38 votes
Accepted

Dividing by zero

You asked: "How do/would you explain why division by zero does not produce a result." Any such explanation that is not rooted in student understanding would be talking to ourselves, not to ...
  • 7,385
28 votes

Dividing by zero

Have the students tell you that division by zero is a non-sequitur. This is possible at any age where division is understood at all. Teacher: If there are eight cookies and four children, how many ...
  • 4,890
22 votes

Dividing by zero

In third grade we taught division using repeated subtraction. To divide 6 by 2, subtract 2 until you get to 0. 6-2=4, 4-2=2, 2-2=0. It took 3 steps so 6÷2=3. This can also be shown on a number ...
  • 7,055
16 votes

Commonly taught method divides by zero

The method is mathematically incorrect, but whether it is wrong or not, depends on whether people know what they're doing or not. The possible division by zero is a mistake, yes. But it is a mistake ...
  • 824
15 votes

Is This Trick Helpful?

I do not think that such tricks are helpful: in fact I believe they are deeply damaging. These types of basic relationships should not be memorized: they should be derived on the fly from an ...
12 votes

Dividing by zero

Dividing $1$ disk into $\frac{1}{n}$-ths ($\frac{1}{3}, \ldots$), leads to $\frac{1}{1/n} = n$ pieces ($3,\ldots$): As $\frac{1}{n}$ approaches $0$, the number of pieces $n$ grows without bound. The ...
11 votes

How is $\frac{a}{b}$ interpreted?

A US specific answer: The Common Core State Standards define $\frac{1}{b}$ by saying it is one of $b$ equal parts making up a whole $1$. $\frac{a}{b}$ is then defined as $a$ of these. Connecting $\...
10 votes

Dividing by zero

This answer is intended for the second category of students: How do/would you explain why division by zero is undefined for Algebra students? Begin by introducing the reciprocal of a real number as ...
8 votes

How to teach quick multiplication and division in head?

"She knows the 10x10 multiplication table in principle, but it takes a huge amount of time till she recalls the result. Sometimes she even needs some seconds to calculate 2x3=6." This is absolutely ...
8 votes

Dividing by zero

Having recently covered this topic in a course for pre-service elementary school teachers, I thought I would write a bit about the somewhat subtle difficulties entailed in tackling this question. I am ...
7 votes

Dividing by zero

Here is my answer to high school underclassmen. If we let $\frac10=n$ for any $n$, we then get $$\frac10 \cdot 0 = n \cdot 0$$ $$1=0$$ This works for any non-zero value divided by zero. Allowing any ...
  • 2,504
7 votes

Is short division taught these days and if not, why not?

If you are tutoring, it's important to value whatever algorithms work. Your frustration with the (new to me) lovely algorithm you show concerns me. It shows why each step makes sense, which is much ...
  • 18k
6 votes

Is short division taught these days and if not, why not?

Not a teacher here, but I noticed when my kids went to school there was far less emphasis than I remember on techniques that require above average insight or intuition. I think there's more pressure ...
6 votes
Accepted

Determining the first digit of the Quotient using hand long division efficiently?

The awkwardness of "guessing" in the division algorithm is an artifact of the base-ten representation of numbers. If you represent in binary, then your only possible "guess" is 1. In binary, your ...
  • 8,093
5 votes

When and Why are different division symbols taught?

1$\;$ When students begin learning arithmetic, they first learn how to do $18+3$, then $18-3$, and $18*3$. Rather than using fraction form to teach division, it's useful to have a symbol that can be ...
5 votes

How to teach quick multiplication and division in head?

In my experience, I feel this is best learned by just practicing. I understand that it may seem remedial for whatever lessons you are giving (or perhaps it isn't) and that it can take away from time ...
5 votes

How is $\frac{a}{b}$ interpreted?

A key idea in maths education, that at least becomes more visible at university level, is that people tend to start thinking of new ideas as processes, but to do more advanced maths they need to move ...
  • 5,576
4 votes

Dividing by zero

Division is multiplication, backwards. These two equations are exactly equivalent, by definition: $$a / c = b$$ $$a = b \times c$$ It's easy to understand division by zero if we look at the ...
4 votes

Dividing by zero

Division itself is defined like this: The result of division (the quotient) of "b divided by a" is defined as the number x in a⋅x = b. For example, let's divide 8 by 4: We look for the number x ...
4 votes

How to teach quick multiplication and division in head?

Permit me to direct you to read an answer to another question by another user Benjamin Dickman first, notably the second part of the answer that begins with the line "Given the above discussion, I ...
4 votes

How is $\frac{a}{b}$ interpreted?

By definition, a rational number is a number that can be expressed as the quotient of two integers. This quotient is called fraction and is written as $\frac{a}{b}$. Hence, division and fraction are ...
  • 1,280
4 votes

What are strategies that a 10-13 year old could use to convert 6/27 into decimal notation?

I prefer method 2, which has the student simplifying first. Even if the student doesn't know that $ \frac{1}{9} = 0.\overline{1} $, the student can still divide 2.00 by 9. If the student uses ...
  • 7,055
3 votes

Determining the first digit of the Quotient using hand long division efficiently?

Round. Check. Correct. What we need is $436\div48$. Rounding to remove the last digit reduces the problem to $44\div5$. This gives us an educated guess of either $8$ or $9$. From here we then ...
3 votes

Dividing by zero

I think it's important to explain students that formally it's not defined, but also give them motivation for why it is not defined. Yes, definitions are in some sense arbitrary, but we have ...
3 votes

How to teach quick multiplication and division in head?

I have 2 ideas for you. First, I have helped a student get number intuition by getting him to building factorisation lattices. For instance, 12 can be represented by a 2D, 2 node by 3 node grid/...
  • 2,588
3 votes

How to teach quick multiplication and division in head?

In fourth grade, our daily math class included a three minute timed exercise. Each student had a page of approximately 50 or 100 math problems. All of the math problems were of one kind -- either ...
  • 3,041
3 votes

Is short division taught these days and if not, why not?

Both of my sons are learning division right now (or rather, just finished the section), one in a public school using Eureka Math (in 4th grade level math), one in a Montessori (in Primary, at 1st ...
  • 161

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