Quiz: Those Frustrating Fractions

[Photo by jimmiehomeschoolmom.]

Fractions confuse almost everybody. In fact, fractions probably cause more math phobia among children (and their parents) than any other topic before algebra. Middle school textbooks devote a tremendous number of pages to teaching fractions, and still many students find fractions impossible to understand. Standardized tests are stacked with fraction questions.

Fractions are a filter, separating the math haves from the luckless have nots. One major source of difficulty with fractions is that the rules do not seem to make sense. Can you explain these to your children?

Start with an easy one…

Question #1

If you need a common denominator to add or subtract fractions…

  • Why don’t you need a common denominator when you multiply?

Question #2

If of a cake is to be added to of a cake, the ...
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When you multiply both terms (the numerator and denominator) of a fraction by the same number, you get an equivalent fraction:

\frac{2}{3}  = \frac{2 \times 4}{3 \times 4}  = \frac{8}{12}

When you divide both terms by the same number, you get an equivalent fraction:

\frac{3}{9}  = \frac{3 \div 3}{9 \div 3}  = \frac{1}{3}

Then why won’t it work to do this:

\frac{1}{2}  = \frac{1 + 1}{2 + 1}  = \frac{2}{3}  \;? No!

  • Why, when you add the same number to both terms, don’t you get an equivalent fraction?

Question #3

To multiply two fractions, you multiply the numerators and multiply the denominators:

\frac{1}{2}  \times \frac{3}{5}  = \frac{1 \times 3}{2 \times 5}  = \frac{3}{10}

To divide fractions, can you divide the numerators and divide the denominators?

\frac{3}{4}  \div \frac{1}{4}  = \frac{3 \div 1}{4 \div 4}  = \frac{3}{1}  = 3\;?  \;


…but it works only if you are careful to keep all the numbers in the right order. Remember that 3 ÷ 1 is NOT the same as 1 ÷ 3.

So then why can’t we do this?

\frac{1}{4}  + \frac{1}{4}  = \frac{1+1}{4+4}  = \frac{2}{8}  \;? \; NO !

  • Why, when you need to add fractions, can’t you just add the numerators and add the denominators?

Question #4

When you divide by a fraction, you can flip the fraction over and multiply:

4  \div \frac{1}{2}  = 4 \times \frac{2}{1}  = 8

When you multiply by a fraction, can you flip the fraction over and divide?

4  \times \frac{1}{2}  = 4 \div \frac{2}{1}  = 2\;? \;

Yes, it works.

  • Then why, when you have to subtract a fraction, can’t you just flip it over and add?

Question #5

If you divide by flipping the fraction over and multiplying, does it matter which fraction you flip?

\frac{2}{3}  \div \frac{1}{3}  = \frac{2}{3}  \times \frac{3}{1}  = 2

But if I flipped the other fraction:

\frac{2}{3}  \div \frac{1}{3}  = \frac{3}{2}  \times \frac{1}{3}  = \frac{1}{2}  \;? \; NO ! !

Only the first equation is correct, so it definitely matters which fraction you flip.

  • But why does it matter?

This is post #1 in the Fraction basics series.

42 thoughts on “Quiz: Those Frustrating Fractions

  1. Those are annoying! It is hard, after a kid has been given a lousy start in fractions, to recover.

    I like to begin with counting numbers, addition, needing an identity (0), needing inverses (negatives)…. then multiplication, needing an id (already have one, 1), needing inverses (unit fractions), and forward from there. But need fairly strong number background for that.

    For other kids, physical models or diagrams of physical models…

    1. I agree. You could use games to strengthen their number skills. Then the properties of fractions can be presented which is what identities, inverses,etc. are.

  2. I’d be very curious to see how you explain these. I get this stuff intuitively but I could not put into words the principles. Luckily my children seem the same way!

  3. Will try to find time to look at all the questions, but let’s start with the 1st. I apologize for spelling and language mistakes. I would do better in Swedish…

    Think about what fractions mean. You have 1/2 pizza and want to remove 1/3 of a full pizza. 1/2 is the same as 3/6 (3 of 6), 1/3 is the same as 2/6. You have 3 slices (1/6 big) and remove 2… that leaves one…

    Now we come to 2/3 times 1/2: What does it mean? It means you have one half pizza, slice it in 3 pieces and keep 2 of them. Close your eyes and you can see that it leaves 1/3 of the full pizza… or you can think about it 1/2 is the same as 3/6, you keep 2 out of 3 (2/3 times 1/2) so now you have 2/6 same as 1/3. (Why? Look at the pizza.)

  4. Thank you for the explanations, Per! I have edited your comments slightly and put a few of your equations into LaTex, to make them easier to read.

  5. Look at the pizza. Let’s say we have toppings on 3/4 of the pizza (3 out of 4 equal slices). If I cut every slice in 3 smaller slices, I will now have toppings on 9 out of 12 (9/12). You can go the other way, if you group many small slices to bigger ones.

    Next… I can’t write it as good as you did so I hope you understand what I am trying to talk about..
    Why isn’t 1/2=(1+1)/(2+1)=2/3 ? One way to explain it is to ask, “Why should it be?” If you can’t understand and show why something in math is true, you shouldn’t use it…the idea doesn’t make sense if you start to think about what it should mean…

  6. Again, think about what it means.
    Let’s think about (2/3) times (4/5).
    (2/3) , This means you have 2 slices, each slice is a third of the pizza. Times (4/5) means you will cut each slice into five smaller slices and keep four of the five. A full pizza would be 15 (3 times 5) small slices, you have 8 small slices (8/15)

    To explain the next part, we need to think about what division really means, that it can mean two different things. 8 divided with 2 can mean… you have 8 books and divide them into two equal piles, how many books goes into each pile? It can also mean you have 8 books and put them into piles with two books in each pile, how many piles will you have?

    You had (3/4) divided with (1/4)… the question is “You have three fourths of a pizza and plan to put one fourths on each plate. How many plates do you need?”

    1/4 + 1/4 …. Think about it, one car plus one car is two cars… why shouldn’t slices of pizza be the same? It must be 2/4… and that is the same as 1/2

  7. 4
    Again think about what it really means. 4 divided with 2/3: You have 4 pizzas and want to serve two thirds on each plate. How many plates do you need? You cut each pizza into 3 slices, (4×3=12 slices), you put 2 slices on each plate (12/2=6).

    Multiplication: 6 times 2/3… You can either think 6 pizzas and slice up every pizza into 3 slices and keep 2… this would give 12 slices, and it takes 3 to make a full pizza (12/3=4). Or you could think, out of every 3 pizzas I will keep 2, where you think “6 divided with 3 times 2.”

    This last bit feels backwards for me… need to think of a better way to explain that…

  8. 4 continue…
    “Then why, when you have to subtract a fraction, can’t you just flip it over and add?”

    Stop trying to think about rules without thinking about what it means… it is just nonsense. You are suggesting that I solve the real problem “I have one half pizza and eat one third of a full pizza. How much is left?” by calculating:

    \frac{1}{2}  - \frac{1}{3}  = \frac{1}{2} + \frac{3}{1}  = \frac{1}{2}  + 3 = 3\frac{1}{2} \; pizza .

  9. 5
    Why can’t you flip the first fraction? Because that gives wrong answers (as you have shown). The real question is why you can flip the 2nd one…

    And I must confess I can’t really show why without better ways of writing fractions.

    \frac{\frac{a}{b}}{\frac{c}{d}}  = \frac{\frac{a}{b}  \times \frac{d}{c}}{\frac{c}{d}  \times \frac{d}{c}}  = \frac{\frac{a}{b}  \times \frac{d}{c}}{1}  = \frac{a}{b}  \times \frac{d}{c}

    That might give you an idea..


    But when it comes to fractions, don’t teach the rules. Teach how to think about what it means.

    1. Argggh! I feel like you’re trying to reinvent the wheel or something. The rules can be understood. There are only 4 operations of arithmetic.
      I will say, though Per is right to show the A’s and b’s- remember you have to state that none of the letters in a denominator can equal zero because it’s undefined.
      Asking why is fine, however, asking why not is troublesome to me, someone who studied number theory and foundations of mathematics. I humbly disagree with this teaching method. I
      I DO think games are the answer, though. They help you learn what you need to know. 🙂

      1. This post is not intended as a teaching method, but rather to force teachers to assess their own understanding (or lack thereof) about fractions. If a teacher cannot answer these questions, she needs to reinforce her own understanding before trying to explain the concepts to her students.

        1. It never hurts to review things. No one has to force me to assess my understanding about fractions.
          I really do like fractions. I never thought they were “hard or tricky”.
          So, all of the Why won’t this work questions are answered by the properties of fractions. Here’s a clear list of them I found online. I am studying it:
          What do you think about this approach of understanding fractions?
          Does it answer your questions?

  10. Who am I? I am a highschool teacher from Sweden, and I have a problem with students coming to my math classes who think the problem is that they don’t remember the rules, but the real problem is that they were not taught to THINK about what it means. Too few students know what question can be asked to give different set up in fractions.

  11. I struggle to teach my daughter this…she’s in jr. high and still can’t seem to figure out why and it’s because *I* don’t really know why. Please give your reponses to these questions. 🙂 Thank you.

  12. As jd and Per said, all of these problems come from trying to apply barely-remembered rules without understanding what is going on in the situation. To answer any of these questions, one has to go back to the beginning and make the foundations of understanding firm.

    The student needs to understand what a fraction actually is (what the two numbers and the dividing line mean, and how they combine to make the single number that is the fraction). For questions #1-4, the student needs to understand what addition and subtraction mean and how they are different from multiplication and division. For question #5, the student needs to understand what division means, and how it is similar to (and different from) multiplication.

    I plan to write several more posts on fractions over the next few weeks, which I am hoping will help to explain these basics. But here is a quick summary:

    Addition and subtraction = counting up how many pieces you have, but that doesn’t make sense unless all the pieces are the same size (common denominator)

    Multiplication = cutting up what you started with into different parts (the multiplication symbol can be read as the word “of”)

    Division = the reverse of multiplication: finding how many of the parts it takes to make what you want

  13. Fractions really aren’t simple “numbers” such as whole numbers are; they are “divisions” in themselves or ratios. So to think about them, one needs to think of the “division” operation that produced them, OR think about the ratio of two numbers.

    If you are able to think of the ratio, then it’s clear that the ratio of 2 to 6 will remain the same if you “expand” or “shrink” both parts equally (multiply or divide both numerator and denominator by the same number).

    However, it’s surely easier for kids to first think in pie parts than to think in ratios.

    See also my article

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  15. Denise, are you sure you didn’t mean that fractions separate the halves from the half nots? 😉
    as for your first point, why you would need a common denominator to add or subtract but not to multiply… let me take a stab at that.
    I teach third-grade math, and one of the points that I really try to hammer home, especially in word problems, is that you can only add or subtract numbers when you have the same units. In other words, you can take away crayons from a group of crayons and you can add chairs to more chairs. But you can’t add 4 tires plus 2 cars. You can add 4 plus 2 and get 6, but what are you going to say for your complete answer — 6 tires or 6 cars? Neither is correct. The tires aren’t going to magically change into cars and the cars are going to transform into tires.
    Similarly, with the fractions, you can only add or subtract them when you have the same thing. Halves of an item are NOT the same thing as fifths of an item. However, unlike tires and cars, you can manipulate fractions to make them equivalent to the same thing.
    I hope that makes any sense whatsoever.

  16. Whoa, dude, you’re blowing my mind! Hehehe.

    Trying to explain “lowest common denominator” over here, and although I know what it means (sort of) I can only explain it by showing examples. Lowest common denominator — common to what?!

  17. Good question, Lydia!

    The common denominator needs to be common to all the fractions you are trying to add or subtract. It is “common” as in, “We have this in common” — that is, we share it. Before you can add or subtract fractions, they need to all share the same denominator.

    This is just another way of saying that it doesn’t make any sense to count up how many pieces of something you have, unless you first make sure all the pieces are the same size. Otherwise, one third and one fourth would be what, two thourths?

    On the other hand, you do not have to use the lowest common denominator. ANY common denominator will do. It’s just that most of us prefer to work with small numbers.

  18. Just a comment about addition mentioned by Mr. Teacher.
    Addition should be taught as a two step process.
    Step 1. Connecting two things together with a + sign (addition).
    Step 2. Combining the two things into one thing.

    So anything can be added together such as a car and a tire. IE car+tire.

    This is critical to teach because later the student will have to add unlike things such as square root of (2) + 1 which cannot be combined.

  19. 1) At an elementary level i.e. grade 2-3, the concept of LCM and HCF is not being introduced. Thus to keep it simple and enable the child to do basic calculations the common denominator is kept same. Also understand that at this stage the challenge is to help them grab the concept i.e. fraction is sum of the whole.
    Say for instance you have a chocolate bar with 12 bars. You ate 8 bars i.e. you have eaten 8 bars out of 12 bars. So the remaining bars are 4. Now split the bar in 2 halves i.e. each half will have 6 bars or one can simply do (1/2 of 12=6 bars each). Say you want to give 1/3rd of 6 bars, it will be 2 bars and the remaining bar will be 4 in one half. So the total no. of bar will be 4+6=10

    2)Lets use the chocolate bar example, take 2 piece out of the 3 pieces. Now divide each piece into 4 equal parts. The total no. of parts will be 12. If I take 8 parts out of the 12 parts, it can be written as 8 out of 12 or 8/12 which is equivalent to 2/3.
    An equivalent fraction is a fraction that is broken into equal parts; sum of those parts will be equivalent to the bigger parts or the vice-versa. In case of multiplication and division the numbers with which the given fraction is multiplied or divided can be cancelled completely which is not possible in case of addition

    3)Let’s take the example of pizza here. The pizza has been divided into 5 equal parts. 3/5 or 3 out of 5 will be 3 parts out of the 5 parts; the remaining part will be 2/5. One further wants ½ of 3/5 part of pizza which will be 3/10 or cut 3/5 into two ½ which again will be 3/10.

    If the kids can be conditioned to imagine these concepts they will be able to tackle the fractions. As such fractions is the gateway for all higher level algebra. Hence use of visuals and imagery can help them grab the concept faster.


  20. These were really great! Have you been peeking into my classroom and listening to my kids talk about fractions? It really helps them if we use concrete examples instead of just paper examples. Once they get the concrete concepts the paper ones make better sense (not total sense – just better sense LOL).

  21. On your question #3 , second part regarding the division as to why it works, you need to include it will also work if you have or find a Common Denominator.

  22. It is true that you can do fraction division by first finding a common denominator, if you are willing to bother with the extra step. My example was chosen with a common denominator so that readers could easily see that the answer made sense.

    But it is not necessary to have a common denominator. You may use the “divide the numerators and divide the denominators” rule with any two fractions:

    \frac{15}{40}  \div \frac{5}{4}  =  \frac{15 \div 5 }{40 \div 4}  = \frac{3}{10}

    It even works with fractions that do not come out even, but the standard “invert and multiply” rule for fractions is much easier for those. It became the standard rule for a very good reason!

  23. Recently I have been dealing with how to teach division of fractions without resorting to mathematical magic. Per’s explanation is the best I could think of for kids in lower school. I would remind students, as you have said, that fractions are numbers and that any number divided by itself is 1 and also that any number times 1 is itself.
    With older students you may want to show that for the binary operators of addition and multiplication you can replace them with the inverse operators of subtraction and division, respectively and the second operand takes the inverse value. This renaming of operators may also help students who are confused by the meaning of PEMDAS and make order of operation errors with problems like 10 / 5 * 2.

  24. Oh, and Mr. Teacher, I think your example would serve to confuse. Perhaps drawing a parallel between adding whole numbers on the number line (rationals with denominator=1) to fractions. You can add them step by step as with whole numbers, but the steps must be the same size which is what I think you are trying to say.

  25. For beginners, I like to break the division up into steps. For instance, if you are dividing by 5/4:

    (1) Find out how many 1/4’s are in your number. It is easy to convince students that, since there are four 1/4’s in every one unit of whatever we are counting, then there are 4n of them in any given number n.

    (2) Then how many groups of 5/4’s do you have? Take the number of 1/4’s you figured out, and divide it by 5 to find out.

    (3) Don’t be scared of fractional answers. Sometimes you come out with part of a group left over — and that’s OK!

  26. Actually if we taught whole numbers in fraction form it starts to make sense. When adding I tell students we are allways adding something (pieces) whether 1/2’s 1/4’s or wholes.

    When we add 3+1=4 what we are really saying is 3/1+1/1 = 4 or 3 ones (wholes) pluss 1 one =4 ones (not 4/2 which is what we get if we were to add denominators)

    Also, adding fractions follows the rules of order of operations (we multiply/divide before adding or subtracting) So in the case of 1/4 + 1/2 we divide 1 by 4 and get .25 and divide 1 by 2 and get .5 for a total of .75 which equals 3/4

    1. Fraction form means rational numbers. Rational numbers means, from Wikipedia:
      n mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number.
      They can be taught this in a way that makes sense, in plain words, with examples.

  27. When you learn to drive, you learn the rules first. Then, you go practice each skill. You don’t say “I will just go drive and figure it out myself” if you want to get your license. Your motivation is to have the privilege of driving.
    What would motivate someone, anyone to understand how to properly multiply, divide, add and subtract fractions?

    1. But when you learn to drive, you don’t just learn abstract rules without understanding the meaning. That would be nonsense!

      Too many people look at the fraction rules as abstract nonsense they just have to memorize so they can pass a test. The rules DO have meaning, but school rarely succeeds in connecting the rules and the meaning in students’ minds.

      See for example this Scientific American article: https://www.scientificamerican.com/article/fractions-where-it-all-goes-wrong/

      1. As for how to teach fractions, I’m a firm believer in the power of story. I try to always relate abstract math relations to a story or word problem, which helps draw out my kids’ intuition and gives them a mental hook on which to hang the rule.

        Even more important, I encourage the kids to make up stories for themselves.

      2. I agree it is a teachers job to answer these questions when they come up. It is not a one minute answer. I would probably tell the student that we are going to go backwards until we see where they got lost in the understanding of fractions.
        It only matters what that one student’s thinking was, at this point. I think this is why schools and teachers of groups resort to just the steps without the reasons. I do think, though, the rules are simple and the questions about why should be asked by the student and answered by the teacher( or resources given to answer it if student so desires).
        There’s no reason not to start with the steps how to add, subtract, multiply and divide, and practice each kind, giving enough real examples of them.If you don’t say what if or why not, they won’t do it!

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