Musings and Strategies From the Teachers Next Door

Archive for the ‘Math Resources’ Category

The Struggle is Good

If you haven’t yet discovered the Open Middle site, may I introduce you.  Welcome to the world of challenging, thought provoking problems.  Dan Meyer introduced this concept, creating problems that have a definite beginning and end, but have an “open middle” that creates the space for students to solve problems multiple ways.  The open middle is what I love about math, why math is so much more than learning a set strategy.  It allows for understanding based on how you think and how you understand numbers.  I have seen kids come up with strategies that I would have never thought of…oh, how I love those conversations.  For that reason, this site is a gift.

This past week, I used a fraction problem with my 3rd graders.  Here is it: (source:  Open Middle)

fraction_number_line from openmiddle

Students were asked to place 3/4 on this number line.

While I anticipated my students to be a little stumped by the fact that 3/3 would not be at the end of the line, I had no idea the level of difficulty this would pose.  I wish I had recorded all the great discussion that ensued.

They justified their position, even tried to form a coup and overthrow me, when I challenged them placing 3/3 at the end of the line.  Guiding students through questioning is an art that I am practicing daily and there were moments that I wanted to just break down and “teach.”  But then it happened….my favorite sound in the world…”Ohhh, I see it.”  First there was one, one brave soul that walked over to the dark side and placed 3/3 on the number line correctly.  She even grabbed a ruler, measured the distance from 0 to 1/3, and proceed to place 2/3 and 3/3 exactly in the right place.  As soon as she began to measure, the kids got so excited as they SAW it.

At this point, I thought it was smooth sailing.  Next came the true task of placing 3/4, so the first thing they wanted to do was place 1/2.  Perfect.  I loved how they were using benchmark fractions, and I was sure that their misconceptions were suddenly gone.  Well….here is how is actually went down.1:2

The black line is where one student placed 1/2 and the red is where another tried to emphasize and defend that 1/2 would indeed be placed there.  I used all the skills I learned in my acting class and asked them to talk about what they knew was true.  The told me that 3/3 was the same as one whole.  We discussed and discussed.  Finally, they did come to realize where 1/2 should be and eventually 3/4, but not without a clear struggle.  These are not kids I teach everyday, but they are bright students with a wonderful teacher.  I don’t know if it was because they have only looked at fraction number lines with ending points of 0 and 1, but this experience reiterated the importance of exposing students to problems in multiple ways.  If they cannot transfer the information they learned from one type of problem to the next, then their understanding is basic and needs to be developed so much further.  I’m reading everything I can get my hands on…I want to be the best teacher I can be…I want to challenge, push, equip and help students make  a rich web of connections.

Feel free to pass on any wisdom you have.  It takes a village…

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Little Debbie Delirium– 3 Act Task

Little Debbie Delirium

A friend of mine posted this picture online and he was so thrilled at the loot he picked up at the Little Debbie discount store. I immediately saw an opportunity!!

 

Standards:  5. NBT.5,  5.NBT.7

ACT 1: Look at the picture. What do you notice? What do you question?

image

Possible questions:

How much was all of that?

How many cakes are there?

How many calories would that be?

*There are lots of additional questions that could be asked, but for fifth graders, these are the questions on which I would focus.

Have students estimate the answer to their question. Write your estimate. Write one that is too high and one that is too low.

 

ACT 2

Included in Little Debbie Loot:

24 boxes Swiss Cake Rolls at $2.50/box

6 big boxes of Oatmeal Cream Pies at $1.99/box

6 boxes of Banana Pudding Rolls at $1.50/box

10 boxes of Devil Cream at $1.50/box

1 box of Fudge Rounds at $1.50/box

2 boxes of Cupcakes at $1.30/box

 

Little Debbie Info

 

ACT 3

Have students present their solution. This is a great place to have students practice SMP #3: Construct viable arguments and critique the reasoning of others.

Do not skip Act 3 even if you think you do not have enough time! The bulk of learning and understanding takes place in ACT 3!

 

 

Have fun with this one! We would love to hear your experiences with this!

 

It’s True! Math Makes Sense!

I had been losing sleep. I had been wracking my brain. I kept coming up with NOTHING.

Even the shower, which is normally where I do my very best brainstorming, was not helping me with this ever perplexing problem.

How in the WORLD am I going to help my students understand division??

Oh, I had tried all the strategies I could think of. I had tried manipulatives, acting it out, actually having the students divide anything I could think of to try to help them “get it”. And they did. For about a minute or until they spotted a double digit divisor and then it was as if the concept of dividing had morphed into a calculus problem. I had a problem. And it was pretty huge. We work a lot with division in fifth grade.

And then….a life line. I hadn’t asked. It just appeared as if out of nowhere and at EXACTLY the right moment. Graham Fletcher came through with this post. What I love about this post is that now I have been introduced to Joe Schwartz and Nicora Placa. Two more awesome educators who share some really awesome ideas, strategies, and experiences aimed at helping elementary students grow into great thinkers, problem solvers, DIVIDERS. I’m pretty sure that isn’t a word, but who cares?? Do you know what they shared? This treasure!!!!

My small groups this week underwent a division intervention. As we went through the lesson, and we’re folding strips of paper, making quotients, and subtracting, inevitably, in EVERY group I heard a collective, “AAAAHHHH!” It was a gasp. The room got a little brighter. They UNDERSTOOD!!! One of my students said, “Oh. So THAT is why you subtract!”

One of my students who has struggled to understand math probably most of her elementary years, was in one of my groups. She diligently went to work. She was very quiet the first day, not sharing much, but always busy. She worked hard. The next day, as we went to groups, I met with those students again. This day was quite different from the previous day. The girl who had struggled, was now the teacher. She took over the group and taught division as if she had been born dividing.

Now THAT is an awesome week! I love it when my kids take over and teach one another! I love it when educators come together and share! So a huge shout of thanks goes to these awesome collaborators. Math really DOES make sense!

 

The Wizard of Oz – 3 Act Task

The Wizard of Oz

3 Act Task

CCGPS1.NBT.1-6, Standards of Mathematical Practice 1-8

ACT 1:

Ask Students:  What do you notice from the video?  Write down students’ observations.

What do you wonder?  Write down responses.

Ideally, you want to guide your students into identifying the problem themselves.  Make sure you validate ALL responses, no matter how silly they seem to you.  We like to write all responses down, on the board, on chart paper, it doesn’t matter where, it just matters that all students feel heard.  For true learning to take place, students must feel safe which means they are free to take risks, free to fail, and free to share.  The beauty of beginning a task this way is that every student has noticed something, which means ALL students have an entry point into the discussion.  All good tasks begin with an entry point for ALL students, not just a few.

Possible questions:  (Be prepared to have students ask ridiculous questions!   Acknowledge and validate all questions, but suggest that we put some questions in a parking lot since we can’t figure out those answers from the clip….and some would require speaking to the Tin Man himself 🙂 )

  • How much oil does it take to limber up the Tin Man?
  • Why are they using oil?
  • Where are they putting the oil?
  • How many squirts did they use? *** (This is the question on which we chose to focus.)
  • How many joints he have?

Have students make an estimate of how many squirts of oil Dorothy and the Scarecrow will use on the Tin Man.  Have them make an estimate that they know is too low and another that they know is too high.  Place these numbers on a number line.  Then have students estimate where they think the actual number of squirts would be on the number and discuss the reasonableness of their answers (SMP 6).

Act 2:

1618_TinMan75yrs_34

Give the students this picture, or pull this up on your interactive white board and ask, “How many joints are there on the Tin Man?” We came up with 14, but don’t feel like you are limited.  This is a great way to differentiate based on your class.  You could get very detailed and include finger joints, or you could keep it to 10 and only deal with multiples of 10.

Now you need to decide how many squirts you heard on the video for each joint.  At times, we heard 2 and other times we heard 3 squirts per joint.  Have your students decide what they heard and be ok with different groups having different estimates for the number of squirts per joint.

Ask:  Now that you know how many squirts Dorothy and the Scarecrow used on each joint and the number of joints the Tin Man has, figure out how many squirts they used on the Tin Man?

Act 3:

This is the reveal.  DO NOT SKIP IT!!! No matter how little time you think you have, do not skip this step!  This is where the real learning begins.  Have students share their answers and their thought process (how they got their answer).  Using your teacher superpowers, help students evaluate their work, learn from their mistakes, and critique the work of others. (SMP 3)

One idea we have found helpful is to post students’ work around the room and allow time to comment together – what we like, how we could improve our answer, and finally compare our work with others.

There’s a Method For That!

method-logo-1

     One of the changes I have decided to make this year in my math classroom, along with probably most of Georgia’s math teachers, is to really help students think through problems so as to be articulate with their thought process. Yes, this decision was SOMEWHAT inspired by the new Georgia Milestones assessment, but not entirely.

I have already been quite vocal about my passion for the 8 SMPs. Focusing on these has revolutionized my classroom. No one ever says they are done! EVER!  We received a new student last week from a county just outside of Atlanta. He is extremely bright! He began class by answering my bell work using just a number. That was it. Nothing else. And there he sat. The students seated nearby took one look at that and audibly gasped. It caught my attention. When I inquired the group as to the problem, the students quickly pointed out that “the answer isn’t good enough!” Inwardly, I smiled as I walked on. I didn’t have to say a word. My students understand that there is more to a correct answer than the number. They also understand that they are never done.

This knowledge, mantra, whatever you want to call it, came about because from day 1, we have talked about the 5 Finger Method of problem solving. I owe a huge amount of thanks to both Mike Wiernicki and Turtle Toms for their hand drawn rubric that I found in Mike’s virtual filing cabinet. I discovered,over the course of the first week, that when presented as it was, my elementary kids had a very hard time recalling all that they needed to include in constructed response questions. So, I made it a little simpler. But the premise is the same. I love that by starting with a “What I know” and a “What I need to Know” Tree Map, students of all levels know where to begin which provides an entry point for all learners.

My absolute favorite portion of the rubric is the “connections”. In my classroom this is where they connect the problem to other math by using the inverse operation and checking, but it can also mean “make a connection between the math you used here and your home or community”…it is SO fun to watch the students make those connections.

So, as a thanks to both Mike and Turtle, and in an effort to pay it forward, here is a 5 Finger Method pack. It includes the visual, a teacher rubric (for grading purposes), and a student rubric (in an easy to understand format for grades 3-5).

While there are other methods to solving constructed response math questions, such as “ACE”…my opinion is that you can “ACE” a constructed math question but yet fail to make connections and ultimately bomb on those standarized tests that require a constructed response.

I hope you find this to be as helpful in your room as I have found it to be in mine! Happy Labor Day!

Five Finger Rubric Slides

You Know What They Say About Assuming…

As a teacher, I have always known that connecting the curriculum to “the real world” is important, but I am just now beginning to understand what a HUGE difference it can make.  I have begun incorporating 3 Act Tasks and other similar activities in my classroom – and I have found that they not only reveal rather large gaps in student learning of the general content, but also in thinking.  Today fascinated me.  I chose to do Robert Kaplinsky’s task “How Much is a 1/3 a Cup of Butter.”  We had great discussion about how 1/4c could be right in the middle of the stick of butter. We modeled how you could take 2 sticks of butter and line them up, so that it would be easy to see that the 1/2c is at the end of one stick, we took time to point out 1/4c, we talked about what we knew about fractions, and we even took time to discuss what we noticed when we looked at the 2 sticks.

Let me stop here and tell you that I really thought I had blown the whole lesson.  I was thinking to myself that I taught them too much and now it would be way to easy.  After all, these kids are gifted…young, but gifted nonetheless.  So imagine my surprise when one student labelled 1/3 cup right after the 3 Tbsp. mark and then justified her answer by saying, “Well, I see 1/4 cup and if you take away one Tbsp. then you have 1/3 cup.  I  asked if anyone agreed with her answer and every kid raised their hand.  So next I had the student point to where 1/2 cup is.  (Remember, it’s labelled at the end and we already had a rather lengthy conversation about it.)  They agreed it should go after the 2Tbsp. mark.

I kept going with their line of reasoning, but I told them I was confused because we now had two different places labelled 1/2 cup.  I then asked which was bigger 1/2 or 1/4.  One girl immediately said 1/2 is bigger, then she drew 2 rectangles and showed 1/2 on one and 1/4 on the other.  When I asked her how she knew, she said her teacher told her last year.

That was a TREMENDOUS moment in our classroom.   I realized how students can know how to solve a problem on paper, but have no idea how it connects to real life.    See, I assumed that if they could determine what fraction was larger – that they would be able to solve this task.  Little did I realize, that their understanding had not progressed beyond solving naked math problems.  That’s a good place to be for a time, and there was even a time in my career when I would have thought that was perfect. But the reality is….they can solve our traditional problems while lacking understanding.  It’s our job to take them from doing math to understanding math.  It’s our job to provide rich opportunities, expose gaps and not merely assume they get it.  It’s an exciting time to be in education.

Number Talks

coffee        I am getting ready this morning to attend “Number Talks” training in preparation for the coming school year. As I am drinking my coffee, I am reflecting over  last year and my attempt at number talks. While my effort was commendable, the outcome was lukewarm. I have pondered for hours (collectively) over why that is. Why can’t our kids talk about numbers? Why can they only explain their strategy as being, “Well, I lined up the numbers in my head. I started in the ones place, and carried/borrowed….” or, “Well, I multiplied the ones and then put my place holder…” We have drilled this type of math for so long that this seems the only suitable/acceptable way of “doing math”. 

     This year, starting at the very beginning, we will focus on how to discuss numbers. Though none of my students are ELL, they still struggle with discussing in ELA class, so interacting with peers about numbers really trips them up. I found this blog post in my early morning web surfing session. Starting simple with what the author calls “string starters” (number talk starters) will help facilitate students in learning to talk about numbers.

So, posters…I need posters.

I think this will really help! What I like about these starters:

“My answer is________________________.”

“My strategy is_______________________.”

is that all students can start here. They will all have an answer whether it is correct or not. They will all have a manner in which they came to that answer. We are on a level playing field. Then we can make it more complex with

I thought about it this way: _____________________________________________________.

I agree with _______________ because _________________________.

I disagree with _______________ because ________________________.

I want to revise my answer because ____________________________.

I have to admit, the most exciting aspect of the latter string starters is the option to revise their answer! When they learn from a peer in a manner that helps them see a mistake or a missed step, that is HUGE! Students making their own meaning, their own connections of how to think about and work with numbers is what Number Talks are all about. And it’s ok to make mistakes. We learn best through them.

Another coffee-induced thought: No one ever mentions that you have to be fast to get the answer to a number string or a number talk. Somehow speed became synonymous with “doing math”. How did that happen? Think about it. How many school math improvement plans include speed drills?

But that is a post for another day.

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