# Tag Info

34

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 students. Therefore both meaning and student understanding are important. Otherwise, what's the point? So I have grounded my response there. Young students (...

28

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 cookies does each child get? Student: Uh, two. Teacher: Yes! This is a division problem. $\frac{8}{4} = 2$. Now, if there are 8 cookies shared by only two ...

26

It's pretty rare these days for anyone to actually do division by hand. Most people reach for a calculator. Given those realities, I would question whether it even makes sense that we spend such a vast amount of time teaching young children even one algorithm for division. Maybe we should postpone it, deemphasize it, or replace long division with a slower or ...

17

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 line, where it takes 3 steps of 2 units to go from 6 to 0. Teaching the concept of division this way is just the inverse of what we have done for multiplication. ...

16

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 one can ignore if in the end you get a solution to the differential equation. To make it full proof, what I do is the following sequence of steps. Possibly ...

15

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 understanding of the meaning of the operations and the meaning of equality. Namely, you should understand that $a = \frac{b}{c}$ is asserting that the number $a$ is ...

11

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 that number that satisfies the property of reciprocals: $$a⋅\dfrac 1{a}= 1$$ and emphasize that $a≠0$ because $0$ times any number is $0$. Next state the ...

11

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 result upon division by $0$, $1/0$, should be this limit. But there is no limit.

11

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 $\frac{a}{b}$ to $a \div b$ requires some reasoning. For instance $\frac{5}{3}$ of a candy bar means you take your one candy bar, divide it into 3 equal sized ...

8

"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 not knowing the multiplication table. As a community college lecturer with lots of remedial courses, I see this a lot; a student will say they know the times ...

8

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 going to use language that may be at the level of undergraduate majors in mathematics or mathematics education, but I believe that the content can be scaled so ...

7

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 value here leads to a contradiction. If we let $\frac00=n$ we get $$\frac00 \cdot 0 = n \cdot 0$$ $$0=0$$ This works for any value of $n$, so which value of $... 6 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 equivalent multiplication. $$a / 0 = b$$ $$a = b \times 0$$ For any real number$b$: $$a = b \times 0 = 0$$ $$a = 0$$ There are two cases with division by zero: If ... 6 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 problem is to divide the six-bit number 110000 into the thirteen-bit number 1000100010000. Scanning the bits of the second number(the dividend) from left to ... 6 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 better than the form of long division I learned. "In fact a lot of kids find it [short division] amazing to learn." If you show an individual a new process at ... 5 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 used like$+, -,$and$*$to teach division. This is where$÷$comes into play. It's easier for them to see$18÷3$than$\frac{18}{3}$. 2$\;$The second method ... 5 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 you need to teach other subjects. I honestly do find that I have certain values (multiplication tables, special fractions) just "memorized", but I still know ... 5 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 these days toward making sure most students achieve a predetermined minimum performance, and less on helping high-performing students stretch their capabilities. ... 5 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 on to thinking of the same things as objects. That is, we are taught to think of$2+3$as an instruction to take$2$and add$3$to it. Most people will write$...

4

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 with: 4⋅x = 8 We find x = 2. So 8 divided by 4 equals 2. Now, if we want to divide for example 8 by zero (!) we have to look for x with 0⋅x = 8. But, you would ...

4

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 would like to make one additional comment" I cannot improve on the linked answer but I can connect it to your question. What you call an "intuition for numbers" ...

4

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 the same, at least in the context of dividing integers and turning them into rationals. This concept is expanded in middle school for dividing irrationals and ...

3

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 multiply and check, either one. Take $8$ for an example. $48\times8=384$ and $436-384=52$. By looking at what we have left, we know that we need only one more $48$ ...

3

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/lattice: 12 6 4 3 2 1 They get really pretty when you get 3D lattices (3 unique factors, like 60). It can help to ...

3

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 adding 2-digit numbers, or subtracting 2-digit numbers, or multiplying numbers between 0 and 12, or dividing numbers between 0 and 144 by numbers between 1 and 12....

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I just realised that this solution is similar to NiloCK but it is different. Teacher: With these 6 pens if I put 2 in each bag how many bags will I have? Students: 3 Teacher: With these 6 pens if I put 1 in each bag how many bags will I have? Students: 6 Teacher: With these 6 pens if I put half a pen in each bag how many bags will I have? Students: 12 ...

3

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 grade age). The public school focused on long division certainly, but (either because of the curriculum, or more likely because of the teacher, as I don't ...

2

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 motivations for defining things certain ways, and I think it is good for high school students to grasp some of those motivations and intuitions behind certain definitions....

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