Category Archives: Math

Combining Like Terms

Since being back in the classroom, I find myself doing activities with my lessons and thinking “I gotta blog about that.”  These ideas may not be profound, but you never know if there’s some teacher out there in the universe that could use this cool thing.

One of my goals every year is for my students to make sense of math.  And recently, one of our units (6th grade) was on equations and expressions.  One of the topics was combining like terms.

I first pass out playing cards (poker cards) to each student.  I made sure to use numbers 1 through 10.  I also make sure I pass out a variety of clubs, diamonds, spades and hearts usually just the numbers 2-10.

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Then I instruct the students to “combine like terms”.  Now this is usually when they give me the most quizzical look.  Then they try to ask a million questions, but I find myself just repeating myself with those 3 words.  “Combine Like Terms!”

When it looks like the students have settled down and found their partner, I take a survey of what cards they put together.  Here’s what they come up with.

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We recorded the pairs of cards when given the instructions to “combine like terms”

Without even looking, I can predict this every year.  And without any direction, I ask the students what their thinking was in pairing up.  The answers I get are usually 1. The 2 cards are the same color.  2. the numbers are the same.  3. The numbers are even or 4. they cards are sequential. 

After this initial pre-assessment, I show them examples of like terms and unlike terms.  Usually I have at least one student who will connect the lesson with the card activity.  

“Hey Mrs. A, what if we matched up the cards by the symbols?”

Bingo!  Now we are onto something.  

I proceeded to turning the  playing cards into algebraic expressions.  We used variables like D to represent for the diamond and H to represent the heart.  Those can’t be combined because they don’t have the same variable.   If you check back on the above picture…the only two cards that could be combined are the 7 of clubs and the 8 of clubs.  Those students partnered up because the cards were “in order,” however they realized during our post-lesson discussion that they were the only pair that could be considered the same terms.

By using manipulatives and contextualizing an abstract algebra topic, it makes it a little easier for the students to grasp.  

Until next time,

Kristen

Ten Percent Increase – Pt.2

During the last blog post, I described the circumstances at the beginning of the year (September 2017).  Teachers were fired up about comments made by our superintendent.   As a matter of fact, there were teachers who attended and spoke at a board meeting.  Let’s just say the board wasn’t thrilled by our superintendent’s words.  

And this brings us back to my school.  My principal called me into her office one afternoon and we had a long chat.  She wanted to make the after school intervention strictly about math. (We already do 4-5 hours of reading intervention each week within the school day).  She wanted to overhaul what was happening.  She wanted the after school program to have direction/purpose.  (I don’t know what the program was like in years prior).  She asked me what I could do for the school.  Um…..where/when do I begin?!?!  Off the top of my head, I bounced some ideas to her.  She liked what she had heard.  I got to work on the preparations.

She was looking for consistency across all grade levels.  She was looking for routines beyond the textbook.  My K-5 colleagues had been dealing with Envision Math for years, so they needed something fresh.  Biggest obstacle of all, we needed to make math engaging and worth a child’s time to come after school for 3 extra hours a week.

Here are questions that I grappled with before I started planning.  How do I hook students into an engaging hour of math afterschool for 3 hours a week?    How can they become active leaners rather than it becoming a study hall/homework help?  Most importantly, how do I convince my colleagues that my program is worth their time and energy?  How do I get everyone on the same page?

And for the next 6 weeks, I was teaching 6th grade and designing a full math intervention (from the ground up).

  • First I built a schedule and it looked like this…..

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Take notice that this was a schedule built on everything I learned in the past 4 years in conferences and via Twitter.  My principal was giving me the chance to train our teachers on all that I had learned (well….maybe not ALL…but certainly a lot).    

Here’s the breakdown of each component.

Number Talks – consisted of a bunch of pictures from my files and other websites.   Each teacher was given a a file of over 200 pictures that included “Which One Doesn’t Belong”, dot images, and powerpoint of any number talk images.  Some pictures were found on ntimages.weebly.com while others were pictures I have collected over the years.  All teachers had to do was find a picture they liked, show the image, and have a conversation.

Fluency Games – rather than practicing our fast facts with worksheets, students would engage in games/activities in which they practiced their facts.  I gathered all kinds of games from Pinterest or TPT (Only the free ones that looked worth while).  

Front Row/Freckle – this is a free website for students to get more experience with problem solving.  It’s engaging (unlike some others I’ve seen).  Students gain coins as they  conquer different topics.  They love it!  

Numberless Word Problems  Brian Bushart introduced math educators to numberless word problems.  Taking the numbers out of a word problem and then “layering” information helps students focus on the context of a problem versus becoming “number pluckers.”   

Performance Tasks  –  In the Envision text my teachers were using, the performance tasks were really weak.  Performance tasks can be effective for students to apply what the skills they have learned.  With a discerning eye, I gathered performance tasks for 7 grade levels (kinder – 6th) for them to practice with the students.  

I wish I had a picture of all the binders of information that was given to the teachers.  7 grade levels worth of all these magnificent routines.  Each binder was at least 1 1/2 inches thick.  Each teacher had number talk pictures downloaded onto their desktops ready to go.  They were thrilled.  Maybe I went overboard with providing  them with each and every single piece of this puzzle, but I didn’t want any excuse as to them NOT doing the program.  The only thing they had to do was figure out which performance tasks or game to use.  No research needed.  And NO EXCUSES.

It was a labor of love to put the program together.  However I would be remiss if I said it was all me.   It wasn’t all me.  I can put together an A+ program, but it achieving a 10% increase was a team effort.  There was buy in from every teacher at my school (including the Transitional Kinder team!)  There was support and leadership from our principal.  It was a school wide effort to help the students.  

  • We were the only school in our entire district to pull a double digit increase in mathematics.
  • 6th grade nearly doubled their scores!  
  • Every grade level increased!

I need to get packing for Palm Springs (CMC – South).  Excited to do not one, but 2 sessions on Clothesline.  Stop by and say hello.  

Until next time,

Kristen

12 Ways to Get 11

As I have said countless times in previous posts, my 6th graders have been math buddies with a kindergarten class.  The other kinder teacher and I spend numerous hours trying to create and plan different activities for both classes to do.  One of the books I came across on Twitter was 12 Ways to Get 11 by Eve Merriam.  

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While planning, we purposely waited to do this activity when kindergarten was much more in tune with their addition facts and when 6th grade needed a break from testing.

I let my collaborative partner perform and read the book.  She’s so expressive with her reading.  The kinder students involved themselves at counting everything to 11.  After the book was read, we counted things around the room to 11.  For instance, students noticed that there were exactly 11 clouds hanging from the ceiling (complete coincidence!).  Students also noticed 11 months that were listed next to the calendar (May was being used on the calendar).

 

Next, we let both sets of students explore combinations of 11.  We explained that more than 2 numbers could add up to 11. My students love working with cuisenaire rods (their manipulative of choice).  

This is what they came up with.  I loved seeing how each set of students would represent the combinations.  Kindergarten mostly just used the rods, where as my students had to represent the combinations with drawings, sketches, or some sort of visual representation.

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Here’s some footage of what we observed.

 

We wish we had more time to do more with this activity.  We meant to go back to it, but other things get thrown at us at the end of the school year.

Until next time,

Kristen

 

Southern Cali. Kinder Conf. 2018

When you hear “kindergarten”, some of my math peeps will shudder and shake. The idea of teaching “the little people” is an exhausting task.  However, this has been part of my universe for a few years now in thanks to my math partner, Stacy.

We have been exploring different conferences with enthusiasm.  We have brought our math story to primary teachers in Northern and Southern California.  At these conferences, I looked forward to meeting different math educators that I kept up with through MTBoS (Math Twitter Blog-O-Sphere).  I would eagerly introduce them to Stacy and tell our story.  They welcomed us with open arms and were intrigued by our partnership. 

This weekend was different.  Instead of Stacy entering my world of math folks, I entered hers of the kindergarten clan.  Rather than be one of thousands of math educators, we were one of three sessions featuring math.   It was a world of songs, bright colors, and construction paper.   Stacy spent the time pointing out different people to me while I tried to “blend” in.  When introducing myself as a 6th grade teacher/math coach, I was repeatedly asked “What are you doing here?”  But then I told our story and why I attend, and they were interested in what we had to say.  

We were scheduled during the toughest time slot–Friday at 5:45.  These primary teachers had been sitting in sessions all day long.  But we persisted.  We showed them “Which One Doesn’t Belong”, clothesline math, and the flipped hundreds chart.   Luckily for us, our participants were enthusiastic.  We challenged their thinking.  And in turn—they challenged us.  One asked, “why are you here for kindergarten?”  I answered her “it was because the little ones are outstanding mathematicians.”   I stand by that.  

We were different from the other sessions in that we didn’t sell our stuff on Teacher Pay Teacher.  We weren’t at the conference with any company selling their goods.  We were there to spread the word of math.  “Math can be fun and interactive,” we told them.  While presenting, we kept telling them we have them covered.  We gave them thumb drives will all kinds of files on it (including all the clothesline cards).  “That deserves a round of applause,” claimed one of our participants.  When I heard that, I remember looking at Stacy who was beaming with her brightest smile.   We did it.  

In a previous post, I had explained how we keep trying to improve our presentation skills.  With this one in the books, we have hit our stride.  We have accomplished what we have set out to doempower more teachers and reach more children with our love of math.  As I keep exploring other conferences and venues for our work, there’s more that can be done in helping the primary teachers.  

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Until next time,

Kristen (& Stacy)

Asilomar 2017

My math partner, Stacy and I were once again fortunate to present this last weekend.  We were invited to speak at California Math Council’s northern conference at Asilomar. We had never been to the northern conference, so we were eager to compare it to our experience in Palm Springs (last year).  I had heard that this was smaller and more intimate, but with better speakers because of its proximity to the Bay Area.  

First of all, the ground of Asilomar are gorgeous.  If you never been, Asilomar was like a leadership retreat-type place surrounded with trees all the while the ocean waves were crashing in the background.  When we were checking into our rooms, there was a random deer waiting for us (no joke).   It was quaint.  It was peaceful and zen-like. 

We were running into all these math heroes every step and every turn.  While backing out the car, I was close to running over Dan Meyer, Zak Champagne and Mike Flynn.  While walking to explore the beach, there was Marilyn Burns taking a stroll in maroon hat.  We helped Ruth Parker get inside our residential building.  It was like being at an  All-Star Math Camp.  

Friday night we went to see and hear from Dr. Jo Boaler.  While waiting to hear from her, I ran into this guy–Chris Shore.  We’ve been planning something for next year.  Incredible guy.  We caught up with each other while Stacy listened in on Dr. Boaler.

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Chris Shore (@mathprojects)  #ClubClothesline

Saturday morning came and it was our time to shine.  Stacy and I presented on the flipped hundreds chart and our work on clothesline math to about 15 teachers.  Really engaged participants.  Lots of conversation.  Lots of sharing.  We felt incredible.  It seems like we keep getting better.  We have found purpose with what we are doing.  

One of our participants stopped us in the hall afterwards and congratulated us on our session.  He said that he couldn’t believe how engaging we were.  We made the session feel really personal.  We were really energetic with our participants and made everyone feel welcome.  Usually there are sessions where there are “talking heads,” but we were the quite the opposite.  And on top of all this, that we made him think about what’s going on in the classroom.

The rest of our Saturday was spent seeing other speakers.  We got to listen in on Annie Fetter and her thoughts on writing for math.  We listened in on Cathy Humphreys as she explained her dissertation on mathematical agency.  Lastly, we attended Ann Carlise and her K-2 number talks with number lines session.

Usually I look for one thing to bring home and use.  I say that if you can just gleam one thing from any professional development, then it’s worth it.  I was lucky enough to have a math partner to talk this through.  We like poking each other with questions and then come to a conclusion. 

When all was mostly done, we questioned what we got out of this conference.  Stacy and I learned something far greater then what some of these great authors and math educators were telling us.  

  • We figured out what kind of speakers we want to continue to be.  We need to be us. The comments from one our participants really was thought provoking.  And that was just our personalities.  We want to be personable in our sessions.  I don’t want to be a “talking head.”  If you know who we are, we are completely the opposite of that.  And we won’t change that.  That’s who we are.
  • One of our expectations is that our participants walk away with something that they could use the next day.  In some of the sessions we attended as participants, that wasn’t happening.  There was lots of theory, but I wouldn’t know what to do with the information in my classroom.  Because of us flying up north, we couldn’t bring our full “show.”   I’ve been expecting our participants to look up all our resources on my blog, but I don’t know if that’s happening. How do we make sure they fully leave with something in hand?  (We have ideas).

We would go again in a heart beat.  We learned more about ourselves then we expected and that was major leap forward.  We didn’t expect that, but we couldn’t pass up processing our thoughts on the subject.  

And so we continue to grow.  Onto the next conference.

Until next time….

keep laughing & keep smiling,

Kristen 

Making sense of conversions

My son, Jared, is in 6th grade.  He has told me he thinks it’s “cool” that I’m teaching 6th grade too.  According to him, we have stuff to talk about.  I can finally help him with his homework.  (Really?  Like I’ve never been able to help you with any of your other homework?!?  What the what?!?!?).  Of course I left that last editorial in my head, but I think I knew what he was talking about.   I think he meant that we are finally at the same grade level as teacher (me) and student.  

On one evening, he was having trouble with conversions of measurements. This was his homework.IMG_6073

I asked him about his notes from class.  “How did your teacher teach this?  Where are your notes?

And this is what he showed me.

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“Memorize the 3 steps.” Uh–no.

 I was not thrilled to see “memorize” on my son’s notes.  It dawned on my husband and I that our son’s teacher has only been teaching a few years.  He’s probably used to just memorizing steps and procedures.  I don’t like to teach that way.  I like to teach for understanding.  I like to teach more conceptually.  I like to have my students make sense of a problem rather than “memorize” steps.

Just for kicks and giggles, I went to page 290 to see what is said.  This is what I found.

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Here are the “steps” my son was asked to memorize

I sat and stared at his paper and at the “steps.” If my son didn’t understand and remember the steps, how could I get him to comprehend what they were asking?

I looked at the 1st question again.  “If 16 C = 1 gallon, then 8 gallons = ________?”  Rather than doing a fancy algorithm or proportion (which he hadn’t done in the curriculum–my husband had an issue with that.), I went back to the basics.  Let’s draw a picture.  

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16 cups are in one gallon (rectangles).  8 gallons with 16 cups in each.   Once I sat and explained the situation to my son, the lightbulb went off in his head.  “Oh mom, all you have to do is multiply 16 times 8 to get the number of cups.”  BINGO!  

And the rest of the hour, we drew pictures, diagrams, and whatever else helped him make sense of the conversions.  And each time we drew a new picture, the lightbulb kept going off in his head.  (Proud Math teacher and Mom!)IMG_9373

The next day after school, I asked my son if his teacher said anything about the homework we did.   My son told me that his teacher said,you should have done it the way I told you too.”   ARE YOU KIDDING ME?!?!?

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I “heart” Tina Fey.

MATH RANT – –   After close to twenty years of teaching math, this just blew me out of the water.  It is no secret that there are many ways to get answers to math problems.  I usually give the anecdote that there are many ways to get from here to New York.  Some ways are faster, some ways are slower, some ways are more expensive and that’s ok.  Pick which way works best for you…..as long as you get to NY.  Same goes for math.  It is our job and soul purpose to teach our students.  It is well known that one size doesn’t fit all.  One approach to solving conversions doesn’t work for everyone.  Why are we still having students memorize procedures if they don’t understand the problem?  What happened to making sense of things? It only happens to be the first standard of math practice!!!    It baffles me that this is from a newer teacher who hasn’t done any conceptual lessons and/or applications.   The whole thing blows my mind! Maybe it’s also the realization that not every math teacher thinks or teaches like me.  

So what do we do?  Do we challenge teachers like this?  Is it worth the fight when they may not understand the importance themselves?  

My son’s situation just proves that there’s more work to be done out there.  Teachers still need training.  And just because we, in our youth, learned to memorize procedures, doesn’t mean we actually made sense of things.  

Math rant over. Disengaging.  

Until next time, 

Kristen

The next term is…

Usually on Thursday nights, I try to squeeze in an hour of professional development by participating in a #elemmathchat on Twitter.  It’s a terrific hour of discourse between coaches, teachers, and all kinds of math peeps.  

Recently, Mark Chubb (@MarkChubb3) hosted a #elemmathchat and posted this.

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My students know that I love throwing them a “challenge” question every now and again.  I use lots of things that I find on Twitter or just online.  I’m just curious as to what their perspective will be and what their conversations will be.  And they are getting used to me bugging them for a picture of their work so that I can blog about it later.  

Here’s a slide show of some of their answers

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These two students had a pattern similar to what I would think would be the next shapes.

What do you think it should be?  Or do we need more info?

 

After this experience, I’m thinking of re-trying this type of exploration with Visual Patterns.

Until next time,

Kristen

Cents clothesline

Recently, I was invited into a 2nd grade classroom to work on money (2.MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately).   The teacher wanted to try out something new to go along with this standard.  My suggestion was to try out the clothesline.  Let’s see if students could put different variations of coins from least to greatest.  It totally make cents (1st bad money pun)

And it went splendidly.  

Each student was given a card to work on.  They calculated the total amount and put their answers on a post it note.  When they were ready, the hung their cards on the clothesline.

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When they were finished, we went over each card and made sure that we added each coin correctly. 

At the end of the session, they students asked why they were doing the activity now (they were well passed their money unit).  I answered by telling them how many skills were involved with this activity.  

  • recognizing each coin and its value.
  • practicing their addition skills with one and two digit numbers
  • comparing and ordering the numbers
  • showing equality with some of the coins (for example- 1 dime is the same as 2 nickels which is the same as 10 pennies)
  • realizing what happens when you have more than 100 cents.

After my answers, I realized how much “bang we got with our buck.” (2nd bad money pun).  This activity had a lot going on it.  And all we did was put up a string and gave them cards.   But the thinking that went on was nothing less than incredible.  

 

Until next time,

Kristen

 

Writing in math

Like I was saying in another blog-post, I’ve been exploring the different ways and types of writing that could go on in a math classroom. Last year,  I was asked to present on the subject.  

This is a topic I dabbled in when I was in my own classroom, so I was pretty excited to share with my elementary teachers.  

It’s been a question that’s been in everyone’s heads for such a long time.  How do we incorporate writing in math?  I know that it should be done, but I wanted it to have be meaningful.  I wanted it to be authentic.  A student’s writing is one way for us to see inside their heads.  What’s going on in that brain?  How is he/she approaching problems? 

As a parent, I’ve seen my own son come home with those “write to explain” questions at the bottom of his worksheets.  Usually his answers are short and blunt.  Or some of us have seen writing like this…

tumblr_inline_nxemap2Y751t2pr7r_1280Yup…this kid is going places.  I do appreciate the humor in this, however this is not what we are going for.

Some elementary teachers have admitted to me that they usually skip the “explain” questions at the end of the homework.  And let’s admit it…what student completely takes ownership of those questions at the end?  How much thinking/reasoning are teachers seeing out of those questions?  It’s not happening. 

We need to get our students’ buy-in.  We need them to take ownership.  We need them to be engaged in the problems.  We as teachers have to be creative.  As William Zessner said,Writing is a way to work yourself into a subject and make it your own.

So here’s a few ways I’ve engaged students into writing.

  1. Performance tasks/PBL – Performance tasks are a perfect way to engage a student into a problem.  It’s a spring board to have them create their own writing.  This is a 4th grade task that one of my teachers tried out. Stone%20Soup  A teacher can cover at least 3 subjects in one task.
  2. Exit cards.  I have used exit cards to ask questions.  I think of it as an extension of a number talk.  For instance, explain to me that 17 x 28 is greater than 16 x 29.  
  3. Error Analysis.  One question I ask students as closure is “how will you know when you’ve learned this?”  Usually I get answers like “when I get an A on the test.”  I’m never convinced.  I’m looking for the student to give me the answer of “when I can show/teach you the concept.”  I’ve created a template that can change with the concept.  For instance, here’s one on division. Div Err Analysis
  4. Start with the answer...I haven’t used this one yet, but I’ve seen a few versions of it. Let’s say that you start with the answer of 6.  The student has to write a math story to go with it.  I see this especially for 1st and 2nd graders who need practice with their addition and subtraction (and also writing).  

I know I don’t have all the answers.  I’m just starting the exploration.  I would welcome others to leave comments as to how they tackled this topic.  

 

Until next time,

Kristen

Clotheslines for Math concepts

Using clotheslines as an interactive number line has been a hot routine this year.  Last year, I slowly and carefully rolled it out into a few classrooms for use of fractions.  This school year, I’ve expanded into more classrooms, but am proud of how my teachers have especially made it work in K-2 classrooms.   It’s been extraordinary to see using a routine where you get so much “bang for your buck.”  There are a good 2-3 content standards that students have been using, let alone multiple Standards of Math Practice.  

The clothesline makes sense of numbers and number placement.  I especially love the fact that it’s interactive, provokes discussion, and gives insight as to a child’s thinking.  Students are actively learning and using multiple strategies to complete the task.  And more importantly, it is a tool and a model for students to see the “big picture”.  

Let’s breakdown each grade level and how they’ve used the clothesline.

KinderI’m a true believer that if you can make a routine work for kindergarten, you can make it work for any grade level.   Ever since I introduced this to my kindergarten team, they’ve come up with MANY different ways to bring clotheslines to life!  It was rough to begin with, but my kinders have now been through the routine 4-5 times and they’ve got it!   Parents are now asking my teachers what kind of math they’re doing because the kids are telling their parents about what they did.  (Score!)  In September, Mrs. Z and I started with number 0-5 first.  Within weeks, we did 0-8.  And by November we did, 0-10.  It’s imperative to point out that kinders are not working on proportionality of the numbers.  They are just working on counting and cardinality (and measurement and data).  We also tried out using the clothesline with weight.  Instead of literally putting each object in order from least to greatest weight, we kept it simplistic with the light items being placed on the left while the heavier items went on the right.  If they were sure of an item, they placed it in the middle.

First grade – tried it out with numbers 0-25.  Lots of conversation.  Teachers got insight into how their students were thinking about numbers.

Second grade – tried it out with numbers 0-50.  Lots of conversation.  What was unique is that students were using their strategies of counting doubles for a few particular cards.

Third grade – have used this with benchmark fractions.  One 3rd grade team just designed a card set with multiple representations of multiplication.  This will be tried out in the new year

Fourth grade – will be using it for fractions.

Fifth grade – one teacher used it for decimals.  The students had been doing all the operations with decimals and wanted to see their number sense when it came to placing decimals on a number line.  What happened was a complete shock to her.  Students grouped the decimals according to number of digits (for instance, .4 and .5 would group together because they have one digit.)  That completely blew my mind.  Surely, we can’t always assume that our students have a true understanding of a concept when we ask them to apply their knowledge elsewhere.  

Sixth gradeteachers will be using clotheslines for integers and integers integrated with decimals, fractions, and percents.

If anyone is interested in downloading the sets of cards for their own use, look here for my sets of cards.

Until next time,

Kristen