Digging into the Common Core: Deep Division!

We are just wrapping up our unit on division, and I wanted to know if they really understood what it is, so I decided to have them create their own problems.  In order to get their best work, I told them we would be compiling our problems into a book to be given to our principal for Christmas.  K-5 Math Teaching Resources  is a fantastic site for problems and activities aligned to the Common Core Standards.  I found a problem aligned with 5.NBT6: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.  The students were asked to:

1. Create a word problem that could be solved by dividing a three digit dividend by a two digit divisor.

2. Estimate the answer to your problem.  Explain your strategy.

3. Solve your problem.  Show all your work.

4. Use a different method of solving your problem to check that your answer is accurate.  Explain your strategy.  

I was really looking to see if they knew the difference between the operations.  We have discussed clue words here and there but not in depth.  Most students were on the right track; only a couple did an addition or subtraction problem.  I made them peer edit their problems before putting them on good paper.  The front showed the problem, while the back showed their estimate and solution.  We had a blast!  They are really proud of their problems, and most included me eating cookies....I wonder where they got that idea? :)

Here are some fantastic examples of our work!

Number Talks: Division

We've been doing a lot of division lately and I began to notice that the preferred method is the traditional algorithm, mainly because that’s what parents are more comfortable with at home when helping with homework.  Because of this, I was noticing that if students made a mistake in following the steps, they really didn't notice they were making a mistake, let alone that their answer was not reasonable.  So, I decided to encourage more mental math.  We did this through some number talks and the use of slates.  In previous Number Talks, I didn't allow my students to write anything down to force them to do work mentally, however, for the division number talks (at least at first) I allowed them to write down their thinking on slates so they wouldn't confuse themselves.  The only stipulation was that they weren't allowed to divide the “normal way.”  No “houses!”  

We started with smaller, simpler numbers: 25 ÷ 4.  Here is a picture of their thinking:

Molly began by saying, "I know that 4 x 5 = 20 , and that was too small, so I added another group of 4 to get to 24.  Then, I knew I just had to add one more to get to 25.  So I had 6 groups of 4 with one left over."

Once some students saw how others were using basic facts to help them, they began to catch on.  More and more students began volunteering their thinking.  This is something we will continue to build a knowledge base of.  My hope is that this will lead my students to a greater understanding of division and answers that are reasonable.

Blake is one of my students that has an amazing sense of numbers: he does almost all of his calculations mentally.  I always try to see what he is thinking to guide the other students.  

This was a great example of two students building from something they already knew: multiples of ten.  I love how one counted up from ten and the other counted down from 20!

Jillian wasn't afraid to say she started with a fact that was too low and built her way from there!  I always encourage them to admit when they make a mistake and tell us how they found it and fixed it!  Chances are, they aren't the only one making that mistake!

We are really improving and they are finally telling me how much they love math!

  

Division!

Oh the joys of division!  For whatever reason, my students are afraid, deathly afraid of division.  From their pre-assessments, I could tell they weren't really thinking of the actual numbers being divided, but more concerned with the PROCESS, and because of this were making silly, careless errors and their answers made no sense!  So, I decided to start at the basics: the concept of division and how it relates to multiplication.  We soon moved on to estimation and using compatible numbers (multiples of ten) to help us solve.  On Friday, we explored three different ways to divide.

We began by using base-ten blocks.  We are pretty good at using them after using them for modeling decimals as well as multiplication.  They knew that if they didn't have enough flats (100's) to fill in the groups evenly, they would have to break one into ten 10's and go from there.  I was extremely happy that they were so fluent in this from adding and subtracting decimals!  They had no idea they could divide this way.  I chose some of my struggling students to model this in front of the class.  They were extremely successful and were proud to show the class even they could divide large numbers.  Here are some pics of Sam and Bobby dividing using models:

Sam had to break a long into 10 ones! He did a great job!

 Next, we moved on to using the distributive property and expanded notation to divide.  The kids are pretty proficient in writing numbers in expanded notation, so we really just had to take it a step further to create the area model.  Here is Allison using her whiteboard to solve.  You can see she broke 248 into 200+40+8, then divided each part by two, and finally added her quotients.  Her area model is below, except I caught her before she had completed it, but you get the idea!

Below: Max bypassed drawing the model and opted instead to just divide each part by three.  Then, he added up the quotients.

Lastly, we learned the partial quotients method for division.  Personally, this is my favorite.  It allows kids who aren't as proficient in their math facts to still be successful in division.  It also really shows the value of the numbers, something my students clearly struggle with.  Here is a shot of Keegan flying through a problem with ease.  I think he's found his favorite way to divide!

Today, after a long, warm weekend of fun, we spent some time going back over each method for division.  I split the kids up into partners: a student who "gets it" with a student who doesn't.  They really worked well together and as I circulated I heard the helpers saying things like "how many times does 3 go into 400-some?  Does it go in at least 10 times?  Do you think we can go higher?"  Sometimes, all it takes is some guidance from a peer rather than teacher lady.  I'm OK with that!

Lastly, we created anchor charts showing the ways for division.  We added a fourth way, the traditional way.  I found that my "high" kids like the traditional way, and I'll let them use it as long as they can explain each step using place value! (Muahahahaha!)  Don't forget to check out our division math rap on YouTube!

Common Core Connection: Standard: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.  Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Mathematical Practices: Make sense of problems and persevere in solving them.

Number Talks

As previously mentioned, our district has the fortunate opportunity to train teachers on the CCSS.  In addition to discussing RICH problems and how they can be used in our own classrooms, we conduct Number Talks.  We use the book Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5, by Sherry Parrish as a resource.  Number Talks were designed to improve students' mental computation skills and number sense.  Inside Mathematics is another fantastic resource that shows how different teachers in varying grade levels apply Number Talks to their lessons.  In fact, this is a wonderful website that allows you to see teachers in action applying the Common Core Practice Standards! Woo! 

Anyway, we have been doing our best to introduce these talks into our classroom.  In my room, students sit together right in front of the board.  I tell them to summon their inner Kindergartner, which they giggle at.  We have only had time for a couple "formal" Number Talks, but we are beginning to apply it in all aspects of our day when answering questions verbally.  Here is an example of how my students were able to solve a basic addition problem using mental math and their process for doing so: 

Are they stellar?  Probably not.  My students were looking at me like I was a crazy person that day.  They didn't quite understand why understanding how to compose and decompose numbers is so important.  Many said, "I put them on top of each other and added.  I also carried the one." My point was that I wanted them to know what "carrying the one" actually meant.  As we do more, they do improve, and I can see it translating in other areas of math.  They are really starting to think about the VALUE of the numbers.  

I plan on doing more complex problems as we go and seeing how they improve.  I told them I would keep their original posters so we can compare them to how they are doing in a few months.  They have already drastically improved!  If you don't do Number Talks in your classroom, I highly recommend you look into incorporating them every now and then, especially in younger classrooms!

RICH Problems: Thanksgiving Table

I am extremely fortunate to be teaching in a district that is forward thinking in their training of their teachers for the CCSS.  We are able to attend professional development once a month (once a month!!!) in the area of Math and English Language Arts.  Even though I currently only teach math, as schools are constantly changing, so are teaching positions, and there may come a time where I will be teaching L.A., also.  I am so thankful that I will be prepared if that day comes!

Because the meetings are monthly, we are able to apply what we are learning about Common Core to the concepts we are currently teaching.  We have been told to continue to teach the Ohio Academic Content Standards, while slowly implementing Common Core where appropriate.  I, for one, love this approach.  I feel I will be better prepared to fully implement Common Core when the time comes.  

One thing we discuss during our meetings are RICH problems: mathematically rich problems that challenge students to investigate math problems and empower them to use their knowledge to come up with different solutions.  These problems also force students to apply the CCSS Mathematical Practice Standards.  We solve the problems ourselves during our meetings and create posters just as our students would be expected to do.  It's a great way to show that even adults come up with varying solutions to the same problem.  We then implement these problems in our classrooms and report back on our results.  

Here is the most recent problem we did in honor of Thanksgiving.  It is aptly named "The Thanksgiving Table Problem."  

 Because we just learned multiplication, this problem really seemed quite perfect.  It forced students to look at numbers based on place value.  They used base-ten blocks to "build" a table with the dimensions of 13 ft x 7 feet and were asked how many ten ft. boards and one ft. boards were used to construct the table.  Students used longs and cubes to create their table, and in doing so, created an area model for 7 x 13.  We have just begun discussing the distributive property, and this was a great way to apply it!  Students saw that they would need seven 10 ft. boards and 21 one ft. boards: 7 (10 + 3).  

One group building their table using base-ten blocks

The beauty of RICH problems is that there are many parts using many different forms of math.  Students complete what they can and can delve further and further into the problem.  The next part of the problem had the students deciding if they needed to add more square footage to make a 100 sq. foot table (the original was only 91 sq. ft.).  They all concluded they needed 9 more sq. ft., but quickly realized that would make their table odd looking: it was no longer a rectangle.  Most groups decided that they would add a few more square feet to either the length or width to ensure the table was at least 100 sq. feet.  

The next task had them calculating the price of their add on, then figuring out if there was a cheaper way to add on to their table.  Here are some of the finished products:

This group did a fantastic job of displaying their information!  They showed their add-on as well as all the math they did to come up with the price of the add-on. 

Love their organization and clear answers!

This particular problem applies many of the CCSS Mathematical Practice Standards:

1. Making sense of problems and persevere in solving them.
4. Model with mathematics.
5. Use appropriate tools strategically.


At the end, each group was able to present their findings.  They really seem to be getting better at doing group work and including all members (this is something we have REALLY been practicing!).  Overall, I would say the problem was a huge success.  I will choose some projects to take with me to my next CCSS meeting next week to share with the other 5th grade teachers in the district.   

A great website for RICH problems is the Ohio Resource Center.  I will follow up with a post about more of the RICH problems we do from this website called "Stella's Stunners".  They are WONDERFUL!

Discovery Learning: Sum of Angles

Rather than just tell students how to do something, I like them to discover it on their own.  It gives me an opportunity to later point out "Hey, remember that day we figured this out?"  On this particular day, we had been talking about angle measures and were going to be moving on to angle measures in triangles and quadrilaterals.  I decided that our Interactive Notebooks would be perfect for discovering the sums of the angles in these polygons.  Students first received two congruent triangles to cut out.  One was glued into their notebooks and the angles measured.  I casually asked the students what the sum of those angles were: 180 degrees.  "Hey!  That's the same number as the measure of a straight line!" piped up Calli.  So, I told them to go ahead and draw a straight line in their notebooks.  Then, we glued the corners of the triangle together to prove that the angles together form a straight line. 

Here is our finished product.  We tested our theory that the sum of a triangle's angles is always 180 degrees.  I found a fantastic interactive website that allows you to adjust the angles in a triangle to show they always add up to 180 degrees, no matter how you do it.  

We then followed the same process for a quadrilateral.  

The kids loved it!  The information stuck with them and they know that a triangle's angles add up to 180 degrees, while a quadrilateral adds up to 360.  One inquisitive student asked if this works for all polygons.  I challenged him to test a pentagon, hexagon, and so on to see if there is any sort of pattern relating to the number of sides in a polygon.  Of course, many students took on this challenge with protractors out and pencils writing.  It was only a matter of time when they excitedly yelled, "There IS a pattern!  You add 180 degrees every time you add a side!"  And my work here is done :)

CCSS Mathematical Practice Standards applied:

2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning

Interactive Notebooks: Algebra

Because both the Ohio Academic Standards and the Common Core State Standards rely heavily on algebraic thinking in 5th grade, Jamie and I knew we really needed to find an effective, yet simple way to get our students thinking algebraically.  We searched for ways that other teachers were teaching variables and stumbled upon another teacher's blog that we found to be extremely useful!  I Speak Math is a wonderful blog written by a 6th/7th grade math teacher in North Carolina.  She created a graphic organizer specifically for writing linear equations from word problems.  We copied the graphic organizer and glued two into our Interactive Notebooks.  Then, we did a whole page of word problems using it!  We quickly saw the relationship between addition and subtraction and multiplication and division and are now able to write two equations using different operations: a fantastic skill!  

This forces students to really think about the unknown: what they are trying to solve for.  They were able to write two equations: 35 + g = 64 and 64 - g = 35 using two operations.  Bring it on, OAA!

   
My students loved it so much that they still follow all four steps when solving ALL word problems, whether I tell them to or not (and we did this over a month ago!).  They are getting really great at solving for the unknown.  Once we get into Order of Operations, they will be able to deal with more complex equations.  I can't wait to see what the future holds for us! 

Magic Number

One thing we started in our room is our Magic Number.  We brainstormed ideas on different ways to make a number: tally marks, addition problems, base-ten blocks, number sentences, etc.  Every couple days, I put a new number on our extra side board.  We started with lower numbers.  Since we had already learned about square numbers, I used a couple of those.  I also used a couple prime numbers since we had been talking about them as well.  When the students finish their work, they may add different ways to name that number.  When we started learning about decimals, our magic number was a decimal.  It was great to see what they came up with!  I can't wait to see what they can do once we get to fractions!  Here is an example of our Magic Number board.  After a few days, the kids decided they wanted to call it our graffiti board: love that idea!

Magic Number 27: I used this because so many kids kept mistakenly  thinking 27 was prime.  Many decided to list 27's factors and realized: nope! Not prime!

It is great to see how much they have improved in their accuracy since we began our graffiti board back in October.  Many are even coming up with their own word problems using the number, and some are even using variables to display the number!  This is one activity that gets them thinking about numbers as soon as they walk in our classroom.  They all can't wait for a chance to show off their ideas! 

Anchor Charts and We Can!

I am a huge fan of using anchor charts in my classroom.  I let the kids help me decide what information to include, then hang them up where they are easily seen.  I remind them to refer to them during tests.  I think of them as a resource and love when I look around and see the kids referring to them.  To me, it's their way of telling me they care and want to do well.  After a while of referring to the charts, the information is just committed to memory!  I don't know what I am going to do in January when we move into our brand new buildings (!!!!!!)  and we aren't allowed to hang things on the walls!  I'll come up with a solution I'm sure!  Here are a couple examples of some of the anchor charts currently hanging in the room:

Using models such as base-ten blocks and number lines to display decimals

Rounding Decimals: CCSS 5.NBT.4

Additionally, I like to really make sure my students know what our goals and objectives are each day.  I want them to be able to say, "Yeah, I did learn that today!"  Each time we learn a new concept, I let them know how it is connected not only to the Ohio Academic Standards, but also to Common Core.   This way, we are all working toward a common goal.  I never want them to be in the dark about what they are learning and what they should be able to understand after a lesson.  Here are two examples of our "We Can" posters:

We Can do so much!

I am really thankful to have so much wall space in our room!

Interactive Notebooks

I am fortunate to have an amazing co-worker to plan and plot alongside.  Her name is Jamie, and just being able to collaborate with her this first few months of school has been awesome!  She only teaches one 90 period of math and the rest science, but it's enough to get us talking to each other every day about lessons and how they went.  She showed me some interactive math and science notebooks she found on blogs, and we both agreed that this would be the perfect year to start them.  We have been given the go-ahead from our district heads that we can now use EDM as a general framework and supplement where necessary so as to cover the Ohio Standards and the Common Core...all at once!  An interactive notebook would be the perfect place to combine all of our accumulative notes and materials. 

Rules page and Table of Contents

After setting up the foundation of our notebooks, we began our first activity and first entry with a game we so creatively call "The Barrier Game."

Barrier Game game boards.  This was a great way to see how my  new students reacted to a challenge!

For this activity, I paired students up (remember, this was the first week of school. I didn't know personalities!) and gave them two different grids: one student received a blank 3x3 grid, while the second received a grid with simple math terms and symbols on it.  We then placed a folder between them so they couldn't see each other's board.  It was the second student's responsibility to communicate what his or her grid looked like while the first student drew it.  Some quickly discovered they weren't communicating efficiently.  They realized they needed to be more specific in their directions to their partner.  One student, Aaliyah, said to me "Ms. McHugh, Cale and I just aren't good partners.  He isn't drawing what I want him to!"  I asked if I could listen in on their conversation.  It turned out that Aaliyah was just telling him things like "Draw a square at the top."  Poor Cale didn't know where to draw his square!  When reflecting on the activity at the end, Aaliyah raised her hand and admitted "It wasn't that we were bad partners, I just need to be more specific when I communicate.  I wasn't giving him enough details!"  That was a wonderful "ah-ha" moment for the class.  Of course, I had some students get extremely frustrated with their partner.  I would like to do this activity again later in the year and see how they do.  They will be doing a LOT of partner/group work in my class, so they need to know how to communicate well!  After we reflected on the activity, we wrote down our thoughts in our notebooks.  Below is an example of an entry regarding the "Barrier Game."

Cale's reflection on his experience with Aaliyah

We also brainstormed ideas on how good mathematicians communicate.  I was really impressed with their ideas!

Back to Big!

Now that I am back to teaching math (to 5th graders!...Sorry Kindergarten! I'll miss you!) I want to do a better job of keeping track of all the awesome things that are going on in our classroom.  I just need a couple more hours in the day...but who doesn't? The last time I taught math two years ago, I started teaching small groups based on ability level.  This year, I want to take that a step further and really work out all the kinks.  I also plan on teaching to the Ohio Academic Standards as well as begin to integrate the Common Core Standards so that my students will be prepared later on.  So far, this hasn't proven to be such an easy task!  But, I'm ready to take on the challenge, and I think my students are ready, too.  Stay tuned for all of the fantastic things we are doing!

 

A Whole New World: Kindergarten

Why no new posts, you ask? Well, I'll tell you.  Last summer, around July, I received a phone call from my former principal.  I had been RIFd that March and he was  frantically trying to find a way to get me back into his building.  Fortunately, a position in Kindergarten opened up.  When I heard the news, I was excited and scared all at once.  Kindergarten was a far cry from experienced 4th graders.  I quickly reviewed my shoe-tying and coat zipping skills and got down to work researching everything I could about that first, and very important year in school.

My main goal was to familiarize myself with best-practice methods to teaching literacy, something I hadn't taught in over two years.  Every evening, I pored over pages in Guided Reading by Fountas and Pinnell, The Next Step in Guided Reading by Jan Richardson, and The Daily 5 by Gail Boushey and Joan Moser.  I took notes and tried to come up with some kind of plan, not knowing at the time what I would be faced with that first day: scared children who weren't sure what "please line up" meant!  

If you've ever seen a Kindergarten teacher after the first week or two of school, you know she looks haggard, tired, and stressed.  My coworkers reassured me that the students would just "get it" one day.  "One day, they will line up without shoving each other.  They will even raise their hands to speak!" they told me time and time again.  Fortunately, they were right.  Soon, that room full of 22 five year olds and their teacher became somewhat of a family.  They understood that the rules applied to everyone, every time.  They understood that they were there to learn, and learn they did!  We were able to set up a consistent literacy block, and they worked hard at building their stamina while their teacher did push-ups each day to show that she could build stamina, too (not my best idea but my arms thanked me!).  We use the "Daily 5" approach to literacy management, which was modified to three rotations in Kindergarten: Read to Self, Work on Writing, and a combination of Listen to Reading, Word Work, and Read to Someone.  This gave me an opportunity to meet with small groups, a feat I was amazed could be accomplished in Kindergarten.  It wasn't always easy, and it wasn't always perfect, but in this way, I was able to teach, really teach, my students how to read.  I watched in amazement as they applied the strategies for figuring out a tricky word.  Wait, they were not only listening, but they understood when they applied a certain skill?  Amazing! I was proud after the first round of benchmarking in December.  So this was why people stayed in Kindergarten.  I was beginning to understand and love my new position. 

While I was busy learning how to teach Literacy, my coworkers were looking to me to strengthen the math program, seeing as the Common Core Standards were soon going to be a huge part of our lives!  I love teaching math and was excited to take on the challenge.  I learned everything I could about the Common Core for Kindergarten and soon began sifting through our curriculum, Everyday Math, to find its strengths and weaknesses.  With the permission of my Principal, I created a Daily Math Journal for all four Kindergarten classes.  Its intent was to reinforce concepts taught in EDM as well as expand on those concepts.  We had a new Journal each month, something I changed to fit the needs of our classes.  I was also fortunate to have the complete support of another Kindergarten teacher in her 6th year in K.  Soon, we decided to begin co-teaching together so that our students had the benefit of two adults in the room, one who was in charge of teaching, one who was able to monitor students and circulate the room to answer questions.  I loved this time of day because I was able to see an experienced K teacher at work and how she dealt with certain behavior issues.  There is no greater benefit than seeing a great teacher doing what she does best: teach!  We bounced ideas off each other and really got our students excited about math time.  Our classes played math games together, and we were able to combine our high students together as well as our low to play games at their level.  It was truly amazing!

Looking back on the year, I know my students learned an immeasurable amount, but I'm not sure they'll ever understand how much they taught me.  I can honestly say that this year I learned something new every day about my profession.  Kindergarten taught me patience beyond belief.  I was able to see my students at their best: eager and willing to learn, something that was much more difficult to come by in 4th grade.  As I look back on this post, I notice a word I used over and over: Amazing.  But I really cannot come up with a better word to describe this year.  It was amazing.  It was amazing that my students practically came in as babies not being able to recognize their names and not knowing too many letters, to six year olds eager to tell you that "that word has 'oo' in it!" or "let me read that to you!"

Now that I'm finished with my 4th (!!!!) year teaching, I'm starting to see that just because I am the teacher does not mean that I am no longer the student.  As long as I am in this profession, I will always be learning something new.  And I can't wait to see what next year brings!