# Category 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.

## Building up a bridge between home and school

I often refer to being on a journey as the author of this blog. I see myself as a lifelong learner, trying to connect my experiences as a parent, as a teacher, as a graduate student in math education. I feel like I am walking in the woods, enjoying the hike, wondering what the next curve may bring.  I pass the curve, and keep on going. Sometimes, I feel like I am getting lost. Sometimes, I reach a clearing, at the top of a hill, that gives me a better view of where I want to go.  Or a reminder of why I started the blog.

I read an article this week, discussing out-of-school learning vs school learning, and how often children do not connect the two of them (Saxe, 1984). It made me think of one of my first posts  : “For every single worksheet my children may bring from School, I want to make sure they know why they are learning these skills” (see post here). Indeed, whatever we do at home, I always try to connect it to Rosie’s or Tom’s school learning. But it might not be natural for everyone.

As you may have noticed with my lastest posts, I was quite inspired by the conference I attended to in June, on Cognitively Guided Instruction. One of the speakers, Tracy Zagger wrote recently a post for new math teachers (here), wanting them “to become addicted to listening to students’ mathematical ideas”. I am not a new math teacher, but it is definitely how I feel.  I think one of the reasons I am so attracted to the CGI approach is that it deeply echoes my vision of  seeing every child as a unique person and my belief that every child, in a supportive environment, can succeed. After the conference, I started following people on Twitter, exploring new blogs. Some are full of activities to implement in the classroom. Others bring math to the home, with discussions on the spot while cooking dinner, or buying groceries. Whether you browse the web as a parent or as a teacher, you can cross the paths of very inspiring people, and the resources are endless. But I see how a piece of the puzzle can easily be left aside, how the link that connects what is learned/done at school with what is learned/done at home can be forgotten.

I will continue my walk in the woods, I even expect reaching out into some deep dark woods as I begin to embrace my doctorate program tomorrow, but I know for sure that I want to keep focusing my effort on working on that bridge. Connecting both worlds can only take us even further.

Reference

• Saxe, G. B.  (1988).  Candy selling and math learning.  Educational Researcher, 17(6), 14–21.

## Solve & Share #1 – Confusing dimes !

Our journey “Solve & Share” will take us to exploring math tasks and sharing some reasoning out loud. I thought the following word problem would be a good way to start as it illustrates, I hope, how much you can learn about your child’s understanding in math by just listening to him/her.

The problem was from the South Dakota Booket I discussed previously (here). As always, my child Rosie, 8, could solve the problem in “any way that makes sense to her” (Carpenter et al, 2014). She could model the problem, with manipulative, a drawing  or a trial error approach, she could use counting strategies, or number facts. As always, it was up to her.

The problem was:

Kenata has 167 coins in her jar. 50 of them are dimes, and the rest are pennies. How many pennies does she have?

I had seen Rosie solve problems that looked similar to me. Using counting strategies and number facts. And confidence.

With this one, she froze.

At some point, she drew the picture shown on the right, with not much conviction though. Used to the Base Ten Blocks and their representation on paper, she drew the 167 coins as 1 Hundred (i.e. the “gridded square” on the top ), 6 Tens (i.e. the tallies), and 7 Ones (the little squares). She wrote an equation with the unknown number : I was kind of expecting Rosie to finish up.

But she froze again.

She tried with smaller numbers, but it did not seem to help.

I suggested another strategy she had been using successfully in the past when she is stuck : change the story. We talked about pets, dogs and cats, instead of coins, dimes and pennies. Rosie did not have any issue to solve it.

But when she went back to the initial problem, she… froze again.

We went back to her drawing. At this point, however, I noticed through her explanation that the Tens in her jar no longer represented 10 coins but… 10 cents i.e. … 1 dime. No wonder why she was confused. Dimes and pennies are so often associated to cents in word problems, that she could not see them as just coins anymore.

I could have helped her, and said “Rosie! Your Tens represent Tens of coins, not Tens of cents!”.

But I did not.

Because I rarely do. Following the steps of Cognitively Guided Instruction (Carpenter et al, 2014), I prefer waiting that it comes from her, even if it requires an additional 5-10 minutes. Or more. But little step by little step, going back and forth from her drawing to the problem, from the problem to the drawing, she saw it. At some point, she saw where her confusion came from. And provided an answer of 117 pennies almost instantly. With a priceless expression on her face.

It may take time to let a child fully make sense of a problem, or a math concept. But, to me, as a parent as much as an educator, it seems so worth it.

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.

## “Exploring the math shelf #2” – Building Blocks of Mathematics

“Exploring the math shelf” is a journey that takes us weekly to our public library to explore their selection of math books. Click here to follow it from the beginning. Whether you are a parent, a teacher, someone supporting a child’s math thinking, I hope you find our books review helpful !

Our weekly trip made us discovered a series of books called “Building Blocks of Mathematics” (by Joseph Midthun and Samuel Hiti).

The series comprises six books

• Numbers
• Subtraction
• Multiplication
• Division
• Fractions

I highly recommend them.

1. The books are amusing, cute, strips books. Once we started with the first one, Rosie, 8,  could not wait to read the next ones.
2. We started with Addition, Subtraction, Multiplication and Division, as Numbers was not available initially, but I do not think it matters. Numbers can be read independently.
3. The book Numbers presents a variety of counting systems, and invites children to create their own. I remember having to create my own Base System as a M.Ed. student, it was quite an instructive process, to say the least ! Rosie started by making random symbols for each numeral she would think of. We started discussing about patterns that usually occurs in counting systems.  Her second attempt was quite close to our decimal system, but in her third attempt, a different logic started to appear. Her reflection is far from being completed, but I can see how the book Numbers could indeed lead to a powerful activity around counting.
4. The book Numbers also goes into place value, presenting how some systems have place value while others do no. I have to say that I had never really thought about it. For instance, the counting symbols used by the Egyptians could be written from left to right, or right to left, each symbol keeping the same value no matter its position. With the Arabic numerals, however, the value of each digit depends on its place in the number (e.g. the 5 in 53 has a value of 5 Tens). Interestingly, it seems to me that the Roman numerals are kind of in-between: V has always a value of 5, but IV and VI have different value, depending on the position of the symbol I (4, when I is placed before V, and 6, when I is placed after V). Place value is a concept so often misunderstood, Numbers provides an opportunity to approach it through another angle that would be helpful even in upper elementary grades.
5. In Numbers, there is even a WHOLE page on Zero, a numeral so often forgotten!!!
6. My hope when I pick a book series, is to find some connections between the math concepts presented in each book (e.g. a link between geometry and fractions, or multiplication and repeated additions, etc). This series exceeded my expectations on that front. The character “+” leads the story in Addition, but is also part of Subtraction aside the character “-” and Multiplication along character “x” . In Division, all characters are present (“+”, “-“, “x” and “÷”) illustrating well the relationship between the four operations.
7. The books contain reassuring words (the character “+”, for instance, saying “It never hurts to slow down when you are doing math”, or “you can always use me to check your work” in the book Multiplication).  I have to say that Rosie is not the most confident person around (the apple doesn’t fall far from the tree), and she found it quite comforting to read that when you are stuck in one operation, you can always go back to another one.
8. The characters “+”, “-“, “x” and “÷” discuss different ways to solve problems, using drawings, number lines, equations, etc. A good review of strategies to discuss with your child.
9. Rosie has not been talking about division at school yet. Still, she was fully engaged in the book “Division”, as the concept is clearly presented, and well connected to the other operations she is more familiar with. I decided not to read the whole book “Fractions”, though. It is well written, but I  want Rosie to keep exploring fractions a little further without going to rapidly into their symbolic representation. I look forward to doing our Time 4 Fractions in the Fall for the third time, I may go back to this book once we are done.
10. Cherry on top: the book Cognitively Guided Instruction (Carpenter et al, 2014) is referred as a resource for educators. If you have been following my blog, you know how highly I recommend this approach of instruction :-)

I could still add to the list,we had so much fun reading them. I hope you do to !

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.