Two more quick games

On weekdays, there isn't enough time to plan a full set of activities.  However, it is still helpful to have a couple of quick games or exercises ready to go.  In some cases, we might find ourselves with surprising free time or, more likely, have a need for some activity to occupy one of them before trouble starts.  Tonight was a bit of the latter.

Jate's Game
We played a couple rounds of another game I got off NRICH.  If I find the link, I will post it in an update to this blogHere is the link: http://nrich.maths.org/6589.  The basic rules:

1. make a number line and label the whole numbers from 1 to 20 (when we started playing, we only went from 1 to 10)
2. The first player chooses two numbers on the number line, crosses them out and circles either their sum or their difference and circles that number.  The two crossed out numbers are now out of play.
3. The next player works with the circled number and chooses another number that hasn't been crossed out.  They then circle either the sum or the difference and cross out their numbers.
4. Each player repeats step 3 until there are no legal moves.
5. The last player to make a legal move is the winner!

It might help to study the picture below.  It isn't especially clear or tidy, but you should be able to get the point:

Jate is working with 17 (on the right of the paper, circled and not yet crossed out)

He has to choose another number. Since so few options remain, he has two choices, either of which causes him to win the game.  See if you can figure out his move. . .

He chose 12, 17-12 is 5, so he circled 5 and crossed out 12 and 17.  That left me with no remaining legal moves.

Now, this hasn't come up yet, but I have seen the opportunity to argue about a special case: what if he had chosen 5 and circled 12 (= 17-5), crossed out the 17 and 5?  Could I have chosen 6 and circled it, because 12-6 = 6?  Seems to me that you have four options for rules to deal with this situation:

• 2n - n = n isn't allowed as a legal move
• 2n - n = n means that you cross out 2n, circle n and the next player uses n for their move
• 2n - n = n means that you cross out 2n and n, the next player chooses 2 new numbers
• 2n - n = n means that you circle n, then cross out 2n and n, and this player automatically wins (because the next player can't make a legal move with the circled n, it has already been killed)

In our case, Jate avoided this ambiguity (did he do it intentionally?)  As always, I invite you to explore each variant.

We played 1 and a half games.  The second game, it turned out Jate had set a booby-trap on one of the numbers, so I automatically lost when I circled that number.  How do I know it was pre-planned and not an arbitrary, last minute rule change?  Because he told Jin and they both started laughing as soon as I circled the trick number.

Jin's Game

Writing up the notes on the tidy-up game (see here) reminded me that I hadn't ever fully explored toetactic (inverse tictactoe.)  Jin was amenable to playing, so we drew a couple of boards and played tictactoe as a warm-up.  Then, I explained we were trying to force the other person to get 3 in a row and we played a couple of boards that way. No deep analysis tonight, but it was fun hearing him say "inverse tic-tac-toe" and seeing the plans go through his mind as he worked out his strategy.

Heads and Feet (and multi-variable equations)

NRICH has a fun little problem on its top page for lower primary: heads and feet.

The challenge: A farmer has chickens and sheep. If the animals all together have 8 heads and 22 feet, how many sheep does he have and how many chickens?

My question: Is this low threshold and high ceiling?

Threshold
The basic task seems to be one unit, but there are a couple of progressions to make it more accessible. My guess is that most children who get stuck will need to be guided toward alternative versions that will let them build strategies for understanding the challenge. Perhaps starting with a version that has fewer animals will be accessible. If necessary, drawing pictures of animals, counting them, counting body parts, etc can help get the exploration started.

Note that drawing realistic animals doesn't necessarily help with the mathematical content (and it can actually get in the way of one solution strategy). However, if you want to practice, here are some instructions that will lead you to a much better animals than I draw!

Plan: have Jane counting objects, Jate drawing pictures of animals and counting body parts, and Jin to get the challenge as originally stated.

Ceiling
The obvious route to a (super) high ceiling is linear equations of multiple variables and the whole world of linear algebra. My idea to open that route is to ask if Jin can write an equation to describe the number of animal heads and another equation to describe the number of feet.

The solution strategy I've seen him use in the past is to draw a picture, starting with the assumption that all the animals are chickens, like this, with heads and feet helpfully labelled:

and then adding feet to make sheep until you get the right total number of feet (the sheep bodies are shaded to make the distinction between sheep and chickens more clear):

I've been wondering about whether this strategy is generalizable, since even three variables seems unclear. My questions, if we get this far:
- can we write an equation or equations that describe the relationships between sheep, chicken and heads? What about sheep, chicken and feet?
- Is always a solution, no matter how many feet are given? Do we notice anything special about the number of feet in this type of problem?
- With the right number of chickens and sheep, can we get all combinations of (heads, feet) = (n, 2m)? Are there any restrictions we can identify?
- Can you think of any problems like this? Maybe chairs (4 legs) and stools (3 legs), bicycles (2 wheels) and tricycles (3 wheels) etc.

Post-Mortem
This activity wasn't loved and didn't capture Jin's attention during the evening rush.  Maybe something to save for a quieter time or perhaps the lesson is to have various investigations available? Mommy's post mortem:  there wasn't time to do this anyway, so nothing lost.

Well, we ended up with another nice chicken picture, at least: