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Common Core

How In the World Do You Teach Rectangular Arrays and Division in 4th Grade?

Well, with large numbers this is something that my fellow colleagues  did not feel comfortable teaching.  And when that happens…who steps in??  None other than The Mathemagician…ta-da!  (which is me of course, but shhh don’t tell anyone!!)…I’ve been off for a few days as I write this, which makes me a little sillier than normal–and probably slightly more interesting! So, on with my lesson!  Now just remember when I post these pictures it is not a beautiful, I spent weeks preparing, colorful, lesson.  This is a practical lesson anyone could use whether you are savvy with a computer or not.  (I may turn this lesson into something more aesthetically pleasing later on.) The part that stumped the teachers was the fact that the standard says “up to 4-digit dividends”.

CCSS.Math.Content.4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends and one-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.

I had to think myself about how to teach this before I spouted off a lesson idea.  (I know I am supposed to be the math expert in my building, but, honestly, I have to look up what exactly common core is expecting before I assume I know what a standard is asking students to do.)  I looked at North Carolina Unpacked–my go to document for what common core expects…

Screen Shot 2013-11-30 at 2.25.01 PM

 

North Carolina shows arrays being broken into smaller arrays similar to ladder division.  Since these students had already been studying ladder division, I thought that they would have an easy time relating arrays to ladder division.  After reading the North Carolina document, I decided to create a large array with base ten blocks and have students find the missing side, but then I realized that students would just count the missing side.  I needed something better, but what?  Then I thought about a training I had been to which had arrays with part of one side covered up.  Students had to figure out a missing side .  However, these type of lessons  were for double digit numbers.
Here is what I decided to do to push students towards solving arrays with large dividends.  I used cm grid paper and cut out different sized arrays.  I cut out a very small one at first to use for modeling and discussion.     I told the kids that the principal had asked me to set up the auditorium with chairs for the 5th grade graduation.  Each cm square represents a chair.  Then I explained that I had spilled coffee all over my seating plans, and I didn’t know what I was going to do.  I told them I needed their help to figure out how many rows there were so I could recreate my plans.
I started  with this one.  Students had no problems figuring out the missing dimension in this array since they knew their facts.Array1
All I did to accomplish this spill effect was to cut out pieces of colored paper in blob shapes and tape them to the array before setting it on the copier.  In this case, the large blob was actually red, but it printed out black on the copier.
Then I showed an array that was larger with a 3-digit dividend.  This array actually had a blob on it, too, but I removed it so that we could discuss different students’ thinking after students figured out the missing dimension.  The blue marker shows my recording of the students’ thinking.
Array2
Then I gave students a much larger array to figure out a missing dimension on their own.  Several students tried to count how many squares there were underneath the coffee spill instead of using more efficient methods.  As you can see, this student drew lines on top of the blob to figure out how many rows there were.  She arrived at 22 rows.  (there were 24)
LargeArray3
This paper above although wrong, was a valuable piece of thinking to tie everyone else’s work together at the end when I had students share because it was so basic.  ( I typically don’t write on students’  work, but since the students were a bit shy about sharing  I asked her if I could write on the work to help other kids make connections.)
LargeArray1.
The array above shows a student who is thinking of each row as a group of 12.  Rather than multiply he chose to add 12 repeatedly.  He realized that 12×12 made 144 and that adding 144 twice would give him 288.  Even though this student is oh so close he isn’t explicitly saying that there were 24 rows of 12 chairs.
I’m still looking for someone who is thinking with groups of 10 so that I can relate this to the ladder method for division.LargeArray2
Here is yet another student who is thinking in multiplication with equal rows.  Surprisingly to me, the students are more comfortable with 12×12=144 than thinking with groups of 10 to break off parts of the array.  The above students is a GT (gifted) student and he had 144 + 144 on his paper for the longest time because he recognized the relationship among 288 and 144.  However, it wasn’t until we had closing discussion that he labeled his other dimension with 24 and columns with 24’s.
Let’s pause here for a brief  teacher reflection moment by Ms. K (soft music playing)next time I prepare an activity like this, I will make sure that I cover up enough rows so that the only rows that show are a group of 10 rows.  This will hopefully push students towards thinking about ladder division.
Ok, finally, someone who thought about the ladder method for division!!!!!!!!! (picture me running through a grassy field to meet this paper).  This student’s paper will let me tie all the other students’ papers together in closing discussion and to relate arrays to everything that the class has been discussing for weeks!!!! (woooooohoooo!)
LargeArray5
This lesson went successfully.  Now for the next lesson, I plan on giving the students another large 3 digit dividend array that is covered after row 10.   After this the next steps are giving students 4 digit dividends and/or rectangles with no lines.
Just so you know…one of the kids asked me if I really did spill coffee on the seating plans.  (so cute) I said, “What do you think?” 🙂

Use This New, Free, Quick, and Easy Classroom Management Tool

If you haven’t already heard of Class Dojo, a cutting edge classroom management tool, let me fill you in.

Our fourth and fifth grade teachers are using it and they absolutely L-O-V-E it!!  Class Dojo is a website which allows you to load all of your students and keep up with their behaviors, positive or negative, based on a point system.  You can upload the behaviors you want that match your classroom rules and expectations.  For example, you can include behaviors such as “off task”, “participating”, “showing respect” etc.  The entire system is on a website and/or it can be used as an app on an ipad or iphone (and android).

The marvelous thing about this is there is no walking over to a behavior chart to change a card or clip etc.  You can carry the ipad around with you and immediately respond to behaviors.  Then when you want to reward or use a consequence, you simply click/touch the child’s name.  Each child has a cute colorful avatar beside their name.

They also have a little red bubble that shows their points accumulated.  Students  avatars can be projected onto a screen or smart board so that they can see their avatars and points.  A whole classroom can be given a point at one time too–say for example all of the students walked quietly in the hall or all students were working hard on an assignment.  I also really like the “random” button.  You can push the random button and different names will flash on the screen until the program finally lands on one name.  Then when it stops on that one name, you can look to see if that student is using appropriate behavior and record his/her point accordingly.  This especially keeps the kids sitting up straight if the names are projected on a screen.

What’s even better is that there can be instant parent communication.  Parents can have a code to view their child’s behavior and see how they are behaving.  Students can even change and decorate their avatar to look like they want it to.  The best part about Class Dojo–it is totally FREE!!

How Can You Make a Life Sized Hundred’s Chart Cheaply?

At the last NCTM conference, I went to a session about “The Learning Carpet” which is a giant 10×10 grid.  This life size grid helps students see number patterns on a 100’s chart.  To actually buy one it costs around $300.  The presenter told us that she started out with a tarp to make hers before she had the carpets manufactured.  Intrigued with the multiple ways the carpet could be used, we set out to teach people about the carpet in our own district and make our own.  Several of us made them with small patterned duct tape.  They took us about 3 hours to make, and it works best if you have help.  Each square is 6 1/2 inches wide.  On the real “Learning Carpet” the lines in the middle of the squares are 1/2 inch wide, but the duct tape we used is over an inch wide.  There are number and letter cards that go with the mat which we made as well on card stock.  One of our teachers took the initiative to get together before school started to make the “learning tarps” so we could help each other.  To buy the tarp and duct tape it costed us about $30. Which is the better buy?  Hmmmm…you do the math! Below are our results.  

Turquoise and pink chevrons…wooo!

Yellow and paint splatters…

Orange and bubble design…

As you can see from this photo, it is easiest to lay all the horizontal stripes first and then go back and lay the vertical stripes.

I tried to make a learning carpet a different way just to see if it would work…my little experiment!  I  built a stencil from a piece of poster board and spray painted the stencil on a full size sheet.  To make the stencil, I cut the 1/2 inch outlines away from the squares while leaving a little piece of poster board attached to the squares so that the stencil wouldn’t fall apart.  The stencil is the green rectangle with paint on it that you see lying on the sheet.

I just moved the stencil and lined it up each time I spray painted to repeat the pattern.  Towards the end I started getting several smudges since the stencil was getting soaked with paint.  Sturdier cardboard may have worked out better.  Two cans of spray paint are necessary to paint all the squares on the sheet.  The total cost for the poster board, the sheet, and the spray paint was about $20.  Although the spray paint/sheet method was messier, it is much easier to create this project alone than stretching the tape exactly straight with the tarp method.  Plus you can easily throw the sheet in the washer when it accumulates too much kid dirt!

A few paint fumes later…my finished product!

 

If you truly wanted to have a learning carpet, this stencil method would work to spray paint a carpet remnant with low pile as well.

 Read my original post about “The Learning Carpet” to find out more about how this can be used in your classroom.

 

One More Fabulous Free Common Core Resource!!!

I will start out with my most favorite state’s common core resources that I have seen so far.  This one ranks up there with the depth of the Georgia State Units.  Not to keep you in suspense, but North Carolina has done a superb amount of work on these units.  Each one has lessons and student tasks ready made to go with each lesson.  These units are not ALL conclusive though.  There are some holes or standards left out, but what is there is excellent quality work!  Thank you North Carolina!

First you will come to this page. Click on the grade level of your choice.

Then you will be taken to this screen below.  There are many resources here including the unpacked CCSS.  To find the unit lessons click where the yellow arrow is, but  the other resources are definitely worth exploring as they will give you an overview of the unit, parent/school connections etc.

 

When you click on the unit link, you will be taken to the cover page of the unit that looks like the following.

 

Then when you look inside the unit, you will see complete lessons….

 

…with tasks and or recording sheets included…

 

I hope these save you time from recreating the wheel :).

 

 

Free Math Common Core Tasks

I just ordered these Battista books to help implement the common core math standards for each grade level at school.  To my delight, the books list a link to extra free resource tasks!  There is a book for place value, multiplication and division, fractions, geometric measurement, and addition and subtraction, hence there are FREE resources for all of these.

These free tasks are at Heinemann, the publisher’s website.  To access the free resources, first click on the book icon of the topic you are interested in.

Next click on the link that says “companion resources”.  This will take you to all of the free tasks for that particular math book.

Here is a sample of one of the tasks:

 

Wondering What the PARCC Assessment Will Look Like?

Now there is less to wonder about.  Admittedly living in disequilibrium and uncertainty all year about how students will be tested in the future, I now see a glimmer of light at the end of the tunnel.  I perused the PARCC assessment items that are up to view.  There are now more questions up than there used to be, so now I have a better understanding of  how students will be asked common core questions.  Now I know how rigorous I need to be in my question asking of students.  I suggest you take a look to if you are one of the PARCC states.

I noticed that students are doing a lot of writing about their mathematical thinking, so it looks like math journals will come in really handy.  Also, students answer questions in which there could be more than one right answer or more than one way to arrive at the answer.  They also seem to love fraction number line questions!

Here is the site.  http://www.parcconline.org/samples/mathematics/grade-3-mathematics  Click on the grade level you want to look at.

 

Then click below to access the tasks.  Also take the time to read about the 3 different types of tasks.

Then click on Elementary School Tasks.

Another Great Common Core Resource with Free Tasks

I’ve just been searching to see if there are new PARCC assessment items out so that I could have a glimpse at the way we will be tested.  I have  checked periodically this year to see what the PARCC items will be like, but there was never much available to see.  Well, while clicking on the tasks I was taken to another page which has lots of sample free common core tasks which are VERY similar to the PARCC assessment   Here is the site:  http://www.illustrativemathematics.org.  Just click on the right sidebar where it says illustrations.

 

You will have to click along the bottom on the blue numbers to go to higher grades.

Fun and Easy Tool to Teach Number Sense for Kinesthetic Learners

Here is my absolute favorite session from NCTM (National Council of Teachers of Mathematics).  The presenter was a lady from Canada who brought The Learning Carpet for us to see.  The learning carpet is a 10 by 10 grid of empty squares that you can use for many things, but it is especially useful for a large 100’s chart.

The  number cards are 6 1/2 inches square and made out of card stock.  Students in groups of five can see how fast they can place the number cards on the carpet.  This can be easily differentiated by giving the easier numbers to the struggling learners and the larger numbers to the students who need a challenge.

Students can also be asked to pick up the numbers whose digits makes sums of 10 or any number.  Students will start to see patterns such as how different sums follow diagonals.  I felt dumb when she showed us this because I had never noticed that the sums make diagonals.

In the above picture you can see the gray squares on the mat.  You could easily make this on a tarp with paint or tape to show the number boxes.  The gray boxes are 6 1/2 inches and the black stripes on the grid lines are 1/2 an inch.    If I made one of these carpets, I would make the squares actually bigger so that feet could more easily fit inside the boxes.  I ordered the book with all the games that you can play so I could make my own if I wanted.  Next year, there may be money in the budget to actually purchase some of the carpets.

The amazing thing about the fact that there are no numbers on the grid actually teaches more number sense.  Students are made to think about number relationships to find spaces on the grid.  If asked to find any number on the blank grid students have to understand the relationships between the numbers.  For example, if trying to find 57 on the grid, students will know that all the sevens are in a column so that 57 will be in the column with sevens.  A marker can be thrown on the grid and then students have to tell what number space that it landed on.  They can walk on the carpet to help them figure it out.

The grid can be used for bar graphs or coordinate grids.   The grid can also be used for area and perimeter like below.

There are so many fun activities you can do with this carpet, and I love the idea of the students actually being able to get up and stand on it to be involved.  If you want to order the resources you can buy learning carpets and resources here.   The kindergarten teacher who designed these is in Canada, and this is the only place you can buy them.  They don’t sell through a larger distributor like Amazon etc.  I have no stock in these, I just think that it is a great idea whether you order the ones she makes or make your own.

Try This Incentive to Encourage Students to Count to 100

The idea of belonging to a club makes kids feel like they belong.  With that said, one of our kindergarten teachers came up with the idea of belonging to the “100 Club”.   What does it take to belong to the 100 Club?  Well, you guessed it…you must be able to count to 100!  I took this idea a step further and suggested that we hang all of the kids pictures on the wall that were in the 100 club.  We will add to this as the remaining students are able to count to 100.  The kids have taken an extra interest in counting to 100 especially if their pictures aren’t on the wall!  This display of the students’ pictures has grabbed students’ attention of course as well as parents and staff members.  We even have a kindergartener that told her teacher, “I counted to 100 in my pillow 3 times last night before I went to bed.” 🙂

Do your students know what a group is? or what a group of ten is?

I’ve been wanting to incorporate counting collections at school for a while, but I haven’t had the understanding of how to organize counting collections effectively.  I recently attended a colleague visit where a kindergarten teacher showed the procedures she used for teaching counting collections.  So, after attending this training, I initiated counting collections at our school with the 1st and kindergarten teachers.  In the meantime, one of the kindergarten teachers shared with me at school that she realized her students didn’t know what a group was– much less know what a group of ten was.  She began her instruction with just discussing groups and what kinds of things can come in groups.  They talked about groups of three, four, or six etc.  They made groups of different amounts in  whole group discussion under the document camera.  Students were able to have a foundation to understand a “group of ten.”  Then the teacher was able to place a different amount of counters underneath the camera to ask if she had a group of ten.  First, she placed less counters under the camera like 8 and asked if she had a group of ten.  After that she placed more counters under the camera, like a group of ten and 3 more, and asked if she had a group of ten.  Doing all of these seemingly common sense-ical counting procedures before hand led to a much more successful counting collections lesson for students to count their collections effectively.  These are the rudimentary things that no college or textbook teaches you!

To read the valuable counting collections article from Teaching Children Mathematics, click here.

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