Geeking Out on Multiplication Strategies

Yesterday, I introduced my students to the new-to-me strategy of multiplying using a ratio table. We did a few whole number examples together, adding this strategy to our multiplication toolboxes along with the standard algorithm (some can use this strategy, but not all...not YET), partial products, and area model. I turned them loose with pictures of oranges I took one day at the grocery store. Some oranges were $.79 each, others were $.55 each, and there were the above bags of 8 oranges that were $2.99 per bag. Their task was to find out how much each option would cost in order to buy an orange for each of the 27 students in the class.

When we gathered at the end of the period to share answers and strategies, I was floored by one of my mathematician's use of the brand new ratio table strategy. First of all, he used $3.00 rather than $2.99, knowing he could take off those extra pennies once he had his solution. Smart! He started his ratio table the way I'd modeled, with his first known information. 3:8 (dollars:oranges). He doubled until he got to 32 oranges and $12. We all assumed you'd just have to buy 5 extra oranges.

But what happened next blew us all away. He showed us that by HALVING instead of doubling, he could get us within one orange (and a half a penny) of the 27 oranges.

It's not very often that a teacher gets to witness what happens when she hands a learner a tool and they take it to the next level all on their own.

It was a geeky-good day in math yesterday!

Math Monday

Thanks to friend and colleague Maria Caplin (@mariacaplin) for tagging me in a tweet with this video. She knows I am always looking for great math ideas and this one looked promising! (And how was it that I was not following Marilyn Burns (@mburnsmath) on Twitter before this--thanks again, Maria!)

The 1-10 Card Investigation on Marilyn Burns' blog looked like a fun thing to try have my 5th graders try.  So last week we gave it a try.  It was our best day of math so far this year.

Marilyn Burns 1-10 Card Investigation

There was so much about this investigation that was perfect for math, especially early in the school year:

• The talk as students struggled with the problem was fabulous. Kids had a partner (one that they'd had for a few days in math) and this investigation invited purposeful collaboration. Kids were truly thinking together and we will build on that using this as an anchor.
• No one was upset about making mistakes as they had been over the. last few weeks. This investigation pretty much assumes you are going to make mistakes-many mistakes.  We used this experience to talk about why these mistakes felt more comfortable for them than other mistakes that they'd gotten upset about previously.
• Everyone wanted to figure this out on their own. They didn't want anyone to share their answer or strategy and they worked hard to think on their own. The fun was in the challenge. 
• There was so much laughing in the room during this 30 minute investigation.  It sounded like a big party. Most of the laughing came when kids thought they had it but then an incorrect card showed up.  The sound in the room is what math should sound like every day and we talked about that feeling of joy you have when you are solving something challenging.

In the 30-40 minutes we worked, about 3 groups figured this out.  (2 could not remember exactly what they did to make it work but they knew they could quickly figure it out again.) Many asked about continuing during an indoor recess one day and some were going to play around with the cards at home. No one asked for the answer and all were excited to get back to it sometime soon.  I looked back at the investigation after we finished and Marilyn Burns also offers a few extensions.

There was so much good about this day in math--I highly recommend this math lesson for upper elementary classrooms.

Math Books for the Classroom

How Many Guinea Pigs Can Fit On a Plane? by Laura Overdeck is a fun new book that I discovered thanks to someone sharing on Social Media!  It was released a week or so ago and I ordered it right away. This book is a full of questions you can answer yourself using math. There are chapters that categorize the questions--Animal Math, Nature Gone Wild, Math for Your Mouth and more. Each two page spread poses a question and gives the info you need to solve the problem. It also answers and explains the reasoning for the answer. For example--How many times do dogs take a bath a year?--and then goes on to investigate. I see this as a great book to use for openers for Math Worksop or in a variety of ways to just have fun with math.

Animals By the Numbers: A Book of Infographics by Steve Jenkins is a book of visuals. The infographics on each two-page spread are all about animals and each infographic is very unique.  Reading each page takes time and attention to detail and I can see doing Notice and Wonder thinking routines with these pages. These are great pages to linger over as you talk about data, displaying information, comparing things and more. And of course this book has Steve Jenkins fabulous illustrations so it can be used in coordination with some of his other books like Down, Down, Down.

Finally I need to thank my friend and colleague Maria Caplin for introducing me to this book.  Mind Boggling Numbers by Michael J. Rosen is another one that is great fun!  This is a book that is similar in concept to How Many Guinea Pigs Can Fit On a Plane? as it also asks questions of readers and then goes on to think through the math.  This book also includes great graphics and. visuals....and some humor!


I am happy to have discovered three books that have great ideas for young mathematicians. And these are definitely for older elementary students. Excited about sharing these 3 books with my 5th graders in the fall.

Math Monday: Estimate 180

Last week, my colleague Kami Wenning and our math coach, McKenzie Zimmerman conducted an informal morning PD session on the site Estimate 180. Kami has been using the site with her 3rd graders and the conversations around it have been astounding so they wanted to share the resource.

Estimate 180 is a website created by Andrew Stadel (@mr_stadel). According to his website, he is a middle school math teacher and coach. He began the site in October 2012 with estimating activities he uses with his students each day of the school year. 

After the PD, McKenzie and I talked about how I could use this site She facilitated the class while I transcribed and listened to her language with students.  She went through the 4 day Lego Estimations and I watched from the back of the room to learn what I could about how best to use this resource and to listen to and record my students' thinking.  The goals for the lesson were from the math practice--explaining your mathematical reasoning and understanding someone else's math reasoning. So that was the focus of the talk over the four days.

The conversations across days went so far beyond the typical estimation activities I've seen. The way that the site is built, the learning builds from one day to another and kids have information to build from.  The talk around numbers was incredible and the engagement was high.  Knowing the standards so well, McKenzie was able to take advantage of the last day's conversation to create a number sentence with a number to solve for.   I am finding that oral language and conversation is such a huge part of math learning and Estimate 180 definitely supports this.

There are so many amazing things about the Estimate 180 site. There is a huge variety on the site. So many math concepts are covered in the over 200 estimation activities on the site. In a few weeks, I am going to use a series of lessons designed around estimating height and I am looking at another that estimates the amount of money in coins.  You can browse the site or search estimations based on math topic.  I also love that these are multi-day activities that are built to help kids think across time and to use understandings from one day to solve the next day's challenge.

Mr. Stadel must think about estimation all day every day because so many of these estimations come from real, daily life and I think kids will start seeing estimation opportunities everywhere after a few weeks of these.

I loved this site so much that I just had to share. I am excited to jump into another estimation with my kids next week (Cheeseball Estimations) and see where the conversations go!

Math Monday: Well Played

I have been reading lots of math professional books as we've implemented Math Workshop in our district.  This summer, I read Well Played:  Building Mathematical Thinking Through Number Games and Puzzles, Grades 3-5. This was a perfect read for me this summer as we move forward with Math Workshop.  I love that Stenhouse provides such an amazing online preview on their site. I was able to dig in and know that I wanted to own a copy of this book. I know it is one I will revisit throughout the school year.

The book is more than just a set of games.  As Kassia Omohundro Wedekind states in her Foreword,
"This is a book about math games and puzzles, but it is also a book about building communities of mathematicians who work together to problem solve, talk about math and figure things out."

The book begins with thoughtful chapters around the use of games in the math classroom. Early on in the book, the authors state, "...many students experience games or puzzles as fun activities or time fillers, but do not consider them as essential to their learning or as an important part of a lesson for which they are accountable."  The authors go on to show us how to make games a more critical piece of our workshop and to help students have ownership of the games, their goals and the conversations they have while playing.

There is a great section about discussions and the authors give lots of practical tips for teaching kids to have productive conversations while playing game.  There are so many examples of these conversations, questions that push thinking and ways to differentiate throughout the book.

Much of the book is organized in chapters by math concept and there are many games that support kids across levels and operations. The authors give great games and give great variations of several of the games.  The games focus on engagement and problem solving and give kids ways to use math vocabulary throughout.

The games throughout the book are introduced in a way that you can really visualize how they might look in your classroom. Directions and materials are given as well as an example of how one teacher introduced the game in a real classroom.  (The appendix is large and provides blackline masters for all of the games, directions, etc.)    The game pages include Tips, What to Look For when observing kids play the game, Exit Card ideas and Extension of ways to change up the game.

An amazing resource for intermediate math teachers!

Math Monday -- In Which the Lesson Doesn't Go As Planned...And is a Success

It's Math Monday! 

Join Mandy at Enjoy and Embrace Learning

for the Math Monday link up!

We were going to begin a big estimation problem (How Many Books Are There in Ms. Hahn's Classroom?), so I chose my math workshop opener from Estimation 180 -- days 28-30, a sequence of toilet paper estimations. I knew exactly what I wanted to get out of this opener, and I expected it to be quick.

What I didn't expect what that my students would get mired down in a dis-remembering of what exactly perimeter, area, and volume are, and why the square footage fact we jotted down from the packaging shown in the answer of day 28 could not be used as the total length of the toilet paper on the roll on day 30. Maybe it's because we were talking about squares of toilet paper that their brains convinced them that square feet would be okay as a unit of length.

I let them struggle through misconceptions like squares and cubes are the same and you use 3D measurement for square feet. It was one student's tentative sharing of a rhyme she learned at her old school, "Perimeter goes around, but area covers the ground" that finally turned the tide away from the confident assertion of another student that square feet is a measure of length. You should have seen the lightbulbs go off above the heads. Boom. They had it back. Area is LxW (2D), volume is LxWxH (3D), and perimeter is S+S+S... (a measurement of length). Whew.

I've written often about the difference between leading the learning and following the learning. The importance of following is something I have to remember over and over again.

Math Monday -- Pans of Brownies

It's Math Monday! 

Join Mandy at Enjoy and Embrace Learning 

for the Math Monday link up!

Dividing whole numbers by unit fractions and unit fractions by whole numbers are 5th grade standards.  (CCSS.MATH.CONTENT.5.NF.B.7)

I've never taught division of fractions, but when you are struggling to understand something yourself, you often do a better job explaining it to someone else.

In my classroom, dividing fractions is all about pans of brownies. 

If you have four pans of brownies and you want to divide them each into fourths, how many fourths will you have?

4 ÷ 1/4 = 16

You will have sixteen one-fourth-sized pieces to share with your friends.

But what if you you share 15 of those one-fourth-sized pieces and realize you forgot to share with 4 other friends? 

If you chop a one-fourth-sized piece into four pieces, what size of piece will each of those friends get?

1/4 ÷ 4 = 1/16

They will get a tiny little piece, but at least you didn't completely forget them!

Creative Commons photo from Wikimedia Commons

Math Monday

It might not happen so much for primary teachers, but I am humbled on about a weekly basis by students in my 5th grade math class who are smarter than I am.

Case in point, this pizza problem. Do whatever you need to do to enlarge that picture. The work you will see there is flat-out brilliant.

In this problem, a class has won the PTO's pizza party for bringing in the most Boxtops For Education™. Each student gets their own personal pizza and eats a different fraction of the pizza. They eat thirds, fourths, eighths, twelfths, and sixteenths. The challenge was to put the fractions in order from greatest to least to find out which student(s) ate the most pizza, and then find out which table group ate the most pizza.

The pair of students who made this poster demonstrate two different ways to create equivalent fractions with a common denominator of 48: the "Bring to 48" table at the top in the center of the page, and the longer version on the right side of the page. (I didn't teach them either of these methods. They came up with them on their own. Brilliant, right?)

On the left side of the poster, they show their work finding an equivalent fraction for each of the children in the problem. They add each column to find out which table group ate the most, and they put all of the fractions/students in order (below the "Bring to 48" table in the center of the page).

Differentiation is important. While these two were engaged in solving this problem and demonstrating their work on this poster, I was working with a group of students who still can't independently make equivalent fractions in order to add and subtract with an unlike denominator. Others in the class were working on solving the pizza party problem, but they never got to the demonstration stage, or else their demonstrations were not nearly as elegantly organized.

It's Math Monday! Join Mandy at Enjoy and Embrace Learning for the Math Monday link up!

Math Monday -- Google Comes to Math Class

I am joining Mandy at Enjoy and Embrace Learning for Math Monday.

I had been struggling with an authentic reason to introduce Google Presentation to my students. I needed a time when we would collaborate on a presentation rather than each student doing his/her own. And I needed a way for collaboration to happen without students revising each other's work.

When we were working on irregular volume in math, I found a way to use Google Preso! I created a slideshow with a page for each student and shared it to their Drives. I demonstrated how to use the drawing tools to make rectangular prisms. Their job was to first build two rectangular prisms using manipulatives, then combine them into one shape, and finally represent them and solve for volume on their slide. If they got finished early, they could add an additional slide and tell the three most important things about volume. For the sake of privacy, I have taken the students' names out.

Math Monday...on Thursday

I am joining Mandy at Enjoy and Embrace Learning for Math Monday. Franki has resolved to join Alyson Beecher's Nonfiction Book Challenge in order to stretch herself to read more nonfiction. I'm going to stretch myself in a different direction and try to focus on what's working (or not working) in my 5th grade math workshop.

This week (in between a snow day and a windchill day) we began working towards a deep understanding of division. Our standards in 5th grade do not require students to be able to do long division with the algorithm. We will be exploring multiple strategies for division.

Mandy wrote this week about the importance of play. What I discovered was the importance of manipulatives...even for fifth graders.

Students were in groups of 4 or 5 on the floor in the meeting area. Each group had different manipulatives (beans, dominoes, pattern blocks, tiles). We modeled what addition looks like (combining groups) and what subtraction looks like (starting with a big group and taking some away from it).

Then we moved to modeling multiplication, which was surprisingly hard for them. After I gave them a problem to model (3x4), they realized/remembered that they needed to make equal groups or an array. We spent a lot of time thinking about what a multiplication problem SAYS -- "Three TIMES" tells you will be repeating a process three times, or making three groups.

Modeling division was as challenging as modeling multiplication. We started with a problem that they could easily solve with mental math so that they could check to make sure their model made sense (22 ÷ 2). Knowing that partial products is one of the first strategies we'll work on once we move to paper-pencil, I also gave them problems like 68 ÷ 5 so we could talk about efficient ways to share 68 into 5 equal groups rather than counting one by one. (Starting with 10 in each of the five groups, for example, and then sharing the leftover 18 into the 5 groups.)

Our math block is cut 10-15 minutes short by related arts, which we have actually come to love, because we can come back to our work and share, or students can complete an exit ticket or formative assessment that will inform my instruction for the next day. I gave each student a sheet of notebook paper and asked them to draw a model for 19 ÷ 3 and then write three things they know about division. What an eye opener! I've got a group of 5-6 who modeled 19 x 3, and another 4 or so who modeled 19 ÷ 3, but didn't demonstrate complete understanding by giving an answer. There were students who could model, but not write anything they know about division, and there were students who could write three things about division but not model.

So, now it's time for differentiation. I need to get some students to that deep understanding of what division means (modeling), and I need to move others along to applying that understanding to various strategies! This is the tricky part! This is the FUN part!

Math Monday!

It's Math Monday!  Join Mandy at Enjoy and Embrace Learning for the Math Monday link up!

The first few days of math are always so interesting as I listen into conversations.  On the third day of school, we used our math time to do a "Numbers About Me" project.  I've seen this often on Pinterest and blogs and wanted to make sure we started the year thinking about math in our world.  It was an interesting conversation as their eyes lit up each time they realized the things in their lives that involved numbers.  They were simple things but making the connection to math made for a good conversation. We combined this with self-portrait work and the kids had a great time creating themselves with their Numbers About Me information.

*Please note that the 3rd boy in the top row made himself wearing an "I Love Mrs. Sibberson" shirt. Hysterical.  Gotta love 3rd grade :-)

Math Monday: Resources for Opening Routines

So excited that Mandy began a weekly time for us, as bloggers, to share our thinking about math teaching and learning.  Today is the first Math Monday!  You can find the round up on Mandy's Blog, Enjoy and Embrace Learning.

Last year, our Math Workshop went pretty well.  But not as well as I had hoped. One area I knew I had to work on was Opening Routines.  I had read Number Talks the year before and used the Number Talk routine daily. But I found that it became very rote when it was the only routine I relied on. So I have really focused on new routines and have found some great resources to kick off quick routines and also to build on those routines through the year.

I started in the spring exploring the Howard County website. There is a whole section on routines for 3rd grade so I read about some new routines that would support math learning.

Then I revisited Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 . This is a book from Stenhouse that I was familiar with but revisited this summer with 3rd graders in mind.  As more of an intermediate teacher, it was helpful to remember all of the math tools that support kids when making sense of number.  3rd is on the upper end of primary but I find so many kids need more support than I think they do at this age. Lots of great ways to support number sense.

Finally, I discovered my favorite new resource for math routines.  It is Minilessons for Math Practice, Grades 3-5 (there is a K-2 version, also) . I bought this book because Mandy had recommended it and I thought it would be filled with mini lesson ideas. But as I browsed through, they seemed more like opening routines to me.  I noticed that the blurb on the back of the book said, "Designed to use during transition times, mini lessons require little or no preparation and take only 5-15 minutes to teach. These activities can be repeated throughout the school year...".  These were the routines I was looking for.

The book focuses on Grades 3-5 and shares 27 routines. Each short chapter focuses on one routine.  Ways to introduce the activity, student examples and ideas for extending the activity are part of each chapter.  This is a great resource! So excited I discovered it!

Professional Reading: Math Workshop

I returned to the classroom last year after 4 years as an elementary librarian. The two years before I became a librarian, I taught only Language Arts and Social Studies and shared my classroom with a colleague who taught the Math and Science. So it had been six years since I'd paid much attention to math.

I've always loved teaching math (which surprises lots of people) and I am actually a better mathematician than I am a reader and writer.  I've always loved math and love to watch the discovery on kids' faces as they explore numbers and problem solving and critical thinking.

So I wanted to jump back in and was happy to see that there were lots of amazing resources out there.   I picked up several professional books on math teaching that I planned to read last summer. Then I got a concussion and my reading life was put on hold.  Over the year, I continued to pick up great books and took recommendations from smart friends and colleagues.  So, my stack has grown and grown.  There are books on my stack that I've already read, books I want to read cover to cover and books that I want to dabble in to get the info I'm looking for.

I moved from teaching 4th grade to teaching 3rd grade this year and the math teaching is a little bit different. I find myself looking more in the K-3 resources these days.

Our district is moving to a Math Workshop and as much as I overall like the way math went this year, there are lots of things I need to change.  I am really thinking hard about better routines, more intentional conversations and the role of student choice in Math Workshop.

Books I've read and loved in the last year or two:

One of my favorite reads over the last few years was  Math Exchanges: Guiding Young Mathematicians in Small Group Meetings by Kassia Omohundro Wedikind. I think I read this one once I learned I was going to be teaching 4th grade. (I blogged about it when I first read it.) It was an amazing read and not only changed my ideas about small group instruction in math, but also my thinking about talk and story in the math classroom.  I'm hoping to revisit the book--I recently got a copy of Kassia's DVD How Did You Solve That?: Small-Group Math Exchanges with Young Students and am excited to watch that this summer.

I spent a great deal of time with Number Talks, Grades K-5: Helping Children Build Mental Math and Computation Strategies two years ago and really learned lots about routines and the importance of these number routines. I feel like I need to revisit pieces of this book now that I've taught a year in 3rd grade. I think revisiting the specific 3rd grade sections will help me be more effective with this routine.

I also spent time with What's Your Math Problem? Getting to the Heart of Teaching Problem Solving which helped me think about not only good problems but how to assess problem solving and how to help my students reflect on their own work.

I spent some time with Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 before I went back into the classroom but it seemed a little primary for 4th grade. Honestly, I forgot about it until I saw the ad from Stenhouse on the accompanying video (Go Figure!) from these authors and I am VERY excited to reread this one from a 3rd grade perspective.  Thinking about routines is definitely one of my biggest goals for next year and this book and video seem perfect to add to my thinking.

Books I'm Most Excited to Read

At MRA this year, we somehow started talking Math and Brian Wyzlic  invented #nerdymathclub.  He recommended 5 Practices for Orchestrating Productive Mathematics Discussions [NCTM] and I ordered it right there and then.  I am excited to read this one and learn more about good math discussions.

Another book that lots of people I trust are talking about is Putting the

Practices Into Action: Implementing the Common Core Standards for Mathematical Practice, K-8.  The focus on The Common Core is an important one for me right now as I want to see what others are thinking about the standards and how best to teach in our current era.  I love that this one focuses on Standards for Mathematical Practice rather than content standards.  I am sure it will give me lots to think about and revise.

A brand new book that I am VERY excited about is Intentional Talk: How to Structure and Lead Productive Mathematical Discussions. Stenhouse has been putting out amazing math professional books so I pretty much trust anything they have on their list. Plus, this book is about talk and I know how important that is. I have read so much about intentional talk in the literacy classroom.
Guided Math in Action

Minds on Mathematics: Using Math Workshop to Develop Deep Understanding in Grades 4-8 looks like one that will help me tweak workshop structure a bit. Even though it is written for grades 4-8, I think the chapters on work time and conferring will help me a lot. I am not sure who recommended this one to me but it is close to the top of my summer stack.

Finally, I picked up Guided Math in Action: Building Each Student's Mathematical Proficiency with Small-Group Instruction because of the focus on small group instruction but looking through it, it will also help me think through workshop in general, observation of students and quality learning opportunities.

There are more on my stack but these are the few I really want to dig into this summer.  I'm open to any other suggestions that will help with Math Workshop in Grade 3!  What are you reading?

Also, we are hoping to have some Twitter Chats around math over the summer. Keep an eye out or the hashtag #nerdymathclub (thanks, @brianwyzlic) if you'd like to join us!