Category Archives: Education

Exploring word problems throughout Summer

Summer break is here, and we are back to exploring word problems regularly.

Here is a good source of word problems if you want to do the same:

South Dakota Booklet

As always with our math journeys (e.g. Time 4 Fractions or WedWordPro), I simply invite my child Rosie, 8, to solve a problem in a meaningful way to her (Cognitively Guided Instruction, Carpenter et al, 2014), and share her thinking out loud. Drawing a visual representation on paper to make sense of the problem, using manipulatives (e.g. buttons, Legos®, Base Ten block, flashcards to fold and cut, etc), writing an equation and solving the problem using a strategy of her choice, it is up to her, I just listen 🙂

Enjoy !


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.

Misconceptions in math

If you have visited this blog for a while, you may have noticed my continuous quest in helping children deepen their understanding in math. I share here what I do as a parent with my own children, hoping that it could help other parents as well. At home, I use games, math discussions,  explorations of word problems, etc. But I also try to identifying any misconceptions they may have in their learning.

Example.

My son Tom, 5, has learned how to count by Fives and Tens, while playing hide-and-seek with older kids.

– “Ten, Twenty, Thirty, Forty, Fifty, Sixty, Seventy, Eighty, Ninety, One hundred ! Ready or not, here I come !”.

I was quite surprised, initially, to hear him count by Ones, Fives, and Tens, as it is not a concept often mastered at a young age. So one day, as he was playing with his cars, I asked him if he could count them.

– “Sure !”, he said, pointing one car at a time, “One, two, three, four, five, six, seven, eight, nine, ten !”.

– “Could you count them by five?”, I asked

– “Sure !”, and pointing one car at a time as he did while counting by Ones, he counted : “Five, Ten, Fifteen, Twenty, twenty-five, thirty, thirty-five, forty, forty-five, fifty !”.

– “Could you count them by Tens ?”, I asked

– “Sure ! Ten, Twenty, Thirty, Forty, Fifty, Sixty, Seventy, Eighty, Ninety, One hundred !”, pointing again one car at a time.

He could root count by Ones, by Fives, by Tens up to a hundred and more. But he did not understand that counting by Fives means counting by groups of five, counting by Tens means counting by groups of tens. For him, it was just three independent ways of counting. You have ten cars when you count by Ones, fifty when you count by Fives, a hundred when you count by Tens. But since you “usually count by Ones”, you have ten cars. Seems logical in Tom’s world.

With Tom, 5, a discussion is usually the best way to assess his understanding, with a “tell me about what you are doing”, or “what does it mean to …”. With Rosie, 8, I like to use pretend-playing : she is the teacher, I am the child. I do not do it often during the school year, but Summer break is a fun time to do so. And since Summer break starts tomorrow… you should hear more about it pretty soon.

Stay tuned !

 


Exploring fractions with “The doorbell rang” by Pat Hutchins

You may know the author Pat Hutchins and her books for young children, such as Rosie’s Thedoorbellrangwalk, Changes changes, Clocks and more Clocks, etc. I bought a few when my children Rosie and Tom were younger, and “The doorbell rang” was one of them. I had almost forgotten about them, until I recently heard Rosie, second grade, say:

“I recognize this book ! We read it in math today !”

So we read it again. The text is attractive as it includes predictable sentences that young children enjoys repeating out loud. And for older kids, the story opens the door to math. Ma made cookies for her two children, Sam and Victoria to share (equally). The doorbell rings, and two more children, Tom and Hannah come and share the cookies. As the doorbell keeps riging, more children come to share the cookies, until twelve children have to share the twelve cookies.

As I was reading the story, Rosie modeled it. She used flashcards to represent the cookies, similarly to what she has been doing with Time 4 Fractions.

  1. At the beginning of the story, Sam and Victoria gets 6 cookies each. How many cookies has Ma baked?
  2. Now that they have to share the 12 cookies among 6 children, how many cookies does each child get?
IMG_0041

6 children sharing 12 cookies equally: each child gets 2 cookies

But a fun activity we added was to twist the story a little, and work not only with whole numbers, but also fraction. You may want to give it a try. I just let my child make sense of the problem, whether using paper to cut, or buttons to count, or the base Ten Blocks. Sometimes, she connects her model to symbols she has learned at school. But the goal is to let her make sense of the problem.

  1. What if Tom does not want any cookie, i.e. three children share the twelve cookies, how many cookies does each child get ?

  2. What if Peter does not want any cookie, i.e. five children share twelve cookies many cookies does each child get ?

  3. What if Tom, Peter and Victoria do not want any cookie i.e. nine children share twelve cookies many cookies does each child get ?

IMG_0042

9 children sharing 12 cookies: each child gets 1 whole cookie, and 1/3 of a cookie (or equivalents)

I always look for opportunities for my children to have fun exploring problems, and make sense of them.  “The doorbell rang” sure is a neat book to create such opportunities.

Give it a try ! I am here if you have any questions !


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).

problem12

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 #11 – Sharing modeling clay

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 another Equal Sharing problem. This week, I am including a few words about how a child may share his/her reasoning, depending on his/her previous exposure to fractions. Hope it helps.


Time 4 Fractions –  Problem #11 – Sharing sticks of modeling clay

Level Yellow – 2 students want to share 7  sticks of modeling clay so that each of them gets the same amount. How many sticks of modeling clay would each get?

Level Orange – 4 students want to share 2 sticks of modeling clay so that each of them gets the same amount. How many sticks of modeling clay would each get?

Level Red – 8 students want to share 3 sticks of modeling clay so that each of them gets the same amount. How many sticks of modeling clay would each get?


Invite your child to either model the problem (with paper (I like flashcards, as you can go back to a whole card to compare what you cut) and scissors for instance) and/or represent the problem with a picture. If your child has learned about fractions at school, he/she may connect symbols to the model or picture. And as always, invite your child to share his/her reasoning with you !

The problem leads to a solution of each student getting 3 sticks and 1/2 (level Yellow) , 1/2  of a stick (level Orange) or 3/8 of a stick (level Red).

Sharing my experience (Fall 2015)

The goal of Time 4 Fractions is really to provide children with additional opportunities to explore fractions at home, so they have stronger foundations to build up on when they study fractions at school. Depending on the level of your child, he/she may share his/her reasoning through (based on Fig 1.18, Empson & Levi, p27)

  1. modeling with concrete object or representing with a picture the situation, without using any terminology related to fraction (e.g. with Level Yellow, the child may say “I am cutting the last stick in 2 pieces, and I give one piece to this student, and one piece to that student”)
  2. using numbers and words. The child solves the problem while modeling / representing the situation, using numbers and words such as halves or fourth (e.g. with Level Orange, the child may say “Each child has one half of a stick”, without writing a fraction symbol)
  3. relating unknown fractions to a well known fractions (e.g. with Level Red, the child may say “each child will have more than a fourth, but less than a half”, without using a fraction symbol of 3/8).
  4. using standard fraction symbols (e.g. with Level Red, the child may say, and write, “each child gets 3/8 of a stick of modeling clay”).

So depending on where your child is on his/her journey of working with fractions, his/her strategy may vary. And that is what Time 4 Fractions is about ! Giving children a chance to explore problems on their own, and have, fun, hopefully !

Sharing my experience (Fall 2016)

With Level Yellow and Orange, both involving halves, it should be fine to follow a child’s reasoning. For instance, with Level Yellow, the child may give 3 sticks to each student, and have one stick left, or give 3 sticks to each student, and realize that the last one can be used and cut in half.

Level Red, however, may open the door to more creativity, before a child has a clear understanding that “a thing shared by b people is a/b” (Empson & Levi, 2011, p25). For instance, my child cut the 3 sticks in halves, then, realizing that she still did not have enough pieces,  in fourth, ending up with 12 fourths. She gave a fourth of a stick to each student, and had a left over of 4 fourths. She cut these fourths in half (which would be eighths of a whole stick), and gave them to each student (picture on the left, each block representing one student). Something I find quite helpful to follow such reasoning is to reproduce what my child does at the same time (“tell me how you cut the sticks first?”): it helps her verbalize what she does, we can keep better track of the pieces, and come back to the whole piece at the end (picture on the right). A child may say say that each student gets 1/4 of a stick and a half of a fourth, noticing eventually that one fourth equals two eighths, and half of a fourth equals… a eighth i.e. each student gets 3/8 of a stick. But it does take time to build up a deep understanding in fraction. No rush !

See you next week for our last Equal Sharing problem !


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 ? 
 Pb10orange

Sharing my experience (Winter 2017)

There is a significant gap between Level Orange and Level Red, so of course, it is perfectly fine if a child decides to explore only Level Yellow and/or Orange. My child tried Level Red by sharing the apples in halves, providing half of an apple to each child but having some left over. We will be back next week to explore further !

Enjoy !


Reference

Empson, S. E., and Levi, L. (2011). Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH : Heinemann. ISBN-13: 978-0325030531.


Update Time 4 Fractions – Problem #9 – Sharing bananas

My daughter and I went on a 12 week-journey last year to explore Fractions. We are doing it again this Fall. I am updating the posts from last year with videos, in case you want to join us this yearClick here if you want to know more about the journey and the previous problems.

Another Equal Sharing problem (Empson & Levi, 2011) to help children make connections with fractions.


Time 4 Fractions –  Problem #9 – Sharing bananas

Level Yellow – 2 children want to share 5 bananas so that each of them gets the same amount. How many bananas would each get?

Level Orange – 4 children want to share 5 bananas so that each of them gets the same amount. How many bananas would each get?

Level Red – 5 children want to share 4 bananas so that each of them gets the same amount. How many bananas would each get?


Invite your child to either model the problem (with paper and scissors for instance) and/or represent the problem with a picture. If your child has learned about fractions at school, invite him/her to connect symbols to the model or picture. And as always, invite your child to share his/her reasoning with you !

Level Yellow involves halves, level Orange, fourths and Level Red, fifths. Level Yellow leads to a mixed number (2 1/2) although it may make more sense to some children to give each child 2 bananas, and have a banana left. Level Orange leads to a mixed number (1 1/4), level Red to a proper fraction (4/5).

Sharing my experience (Fall 2015)
My child got a little frustrated with Level Red. If it happens to your child, you may want to start the problem with 5 children sharing 1 banana.  It was a good alternative for us.

Sharing my experience (Fall 2016)

We continued modeling the different levels with flashcards, through folding/cutting paper similarly to what we did last week, each flashcard representing a banana. It is a good way to explore half, fourth, or fifth, depending on the level, with the option to going back to a “whole” banana if need be. We also took the chance to compare a fourth of a “banana “to half of a “banana”, or “two fourth” of a banana to half of a “banana” (“it is the same!”), etc.

Enjoy !


Reference

Empson, S. E., and Levi, L. (2011). Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH : Heinemann. ISBN-13: 978-0325030531.