**Sara and Maya find Patterns in division, and
AnnEmily adds and multiplies fractions**

**Don had Sara do some
division problems the week before her work below. One he gave her was 10 divided
by 3, which Don, as usual, talked about sharing 10 cookies between 3
people, using 3x5" cards for cookies (see sample problems from chapter
2 his worksheet book at www.mathman.biz/html/prob2.html
). Sara said each person got 3 cookies with 1 left over. She cut the 1
leftover-cookie into 3 pieces, each 1/3 of the cookie- see the diagram below. So
each person received 3+ 1/3 cookies. He also showed her other ways to write a
division problem. And Sara had to figure out that when one cuts a cookie into 3
pieces, each piece is 1/3 of the cookie, a key idea! She tried the same problem on a calculator and got
3.3333... Instead of trying to show her how decimals work, Don had the idea of
seeing what would happen if she did 10/1, 10/2, 10/3, 10/4, 10/5, 10/6, 10/7,
10/8, 10/9, and 10/10. Don wasn't sure what would happen, but he was almost sure
there would be a PATTERN! **

**After
finishing 10/7, her time had run out, but Sara insisted on finishing the last 3.** **When we got to ****10/9
below, Don asked Sara if there was a Pattern in here:**

**Sara saw that going from right to left,
the top numbers in the fractions got bigger by 1 and the bottom number got
smaller by 1!! Now Don had never seen this before, in 54 years of teaching! Then
when they got to 10/5 = 2 the pattern wasn't there; so they changed the
remaining answers to 1 and something. The 10/5 = 2 = 1 5/5, then it fit in the
pattern! After a lot of work in renaming numbers by Sara, even knowing
what the answers would look like from the pattern, the final results ended up looking like this:**

**Maya's
work dividing 9 by different numbers**

**This
shows the power of PATTERNS in mathematics! It also shows Don's willingness to
try new things, his fearlessness, and his being excited about the unknown and
what will happen. Normally Don aims for students finding different ways to solve
a problem; for example, if one shares 10 cookies between 3 people, a student
might share 3 each, then cut the left-over cookie into 4 pieces, each getting
1/4, then cutting the left-over 1/4 into 3 pieces, each 1/12, so each person
would get 3 + 1/4 + 1/12 (which is equivalent to 3+1/3 which Sara ended up with
above). OR students, sharing when they can, then cut left-over pieces into 2
equal pieces every time, and end up with an infinite series 3 + 0/2 + 1/4 + 0/8 +
1/16 + 0/32 + ... which is also equal to 3 1/3! **

**Notice
that Maya worked on The Peg Game and The Tower Puzzle (ch.6) before the
division. Maya knew how to do these divisions by cookie-sharing also. Don
left 9/6 = 1 3/6 without reducing the 3/6, in order to continue the pattern. Too
much emphasis is placed on reducing fractions to lowest terms, because you also lose
the patterns if you do that- which is more important.**

**Great
work Sara and Maya !**

**AnnEmily, 2nd grader, adds fractions using
complex fractions and multiplies mixed numbers-WOW!**

**Don gave AnnEmily the problem
2/3 + 1/4 = ? He walked away to work with other students. When he came back, he
saw this picture (without 1/8 and so on in black print which he put in after to
help him and other people understand what she did).**

**AnnEmily explained the picture
to Don (as he remembered it): The whole is a 12x12 square. The red section is
2/3 of that. The blue part is 1/4 of it.**

**The 3 vertical rows at the
left, make up 1/4 of the whole. Half of 1/4 is 1/8. So she counted 4/8 for the 4
sections. Looking at the 2 red vertical rows to the right, that was 2/3/8
(2/3 of an 8th) on top and 2/3/8
on the bottom. AnnEmily added the 1/8's to get the red total: 1/8 + 1/8 + 1/8 +
1/8 + 2/3/8
+ 2/3/8
= 4/8 + 4/3/8
= 4/8 + 1 1/3/8
=**

**5 1/3/8
([5 and 1/3] eighths). The blue section is 1/4 = 2/8, so the total 2/3 + 1/4 = 7
1/3/8
(on the right above), which was her answer. Don was really impressed with
AnnEmily's use of complex fractions.** **He
checked her answer for himself, by changing the complex fraction to a simple
fraction by multiplying top and bottom by 3 to get 22/24 = 11/12. Then he looked
at her picture and got 11/12 also.**

**AnnEmily multiplies 3 1/2 x
2 1/3**

**See her picture below: First
she did 3 1/2 x 2 = 7, then 3 x 1/3 = 1, then 1/2 x 1/3 = 1/6, then added these
to get 8 1/6. Simple!**

**Fine job AnnEmily!**

**Don emphasizes using
rectangular cookies to share with people, but AnnEmily shows she can handle the
circular cookies to share and multiply! One keeps learning from the students,
which is great!**

To order Don's materials

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