# Category Archives: Math

## 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 year

Here comes our last Equal Sharing problem !

Time 4 Fractions –  Problem #12 – Sharing cereal bars

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?

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?

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).

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 year

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

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?

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?

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 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 year

Here is the problem for the week.

Time 4 Fractions –  Problem #10 – Sharing apples

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

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

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 !

Yellow leads to 1 apple and a half, Orange leads to 2 apples and a 1/4 of an apple, and 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 ?

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.

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

Time 4 Fractions –  Problem #9 – Sharing bananas

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

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

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 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.