Proportional Reasoning – Capture Recapture with Goldfish

IMG_6411You know a lesson is awesome when the Goldfish that the kids get to eat aren’t even their favorite part of the lesson!  I had done the Capture Recapture lesson a few years ago and it didn’t go very well, so I abandoned it.  However, after seeing a video about the concept (thanks to Alisan’s presentation at NCCTM), I decided to revisit it.

I introduced the concept with a question from NRich, then we watched this video on YouTube.  It is imperative that you show your students the video.  If a picture is worth a 1,000 words, a good video can be worth 1,000 explanations.  The video starts with ping pong balls, then moves on to black cabs.  I only showed my student the ping pong portion with my students, and then stopped it before he showed the math.  I had them calculate the estimated number of balls, and then showed them the result.  They were hooked!

We moved on to Goldfish.  I gave each pair of two their own “pond” full of about 150 yellow Goldfish crackers in a container.  I also gave them a small sample of colored Goldfish in a dixie cup.  I first had them estimate the number of fish in their pond.  I would not let them dump the fish out  for the estimation as biologist do not dump the fish out of the pond.  We then counted the colored fish (our tagged sample) and replaced Goldfish with the tagged colored Goldfish.  I let them eat the fish they replaced.  After mixing the tagged fish into their pond, they took a sample and calculated the proportion.  To have accurate results, we repeated this 4 total times and then took an average.  After they calculated their average, they counted their fish and we compared results.  This was the best part!  They were shocked to see how close their calculated proportions were to the actual number!  Most groups were only off the actual count by 10 or less Goldfish!  I even  had groups come within 1, 2,  and 3 of the actual number of fish in their pond!


  1. Estimate number of Goldfish in pond and record.
  2. Count the number of tagged fish, record.
  3. Replace Goldfish with tagged fish.
  4. Mix tagged fish into pond.
  5. Take a new sample.  Count total sample and tagged fish, record.
  6. Calculate proportion to find estimated number of fish in pond.
  7. Repeat this three more times.
  8. Find an average.
  9. Count actual fish in the pond and compare.


  1. Pdf – Capture Recapture Data Sheet
  2. Powerpoint – Goldfish Proportions Capture Recapture
  3. Slideshare File (same as ppt, but in slideshare)

Goldfish Capture Recapture Worksheet

Algebraic Expressions with 6th Grade

At Twitter Math Camp, Nicole Paris introduced us all to the “orangamallow“.  Wow.  So simple, yet so brilliant.  How many times have I SAID, “a + b doesn’t make an applebanana“?  But Nicole took it one step further and SHOWED this to her students, bridging the gap from the concrete to the abstract.  Never underestimate the power of manipulatives, or visuals.

My students loved this so much, and got so excited about it, that I even recorded them today.  This is big for me because I have never recorded myself teaching.  But, I wanted Nicole to see how her idea inspired my class.  THIS is why you all should be blogging and sharing, even if you don’t think you have that much to add.  OR, if you think that what you have to add is obvious and couldn’t help much.  Please find the time and blog your lessons!  Just watch my students, who got so excited when I brought out the bag today that I was instantly sad I wasn’t already recording.

I also didn’t really want to record myself, so I hid behind the table as much as I could.  I should record my lessons more as I noticed so much that I can improve upon in just one short video.  Wow.  I noticed that I talk way too much, no surprises there.  But, when I talk too much, I say stupid things.  I actually said algebraic equation instead of expression twice.  Oh my.

Kinesthetic Algebra – Making Abstract Variables Concrete

If you haven’t signed up for Explore the MathTwitterBlogosphere initiative, YOU SHOULD.  Because, as Tina said best,

No matter how much time I spend online, I’m always happening upon new parts of our awesome math teacher community. Even yesterday, when we were planning the first few missions for this oh so exciting event, I learned about some sites I didn’t know existed. This is exactly why I am inviting you to join me on an exploration of the best parts of the internet – the Math Twitter Blogosphere.

Tina Cardone

Yes, this exact same thing happened to me on Sunday.  I got this idea from Emma McCrea.   I tweeted her immediately to tell her how much I enjoyed her lesson, and she tweeted back that she has signed up for Explore MTBoS!  Win-Win!

I gathered spaghetti, yarn, and dental floss cut into different lengths, colored index cards cut into squares and rectangles, foam cubes, and then more of these pieces taped to other pieces.  I dumped them into five bins.  This was actually pretty fun for me.

Notice and Wonder:

When the students came in, I gave each table group a Notice and Wonder form and a bin of stuff.  I told them nothing except, “Fill out the Notice and Wonder form with your group.” We have done Notice and Wonder many times before, so they got right to it.


They were FABULOUS.  Groups had great questions for me such as, “Is this 2D or 3D?”.  They noticed the different size shapes, lengths, and combinations.  They wondered if they were 1D, 2D, 3D or all 3D?  They wondered if they combined shapes were 4D.  Many groups even started sorting them.  Great idea.


I then had the groups sort their objects however they wanted.  I did get a couple of interesting sorts, but almost almost all groups sorted the same way.  So they could see what I was seeing, I had all of the groups walk around and look at all of the sorts.  My crazy sorters were able to see how most of the groups had sorted.



After sorting, they shared their notice and wonder.  I explained to them that ALL of the objects were actually 3D, but we were going to look at them as 1D (length), 2D (flat paper), and 3D (3D objects).  At this point, the kids all said, “Like Flatland!” and a connection was born.

After that I drew pictures of the objects on the board, starting with 1D and moving up to 3D.  We talked about how the lengths of all of our green string were close, but different sizes, and how we can use a variable to name an unknown length.  To make this more interesting, I let them name the unknown lengths.  This was a HUGE hit, and they picked funny names like Wormy (w) and Doofy (d).  As we picked more lengths, it was a great chat when they wanted to name a new length Wanda (w), but could NOT because we already had a w and it was NOT the same length.  It was a blast!

When we had progressed all of the way to the cubes, the kids noticed a pattern.  They noticed that the variables for the 1D objects had 1 as an exponent, some 2D objects had an exponent of 2, and some variable of the 3D objects had an exponent of 3.  I love it when magic happens.

To end up the first day, we discussed the difference between wormy + wormy and wormy times wormy.  I held two wormy strings together to illustrate addition of two lengths.  I also drew a 2×3 multiplication array to illustrate multiplication and make sure they were getting the difference.

They seemed to get it (and seemed to enjoy it).  Tomorrow, I am going to let them cut their own lengths from spaghetti and draw their own 1D, 2D, and 3D shapes.

The next day, I will do Emma’s great sorting activity!  It is at the bottom of the page.  I will let you all know how it goes!

Snowball Fight Icebreaker

I have never needed to lead an icebreaker – until #TMC13 when I facilitated the Middle School Math Morning Sessions.  I teach at a small private school.  Almost all of our kids have known each other since early elementary school.  The class size goes from 16 to 32 from 5th to 6th grade.  However, they have tons of bonding time and activities together before they go to math class so I don’t really need to do an icebreaker.

I was nervous about leading the MS Morning session, especially since we had almost 40 people signed up!  Several participants had told me they were nervous too so I wanted an icebreaker that would be fun and get us moving.  Since there were too many people in the room to do musical chairs, we had a snowball fight instead!  This was my first time every doing a “Snowball Fight” but I think it turned out pretty well.  I sure had fun and people were laughing, so I think it made us all loosen up a bit that first morning of TMC.  I wish I had taken a picture!

SNOWBALL FIGHT! – How to Play:

  1. Don’t tell them you are having a snowball fight!
  2. Give everyone a half sheet of paper.  You should play too!
  3. Have everyone write their name and three things about themselves on the paper.  You can let them write whatever or chose things for them to write about.  @pegcagle suggested two truths and a lie, which incorporates TWO icebreakers!
  4. After everyone is finished, tell them to crumple their paper up.  You will get strange looks.
  5. Then, tell them to have a snowball fight!  We kept picking up the paper and throwing it for a few minutes.  I would suggest this as the first throw is someone curious/skeptical/reserved and people have more fun as they throw more.  Don’t let them throw too much or it could get crazy.  😉
  6. Call cease snowball!  Then have everyone grab a snowball.
  7. I start to model what to do.  I called out the name on the paper and then read their three things.  Then they read and so on.

** Warning – You may lose a snowball!  We did!  In that case, you will have to improvise.  This ended up funny and then even more funny when we found their snowball later.  I would tell the lost snowball that their snowball has magic hiding powers and then offer them a treat so they won’t feel bad.

Variations – 

  • You could leave the name off and let students guess who it was after reading the things?  I’m not sure I like this as much though bc I really liked looking at the people when they were being talked about.  However, it may be less embarrassing for students if people aren’t looking at them when their items are read.  Thoughts?
  • It would be fun for everyone to get in a line according to whose snowball they have.  Then they would all have to ask names and talk to each other (if they didn’t know each other) in order to line up.
  • I think this could also work really well as an activity instead of an icebreaker using math questions.  Students could make up a math question, have a snowball fight, and then solve the problem on a snowball they pick up.  They could then get with the person to see if their answer’s matched.

I have only done this once and never in a class of students.  I think that it would be fun as an activity in addition to an ice breaker.  I would love to hear any variations that you have done – or can come up with – in the comments!

Quadratic Frames – Totally Nguyening

If you teach middle school math or Algebra 1 and you are not reading Fawn’s blog, then you should.  I get all of my ideas and inspiration from her!

She posted a great activity about Quadratics and framing.  Please go read her post for instructions.  I wanted to post how I modified her activity for Pre-Algebra (using simple factoring instead of the quadratic formula).  I followed her instructions exactly, and modified four things.

  1. I teach Pre-Algebra so I made the problem easier for my students so that the numbers would be easily factorable and not need the quadratic formula.  As a result, my students were more easily able to discover the frame dimensions (sadly, they did not beg).  It did take most of them quite a while however and they appreciated when I finally showed them the math.
  2. I had my students draw their own picture on a 3×5 index card.  They could also take a picture and bring it in.  This way I didn’t have to find a picture for them, or print it and make copies, or cut it out.  (Yes, I’m lazy and they love being creative so it was win-win.)
  3. I used index cards so I wouldn’t have to cut up a bunch of paper to the perfect size.  Really Fawn, you are a saint.  I had my students draw the picture on 3×5 index cards, then I used 4×6 index cards for the frame.  I cut them down to 4×5 to make the border an even 1″ (see #1).
  4. I had them post the finish product on a half sheet of paper.  This way they could glue down the picture and the frame instead of using tape.  I am always short on tape.  Also, on the back of the picture is where we all “did the math”.

I am posting mainly to say THANK YOU for Fawn, and to share my students creations!  I am really into notebooks so I also included an idea for adding the work to the students notebook.

The Process:

The finished product:

On the back of their picture, we did the math.


The Notebook Entry:

Notebook Entry

It's like a foldable!

It’s like a foldable!

Oreo Cookie Math – Double Stuf Steals the Show

When I saw Dan’s post on Oreo cookies I knew this was meant JUST FOR ME. It was especially interesting because Chris Danielson in part 1 2 and 3, and Chris Lusto in part 1 and 2 also investigated it. I had to jump into the fun. I mean, WHO doesn’t love Oreos?

I have math club once a week and always bring them treats to eat. I’m not below bribery. So, when I saw Dan’s post I thought, “Awesome! I can feed Math Club AND make the food about math at the same time!” #Nguyen for sure.

Woodlawn Math Club

I pulled the four 4 packages of Oreo cookies I had bought to the “Oohs and ahhs!” of my math club. My 16 kids did the math in their heads instantly, calculating how many Oreos each of them were going to get to eat today. I opened the packages and pulled one out from each, original, double stuffed, chocolate, and triple stuffed. (Side note: I was very disappointed I couldn’t find a mega-stuffed package at the grocery store because I KNEW they would have gone nuts over that.) I then lined the cookies up next to each other. “Any Questions?” I asked as I handed them each a post-it note.

Excitement buzzed through the room as they posted their questions up on the board and then we sorted them.
IMG_4905 IMG_4914IMG_4911 IMG_4913After talking through all of the questions they decided to work on “Is double stuf REALLY double the stuffing?” I offered to let different groups work on different questions but double stuffing won out! Then, they brainstormed how we could find the answer to our question. Here is what they came up with.

IMG_4899They then decided to break up into teams. Even though there were four teams, it ended up being girls vs. boys. The girls decided to weigh the wafers and the stuffing and the boys decided to build a double stuffed Oreos from the stuffing of two single stuffed Oreos.

The room was a bustle of activity as the students went to work. I was the “official” weigher so I could ensure all of the results were consistent for a good comparison. All groups decided to weigh and measure several cookies (or cookie parts) so that we could get more accurate results. I was very busy with all of the weighing so unfortunately I do not have pictures of this stage. But I promise, it was fun, messy, and mathy!

After they were all finished they shared their results with each other. It was great because all four groups came to the same conclusion. Based on weight, double stuffed Oreos are MORE than double stuffing compared to regular Oreos!


In the end I videoed their conclusions, one representative from the girls side and one from the boys. Unfortunately, my iPhone went wonky and did not record the boys conclusions. this was tragic as they had made their own mega-stuffed Oreo and even had an Oreo dance! Ah, technology!

Thanks so much to the MathTwitterBlogoSphere for making my job so easy! I don’t even have to come up with the ideas – they come to me in my Twitter Feed and GReader (RIP – sigh).

** Edited to say Double Stuf – one f.  Seriously, who knew that?

Life Sized Human Graphing

Today I did a lesson with 6th grade called Linear Graphs:  Life-Sized Coordinate Pairs  that I found on the Teaching Channel.  Definitely go there and watch the video.  It explains it in great detail!

To do Life Sized Human Graphing put a giant coordinate graph on the floor and then give the kids a rule.  They must come up with an ordered pair that fits the rule and then go and stand on their ordered pair.

I taped a coordinate plane on the floor with painters tape.  I also had each student wear a dry-erase necklace to write the ordered pair on so everyone could see their ordered pair.  They kept saying, “Does my ordered pair work?”   I kept saying, “Go graph yourself and see!”.  The class stayed engaged the entire time, and were even “overly engaged” arguing about the math at points.  I love that. After the first few examples I let them ALL come up and get in line instead of just four or five.  I did this mostly because they begged me.  They couldn’t wait to get up and see if their point fell on the line!

Whole Body Graphing ispeakmath

I used Max’s “noticings” in this as well.  They first noticed that they were all in a line.  Then they noticed that the line was steeper when it was 2x or 3x.  They noticed that 2x made the y-value go up by 2, 3x by 3, and so on.  The connections that were made today were amazing and I hope that they carry over tomorrow when I introduce slope (aka rate of change)!

I also made a powerpoint so that the game would be easy to manage.  Just click to follow the link and download the file from the Math Wiki.  After we start graphing linear equations I want to try Graphing Full Body Style!

Life Sized Graphing ppt


Graphing Linear Inequalities – Day 2 and Foldable

I started graphing linear inequalities at the end, with our Knittng Business.  For day two, I gave them a foldable that we went over.  This day we worked on graphing the lines, then picking and testing their “test points” in class.  It went very well as I could refer back to the knitting example with almost every question.

7th – Graphing Inequalities Foldable pdf

Screen Shot 2013-02-22 at 2.40.25 PMThe next day I wanted them to learn how to write inequality equations.  I used Chick-Fil-A for this example because they have all of their prices online.  I showed them this slide and set them to work with a partner.

Once they came up with their food and drink combinations, I had them go to the board and enter their information onto a chart.

IMG_4797 Once their information was all on the board, I asked them to try and write an inequality for this information.  Some groups got it right away.  For those that struggled, I gave hints.  First, I put the <= 15 on the board between chicken and total.  That was all most groups needed.  I then asked them HOW they calculated their total.  Almost every student was able to write the equation.  It was fantastic!   I think that the table really helped them visualize the equation.  I had not put in the total column before today – but it was key! IMG_4798


After we wrote the equation we went to and entered the equation with the shading.  They love that.

Screen Shot 2013-02-21 at 2.07.22 PM

After that, we added a table and then entered all of their food and drink combinations!  They noticed that all of the combinations that we entered were actually “test” points.  They also saw what it really meant to use a test point and why it was necessary.  One group in each class had a wrong answer so they even got to see what happens when you shade one side and then pick a combination that doesn’t work!  It was another day of “math-a-magic”!

Screen Shot 2013-02-21 at 2.11.06 PM Screen Shot 2013-02-21 at 2.11.35 PM


MathCounts in Hickory, NC

I’m at MathCounts in Hickory, NC today.  I brought 7 6th graders and 3 7th graders.  My team consists of four 6th graders.  None of these students are even in Algebra.  But, they have worked so hard and are so excited about coming today!  None of us could sleep last night.  I don’t know how they will do since they are so young.  These problems are tough for them.  They were all quite intimidated when they saw that everyone but them has a graphing calculator.  But, none of that matters.  They are fabulous, brilliant kids!  They are excited to be here and I am so excited for them!  Best of all, they requested that we continue math club even after MathCounts!  Win for math!