How we play math games, an example

Who: J1
Where: dining table
What materials: a pack of playing cards, a 100 board
When: mid-afternoon

---

Some conversation snippets that illustrate what we talk about when playing math games.

For background, we were playing our 35 game from class, with some slight modifications. Each player gets 5 cards and can choose what card to play on their turn. After each play, you draw a new card so your hand should always have 5 cards. In this case, I was just picking cards at random from my hand, not choosing the cards to play.

Conversations:
J0: 26 + 3 = 29. Now, what do you need to get to 35 this turn?
J1: 6 <looks at cards and makes a sad face>.
....
....
J0: What's the count?
J1: 21, can you make 35 on this turn?
J0: <looks at his card and thinks> Oh, no, I can't.
J1: How did you know?
J0: I looked at my highest card and added  to 21.
....
....
J1: <playing a card> 11 + 5 = 16
J0: Can I get to 35 on this turn?
J1: <thinking for a bit> No, the highest you can get is 29.
....
....
J1: <playing a card> 30 + 8 is... oh I have to subtract. 30 - 8 is 22
J0: What is the lowest you can get by bouncing back and subtracting?
J1: <thinking for a while, looking at the 100 board> Oh, we could get back to 10.
J0: How do you get to 10?
J1: We start on 23 and play a king (value 13).
....
....

What's the point?

There are two things I was trying to do: get him to explain and verbalize his thinking and explore the structure of the game. For almost all activities, there will be "what is the largest..?" or "what is the smallest ..?" investigations worth pursuing.

What did I miss?

At least two things I could have added to our conversations:

(1) talking more about my own thinking when doing calculations

(2) Asking for or introducing alternative ways to do a particular calculation.

What else do you see? Feel free to criticize (constructively).

Some shapes

We recently had another round of flexagon making. In our attempts to practice making hexahexaflexagons, we ended up with some spare paper with a triangular tiling. These were good for making little hexahedrons and octahedrons:

Ever since reading Christopher Danielson's class activity on the taxonomy of hexagons, I've been on the look-out for natural occurrences of irregular hexagons. In my experience, they are actually pretty rare. Here is one, captured in its native habitat:

Obviously, a close cousin to the regular hexagon