Wednesday, October 26, 2016

The importance of concrete models: Multiplication Arrays

concrete models multiplication
We are moving into our mini-unit on factors, multiples, prime numbers, and factoring--SO much vocabulary and such complicated ideas for students who may or may not be secure with their multiplication facts.  To really try to help students "SEE" the concepts, I believe super strongly in using concrete models.


Chocolate models--let's be honest.  

Seriously though...I really want my students to deeply understand the concept of arrays and how they relate to multiplication, factors, products--and eventually prime and composite numbers.

So the other day I asked my students to tell me what they knew about arrays--and they shared out their ideas.  They DID have a bit of a hard time using precise mathematical language so I flipped on the projector and document camera, whipped this Dove monster out of my school bag, dramatically wafted the smell toward them, and threw it under the camera asking, "Will THIS help you describe what an array is?"  After a few moments of whining about how it wasn't fair that I had one and they didn't, we got to work using clear math language to describe this array.  We used the words factor and product and array.  We talked about directionality (Is 6 rows of 3 the same as 3 rows of 6? Is 6 bags of 3 apples the same as 6 bags of 3 apples?)
multiplication arrays
 I then pulled out a bag of these to a roar of delight from the crowd and asked to practice that mathematical language to describe THIS array.  The results were much more clear!
teaching multiplicationmultiplication
 After consuming a little "Brain Food", we got to work on one of my favorite hands-on lessons of the year--the Candy Factory!  In this project, we arrange candy "tiles" in all the possible arrays trying to look for patterns...we uncover the "double/half" strategy, started to notice some thing about arrays with odd numbers, and more!

As students moved forward from making "boxes" of one candy...then two candies...then three, you could feel their confidence growing--and their willingness to model with the tiles diminishing.  I continued to encourage them to use them...but some were insistent that they could do it in their head.  I was noticing everyone getting the "one by..." arrays (1 x 12... 1 x 13...and so on) but were starting to miss some of the "short, chubby" rectangles.  Some students had even made a claim that "All odd products only have 2 factors."  Hmmmmmm
multiplication arrays
 But it was when I came upon this example that I knew we had problems.

Big.  Problems.
multiplication arrays

 So at this point, I INSISTED that he break out the tiles and SHOW me the array he had drawn.  As you can imagine, he wasn't able to do so and a little light bulb went off.  I asked him how he thought he COULD arrange the 9 tiles and, after some experimenting, figured out that, in addition to the 1 x 9 array, he could also build a 3 x 3.  He could NOT do it in his head...he needed that tangible tool--FOR NOW.  I asked him if I could share his story with the class so we could all learn from it and he agreed.
multiplication arrays

All in all, we have been really building some solid conceptual understanding of factors, products, and arrays--and we are ready to dig into the second part of the activity--determining the difference between prime and composite numbers!  We are on our way!  If you are interested in seeing more, just click the image below to see more about this resource. 

teaching factors

Thanks for stopping by!

Sunday, October 23, 2016

Helping Students "Make Sense" of Problems

math practice standards
Teaching math is a complicated venture--to say the least.  Between figuring out how to meet the needs of all our students, balance a jam-packed curriculum, worry about interventions and enrichment--and then throw in fire drills, strep throat epidemics, students gone on vacations,'s a wonder we get anything done!

The one thing I always use to "center" myself is the Standards for Mathematical Practice--or "Math Behaviors", if you prefer.  These "ways of being" in math class are the glue that holds my  math class together...whether my students are above or below grade level--and no matter if we are doing addition or fractions or measurement.

One of these standards is often called the "perseverance" standard...and people do a GREAT job of teaching students that word and stressing that "I can" spirit in their classrooms.  That being said--"perseverance" is only a small part of that standard and we cannot forget the rest.  This standard talks about sticking with problems--but perhaps more importantly talks about MAKING SENSE of problems...using a thoughtful, logical, and organize process to dig in to the information presented and to tackle it.  It's hard to persevere on a problem until you have "decoded" it!
This is one of the posters in my "kid friendly" math standards posters resources.
So the other day I decided to give my students one of my open ended challenges--with NO preliminary work.  I didn't read it to them.  I didn't give them any hints or talk them through it.  I told them--your goal today is to figure out WHAT this problem is asking, think about how to get started, and how you will organize your work.  I told them I would read them any word they struggled on (not reading class!) but it would be up to them to try to "decode" the problem.  I built up the energy...told them I KNEW they could do it and explained we would start tackling it alone but then would move into partnerships to compare notes and to get started.

So off they went!  I walked around and circulated...asked a few questions here and there--mostly things like, "What have you figured out so far?" and "Tell me what you notice." to try to get a sense for what they were gleaning from the problem.
math challenges
After five minutes or so, I then partnered students up to compare notes.  I started to see some great stuff...highlighters came out.  Students were pointing out information and underlining it.  I heard things like " I get it!" and "I totally missed that part!".  After a while longer, we came back as a class and talked about our findings--and also talked about the power of partners!  It was pretty evident to students that two heads were definitely better than one in this case!
math workshop
At that point, I put a copy of the problem under the document camera and we shared out our findings.  I used highlighters and my pen to showcase what students told me they had found in the problem and what they were noticing about getting it started.  After a few more minutes of processing, I was pretty sure that partnerships were ready to go tackle it! They were chomping at the bit to get going, and I am pretty sure that taking the time to process on this problem was just as valuable as actually working the problem itself!  I quickly reminded them about working precisely and in an organized fashion (our recent goal) and sent them off on their way.
math practices
From beginning to end, my students worked for 45 minutes on this problem this day--and could have done another 15.  We finished up the next day and had a ton of fun sharing our different solutions.  I love hearing the students' logic about which solution made the most sense for them.  I think I have some entrepreneurs in training in my class!
math challenges

So today's food for thought?  Think about how much work YOU are doing for your students by helping them get started...and consider how many opportunities you are giving them to try to learn how to make sense of tricky problems on their own.  Want to try 
the Marco Problem with YOUR class?  It's one of the three challenges in Set 1 of my "Open Ended Challenges" resource..  Click the image below if you want to see more!

math word problems
 Also available in this bundled set of 9 challenges!
open ended problems

Thanks for stopping by!

Thursday, October 13, 2016

The importance of watching students work...helping with place value misconceptions

expanded and standard form

We recently finished our unit on place value and I KNEW that not all students were solid.  As I usually do, I continued to give occasional entrance slips to try to keep my finger on the pulse of the class.  When I do this, I try to work in time to meet with strugglers--and I try to do it in groups of no more than 3 so I can WATCH them.

Here's what I mean...I gave my students an entrance slip on Wednesday as class started.  As they turned them in, I did a quick check of them and sorted them into two piles--"no problem" and "better check it out".
formative assessment
These assessments are from my "Formative Assessment Toolbox:  Place Value Edition" resource.  CLICK HERE to see it!
Looking at student work can tell you certain things...but I firmly believe we need to take things a step further and WATCH students to see HOW they are making their mistakes.  For example, I was looking at some addition with regrouping work a student had done--and time after time there was an error in the "regrouping".  I could NOT for the life of me figure out what she was doing so I called her over and asked her to do a problem.  Within ten seconds I figured it out--she was starting on the left and regrouping to the right--showing a HUGE misunderstanding.  Turns out she had been pretty solid with partial sums (which can be done left to right or right to left) and was overgeneralizing that algorithm.  Without sitting right next to her and watching, I wouldn't have figured it out and I wouldn't have known to pull out the base ten blocks and model with her.

The same was true with a few of my kiddos who were making some mistakes with expanded and standard form.  I need to sit right with them and watch them work--and ask their thinking.
I worked with a few students at a time with my expanded form task cards (actually, to make it more fun, I put the cards in piles and the students took turns picking cards--it's AMAZING how something as simple as letting students flip a card can keep them more engaged!)--and I watched them work on their white boards, asked them questions, and asked them to explain their thinking.
place value
As I was reassured that misconceptions were being fixed, I would send students away and fill their spot with new ones.  Other students were busy working on their math workshop options and I had a glorious 40 minutes to work up close and personal with these students.  The light bulbs kept going off-and I know the WATCHING was a huge part of it...putting myself in their shoes and trying to be a scientist figuring out where their thinking was going wrong.  It is SO validating for a child to hear, "I can TOTALLY see what you did it that way!  Let me show you something..." instead of only seeing problems marked wrong or sitting back in a large group discussion feeling confused and lost.

Watching students can help you see exactly what misconceptions students have--and for these place value concepts, several things come to the forefront and need coaching:

  • A lack of understanding of our base 10 system (especially once students get past 1,000 numbers get much harder to visualize)
  • A misunderstanding of "0" and how having "no" thousands gets represented
  • An inability to understand the organization of our number system...that we need three "places" before a comma, then three more and a comma, and so on--and that each of those "groups" or "periods" has a name and follows the pattern of one, ten, hundred.
  • The inability to recognize expanded form terms out of order (they might write 50 + 3 + 600 as 536)
  • Difficulties reading and writing big numbers and "hearing" the parts.  For example, if I say "four hundred thirty-two thousand", students should hear that "432" and know it will come in front of the thousand comma.  If I say, "four hundred thirty-two thousand, seven", they should know that they need to write a 0 in the hundreds and tens spots because there are no hundreds or tens.  Asking students to read and write big numbers is a great way to check for understanding.
So much of this is tricky whole class--you can present information this way, but when you see how many different ways place value understanding can break down, it becomes more and more clear how our interventions need to be much more personalized.

standard form expanded form
An added bonus of working next to small groups is the ability to "coach" them through some partner work.  After solving a card, I would ask students to compare work and try to reconcile any differences.  Learning how to check over their work, look for errors, and explain thinking is such a key part of the standards for mathematical practice.  As they got better at catching each other's errors, my role became much more of an observer than a teacher--and the power got turned over to them.  Hearing things like, "Wait--you wrote 50,000 but that 5 is in the one thousands place" or "Remember that if there aren't any tens you need to write a 0." is enough to make THIS math teacher melt.  Seriously.
teaching place value and expanded form
Interested in checking out the task cards I used with my groups?  Just check them out below!  There are 36 cards with three different types of questions.  Each one has a "bonus" question too--so I used them with my entire class and then pulled my intervention kiddos and used them again.  See what you think!
expanded standard form task cards

I have a ton of place value activities in my store so check them out if you are looking for games, lessons, and more.  Just type "place value" into the search bar of my store!

Sunday, October 9, 2016

Why do we estimate?

learning to estimate
With our current unit in math, students are working on addition with regrouping and estimating.  Over the years, I have found that students really see estimating as a task that is a "fill in the blank" activity--not a meaningful math "thinking" experience.

For example, we have pages in our practice book where students are supposed to estimate and then find the exact sum.  Reasonable, right?  After watching a few students working on a page this week asking them to estimate, then add and instead solved the addition problem and then wrote an "estimate" for the exact sum they had found (rather than estimate the two numbers before adding), they shrugged and said that "it was faster".  It was pretty clear that they weren't understanding the purpose of estimating at all...

So--time to back up.

It was time to come back as a whole class and talk about the "why" we estimate--because we DO estimate for different reasons--and I really want my students to NOT think they estimate just because there was an answer blank on a textbook page!

First of all, I asked my students to tell me how many people went to the last Packer game.  They had a few guesses...maybe 70,000.  Maybe 80,000.  I asked them if it was important to know exactly how many, and they thought probably not.  We talked about the food vendors...and the ushers...and the other employees--and that they prepare for a game for "about" how many people will be there..

We then came up with a whole bunch of other times when an "exact" number isn't necessary.  However, this still doesn't really address WHY our textbooks ask students to estimate at seemingly random times.

The simple fact is, one of the BEST uses for estimating involves "checking for reasonableness"--and it's something we should be training our students to do constantly--no matter the content area.

In fact, the Standards for Mathematical Process explicitly state the following as a part of the "Uses Appropriate Tools Strategically" standard.
use appropriate tools
This is a poster from one of my Kid-Friendly Standards for Mathematical Practice posters...available in a number of different designs!  

"They detect possible errors by strategically using estimation and other mathematical knowledge."

This standard is far more than using rulers and's all about making smart mathematical decisions--and estimation is one such way to evaluate those decisions.  Being able to ask the question, "Does this make sense?" is such a critical question we want our students to be able to have in their minds constantly--no matter what unit we are studying!  We can really build upon the Standards for Mathematical Practice by asking students to talk about their estimates and to explain their thinking. This is a huge part of the standard "Construct viable arguments and critique the reasoning of others" as stated below:
constructs reasoning and critiques reasoning

"They justify their conclusions, communicate them to others, and respond to the arguments of others."

That being said, being able to quickly and reasonable estimate with the four operations is critical and provides a great opportunity for students to practice this important skill.  Helping students understand the difference between estimating and rounding (rounding means that you take an exact number and make it "less precise"--by changing to to be the "nearest ten" (or nearest hundred, thousand, etc).  Rounding is a way to estimate--but it isn't the ONLY way to estimate a number!  Rounding can be tricky for students...but it really helps if they understand that it is a way to make a number easier to work with--and that it totally relates to skip counting by a certain place...which "10" is it closest to...or what "100", and so on.  Number lines are the BEST way to show this process, in my opinion!  After teaching it this year, we practiced with these cards in pairs where they had to really work on that math talk where they explained their thinking and had to defend their ideas to their partners.
rounding task cards

So because I want to continue to refine my students' ability to estimate numbers and explain their thinking, I wanted students to have a different type of estimating experience--not based on rounding, but purely based on place value understanding and this sense of "reasonableness".  If you have followed me for any amount of time, you know that concept sorts are a huge part of my math program--for the higher level thinking they require AND for the math talk they generate.  These cards are designed to promote talk--because there are often multiple categories that cards can fit it--and it's up to the students to engage in the discourse that will allow them to make decisions about sorting the cards. I put it together in a resource in case you want to try it with your students...all directions are included.  Just click the image below  if you want to check it out.
estimation lessons

So my challenge to you is this...ask your students some of the following questions:

Why do we estimate?

What is the difference between rounding and esimating?

How do you explain how to estimate?

When do we estimate?

How can estimating help me as a mathematician?

Let's see if we can't get our students really incorporating estimating and checking for reasonableness into their mathematical habits--not just when they are assigned to do it.  I'd love to hear your thoughts!

Saturday, October 1, 2016

LOVING our narrative writing unit!

narrative writing unit

Teaching narrative writing isn't easy...writing a story from start to finish can seem like a pretty overwhelming task--and it is!  I've been working for years on how to make it more accessible and meaningful to my students because, let's face it, writing stories for a living isn't what 99% of them will be doing.  That being said, the PROCESS of writing a story and the creative energy of planning one is super valuable--and, of course, there are many writing skills that cross genres.  I thought I'd share what we did this year...and I'm trying to consolidate it without losing too much of the detail.  Hope this all makes sense!

For the last few years I have used a planning tool that I turned into a resource which helps kick off the unit of narrative writing. It has been SUPER helpful for me as I try to set the tone for a writing community--and to get them started planning their stories.  I still use it and it really helps my students with their planning.

As our curriculum has evolved a bit, I started looking for ways to really get my students digging in to character development so that their narratives were richer and more thoughtful.  We worked hard to try to get students to really think about a main character...who this person would be.  What they would be like as people.  What they enjoy--and what they don't.  What their strengths are--and their weaknesses.  As I started to jot down these ideas, I used our read aloud to try to see if we could see how a "real" author developed those characters.
graphic organizers for writing

Because we are reading Fish in a Tree, there are PLENTY of opportunities to talk about characters--both Ally as a main character and the several other important secondary characters.  I started tracking what we knew about two of the characters (Ally and Shay) in terms of their "external" characteristics and their internal traits, interests, strengths, and more.  We tracked the characters on these forms and learned some things...we learned that authors do NOT spend much time describing what characters look like (I was thrilled when my students said, "That doesn't really matter!") and most of the clues that are given are HINTED at (ah, "inferring"!).  This led to some pretty interesting discussions about what WE needed to do as writers--that we really needed to focus on creating characters that we could describe by giving our readers clues.

I decided that the best way to help show my students how to start creating such a character would be for us to work together to create one...I made new copies of these forms and, together we worked to "invent" a character under the document camera.  This is really one of my favorite things to do--to write WITH my students.  It's a great way to get ALL students involved in the writing takes the pressure of the actual WRITING away and lets even struggling writers be active participants.  We did lots of "turn and talk" and then I let students share ideas and then I made final decisions and recorded our thinking.

After spending about half a class period "creating" this character, I had my students work to talk with small groups about the types of characters THEY wanted to create.  I am kind of a stickler--I tell students that writers write about what they are experts in--so their characters need to be similar in age to them...they simply don't know what it is like to be a teenager and deal with teenager problems.  This keeps the stories much more realistic and saves a lot of tough discussions!
writing graphic organizers

After this, it was time to get to work.  I broke this up into two days--where we worked through exterior "looks" of our characters and the trait list.  It is always amazing to me how many students don't know many of these vocabulary words...words like "conceited" and "compassionate"--so we work on those.  I then tell them to pick ONE key trait that they will work to build...and one or two others that might show up in the story.  We looked back at our study of Ally and realize that although the author paints her as a bit defiant and difficult, she also has her creative and compassionate that was good food for thought as they picked.
teaching prewriting
The next day, we continued "getting to know" our new characters (I told my class that they should "know" their character so well that they would recognize them if they walked through the door!) and expanded our planning.  We got back to work inventing "Danny", my character by coming up with his likes and dislikes as well as special items or treasures he might have.  We talked about characters we know and love--and went back to Ally to talk about her likes, dislikes, and special objects.  I could really see that my students were starting to "get it" and they were itching to get to work on their own off they went to get creative.  I kept this a pretty relaxed time...some worked alone, some brainstormed together, and then we came together at the end to talk about what we had accomplished.

teaching writing
The next step of our journey started to get us deeper into story planning--stories don't tend to have just ONE character, so we talked at length about some of the secondary characters in Fish in a Tree and how Ally interacts with them.  We talked about their traits, likes, and dislikes, and how they were important to the story.  Then what?  That's right--time to invent some secondary characters for Danny!
planning writing

I love the idea of having students think about characters who can HELP the main characters, and those who make life you know, this is what stories are made of!  We had TONS of ideas for who might make life difficult for Danny...but we kept it simple.  These are meant to be short stories, not novels after all!  It was time for students to get to work...and I asked them to work with a partner to make sure their new characters seemed to "fit" with what they had already invented about their main character.
narrative writing unit

Next, it was time to decide what types of problems our character could have...all the while making sure that these problems fit in with what we already had created.  We pulled Fish in a Tree back into the mix and talked about how Ally and her problems were directly related to other characters (Shay), and her own dislikes and traits (her unwillingness to ask for help).  We created a few situations for Danny, then students went off to work on designing some perfect problems for their characters and secondary characters.

As our stories began to take shape, I knew I wanted to get a little bit more background information for the students so we dug into a few books we had read looking for setting clues.  I wanted them to continue thinking of this idea that authors "give hints" without just TELLING the reader...and then we started creating a chart to brainstorm ideas for the setting of "the ballpark"--and how to come up with descriptive words and phrases that could help a reader know we are there without really telling them.
writing setting details

They had SO much fun...and got better and better as our sharing continued.  When we were finished?  Off to our writer's notebook to start creating our settings!

The two other elements of writing that we tackled before we started writing were working on writing dialogue and planning a narrative with a story map.  I won't bore you with all the details...but if you are interested, I do have an old blog post that you can find HERE all about what I do!
writing dialogue

So anyhoo...hope this helps you see how I tried to raise the level of writing AND the level of excitement about students loved it! The fun part is, you don't really even have to write complete could just write short "scenes" that their characters might might be more fun to write several shorter scenes than one big story--and all great ways to get to be better writers!  I did ask my students to do a lot of self-assessing as we went...
teaching narrative writing

These lessons are all super easy to can do them in their writer's notebook or, if you are like me, I wanted the students to do their creative work outside their notebook and then use their notebooks for crafting their scenes.  I put all the brainstorming sheets together in a little resource that might be helpful for you...and also made a bundle with a few other narrative writing products as well!  Hope you find them valuable...and have fun writing!  Want to check out this Narrative Writing Toolkit or the bundle?  Just click the images below!

Tuesday, September 27, 2016

Rethinking Problem Solving: More Than a Word Problem

addition and subtraction activities
For years I have used the phrase "problem solving does not equal word problems", but I love it when I have a real world "math experience" that helps prove this point.  Today was one of those days--so I thought I'd share!

If you have followed me for any length of time, you may have seen me post about "Mind Boggling Math", a resource I created about three years ago to help provide me students with a challenging way to practice their addition, subtraction, estimation, and perseverance skills.  I have used it in different ways--sometimes as an activity for fast finishers, sometimes I have done it cooperatively, and sometimes I introduce it to the entire class and just see what happens!

That's what happened this year.  I explained the tasks to my class...explained that the goal was to improve their math organization, to work on their addition with regrouping fluency and accuracy, and to persevere with challenging tasks.  I was kind of anticipating using it as a part of math workshop...where students could be working on something meaningful while I pull small intervention groups.  That's how it started.

Until today.

I had a group of four girls working together when one of them came up to me and said, "Mrs. A, we have been working really hard on this page--but we have four different answers."

Yikes, right?

I asked them what they thought they could do to try to find their mistakes...and one suggested the start with one column and see if they could get agreement on that one.  I thought that sounded like a great idea--and let them get back to work.
math workshop
I walked past another student who told me, "I keep getting screwed up on what numbers I have used."

I sat down next to him and asked what he thought he could to to organize his thinking.  He thought for a minute and then said, "Maybe I should cross off numbers I use so I can easily see them."  I thought that seemed like a great strategy--and let him get back to work.
problem solving
So as I was circulating around the room getting ready to pull my next intervention group, I saw a group of three boys huddled up--one with his arms crossed and a pretty sour look on his face.  The other two had a packet on a clipboard and were pointing and looking at their own I asked what they were doing. (First thought--can't like--they were copying off each other).

One replied, "---- has a different answer than we have so we are trying to help him find where he went wrong."

The "victim" chimed in with "I think THEY are wrong and I am right!".  I asked how they planned to figure out the correct answer...and they thought for a bit.  One said, "Let's try adding them up in a different order and see what we get."  I smiled and asked them to let me know what they found out.
math workshop
So...despite the fact that I am a HUGE believer in word problems, I think it's important for us to remember that "solving problems" is more than solving word's about solving "problems" that occur when processing math.  These three examples literally happened within 15 minutes of each other...and within 20 feet of classroom space!  I shared with the students what I had noticed--and how that type of thinking is TRULY what builds mathematical minds.  This is way more than solving these addition tasks--it's about developing strategies (like solving a problem a different way...or marking off numbers to keep organized...) that can apply to MANY other math situations.  This is why the Standards for Mathematical Practice are so critical--it's more than the content.

I watched the students start to understand this...and to be proud of the struggles they had.  This is why I teach.  I. Love. These. Moments.  I reminded my students what we had learned about the brain and how it grows from mistakes--and how proud I was that they were sticking with the task despite it being challenging and frustrating--and that their hard work truly was paying off...even if they NEVER find the correct answer.  #teachablemoment
addition and subtraction problem solving
If you are interested, I have this resource in the version I have pictured (The "original" Mind Boggling Math) and also a "Money" edition to add decimals and also a 2 digit version...I may replace this one with the two digit version for some of my struggling students.  I may also have a few use a calculator to check their work as they go so they don't get too frustrated...the activities are so valuable and I want all my students to have access.  See what you think!