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

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 !

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?

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  :-)

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 #7 – Reviewing with buttons

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.

This week is going to be a little different, as it is our last problem before starting Equal Sharing problems and fractions (Yeah ! Finally !). You may have noticed your child being more comfortable with some of the problems posted for the past 6 weeks, and less comfortable with others. This week is your chance to review what we have done so far.  I hope you do not feel too overwhelmed with all the options. If so, just come back next week, we will be back to our 3 Levels :-)

Time 4 Fractions –  Problem #7 – Reviewing with buttons

Review multiplication problems

• Level Yellow – Mr. Needle is making 2 jackets. On each jacket, he puts 3 buttons. How many buttons does Mr Needle need in total?
• Level Orange – Mr. Needle is making 5 jackets. On each jacket, he puts 4 buttons. How many buttons does Mr Needle need  in total?
• Level Red – Mr. Needle is making ___ jackets. On each jacket, he puts ___ buttons. How many buttons does Mr Needle need  in total? Complete the problem with the numbers of your choice.

Review measurement division problems

• Level Yellow – Mr Needle has 5 buttons. He wants to make jackets with 2 buttons on each jacket. How many jackets can Mr Needle make?
• Level Orange – Mr Needle has 15 buttons. He wants to make jackets with 4 buttons on each jacket. How many jackets can Mr Needle make?
• Level Red – Mr Needle has ___ buttons. He wants to make jackets with ___ buttons on each jacket. How many jackets can Mr Needle make?  Complete the problem with the numbers of your choice. (e.g. 27 buttons, 8 buttons on each jacket)

Review partitive division problems

• Level Yellow – Mr Needle has 8 buttons. He wants to make 3 jackets, using the same number of buttons on each jacket. How many buttons can Mr Needle use for each jacket ?
• Level Orange – Mr Needle has 13 buttons. He wants to make 4 jackets, using the same number of buttons on each jacket. How many buttons can Mr Needle use for each jacket ?
• Level Red – Mr Needle has ___ buttons. He wants to make ___ jackets, using the same number of buttons on each jacket. How many buttons can Mr Needle use for each jacket ? Complete the problem with the numbers of your choice. (e.g. 31 buttons, 7 jackets)

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.

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

We did not review these problems this year, but here is a quick summary of what we did a couple of years ago.

My child started with the multiplication problem, and what I found interesting is that, compared to what she did with Problem #1 and Problem #2 (which were also multiplication problems), she wrote, as an equation, a multiplication. Apparently, she learned about multiplication recently (through playing on Starfall.com, from what she said) and made the connection with the buttons on the jackets. As I told you before, our T4F journey is about exploring, but symbols do come in time ! My child was not ready to write a multiplication equation a few weeks ago, but it looks like now, she is.

Time to move on to the next leg of our journey: equal sharing problems, here we come !

## 3rd edition – Time 4 Fractions – Problem #6 – Stacking blocks

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 #6, a second partitive division problem.

Time 4 Fractions –  Problem #6 – Stacking blocks

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?

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?

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?

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.

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 ?

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

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.

Reference:

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

## 3rd edition – Fractions – Problem #5 – Peg dolls

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 #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

Level 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?

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.

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 ?

With level Red, I left again the option open to pick the number of peg dolls and the number of bags. 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

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).

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).

Click here to see a video of the exploration of 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.

## 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 Toys

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 Level 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 problem.

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.

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).

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.

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’ ?
•  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?
• Making sense of the problem with a different visual representation, an array. The child wrote then both a repeated addition and a multiplication.

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.

• Making sense of the problem with buttons. (7 rows, 5 seeds). The child also wrote, as an equation, a multiplication 7 x 5 = 35.
• 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.

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

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.