Tag Archives: Multiplication

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

Please click here to follow Time 4 Fractions.

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.

Mastering multiplication facts

A friend of mine asked me an interesting question about multiplication facts. I thought I should share it with you.

How do you make sure that when a child learns multiplication facts by heart, he/she understands the underlying calculation?

I recently read two articles related to learning combinations of numbers (Baroody, 2006), and multiplication of one-digit numbers in particular (Kling & Bay Williams, 2015). According to these articles, it seems that mastering multiplication facts should be seen more as a journey than a step of memorization.

  • Phase 1 – The child uses modeling and/or counting strategies to solve a multiplication. He/she may use manipulative, fingers, a visual representation (e.g. to solve 6 x 4, a child draws 6 groups of 4 dots, and skip counting the dots; Fig 1, Kling & Bay-Williams, 2015)
  • Phase 2 – The child starts using reasoning strategies, deducing an answer from known facts and relationships (e.g. to solve 6 x 4, the child may solve 5 x 4 = 20, and add one more group of 4; Fig 1, Kling & Bay-Williams, 2015)
  • Phase 3 – The child provides fast, and accurate, answers (e.g. the child knows that 6 x 4 = 24; Fig 1, Kling & Bay-Williams, 2015)

I like the approach of a journey. I remember as a child being pretty stressed out by learning all the time tables by heart. But building it up from reasoning strategies sounds so helpful.

Solving 7×8? Well, I am not quite sure. I could skip counting by 7. Or since I remember 7×7 = 49, I just have to add another  7. The answer is 56 !

So at home, when my child starts remembering some multiplication facts (my child is not there yet, it is by grade 3, that she will have to “know from memory all products of two one-digit numbers”, CCSM 3.OA.7), I will likely

  • keep providing opportunities to explore multiplication with modeling and representing word problems, so that she can strenghten her own reasoning strategies.
  • come up with some games to practice time table. Two quick examples, I just like the idea of practicing multiplication facts often, but for a very short period of time
    • Rolling 2 dice, and multiply the 2 numbers, just a few times in a row but several times a week
    • Asking for “Passwords” on the door worked very well for us to practice pairs to 10 (see my previous post here), so we may come up with something similar with time tables.

While implementing these activities, I would make sure she sees some combinations as “safe bases” (e.g. multiples of 5, multiples of 10), and check regularly that her reasoning strategies are solid and in place.  According to Kling & Bay-Williams (2015), it is fundamental to help the child retain the facts.

I am such a fan of letting children exploring concepts on their own, and allowing them to build up mastery from there. And home is a great place to do so, and support all the good stuff they learn at School.


References

  • Baroody, A. J. (2006). Why Children Have Difficulties Mastering the Basic Number Combinations and How to Help Them. Teaching Children Mathematics13(1), 22-31.
  • Kling, G., & Bay-Williams, J. M. (2015). Three steps to mastering multiplication facts. Teaching Children Mathematics21(9), 548-559.

Time 4 Fractions – Problem #1 – Walking along a pond

This is the second post related to our Time 4 Fractions journey. Please click here to start from the beginning.

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 one 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. When children receive their formal 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.

One last thing, our journey will start slowly. It has to, if we want as many children, and as many parents, to benefit from it. We can’t just jump into fractions. Children must explore a variety of problems, and build up strategies they can use with fractions later on. Slowly, but surely.

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 ?

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. 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 !

I purposely stepped away from grade level, or an obvious nomenclature such as Level 1 to 3 or A to C. Each child should pick the problem that he/she feels like exploring.  It is not a test, I am not giving grades. In the coming weeks, some upper graders may decide to pick a Level Yellow to model, represent, and write the equation. Some lower graders may decide to pick a Level Orange but model it only.

I thought the Level Red could be great for children who are already used to explaining their thinking, representing the problem with a picture or a diagram, etc.  Just make sure your child does not get frustrated by picking numbers that may be too high !

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. Fractions are key ! 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.

Sharing my experience

  • I did my best to pick the right time for my child to solve the problem, and I stayed with her the entire time (5-10 min max). A special one-on-one time that we both enjoyed.
  • Most of the time, I just observed what she was doing, and learned a lot from it.  IMG_0470For instance, she stayed focused on Math while using buttons and cups, and solved the problem quite rapidly, counting the buttons by 5. With the visual representation, though, she got distracted with adding a flower on the water lilies, the water, etc. I have to keep that in mind next time she brings a worksheet from School. Does she do it there too?
  • She was not sure which kind of equation to write, and that is totally fine. Eventually, she will.IMG_0471
  • Doing Level Red was quite fun ! My child picked 2 leaves and 5 flies. How about yours?
  • I know we are all busy, having to be here and there, but I have found it so rewarding as a parent to just sit down for a few minutes and observe my child’s reasoning. I hope you do to. It can be a very special bonding time !

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  2-4 weeks on Multiplication problems (see problem #1) above. Then, we will continue with 2-4 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.

Questions ? Comment or email (journey2helpchildrenwithmath@gmail.com) ! Any feedback would be highly appreciated. This journey is our journey !


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.

Opening the door to multiplication

My daughter seems to enjoy exploring word problems, at her pace, at home. The more seahorses, fairies and cardinals involved, the better. We read a problem. Then, off she goes with her legos, marbles, base-ten blocks, paper, markers, crayons, any tools she may feel like using. It is like observing at home, one of the kids from the books and literature I read as a graduate student in Math Elementary Education. I sure enjoy that.

Young children can intuitively solve many different kind of problems. I have read it, I have seen it. Give two cookies to my 3 year old son to share equally with his sister, and he will do it : “One for me, and one for you !”. If you give him more cookies, he may not be able to count them all, but still he will be able to apply the same strategy “one for me, and one for you. One for me, and one for you. One for me, and one for you”. So at home, I try to give my children all kind of opportunities to explore problems. But this Summer, focused on the K standards with Rosie, I somehow forgot to do so.  A few weeks ago, though, I had a good reminder.

Working on an assignment for my Master, I was watching a video of a teacher discussing a word problem with her class. A 2-step problem involving a multiplicationRosie, who had mostly explored addition and subtraction problems, was nearby, doing her things. She stopped, came to me, and whispered the answer in my ear. Intrigued, to say the least, I asked her to solve another, similar, problem.

Emily has 5 bags of sparkly marbles. In each bag, there are 10 marbles. Under her bed, she also finds 3 more marbles. How many marbles does Emily have altogether?

IMG_0391

Representing the problem, with a mixture of drawings and symbols, she solved it (see picture on the right). A first bag with 10 dots representing 10 marbles, four more bags with the number 10 “because it is faster than drawing”, a representation of the bed with the 3 additional marbles. Count by 10, add 3 more. Solved. Did she write a multiplication equation? Of course, not. Symbols associated to multiplication will come later on. Still, she came up with a strategy to solve the problem, having fun exploring another kind of operation.

Time to extend my range of problems ! And share them with you. Because if my child is ready for more, your child may very well be too.

Come back on Monday, 8/24, for the beginning of another exciting journey. Spread the word, and bring a friend !