# Tag Archives: cognitively guided instruction

## 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.

## 3rd Edition – Time 4 Fractions – Problem #1 – Walking along a pond

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, 2015, 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 ? (e.g. 10 pads and 5 flies; 12 pads and 8 flies; 13 pads and 21 flies, etc.)

What to do as a parent ?

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. I purposely stepped away from grade level. 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 ?

If your child is not used to solving multiplication problems, you may have to 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. Just resist to showing him/her how you would solve the problem.

I am including a link to 2 videos that we did a while ago. Just bear with the French accent, the camera made me quite uncomfortable… :
• 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 can be done 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.

One more thing: you are right, there is no fraction involved in this problem. 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, 2015).

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).

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?

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 !

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.

## It sure is a journey !

When I decided, a couple of years ago, to start this blog, I saw it as a journey, my journey as a parent helping my children with math at home, willing to share all the good stuff I would learn as a M.Ed. student in math elementary education to, hopefully, inspire other parents.

It has, indeed, been an instructive journey. And attending recently a conference dedicated to Cognitively Guided Instruction (CGI) makes me feel like embracing the journey even more. In the past two years, I have enjoyed listening to my children solving problems “a way that makes sense to them” (Carpenter et al, 2014) and meeting the CGI community deeply confirmed my beliefs in such approach. CGI can be complex to describe, but in the context of this blog,  I would define it as a math instruction focused on how children think in math i.e. children’s mathematical thinking: children are invited to explore problems prior to receiving any formal instruction, prior to being introduced to any symbols or procedures. While children make sense of a problem, adults listen. In a CGI classroom, as children share their work, teachers embrace opportunities to build up their math instruction. At home, of course, one may not expect a nurturing classroom discussion. Still, I believe the exploration as such, without time pressure or peer pressure, the “out-loud” thinking is quite valuable. I have opened the door of my house to CGI, and I have enjoyed sharing my experience as a parent on this blog.

I often wonder, though, how I can reach out to more parents. Because it was, of course, my main goal in blogging: helping other parents. I use pen names, which makes it trickier to use social media as a megaphone, and I am more of a “behind-the-curtain” kind of person. So I guess the key will be to get back to more regularity  in my posts. Luckily, I came back from the conference with plethora of new activities to do with my kids, new blogs and resources to explore. So from now on, you can expect to find, once a week, at least one of these posts :

• “Solve and Share”:  I will continue to post word problems to explore across elementary grades but I will include additional children strategies from the literature to complete the experience I have with my own children. Hopefully, these posts will inspire you to welcome CGI at home.
• “How many?”: I am super excited about sharing this activity as I have done it a couple of times with my kids, and they loved it. I will dig further to let you know the genesis of it and explain it in further details when I write our first post but basically, you show the kiddos a picture, and ask them “how many?”. And they can count … whatever they want. They may start with counting items one by one, but the picture can open up to counting by groups, finding arrays, discussing fractions, etc. Here is a picture to help you start thinking about it.

• “Exploring the math shelf”: I naturally add math questions to any book I read to my kids, but I discovered an entire shelf dedicated to math books at our library (I know, it is about time). I started reviewing them, I have to say that some are much better than others. No wonder why kids get easily confused with math ! So each time we go to the library, I will bring a few math books and share my thoughts with you.

Of course, I will continue to post about any relevant matter for parents I read  as a doctoral student. Please, do not hesitate to share your thoughts as well, and raise any questions you may have. Time to fully connect with the math e-community !

Reference:

• 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.

## 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.