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The ideas in this division lesson are taken from Multiplication Division 2 ebook. Only a few examples of each problem type are shown; you should make more problems of each kind for the student. Long division and why it works
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The standard long division algorithmWe compare here the repeated subtraction of the previous lesson and the conventional long division 'corner'. The
steps are the same, just written out differently. For clarity's sake, we
will initially write out the subtracted numbers with all the zeros included. Also,
for clarity and for easy comparison, we will write the parts of the quotient
above each other. As an example, we study 789 ÷ 3. You can
think of it as 789 apples that you are bagging in bags of 3 apples, wanting to
know how many bags you need.
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Why it worksComparing the division to the continued subtraction probably has already let you see why it works. In the conventional way of writing the long division, it's not so easy to see the process. The key is that in each step, one does NOT actually divide by the actual divisor but by a multiple of it. Just like in the apples/bags examples, you don't start out by subtracting 3 apples each time, but first 'hit it hard' by subtracting multiples of 300 apples if possible, then multiples of 30, then 3. In essence, you first divide by 300, then by 30, then by 3. Also, in the conventional long division, you only place one digit into the quotient in each step, not with all the zeros. The digits shown in gray are not usually written out in the conventional long division algorithm.
To get the hundreds digit in the quotient, one asks the question: "How many times does
300 go into 789", or the
division 789 ÷ 300! You are not dividing by 3 because you try to 'hit it
hard' and subtract as many multiples of 300 as possible. Since 300 is a whole hundred, the tens and ones
digits in the 789 won't matter when you are finding how many times 300 goes into
789. So the thing can be done easier by calculating
7 ÷ 3, or thinking "How many times does 3 go into 7". To get the tens digit, similarly one asks the question: "How many times does 30 go into 189", or does the division 189 ÷ 30. Again, since you're dividing by a multiple of ten, the ones digit '9' in the 189 does not affect the division at all. The important thing is to look at the whole tens in the number 189, which is 180. So to find the answer to the division 189 ÷ 30, you can think of the division 180 ÷ 30, which is the same as thinking 18 ÷ 3: "How many times does 3 go into 18?" The last step is simple since it is dealing with ones digits, how many times does 3 go into 9. |
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These examples show how long division is done, with all of the dropping down of digits and such. It is important to keep the rows and columns lined up.
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850 ÷ 2 = ?
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1. Divide using long division. Check by multiplication.
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