Clear It Addition is a fun game to practice and extend math fact fluency. Players must create number sentences that add up to the target numbers provided. Earn stars for extra points and clear the board before the timer runs out for extra points. The multiplier and the multiplicand are also called factors. The answer to a multiplication problem is called the product. Another way to look at the equation is in terms of addition: 2 + 2 + 2 = 6. So multiplication is a handy shortcut to adding groups of a number together. Multiplication has properties that are unique to its operation. Multiplication tables are important and there are not many places where you can learn them quickly and easily, so we have come up with Timestables.com! Practicing your tables online is really easy at Timestables.com. The multiplication games are clear and simple so you can get started right away. Click on one of the tables to get started right. Multiplication Chart for Kids is created in a way that can be easily understood and learnt by students of all stages. The list of Blank Multiplication Chart will enable students to learn and fill the chart online and this is an interesting way of making students learn the multiplication chart.
I want to close this series with a topic that arises constantly, both in classrooms and on social media: How do you evaluate an expression like (adiv bc) or (8div 4(3-1)), where the multiplication is indicated without a specific symbol? There are several reasons one might want to interpret this differently than the rule we’ve discussed, that multiplication and division are done from left to right. We’ll look at this first from the perspective of students and teachers, and then (next time) investigate some historical issues to close out the series.
Two ways to evaluate ax÷by
Let’s first look at one of the earlier questions we had about this issue, in 1999, to set the stage:
(Note that at that time, the only way to type division in our email was to use the slash, (a/b), which I generally assume represents an expression actually written as (adiv b). I will occasionally be inserting an obelus, ÷, where we made rough attempts to simulate it.)
The first way follows PEMDAS literally, as usually taught and as I’ve presented it here, by evaluating from left to right as (acdot xdiv bcdot y = ((acdot x)div b)cdot y).
The second sees it as (axdiv by = (ax)div (by)). This isn’t explained as following any taught rule, but just as doing what looks right, either because the division is read as if it were a fraction bar, or just because “by” looks like it belongs together as a unit. We’ll be seeing several reasons students have given for doing this.
Though I had been with Ask Dr. Math less than a year, this was already a familiar question, which I wanted to answer thoroughly for the sake of the archive:
Note that it is not only students doing what feels right, but also some textbooks and calculators that follow the second method.
A new rule, or what looks right?
I elaborated on the two methods, taking the PEMDAS version as correct (though I’ll have some second thoughts on that):
I think I had made up the term “implied, or implicit, multiplication” when I answered my first question on the topic a few months before, to refer to multiplication indicated by just putting two numbers or variables or parenthesized expressions next to one another – “juxtaposition“, as others call it – like ( ab) or ( 2b) or ( a(b+c)), as opposed to explicitly writing ( atimes b) or ( acdot b).
We had seen some questions from students whose textbooks taught only the usual PEMDAS, yet evaluated the second way in examples or solutions, without comment. This might have been due to the answers in the back being written by someone other than the author, but it is an inexcusable inconsistency.
Why would an author make this extra rule? I have had different opinions at various times about whether the rule is a good idea, but have always recognized that it is not what is usually taught:
A rule that is not a rule is worthless, no matter how reasonable it is. Yes, the “new rule” is the natural way to read (axdiv by) because (by) looks like a single entity; but until everyone teaches that, we can’t do it and expect to be understood by all readers.
In particular, many students assume that it represents a horizontal version of (displaystylefrac{ax}{by}):
By using parentheses, we can avoid writing something that people who were taught different rules, or who ignore the rules they were taught, might take differently than we intend.
Clear It Multiplication Abcya
Calculator issues
The link there went bad long ago; but when a specific question about a calculator came up in 2008, I quoted from what TI said in their Knowledge Base: Is codesall engineering e books for free .
This makes it clear that calculator designers have to decide on their own rules, which don’t have to be the same as rules for writing on paper; but educators seem to have convinced them to keep things as much the same as possible for students’ sake. Xjz survey remover crack.
In conclusion (back to the 1999 answer):
Subsequently, we had many more questions about this; I’ll just quote a few unique bits from some of those answers.
Old-fashioned math?
Here is a typical example of a school conflict, from 2000:
The problem is (Ndiv ml), and the parents are doing the multiplication first. I replied, in part:
I explained the standard rules, and added:
I’ll have more on the history next time.
More recently, the fights tend to be on social media!
Misapplying the distributive property
I’ll close with the most recent archived discussion. This question is from 2017:
I answered with a collection of my standard answers to this sort of question; even my first archived answer on the topic in 1999 was largely a standard response I had given to others before. Here, I’ll just look at a few points I made that haven’t been fully covered above.
I first summarized what was going on:
Clear It Multiplication Abcya Games
If you recall earlier statements that PEMDAS is (a) in harmony with the properties of operations, and (b) fitting with the visual impression of our notation, then some alarm bells should already be going off!
It is not really the distributive property that led to the “wrong” result, but the fact that in distributing, the 4 was seen as the multiplier.
I have observed that many students learn to distribute so well that they automatically do it when it is not helpful, as here, and even when it is not applicable, as when there is no addition at all!
Why students do it
In an unarchived answer in 2012, I listed reasons students have given for doing the multiplication first; on this occasion the question was about the expression (a^2div 4b + c):
For an example of “sticky parentheses”, see
For an example of seeing the division sign as a fraction bar (and a long discussion of not being swayed by visual appearance), see
Back to the 2017 answer …
Avoidance is the best policy
Abcya Multiples
I closed with a plea for peace:
Clear It Multiplication Abcya Games
Arguments on social media about this sort of thing are a waste of time. But thinking about our conventions can be very enlightening. Next time, I’ll close everything out with a look at history, and some solid reasons to think the “new rule” is in fact correct.