Category Archives: Time 4 Fractions

3rd edition – Time 4 Fractions – Problem #4 – Making toys

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.

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?


What to do as a parent ?

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) or drawing a picture. He/she may write an equation. Each child should pick the problem that he/she feels like exploring.

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).

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 ?

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

It is quite interesting to me, so curious about children’s mathematical thinking, to follow my child’s reasoning over the years.

A couple of years ago,  she explored Problem4Level Orange with drawing 14 wheels, taking away groups of 4. 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 drew 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”. It is good for kids to have opportunities to discover on their own that some representations may work better in some situations, and less in others. Indeed, it is going to be up to them to select the most useful one depending on the problempb4.

 

Last year, she explored Level Red as well, by but she added the groups of wheels needed for one vehicle (e.g. 6 wheels to make a truck) until she reached the total number of wheels available.

Now this year, she went back to an approach similar to what she did 2 years ago, drawing 31 wheels, and grouping them by 6, to make 5 trucks, with one wheel left.

I just found it fascinating to see the various ways a child may solve a problem, leading him/her to exploring  the relationship between all operations. Who knows what next year will bring in Rosie’s world.

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

Enjoy !


Reference:

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



3rd Edition – Time 4 Fractions – Problem #3 – Baskets of eggs

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.

Hope your child had fun exploring Problem #1, and Problem #2, two multiplication problems. Here is Problem #3, a measurement division problem (also called quotative division problem), our second step towards Equal Sharing problems (Empson & Levi, 2011, p 9).


Time 4 Fractions –  Problem #3 – Baskets of eggs

 Yellow : Mr Moose has 4 eggs and some baskets. He wants to put 2 eggs in each basket. How many baskets can he fill?

 Orange : Mr Moose has 12 eggs and some baskets. He wants to put 3 eggs in each basket. How many baskets can he fill?

 Red : Mr Moose has 20 eggs and some baskets. He wants to put ___ eggs in each basket. How many baskets can he fill?


What to do as a parent ?

As with Problem #1, 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) or drawing a picture. He/she may write an equation. Each child should pick the problem that he/she feels like exploring.

With Level Yellow and Orange, all eggs will be dispatched in a basket, and Mr Moose will have no egg left. With Level Red, invite the child to pick the number of eggs he/she wants to put in each basket. Depending on the number he/she picks, though, please note that Mr Moose may have some eggs left.

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 ?

 

Enjoy following his/her way of thinking !

Sharing my experience

  • Click here to see a video we did last year.  Just remember it is just an example of how a child may explore the problem. Your child may approach it differently!
  • An observation I found quite comforting regarding our journey is my child saying “You see, the more eggs you put, the less baskets you need !”, noticing the relationship between the number of items, the number of groups of items and the number of items in each group.  Do you see how this kind of connection relates somewhat to fractions, and the fact that sharing an item in 8 (1/8) provides smaller pieces than sharing the same item in 2  (1/2), i.e. the number 1/8 is smaller than the number 1/2 ? It is all about mathematical relationships.
  • I am also including a example of how a child, like…. Rosie,  may represent her thinking on paper. The picture on the left may look “messy” for some,  but I think it illustrates well what may be going on in a child’s brain while making sense of a problem. The twenty eggs are presented in four groups of five before an equation is written (a division, but also a repeated addition (making group of 5s from the 20 eggs), a repeated subtraction (taking away groups of 5s out of the 20 eggs).

Problem3

Have fun, and see you next week for Problem #4 !


Reference:

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



3rd Edition – Time 4 Fractions – Problem #2 – Gardening

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.


Hope you had fun with your child exploring Problem 1. Here is Problem #2, a second multiplication problem, before introducing division problems next week. Please remember that the goal of our journey is to provide children with plenty of opportunities to explore fractions through Equal Sharing problems (Empson & Levi, 2011), and solving multiplication and division problems will prepare them to do so (Empson & Levi, 2011, p 9).


Time 4 Fractions –  Problem #2 – Gardening

Level Yellow : Mr. Purple loves gardening. He planted 3 rows of pumpkin seeds. In each row, there were 2 seeds. How many pumpkin seeds did Mr. Purple plant?

Level Orange: Mr. Purple loves gardening. He planted 5 rows of pumpkin seeds. In each row, there were 4 seeds. How many pumpkin seeds did Mr. Purple plant? 

Level Red : Mr. Purple loves gardening. He planted ____ rows of pumpkin seeds. In each row, there were ____ pumpkin seeds. How many pumpkin seeds did Mr. Purple plant ?


What to do as a parent ?

As with Problem #1, 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) or drawing a picture. He/she may write an equation. Each child should pick the problem that he/she feels like exploring. With Level Red, invite the child to pick numbers he/she feels like comfortable using. For instance, if your child picks 5 pumpkin seeds, he/she may end up counting the seeds by 5, or he/she may use from memory the 5s times table (i.e 5 x 12 if he/she picks 12 rows of seeds).

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 ?

Sharing my experience

I thought it would be helpful this week to provide some work samples I gathered in the past 3 years from Rosie and the daughters of a dear friend of mine.  No teaching was involved, the girls were just invited to solve the problems in a way that made sense to them. It may give you an idea of strategies a child may use. Please remember that I am sharing these samples to help you see what a child may come up with, not as examples of what a child should come up with :-)

Level Yellow : Mr. Purple loves gardening. He planted 3 rows of pumpkin seeds. In each row, there were 2 seeds. How many pumpkin seeds did Mr. Purple plant?

  • Making sense of the problem with a picture. The child wrote then both a repeated addition and a multiplication.

FullSizeRender-1

Level Orange – “Mr. Purple loves gardening. He planted 5 rows of pumpkin seeds. In each row, there were 4 seeds. How many pumpkin seeds did Mr. Purple plant?”

  •  Making sense of the problem with marbles and paper.  The child counted the marbles by 1s’. Your child may count by 4s’ ?Problem 2 - 5 groups of 4
  •  Making sense of the problem with a picture representing the rows of pumpkin seeds. The child wrote, as an equation, a repeated addition. Your child may write a multiplication (4 x 5 = 20) instead?Problem 2 - 5 groups of 4
  • Making sense of the problem with a different visual representation, an array. The child wrote then both a repeated addition and a multiplication.

FullSizeRender

Level Red – “Mr. Purple loves gardening. He planted ____ rows of pumpkin seeds. In each row, there were _____  pumpkin seeds”.

  • Making sense of the problem with Duplos® (5 rows, 8 seeds). The child counted the blocks by 1 up to 15, and noticed that she was counting by 5. She started over, counting by 5, and answered 40 pumpkin seeds. This sure was fun to watch a child, noticing a pattern of counting, changing her strategy to a more efficient one.

Problem 2 - 8 groups of 5 (Duplos)

  • Making sense of the problem with buttons. (7 rows, 5 seeds). The child also wrote, as an equation, a multiplication 7 x 5 = 35.
    IMG_0525
  • Making sense of the problem  (8 rows, 5 seeds) with a picture representing the rows of pumpkin seeds, drawing the seeds in the first row, and writing the number of seeds instead on the next rows. Always interesting to see how a child may switch from a drawing to a more symbolic representation.

Problem 1 - 8 groups of 5

  • Making sense of the problem  (2 rows, 4 seeds) with a picture representing the rows of pumpkin seeds, writing, as an equation, an addition.

IMG_0527

 

Also, here are some examples of questions I asked to follow the child’s reasoning:

  • Tell me about what you did.
  • Could you tell me about the marbles you used ?
  • I see you wrote the equation 4 + 4 + 4 + 4 + 4 = 20. Could you show me on your drawing where the 4 comes from? The 20 ? Why did you add 4 five times? What does the symbol “+” mean? And the symbol “=”?
  • You said “20”. 20 what? Could you tell me the unit?

No video this week, as the problem is similar to the one explored last week.

Have fun, and see you next week for Problem #3 !


Reference:

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



Our 3rd edition of Time 4 Fractions is ready to start: all aboard !

Please click here to follow Time 4 Fractions.

I am quite excited about starting our 3rd edition of Time 4 Fractions in the coming weeks. I should be able to update the posts from last year significantly, since I went to the conference dedicated to Cognitively Guided Instruction in June and I am taking a course on Children’s thinking this semester as a doctoral student.

As you may remember, I started Time 4 Fractions two years ago, after I read the book “Extending Children’s Mathematics – Fractions and Decimals” (Epson & Levi, 2011) as a M.Ed. student, thinking “This IS the way I would have liked to explore fractions! “. An ah-HA! moment, a true eye-opening: building up meaning for fractions through equal sharing problems. A wonderful approach to pursue at School. But also at home, I believe: the more opportunities to extend math reasoning, the better.

Over the twelve coming weeks, I am going to post a word problem that will take the kids to slowly, gradually, explore the concept of fractions. We will start our journey with multiplication problems (yes, even with lower graders, click here if you are not sure why !), division problems, then, finally equal sharing problems, the core of our journey, and the true beginning of our fraction exploration. The sequence of problems is based on the reading of two books, Children’s Mathematics (Carpenter et al, 2015) and Extending Children’s Mathematics – Fractions and Decimals” (Epson & Levi, 2011).

Whether your child is in lower grade or upper grade, I hope you join us. I share what I do with my own child as a illustration of what a child may do, but by no mean as what a child should do. It is not a test, it is not a race. Week after week, problem after problem, children practice their reasoning skills by creating their own strategies to solve problems.

In the previous year, I found it quite convenient to put together a “math box”. You may want to do the same before we start !

  • paper and pencils. Markers are also helpful to connect a visual representation to an equation.
  • manipulatives to model the problem.  You do not need the base-Ten blocks. Marbles, buttons can do the trick. I like Legos® and Duplos®, as you can stack them in Tens.
  • Containers (e.g. paper cups, Tupperware®), to model problems involving groups of items.
  • A stack of paper (e.g. blank flashcards), to explore fractions, by cutting parts of a whole, and putting them back together.Our math box

 

Most important, I will be here to support you in the journey. Please, feel free to comment or email at journey2helpchildrenwithmath(at)gmail(dot)com if you have any question about our journey. The more feedback I receive, the more complete the next post will be ! Let’s build up a community of people supporting at home what our children learn during Math instruction !

Off we go !

References:

  • Carpenter, T., Fennema, E., Franke, M., Levi, L. and Empson S. (2015). Children’s Mathematics, Second Edition: Cognitively Guided Instruction. Portsmouth, NH: Heinemann. ISBN-13:978-0325052878.
  • Empson, S. E., and Levi, L. (2011). Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH : Heinemann. ISBN-13: 978-0325030531.

Update Ending Time 4 Fractions – Problem #12 – Sharing cereal bars

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 comes our last Equal Sharing problem !


Time 4 Fractions –  Problem #12 – Sharing cereal bars

Level Yellow – 2 people want to share 1 cereal bar so that each of them gets the same amount. How many cereal bar would each get?

Level Orange – 3 people want to share 1 cereal bar so that each of them gets the same amount. How many cereal bar would each get?

Level Red – 5 people want to share 3 cereal bars so that each of them gets the same amount. How many cereal bar 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 !

The problem will lead to a answer of each person getting 1/2 of a cereal bar (level Yellow) , 1/3 of a cereal bar (level Orange) or 3/5 of a cereal bar (level Red).

problem12

Level Red – Child’s sample

This is the last problem. What can you do now ?

The goal of T4F was to provide children with opportunities to explore fractions at home, so they have stronger foundations to build up on when they study fractions at school. This is our last problem, but it does not have to be the end of our journey. The set of problems was designed to provide a wide range of answers, to explore halves, fourths, thirds, fifths and so on, so do not hesitate to go back to these problems and provide one regularly to your child, until your child figures out that “a thing shared by b people is a/b” (Empson & Levi, 2011, p25).  For instance, Problem 12, Level Orange, leading to an answer of 1/3 would be an instructive step towards Problem 8, Level Red, that leads to an answer of 2/3.

The level of difficulties can be seen as follow (Epson & Levi, 2011):

  • Equal Sharing problems that lead to a whole number (i.e. Problem 8, Level Yellow)
  • Equal Sharing problems that lead to an answer that is more than one, with the children having to decide what to do with any left over they may have (first in halves, e.g. Problem 8, Level Orange, or Problem 9, Level Yellow, then fourth e.g. Problem 9, Level Orange)
  • Equal Sharing problems that lead to an answer that is less than one (first with halves or fourths e.g. Problem 12, Level Yellow, then thirds, e.g. Problem 8, Level Red, Problem 12, Level Orange, and so on)

I am including a table summarizing the problems and set of numbers we have exploring so far, I thought it might help.

Problem Level Number involved
Problem 8 – Sharing paper Level Yellow 2
Level Orange 2 1/2
Level Red 2/3
Problem 9 – Sharing bananas Level Yellow 2 1/2
Level Orange 1 1/4
Level Red 4/5
Problem 10 – Sharing apples Level Yellow 1 1/2
Level Orange 2 1/4
Level Red 4/6
Problem 11 – Sharing clay Level Yellow 3 1/2
Level Orange 1/2
Level Red 3/8
Problem 12 – Sharing cereal bars Level Yellow 1/2
Level Orange 1/3
Level Red 3/5

Hope you enjoyed our T4F journey ! As always, I  appreciate any feedback you may have. Comment, or email at journey2helpchildrenwithmath(at)gmail.com.


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 #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.