*This is the eighth post related to our Time 4 Fractions journey. Please click here to start from the beginning.*

Finally, our first Equal Sharing problem (Empson & Levi, 2011) is here !

**Time 4 Fractions – Problem #8 – Sharing sheets of paper**

*Level Yellow*– 2 children want to share 4 colorful sheets of paper so that each of them gets the same amount. How many sheets would each get?*Level Orange*– 2 children want to share 7 colorful sheets of paper so that each of them gets the same amount. How many sheets would each get?*Level Red*– 3 children want to share 2 colorful sheets of paper so that each of them gets the same amount. How many sheets would each get?

Invite your child to either model the problem (with paper and scissors for instance) and/or represent the problem. If your child has learned about fractions at school, invite him/her to add symbols to a representation. And as always, invite your child to share his/her reasoning with you !

Level Yellow leads to an whole number answer, level Orange to a mixed number (3 1/2), and level Red to a proper fraction (2/3 or* equivalents)*.

**Sharing my experience
**

We used flashcards to model the problem. It worked very well, as my child was able to go back to a whole colorful sheet of paper, to explain her reasoning, compare what each child would have to a whole piece of paper, or… start over. Indeed, creating fractional parts by cutting paper does support children’s understanding of fractional quantities (Empson & Levi, 2011, p22).

As often, my child started with Level Yellow (she drew it), and moved to Level Orange (she modeled it with paper). Then she decided to try Level Red, and I thought I should share her reasoning in more details. Not as an example of what *my* child could do, as an example of what *a* child can do. Indeed, children’s brains will never stop surprising me.

So with level Red, she quickly saw that each child could not have a whole sheet of paper, so she started cutting each sheet into halves (4 halves in total). She gave one to each of the 3 children, and had one half left. She cut it into 2 more pieces, give one to one child, cut the other one into two more pieces, and so on until she had this pile of little pieces. Then she stopped, and said “well, I am not sure”.

Later that night, while she had been in bed for 20 min or so, she got up, came to the living room and said “I think I got it. You know, the problem with the 3 kids? I think I know”. So I could not resist, I gave her two more flashcards.

“You see, they cannot have a whole piece, so I am going to cut it in half. But then, I am going to have to cut the half into 3 pieces, so they can all have one. Because if I cut it into 2 pieces, it doesn’t help, there are 3 people !”. As a way to help her cut the half into 3 equal parts, she drew 3 squares on the top, and cut them out, as well as the rectangles that would represent a 1/3 of the 1/2 of the sheet (i.e 1/6 of the sheet… following?). Then, she dispatched the 3 pieces from the first half, then 3 pieces from the second half from the first sheet of paper, the first half, the second half from the second sheet of paper. “Here you go. See? They all have the same amount and I do *not* have anything left”.

Overall, she ended up cutting the 2 sheets into 6 equal parts, and gave 4 parts to each child (i.e 4/6, an equivalent of 2/3). Why didn’t she cut the sheet into 3 pieces right away instead of in halves first and then 3 pieces? I am not sure. But she solved the problem, in a way that “made sense to her”. And with her explanation, it made sense to me as well. And that’s what our journey is about 🙂

My child has not learned symbols related to fractions yet, so we did not write anything on paper. If your child is in upper grade, though, you may see neat connections to make between models and symbols. Keep me posted!

Enjoy !

Reference:

- Empson, S. E., and Levi, L. (2011). Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH : Heinemann. ISBN-13: 978-0325030531.

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