Month: March 2015

Unlikely Students in Unknown Places

I recently got back from Santa Fe.  I was attending a conference there for a few days last week and afterward, I drove to El Paso to visit my brother’s family (he’s currently stationed in the middle east so I didn’t get to see him – unless you count face time) for a day before flying home.

Let me preface this story by saying that we all probably have a story similar this, but how we handle it can be a possible game changer.

Somewhere on my long drive, I stopped at a fast food restaurant to grab a quick bite.  So, I went inside and got in line.  The following outlines the beginning of our interaction as I stepped to the counter:

Cashier:  May I take your order?

Me:  Yes, please.  I’d like a number 2.

Cashier:  Large or medium.

Me:  Medium, please.

Cashier:  (after pushing more buttons than is conceivably necessary to enter my choice of “medium”): Your total will be $6.05.

I dug through my wallet (receipts from the trip and everything) and found that all I had was a $10 bill, so I handed it to her.  She entered $10.00 correctly and the correct change of $3.95 showed up on the little screen.  At just about that point, I remembered that I had a bunch of change in my pocket and said quite enthusiastically, “Oh, wait, I think I have a nickel.”  Who wants to carry around $0.95 in change in their pocket.

The cashier didn’t miss a beat, and said, “So, your change will be $4.00 even.”  I kind of smiled as I continued to look through my change, proud that she had a mental strategy to adjust to the situation and that she seemed quite confident and comfortable using it in this situation.

Unfortunately, I didn’t have a nickel, but I still didn’t want change falling out of my pocket into the depths of the rental car, never to be seen by me again.  So, I told her, “Oh, I’m sorry, I don’t have a nickel, but I do have a dime.”

As I handed her the dime, I saw her face morph from a confident smirk to a confused, almost terrified look of despair .  I had just taken her from a mathematical point of “Yeah, I can do this math stuff.  I may not use the computer for the rest of my shift” to “Holy $#!+, what the #=|| just happened!”

I went into math teacher mode and waited patiently for her to begin breathing again.  And then I waited for her begin thinking.  She adjusted my change with my introduction of the idea of a nickel, why not a dime? After what seemed like 5 minutes (probably more to her), it was painfully obvious to all around that her anxiety in this situation was taking over her ability to tackle this problem. So, I tried to think of a “least helpful question” to ask.  Now I put myself on the spot.  If she only knew that we were both now feeling some of this pressure.

So, I finally asked her my question and she gave me the correct change within a few seconds.  She smiled as she gave me my change and my new “unknown” student and I parted ways.  I know I felt good about helping someone develop a strategy outside of the classroom.  I hope she had a similar feeling about learning to make sense (no pun here) of making change.

Being a math teacher is a 24-7 job sometimes and we can find our students anywhere – even in a fast food restaurant in New Mexico!

What you would have asked the cashier in this situation.  I’d love to hear what your “least helpful” question would have been. No pressure, take as long as you like.  No one is waiting in line behind you!

Feed the hungry!

Oh, here’s my question:  “If you could change the dime into some other coins, what would you change it for?”

Math: A Fun After Homework Activity

All week long I’ve been asking Connor, my 9th grade son, what he has been working on in coordinate algebra.  Here’s a snippet of a recent conversation:

  • Me:  So, Connor, what have you been working on in your coordinate algebra class?
  • Connor:  We’ve been graphing.
  • Me:  Graphing what?
  • Connor:  Graphing different lines.
  • Me:  What kinds of lines are you graphing?
  • Connor:  Ummmm…
  • Me:  Are they linear functions.
  • Connor: Yeah, there are linear functions, but we also do curves…
  • Me:  Like what kind of curves?
  • Connor: Umm… exponents
  • Me:  Ok.  Anything else?
  • Connor: Umm…
  • Me:  Hey, I want to show you something. . .

Versions of this conversation happened several times this week.  Due to soccer practices, games, homework, and Life in general, we never got much past Connor’s last “Umm…”

Until yesterday!  The conversation changed a bit:

  • Connor:  We did something cool in class today.
  • Me:  Oh, yeah?  What was it?
  • Connor:  We had to build a picture using graphs of different lines.  We built a shamrock.
  • Me:  That’s what I’ve been meaning to show you all week.  Go grab my laptop.
  • Connor:  (playing game of war on an ipad) But I finished my homework.
  • Me:  Just take a look at this for a few minutes and see what you think.
  • Connor: (heavy sigh)

Enter Des-Man from Desmos.  Once he had gone through the tutorial, he was hooked. . . for a while!  He engaged in this for about 2 1/2 hours.  When he wanted to make something happen, but didn’t know how, he would come to me and ask.  We’d figure it out together.  The best part of this whole experience was when he realized he knew how to create something on his own and went to his math work from class as a reference.

Fast forward to 2 1/2 hours later, when Connor finished his Desman.

Des-man

 

 

 

 

 

 

 

To see the picture in detail along with the equations Connor used to create this graph, click Connor Face Graph.

It didn’t stop there.  I had some tabs open and clicked on one with the In-N-Out Burger task from Robert Kaplinsky.  He was curious enough to work through it even after all of the Des-man work.  So, I showed him more by clicking on the Open Middle tab (also from Robert Kaplinsky).  I selfishly pulled up the task that I wrote in collaboration with Graham Fletcher called The Greatest Difference of Two Rounded Numbers.  After making sense of the problem, and a lot of eye opening moments that led to phrases like “Oh, I can make it larger!” He got what he thought was the final answer and we validated his reasoning by clicking on the answer.  A slight smile!

So, we’re looking at close to 3 hours of after homework math investigation that ranged from rounding numbers to graphing equations, and solving problems.  Sounds like a great evening to me.  Great conversations and fun while learning and reinforcing mathematics understanding!  What could be better?  Talking Math With Your Kids – High School Edition.

Feed the hunger of all ages!

More with Connor:  Real Math Homework and Real Learning

Connecting Percents and Fractions

Not understanding mathematics can be extremely frustrating for students.  As a teacher, figuring out how to help students understand mathematics can be just as frustrating.  My primary go-to resource for these situations is Teaching Student-Centered Mathematics, by John Van de Walle et. al.  because it’s all about focusing on big ideas and helping students make sense of the math they’re learning in a conceptual way.

Recently, I was asked to model a lesson for a 6th grade class who was having difficulty working with percents.  So, I turned to my go-to resource, and during planning, I realized that I didn’t know anything about these students other than that they were struggling with percents.  So, I couldn’t assume anything.  I ended up creating three separate lessons and combined them into 1.

First, I handed groups of students a set of Percent Cards and Circle Graph Cards.  Their task was to match the percent with the corresponding circle graph.  As students were working on this, I heard groups reasoning about how they were matching the cards.  Many started with benchmarks of 25%, 50%, and 75%, while others started with the smallest (10%) and matching it to the graph with the smallest wedge.  As groups finished, they were asked to find pairs, using the matches they made, that totaled 100%.  Once finished, a discussion about their process for completing these tasks revealed a solid understanding of percent as representing a part of a whole.

Now to shake their world up a bit.  I asked them to leave their cards because they would be using them again shortly.  I introduced these Percent circles and asked them what they were.  A brief discussion revealed some misconceptions.  Some students said they were fractions, others said they were wholes because nothing was shaded.  I altered my planned line of questioning to questions that eventually led to a common understanding of what fractions were and how the pictures of the fraction (percent) circles really showed wholes and parts (fractions).

Their next task was to match their cards with the equivalent fraction circle.  This was incredibly eye-opening. Groups began to notice that some percent card matches could fit with multiple fraction circles (50% could be matched with the halves, quarters, eighths, and tenths).   Thirds and eighths were the last to be matched.  But their reasoning didn’t disappoint.  One group noticed that the percents ending in .5 all belonged with the eighths because they were too small to be thirds (the other percents with decimals).

Students were eager to share their thoughts about what they learned about fraction circles and percents:

  • Fractions and percents are the same because the pieces look the same.
  • 1/4 is the same as 25% and 2/8
  • I don’t get why the eighths end in .5.
  • The percents all can be fractions.
  • 1/8 is 12.5% because it’s half of 25%

Finally, I asked students to solve a percent problem (now that they’ve all realized that fractions and percents can be used interchangeably).   I gave them the m & m problem from this set of percent problem cards.  The only direction I gave was that they had to solve the problem using some representation of the percent in the problem before they wrote any numbers.

My bag of M&M’s had 30 candies inside.  40% of the candies were brown.  How many brown candies is that?

While this was problematic at first, students looked at their fraction circles and percent cards and realized they could use four of the tenths since each tenth was the same as 10%.  Most students needed just one “least helpful” question to get on the right track:  Where do the 30 m & m’s belong in your representation?

Most groups were able to make sense and persevere to solve the problem correctly, and explain why they “shared the 30 m & m’s equally among the ten tenths in the fraction circle” and why they “only looked at four of the tenths because that’s the same as 40%.”

My beliefs that were reinforced with this lesson:

  • We can’t assume understanding from correct answers alone.  We need to listen to students reason through problematic situations.
  • Students really want to share their thinking when they realize that someone is really interested in hearing it.
  • Students crave understanding.  They really want to make sense.
  • Procedures are important, but not at the expense of understanding.
  • Empowering students by allowing them to build their own understanding and allowing them to make connections allows students to feel comfortable taking risks in problem solving.

Please take a look at Jenise Sexton’s recent blog about percents with 7th grade students for some fantastic ideas about students using number lines and double number lines to solve percent problems.  It’s SWEET!

 

Empowering Students with In-N-Out Burger

The following is a reflection on a 3-Act task I modeled for an 8th grade teacher last week.  The 3-Act is In-N-Out Burger from Robert Kaplinsky and the plan I followed I completely stole from the amazing @approx_normal ‘s blog post on her work with the same 3-Act with administrators last year.

This past Thursday was the day we agreed on to model the lesson.  So, this group of 8th grade students, who have never even seen me before, are wondering who this guy is that’s about to teach their class.  And, just as planned, they were giving me weird looks when I showed them the first cheeseburger picture and asked them what they noticed.  I believe one of them even asked, “Are you a teacher?”

Fast forward through to the “What do you wonder?” piece and the questions were amazingly well thought:

  • “How much weight would you gain if you at that whole thing (100×100 burger?)”
  • “How much do the ingredients cost for it (100×100 burger)?”
  • How much does it (100×100 burger) cost?
  • “Why would someone order that (100×100 burger)?”
  • “Did someone really order that (100×100 burger)?”
  • “How long did it take to make the (100×100 burger?)”

There were just a couple more, and they all came up very quickly.  The students were curious from the moment we started the lesson.  They are still working on precision of language.  The parentheses in their questions above denote that this phrase was not used in the question, but was implied by the students.  We had to ask what “it” or “that” was periodically throughout the lesson as they worked and as time went on, they did become more consistent.

The focus question chosen was:

  • How much does it (the 100×100 burger) cost?

Students made estimates that ranged from $20 to $150.  We discussed this briefly and decided that the cost of the 100 x 100 burger would be somewhere between $20 and $150, and many said it would be closer to $150 because “Cheeseburgers cost like $1.00, and double cheeseburgers cost like $1.50, so it’s got to be close to $150.”  That’s some pretty sound reasoning for an estimate by a “low” student.

As students began Act 2, they struggled a bit.  They weren’t used to seeking out information needed, but they persevered and decided that they needed to know how much a regular In-N-Out cheeseburger would cost, so I showed them the menu and they got to  work.

I sat down with one group consisting of 2 boys (who were tossing ideas back and forth) and 1 girl (Angel) who was staring at the menu projected at the front of the room.  She wasn’t lost.  She had that look that says “I think I’ve got something.”  So, I opened the door for her and asked her to share whatever idea she had that was in her head.  She said, “Well, I think we need to find out how much just one beef patty and one slice of cheese costs, because when we buy a double double we aren’t paying for all of that other stuff, like lettuce and tomato and everything.”  The boys chimed in: “Yeah.”  I asked them how they would figure it out.  Angel:  “I think we could subtract the double-double and the regular cheeseburger.  The boys, chimed in again:  “Yeah, because all you get extra for the double double is 1 cheese and 1 beef.”  “Well done, Angel!”  You helped yourself and your group make sense of the problem and you helped create a strategy to solve this problem!  Angel: (Proud Smile)!

We had to stop, since class time was over.  Other groups were also just making sense of the idea that they couldn’t just multiply the cost of a cheeseburger by 100, since they didn’t think they should have to pay for all of the lettuce, tomato, onion, etc.

They came back on Friday ready to go.  They picked up their white boards and markers and after a quick review of the previous day’s events and ah-ha moments, they got to work.  Here is a sample after about 15 minutes:

IMG_0940

 

 

 

 

 

Many groups had a similar answer, but followed different solution pathways.  I wanted them to share, but I also wanted them to see the value in looking at other students’ work to learn from it.  So I showed this group’s work (below-it didn’t have the post-its on it then.  That’s next.).  I asked them to discuss what they like about the group’s work and what might make it clearer to understand for anyone who just walked in the classroom.

Shared Student Work

 

 

 

 

 

 

 

Here’s what they said:

  • I like how they have everything one way (top to bottom).
  • I like how they have some labels.
  • I’m not sure where the 99 came from.  Maybe they could label that.
  • Where’s the answer…

During this discussion, many groups did just what @approx_normal saw her administrators do when she did this lesson with them.  They began to make the improvements they were suggesting for the work at the front of the room. It was beautiful.  Students began to recognize that they could make their work better.  After about 5 minutes, I asked the class to please take some post-its on the table and do a gallery walk to take a close look at other groups’ work.  They were to look at the work and give the groups feedback on their final drafts of the work using these sentence starters (again, from @approx_normal – I’m a relentless thief!):

  • I like how you. . .
  • It would help me if you. . .
  • Can you explain how you. . .

Some of the feedback (because the picture clarity doesn’t show the student feedback well):

  • I like how you showed your work and labeled everything.
  • I like how you broke it down into broke it down into separate parts.
  • It would help me if you spaced it out better.
  • I like how you explained your answer.
  • It would help me if it was neater.
  • I like how you explain your prices.
  • I like how you wrote your plan.
  • I like how you explain your plan.
  • I like how you told what you were going to do.
  • Can you explain how you got your numbers.
  • I like how you wrote it in different colors.
  • It would help me if you wrote a little larger.

Some samples with student feedback:

Student Feedback 1 Student Feedback 2 Student Feedback 3 Student Feedback 4 Student Feedback 5 Student Feedback 6 Student Feedback 7 Student Feedback 8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Not only was the feedback helpful to groups as they returned to their seats, it was positive.  Students were excited to see what their peers wrote about their work.

Now for the best part!  Remember Angel?  As she was packing up to leave, I asked her if her brain hurt.  She said, “No.”  After a short pause she added, “I actually feel smart!”  As she turned the corner to head to class, there was a faint, proud smile on her face.  Score one for meaningful math lessons that empower students.

Please check out the websites I mentioned in this post.  These are smart people sharing smart teaching practices that are best for students.  We can all learn from them.