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48 votes
Accepted

Which product of single digits do children usually get wrong?

https://www.theguardian.com/news/datablog/2013/may/31/times-tables-hardest-easiest-children There are links to a dataset in the article. As far as I can tell, this isn't a formal study: But some new ...
Adam's user avatar
  • 5,733
33 votes

Is this primarily a "rote computational trick" for multiplication by 9?

Anything that is just a trick leads to students having wrong ideas about what math is. But methods that help students see the patterns can help them learn the multiplication facts, along with getting ...
Sue VanHattum's user avatar
  • 20.8k
20 votes

Is this primarily a "rote computational trick" for multiplication by 9?

Yes. This is also a trick that you can do on your fingers, too. For instance, let's say you wanted to calculate $9\times3$. Hold out your hands and bend your third finger down as shown. So nine ...
Matthew Daly's user avatar
  • 5,619
20 votes

What are the arguments for and against learning the multiplication table by heart?

Let me add my 2 cents in. As someone who grew up in Latin America (where the tables were drilled into us, or else) and who now tutors mathematics, more than once I've had the experience of students ...
Jorge Medina's user avatar
17 votes
Accepted

How do I explain the distributivity of multiplication to a student without using the analogy of areas?

It depends on how the student conceptualises multiplication. First, find that out. Then design an approach that uses it. For example, if they think of multiplication as repeated addition, then you ...
Nullius in Verba's user avatar
17 votes

How do I explain the distributivity of multiplication to a student without using the analogy of areas?

This is based on a framing device I used in a Khan Academy comment helping students get an intuitive appreciation of the distributive rule. It is just a description of $a(b+c)=ab+ac$, but it might be ...
Matthew Daly's user avatar
  • 5,619
11 votes

Is this primarily a "rote computational trick" for multiplication by 9?

To add on to the other answers, the reason this works is because we use the decimal system, a.k.a. the base-10 system, for our everyday maths. The multiples of the number that is one less than the ...
Abion47's user avatar
  • 271
8 votes

Arguments against multiplication by 'stacking'

Some context is missing here. But I'll go ahead and speculate scenarios for why a math teacher would (temporarily?) prohibit multiplication by stacking [and in writing? --- nothing stops the student ...
Math Misery's user avatar
7 votes
Accepted

Is this primarily a "rote computational trick" for multiplication by 9?

Note $9 = 10-1$ so: $$ 5 \times 9 = 5 \times (10-1) = 50 - 5 = 45, $$ and the same for all the others: $$ 8 \times 9 = 8 \times (10-1) = 80 - 8 = 72. $$ This works for $k \times 9$ where $1 \le k\le ...
Gerald Edgar's user avatar
  • 7,607
5 votes

How many hours / school years does it take for the average child to memorize the $10\times 10$ addition and multiplication tables?

When/where I grew up in California, ordinary second-graders (7 - 8 year-olds) were expected to learn the times tables well enough to answer randomly chosen problems without writing anything down or ...
Jasper's user avatar
  • 3,178
5 votes

Which product of single digits do children usually get wrong?

In the comments, it seems some people are surprised that $4 \times 8$ and $6 \times 8$ have such low accuracy (as shown in the table in the accepted answer). There's actually a cognitive principle ...
Justin Skycak's user avatar
4 votes

How do I explain the distributivity of multiplication to a student without using the analogy of areas?

If you want to avoid area, then the next closest thing is arrays. See: https://www.khanacademy.org/math/7th-grade-foundations-engageny/7th-m2-engage-ny-foundations/7th-m2-tc-foundations/a/distributive-...
Aeryk's user avatar
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4 votes

How do I explain the distributivity of multiplication to a student without using the analogy of areas?

Reteach multiplication of multi-digit numbers (eg $12 \times 34$) by pen and paper in terms of $(10+2)(30+4)$. You should reconcile their early memories of arithmetics with this somewhat new concept ...
okzoomer's user avatar
  • 341
4 votes

How many hours / school years does it take for the average child to memorize the $10\times 10$ addition and multiplication tables?

Assuming that the German "Kerncurriculum" (actually, there are 16 of them, one for each federal state. I'm referring to the one from Lower Saxony) for primary school is at least somewhat tailored for ...
Jasper's user avatar
  • 2,699
3 votes

Is this primarily a "rote computational trick" for multiplication by 9?

One neat trick that might help for some of your students is the digital root of a number. For a number, the sum of its digits is taken. For example, $$ sumDigits(12345) = 1 + 2 + 3 + 4 + 5 = 15 $$ ...
Daniel Soutar's user avatar
3 votes

Resources for Learning Multiplication Facts

As for 3's, I always found it easy as a student to think it as $3=2+1$. So, in order to multiply by 3, one has to double and then to add what they've found. For instance: $$3\times7=2\times7+7=14+7=...
Vassilis Markos's user avatar
3 votes

Multiplying two decimals using (camouflaged) binary representation

I am someone (in a country), and I teach it in certain (college) classes. In a discrete math class I teach it and then use the fact that if you multiply $n\times 1$ with this method (halving $n$ ...
paw88789's user avatar
  • 655
3 votes

Arguments against multiplication by 'stacking'

Although anecdotes are usually badly reported samples of size one, I'll answer with one. Trying to diagnose his problems with multiplication, I tried giving my grade school son variants of the same ...
Dan Fox's user avatar
  • 5,869
3 votes

Arguments against multiplication by 'stacking'

Maybe instead of the "stacking" method we learned in the last century, they are supposed to use the "diagonal" method like this
Gerald Edgar's user avatar
  • 7,607
2 votes

When should a kid have memorized the multiplication table?

The answer of this question might be found in the question: "When are multiplications used in primary education?", and for answering that question, let's have a look at following example: the ...
Dominique's user avatar
  • 2,119
1 vote

How do I explain the distributivity of multiplication to a student without using the analogy of areas?

If you do not want to use area, here is another way: You have 9 bracelets. Each bracelet has 7 beads on it. Now you put 4 bracelets on your left hand and 5 on your right. How many beads you have in ...
BKE's user avatar
  • 1,292
1 vote

How do I explain the distributivity of multiplication to a student without using the analogy of areas?

I recommend Cuisenair Rods. While one could argue that their use essentially breaks down to area it's very intuitive and accessible. As always, a physical involvement like manually handling the rods ...
Peter - Reinstate Monica's user avatar
1 vote

How do I explain the distributivity of multiplication to a student without using the analogy of areas?

My highschool teacher would say If you have two times three apples and five plums, you have six apples and ten plums. That is, $$ 2(3a+5p)=2\times3\,a+2\times5\,p $$
Martin Argerami's user avatar
1 vote

How do I explain the distributivity of multiplication to a student without using the analogy of areas?

Despite your requirement, I recommend using array and area models. I suggest engaging them in tasks such as the following: Write as many algebraic expressions as you can (using integers and the 4 ...
Steven Gubkin's user avatar
1 vote

Is this primarily a "rote computational trick" for multiplication by 9?

I was taught the multiplication table for single digit numbers multipled by $9$ based on two observations: a. the digits of the result sum to $9$ b. the first digit of $k \times 9$ is $k-1$. This ...
Dan Fox's user avatar
  • 5,869
1 vote

Is this primarily a "rote computational trick" for multiplication by 9?

In answer to the question "I've never seen this before" I knew this 'trick' with the 9x table and the one with 11x table also mentioned here when I went to primary school about 50 years ago. I will ...
Anthony Sach's user avatar
1 vote

Arguments against multiplication by 'stacking'

I've never heard of 'stacking', the method shown appears the standard way to multiply digit by digit. 37 squared. For those who wish to do this in their head, I'd go up to 40, and the same 3 down to ...
JTP - Apologise to Monica's user avatar

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