Tuesday, February 17, 2015

Dominoes and Damult Dice (math class 10)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

Warm-up patterns

Today, we started skip counting by 3, 4, and 6. This was because I wanted to play a game involving multiplication and I wasn't sure whether the kids were ready. They are familiar with skip counting and understand the linkage with multiplication, but clearly need more practice to become fully comfortable.

New Game: Damult Dice

The new game comes from Math4Love. Like NRICH, they are a consistent source of good material for our activities. I wanted this to be a more structured game than usual, so we established the following set-up:
  • Class is split into two groups (3 and 4 students)
  • One player in each group rolls three dice
  • that student chooses two dice to add together. The third is for multiplying
  • By skip counting, the team works together to multiply the two numbers.
  • that result is their team score for that round and gets added to their total.
This game worked very well. It was especially encouraging to see them thinking about which combinations would yield the largest result and remembering prior results.

Old homework: The Triangle Puzzle

Homework from last time was to work on the triangle puzzles. I was curious to hear what they found. Nearly everyone found a solution to at least one of the sizes (6 gaps or 9 gaps), so they all got to share something about what they found.
  • For the 3 space triangle, were there any solutions? No one had really thought about this, but during the discussion a couple of second grade students realized that all the numbers had to be the same for the sum of sides to be the same. I let them share their ideas about why.
  • For the 6 space triangle, students found answers with sides summing to 9, 10, 11, and 12. 
  • We talked about why 9 was the smallest (6 has to be linked with numbers at least as big as 1 and 2) and why 12 is the largest (1 has to be linked with numbers at most as large as 5 and 6).
  • For the 9 space triangle, students claimed to find answers with sums 17, 20, 21 and 23. For an extra investigation, they can confirm these and try to find solutions with sums 18, 19, and 22.
For the 9 space triangle, I noticed a relationship between two of the students' answers, so drew these on the board:

Admittedly, I made some adjustments to highlight the relationship. You see that the numbers in corresponding slots add up to 10, right? In a sense, these are complements. If
you look closely, there is one other relationship, but I'll leave that for you to discover.

A dominoes version

I was lucky to find a direct complement to our triangle activity on NRICH: 4 Dom. This time, the challenge is to arrange 4 dominoes into a square with three numbers on each side where the sum on each side is the same. We got the kids to make their own dominoes by cutting out and coloring little strips of paper. That gave them a hands-on tool to explore three questions, all assigned for homework:
  1. Allowing sides to have different sums, what is the smallest sum you can make with these 4 dominoes? How do you know it is the smallest?
  2. Allowing sides to have different sums, what is the largest sum you can make with these 4 dominoes? How do you know it is the largest?
  3. What arrangement makes all the sides have the same sum? What is the sum? How many solutions are there?


We gave them four pieces of homework. This seems like a lot, but the game is very short and most of the kids answered several of the dominoes questions already in class:

  1. Damult dice: play the game with someone in your family, first to 200 wins. For each roll, write down the equation you are calculating, for example (6+1) x 2 = 14
  2. Dominoes puzzle: answer the three questions listed above. In short, what is the smallest sum that a side of the square could have, what is the largest, what sum and arrangement works so that all sides are the same?

Congratulations for getting this far. Here is something pretty for your efforts. Please post in the comments any mathematical ideas this picture gives you!

1 comment:

  1. In first grade, we did the activities in the order listed above and it seemed to flow more smoothly. In the second grade class, we did the triangle and dominoes puzzles first and it seemed a bit disruptive to switch gears in this way.

    Also, some extensions i want to record, but not get in the way of the class notes:
    - In a set of double 6 dominoes, can you find 4 that cannot be arranged in this square configuration (i.e., the sides always have different sums)?
    - Thinking recursively/inductively, can you find a solution to any equilateral triangle puzzle with the integers 1 to n arranged in equal numbers of open slots on the sides (and 3 on the vertices)? Yes! What is it?