My daughter and I went on a 12 week- journey the past two years to explore Fractions. We are doing it again this Fall. I am updating the posts from our previous journeys, in case you want to join us this year. Click here if you want to know more about the journey.
Finally, our first Equal Sharing problem (Empson & Levi, 2011) is here !
- 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?
As always, invite your child to solve one of the problems, and listen to his/her way of solving it. He/she can make sense of the problem while using small objects (such as buttons, marbles, etc, and small containers; flashcards to cut and fold work well too with fractions) or drawing a picture. He/she may write an equation. Each child should pick the problem that he/she feels like exploring. If your child calls out the answer right away, remind him/her that the answer is fine, but how it was obtained is even more important in this journey. How would he/she explain it to a younger child? Could he/she represent the problem with a drawing? a diagram? Using small objects ?
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 (Fall 2015)
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 between models and symbols. Keep me posted!
Sharing my experience (Fall 2016)
Last year, we did a review of Problem 1 to 6, but we skipped it this year. Here it is, if you want to (here!).
With Problem #8, Rosie started with Level Yellow, drawing the situation, and writing an
equation . It is something I have encouraged her to do this year, write an equation that would match her drawing. She does not have to, but it helps me see her reasoning at a more symbolic level.
She explored Level Orange similarly. I just had to remind her, after she wrote 6+1 = 7, 5 + 2 = 7, that the goal is to have the equation matching the picture :-)
With Level Red, she used the flashcard. She started with cutting both cards into halves, to give a half to each
bear child. She then kept cutting the last piece into halves until she realized at one point that she had to cut into third i.e. 3 equal parts.
Then, she started over, and cut each sheet into “thirds”, to come up with the answer of 2/3 of a sheet. She noticed that the “thirds” she cut ended up not being 3 equal parts: “Maybe later, I should use a measuring tape”. Our flashcards being 3 x 5 inch, it sure could lead to another interesting exploration :-)
- Empson, S. E., and Levi, L. (2011). Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH : Heinemann. ISBN-13: 978-0325030531.