Countdown to the Holidays!

You know how I love nrich.maths.org for rich problems to solve.  Well--today I saw that they have an advent calendar for elementary students.  Each day has a rich task that accompanies it.  The problems are not holiday based; only the design of the calendar is.

You might just refer to it as a holiday countdown calendar, but no matter what you call it, it should lead to some good student thinking!

https://nrich.maths.org/14060

3 Act Tasks: Have you tried them yet?

3 Act Tasks offer our students such an engaging opportunity to make sense of math! However, as teachers, sometimes we are afraid to try something new not knowing where it might take us and scared that something bad might happen.  I just heard an analogy recently about this being like being at the top of a rollercoaster:









via ytCropper
And, while it might have twists and turns and dropoffs, not to mention bugs in the face, it also brings an excitement and thrill that we rarely find doing a traditional math lesson. We need to think about putting ourselves in that precarious position at the top of the rollercoaster; it's what we ask our students to do on a regular basis so that they can grow.  We should try it, too!

3 Act Tasks are real-world problem-solving scenarios which require students to make sense of what to do.  During Act 1, they use the reading skills of visualizing, predicting, and inferring in a math context.  They figure out what they need to know to solve the problem. In Act 2, students work to solve the math question in a way that makes sense to them. They discuss their thinking with a friend and compare their answer to their estimate.  Act 3 is exciting because that is when they find out if they were on the right track!  

While they are working during Act 2, you are monitoring (and asking questions that help them understand--not helping too much).  You are monitoring to see the methods that students used in order to solve the problem.  During Act 3, it is your job to have students share ways that they solved the problem.  These should be sequenced so that you can show connections between different methods. Be clear on what your math target for the lesson was and be sure that your models and discussion help that math target to be evident to everyone.  You synthesize the learning at the end of the lesson.

3 Acts are very visual and often use videos to help students better understand the situation. They follow more of the format of "you do, we do, I do" rather than the traditional layout of "I do, we do, you do."

Typically, a 3 Act Task in K-2  about 20-30 minutes.  In 3-5, a task usually takes about 4o minutes.  As students get stronger and more comfortable, the time needed for a 3 Act might decrease.

I am happy to come and model a 3 Act Task for you and your class, but I know that many of you can do them without my support.  You just need to put yourself on the rollercoaster!

At the top of this blog, you will find a tab labeled 3 Act Tasks.  This will take you to a large number of standards-aligned tasks that I have organized using SMART Notebook (and sometimes Google slides).  They are by a variety of mathematicians; I just put all of the pieces together into one format.  You can also find more great tasks by clicking here or here or even by Googling 3 Act Tasks.

As always, share with me your questions, struggles, or successes and let me know how I can help!

Would You Rather? -- Holiday Edition

If you are looking for some problem solving to get your students thinking and proving themselves while mixing in some holiday cheer, maybe these Would You Rather problems will work for you!

Would You Rathers are something that students are very familiar with, and these just include a mathematical twist.  Students can choose whichever option they want, but they need to have a mathematical explanation of why that is what they chose or didn't choose.  They can be used with students of all ages and offer great opportunities for collaboration.

If you like these, you can find more like them at this site, but I am betting you can come up with some good ones of your own!

Holiday Would You Rather 1

Holiday Would You Rather 2

Holiday Would You Rather 3

Holiday Would You Rather 4

Holiday Would You Rather 5

A Stepping Stone to a More Student-Driven Workshop

If you are still looking for ways to make your math workshop more about collaboration and problem-solving rather than about independent work and computation, a Week of Inspirational Math by Jo Boaler/Youcubed might just be the ticket!  If you have done any of these lessons before, you know how awesome they can be!  Well--she has recently posted her WIM #4!  That's right--4 weeks of math explorations and engaging lessons that you can use to build your mathematicians.

These engaging lessons are organized by grade level strands and are grouped into a week's worth of lessons.  They are great to use a week at a time, but you certainly could use them independently if you needed to.  They embody the paradigm shift in math education with a focus on growth mindset, visual math, patterns, and collaboration.

Besides that, each lesson will really engage your students for at least math period.  (Oftentimes, students keep working on the problem after class...) Each lesson begins with a video that you can show if you wish.  It helps to build the growth mindset in your students and presents them with mathematical thinking that helps them to see math is all around them.  After the video, there is a lesson (with full lesson-plan) that you can have students work on in small groups and share out with the class.  You can learn so much about your students as you circulate the room listening to their thinking, and they will learn so much about themselves as mathematicians!

With the craziness of the holidays, you might find a few days where some WIM activities are just the ticket, and they might cause you to begin rethinking your workshop!  They might be a good way to spend your math time during those days right before holiday break, but they might also be an awesome way to set the tone for 2019!

I have blogged about WIM before, but I can't tell you enough how much they can invigorate and change your classroom, your student's thinking, and your own thinking!  Time well spent!

Building Collaborative Thinkers

Are you giving your students enough time to work with each other to solve problems?  This is an area of growth for many of us.  Consider the following problem.  How could you build it into your math workshop?  In what ways would you anticipate that your students would be able to prove their answer?

Thanksgiving in the US has been declared to be the fourth Thursday in November.  This year, Thanksgiving is the earliest date it can possibly be.  What is the latest date Thanksgiving can possibly be? Be prepared to prove your answer.

I'd love to hear your results!

The Last Number--an exploration

Just found this problem recently, and thought about what a great exploration it would be--for nearly any age!

Consider the string 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Cross out any two numbers in this list and add the difference to the end of the list. This new number is now part of the list. Continue the process of crossing out two number on the list and adding the difference until there remains only one number. What can you say about the last number? Explore. [from Richard Hoshino]

This problem offers a rich exploration of number while practicing basic subtraction facts. (Making it great for 2nd grade!).  

Once your students have a conjecture about the final number, can they try that conjecture out on a different string of numbers?  What happens when your string goes from 1-15?  How about a string from 3-12?  Can they figure out what the pattern is and why the difference ends up like it does?  Even if they can't, they should have some rich time practicing subtraction, working together, and looking for patterns!

Can you figure out what is happening?  If you are like me, you might spend a lot of your free time working on this problem to see if you can make sense of it!

Building Conceptual Understanding of Fractions

In 3rd, 4th, and 5th grades, fractions dominate much of our instructional time, and it is important that we work to help our students gain a firm grasp on the concepts of fractions. Going deep now will help them down the road as they use fractions in more complicated mathematical situations.

A reminder of the progression of fractions in elementary school by Graham Fletcher allows us to remember how important our models are:



Sometimes I'm afraid that teachers steer clear of fraction exploration because they are uncomfortable with them themselves.  Imagine all of the learning that could occur if we went out of our own comfort zone with our students?  

I have shared Fraction resources before in this blog.  I have also shared an exploration or two.  Here are some more resources that might help your students (and you) to make more sense of fractions.

When and when not to give the answer:  This Marilyn Burns' post offers an opportunity for your students to build their own understanding of fractions.

Fractions, Decimals, & Percentages:  A number talk which begins with fraction addition.

Exploring Fractions:  An article from nrich.maths.org which offers links to rich tasks that develop a deep understanding of fractions.

Illustrative Mathematics also has some resources that will help you better understand the horizontal progression of fraction skills.

Maybe you find one or two ideas above to help you stretch during your fractions unit--that's great!  Don't try to change everything all at once. As always, let me know if there is something I can do to help!

Does it look right? Does it sound right? Does it make sense?

These words were always such a big part of my instruction of reading workshop.  Students used hand signals to help them remember the three questions as they reviewed what they had just read.  Many times, they realized the word they had just read did not sound right or make sense in the context they had just read.  So---they worked to figure out what the new word was that they were struggling to read.

Do we teach our students to do this same kind of thinking when solving a math problem?  In my experience, it is a less common practice.  Sometimes, a teacher will ask the students if the answer makes sense, but really, shouldn't we be teaching our students to ask themselves that question?

Tying our math instruction to real world situations allows for the students to better make sense of their answer.  We need to be intentional in our plans to teach students to make sense of situations and solutions. One of our biggest goals during math instruction is for our students to EXPECT that their answer will make sense.

Rethinking Grouping in Math Workshop

As we continue to work to grow in our role as leaders of mathematicians, we need to begin to rethink our use of grouping in the elementary classroom. Ms. Boaler gives a strong argument for us to consider why we should not use ability grouping in our classroom.

For a long time, we thought that grouping our students by ability during math workshop was a great way to meet everyone's needs.  I know. I did it for a long time, too-- many different groups, many different activities.

But, as we learn more about best practices of mathematical instruction, we discover that providing ALL of our students the opportunity to be engaged in rich, challenging, mathematical activities and tasks. 

This does not mean that reteaching skills to some of your students is not the right thing to do.  Of course, there are times that reteaching and scaffolding are necessary for our students.  Having flexibility in our grouping is important and should not be so difficult as we learn more about our students

In the past few years, I have read articles and books about this.  I have participated in professional development that discusses this.  I have listened intently to members of my PLN discuss the importance of eliminating tracking and the impact it can have on our students.  Here are some of the best resources I have used to gain this mindset:

Another great article to read is this one by Nick Tutolo.  Although it is focused on middle school/high school math, its points are still important for us, as elementary teachers, to consider.

This change in our instruction is not without some challenges.  However, I believe that we will be able to see more growth in ALL of our students if we begin to use some of these best practices:

• Low floor-high ceiling activities
• Use of CRA instruction in order for all students to build their conceptual understanding
• Number talks that encourage and embolden students to use methods/strategies that help them understand better
• Use of visuals for all math concepts allowing all students to gain an understanding
• Building time into your schedule for your students to participate in explorations and other tasks to grow their critical thinking
• Continued modeling of growth mindset in our words and actions
• Eliminate referring to our students by ability 

What changes can you make in your classroom to allow all of your students to grow as mathematicians?

Continuing to Find Ways to Build Math Talk

I have participated in a number of professional discussions in the last week around the concept of Number Talks.  You may remember that I blogged about Sherry Parish's book before. It is an excellent resource to help you get started with Number Talks.

Many classrooms use Number Talks every day.  It is a great tool for building mathematical discourse, exposure to new strategies, and strengthening students' flexibility and fluency.

However--some people are a little intimidated by Number Talks.  It has some elements of the unknown, and this makes teachers a little leary to try it with their own students.  This is totally understandable.  One way to make this work is for the teacher to work on the anticipation part of the number talk fully.  By anticipating all of the possible ways a student may respond, it will give you the preparation to feel confident as you step into the Number Talk.  It will also allow you time to prepare visual images to represent ways that the students might explain.  It is important that we show the students' ideas using visual representations as this will allow us to reach more students.

I 

Here is a great example of ways that students might see 18 x 5.

I do believe that the term Number Talk is broader than just the use of functions with numbers as is found in the Number Talks book.  Of course, traditional number talks often involve quick-look cards and other visuals to help students visualize the math, but they are still considered to be from the basic concepts presented by Sherry Parrish in her book.

If you click on the tab labeled Routines located at the top of this page, you will find links to resources and videos of a large variety of mathematical routines that will encourage students' math talk.  There are many low-floor/high-ceiling activities which are good to put the students (and sometimes the teacher) at ease when doing a number talk.  In these types of tasks, nearly everyone can find an answer right away, but because these routines lend themselves to multiple answers, we find students recognizing all kinds of things besides the obvious.  Using these type of routines in addition to Number Talks will help your students to grow in ways you won't believe!

Challenge yourself to something new.  Try to add routines to your math class each day--you will be glad you did!

Genius Hour Podcast

Whether you are a beginner to the Genius Hour or are just playing around with the idea of using it in your classroom, I recommend that you listen to this VrainWaves podcast where they interview AJ Juliani.  It has so many great ideas for getting started (big or small), organization, and accountability!

And... you can listen to it as you work in your classroom.

https://itunes.apple.com/us/podcast/vrain-waves-teaching-conversations-minds-shaping-education/id1365316994?mt=2&i=1000413447686

Virtual Concrete Manipulatives

We know how important concrete manipulatives are for our students as they build a deeper understanding of mathematical concepts. Being able to see and manipulate objects enables students to visually represent problems, see patterns, and make connections.

Hopefully, you have a variety of manipulatives in your room that students can easily access at all times.  Virtual manipulatives are important to have easily accessible, too!  Now is a good time to download some of these apps onto your iPad, bookmark on your computer, links in your SMART Notebooks or PowerPoints and/or add to your Symbaloo!

There are a lot of great FREE virtual manipulatives out there, but today I want to focus on the apps by Math Learning Center.  I have good success with many of them. The site has a lot of other good resources, so when you have some time, check them out as well!

Fractions: This app not only allows you to create bar or circle models of fractions, but it also to layer fraction models to see if they how they compare to one when you add them.

3/5 + 1/3 < 1

The app allows you to place number models or write directly on the screen as well. Take some time to explore the capabilities of this app!

Vocabulary Cards: Another great app that you can probably find multiple uses for! There is a large database of words in this app divided into grade bands K-2 and 3-5.  You can choose to see them all or only certain words.  Each card has 3 parts: the word, examples, and a definition.  You can choose which part you want hidden.  You can also choose the language for the card. I can see these being used as individual review for students, but I can also envision one of these on the board as an intro activity or even a quick exit task.

Money pieces:   The money pieces app allows you to display money with or without the accompanying base ten blocks depending on student needs.  It also has a variety of tools similar to the games Bears in the Cave or Pennies in the Hand where you can put your coins up and then hide some.  For example.  I have a pocket in the above screen.  If I told you that I have 35 cents altogether, can you tell what I have in my pocket?

With the click of a button, I can remove the pocket to show that there is a quarter in it.  You can do this similarly with a hand or a bank.  

Money seems to be difficult for children anymore because they have less interaction with it than we did. Our students don't get the same opportunities we did to spend cash, but it is still important to understand, and this app will give them some basic experiences with it.

Number Frames:  This app not only allows the important 5 frame, 10 frame, and Double 10 frame, it allows you to customize the frame you want up to 120. As with many of the apps, it allows you to enter number sentences and to write on the screen.  This app is not just for Littles!  Teachers in 3-5 can use it to explore place value as well as patterns in multiplication.

Number Lines:  I love this app!  It has so many options for you to customize the number line including fractions, decimals, hidden numbers, hidden tick marks... I think it really helps students to represent their work, and it marries nicely with the beaded number line for moving from concrete to representational.

Number Pieces:  There are two versions of this app.  The one I am showing above is more advanced than the version called Number Pieces Basic. It is base-ten blocks, but you can break the large pieces apart to show number relationships. You have choices in color and orientation of the pieces.  Again, you can enter number sentences or write on the screen.

Number Rack:  You know how much I love this one!  It comes in handy as a teacher model on the SMARTBoard as students manipulate their own Rekenreks.  It is customizable by sets of 10 up to 100. It allows for teacher annotations like the others, but it also allows for teachers to hide beads as below.  Well worth your time to explore this one--not just for Littles!

Do you know how many beads I have hidden?

Pattern Shapes: It is so important that we give students time to play with pattern blocks!  This offers a blank mat for students to create their own patterns.  It also has templates (as above) for students to fill in with shapes.  (Very good for sharpening visual skills)  For older students, it has two different grid backgrounds to allow exploration of area and perimeter.

Geoboard:  The geoboard app has different sizes of geoboards and allows for customization in many ways.  Another great one to have up on the SMARTBoard as students manipulate their own geoboards.

Partial Product Finder: While still in Beta form, this is an awesome app to help your students better understand partial products as well as the distributive property.  You can customize the rectangle up to 30 x 30 and then decompose one or both sides.  The matching equation shows up at the bottom of the screen. I have blogged about this one before--it is awesome!

I highly recommend that you take some time to explore these!  Hopefully you will find some that you make available for student use just as you do other manipulatives.  Maybe you'll find ways to use them within your instruction.  Whatever works best for you and your students!  

Let me know if you have other virtual manipulative apps that you would recommend!

Culture-Building for the New School Year

How do you plan to transform the culture of math (teaching and learning) in your classroom or building this year?  Have you had a chance to think about this?  It can be easy to get caught-up in the content that will be covered and forget the importance of building strong mathematical communities in our classroom.  However, the time spent building your students into teammates in math will be worth it.  Go slow to go fast.

Let's think about Ron Ritchhart's Cultural Forces that Define a Classroom and how they can impact our building of a math community.

1.  Physical Environment:  Is your classroom arranged in a way to promote collaboration?  Are the spaces clear where students can gather? Have you thought about the places where students will be able to visually share thinking?

2.  Interactions & Relationships: What steps can you take to build a feeling of respect which will allow students to be confident enough to share their ideas and strategies?  What will you do to be sure all students and their ideas are valued in your classroom? How will you encourage collaborative inquiry for your students?  What methods will you employ to build  a growth mindset in students? How will emphasize growth and celebrate success?

• This sample chapter from Thinking Together: 9 Beliefs for Building a Mathematical Community by Rozlynn Dance and Tessa Kaplan supports these concepts.
• More ideas can be found in this sample chapter of Count Me In!: Including Learners with Special Needs in the Inclusive Classroom by Judy Storeygard.

3.  Expectations: What are the cornerstones for your mathematics community?  As your class determines the classroom norms, which ones do you feel MUST be part of the list? How will these be enforced?

• Youcubed article on setting up positive norms in your math classroom

4. Time:  How will time be structured during your workshop?  What will you put in place so that students don't feel pressure to work through concepts quickly? On the other hand, how will  you build efficient use of time for your students?  What will you do to be sure you offer enough thinking time for students?  In what ways will you support your students to show perseverance?

5. Language: What do you think is the key mathematical language for your students to learn during the year?  What will you do to build their mathematical discourse?  What language will be modeled for them to use when working with partners or small groups?  Will your classroom contain a word wall or other location where students can easily refer? What growth mindset language will you be sure to include?

• Scholastic blog post about math talk
• Sample Accountable Talk anchor chart
• Blog post focused on Number Talks

6.  Routines & Structure: What daily mathematical routines will you put into place, and how will they help build mathematical discourse in your classroom? How will your math block be structured?  What management routines will you have in place to help your classroom run smoothly?

• Dr. Nicki Newton's First 20 Days outline might help with some of these questions
• My blog page with links to best-practice routines

7.  Opportunities:  How will you regularly opportunities for all students to interact with rich math tasks? In what ways will students grow in the math practice standards as well as the mathematical content standards?  What types of explorations and problem-solving will you use in your classroom?  Will they promote perseverance in your students?  How will students be encouraged to find and explore their own mathematical questions? 

• My blog post on exploration resources

8.  Modeling:  How will you model creativity and risk-taking?  In what ways will you provide examples of collaborative talk and respectful debate? How will students know that this is a safe classroom to take risks? What will you do to share your own wonderings and questions with your students? How can you be intentional about modeling perseverance?

How can you take risks this year in order to grow as a math teacher?  What resources can you use to help you to learn more about best practices in mathematical instruction? Using these 8 ideas as a starting point should help you on your way!

Looking for Explorations/Investigations to do with your class?

Explorations and investigations help your students to take ownership of their own learning and are a great way to get students excited about math!  They encourage critical and creative thinking skills.  They build a sense of math community, and make us all better mathematicians.

If we get started with predetermined explorations, and our students become comfortable with the format, they may begin to come up with their own explorations that you can embed into your instruction!  

I have posted about a number of explorations either that I have done with students or have seen others do. However, I wanted to remind you of some good places that you can go to find an exploration that works for you and your students!

WIM:  The weeks of inspirational math from Youcubed are all set up and ready to go for you.  I have blogged about them before, and I can't say enough about how they not only encourage a growth mindset, but also that they are a lot of fun!

Math Solutions:  This location is full of exploration options for you.  Many have a great connection to literature.

Math for Love:  The free lessons on this site typically involve investigations.

100 Numbers to get students talking: This task has step by step directions and examples of how to use it to build your students' group work abilities this year.  

Finally, I have blogged about some different explorations that you could try in your room.  You might find one that will be a great review or introduction for your students this year.  To find the blog posts, look over at the right side of my blog at the labels.  Click on the explorations label, and it will show you all of my posts about explorations.

If you have other great explorations to share, please post in the comments below.

What a great way to start your year of math learning!  Beginning with some explorations will give you plenty of time to get to know your students, build your classroom culture, and develop your routines.  Let me know if I can help in any way!

Own your mistakes. Strive to be better.

I recently had this conversation with my 16 year old son as I investigated what had happened to our mailbox. I didn't even need to see the ding in our new second car's hood to have a pretty good idea what had happened...

My frustration in this incident was not that he broke the mailbox,but rather that it took him so long to admit that he had.  After an admission, we discussed what had caused the accident, and ways we could work to help it not to happen again. 

Owning our mistakes helps us to grow.  As I have transitioned from the classroom to my current coaching role, I have had the opportunity to put all of my focus into math educational research and resources.  I learn something new most days, but one of the things I have learned that I must do is own my own mistakes as a teacher.

I have read dozens of math books in the last two years, but three have really had an impact on me.

These books have helped me to better form in my mind the necessary shifts that need to happen in our classrooms to help all of our students truly grow as mathematicians. As I read each book, I recognized myself--not always in their examples of best practices.  I saw all of the mistakes I made as a classroom teacher.  Over and over again. It hit me hard.  

Thankfully, I also saw myself in some of the good practices, but that did little to make me feel better about the ways that I felt I had failed my students.  After owning these mistakes, I identified the causes (fixed mindset, poor models, little time to keep up with best practices...) and now I am working to help it not continue to happen.  

I talk often with teachers about the shifts in math instruction that need to happen, but I find it hard for most of them to understand and visualize what this should look like. I continue to share these books with teachers all of the time, but I know that not all of them have the time or desire to read them (especially those who don't see themselves as mathematicians).  I get it.  

So--I have set some goals. My first goal is to put the concepts from these books in front of teachers through the PD we offer.  Not just workshops and presentations--but also through preparing PLC modules that will highlight some of these concepts in ways that teachers can try in their own classrooms and then discuss with their colleagues.  

My second goal is to continue to grow my own understanding through practice, practice, practice! (and not practice of 25 problems on a page...) As I go into classrooms to help or lead a lesson, I am going to do all I can to model the ideas I have read about in these books, discussed on Twitter, and witnessed in classrooms and webinars. I know that I will continue to make mistakes, but I also know that these mistakes will only help me to grow stronger in my practices.

Finally, I know I still have plenty to learn!  If I want to make changes that benefit our students and teachers, I need to continue to push myself in my own learning. Through reading new books, participating in Twitter chats, and attending math conferences, I believe I can continue to learn more and grow as a teacher and mathematician.

Using Counting Collections in the Classroom

Have you tried counting collections? 



Counting Collections: Kindergarten - a common core classroom friendly exercise from Luna Productions on Vimeo

This activity is a great opportunity for our primary students to gain a better understanding of counting and number, and with some modifications, I think it could be used, at some level, in the upper elementary classroom as well.

You will need to begin by creating some collections.  In the link above, they give some examples of objects you can gather for counting, but I'm sure you can find other items around your home or classroom that will work as well.  You will probably want them to be smaller in size so that storing them doesn't become much of an issue.  Hopefully, you can find other teachers in your classroom to join you, and then you can find a common space to share your different collections. This handout will also be able to help guide you as you plan for your collections and fine-tune the activity.

I think that these would be a great way to kick off your math habits to start the year.  You would be able to learn a lot about your students by interviewing and talking to them as they work on organizing and counting their collections.  

For older grades, I have thought that you could have them count objects in multiples or fractions to get a total.  You could also have them count by sets.  Packs of items...Can they count by 24 or 36? What if you offer them decks of cards? Can they count by 52s?  While they may not be fluent as they move through these unusual multiples, it will certainly aid them in developing mental math strategies.  Introducing fresh concepts through counting offers all students an access point, so students can practice new concepts using the math routine of counting.

Here is an example of a Counting Collection in a 3rd grade classroom:
Counting Collections: Third Grade - a common core classroom friendly exercise from Luna Productions on Vimeo.


Counting collections can adjust as your students develop their number sense. It is a routine that will allow students to think about better ways to organize, more efficient ways to count, and concepts of number. It can be used all year long.

What do you think about this?  How can you make it work in your classroom?  Please share your ideas!

I'd love to join in on the fun as your class does a counting collection!  If you are okay with that, let me know when you think you would like to do one, and I will see if I can join you.

Rethinking Homework

As we rethink the role that homework is playing in our students' learning, we should look at new ways to have our students practice their learning rather than just a set of practice problems each night.  

One idea to consider is a math reflection question for the students to respond to.   Building their metacognition through these written responses has proven to build better mathematical thinkers.  

You might decide to have a menu of questions for students to answer, or you might begin by having them all answer the same question--whatever works best for your students.

Click below to a link of some potential questions.  Reflecting on these ideas each day will help to build the types of thinkers we hope to cultivate.

Building Understanding and Developing Culture

I have done this problem with first graders for the past couple years.  The first time I offered it to students, I was surprised by how difficult they found it (due to the majority of my experiences being with older students).  When I did it subsequent times, I offered a lot of "up-front" guidance to help them think about it, and I highly encouraged them to use manipulatives and images to help them make sense. However, they still struggled...

I have decided that if  I work in a first grade classroom next year, we will again do this problem. I have been pondering some different ways to approach it.  

I think it will make a great numberless word problem that we can do with a slow reveal in hopes that students work to make sense of it rather than just trying to solve it. Using the slow reveal will offer a great way to directly instruct the students in the process of notice/wonder and making sense of the problem before trying to solve it.  This use of numberless problems should help them as we continue to build understanding throughout the year.

Doing a general overview as a class, offering the students manipulatives and encouraging them to work with a partner to solve it.  Once they think they have solved it, they must find another pair who has a solution, and each group must convince the other that their solution is correct.  I like this idea, too, as it should help to build mathematical culture int the classroom--collaboration, sharing methods, convincing others of our solution... I think that the deliberate teaching of convincing others with math talk will need to come before this lesson, so that both sets of students don't just say the answer they got and move on.  

I could provide a picture of two spotless ladybugs and then have the students work to put the spots on to match the words in the problem. This one seems very direct, but it does emphasize the importance of using visuals to help us make sense of the problem. Again, I think partner work is a great way for them to approach this.

I think the sharing out of this problem is important and recognizing the different ways that students went about solving it. Determining as a class what we think the correct answer is and acknowledging how we worked with our partners to solve it.  Maybe a gallery walk of our thinking?  

A possible follow-up activity could include students writing their own kind of problems like this and then switching problems with others.

This problem is written with 1st grade in mind.  Certainly, this type of problem could be adapted for different grade levels:
* I have six pieces of candy in my purse.  There are four more pieces of gum than mints. How many pieces of gum do I have, and how many mints do I have?
* We have 27 students in our class.  There are 5 more girls than boys in our class.  How many boys and girls are in our class?
* There are 114 vehicles in the parking lot.  There are 72 more vehicles with 4 wheels than vehicles with 2 wheels.  How many vehicles have two wheels?  How many vehicles have four wheels?
* In my closet, there are 18 shoes on the floor. However, there are 2 more right shoes than left shoes.  How many complete pairs of shoes do I have in my closet?
* I found thirty-five coins in my car.  There were four times as many pennies as all of the other coins combined?  How many pennies did I find in my car?

What do you think?  How might you approach a problem like this with your students? Do any of the ideas I shared seem to be better than the others?  I'd love to hear your thoughts.

How many equations?

I saw this image recently on Twitter with the question: How many ways can your find to make 7/7? Students can use pattern blocks to represent fractions, and then write number sentences and draw a representation to match their blocks.  This could be a great exploration for beginning fraction concepts. It could be made with different wholes depending on your grade level.

It could also be asked as "How many ways can you find to make 7/6?...

What question might you ask to go with this pattern?  Goals in our classrooms should include making the math visual, allowing for all students to be able to access the math, and to provide rigor for our students.

Open-ended problems like this provide all of these things for our students, and are relatively easy for us to embed into our instruction through workshop.  The problems above are fraction problems, but couldn't primary students do the same with 5 or 13 cubes?  How many ways can they find to make 5?  Is that all of the ways?  Can they prove it?

Putting our students into situations where they get to work with manipulatives and make sense of numbers is an important part of their learning.  Can you think of ways that students in your classroom can explore number, whether it is whole numbers, fractions, decimals or something else?  How many equations can they make, and can they be sure that they have found them all?

Multiplying by 11 Exploration

Multiplying by 11 has an evident pattern for our beginning multipliers.  They love the fact that they know how to multiply by a number larger than 10! We need to be sure that they are building a conceptual understanding of what 11 times another number looks like.

In addition, we want to give our students an opportunity to identify patterns.  Why does 11 times a number less than 10 ten give a double digit number of the other factor? Help them use visual pieces to make sense of it.  I think these 10 frames +1 make it easy to see, but maybe they have another idea.

Once they can identify why 11 x 3 = 33, take them on another exploration.

Can they find the pattern when you take 11 x a 2 digit number?  Can they figure out why it happens?

11 x 12 = 132

15 x 11=165

When multiplying 11 by a 2-digit number, the tens digit is the same as the hundreds digit in the product and the ones digit in the product is the same as the ones digit in the 2-digit factor.  The tens digit in the product is the sum of the hundreds and ones digit.

Don't tell your students this pattern.  Give them the tools to uncover it, and then let them use the visuals to see if they can find an explanation.  Many 4th and 5th graders should be able to explain.    What happens when the digits in the 2 digit number have a sum greater than 10?  Looking at the groups of 11 can they figure out what is happening?
Will this pattern continue into 3 digit numbers?  What stays the same? What changes?

This is not to be taught or used as "a trick." This exploration is to help students find patterns in numbers and place value and for them to be able to see why it is happening. It is important that we give them lots of opportunities to explore and make sense of the things that they find.