# Category Archives: Operation

## 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.

## 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.

## WedWoPro #14 – Last one !

Every Wednesday, I give a chance to my child to explore a word problem a way that makes sense to her. And every Wednesday, I share the word problem, and my experience with you, so you can do the same ! Click here  to start from the beginning ! Hope you join us !

Here comes WedWoPro #14, that will end our journey for a little while. I don’t know if Rosie is getting tired with the end of the year but I think it is time to take a little break in word problems to come back even stronger once Summer break starts.

WedWoPro #14 – Last one

Today, you are the teacher ! Write a word problem, and I will solve it !

Sharing my experience:

Here is what I had to solve :

“Once upon a time, 6 butterflies came to have an ice-cream at a friend’s house because it was very hot outside. 4 more came at the friend’s house to have ice cream.

How many butterflies were there in total?

One butterfly left. Then, 3 more left. Then, 3 more left again.

How many butterflies were still at the friend’s house?”

I love doing this kind of task, and opening the door to creativity. It always lead to fun discussion. Today, it gave us a chance to discuss again how to select information that was necessary to solve the problem (e.g. 6 butterflies, 4 more, etc)  vs the information that was not (e.g. it was very hot outside). It was also a fun way to model how I would solve the problem, and share my reasoning, by pretending to be the kid.

Until next time !

## WedWoPro #13 – Few more days !

Every Wednesday, I give a chance to my child to explore a word problem a way that makes sense to her. And every Wednesday, I share the word problem, and my experience with you, so you can do the same ! Click here  to start from the beginning ! Hope you join us !

Here comes WedWoPro #13. We will be working with time for another week. Just to show Rosie that time is not only about minutes and hours. A one-step problem.

WedWoPro #13 – Few more days !

Rosie was very excited. Soon, his grandma would come to visit.

• “I can’t wait to see her tomorrow ! “
• “I can’t wait to see her  ! Just 2 more days !”
• “I can’t wait to see her ! Just 3 more weeks !”

If today is Wednesday, April 13, when is Grandma coming ?

Here is the .pdf if you want to print it out (WedWoPro13).

• Level Green:  Grandma is coming on Thursday, April 14.
• Level Orange: Grandma is coming on Friday, April 15.
• Level Red: Grandma is coming on Wednesday, May 4.

As always, invite your child to solve the level of his/her choice a way that is meaningful to him/her (Carpenter et al., 2014) !

Sharing my experience:

Well, Rosie’s grandma actually arrived TODAY from France. Too much excitement around here… Will have to update the post later !

A bientôt !

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.

## Making sense of subtracting in column

I thought I should complete my latest post,  making sense of adding in column (here), with a quick post on using Base Ten blocks to make sense of subtracting in column. My daughter is not there yet, but your child may be.

The picture above presents a concrete illustration of 43 – 15. It is quite helpful for kids to visualize that, when subtracting 15 to 43, they trade a Ten from the Ten column into Ones. Then, they can subtract 5 Ones to 13 Ones, 1 Ten to 3 Tens, and end up with 2 Tens and 8 Ones i.e. 28.

I just spent the morning in my son’s classroom. He is 4 and attends a Montessori school. Their approach to teaching math is amazing. Indeed, using concrete objects, little 4-5 years old kiddos solve 4-digits addition without even thinking about it.  I should write a special post on the Montessori approach to Math. Quite inspiring, indeed.

## Making sense of adding in column

Adding in column 23 + 14 ?  3 +4 = 7, 2 + 1 = 3… so the answer is 37. Adding 37 + 44 ?  7 +4 = 11, 3 + 4 = 7… so the answer is 711. Wait, Mom. It does not make sense, doesn’t it?

Nope, Rosie, 711 doesn’t seem to make sense. So let’s step back an inch, with the Base Ten Blocks (click here if you want to read more about these blocks).

Here is an example of 32 +23, and the connection between the blocks, and the addition in column. While adding in column, you add the Ones, then the Tens, then the Hundreds, and so on, and the blocks provide a neat concrete representation of such process. Indeed, it shows why you have to “align” digits (because if you don’t, you end up adding Ones to Tens !).

But what I like the most with these blocks is how they  help children  make sense of carrying an over to the next column. Here is an example with 37 +44.

From the 11 Ones you get from the right column (i.e. the Ones column 7+4), you trade 10 Ones from 1 Ten that you carry over to the left column (i.e. the Tens column).

Here you go, Rosie, 711 does not make sense, but 8 Tens 1 Ones aka 81 does.

## WedWoPro #12 – You have few more minutes !

Every Wednesday, I give a chance to my child to explore a word problem a way that makes sense to her. And every Wednesday, I share the word problem, and my experience with you, so you can do the same ! Click here  to start from the beginning ! Hope you join us !

Here comes WedWoPro #12. We will be working with time this week !

WedWoPro #12 – You have few more minutes !

Rosie was reading a book about unicorns and fairies. Soon, it will be time to get ready to go to School.

• “It is 7:00. In 30 minutes, it will be time to go to School.”
• “It is 7:00. In 15 minutes, it will be time to go to School.”
• “It is 6:50. If you read for 20 min, you will have 15 min left before it is time to go to School.”

What time does Rosie need to go to School?

Here is the .pdf if you want to print it out (WedWoPro12).

• Level Green:  Rosie needs to leave at 7:30.
• Level Orange: Rosie needs to leave at 7:15
• Level Red: Rosie needs to leave at 7:25

As always, invite your child to solve the level of his/her choice a way that is meaningful to him/her (Carpenter et al, 2014)!

Sharing my experience:

Here came a loud “7:30!” as soon as Rosie finished reading the first level. OK, Rosie, but I want more… How did you figure it out?

So she started drawing a clock, the 2 hands, writing 1 to 12 on it.

What are these numbers for ? What about the hands? How would the big hand move in Level Green? How about the little hand? In Level Orange? How could you figure it out Level Red?  And so on. And so on.

Indeed, these 3 levels took us to a fun discussion on how clock works. So engaging that I will come up soon with a post dedicated to time…. Stay tuned, I should finally have more time to write on my blog  very soon.

Until next time !

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.