- What are the different lengths that Kimie can make using only her blue sticks? What do you notice?
Kimie can make multiples of 2
- What are the different lengths that Sebastian can make using only his red sticks? What do you notice?
Sebastian can make multiples of 3
- Can they make their lines the same length? How many sticks could Kimie use? How many would Sebastian put down? How long is the line altogether?
The length needs to be 6
- Can they make any other same lines using only blue sticks or only red sticks ? What do you notice? I notice that it is the multiple of 6
- What lengths can we make using 2 and 3 together? We can make any number except of 1 because we can make even numbers with 2 and odd numbers except of 1 with 3 and a bunch of 2
- Explain your findings.
The length needed for the blocks to be equal is supposed to be a multiple of 6
- How many ways a number can be represented?
is according to how many 2s it is made of for example 8 can be represented by 2+2+2+2 or 3+3+2
But for a number like 36 you will have 2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2 then there will be 2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+3+3, 2+2+2+2+2+2+2+2+2+2+2+2+3+3+3+3, 2+2+2+2+2+2+2+2+2+3+3+3+3+3+3, 2+2+2+2+2+2+2+3+3+3+3+3+3+3+3….
…
3+3+3+3+3+3+3+3+3+3+3+3
But the steps for even numbers is
Step 1: divide by 2
Step 2: subtract 0,1or2 to make a multiple of 3
Step 3: divide by 3
Step 4: add 1
But the formula for odd numbers is
Step 1: subtract 3
Step 2: divide by 2
Step 3: subtract 0,1or2 to make a multiple of 3
Step 4: divide by 3
Step 5: add 1
For ex:
12=2+2+2+2+2+2=2+2+2+3+3=3+3+3+3
3 ways and the formula is 12/2=6-0beacuse 6 is a multipule of 3=6/3=2+1 =3 ways correct
13=3+2+2+2+2+2=3+3+3+2+2
13-3=10/2=5-2 because then it will become a multiple of 3=3/3=1+1= 2 ways correct