**Erin's math, Summer after finishing 3rd grade
and Fall in 4th grade**

**1.**
**Solving the equation 2x + 3 =
18, trying to find the number for x that will make this sentence true, in ****Erin's**
**words:**

**"One day I tried to figure out an algebra problem that said
2x + 3 = 18. I tried 8 but that was too big and 7 was too small. I
drew a number line --7---------8--
. But
I did not know that there was a number between 7 & 8, but then I thought of
7 1/2. So I tried it and it worked! That gave me an idea: I wanted to see
how many numbers I could find between 7 & 8. I found 29 of them and they
are: 7 1/2, 7 3/7, 7 9/10, 7 4/10, 7 3/9, 7 8/9, 7 5/6, 7 2/9, 7 3/4, 7
3/6, 7 2/9, 7 6/9, 74/9, 7 5/9, 7 1/3, 7 1/4, 7 1/5, 7 1/6, 7 1/7, 7 1/8, 7 1/9,
7 2/32, 7 6/8, 7 4/6, 7 6/10, 7 4/5, 7 2/11, 7 3/5, 7 5/8".**

**2.
Sharing
2 cookies between 3 people (the cookies were 3x5" cards). **

**Erin decided right away that if she couldn't
share, she would cut the remaining pieces into 2 equal pieces [a common and
simple way of solving a problem like this]. Erin & Don discussed how many of
each piece would make a whole cookie, then she could name that piece. [See the
sample problems for chapter 2]. **

^{Each person got}**
^{of
a cookie!}**

^{ Don then asked Erin to do the}
^{ same problem, but cut
the cookies in a }

**
^{ different} way, like cut into 3 pieces. When
she
did this, each person got
of a cookie. So ^{ }^{the infinite series above goes to (=)} , ^{ }**

**Then Don asked Erin "if 2/3
equals this infinite series, 1/3 would equal what infinite series ?"
Don left Erin to work this out and when he came back she had this:**

**
****which couldn't work!**

**Later Don asked
Erin to do 1/3 and 1/3 of 1/3, and so forth**

**Don looked at what Erin did and was really
confused. What Erin did was to take 1/3 of the 9x9 square (orange piece); that
was fine. Then instead of adding 1/3 of 1/3, she added 1/3 of what was left!!!
The red part was 1/3 of 2/3 = 18/81 = 2/9 of the big square. Don asked Erin to
keep going using her method and see what was going to happen. Meanwhile Don proceeded to continue her method by himself,
to obtain the interesting result.**

**Later in December Don sat down with Erin
during class, to
help her get the following series**

**Do you see a pattern? See if
you can continue this pattern and get the sum of this series. Note: this is a
classic case of not writing the answer as one number, but writing it to let one
see a pattern!**

**Don will continue this with
Erin next time, perhaps; she needs to get **

**2/3 + 4/9 + 8/27 + ... before
we can do the one above.**

Mathman home