In
order to solve this hand puppet problem, I gave each of my 4th graders a sheet
of 2-cm graph paper and 18 snap cubes.

The cubes were arranged in sets of 6 and
included 3 different colors to help each child connect with the fabric colors in
the story problem.

Immediately, the discussion began with “How much is 5/6

^{th}of a yard?” After Eligio determined that he could represent this with 5 of the 6 snap cubes, then Fatima struggled with “How much is 3 times 5/6^{th}of a yard?”
Patricio
said that 3 x 5/6 was 15/18, but Rigoberto disagreed. Rigo showed the class
that “three times a number” was the same as adding it 3 times. Therefore,

5/6 + 5/6 + 5/6 = 15/6.

5/6 + 5/6 + 5/6 = 15/6.

Martha saw
that each cube (1/6) was worth $2, because there were 6 cubes (6/6) in a yard
and each yard cost $12.

Esmeralda
wrote, “The girls need to buy 2 ½ yards of felt. It will cost $30. I know this
for sure, because 1/6 = $2 and $2 x 5/6 = $10. So $10 x 3 yd. = $30 and that’s
how I know.” She confused the 3 times 5/6 with 3 yards, but was definitely making progress in her understanding.

Patricio showed the class that he could move 2 cubes from the third
5/6 of a yard to make 2 complete yards, plus ½ a yard. “That equals 2 ½ yards
of felt,” he beamed. “Each block costs $2 and 15 x 2 = 30. I got the 15 from
each block.” It was obvious that he had clarified his earlier thinking that
multiplying 5/6 by 3 was not the same as multiplying 5/6 by 3/3. He also
realized that ½ a yard was $6 and not ½ a dollar or 50 cents.

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