I am finding it difficult to convince my kid that 2/4 and 1/2 are same. As per the kid, 2/4 is more than 1/2 since in first case the boy gets 2 candies out of 4 and in second case he gets 1 candy out of the 2.

Any ideas or any books I can buy which can help me to explain this?

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    Your kid is right, half of four is more than half of two. Try keeping the overall size constant. For example, by talking about breaking a single cookie into two or four pieces. – Adam Nov 27 at 0:37
  • Isn't it that 2/4 is also ultimately 1/2 and hence 2/4 and 1/2 are same. In the sense that they both are half only? I mean it is 50% in both cases and hence 2/4 = 1/2. – NotAgain Nov 27 at 0:49
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    I agree with the keeping it the same size. Also, realize your boy is not behind. It is normal for kids to struggle a little with fractions and 7 is about when you start with that. Maybe work on some other stuff for a while (multiplication table, etc.) A fraction is really a bit of an abstraction (quotient). Work with just adding two halves into 1. Four halves into two, etc. simpler examples. Kids really don't do abstractions the way math profs do. Just give him some time and some playing around with simpler ideas. – guest Nov 27 at 1:58
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    Cut one of the candies in three, give the kid two of the pieces and ask the kid if 2/3 is bigger than 1. – Dan Fox Nov 27 at 16:47
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    The most important idea is that a whole is fixed during all manipulations with fractions within a particular exercise. It is one half of a whole, or two fourth of THE SAME whole. Ok, pizza. It is the same pizza. Same as with percent, people are making exactly the same mistake adding percent without realizing that they are just a portion of a whole, and the whole must be the same for operations with percent to be meaningful. Students "study" the same concept several times - as regular fractions, decimal fractions, percent - without grasping the idea. – Rusty Core Nov 27 at 17:39

Comparing fractions only works when the whole is the same size. Here's a few examples to get help the 7 year old understand what happens when things aren't the same size:

  1. 1/2 will be greater than 2/4 when you compare 1/2 of a watermelon and 2/4 of an apple.
  2. 1/2 will be less than 2/4 when you compare 2/4 of a watermelon and 1/2 of an apple.
  3. If you take 1/2 of 4 pieces of candy, you will take 1 from the first 2 and 1 from the second 2 which is the same 2 out of 4 or 2/4.
  4. Furthermore 1/2 of 6 pieces of candy is 3 candies and 2/4 of 4 candies is 2 candies. In this case 1/2 is greater than 2/4.

By showing the examples the 7 year old will start to understand that comparing fractions is can only work if the wholes that you take the fraction of are identical.

My suggestion is to show all these examples, and then explain that since 1/2 is sometimes greater than 2/4 (see 1 and 4), sometimes equal to (see 3), and sometimes less than 2/4 (see 2)- the mathematicians decided we would only compare 1/2 and 2/4 (or any fractions) when we are taking 1/2 and 2/4 of the same thing.

Typically, it is better to introduce mathematical ideas after the need for that new idea is obvious. For example:

  1. How should 2 children share 8 pieces of licorice?
  2. How should 4 children share 8 pieces of licorice?
  3. How should 3 children share 15 pieces of licorice?
  4. How should 3 children share 2 pieces of licorice? And why is this different from the prior cases?
  5. How should 5 children share 4 pieces of licorice?
  6. How should 4 children share 5 pieces of licorice?
  7. etc.

Challenge them to show and assess all of their work according to three main criteria:

  1. Everyone must have the same amount.
  2. There should be no leftovers.
  3. They can name the amount everyone has. [This is where formal fractional language comes in handy.]

Ensure you compare and contrast whole numbers vs fractional numbers. That is really the heart of the matter - that fractions are numbers, most of which are used to name quantities between consecutive whole numbers.

Bonus criteria and tasks can include:

  • How can you do the sharing with as few cuts as possible?
  • Is there some other way you can name the amount everyone received?
  • Is there a systematic way to solve all of these problems?
  • Generate some sharing examples that do require fractions, then generate some sharing examples that do not require fractions. How do you know when fractions are necessary?

Eventually you can guide them toward the situations of 2 kids sharing one unit of food vs 4 kids sharing 2 units of food. When they have to name the amounts, you can show them the amount of food is exactly the same and then show them that even the cuts can be the same.

You can also show them counting by fractions on a number line to show that 1/2 and 2/4 occupy the same point on a number line.

7? I work in the math dept at a local high school, and students 14-18 struggle with fractions.

I've found 2 things that help.

  • Money Math

Somehow, the same student struggling to take 1/2 of 2-1/2 can easily tell me that half of \$2.50 is \$1.25. For your son, 1/2 of a dollar is 50 cents. No need to stack the pennies, he knows (right?) that 50 cents is 2 quarters. But wait, we just said two quarters, 2/4. Half of a dollar is literally 2 quarters, assuming, of course, that your currency works this way. I am in the US. After this discussion, ask him what half of \$4 is, and then what 2 fourths of that same \$4 is. Further on, you can show how this all changes if the second item, the thing we're multiplying, changes. Of course, he'd prefer to have 1/4 of \$10 than half of \$1.

  • Pizza/food Math

I'm a fan of the circular pizza cut into 8 equal sectors. It makes discussing fractions easy for any student, as the fractions 1/2, 1/4, 1/8 are right there. Similar to the money discussion, you can start with what 1/2 means, and how 1/2 of 1/2 is 1/4, and then go into the above explanation of how the whole is our 8 slice pizza. When we ask "how much pizza would you like?" it's essential to know how much is there to begin with. 1/4 of one pizza is not the same as 1/4 of the 10 pizzas I bought to treat the class at year end. As far as 1/2 vs 2/4 is concerned, the answer is right there, they are both the same 4 slices of our standard 8 slice pizza.

In both cases, money and food lend themselves to divisibility. And having them right there avoids the issue with things being too abstract. Even in high school, I find that numbers and variables create a struggle for some students, and the faster I can move the conversation to solid objects or real life examples, the easier they get it.

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    +1 Because the pizza/pie is the key. On the other hand, children do not always fully understand money. Food is much more visual. Make sure you draw a big picture of a pizza. – FGSUZ Dec 7 at 23:27

OP (@NotAgain): "Any ideas or any books I can buy which can help me to explain this?"

May I suggest games (rather than books)? For example, BrainPop's Refraction:


          Refraction


Or the Fraction Game:


         


Or MathGamesFractions:


         
or 7 Fun Fraction Games for Kids, or 33 Fraction Games, ... or many other online fraction games.

I think this is the current locus of creative ideas, not books.

  • Oh thanks. Kid does have an inclination towards games rather than the dry textbook text. Thanks again. If you have more links to games covering the whole spectrum of maths please do add to the answer. Whenever I try to explain I end up making it even worse. – NotAgain Dec 3 at 23:11
  • @NotAgain I take it, you did not watch videos I suggested. – Rusty Core Dec 6 at 6:05
  • @RustyCore Yup have to watch. Sorry for having so little time at hand to do so many things. – NotAgain Dec 7 at 1:48

With regard to this question:

Any ... books I can buy which can help me to explain this?

Here is a specific book recommendation, which is a source that I have personally found helpful:

National Council of Teachers of Mathematics. (2010). Developing Essential Understanding of Rational Numbers for Teaching Mathematics in Grades 3-5. ERIC.

Here is the table of contents:

enter image description here enter image description here

The numbers in a fraction are different parts of speech. Denominators are nouns. Numerators are adjectives. Denominators say 'what' Numerators say 'how many'

This becomes somewhat more apparent in words. I have three quarters of a pie.

But you have to take it a step further: A fraction needs a prepostional phrase: It's always a fraction of something When we go abstract, the something can be thought of as being just 1.

So a half of 4 candies is 2 candies. Two quarters of 4 candies is 2 candies.

A half of 12 m&m's is 6 m&m's A third of 12 m&m's is 4 m&m's A third of 6 m&m's is 2 m&m's.

When you are doing these, you can lay them out in rows and arrays.

An array of 3x4 m&m's with one row being yellow:

What fraction of m&m's are yellow. Four twelfths.

Divide into columns each with 2 browns and a yellow.

Are the columns the same?

Yes

What fraction of the column is yellow?

One third

Play with this for a while. At the end let him eat the manipulatives.

If you are on a no-sugar regime, coins, cards and poker chips also work.

Note that this prepositional phrase requirement is what gets people screwed up with percents.

"The incidence of measles fatalities has risen 70% due to anti-vaxers" Sounds like an epidemic. But when you find out that it went from 10 to 17 out of 60 million kids it's not as alarming.

You can illustrate this with a puzzle:

"I'm sorry Mike, I have to give you a 50% pay cut. Instead of $12/hour I can only pay you $6 and hour. Maybe next month..."

A month goes by. "Mike, things are looking up. I'm giving you a 50% pay raise to $9/hour"

WHERE's my other 3 bucks!

7 years old seems quite young, so my answer is not specifically thought for this age (it is much inspired by a recent twitter thread about these questions for 12 yo), but I hope it can still help.

The example you give shows precisely how using "real world" interpretations can be misinterpreted, therefore I do not think that coming up with more real-world interpretation helps (such interpretation are needed in the course of learning fraction, but they are in my opinion not the solution to your particular problem).

What seems is missing is the very meaning of the symbols $\frac12$, $\frac24$. First and foremost : $\frac12$ is a number, not an operation. This is a common belief that needs to be dispelled, but is very understandable given the way we write. In your example, $\frac24$ is interpreted by the pupil as "2 out of 4", which it is not. I think that it might actually be easier to stick to the maths here in order not to confuse these things.

So what is $\frac24$ ? By definition, it is the unique number that, when multiplied by $4$, gives $2$. Nothing less, nothing more. You can lean on the relative integers to explain this:

"Do you remember that we could not perform the operation $3-5$ at some point, because we would have needed a number that, when added $5$, gives $3$? And we only had non-negative number, none of which would do? Then we introduced new numbers, e.g. $-2$, that yields solutions to such problems. Here it is the same: with only integers, we cannot find any number which, when multiplied by $4$, would give $2$. So we just added numbers to that effect, and denoted them with this somewhat weird notation that, you'll see, will prove very convenient in computation."

Note 1. If you have not yet dealt with negative numbers, maybe it would be easier to do so first. You can use level buttons in an elevator for concreteness, find this much better then the usual money/debt considerations.

Note 2. This is a fantastic opportunity to recall that the division operation is performed by actually solving an equation.

Now, how can one see that $\frac12=\frac24$? We should use the definition, and e.g. look at what happens to $\frac12$ when we multiply it by $4$. Its own definition only says what happens when we multiply it by $2$, so let us use that $4=2\times 2$ to observe that multiplying $\frac12$ by $4$ is the same as multiplying it by $2$, and then multiplying the result by $2$. By definition $\frac12\times 2 = 1$, so that $\frac12 \times 4 = \frac12\times 2\times 2 = 1\times 2 = 2$. Now by definition, the number that, when multiplied by $4$, gives $2$, is $\frac24$. So $\frac12 = \frac24$. That seems long and hard only because it is very detailed; this motivates the learning of computation techniques that will simplify these kind of reasoning and "factor" them quite a lot. Because we are used to these computations, we easily forget where they come from, but they need to be grounded in the definition in order to avoid them being merely magic formulas.

Other erroneous conceptions that will probably need to be dispelled along the way:

  • "two different sets of symbols must represent two different numbers": here it can be a problem to even imagine that we can use both $\frac12$ and $\frac24$ to denote the very same thing; this is unavoidable from our definitions: in our words, we want the number we added to be a solution to $4x=2$ to be unique, but it may well have been added as solution of another equation (here, $2x=1$),

  • "equality is like the = button of my pocket calculator: it is an oriented symbols that tells what happens when I perform the operation mentioned in the left term": very common conception, even among undergrads. Often $\frac12 =\frac24$ will puzzle students much more than $\frac24 = \frac12$, but in any case to understand what we mean by this they need to be told that '=' merely expresses that the terms on both sides are different guises for the very same object.

  • :'-( I will never teach my kid maths now. – NotAgain Dec 7 at 13:19
  • "First and foremost : 12 is a number, not an operation." - For the Greeks it was an operation, moreover it is still customary and often much easier to write division in a form of a fraction. I would stick to "2 out of four" or to "2 of 1/4" when learning fractions, then moving on to them being numbers by themselves. New Math blew up in flames for a reason. – Rusty Core Dec 7 at 17:09
  • @NotAgain: sorry to read that, could you point out what triggered this? My post is long, but I though it was so because of taking time to discuss each point. I obviously have failed, but would like to know where. – Benoît Kloeckner 2 days ago
  • @RustyCore: certainly there is a division operation, but when one writes e.g. "2 divided by 4 gives 1/2", then one thinks of 1/2 as a number (the result of the operation being performed). This seems universally agreed on, won't you agree? There are thus two things, an operation and a number, and one could prefer to have different notations for both (I do); even if one does not, then one should be aware of the two interpretations of the symbols. – Benoît Kloeckner 2 days ago
  • @RustyCore: Saying 2/4 is "two out of four" is problematic, because it can be interpreted wrongly as in the question (the kid prefers 2 candies out of a total of 4 than getting 1 out of a total of 2). Saying 2/4 is two fourth, on the other hand, is ok. You seem to misrepresent what New Math was; at least in France a large part of it was about introducing set theory and algebraic structures in secondary education. – Benoît Kloeckner 2 days ago

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