in the criss-cross way. I try to never use cross-multiplication when doing operations on fractions.

  3. Geneva says:

    Does this method work when you have mixed numbers? If so, how would this work in a subtraction problem when the first mixed number is smaller than the second mixed number. For example, 1 and 2/5 minus (-) 1 and 7/8. In the method I use, if this occurred, you would have to borrow from the whole number.

    • Miranda says:

      For you it would be 7/5 take away 15/8. Which would be 7*8 – 15*5 which is 56 – 75 which equals -19 and then 8*5 is 40. So the answer would be -19/40.

  4. veronica says:

    great idea. NOw how about neg and postivie?

  5. we are doing multiply mixed numbers by distributive propertys. help

  6. Laurie says:

    I work with various levels of secondary math students and it always seems to be the memorization tricks (magic math) which mess students up in the long run (they can’t remember what trick goes with with siutation). This may make sense to them now but teaching, reinforcing and assessing the concepts behind the process – the why of LCDs, invert and multiply, etc. – would stick longer and possibly make it easier for students to retain for the future.

    • Anonymous says:

      This is so true. I teach 6th graders that were taught to cross multiply when dividing fractions. So they can get a correct answer when following the trick but can’t find an error when looking at a traditionally solved problem. For example, a problem solved by flipping the first fraction and then multiplying results in an incorrect answer but they don’t know that. And our standards ask students to find errors in problems.

      • Anonymous says:

        Very true. Long gone are the days where students simply answer fraction problems. They may be asked to create the equivalent fraction number sentence before they can add/subtract or write the multiplication number sentence for the division of fraction problems. And like previously stated students have to be able to find which step an error is made in a sample problem. I teach fractions the “old school” way but I teach students to create reference examples for the various operations using a half and a fourth. I cant add/subtract a half and a fourth because the denominators are not alike so I have to create an equivalent fraction…

  7. Anjali says:

    Criss cross is a good technique , but when the number of digits are more you can also use base multiplication which is also equally good

  8. Anonymous says:

    This is why we have a society that freaks out when they see fractions. They think of them as a set of confusing rules & don’t remember when to use which trick. Absolutely no conceptual understanding here. Very disappointed to see this being shared around on Pinterest.

  9. Anonymous says:

    This ‘trick’ is actually only useful for adding and subtracting fractions if you have very small denominators (like 3/8 +4/5) that are not mixed numbers–for example, if you have 19/25 + 4/5 it’s much easier to first give them a common denominator, then add, rather than having huge numbers to multiply, add, and then convert back into a mixed number using long division. (I also teach math and I can see how this would be helpful, but again, only in the case where the denominators are smaller.) It’s important to understand the common denominator ‘rule’, as you need to add apples with apples, and fourths with fourths. You cannot add eights with fifths, as they are different units of measure. 🙂

  10. Anonymous says:

    When you teach kids “tricks” to do fractions they can solve the problem but still don’t understand fractions. This will greatly hurt them later when they get to highschool, college and life. Let’s come together and start teaching for understanding and not teaching to memorize a bunch of tricks!

  11. Sandra says:

    Thanks I’m glad I found this

  12. Sue says:

    Tricks are great when students come up with them on their own. It shows they have a strong level of understanding and are finding quicker ways of doing the mental math just like we as teachers do. Otherwise, teaching tricks is more confusing than helpful, and I avoid them.

  13. Naomi K says:

    These are great tricks. How do you teach them why it works? I am curious how to explain it to kids.

  14. Anonymous says:

    I love this fast fraction method. I am a special education teacher and know that students have different ways of learning. I had struggled trying to teach the traditional method that requires like denominators to add and subtract, but some students got lost trying to form equivalent fractions. This method avoids that and allows students to be successful. I plan to teach both methods and allow students to choose whichever one they understand.

  15. jayson says:

    Very true. Long gone are the days where students simply answer fraction problems. They may be asked to create the equivalent fraction number sentence before they can add/subtract or write the multiplication number sentence for the division of fraction problems. And like previously stated students have to be able to find which step an error is made in a sample problem. I teach fractions the “old school” way but I teach students to create reference examples for the various operations using a half and a fourth. I cant add/subtract a half and a fourth because the denominators are not alike so I have to create an equivalent fraction…

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