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How to convince a middle school student that $0.50=0.5=0.500=\cdots$?

I used the fact that $0.50=\frac{5}{10}+\frac{0}{100}=\frac{5}{10}=0.5$ but that far from intuitive.

Then I tried to explain that $50$ apples to $100$ apples is the same as $5$ apples to $10$ apples (the half): $0.50=\frac{50}{100}=\frac{1}{2}=\frac{5}{10}=0.5$. It worked in this simple example but I'm seeking an intuitive way to use for all numbers (e.g. to explain why $520.100500=520.1005$ and not $52.15$ or $520.15$).

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What was unintuitive about $0.50=(5/10)+(0/100)$? This should appeal to what decimal notation actually means, so if the student doesn't follow that explanation, this might signal a misunderstanding of decimals. –  brendansullivan07 Apr 2 at 22:36
    
Or if you've already taught how to simplify fractions tell them to put it in simplest terms so $0.5=\frac{5}{10}=\frac{1}{2}=\frac{50}{100}=0.50$ –  ruler501 Apr 2 at 22:45
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In physics and similar, 0.5 and 0.500 can really make a difference when the value means that is it some measured quantity. The latter one means 0.500 is measured (up that accuracy), but first one menas that the measured quantity is 0.5?? (only one digit of accuracy). –  Markus Klein Apr 3 at 6:13
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@MarkusKlein That is wrong. Accuracy should never be given by the number of digits, but by an explicit uncertainty e.g. $0.5\pm0.05$ or $0.5\pm0.0005$. –  Toscho Apr 10 at 21:11
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10 Answers 10

When it comes to convincing younger students of something, I find that analogy can be quite useful, even if you have to squint a bit to make it technically rigourous.

I imagine a younger student reasoning something like this: "Hey, I know 5 is different from 50, and 50 is different from 500. When you add zeros at the end of number, it changes the number. So why shouldn't this be true after the decimal point?"

Here's a potential antidote to this line of reasoning: "It's true that 50 is different from 500, but what about 050, or 0050? What are those numbers? We usually don't write zeros before the leftmost number, right? Because it doesn't change anything. Really what's happening is that we just assume it's all zeros to the left. Similarly, after the decimal point, we don't write zeros after the rightmost number because it doesn't change anything---we just assume that it's all zeros to the right."

Let me say right away that I concede this explanation is somewhat less than satisfying from a technical perspective. It's not incorrect, but it doesn't exactly shed light on every aspect of decimal notation that's relevant here. Nonetheless, I think it's a reasonable first step at an explanation, something that might be revisited and fleshed out in future lessons, or in later years. My thinking is, if you hit them with too much detail while they're skeptical, no matter how good your intentions, you stand a good chance of (effectively) just making an argument from authority. They might not really understand your explanation, but decide to just accept it's true for mysterious reasons. I tend to prefer to offer them a simple analogy that they can use to overcome their initial skepticism, opening the way for a more detailed explanation later on.

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My own view is this sort of "analogy" might work as a mnemonic, but it can also lead many to feel like mathematics is a particularly illogical area of study. Sometimes writing down zeros changes the number; sometimes it doesn't. I'll just have to remember the rule that says 050 = 50 just like 0.5 = 0.50. To me, this method of teaching does the students a disservice. Better to, for example, start by explaining 1 digit numbers. Then explain 2 digit numbers. Then talk about what the decimal system means. Then 3 digit numbers... and, someday, tenths, hundredths, etc. –  Benjamin Dickman Apr 3 at 9:47
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I'd have thought that phrasing it as "Adding Zeros only changes the number if they come between a different digit and the decimal point." –  Chris Apr 3 at 11:02
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@BenjaminDickman I agree that this way of teaching might cause trouble if it's how you approach teaching numbers to begin with. But I was imagining students who are already quite comfortable with the idea that the first "0" in "050" is redundant, taking that comfort, and transferring it to a less familiar situation. The goal isn't to show that math is logical as opposed to illogical, but rather to help them think of it as familiar rather than mysterious. To be honest, for most kids, I think the latter needs to come before the former. –  Adam Bjorndahl Apr 3 at 14:16
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This seems like a very good way of getting the student to pause and go "Well, I... hmm". Then you can explain in detail why this is the case. –  Jack M Apr 4 at 3:16
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This is especially good because it reflects how decimal notation is just that: notation. It's fairly arbitrary, but we all agree on it. We all agree that .5 == .500, but it makes alot of sense that 050 == 50. –  br1ckb0t Apr 4 at 12:23

Check out Berkeley mathematician H.H. Wu's homepage.

In particular, see his textbook drafts for Pre-Algebra (pdf) and Introduction to School Algebra (pdf).

For example, see p. 20 and the discussion of decimals as "a class of fractions" in the former text.

Note that the book is intended for teachers of 6th-8th graders and not for students.

The exposition can, at times, be more rigorous than what you might be used to or expect.

At the least, I think this can help clarify some of the instructor's thinking around topics like decimals; after that, it will be up to you how best to adapt the presentation so as to make it useful for your students.

Specific example: See p. 28 in the Pre-Algebra text; I will paste a screen-shot below.

enter image description here

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Thanks for the references, but H.H. Wu's method is just what I showed in my question. I don't want a proof of this fact (I already proved it), I want them to understand it. Like understanding why $4\times 7=7\times 4$, you don't need an axiomatic approach, you just need some examples and maybe analogies. –  metacompactness Apr 3 at 9:45
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@metacompactness My guess is Wu would write $0.50 = \frac{50}{100} = \frac{5 \times 10}{10 \times 10} = \frac{5}{10} = 0.5$, but this relies on a familiarity with the earlier material as he has set it up. Note that he is decomposing a product and then using a cancellation law, whereas you rewrote it as a sum, i.e., $0.50 = \frac{5}{10} + \frac{0}{100}$, etc. –  Benjamin Dickman Apr 3 at 9:54

Turn the problem around. Say to the student:

If "5.00" is not the same number as "5", then one of them is larger than the other. Which, and by how much? You're the one who claims there is a difference. Then tell me, literally "what is the difference"? Does a movie ticket priced at "five dollars and no cents" cost more or less than one priced at "five dollars", or are those the same price expressed two different ways?

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Essentially the same idea: Let them calculate $0.5 + 0.\overline{3}$ and $0.50 + 0.\overline{3}$. –  Wrzlprmft Apr 3 at 21:06
    
This works well for good students (who might not even have this problem at all). But for weak students, it adds an extra level of abstractedness. –  Toscho Apr 10 at 21:23

Let me play devil's advocate. They are not identical. In situations where complex calculations result in a number such as 6.3856, and the science teacher is asking for 4 significant digits after decimal, one wouldn't round at all. If the answer happened to be 5.0000, the student should keep the zeros to indicate the level of accuracy, that no rounding was involved.

My middle schooler learned about significant digits in a science class, and I could see a potential conflict as she applied this reasoning when it wasn't really necessary.

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Significant digits is a notation issue I believe. Math teachers should be responsible to teach what the notation means, and they represent the same number, so I think that is still an important lesson to teach. –  ruler501 Apr 3 at 15:17
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This is why math should not be taught by engineers. –  jwg Apr 4 at 10:10

If the student struggles with this task, what do you think he (or she) thinks of a decimal number? What is .37 for him? Let him try to explain the meaning of .37 in terms of examples or calculations where this number is used. A number must have a deeper meaning than just being a sequence of digits in order to understand how we use them. One of your examples ($520.100500=52.15$) would indicate even problems with the naturals ($520=52$?)! The student should also be able so say that $520 = 5\cdot100+2\cdot 10+0\cdot1$. Based on this description of a natural number, fractions like $1/10$ in decimals are not too far away. You may convince your student and it may work for some technical tasks. However, for understanding, there is no way but fractions. Thus, I recommend Benjamins answer.

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This would appear to be an essential problem with understanding the place value system. If the student hasn't already understood place value, showing and explaining is unlikely to be effective. First off, you may be unaware of what the student's conception of the value of these numbers is. An activity might help you get to the heart of that, if not help the student see through the misconception.

A number line representation, if the student already understands it, can be used to scaffold this place value understanding.

But first, it might help to have a representation that shows the symmetry of naming of places around the one's place.

diagram of place names

Ask the student to consider, what does a zero mean in any of those places? The answer may be helpful to you.

Next, draw a number line like so:

enter image description here

Have the student place numbers on this line to illustrate relative positions. Make sure that 5.0 and 5.00 are among the numbers, but try also numbers with values in the tenths and hundredths place.

What does it mean if numbers (5.0, 5, and 5.00) are in the same spot on this number line?

Of course, a student could place these numbers at juuuuuust before 5.1, for example.

Then it would appear you need to have the student focus more heavily on each place of place value in the questions he answers. He may not see 105.3 as One hundred(s) AND 5 ones AND three tenths. Tell him that you want him to talk a while using this long name of this type of number.

What is 5.0? What is 5.00?

What does it mean to have no tenths? No hundredths? What is five (plus/and) no tenths?

Cheers.

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Perhaps are the student more used to real world rounded values where 0.5 and 0.500 are different. The former just mean some number between 0.45 included and 0.55 excluded while the latter is more precise, between 0.4995 and 0.5005.

You should then tell them this being a Math course, you are not talking about the rounded approximations used for example in Science and Engineering but about mathematical unambiguous representations.

The next and tougher exercise would be to convince them 0.5 and 0.4999999999999999999... represent strictly identical numbers ... ;-)

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Make sure if you teach this you emphasize that it is only the meaning of the lack of zeroes in the sciences and engineering as far as I know. In math you generally wouldn't say that 0.5 can have multiple values. –  ruler501 Apr 5 at 1:01
    
@ruler501 Thanks for your comment, answer updated to clarify . –  jlliagre Apr 5 at 1:07
    
+1 - your answer adds to mine in the implication that the unwritten digits may have been rounded off. –  JoeTaxpayer Apr 8 at 14:31
    
Giving uncertainty by the number of decimals is non-standard in sciences and engineering. The uncertainty has to be given explicitely. bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf –  Toscho Apr 10 at 21:26
    
@Toscho It should indeed but rarely is. It is interesting the very same document you provide a link to doesn't follow this rule by containing this very sentence Thus the measurand should be specified as, for example, the length of the bar at 25,00 °C –  jlliagre Apr 10 at 22:00

It seems to me there is a conceptual mistake hidden in this misunderstanding: many students (and, sadly, probably many educators) misidentify numbers with their decimal expansions. Number are represented by, not identical to their decimal expansions.

Can you convince someone of this? No. It's rather like the plus/quus-type problems in the Kripkenstein literature. On the other hand, finding a student making this type of misidentification is the chance for a great learning opportunity; I think the previous responses have identified many ways to bring out the consequences of this mistake: integers become ambiguous; applications cease to make sense, etc. In this way, you are confronting the student with the cost of maintaining their point of view. Hopefully they find it too expensive.

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I developed my personal understanding of zeros similar to how Adam describes it only a bit different. I understand zeros in the following way:

The number 1.1 can be represented as:

01.1
001.1
0001.1

They are the same number therefore the leading zeros can be discarded. Similarly, it can also be represented as:

1.10
1.100
1.1000

They are the same number therefore the trailing zeros can be discarded.

The important part is the dot. Zeros close to the dot mean something since they tell you the position of the digit and therefore its magnitude. Zeros far away are redundant and can therefore be discarded.

I'm not entirely sure if others find the above notion intuitive but that's how I understood it when I was a kid.

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Use this opportunity for revisiting place-value-systems and other systems. The 0 in the decimal system was originally just a gap, an empty space. The eastern arabic numerals still only have the zero as a centered dot.

Another example are roman numerals. What's the difference between

XVI and X VI?

What's the difference between

123 and 1_23?

123 and 1023?

123 and 0123?

123 and 1230?

12.3 and 12.03?

12.3 and 12.30?

Answer a 0 has no value in itself, but pushes other digits outward onto different values and thus changes the value of a numbers. But if the 0 is at the outward end of a number, it can't push other digits outward. So it doesn't change the value.

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