Leftorvers with 100 game

In Grades 3 and 4, we played a nice game that (I think) we got from Marilyn Burns. Looking for a reference after the fact, I see it explained in her book Lessons for Extending Division.

Basic play

• Start with a target number (we used 100) and collection of available divisors (we used integers 1 to 20)
• Players take turns choosing a divisor from the remaining available options. They divide the current target by that divisor and keep the remainder as their score for the turn. They also subtract the remainder from the target to create a new target for the next player.
• Each divisor gets crossed out when it is used, so it can only be used once.
• The game ends when the target is reduced to 0 or when all available divisors are exhausted.
• We played as a two player game.

Here's an example of a game play:

Player 1 chooses 17. 100 = 17 * 5 + 15, so player 1 scores 15 points, the target is reduced to 85, and 17 is no longer available as a divisor.

Player 2 chooses 20. 85 = 20 * 4 + 5, so player 2 scores 5 points, the target is reduced to 80, and 20 is no longer available as a divisor.

Player 1 chooses 14. 80 = 14 * 5 + 10, so player 1 scores 10 points, the target is reduced to 70, and 14 is no longer available as a divisor.

Player 2 chooses 18. 70 = 18 * 3 + 16, so player 2 scores 16 points, the target is reduced to 54, and 18 is no longer available as a divisor.

Player 1 chooses 19. 54 = 19 * 2 + 16, so player 1 scores 16 points, the target is reduced to 38, and 19 is no longer available as a divisor.

Player 2 chooses 13. 28 = 13 * 2 + 12, so player 2 scores 12 points, the target is reduced to 26, and 13 is no longer available as a divisor.

Player 1 chooses 15. 26 = 15 * 1 + 11, so player 1 scores 11 points, the target is reduced to 15, and 15 is no longer available as a divisor.

Player 2 chooses 16. 15 = 16 * 0 + 15, so player 2 scores 15 points, the target is reduced to 0 and the game ends.

Player one wins 52 to 48.

Our experience

We found this to be a fun, interesting, and engaging game. The practice with dividing and remainders was pretty obvious. In addition, it opened up some opportunities for strategic thinking, particularly at the end-stage of the game. I think there are also several good extension explorations.

Extensions

First, I created a simple pencilcode program for two players to play this game against each other. Here's a playable version (and here's the code).

Second, you'll notice that the first player in our sample game followed a "greedy strategy."  At each stage, that player chose the divisor that would give the most points on that turn. If you look closely, that isn't the best strategy at the end of the game.

So, a natural exploration is to find the best strategy for different starting targets. One specific point of about which we're curious: is it ever desirable to skip your turn (choosing 1 as the divisor is effectively a turn skip)? 

Some other areas for investigation:

• must the game always end on 0 or can we run out of divisors?
• given a target and collection of starting divisors, what is the shortest (number of turns) game possible? What is the longest game (number of turns) that does end at 0?

Class summaries

Quick notes on the activities for the classes today.

Grades 1 and 2

Mobius loops
We are splitting up games with other activities. This week, we are introducing several geometry explorations. First up are Mobius strip activities, nicely shown in this Matt Parker second favorite shape:

1. cylindrical loop: using 2 colored pencils, draw a line along the center of the loop on the inside and the outside. We can see that there are two sides, no big surprise. Cut along one of the lines and the loop splits into two new loops
2. Mobius strip: give the paper a twist. Now, draw along the middle of the paper and see that there is only one side. Now, cut along this line and see what happens. Repeat this, drawing another line along the center of the new strip. Do you have one side or two? Again, cut along the central line. What do you get?
3. Two connected loops: tape two,  untwisted loops, together in perpendicular directions. Now cut along the center lines of each loop. What do you get?
4. Two connected Mobius strips: tape together two mobius strips and cut along their center lines. What results now? Did everyone get the same result?

There are some natural extension explorations:

• try these with more twists (as per Matt's video)
• keep cutting the center lines
• Connect a Mobius strip and an untwisted loop (half-way step between 3 and 4). Now, cut along the center lines. What happens?
• Inspired by the thinner and thinner loops, kids can explore ways to cut paper so that they get longer and longer strips or loops

Note: these activities can be even more rewarding when something goes wrong. For example, what if there isn't enough tape connected the ends of the loops? These mishaps make everyone pause and consider more carefully what is actually happening.

Also, in the class, we only had time for the first two make-and-cuts, then demonstrated the two connected loops.

Punch (fold and cut)
All this cutting fits nicely with our second exploration: the punch activities from Joel David Hamkins' post punch, fold, and cut from Joel David Hamkins.

Grades 3 and 4

We started with the Shapes x Shapes puzzle from NRICH:

We added a couple of extra questions to this challenge:

• Before completing the puzzle, which numbers do they think are excluded? Why?
• Make extra equations that allow us to include those missing numbers. Are they easier to incorporate using multiplication or addition equations? What about equations that combine multiplication and addition?

Observations: 

Once again, this appears to be a very simple activity, but gave us a lot to talk about. In particular, it was very helpful for highlighting a lot of misconceptions and gaps in understanding. Examples:

• "identity" relationships were still unclear: 1 x n = n, 0 x n = 0
• Several students thought the first equation would be 4x4x4 = 12 (confusing multiplication and addition) 

 

Division Dice move to Cards
For our core activity, we are extending the Division Dice game. This time, we use playing cards, A through 10, instead of dice rolls to generate the random components of their equations. In this case, the aces are wild and can be any number from 2 to 10. When they form a multi=digit number with a 10, the 10 counts as two digits. For example, 3, 5, 10 could form 105 ÷ 3.

With cards instead of dice, we lack the natural move of flipping the dice to the opposite side which we used to make sure all throws could give us whole number divisions. In this version, we allow division with remainder. However, the twist is that the remainder becomes points for the opponent.
For example, if I draw 3, 3, 5, I can form 53 ÷ 3 to score 17 points for myself, but the opponent gets 2 points.

Question: Are there cases where the best play is not to form the largest possible number divided by the smallest number?

An interesting game variant: swap the scoring so that the active player scores the remainder and their opponent scores the quotient.

Improv Math and Division Dice follow-up

We had a really good experience playing Division Dice, the game that we introduced a couple of posts ago.  Mainly, I want to illustrate something fun that came out of really listening and paying attention to what the kids are doing and saying. I like to think of this as "improv math," as a way to credit my improv comedy experiences for heightening my awareness of how important this is.

Division Dice for number sense

I was really pleased about the quality of thinking stimulated by the game. We played with the most loose rules (1s are wild, the components of the 2 digit value can be flipped to their 7s complement). That gave a lot of opportunity for the kids to think through options to (a) make whole number divisions and (b) maximize values.

For example, rolling 3, 4, 6:

• what are the allowed groupings that give a whole number division? Remember, in the 2 digit number, we can use any of the values 1, 3, 4, 6, and it is possible for us to use two 3s or two 4s in our calculation.
• What is the highest scoring choice?

Division Dice for arithmetic exercises

As a way to create virtual worksheets, this game is mediocre. The basic structure means that students are never dividing by a divisor larger than 6. This leave out a lot of fact families. However, because the kids are trying to maximize their scores, they quickly realize that they can almost always get away with division by 2, occasionally must divide by 3, and rarely get stuck dividing by 4 or 5. I haven't yet seen a case in a live game where division by 6 was necessary.

Fun exploration: what scenarios will require division by 6?

Using playing cards or other dice shapes allows us to extend the possible values and reduce the likelihood of dividing by 2 or 3. However, it also increases the number of cases that don't have a whole number division relationship. We are thinking about ways to incorporate division with remainder and will try out a variant tomorrow.

Improv Extension

Playing at home, the 3, 4, 6, case led J1 to consider: how do 63 ÷ 3 and 64 ÷ 4 compare?
As he contemplated that, I realized that we had a nice sequence of multiples, meaning all of these are whole numbers:

There were several cool things for J1 to observe here:

• 4 of the 6 quotients end in 1
• The quotients are all decreasing
• The drops between successive quotients are themselves decreasing
• the dividends are equal to the divisors + 60

We pursued this in two ways:
Extension 1: what if we add something else to the dividends?
We tried three versions.

1. starting with 60 and adding 6 at each step
2. Starting with 60 and adding 60 at each step.
3. starting wit 1 and adding 7 at each step

You can see our notes mid-discussion below:

Later, when J2 was also involved, I offered them another sequence: starting with 66 and adding 6 for each increment:

66 ÷ 1

 72 ÷ 2 

78 ÷ 3

84 ÷ 4

90 ÷ 5

96 ÷ 6

120 ÷ 10

132 ÷ 12

150 ÷ 15

180 ÷ 20

240 ÷ 30

420 ÷ 60

3660 ÷ 600

36060 ÷ 6000

We're breaking the rule about the dividends being multiples of the divisors, but the last two calculations are still easy and nicely illustrate the limiting behavior.

Extension 2: can we find other chains of whole number division equations?
We started this by thinking more simply: for chains shorter than 6. For example, what are the smallest K, L, M, N larger than 1 such that all of the following are whole numbers:

K ÷ 1

 (K+1) ÷ 2 

L ÷ 1

(L+1) ÷ 2

(L+2) ÷ 3

M ÷ 1

(M+1) ÷ 2

(M+2) ÷ 3

(M+3) ÷ 4

N ÷ 1

(N+1) ÷ 2

(N+2) ÷ 3

(N+3) ÷ 4

(N+4) ÷ 5

After getting the shorter cases under our belt, we then went for a chain of length 7. J2 worked by himself for a while, then came back and announced that no chain with dividends smaller than 100 would work.  He went away and then came back quickly with the idea that maybe we could add 7! to each divisor.

Division Dice (math games class)

Who: grades 3 and 4
Where: in school

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A dice division game

We created a simple dice game to practice division. Here's a description of the basic element of play:

1. roll three dice: for example, 3, 4, 5
2. group two of them into a two digit number: for example, 45
3. Divide the two digit number by the remaining single digit: for example, 45 / 3 = 9
4. This value is your score for the round
5. First player to 200 or more points wins (we used 100 for the initial game)

Key constraints

• You can only score points if the single digit is a factor of the two digit number (remainder must be 0)
• Where there are multiple options, the player can choose the combination that gives them the maximum score

This pencilcode program (see code) analyzes this basic game structure, identifying how often there will be no legal scoring arrangement and showing a histogram of the largest scores.

Modifications/Extensions

I wasn't satisfied with three elements of this game: (a) any time a 1 occurs, the division calculation is too easy, (b) too many combinations don't allow a score (about 15%) and (c) there aren't many decisions for the students to make (just six combinations to investigate).

We addressed these by adding two extra rules:

• 1 is a wild that must be replaced by a value from 2 to 6 (cannot be left as a 1)
• On your turn, you can flip the over the dice in the two digit number. For example, a 6 can be flipped to a 1, 5 to 2, 4 to 3, etc.

The first point removes the division by 1 cases, the second one allows more choices and reduces the number of non-scoring cases.

*UPDATE*
Allowing the dice flip and wild 1s seems to make the game too loose. Instead we dropped the wild 1s rule and added these two:

• Division by 1 is not allowed in the game
• If you roll triple 1, re-roll

Factors and division

who: grades 3 and 4 at Baan Pathomtham
where: in school

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Sorry about the lack of pictures. This is a short and sweet note.

Dots & Boxes and Factor Game Mash-up

To start the year, we played a version of dots & boxes that integrates the factor game (here is one example). This is based on the game template from Mathified Squares Game that we used last year.
Instead of using dice to determine where each player can play, we introduce factors 1 to 6 at the bottom of the page and selectors.

As with the basic factor game, this version creates multiplication and division. This is the point we want to draw out for the game.

Homework

Play the game at home and write down 15 division equations that come up in the course of play.

*UPDATE* Having now played through this game fully, I really like this structure. Using the factor selectors drives some interesting thinking about common factors, especially during the middle and end-game phases.

We did find that it starts a bit slowly as players can make moves on distant parts of the board and decisions don't have clear connections to capturing squares. For the young kids, we recommend just pushing past that stage. For older kids, that can be an interesting (and difficult) strategic analysis.