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!

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.

Four Fours Game 1-10

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.

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!

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!