Update Time 4 Fractions – Problem #11 – Sharing modeling clay

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 yearClick 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)

  1. 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”)
  2. 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)
  3. 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).
  4. 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 !


Reference

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


Update Time 4 Fractions – Problem #10 – Sharing apples

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 yearClick here if you want to know more about the journey and the previous problems.

Here is the problem for the week.


Time 4 Fractions –  Problem #10 – Sharing apples

Level Yellow – 2 people want to share 3 apples so that each of them gets the same amount. How many apples would each get?

Level Orange – 4 people want to share 9 apples so that each of them gets the same amount. How many apples would each get?

Level Red – 6 people want to share 4 apples so that each of them gets the same amount. How many apples would each get?


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

Level Yellow leads to 1 apple and a half, Level Orange leads to 2 apples and a 1/4 of an apple, and Level Red leads to 4/6 of an apple, or its equivalent 2/3, depending on the strategy the child may use.

Sharing my experience (Fall 2015)

My child went with Level Yellow and Level Orange. I was surprised to see her writing a fraction symbol (1/4). She apparently learned the symbol on her own while playing an education game on the tablet, through a short video, showing a pizza, cut into halves, fourths, and eights. Pretty neat, but at one point, the video talks about 3 fourths of a pizza (3/4) left to eat showing … 6 eights of a pizza (6/8). The 2 fractions are equivalents, but how puzzling to hear 3/4 and see 6/8 of a pizza ? 
 Pb10orange

Sharing my experience (Winter 2017)

There is a significant gap between Level Orange and Level Red, so of course, it is perfectly fine if a child decides to explore only Level Yellow and/or Orange. My child tried Level Red by sharing the apples in halves, providing half of an apple to each child but having some left over. We will be back next week to explore further !

Enjoy !


Reference

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


Update Time 4 Fractions – Problem #9 – Sharing bananas

My daughter and I went on a 12 week-journey last year to explore Fractions. We are doing it again this Fall. I am updating the posts from last year with videos, in case you want to join us this yearClick here if you want to know more about the journey and the previous problems.

Another Equal Sharing problem (Empson & Levi, 2011) to help children make connections with fractions.


Time 4 Fractions –  Problem #9 – Sharing bananas

Level Yellow – 2 children want to share 5 bananas so that each of them gets the same amount. How many bananas would each get?

Level Orange – 4 children want to share 5 bananas so that each of them gets the same amount. How many bananas would each get?

Level Red – 5 children want to share 4 bananas so that each of them gets the same amount. How many bananas would each get?


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

Level Yellow involves halves, level Orange, fourths and Level Red, fifths. Level Yellow leads to a mixed number (2 1/2) although it may make more sense to some children to give each child 2 bananas, and have a banana left. Level Orange leads to a mixed number (1 1/4), level Red to a proper fraction (4/5).

Sharing my experience (Fall 2015)
My child got a little frustrated with Level Red. If it happens to your child, you may want to start the problem with 5 children sharing 1 banana.  It was a good alternative for us.

Sharing my experience (Fall 2016)

We continued modeling the different levels with flashcards, through folding/cutting paper similarly to what we did last week, each flashcard representing a banana. It is a good way to explore half, fourth, or fifth, depending on the level, with the option to going back to a “whole” banana if need be. We also took the chance to compare a fourth of a “banana “to half of a “banana”, or “two fourth” of a banana to half of a “banana” (“it is the same!”), etc.

Enjoy !


Reference

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


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

My daughter and I went on a 12 week-journey last year to explore Fractions. We are doing it again this Fall. I am updating the posts from last year with videos, in case you want to join us this yearClick here if you want to know more about the journey and the previous problems.

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


Time 4 Fractions –  Problem #8 – Sharing sheets of paperT4F_Pb#8

  • 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 (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. IMG_4131

“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.

fullsizerender-5She 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 🙂

fullsizerender-6

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.

fullsizerender-3Then, 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  🙂

fullsizerender-4

Enjoy !

Reference:

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

Update Time 4 Fractions – Problem #6 – Stacking blocks

My daughter and I went on a 12 week-journey last year to explore Fractions. We are doing it again this Fall. I am updating the posts from last year with videos, in case you want to join us this yearClick here if you want to know more about the journey and the previous problems.

Here is Problem #6, a second partitive division problem.


Time 4 Fractions –  Problem #6 – Stacking blocksT4F_Pb#6

Level Yellow – Emmy has 5 wooden blocks. She wants to make 2 towers as tall as possible, using the same number of blocks in each tower. How many blocks should she use in each tower?

Level Orange – Emmy has 13 wooden blocks. She wants to make 4 towers as tall as possible, using the same number of blocks in each tower. How many blocks should she use in each tower?

Level Red – Emmy has 23 wooden blocks. She wants to make ___ towers as tall as possible, using the same number of blocks in each tower. How many blocks should she use in each tower?


Invite your child to solve one of the problems  by

  1. modeling the problem with manipulatives (such as buttons, marbles, etc, and small containers),
  2. representing the problem on a piece of paper, and/or
  3. writing an equation.

When your child is done, invite him/her to share his/her reasoning with you. If your child only writes an equation, encourage him to represent or model the problem as well, and connect the parts of the equation to the model/representation.

This week, all levels involve a remainder (Level Yellow: 2 blocks/tower, 1 block left; Level Orange: 3 blocks / tower, 1 block left) .

Sharing my experience (Fall 2015)RepresentationProblem6

At week 6 of our Time 4 Fractions journey, it seems that my child has her own routine to solve the problem, through at least 2 Levels. She starts with modeling level Yellow, and usually draws a picture to solve level Orange and/or Red. Then, she adds an equation that would match her drawing. This week was no different. She modeled Level Yellow, and drew the blocks, one at a time, in 4 towers, to solve Level Orange.

Sharing my experience (Fall 2016)

Our experience this week was quite similar to last year. Time to move to fraction problems!


Reference:

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


Update Time 4 Fractions – Problem #5 – Peg dolls

My daughter and I went on a 12 week-journey last year to explore Fractions. We are doing it again this Fall. I am updating the posts from last year with videos, in case you want to join us this yearClick here if you want to know more about the journey and the previous problems.

Here is Problem #5, a partitive division problem. Last week, with the measurement division problem, children knew the number of items in each group, and needed to find the number of groups. This week, children know how many groups they have, and have to find out how many items are in each group. Just another way to keep exploring division and mathematical relationships.


Time 4 Fractions –  Problem #5 – Peg dolls

PegDollsLevel Yellow – Peter and Julie made 6 peg dolls. They put them into 3 gift
bags with the same number of peg dolls in each bag. How many peg dolls are in each bag?

Level Orange – Peter and Julie made 18 peg dolls. They put them into 6 gift bags with the same number of peg dolls in each bag. How many peg dolls are in each bag?

Level Red – Peter and Julie made ___ peg dolls. They put them into ___ bags with the same number of peg dolls in each bag. How many peg dolls are in each bag?


As always, invite your child to solve one of the problems by

  1. modeling the problem with manipulatives (such as buttons, marbles, etc, and small containers),
  2. representing the problem on a piece of paper, and/or
  3. writing an equation.

When your child is done, invite him/her to share his/her reasoning with you.

With level Red, I left again the option open to pick the number of peg dolls and the number of bags, as my child seems to enjoy the freedom. You may want to invite your child to explore Level Yellow or Level Orange first, though, with modeling the problem with manipulative or a picture. Be aware though, that depending on the numbers the child picks, Peter and Julie may have some peg dolls left (e.g. 13 dolls to put into 5 bags), or may not have enough dolls (e.g. 6 dolls, to put into 12 bags). Let me know how it works !

Sharing my experience (Fall 2015)

My child solved Level Yellow first by modeling it, though dispatching 6 marbles into 3 containers, one marble at a time. She also did a representation of the problem, and wrote an equation (repeated subtraction). Problem#5

For Level Red, she picked 20 peg dolls, and 4 bags. Then, she asked me to solve it. But I am glad she did, as we ended up talking about how different people may use different ways to solve a same problem, and how she will learn additional strategies and symbols at school (i.e. division instead of repeated subtraction, multiplication instead of repeated addition).

Sharing my experience (Fall 2016)

Here is Rosie exploring Level Orange with buttons. As always, it is just to provide an example of how a child may explore the problem.

Enjoy !


Reference:

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


Update Time 4 Fractions – Problem #4 – Making toys

My daughter and I went on a 12 week- journey last year to explore Fractions. We are doing it again this Fall. I am updating the posts from last year with videos, in case you want to join us this yearClick here if you want to know more about the journey and the previous problems.

Here is Problem #4, a second measurement division problem.


Time 4 Fractions –  Problem #4 – Making ToysMsButternutt

Level Yellow : Ms. Butternut makes wooden toys. She has 5 wheels. She needs 2 wheels to make a motorcycle. How many motorcycles can she make?

Level Orange : Ms. Butternut makes wooden toys. She has 14 wheels. She needs 4 wheels to make a car. How many cars can she make?

Level Red : Ms. Butternut makes wooden toys. She has 31 wheels. She needs ____ wheels to make a heavy truck. How many heavy trucks can she make?


As always, invite your child to solve one of the problems by

  1. modeling the problem with manipulatives (such as buttons, marbles, etc, and small containers),
  2. representing the problem on a piece of paper, and/or
  3. writing an equation.

With all Levels, Ms Butternut has a left over of wheels. (Level Yellow: 2 motorcycles can be made, with 1 wheel left, Level Orange: 3 cars can be made, with 2 wheels left).

When your child is done, invite him/her to share his/her reasoning with you. By now, you know the routine, right ?  🙂

Sharing my experience (Fall 2015)

Last week, my child decided to mostly model with manipulative the problems. I think she was not sure how to represent the problem, and confused with the equation she could use. This week, she seemed more confident in her exploration. Problem4

With Level Orange, she started with drawing 14 wheels, and took away groups of 4 one at a time. With such strategy, she quickly saw the equation that could be associated to her reasoning: a repeated subtraction (which is how division can be seen). She used the left over to make a bicycle, but your child may state that Ms Butternut has 2 wheels left.

With Level Red (31wheels, 6 wheels / truck), she decided to draw tallies (by groups of 5) to represent 31 wheels. Then, as previously, she took away groups of 6, to end up with 5 trucks (and a tricycle i.e 3 wheels left). Now, I do not know how she did not get confused with taking groups of 6 out of her tiny groups of 5 tallies, but she did say along the process that  “maybe using tallies was not such a good idea”. I enjoyed watching Rosie discover on her own that some representations may work better in some situations, and less in others. Indeed, it is going to be up to her to select the most useful one depending on the problem.

I am also sharing below the work of a friend’s child, a 5th grader solving Level Red. In parallel with writing the equation, and labeling each part of it, the child also explained the model she could use to solve the problem.

Pb4_LevelRed

 

Sharing my experience (Fall 2016)

Funny how, from one year to the next, a same problem could lead to another exploration. Last year, Rosie drew all the wheels, and took away groups of them (e.g. a group of 4, while solving Level Orange). This year, she added up the groups of wheels needed for one vehicle (e.g. 6 wheels, like last year, to make a truck) until she reaches the total number of wheels available. Such an interesting way to explore the connections between all operations. Also, I think she enjoyed adding equations afterwards, as she could fully connect every part to the picture.

pb4


Reference:

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