My daughter and I went on a 12 week-journey last year to explore Fractions. We are doing it again this Fall/Winter. I am updating the posts, in case you want to join us this year. Click here if you want to know more about the journey and the previous problems.
Here is another Equal Sharing problem. This week, I am including a few words about how a child may share his/her reasoning, depending on his/her previous exposure to fractions. Hope it helps.
Time 4 Fractions – Problem #11 – Sharing sticks of modeling clay
Level Yellow – 2 students want to share 7 sticks of modeling clay so that each of them gets the same amount. How many sticks of modeling clay would each get?
Level Orange – 4 students want to share 2 sticks of modeling clay so that each of them gets the same amount. How many sticks of modeling clay would each get?
Level Red – 8 students want to share 3 sticks of modeling clay so that each of them gets the same amount. How many sticks of modeling clay would each get?
Invite your child to either model the problem (with paper (I like flashcards, as you can go back to a whole card to compare what you cut) and scissors for instance) and/or represent the problem with a picture. If your child has learned about fractions at school, he/she may connect symbols to the model or picture. And as always, invite your child to share his/her reasoning with you !
The problem leads to a solution of each student getting 3 sticks and 1/2 (level Yellow) , 1/2 of a stick (level Orange) or 3/8 of a stick (level Red).
Sharing my experience (Fall 2015)
The goal of Time 4 Fractions is really to provide children with additional opportunities to explore fractions at home, so they have stronger foundations to build up on when they study fractions at school. Depending on the level of your child, he/she may share his/her reasoning through (based on Fig 1.18, Empson & Levi, p27)
- modeling with concrete object or representing with a picture the situation, without using any terminology related to fraction (e.g. with Level Yellow, the child may say “I am cutting the last stick in 2 pieces, and I give one piece to this student, and one piece to that student”)
- using numbers and words. The child solves the problem while modeling / representing the situation, using numbers and words such as halves or fourth (e.g. with Level Orange, the child may say “Each child has one half of a stick”, without writing a fraction symbol)
- relating unknown fractions to a well known fractions (e.g. with Level Red, the child may say “each child will have more than a fourth, but less than a half”, without using a fraction symbol of 3/8).
- using standard fraction symbols (e.g. with Level Red, the child may say, and write, “each child gets 3/8 of a stick of modeling clay”).
So depending on where your child is on his/her journey of working with fractions, his/her strategy may vary. And that is what Time 4 Fractions is about ! Giving children a chance to explore problems on their own, and have, fun, hopefully !
Sharing my experience (Fall 2016)
With Level Yellow and Orange, both involving halves, it should be fine to follow a child’s reasoning. For instance, with Level Yellow, the child may give 3 sticks to each student, and have one stick left, or give 3 sticks to each student, and realize that the last one can be used and cut in half.
Level Red, however, may open the door to more creativity, before a child has a clear understanding that “a thing shared by b people is a/b” (Empson & Levi, 2011, p25). For instance, my child cut the 3 sticks in halves, then, realizing that she still did not have enough pieces, in fourth, ending up with 12 fourths. She gave a fourth of a stick to each student, and had a left over of 4 fourths. She cut these fourths in half (which would be eighths of a whole stick), and gave them to each student (picture on the left, each block representing one student). Something I find quite helpful to follow such reasoning is to reproduce what my child does at the same time (“tell me how you cut the sticks first?”): it helps her verbalize what she does, we can keep better track of the pieces, and come back to the whole piece at the end (picture on the right). A child may say say that each student gets 1/4 of a stick and a half of a fourth, noticing eventually that one fourth equals two eighths, and half of a fourth equals… a eighth i.e. each student gets 3/8 of a stick. But it does take time to build up a deep understanding in fraction. No rush !
See you next week for our last Equal Sharing problem !
Empson, S. E., and Levi, L. (2011). Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH : Heinemann. ISBN-13: 978-0325030531.