# Tag Archives: Time 4 Fractions

## 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).

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 #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 ?

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 #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 blocks

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)

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

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?

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).

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 #1 – Walking along a pond

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, in case you want to join us this year. I am adding videos this time 🙂 Click here if you want to know more about the journey.

Welcome to our first problem ! This week will be a warm-up, as I want to make sure we are all aboard and comfortable with pursuing the journey from home. Bear with me with the length of this post, next week will be much shorter.

The goal of this journey is to provide opportunities for children to explore word problems in “any way that they wish” (Carpenter et al, 2014, page 80), extend their reasoning skills, and gradually strengthen their foundation in fractions. Each problem is differentiated to target all elementary grades and is quite short. A child may be done within 5-10 min, or may decide to take more time to fully explore it with a visual representation and manipulatives. It is not a test, it is not a race. Week after week, problem after problem, children strengthen their reasoning skills by creating their own strategies to solve problems.

When children receive their formal fraction instruction in class, they will have a stronger background to build upon. If you decide to take the journey with us, from home, I hope you will enjoy observing your child’s thinking as much as I do with mine. It is fascinating. They explore. We listen.

So, here we go:

Problem #1 –  Walking along the pond

• Level Yellow : Mr. Wood is walking along a pond. He sees 3 waterlily pads. On each pad, there are 2 frogs. How many frogs does Mr. Wood see ?
• Level Orange: Mr. Wood is walking along a pond. He sees 4 giant waterlily pads. On each pad, there are 5 frogs. How many frogs does Mr. Wood see ?
• Level Red : Complete the problem with the numbers of your choice. Mr. Wood is walking along a pond. He sees ____ giant waterlily pads. On each pad, there are ___ flies. How many flies does Mr. Wood see ?

What to do as a parent ?

1. Invite your child to solve one of the problems. He/she can model the problem with some manipulatives ( such as buttons, marbles, etc, and small containers), represent the problem on a piece of paper, write an equation.
2. I purposely stepped away from grade level. Each child should pick the problem that he/she feels like exploring. In the coming weeks, some upper graders may decide to pick a Level orange to model, represent, and write the equation. Some lower graders may decide to pick a Level Red and model it only.
3. When your child is done, invite him/her to share his/her reasoning with you. If he/she writes only an equation, encourage him to share his/her strategy another way (with a visual representation, or with manipulatives). Enjoy following his/her way of thinking. Just resist teaching him/her symbols ! They will come in time !
4. Our journey starts slowly, as children must explore a variety of problems, and build up strategies they can use with fractions later on.  If you child solve Level Red, providing an equation and another way of representing his/her strategy, you can always invite him/her to invent his/her own multiplication problem.
5. One more thing, you may want to find an example on how fractions could be useful in your child’s life. For instance, my child loves working with me in the workshop, where we measure, add length, etc. That way, even if the problem is on a topic that may be less appealing to her, I can remind her why, in the long run, it can contribute to her learning all the same.
I am including a link to 2 videos this  week:
• Video Level Yellow : this short video (2 min) shows the material we use at home, and how a child may solve Level Yellow with a drawing
•  Video Level Orange : this one (3 min) is an example of a child solving Level Orange with manipulative
These videos are just examples, but I hope they help you see what we do at home. It is all about the exploration. Your child may not use the same approach, but as long as he/she solve the problem a way that makes sense to him/her, it is all that matters.
Questions you may have ?

I do not see any fraction in this problem !

Good point ! Just remember that we are going to explore the concept gradually.

We will start with  2 weeks on Multiplication problems (see problem #1) above. Then, we will continue with 2 weeks on Measurement Division problems (Carpenter et al, 2014).

E.g. An elf has 10 berries and some bags. He wants to put 2 berries in each bag. How many bags can he fill?

Finally, we will explore Partitive Division problems and Equal Sharing problems, the core of our fractions exploration (Epson & Levi, 2011). When we reach that step, our children will have developed/reinforced the habit of modeling directly a problem with manipulative, or representing the problem on paper,  and will naturally continue to do so with fractions.

E.g. An elf has 15 berries. He puts the berries into 3 bags with the same number in  each bag. How many berries are in each bag ?
E.g. Two elves want to share 5 berries so that each of them gets the same amount. How many berries would each get?

My child does not know how to start

If your child is not used to solving multiplication problems, it is to be expected ! You may read the problem again, and say things like “I am wondering if these cups and buttons could help us solve the problem” or “Do you think it would help to draw the situation? What should we draw?”. Level Yellow is great for that.

My child provides the answer straight away

Remind him/her that the answer is important, but how it was obtained is even more important. That’s when learning happens ! Invite him/her to model and/or represent the problem with a drawing or a diagram. Invite him/her to write an equation and connect each part of the equation to his/her model and/or representation.

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 !

References:

• Carpenter, T., Fennema, E., Franke, M., Levi, L. and Empson S. (2014). 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 Time 4 Fractions : welcome aboard !

My daughter and I went on a 12 weeks journey in the Fall 2016 to explore Fractions. We are doing it again this Fall. I am updating the posts from last year, in case you want to join us this year.

I have little memory of studying fractions at School. I remember adding fractions, multiplying fractions, finding the least common denominator, but nothing about exploring the concept as such. It was last year, while I was taking a class about fractions, reading “Extending Children’s Mathematics – Fractions and Decimals”(Epson & Levi, 2011)  that I started thinking: “Ah ! This IS the way I would have liked to explore fractions !”. An ah-HA! moment, a true eye-opening. Using word problems to build meaning for fractions. Then, incorporate symbols and equations. A wonderful approach to pursue at School. But also at home, I believe: the more opportunities to extend math reasoning, the better.

How is it going to work :

• Once a week, I will invite my child to explore a word problem and share my experience with you. We will start our journey, labeled as “Time 4 Fractions”, 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, 2014) and Extending Children’s Mathematics – Fractions and Decimals” (Epson & Levi, 2011).
• Each problem is differentiated to target all elementary grades and is quite short. A child may be done within 5-10 min, or may decide to take a few days to fully explore it with a visual representation and manipulatives. It is not a test, it is not a race. Week after week, problem after problem, children strengthen their reasoning skills by creating their own strategies to solve problems.
• 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.
• Most important, I am 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. (2014). 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.