Where: In school

When: after science and before lunch

# Warm-up patterns

Some students requested we talk about the Fibonacci sequence, so I decided to start with some number patterns. If I were going to do these again, I would stop after the second and third terms and ask for a range of different ideas about how the patterns might continue. Once we have 4 or 5 terms, though, it becomes pretty difficult to identify a non-obvious pattern that fits the available data.The point we want to communicate is that there are many possible answers and the only condition is that they should fit the existing data.

Patterns we used today:

- 0, 1, 2, 3, 4, 5.... universally extrapolated as counting up by 1
- 3, 10, 17, 24, 31, .... best fit was counting up by 7 starting with 3
- 18, 14, 10, 6, 2, -2, -6, ... Subtracting 4, starting with 18
- 1, 2, 4, 8, 16, .... powers of 2 or doubling the prior element of the sequence
- 1, 1, 2, 3, 5, 8, 13, .... Fibonacci sequence

One idea was 18, 14, 12 which we explored a bit more and came up with the following:

- 18, 14, 12, 12, 14, 18, 24 .... the difference of the differences is +2
- 18, 14, 12, 12, 12, 12, 12, 12.... hits a floor and stays at 12
- 18, 14, 12, 11, 10.5, 10.25, .... the differences are getting halved at each step

# Triangle Puzzles

Mathematics Mastery sent a Christmas card that inspired this exploration. As a basic set-up, you have a triangle with circles on the vertices and some number of circles on the sides. You have to place distinct digits in the circles so that the sum of the numbers on a side is the same for each side. You can see the pictures for triangles with 0 extra circles (only the circles on the 3 vertices), 1 extra on each side and, if you look closely, 2 extra on each side:The kids had a lot of fun with this puzzle. Here are some notes:

- 0 extra circles/3 total circles: Is it possible with three distinct numbers? If not, why not? Can you convince your friends you are right?
- 1 extra circle on each side/6 total circles: fill in with the integers 1 to 6. How many solutions can you find? How do you know you have found them all? Given a collection of 6 distinct integers, can you always find an arrangement that works? If not, are there any conditions that must be satisfied? Are there any conditions which are sufficient?
- 2 extra circles on each side/9 total circles: similar questions to the 1 extra circle
- non-equilateral triangles: with n-circles, when can you fit the numbers 1 to n into the triangle to meet the condition of equal side sums?

If you get tired of triangles, of course, you can explore square figures, polygons with more sides, and stars (put circles on the points and internal intersections).

# Pico/Fermi/Bagel

This game was also a student request. I've talked about this game before (here), so won't bother to repeat the rules. Today, we did two rounds with 3 digit numbers, then let the kids play in pairs for a couple minutes. For playing in class, we found this a good game because:- It was fun!
- Everyone could guess a number and stay involved in the game
- Students got a chance to explain their thinking and try to reason logically through the evidence

We also experimented with having one teacher score guesses for different target numbers for individual students. This was workable, but it significantly reduces our ability to have a deeper conversation with the students about what they have learned from each clue and why they are choosing a particular guess.

# Homework

The kids are assigned more exploration for the triangle puzzles:- Find multiple solutions to the 6 space triangle
- Find a solution to the 9 space triangle
- Look for solutions to the 3 space triangle and talk about what you find

Rather than go to sleep, J1 wanted to work on some of the triangle puzzles. He had written down that other classmates found 6-circle triangles with sides summing 9, 10, and 12. I asked if that seemed strange and he asked, "what about 11?" He found an answer for 11, one for 10, then one for 12 and one for 9. After that, he noticed that the answers were in pairs:

ReplyDeletetriangles for 11 and 10 just had their vertices swapped with their sides as did the ones for 12 and 9 (they are dual).

Also, he figured out that 9 is the smallest side sum possible because 6 has to be on a side and the smallest numbers that can join that side are 1 and 2. Following a similar argument, he saw that 12 was the largest side sum.