How to Conquer the Times Table, Part 2

Photo of Eeva times 6, by Eric Horst, via flickr.

The question is common on parenting forums:

My daughter is in 4th grade. She has been studying multiplication in school for nearly a year, but she still stumbles over the facts and counts on her fingers. How can I help her?

Many people resort to flashcards and worksheets in such situations, and computer games that flash the math facts are quite popular with parents. I recommend a different approach: Challenge your student to a joint experiment in mental math. Over the next two months, without flashcards or memory drill, how many math facts can the two of you learn together?

We will use the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible.

Make a Monster Times Table

If she is willing to take the challenge, you will need a way to keep track of your progress. Use a manila file folder or tape two pieces of copy paper together, long sides touching. With a ruler, draw a large, blank times table. Take the chart up to 12 × 12, or even higher (and include a column for zero, if you like). One online friend credits his math skills to working out a 30 × 30 times table as a child.

Your daughter has almost certainly seen a chart like this before, but even so, she will probably find it intimidating. Help her fill in the chart for herself by adding or skip counting the numbers in each row and column.

For instance, to write the times-7 facts, she can imagine using the multiplication ray gun’s Replicate setting to copy a row of 7 blocks, over and over again. As she moves from one row to the next, she will write the total number of blocks, adding seven and seven and seven more…

Be patient. If your daughter is afraid of math, it may take several days to fill in the chart, a little bit at a time.

When the chart is finished, hang it on the refrigerator or some other prominent place. Your daughter will be able to find any particular math fact by looking where the appropriate row and column meet. The answer to $6 \times 7 = ?$ is found where the 6th row meets the 7th column (or the 7th row and 6th column), because it is like counting up six of the sevens.

At first glance, that chart looks like a real pain: 144 math facts to memorize. But we are not going to memorize hardly anything — we’re just going to look for number patterns.

No Worksheets!

Our goal in these times table conversations is to build up your child’s mental math skills. Therefore, do not resort to worksheets in an effort to teach the math facts. Instead, take the time — and it does take time — to talk through these patterns and work many, many, many oral math problems together. Discuss the different ways you can find each answer. Notice how the number patterns connect to each other.

When you are practicing each family of rules, be sure to experiment with larger numbers, too. Make it into a game, and take turns quizzing each other for just a few minutes at a time. Students love the chance to stump their parents. Try working mental multiplication puzzles while you are doing dishes, or on the way to soccer practice, or whenever you can find a spare moment of time.

Even the patterns which seem easy are worth spending time to master. For instance, the magic power of one — that it can multiply a number without changing the value — is essential to working with fractions and can simplify a multitude of high school chemistry problems.

The Ones Family

Sit down with your daughter and the chart. Use a highlighter to color in the facts she already knows, making a bright, dynamic statement of how good she is at math. You can mark out the times-1 facts very quickly. Surely your student knows that anything times one is itself, right? With the scale factor set at “1”, the multiplication ray gun won’t change anything.

7 ones = 7
359 $\times$ 1 = 359
etc.

Practice with large and small numbers, fractions, and more:

1 $\times \frac {1}{2}$ = $\frac {1}{2}$
4.6 $\times$ 1 = 4.6
1 $\times$ 897 = 897

Now have your child color in the times-1 row and column with the highlighter.

The Tens Family

Remind your student of how easy the times-10 facts are:

2 tens is 20.
5 tens is 50.
9 tens is 90.
11 tens is eleventy, or 110.
12 tens is twelvety, or 120.
17 tens is 17-ty, or 170.
156 tens is 156-ty, or 1,560.
4,000 tens is 4,000-ty, or 40,000.

Practice the pattern with small numbers, big numbers, and anything in between. Then mark times-10 as mastered, coloring both the row and the column.

More Easy Facts

You can get rid of nearly half the chart in one sweep of the marker. Does your student know — really, truly, thoroughly understand — that $3 \times 4$ is exactly the same as $4 \times 3$ ? Multiplication is commutative, which means you can move the numbers around without changing the answer: 3 rows of 4 blocks have the same total number as 4 rows of 3 blocks.

Spend a day (or a week, or however long your child needs) practicing, to make sure this principle sticks thoroughly in mind:

Q: What is 7 $\times$ 8?
A: The same as 8 $\times$ 7.

Q: What is 115 $\times$ 6?
A: The same as 6 $\times$ 115.

78 $\times$ 49 is the same as 49 $\times$ 78.
$\frac {3}{5} \times \frac {4}{13}$ is the same as $\frac {4}{13} \times \frac {3}{5}$.
192 $\times$ 7 is the same as 7 $\times$ 192.

Then you can mark out all the facts on the lower, left-hand section of the chart that have duplicates on the upper, right-hand section.

Don’t try to do too much at once. So far, you have marked off nearly two-thirds of the chart. Good job! Now would be a great time to take a break and do some fun math — like multiplying a batch of cookies.

This is the third post in my Times Table Series. To be continued…[Go to part 3.]

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15 thoughts on “How to Conquer the Times Table, Part 2”

1. Have you considered drawing the multiplication table to scale, on a fine grid so that each number along the edges has a length of that number of boxes, and the products are actually the areas?

I find that drawing it that way awakens all kinds of connections and makes a lot of things easier to remember for some people.

2. What a cool idea! It would probably take a few sheets of poster board, but it would make so many connections: the commutative property would show clearly, and the square numbers would be obvious, and it would make an easy transition into the area model for multiplying polynomials…

3. A rather interesting alternative. Its not just young kids who struggle with their times tables, at times even my teen students are not able to produce answers to basic multiplication problems and simply rely on their calculators to do all the work for them. I reckon that is pretty unhealthy. In your professional opinion, what would you recommend to troubleshoot such “issues” faced by these 17-18 year olds?

4. I know it is not politically correct, but I still see value in students being able to chant their tables and give a reflexive response to a question

5. Being able to give a reflexive response is pretty useful, just to be that quick. But does the chanting really help you get there?

It’s sort of like the way most people only know the alphabet by the alphabet song. So if you ask them “What’s the 20th letter” or “what comes before p?” or “recite it in reverse order”, they might be very, very slow. Someone who’s a good filer will know the alphabet in a much more rich way.

I think something more like a You Can Count On Monsters approach might be good here, if only I could figure out what it should look like.

1. How funny! I was just starting to type the same thing, when I saw Joshua’s post. I agree with Steve that a reflexive response is the ideal for math facts, but I don’t think chanting the tables helps us get there effectively. For instance, in order to remember whether J comes before or after L, I have to go back and recite half the alphabet. I don’t want my students to have to do that with times tables.

6. Whitecorp, my recommendation for teens who are struggling with this kind of math is to find them a group of very young students just barely beginning to learn about multiplication that they can “teach” (with support and scaffolding) via fun discovery conversations like the ones Denise describes above.

The best way to learn something is to teach somebody else, and there is far more dignity in teaching an older student to help a younger student in this way than in directly teaching the teen herself.

7. Thanks Mary for your suggestion-indeed one learns best by teaching others, but many Singaporean teens (who are bitingly practical and exceedingly impatient) simply want to get over and done with asap when it comes to maths, so peer tutoring to them would only be a redundant, senseless chore in their eyes unless they are rewarded with certain incentives. When done unwillingly and with the wrong set of motivations, I doubt much benefits can be reaped from such coaching of little juniors.

The approach of educating them on the merits of helping oneself by helping others won’t quite work over here given the current education dynamics and trends prevalent in Singapore; honestly I would be more than relieved if I can somehow get my students to improve their personal mental sums skills. I wouldn’t even dream of getting them to teach younger kids.

8. I love this idea! Searching for patterns in the multiplication table demystifies it, for sure. I wish I’d learned it this way.

9. My friend David Millar drew up some nice versions of the to-scale multiplication table that I mentioned in the first comment here: http://thegriddle.net/educators

I also wrote a blog post about one way I use those drawings as the summary of a problem-solving activity, http://uncoverafew.wordpress.com/2011/11/07/the-multiplication-table-part-1/ . The point is to give a new way of seeing the table, understanding as a whole rather than as 100 individual cells, emphasizing the distributive property and reminding students of some of the relationships in the table by having them view it with a different goal in mind.

10. I will definitely add these to my file of multiplication ideas. Thank you, Joshua, and David too!

11. I have spent a great deal of time lately thinking about multiplication after I observed a 2nd grade class, in a district implementing CGI math, create and solve multiplication problems while developing great number sense – http://suedowning.blogspot.com/2012/03/marvelous-math-play-date.html

Also, while developing a multiplication game using a 100s chart, I discovered just how few possible products there are when multiplying 1 through 10. I decided to use gray for all of the non-answers and realized that kids spend countless hours of frustration trying to learn just 42 numbers, half of which are either one-digit or multiples of ten – http://suedowning.blogspot.com/2012/03/is-that-all-there-are.html

1. I like your grayed-out hundred chart, Sue. The math facts definitely looks less intimidating when viewed like that. I’m often amazed how many parents stress out over memorizing facts, when it’s ever so much more important to spend time playing with numbers and developing good number sense.

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