# Tag Archives: multiplication

## Multiplication Practice Plus Creativity

I hope this post is particularly helpful for parents and teachers of elementary school children.  Through my Twitter network last week I found via @Maths_Master‘s Great Maths Teaching Ideas ‘blog a 2010 post summarizing Dan Finkel’s Damult dice game. Recognizing that “practicing times tables can be unmotivated and boring for kids,” Damult is an attempt to make learning elementary multiplication facts more entertaining. I offer some game variations and strategies following a description of the game.

Here’s Dan’s game:

image via Wikipedia

Each player takes turns rolling 3 dice. First to break 200 (or 500, etc.) wins. On your turn, you get to choose two dice to add together, then you multiply the sum by the final die. That’s your score for that turn.

Simple; no bells, no whistles. For example, I roll a 3, a 4, and a 6 on my turn. I could either do (3+4) times 6 for 42 points, OR (3+6) times 4 for 36 points, OR (4+6) times 3 for 30 points. I’ll take the 42 points.

I spent some time playing this with kids the other day and I saw that (1) it was genuinely fun, and (2) it gives you almost all the multiplication practice you could ask for. In fact, it gives even more, because the choice of which dice to add and which to multiply reveals some interesting structure of numbers. Seriously, get a kid hooked on this game, and it’s the equivalent of dozens or hundreds of times table practice sheets.

It’s a fun activity idea by itself.  Damult combines a bit of luck and memory, and rewards the ability to recall multiplication facts.  As an added bonus, it requires players to be able to manipulate objects in their heads–how many different ways can the three given dice be manipulated in summation stage to create unique products? How can a player ensure that she has found the biggest product for her score?  Try the game!

CONNECTIONS AND EXTENSIONS:

This is a great opportunity for parents to engage with their  children as they learn multiplication facts.  Parents and teachers could play along, or the learner might be the only player, talking out loud so that the teacher or parent can “hear the thinking.”

I love that the game completely randomizes the multiplication tables.  This significantly enhances recall as memory is not tied to particular patterns or positions on fact pages. Players must adapt to each random roll.

In any variation, there obviously should be a discussion among all players about what products were found to confirm the results. Make the game more formative or more competitive, depending on the experience level of the players.  In more competitive variations with experienced learners, if a product was miscalculated and claimed, you might decide that no score should be recorded for that round.

If you’re guiding someone on this it is critical that you DO NOT give answers.  Students need to explore, hypothesize, discover errors, learn how to communicate their conclusions in clear and concise language, and to learn how to defend their findings while also learning how to admit flaws in their reasoning when faced with contradicting data.  Experimenting and discovery is always deeper, richer,and more long-lasting than just being told.  Remember the Chinese Proverb: “I hear and I forget. I see and I remember. I do and I understand.” Always seek understanding.

The first comments on Dan’s post noted that while one player was summing and multiplying, the other player(s) were largely disengaged. Also, the game could drag on as unconfident players tried to make sure  they had explored every possibility.  To address that and several other possibilities, I offer the following Damult variations.  Some more complex variations are toward the end. Read on!

Finally, if you’ve read my ‘blog much, you know that I’m a huge fan of leveraging technology for math learning, but this is one of those situations where I think you should 100% unplug. To learn multiplication facts is to learn some of the basic grammar and vocabulary that makes the language of mathematics work.  You simply can’t communicate mathematically with an underlying awareness of how the structure of the language works.

GAME VARIATIONS:

Variation 1: Adding a timer to the game could cure the slow-down issue. Depending on the age of the child and his/her familiarity with multiplication, the timer can be longer or shorter.  If the skill levels of the players are unequal, make the timer unequal.  (I love the adage, “Fair is seldom equal, and equal is seldom fair.”)

Variation 2: Why must only one player be active? The players could take turns rolling the dice while both record scores based on what they find.  If a particular combination was not noticed by one player, that player doesn’t get to consider it for his/her score.

Variation 3 – As an aside, notice that Dan implicitly claims there are only 3 possible sums from a 3-dice roll. Will that always be the case?  Can you convince someone why your solution is correct?

(For 3 dice the maximum number of possible sums is 3. When and why would there be fewer products?)

Variation 4 – How many multiplication facts are possible using only 3 dice?

This would be a great number sense exploration.  Some may try it by gathering lots of data, others may have more sophisticated reasoning.  I suggest that you or your students hypothesize an answer first along with some reason why you think your hypothesis is correct.  Different answers are OK, and you can always revise your hypotheses if you get evidence leaning in another direction. No matter what, have fun exploring and learning.

(Middle School extension: Damult creates products of axb where a can be any integer 1 – 6 and b can be any integer 2 – 12.  That gives 66 different products if you count different arrangements (3×4 and 4×3) as different products. Can you or your student see why? How many outcomes are possible if you look only at the product result and not at the factors which created it?)

Variation 5 – After discovering or just using the answer to the last variation, you could use a table of multiplication facts and see how quickly different facts and be “discovered” from rolls of the dice.  After rolling 3 dice, mark off all multiplication facts you can using the sum-then-multiply combination rules posed at the beginning.  This might be a fun way for early learners to familiarize themselves with multiplication patterns.

NOTE: If you play variations 4 or 5 as a game, you’ll likely want (or need) to stop before all possibilities are found.  Some (eg, 6×12 and 1×2) will be pretty uncommon from dice rolls.

Variation 6 – You could make a Bingo-like or a 4 or 5-in-a-row game.  The first person to mark off a certain number of facts or the first to get a certain number in a row would be a winner.

Variation 6 – If you try the last few variations, you’ll see that some products occur much more frequently from the dice rolls than others.  This could be used to introduce probability. Which products are more likely and why?

As an example, I suspect 3×7 could happen six times more often 1×2.  Can you convince yourself why 3×7 is so much more likely?  Can you see why 3×7 is exactly six times more likely than 1×2?

Variation 7 – Why restrict yourself to 3 dice? When just starting out, using more than 3 dice would definitely be a frustration factor, but once you’ve got a good grip on the game, consider rolling 4 dice and allow players to multiply the sum of any 2 or 3 of the dice by the sum of the remaining dice.

By my computation, using 4 dice means there are up to 7 possible combinations in a given roll.  Can you prove that? Being able to consistently find them all is likely to be a very difficult challenge, but it is a phenomenal and early opportunity to stretch a young person’s mind into considering multiple outcomes and reliable ways to guarantee that you’ve considered all possibilities.

Variation 8 – Why go for maximum products and being the first to get to 200 or 500 points?  Why not try for a low score (like golf), seeking minimum products  and being the last to exceed 100 or 200?

Variation 9 – Stealthy Calculus:  OK, my analysis on this one goes way deeper than is necessary to play the game, but sometimes knowing more than is necessary can give insights and can help you lead others toward developing “math sense”–a truly invaluable skill.

LOW LEVEL – After you’ve played this a few times, ask the player(s) if there is some strategy that could be used to guarantee the biggest (or smallest) possible product for any roll.  This could be a great mathematical experiment for which the solutions are not at all intuitive, I think.  Some might figure it out quickly and others might need to gather lots of data, comparing products from lots of rolls before distilling the relationship.

If you’re guiding someone on this it is critical that you DO NOT give the answer.  Students need to explore, hypothesize, learn how to communicate their conclusions in clear and concise language, and to learn how to defend their findings while also learning how to admit flaws in their reasoning when faced with contradicting data.  If you don’t know the answer, stop reading now and figure it out for yourself. I provide an answer in the next paragraphs, but experimenting and discovery is always deeper, richer,and more long-lasting than just being told.  Remember the Chinese Proverb: “I hear and I forget. I see and I remember. I do and I understand.” Always seek understanding.

MUCH HIGHER LEVEL – As a calculus teacher, the very first fact that struck me was Damult’s implied goal: Getting the largest possible product from any roll of three dice.  That’s an optimization problem, and I knew from calculus that the greatest possible product of two numbers whose sum was constant happens when the two numbers are as close as possible to being equal.  Likewise, the smallest possible product happens when the two factors are as far apart as possible.  (If you recall some calculus of derivatives, I encourage you to prove these for yourself.  If anyone asks, I could write a future post with the proof.)

In Dan’s initial example above in which 3, 4, and 6 were rolled, I stopped reading after the first sentence of paragraph 2 (pausing to think and draw your own conclusions is a great habit of the mind) for a few moments as I thought, “I know 3+4 and 6 are as close to equivalent as I can get, so 7*6=42 is the greatest possible product.”  I didn’t even look at the other possibilities, I knew they were less. This fact was established (unnecessarily for me) in the end of the paragraph.

Without calculus, I propose students try making tables of their data.  They’ll have up to three unique products (Variation 3) and will need to explore the data before hopefully discovering the relationship. If a young person doesn’t discover the relationship, Don’t tell him/her! it is far better to leave a question as unanswered to think on and answer another day than to have a relationship given unearned.  Value comes from effort and discovery. Don’t cheat young learners out of that experience or lesson.

Conclusion: Don’t just play a game. Be creative! Strategize! Encourage young ones not just to play, but to play well. Children are quite creative in free play as they continually make new and adapt old “rules”.  Why should intellectual play be any different?  I’d love to see what variations others discover or have to offer.

## Extending Multiplication to Algebra and Calculus

In my earlier post on a multiplication trick for young learners, I forgot to include a higher level connection, so I’ll do that here.

LEVEL 6:  Advanced Algebra or Calculus.
Part of what makes this problem work out so nicely is that the sums of the numbers involved in the lists are constant, an omnipresent property of position-symmetric terms from any arithmetic sequence.  That is, given any arithmetic sequence of sufficient length (e.g., {1, 2, 3, 4, 5, 6, 7, 8}), the first and last terms (1 & 8) will always have the same sum as the terms just inside those (2 & 7), the next terms inside (3 & 6), and so forth until you run out of terms.  This property can be proved for any arithmetic sequence with introductory algebra.

Here’s the advanced math connection.  Given all pairs of numbers which add to some constant sum, k, what is the relationship between the pair of these numbers whose product is maximum?

Let the two numbers be X and Y where $X+Y=k$.  We want to find the maximum of the product $X\cdot Y=X\cdot (k-X)$.  If you recognize this last expression as a quadratic function, $f(X)=X\cdot (k-X)$, then you can use the location of the vertex of a parabola (algebra) or some calculus (find the zero of $\displaystyle\frac{df}{dX}$) to see that the maximum of f occurs at $\displaystyle X=\frac{k}{2}=Y$.

Complicated math aside, this means that when you have a list of pairs of numbers that all add to the same constant, the pair with the biggest product is always the pair of numbers closest in proximity to each other.  In the context of this problem, this explains why the inner products of any arithmetic sequence are always larger than the products of the first and last terms.

## Multiplication Puzzle 2 for the Very Young

A former student of mine, Paul Sperduto, is currently teaching 5th grade math in Houston as part of Teach for America.  After reading yesterday’s post on inspiring multiplication in young learners, he responded with the following story.  I reprint it here (with Paul’s permission) for two reasons:  to give another fun math game for parents and teachers to use with eager young learners, AND to show what can happen when young people have the freedom to think and explore (whether explicitly designed or personally re-claimed as in Paul’s case).

Here’s Paul’s story in his own words with occasional commentary from me in brackets.

I use a similar multiplication “trick” with my students, and I think it is setting them up very well to learn and understand applications for the difference of squares down the road. It’s actually something I randomly happened into in about 3rd grade (I was one of those elementary school kids who knew his times tables before I even started 1st grade, so I had a lot of math class time to think about stuff like this…), but I didn’t really understand why until I was much, much older.

[As I’ve noted earlier, it is far more important for younger learners to play the game.  Finding a pattern you think others have overlooked is far more motivating for young people than knowing why the trick works.]

I told my students that I could mentally multiply any two numbers between 1 and 31 in under five seconds as long as the numbers have an even difference. For example, I can multiply $23\cdot 27$ in under five seconds since $27-23=4$, but I can’t do $23\cdot 26$ that quickly because $26-23=3$.

[This is a glorious hook!  From experience, many students will jump at a claim like this.  Part of the game becomes trying to find two numbers the teacher can’t handle.  Learning = game = fun.]

When they asked me how I do it, I told them that if they memorize their squares (I have mine memorized through 30), they can do it too. This was surprisingly motivational, and I quickly had many students with a lot of squares memorized, eager to learn how to use them.

[This is what happens when you make learning FUN.  Even seemingly dry topics like memorizing squares of numbers becomes a worthwhile endeavor when it has an entertaining purpose.  Students are willing to engage in what they perceive to be drudgery if there is a payoff.]

The trick lies in the difference of two squares formula, $A^2 - B^2 = (A+B)*(A-B)$. Given the example of multiplying 23 and 27, it is easy to see that 25 would be the mean of the two numbers–each is 2 away from 25. So if A is the mean of the numbers, and B is the distance to the mean from each, the difference of squares formula gives us the answer to the problem:

$27\cdot 23 = (25+2)\cdot (25-2) = 25^2 - 2^2 = 625 - 4 = 621$

That last step is easy once you’ve memorized the squares.  So as long as you do this and are good with generally simple subtraction, this trick is very easy.

It’s been fun trying to help them discover this one. We started by noticing patterns near the squares on a standard multiplication chart. They noticed that all squares have another multiplication that gives one less than the square. For example $6\cdot 8=7^2-1$ and $10\cdot 12=11^2-1$. Working our way along the diagonals of the chart, they also discovered that each square has a multiplication that gives 4 less than the square ($5\cdot 9 =7^2-4$), 9 less than the square ($4\cdot 10= 7^2-9$) and 16 less than the square ($3\cdot 11=7^2-16$). It took a little prodding, but they figured out that the differences are just the squares, and it was all downhill from there.

[Here’s another thought:  I suspect participants typically make these problems easier on the “performer” while thinking they’re doing just the opposite.  Many likely think giving two bigger numbers to multiply would be a harder task because the algorithms they typically are required to use in school become longer, if not more challenging, when longer numbers are employed.  But using two larger numbers under Paul’s approach probably makes the value of B smaller in most cases, simplifying the final subtraction.

Thank you, Paul, for sharing.]

## Multiplication Puzzle for the Very Young

I just read a recent post on NRICH Mathematics that asked readers or students to list four consecutive whole numbers and compare the products of the outer pair of numbers in the list to the product of the inner pair.  For example, if you used the list {4, 5, 6, 7}, you would have $4\cdot 7=28$ and $5\cdot 6=30$.  Nothing particularly exciting seems to be here, but try another list of four consecutive whole numbers.  Grab a calculator if you want to be particularly daring or obnoxious with the members in your list.  Do you notice anything now?

I argue the beauty of mathematics as the “science of patterns” kicks in after you find these products for a few different lists.

LEVEL 1:  For the very young who are just learning to multiply, I think this is a GRAND problem.  No proof required.  It’s just crazy cool that those two products always have the same relationship.  Allowing calculators to permit young explorers to try lists beyond their ability to hand or mentally compute enhances the mystery, in my opinion.

I just played this with my eldest daughter.  She first wrote {19, 20, 21, 22} when I asked her for a list of consecutive numbers.  When I then asked her for the products, she asked if she could use a smaller list.  She opted for {3, 4, 5, 6} and {1, 2, 3, 4} without seeing the pattern.  When I offered a calculator for her original list, she got 418 & 420.  Surprised that they were so close, she said, “Wow, they’re only 2 apart!” I asked if that happened other times.  She looked at her simpler two lists and exclaimed, “Cool!”  I asked if that always happened.  She said, “No.  It couldn’t.”  When I asked for a list where it wouldn’t, she suggested {401, 402, 403, 404}.  The outer product was 162004.  You should have seen her face after she pressed enter on the inner product to get 162006.  “Maybe it does always work!”  Then she asked if she could move on to clean her desk.  Game over … for now.

Part of the power and beauty of mathematics lies in showing that patterns are universal and aren’t limited to numbers we can manipulate quickly in our heads.  I think calculators added to my daughter’s wonder.  I’d love to see my daughter going up to one of her teachers, posing the problem, and predicting the answer without ever knowing the numbers the teacher (or anyone else) had picked.  I think I’d smile even bigger if she had a calculator at hand to offer the adult some “help” if needed!  Math is magical.  Play it up!

LEVEL 2a:  Extend to all integers.  NRICH suggests that the lists need to be whole numbers.  That just isn’t true.  You can start with any integer.  My eldest has been playing with adding negative numbers lately, so I may see if she’s interested in multiplication of negatives.  I’ll think about how to make that idea make sense to her.  At some point in the future, I’ll bring this problem up again and she’ll get an even bigger kick out of seeing that it doesn’t just apply to ordinary and ridiculously large numbers, but negatives, too.

LEVEL 2b:  Proof for the very young.  The NRICH site offers two solutions from “students”.  Whether she’s real or fictional, the approach “Alison” uses is one that I think some sophisticated young learners could grasp long before they learn what a variable is.  Granted, the geometric understanding of multiplication technically works only for specific (not generic) products, but if you set up a few of these, your young one might start to see how the areas grow as the list numbers grow, but the differences in the areas remain constant.

NOTE:  LEVELS 2a and 2b, in my mind, are pretty interchangeable, depending on the readiness and interest of your young learners.  As with all things for young people, throw out the line.  If the interest isn’t there, save the idea for another day.  If you get a nibble, prepare to play!!!

LEVEL 3:  Extend to any arithmetic sequence.  The suggestions NRICH makes for extending the problem all dance around the idea that this property works for any list of four consecutive elements of any arithmetic sequence.  The difference between the two products depends solely on the common difference of the sequence and is completely independent of the initial term in the sequence.  Try {1.1, 1.2, 1.3, 1.4}.  The difference in the outer and inner pair products will be the same as for {98.8, 98.9, 99.0, 99.1} simply because both lists increase by 0.1.

LEVEL 4:  Algebra.  Those who remember their algebra classes may have jumped right to an algebraic justification.  That’s what I did, and that’s the solution “Charlie” gives on the original NRICH post.  In a way, I think I cheated myself out of seeking the pattern as my daughter discovered it.  Whenever your young ones are ready to deal with the magic and power of variables, try out proving this for integers.  When they’re ready for more, prove it for all arithmetic sequences with any initial term.  You’ll know they’re strong when they can argue on their own why the initial term is irrelevant.

LEVEL 5:  More Algebra.  This “trick” extends to to any arithmetic sequence of any length.  With algebra, one can determine a formula for the difference between the products of the last terms and the next-to-last terms.  I think a talented middle school student or young high school student who knows how to handle very generic cases could find that formula.

And it all starts with playing with some little numbers.

## Elementary Multiplication

One of my daughters is now in 2nd grade and I’ve always been interested in keeping her curiosity piqued–whether in math or any other discipline. I never want to push her to memorize anything or accelerate her learning beyond what she’s ready to engage.  But she has always enjoyed games and has been intensely interested in art.  Following are some ideas I’ve been playing with my daughter during our recent conversations.  Perhaps some of parents out there can benefit from my ideas or others can give me some additional leads on other good ideas

I always play number games with my daughter.  A few years ago I asked her how many apples (or dolls, or crackers, or whatever was in front of her at the time) she would have if she had 2 and I gave her 2 more.  There were many variation on this theme.  Eventually the numbers grew larger and then I asked her how many I would need to give her if she had 2 apples now and would have 5 after my donation.  It was my attempt at introducing subtraction without needing to name a new concept.  From my end, this has worked well.  My daughter likes playing with numbers and I keep pushing the window of what she can handle.  I make it clear that she can always ask for hints and that I’m never disappointed if she can’t handle a question I give her so long as she tries.  It’s a delicate balancing act, reading my daughter’s readiness and trying not to overburden her.  When I misjudge, her blank face tells me to go in another direction.

Today, she mentioned negative numbers and I jumped on her exploration of commutativity.  She told me that she knew “somehow” that a subtraction gave a negative result if the “second number was bigger.”  I told her that the only difference between 10-13 and the 13-10 that she already knew was that 10-13 gave a negative answer.  Further details can happen later, but for now, I jumped on a moment of interest and continued a game that we’ve been playing for months. Her face lit up when she realized that negative numbers really aren’t that hard!  It’s never about memorizing facts and I’m always ready to back off.  My mantra:  Keep reading your audience and keep it fun.

Here’s another set-up I started a week ago.  I’ve never seen multiplication started from this angle (but I’ve not been trained as an elementary teacher either).  Nevertheless, I was thinking about how to introduce the concept of multiplication without making it a chore or making it a new idea, so I tried tapping into her art interest.  Two weeks ago, I asked her how many ways she could arrange 6 dots into rectangles.  Grabbing some paper, she quickly made an arrangement of 2 rows and 3 columns and a short time later, 3 rows and 2 columns.  It took some prompting to get her to see a line of dots as a rectangle 1 unit high (or 1 unit wide), but the hook was set.  What follows is a sampling from a journal she keeps for playing around with shapes or math ideas.  I had asked her to try this rectangle arrangement of dots for every number from 1 to 20 using what she had learned from arranging 6 dots.  I asked her to list beside each arrangement the dimensions of the rectangles she could find.  She missed a few, but I’m a pretty proud dad right now.

While she’s just scratching the surface of multiplication right now, I’m pretty psyched that she has written multiplication while thinking that she was just describing the dimensions of rectangles.  What I really think is cool is that she is learning multiplication in reverse–starting with the products and learning the factors.  Eventually, we’ll do this in the “normal” order, but for  now, the art connection has her totally hooked.

Also, she already knows about even numbers and odd numbers.  For now, I have future plans to introduce prime numbers as those that can form exactly 2 rectangles.  We’ll also explore commutativity further once she gets more comfortable with multiplication.  Down the road, I see showing her that 1 is a special number because it is the only number that has only 1 rectangle.  Also, perfect squares (the square numbers that we’ve also discussed) are the only numbers with an odd number of possible rectangles–another consequence of commutativity.

I’d love any feedback on these rambling musings.

## “Digit”al Multiplication

So here’s another musing I had on a beach visit. I don’t recall where I learned this trick, but I’ve had it for decades.  I suspect most of you already know how you can multiply by 9 using your fingers, but I’ll briefly explain just in case.  An extension follows.

Start by laying out both hands with all fingers outstretched.  Number your fingers from 1 to 10 from left to right.

To compute $9*n$ for integer values of $n$ between 1 and 10, fold down the $n^{th}$ finger and count the number of still-extended fingers before and after the folded finger. Thinking of those two numbers as a two-digit number gives your answer. For example, to compute $9*9$, fold down the $9^{th}$ finger as shown below.

Because there are 8 fingers before the fold and 1 after, $9*9=81$. Simple.

On the beach, I recalled this little trick and first extended it to two other simple multiples of 9. Computing $9*1$ is simple, but folding down the $1^{st}$ finger confirms the answer by showing 0 before and 9 fingers after the fold, so $9*1=09=9$.

Computing $9*10$ works the same way.  Folding down the $10^{th}$ finger shows 9 before and 0 fingers after the fold, so $9*10=90$.

As I lay on the sand, I marveled at how nice this worked, but was saddened that such a cute approach had such limited applicability.  Thinking about what was going on in this problem, the obvious observation was that all were products of 9 and worked from 10 initial fingers.  So what would happen if you used a different number of fingers?

Starting with 7 fingers, numbered as before, I thought I might be able to multiply by 6.  As a first attempt, I tried $6*6$, but the next image shows that my approach gives $6*6=51$–obviously not the correct result.

Then inspiration struck.  Perhaps the answer was good, but my interpretation was off.  If multiplication by 9s (using 10 fingers) gave an answer in base-10, perhaps multiplication by 6s (using 7 fingers) needed to be interpreted in base-7.  That is, $6*6=51_7=(5*7^1+1*7^0)_{10}=(35+1)_{10}=36_{10}$.  Eureka!  But does it always work?

Testing once more, I tried using 5 fingers meaning I would be multiplying by one less (4) and getting an answer in the base of the number of fingers I was using.  The next image shows $4*2=13_5=(1*5^1+3*5^0)_{10}=(5+3)_{10}=8_{10}$.

Generalizing, imagine that you could hold out any number of fingers.  The image below suggests that you extended $n$ “digits”, so you could use this to multiply by $(n-1)$.

The next image supposes that you hold down the $k^{th}$ finger leaving $k-1$ fingers before and $n-k$ fingers after.

That suggests $(n-1)*k=(k-1)(n-k)_n$ where $(k-1)(n-k)$ is a 2-digit number whose left digit is $(k-l)$ and whose right digit is $(n-k)$.  Expanding, $(k-1)(n-k)_n=((k-1)*n^1+(n-k)*n^0)_{10}$ and $(n-1)*k=(k-1)(n-k)_n=(k*n-k)_{10}$.  QED

So, if you just had enough “digits” (pun intended) and didn’t mind working in different number bases, the result of every single multiplication could be known with one simple finger fold!