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.

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2 responses to “Multiplication Practice Plus Creativity

  1. Good stuff.
    I like the unplugged approach. But my mind quickly jumps to having the students write a program on a graphing calc. It could be as simple as generating the three numbers each time up to…
    I like the idea of providing a big blank grid matrix and they write in as they play.

    • Thanks, Travis. I had been thinking all along about the Elementary side of this, but certainly if a student could conceive of programming as a way to save time and energy, then I’d say charge right on ahead. NIce extension! Thanks.

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