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I am teaching math to a 10 year old student. He learned that $$\frac{300}{100}=\require{cancel}\frac{3\cancel{00}}{1\cancel{00}}=3$$ and $$\frac{6000}{300}=\require{cancel}\frac{60\cancel{00}}{3\cancel{00}}=\frac{60}{3}=\frac{20+20+20}{3}=20$$

But now I want to teach him how to compute $\frac{300}{200}$. Before I explain this he responded quickly that first we cancel out zeroes:

$$\frac{300}{200}=\require{cancel}\frac{3\cancel{00}}{2\cancel{00}}=\frac{3}{2}$$

Well, his argument is correct but I want to teach him that $300=1\times 200+100$ (using this method)

and remainder is $100$. But in previous solution its remainder is $1$ ($3=1\times 2+1$). How can I explain this without fooling or tricking him?

P.S. Note that he started his education at the age of seven.

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    $\begingroup$ "he started his education at the age of seven" So he had no formal education at all before 7 years old? $\endgroup$ – Joel Reyes Noche Sep 21 at 10:31
  • $\begingroup$ Yes. It is usual here that most children start at the age of seven. Before 7 they just learn paint, and kids related games-art. $\endgroup$ – C.F.G Sep 21 at 11:18
  • $\begingroup$ What age kids start to learn math in your country? $\endgroup$ – C.F.G Sep 21 at 11:25
  • $\begingroup$ When I was young, I entered kindergarten at 4 years old, but most of my classmates were 5 years old. Now, we have nursery (pre-school 1), so formal education can start at around 4 years old (in the Philippines). Kindergarten is pre-school 2, Prep is pre-school 3. I remember thinking that one of my Prep tests was very difficult. The questions I remember were 2-0=? and 2-2=? $\endgroup$ – Joel Reyes Noche Sep 21 at 13:07
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    $\begingroup$ Just like in a decimal system units are added to units and tens to tens, with regular fractions you can add twohundredths to twohundredths. You have 300 of them, but you can split them any way you want, like 200 of them and 100 of them. 200 twohundredths happens to be one, and 100 twohundredths is one half. There nothing special here, the same associativity rule as for integers. And if the denominators are different, then you cannot add them as is, here the common denominator comes into play. Here: Math-U-See: Fractions $\endgroup$ – Rusty Core Sep 21 at 17:06
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...his argument is correct but I want to teach him that $300=1\times 200+100$ and reminder is $100$. But in previous solution its reminder is $1$ ($3=1\times 2+1$). How can I explain this without fooling or tricking him?

If I understood it correctly, the question is how to explain the fact that equivalent fractions leave different remainders. Well, you could try this:

  • Explain that a fraction represents a part of a whole.

  • Explain that an equality between two fractions means that both fractions represent the same part.

  • Illustrate using pictures: enter image description here

  • Conclude that the remainders of equal fractions are possibly different because the whole was divided into different parts: remainder $100$ for $\frac{300}{200}$ means $100$ parts of $200$ small pieces while remainder $1$ for $\frac{3}{2}$ means only $1$ part of $2$ bigger pieces.

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Unless I'm misinterpreting the question, is this about division with remainders?

I'd probably ask with "how many X are in Y?" and use composite numbers as my Y's to get lots of results

For example, "How many 10's are in 60?", "how many 2's in 60?", "how many 15's in 60?"

Then I'd ask, "how many 25's are in 60?" And then you might get something like "there are 2 full 25's in 60 but then you have some left over" And then you can ask, "how many are left over?"

If you want applications of this, you could ask, if you have a 60 books and you have a bookshelf with 6 shelves, how many books on each shelf if you want the same number on each shelf. (I think with books you'd prevent decimal or fractional answers)

Although I've never taught remainders, 300 and 200 seem to be big numbers? Not like I can recall learning this, but I'm sure I must have learned remainders by drawing something like 13 dots and then drawing circles around them so that there were five dots in each circle, and then realizing 3 were outside of any circle. If the student is new to remainders, maybe try numbers that could be dotted out?

Continuing with that, then I'd have 2 groups of 5 and 3 left over, and connect that to the idea that $2 \times 5 + 3 = 13$ is "the amount of groups" times "the amount in each groups" add "amount left over".

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  • $\begingroup$ Yeas my question is about division with remainders. I know that ways you encountered. But how he can come up with these two different approach? $\endgroup$ – C.F.G Sep 21 at 11:22
  • $\begingroup$ Also he is learned counting to 1 billion. and 300 is nothing for him. $\endgroup$ – C.F.G Sep 21 at 11:22
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    $\begingroup$ @C.F.G Counting to one billion? He should be over 30 years old now, counting one per second without taking time to eat or sleep. $\endgroup$ – Rusty Core Sep 21 at 17:09

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