# “I Feel So Accomplished!” – Problem Solving, Noah’s Ark

“I feel so accomplished!” is what I overheard a student say after doing the Noah’s Ark problem solving activity in my class today.  I also heard, “You will REALLY like math class today!  It was so hard, but so much fun!”

I gave them the Noah’s Ark problem (thanks Fawn)!  I followed Fawn’s suggestions for problem solving because she is the expert.

1. Give them all their own copy of the problem.  (And a sheet of animals to cut out if they wanted it).  Noah’s Arc PS
2. Let them read it silently
3. Let one of them read it aloud while other students read along silently.
4. Let them work silently for a few minutes.
5. Let them work together.
6. Discuss solutions.

While working silently, many students came up with a solution.  However, once they started working with others, they realized that their solutions may not be correct.  I loved how they were explaining their solutions to their classmates, and their classmates would say, “Oh, but then you would have too many zebras.” and I would hear the, “Ohhhhh…”  My favorite had to be “But WHAT is kangaroo?  WHAT ARE YOU?”  Several students asked for another copy because they had written all over theirs and wanted to start again.

Some students let the animals equal animal values.

Most students had to work together to come up with the correct solution.  A cluster of my students decided to assign point values to some of the animals, and variables for others.

This was an altered version of substitution, and so interesting for me to see.  I loved how they molded the problem to fit their level of understanding.  Their minds work in fascinating ways!

Student Solutions:

Here the animal cut-outs are used to visually show the substitution.  This was helpful for the students who were not convinced by the algebraic solution.

Noah’s Arc PS

# Proportional Reasoning – Capture Recapture with Goldfish

You 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!

Procedure:

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.

Files:

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

# Problem Solving – MS Sunday Funday

For the next eight weeks, MS Sunday Funday will be participating in Explore the MTBoS!  Instead of blogging about MS Sunday Funday topics, I would like to encourage everyone who currently blogs (or would like to start blogging), to come and Explore with us!

October is problem solving month at Global Math and they have many great speakers this month.  Be sure to check it out!

In honor of Global Math, this week’s topic was Problem Solving.  A special thanks to Beth who so consistently participates each week in MS Sunday Funday!

# Problem Solving Webinar – Global Math

Tonight on Global Math, we will be discussing problem solving and our favorite problem based lessons and tasks.  I will be presenting, along with Alisan Royster and Justin Aion.  We would love to see you there!

To see the recorded webinar, click here to watch it on the Global Math site.

# Bedtime Math for Math Club

Sadly, my math club has been cut down to 25 minutes this year, and that includes students eating lunch.  So, I need fun, engaging activities that are shorter in nature and are great for both 6th and 7th grade students.

Today I decided to try an interesting problem I saw in my email this morning from Bedtime Math.

I like these problems because not only are the brief, interesting and relevant, they have multiple questions, depending on your level of student.  For us, all of these questions made us think of even more questions!  My students extended the question to ask, “Do you make more money doing this study than working a minimum wage job for the same time period?”

They went right to work and loved it!  We talked about the minimum wage amount (\$7.25), and how many work hours were in a typical week.  We also had an interesting decimal discussion when one group of students divided the 10,000 by 70 days but then ROUNDED , coming up with only \$994 a week instead of \$1,000 per week for the study participants.  After we all decided that you make much more money over 70 days for the study, one student pointed out that you are actually making less per hour since you are really working 24 hours a day in the study.

I have amazing math club students that make every day a joy!  Thank you so much Bedtime Math for your amazing free resource that helps me give them interesting and relevant problems!

# Four Fours Problem Puzzler/Game

Problem solving has been the best thing that I have incorporated this year!  It’s more than one good thing.  It is actually many good things that happen every day that I MAKE the time to do problem solving with my students.  And yes, I do have to make the time.

The most important thing that I learned from this activity is that I also need to have students do individual problem solving, so that each student will learn to persevere in mathematics, and not rely on a group member.

I first heard about the Four Fours Problem while taking Jo Boaler’s “How To Learn Math” class this summer.  It sounded like a fun activity and I couldn’t wait to try it with my classes.

After doing Fawn’s PEMDAS Relay earlier in the week, I knew we still had some work to do with the order of operations.  I decided to give them the Four Fours for their problem solving brains.  However, I also wanted their work checked and corrected – but not by me.

Enter the Four Fours Problem GAME.  I reduced the assignment to make the numbers 1 – 10.  I explained the Four Fours, and then had them brainstorm answers for the number 1.  I then showed them 44/44, and they loved that!  Then, they had to fill out the top half of a sheet that I made with the remaining Four Fours problems.

After completing their Four Fours, I had them randomly assign the letters a-j to each problem.  Then, they wrote the problem, but not the solution, on the bottom half of their worksheet.  Once that was complete, they cut the sheet apart and swapped with another student.  That student then worked out their problems to see if they could find which number it equaled.  If they did not get a number between 1 and 10, or ALL of the numbers 1 through 10, someone made a mistake.  Then, they were to get with their partner and figure out who made the mistake.

When I first introduced this problem, there were many groans around the room.  I am not used to this so I was a taken a bit off guard.  They did not think they were going to like this lesson.  “Do we HAVE to use four fours?“, “Why are we doing this?“, and “Really?”  I was worried, but, I had planned this lesson so I plowed on.  After a few minutes however, my students were totally into it!  All it took was a friend to go, “I found five!” to get their competitive juices flowing and see this as a challenge.  They worked so hard!  After they started coming up with their problems, they really got into it.  I even had to quiet them down several times with “Other students cannot think if you are screaming!”

Not every student was ecstatic however.  They have been doing problem solving in groups so far this year.  Since they had to trade papers, their problems had to be kept “secret”, so this was an individual assignment.  It was frustrating for several students.  They wanted to quit, constantly wanted me to check their work, or kept begging me to give them “just ONE hint for the number ____ !!!” They even begged me to let them switch solutions for just one number with another student.  “I need two and she needs five.  It’s just one number!”   “NO, you CAN do this!” was my answer every time.

Overall, it was a thrilling success.  At the end of class, I had several students ask me for an additional copy so they could take it home and give it to their siblings or parents, or come up with more Four Fours.  One student wanted to come up with negative answers.  This would actually be a great twist for 7th grade!

After doing this activity in class, I changed the Four Fours sheet to include instructions for the students and I made it a whole sheet of paper so they would have more room to work.

Side Note:  This is more of a puzzle game than an actual competition and students weren’t really racing each other to finish first.  Of course, some students will finish this much more quickly than others.  That actually works out very well.  As they finish, they trade and work each other’s problems.  Then, they work on something else while the other students finish.  I use this time for in class project work but you could also have a row game ready for pairs who finish more quickly.

# Pizza + Linear Equations = Fun, Free Mathalicious Lesson

Every kid loves pizza, right?  Well, my students do, and they LOVED their first Mathalicious lesson, Domino Effect.  The best part is that it is one of the sample lessons from Mathalicious so you can try it for free!

This lesson is intended for students in Algebra 1. However I did this lesson with 6th grade students who had not learned linear equations yet. After introducing the lesson, I gave them the student handouts. The student handout was self-explanatory and easy for my 6th grade students to follow.  I gave each student their own set of handouts but let them work in pairs so they had someone to talk through their work with.

I appreciated that there were several related questions.  This made students aware that they had made a mistake in a previous question so they knew they had to go back and rethink it.

The questions were leading enough so that my students (who had not written linear equations yet) were able to summarize their findings.  I was pleasantly surprised that several students even came up with an equation.

Many students ran into trouble in Act 2 on question number 2, when they assumed that the medium topping price was the price of toppings for all pizza sizes.  The layout of the chart alerted most of these students that they had made a mistake when they noticed that a 2-topping small pizza was \$10.48, but a 3-topping small was only \$10.99.  Other students discovered this when graphing the points.  I noticed that some students did not notice this discrepancy at all, but and let them continue to fill out the chart incorrectly.  To help students discover their mistake, I asked them if they could figure out how much a small, medium, and large 10-topping pizza would be.

Some students went straight to their calculators.  Other students wanted to use their graphs but were not sure that they were allowed to extend the lines.  This was another opportunity for discussing lines and patterns.  They loved this “shortcut” and that it quickly helped them price their 10 topping pizzas.

After all of the groups had come up with their 10-topping pizza prices, I drew a large chart on the board and had the groups fill in their prices to illuminate the differences.

Once they saw that other students pizza pricings were different, they had a heated class discussion about who was right.  “Proving” their calculations to each other helped all of the groups realize that the topping price was supposed to be different for differently sized pizzas.  After everyone had time to find and fix their mistakes, they updated the board chart to make sure everyone was now on the same page.

I liked having them find the price of a 10-topping pizza because it prepared them for the big reveal.  My students pretty much lost it when they saw Domino’s actual pricing, since it was so much less than they had calculated.  They all wanted to fix their graphs, which of course, I was thrilled about, so I let them.  Notice the key at the bottom on the picture.  (That is supposed to be thought).

I loved this lesson because it was fun, self-explanatory, and easy to implement.  It was written so that I was able to walk around the room and support students that needed extra assistance or were going down the wrong path.  I did not “teach a lesson”, but at the end of Act 1 several of my students had figured out the equation by going through the questions.

This can be done with younger students, even if you haven’t taught linear equations yet.  In fact, this is a great introduction to the topic of linear equations.  Next time I do this lesson with 6th grade students, I will do this over two days.  The first day I will do Act 1, and then finish with the chart on Act 2.  Then, on the second day I will explore Act 2, questions 3 and 4 in more depth.  Only a few students were able to come up with the equation in Act 1, so I will need more in class time to help them discover this in Act 2.  I anticipate that I can teach students about writing equations of lines and even explore what the y-intercept really means with this lesson.

Pictures of my students finished projects:

PIzza Project Rubric

# Factor Craze – Success With Problem Solving

The first week of school I tried out Factor Craze on my students.

It’s great timing for the beginning of school as we are reviewing prime factorization, exponential notation, and exponents in Pre-Algebra.  We had just finished working on the Sieve of Eratosthenes the day before, and this proved very helpful for the problem.

I made this slide to show them the problem, put them on the giant whiteboards, and off they went!

I gave zero directions, and most students started out the same way, listing the numbers that go with each rule.  I let them work and work.

During this work I had some questions and heard many great comments.

• Does one count as a factor?
• There has to be an easier way to do this.
• Guys, let’s look for a pattern.
• There’s got to be a pattern here!

There were having an impossible time finding any pattern (well, any pattern that actually worked).  They were very creative however, and came up with many patterns that worked for some numbers.

Finally, I couldn’t take it anymore and asked the class, “What have we been working on the past couple of days?”  “Prime factorization.” they answered.  I gave them the hint, “Well, maybe you should look at the prime factors of all of those numbers you have listed.  They looked at me doubtfully, but were happy to have any hint at this point in time so they went to it.

Not long after, students actually started screaming, “I found it!  I see a pattern!”  Well, they thought they had found part of a pattern, but weren’t quite there.  I told them how excited I was that they had found a pattern!  Then I asked them to test the next number and see if their pattern worked!  “Oh no! The three’s rule doesn’t work for 4! Maybe it’s all ODD numbers!”  After they tested AGAIN, “Oh no, it doesn’t work for 9!”  It took each group a while, but eventually they all discovered the rule for exactly three factors.  I encouraged them to write down their rule in words.  Then, I asked them if they could make it more, efficient or algebraic.  They were all able to make “a prime number squared”  into p^2.

Once this happened, they were inspired!  Most of them quickly discovered that this worked for each question.  And then I said, “That is sooo awesome!  Hey!  Look at your 4’s, does your rule work for all numbers listed?  What about 6, 10, 14, 15,… ?”  I was very excited when someone asked me if there could be more than one rule that worked, because I really didn’t want to give them another hint.

At the end of the class, all students had discovered at least one rule.  I had them all take notes of what they had done on the giant whiteboards into their graph books.  For homework, I told them that I wanted them to continue to work on the problem, and see if they could discover any more rules.  Then, I told them that they could work together tonight, with anyone they wanted in both 7th grade classes.  At that point, students started making plans to chat on Skype and Google + that night.  I’m not going to lie, that was a pretty exciting moment for me.

The next day they all came in excited to tell me the new rules they had found.  I made a chart and posted it up on the board, and let all of the students come and write their rules.

The students had discovered all of the rules, although they did need just a small bit of questioning to help realize that they couldn’t write prime times a different prime as prime squared.  Luckily, they remembered their subscripts.  So, we used p sub 1 and p sub 2.

After they finished pulling all of their ideas together, I went over each rule again as a summary.  This helped clear up remaining questions for some students.  Problem solving for the win!

# 7th Grade – Problem Solving Goal 2013-2014

Much of my teaching is discovery lessons, interactive games/whiteboards, and project based learning.  This year, I want to add problem based learning to the mix.  I do a little of this now, but not nearly enough.  This year, it is going to be a focus, especially for 7th grade.

7th grade has always been tough for me.  I teach them two years in a row, so I’m not new to them anymore.  Plus, the math becomes more abstract as we segway into algebra.  I am trying to make 7th grade math just as (if not even more) exciting than 6th grade math.  As 7th grade math is an extension of the topics they learned in 6th grade, I also want to be sure I am challenging them, and that they are growing.  I changed homework workbooks again this year (third time in four years).  I change up the projects every year.  Sometimes, we hit gold and I find something they really love.  But 7th grade can be a tough crowd, and lessons that thrill 6th graders can fall flat in 7th.  So, I’ll pretty much do anything to get them excited about math!

This year, over half of my rising 7th grade participated in my Math Club last year.  They willingly gave up their lunch and recess time (even in the last weeks of school) to learn more math!  They loved playing with the pythagorean theorem and even the quadratic formula.  They gave up an additional recess to help me record the Draw-It game.  And, they had a fit over the Oreo question.  This class loves a challenge and I believe they are ripe for problem solving.

I am excited about my plan for this year and I hope I am up to the challenge!  I will keep combing the MathTwitterBlogosphere for resources.   I plan on starting with the Math Forum’s Problem of the Week, everything Fawn says, Exeter Problem Sets, NRich, and incorporating Mathalicious material.  There is also a Middle School Problem Solving page on the TMC wiki.  Right now it is pretty empty, but I plan on filling it up as I find more resources.

If you have a favorite problem solving site (or even a favorite problem).  Please put the link in the comments!  I could use the help!

# 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.

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.
After 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.

They 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?