Playing with Numbers

After reading the book Math Recess by Sunil Singh and Dr. Christopher Brownell, I realized the importance of giving students time to explore numbers by playing with them.  This post shares a couple of ideas for these explorations.

This book is a great read and will make you rethink your instructional practices!

Abundant Numbers:  Have students search for ABUNDANT NUMBERS.  A number is considered abundant if the sum of its divisors is greater than the number.  For example, twelve is abundant because its divisors (1,2,3,4, and 6) is greater than 12.How many can your students find?

Circular Primes:  A circular prime is one that remains prime with the relocation of the first digit to the end.    So for example, 113 is a circular prime:  113 is prime.  When I move the 1 to the end of the number, my new number is 131, which is also prime.  When I again  move the first digit to the end, I get the number 311.  It is also prime; so it means that all 3 of those numbers are CIRCULAR PRIMES.

Happy Numbers: 19 is a HAPPY NUMBER.  How do I know?  To find a happy number, square each digit and find the sum. Continue doing until you find the final number.  If it is 1, then the number is happy.  

Click here to see how I know 19 is happy!

Make it a goal to give your students some time with these ideas.  Can they find more of any type of number?  How many can your class find this week?  before winter break?  this school year?  Can they prove that the numbers they found fit the definition provided?

Let your students spend time playing and thinking about numbers.  

Building Number Sense through Choral Counting

Another routine that we can use in our classrooms to help our students better understand number is called Choral Counting. This routine, as modeled above, is done whole class, and can be differentiated by grade level.

It is a deceptively easy routine to put in place; however, the teacher's intentionality in planning can make it more powerful and effective than if it is just done at the spur of the moment.

Counts can range from Counting by 1s to counting by 3/8.  The beauty is that you can decide what is best for your students.

When planning a choral count, the teacher should decide what s/he wants to count by and then and then whether that count will go forwards or backwards.  After those decisions have been made, it is important for the teacher to determine how the numbers will be recorded. From left to right? From top to bottom?  How many numbers in each row or column?  The recording of the numbers is important in highlighting the patterns that can be found in the counting sequence.  Stenhouse Publishers has a free online tool that you can use to help with your planning of your choral count.  It is accessible here.

After the students and you complete the choral count and recording of the numbers, then the students begin the process of noticing and wondering.  What patterns do they see?  What will come next in the count sequence? How do the patterns they see help them to know what comes next?

As your students become more adept at this routine, you might decide to have them do the count while including a clap or a stomp on benchmark number.  "Stomp on every multiple of 10." "Clap when we reach a whole number." "Snap when the number is even." These little ideas may enable your students to find patterns more easily.

If you are interested in learning more about Counting Collections, consider purchasing this book from Stenhouse Publishers.  

Let me know if I can help with implementing choral counts in your classroom!

Using Puzzles to Create Persevering Problem Solvers

Helping our students to grow through productive struggle is often a challenge for us.  One way to consider as you work to build your students' problem solving skills is through the use of puzzles.  Here are some ideas for you to consider:

Yes--it can be as simple as having a jigsaw puzzle in the back of your classroom.  The visual acuity needed to solve jigsaw puzzles is important for students to build, and who doesn't love to finish a jigsaw puzzle?  Oftentimes, students do not have this opportunity at home, so having one set up in your classroom could help them to develop a new skill or hobby.

Besides jigsaw puzzles, tangram puzzles and pattern block puzzles also help students to build visual and geometric skills.  Allowing students time to explore with tangrams and pattern blocks is important, too, and it allows them to find the creativity in math as well as in themselves.  You can find other resources to help with tangrams in this post as well.

Kenken puzzles are an awesome tool to help build logic, fact fluency and number sense in all students.  The website also has an education portion, and you can sign up to have new sets of these puzzles sent to you weekly.  These puzzles are one of my favorites, but you do need to take a little time being sure your students understand how to do them and reminding them to explain how they know where numbers go in the puzzle.  Too often, students will guess where numbers go.  This method will work for a short time, but as the puzzles become more difficult, guessing will not lead to success.   While Kenkens have some similarities to Sudoku, I think they are better in the classroom because of the decomposition of numbers that is involved.  Many students are familiar with Sudokus, and they are a great type of logic puzzle that can be easily found. More information about Kenkens can be found here.

The website Math Pickle also has a large puzzle bank for you to choose from.  This site offers students the opportunity to play with their work in a way that allows them to better develop math concepts.

These puzzles from the Julia Robinson Mathematics Festival also allow students an opportunity to persevere through problem solving in  a fun way.  Although developed for a building-wide math night, many of the puzzles could be used in the classroom.

Let students decide how they want to do their puzzles.  Sometimes students prefer to solve them alone.  Others need to talk it over and think it through with a classmate.  Neither way is wrong.  Let students build their puzzle-solving skills in the way that feels best to them!

Math is fun.  It is puzzles.  It is visual. It is cooperative.  Sometimes, we get so caught up in teaching a curriculum or a standard, that we forget these important pieces of math.  There are so many good puzzles and problems out there for students; this is not even the tip off of the iceberg.  In what ways can you work to incorporate some of these ideas/resources into your classroom.  How can you help your students to not only become more persevering, but to ENJOY doing math?  

Summertime Ideas

As summer approaches, we need to remember that this important time of rejuvenation offers students time to explore, play, and be a kid.  It is not about daily academic programs or workbooks to complete.  

It is good for parents to continue to offer their child ways to maintain academic skills while also helping them to enjoy the summer. Besides the list of ideas I have shared before, parents may want to consider using Bedtime Math. It offers short interesting math practice that parents and children talk through and discuss together.  The discussion is the important part of this app; not the answer.  It allows parents to engage children with math in a non-threatening, fun way. Another good site that is geared towards parents is called Table-talk math. The site is in conjunction with a book of the same name by John Stevens, but this site offers ideas and examples of concepts for parents to talk with their children about.

Consider sharing these sites or the list of the ideas from my earlier blog post with parents.  As a parent, I know I always appreciate when teachers give me ideas of things that I can do to enrich my children.  I'm sure yours will, too!

Primary Exploration of Multiples

Time for our students to explore number concepts is important!  It allows our students to play with numbers in a non-threatening way and to make sense of the math in a way that works for them. This primary exploration allows students to play with numbers in a cooperative way.

Partner your students up.  Each has a different color of unifix cubes.  One is responsible for the sticks of 2, and the other for sticks of 3.  

 After they have created their "sticks," they begin to hook their colors together to see if they can get them to match up.  In this case 3 + 3 = 6 and 2 + 2 + 2 = 6.

 Have them continue in this way, finding how many times the two groups will match up.  Encourage them to write number sentences to match their blocks.  3 + 3 + 3 + 3 = 2 + 2 + 2 + 2 + 2 + 2

Can they begin to see the pattern on when the blocks match up?  When they run out of blocks, can they figure out how many they would need to make the next match? You might take this opportunity to introduce the word multiples in 2nd or 3rd grade.  You could differentiate/extend this as you wish for your students by changing the lengths of the sticks...

When you are done with the exploration, be sure to have a Notice and Wonder session.  Write down the things that your students find so that you can refer to it at later times. 

What other ideas do you have for this activity or one similar to it?

What do I do with all of these ideas?

I recently completed providing some professional development to teachers in our district.  At the end of the final session, I asked teachers to come up with some goals of things they would like to try.  Hopefully, they will get a chance to try it now before the end of this school year instead of waiting until next fall.  Here were some of their takeaways:

Goals are important as they help us and our growth mindset. It is good practice to after a training to take one big idea back with you to help you grow as an educator! 

Of course, while participants found resources and ideas to put into their classrooms, questions remained.  These are some that were left on my question board at the end of the session:

How do I fit all of these ideas into my workshop along with my activities?
  With everything that a teacher has to fit in during the day, I see where this can be a concern.  I understand how difficult it might be to get in your lesson each day.  Some of the activities/resources that I have shared with you might be good opening activities in lieu of or in addition to the mini-lesson.  In addition, once you have taught the routine, the activity might be an activity that students can complete independently or with a partner.

  In all, we need to be intentional as we plan our lessons to determine what pieces of our text we want to use--maybe doing all of the math journal problems won't be necessary?  Maybe students can practice a few of the problems instead of the whole page, and that will leave you extra time for some of the extra resources you are getting.  I might also recommend that a technology station that students use, like IXL, is not necessary every day.  The rich problem solving activities and thinking routines will provide them with far more realistic practice than the rote practice of problems that some online sites might offer.  Mix it up!  Every day doesn't have to look the same, and building our students' conceptual understanding and problem solving skills is worth finding the time for.

  If you continue to struggle finding ways to make it work, consider a coaching cycle or meeting to help you wrap your brain around all of your options.

How do I use the number rack and beaded number line whole class?  How can I teach them when and how to use that tool?
  The rekenrek and beaded number line are great tools for our students to use to make sense of problems.  We should encourage their use at all elementary grade levels.  The more that these tools are used as part of our instruction, the more likely that the students might be to use the rekenrek independently.

   There are many online resources to help you better understand the use of the rekenrek and beaded number line.  
Here are a few:  Rekenrek Activities 
                              The rekenrek as a visual model  
                              Top 5 Rekenrek Activities
                             Beaded number line for 3rd grade
                             Beaded number line for 4th grade
                             Beaded number line for 5th grade

How do I set a growth mindset in math? I have kids that give up before even attempting a problem.
   Great question! I believe that setting this tone from the beginning of the year is key.  Jo Boaler's youcubed website  has some great resources about this topic.  If you look through her Weeks of Inspirational Math activities, growth mindset is prevalent, and it will help guide your classroom learning.  She also has other resources available that focus on the growth mindset.

   This topic has many resources available online with just a little searching.  Certainly, praising effort over intelligence is a key change that can have a big impact!

If you look at the word cloud on the right side of the screen, you may notice that I have written about these topics in the past as well.  Maybe following one of those links will give you other ideas or resources, too.

I enjoy providing PD for our district, and participant feedback is important to me.  I want to be sure that I am meeting the needs of our teachers.  I hope some of these resources will help!

Odd and Even: A Visual Exploration

Sometimes we need reminders about how important some things are.  I know our math should be visual, but sometimes, when you see someone teach something visually, it hits you upside the head!  How had I not seen that before?

I recently participated in a great webinar with @MonicaNeagoy entitled "Planting the Seeds of Algebra."  It was a very interesting and informative webinar.  If you would be interested in listening to her ideas on ways we can make our math come alive for our students and prepare them for increased math connections as they progress through math curriculum, you can access the webinar here.

Early in the webinar, she discussed an exploration she does with students where she discusses odd and even numbers.  She had the students create snap cubes models to match the story of double-decker buses and car trailers she was talking about.

6-A double decker bus with 3 windows on each level

3-A car trailer with the cab and two cars on back

After the models are created, students sequence them

And a pattern begins to emerge

Students then looked at the models, which were visually clear in the difference between odd and even numbers.  Students were then encouraged to continue their thinking about odd and even numbers by exploring the sums of these numbers, and hopefully finding and testing some common patterns in these additions.  Again, Ms. Neagoy goes into much more detail in her webinar, if you are interested.  (It is worth your time!)

EVEN + EVEN:  4 + 8 = 12

EVEN + ODD: 2 + 9 = 11

ODD + ODD:  5 + 11 = 16

Well--this discussion and model got me thinking about a similar exploration for older students.  Using the same odd/even models, what happens when you multiply by an even number an even number of times or an odd number an odd number times?  What can our students determine through this exploration?  How will it change as the numbers increase, or will it?  Can students determine and odd/even rule that accompanies the multiplication of whole numbers?  

EVEN x ODD:  4 x 3 = 12

EVEN x EVEN: 4 x 6 = 24

ODD x EVEN: 7 x 4 = 28

ODD x ODD: 7 x 7 = 49

What a worthwhile way to spend a class period!  The primary version of the odd/even exploration and the older student exploration may be subjects that you discuss in your classroom already, but do you make a point to do it in such a VISUAL way?  And are your students figuring this out on their own, or are you telling them the patterns that exist?  

The shifts in math instruction show us that students will learn and understand better if the math is visual and open.  Taking these basic ideas of instruction and creating a visual tool for our students to use to figure concepts out on their own are steps that we want to be taking as teachers

Let me know if you try this with your students and how it goes. I would also love to hear about other odd/even explorations that you have found for elementary school students!

Start with Counting

Counting is usually our introduction to numbers. From an early age, we heard people around us counting.  It is such an important skill, yet sometimes we look past it as too primary.

After the NCTM Regional Conference last fall, I came back with tons of ideas I wanted to put into action.  One of the biggest concepts I decided to do more of is counting.  I often work with students who struggle with basic number sense, and since that conference, we begin nearly every day with counting.  Sometimes we count by 1s, sometimes we count backwards, sometimes we count off decade....you get the picture.

I always try to use visuals/concrete objects as we are doing our counting, but sometimes my fingers won't move fast enough to keep up with the students pace.  Recently, I found this AWESOME site full of math visual videos that we can use as we count.  

Using these visuals and asking questions about what is happening is important to helping our students gain a better understanding of numbers.  This link takes you to the counting and place value videos, but Berkeley Everett has created visuals to go with many other math concepts.  Using these videos will really help us to make our math more visual!

Counting isn't just for our littles!  It is a great idea for all classrooms to begin class with counting.  Counting by multiples, fractions, whatever will strengthen your class's thinking and understanding of number!

Numberless Word Problems

In our ever-increasing efforts to get our students to UNDERSTAND a story problem and what is happening, we want to stress to them the importance of that movie playing in their heads.

Numberless word problems help to encourage just that!  These problems ask students to THINK about the problem as a reader before getting caught up in getting an answer.  

These problems can be made "on the spot" by covering or eliminating the numbers in a problem that your class is contemplating, or you can be even more intentional by building the problem through a "slow reveal" to increase their wondering and thinking about the problem.

A slow reveal might look like this:

1) TERESA IS PREPARING DINNER AND WANTS TO HAVE ENOUGH MEAT FOR ALL OF HER GUESTS.

• What math do you see in this problem?
• Do you see any math relationships?
• What are you wondering?

2) TERESA IS PREPARING DINNER AND WANTS TO HAVE 1/4 LB. OF MEAT FOR EACH OF HER GUESTS.

• How does this change what you know or what you were thinking
• What do we know now?
• What do we still need to know in order to have the full picture?

3) TERESA IS PREPARING DINNER AND WANTS TO HAVE 1/4 LB. OF MEAT FOR EACH OF HER 12 GUESTS.

• What information has been added?
• Do we need any more information?
• What question could we ask about this story?
• How could we use the information that we are given?

Numberless word problems are a good first step for us to take towards better math instruction!  Throughout the lesson, offer many opportunities for students to think/pair/share. Model good math vocabulary as you discuss the math that is presented. Encourage your students to be THINKERS!  Good luck!

Rekenreks--Ideas for use

The rekenrek is showing up more and more in our classrooms.  Hurrah!  They are great tools for us to use to help our students build number sense.  It is also a nice concrete way for students to model what they are thinking.

Christina Tondevold is an educator who provides great professional development for teachers.  She calls herself the Recovering Traditionalist and often discusses her transformation as a math educator.  She recently did a vlog post about her top 5 ways to use the rekenrek.  You can find it here.  It is only about 12 minutes and totally worth your while.

Here are her Top Five Ways to Use the Rekenrek:

5.  PLAY--Just let the students play and discover on the rekenrek independently.

4.  NUMBER STRINGS--Build student understanding through number strings talks.

3.  QUICK LOOKS--Grow students' subitizing skills through quick looks on the rekenrek and discussion afterwards.

2.  SHOW ME--Giving the students prompts and seeing the different ways that they "show me eight."

1.  MODELING STORY PROBLEMS--Having students use this concrete tool to help them make sense of story problems they encounter.

I have shared with many of you an online version that works well on the SMARTBoard.   Here is a link to that site.  You might even want to make it a tab in Chrome so that it is always handy!

It is a good idea to try to build in times when students take their concrete representation on the rekenrek and make it pictorial.  You can find many examples of rekenrek recording sheets online.  Beware, however!  This is much more difficult for young students to do than expected, but a great way to build their understanding.

Finally--I'll end with another great video about the rekenrek by Graham Fletcher.




Let me know if I can help in any way!