A Multiplication Based Logic Puzzle

Magic mirror on the wall.
Am I teaching one and all?

Teachers reflect. They often ask themselves how their lesson went, what went well, and how they could improve.

Many years ago I taught an algebra class. The textbook suggested I use algebra tiles to teach a lesson on adding negative and positive integers. I had never heard of algebra tiles before. The school didn’t have any, and there wasn’t time to order some online. Later that September day, I looked at some Halloween candy in a store. When I saw a package of Pumpkin Mix m&m’s, I knew I had found the perfect algebra tiles. All the m&m’s in the package were brown or orange and had pumpkin faces with an “m” for the pumpkin’s nose, but this is how I saw them:

The algebra students learned about adding and subtracting positive and negative integers without any problems, and they LOVED it.

Pumpkin Mix m&m’s have been replaced with other varieties. The colors don’t matter. You could have the sides with the “m” be positive and the side without the “m” be negative.

That summer I enrolled in a Teaching Secondary Mathematics class at the university. I needed to do some volunteer work in a school, reflect on the experience, and write a paper about it. I share that slightly edited paper with you today:

I worked with Mark’s classes. Shon and Serena volunteered there as well. Mark’s students are adults many of which are learning English as a second language as they prepare for the GED test.  Every student I observed was motivated to learn.  One of the students struggled with basic addition facts.  Another understood the concepts but wrote the symbols for algebraic sentences in a different order than we use.  Most of the students are learning Pre-Algebra concepts and getting individualized instruction from the computer program “Classworks.”  If students don’t pass a pretest, they can read a brief lesson on the computer, use some virtual manipulatives to learn the concept, and demonstrate what they have learned.  Several students worked on a lesson that required them to solve for x by balancing equations.  Some of the students seemed confused.  I thought it might be helpful if they had physical manipulatives that required them to do the balancing rather than the computer.  Mark allowed me to plan and prepare a lesson for the twelve students in his first-period class.

I typed and printed a worksheet that consisted of four equations and two large rectangles.  I purchased twelve 1.5 oz packages of Reece’s Pieces to use as Algebra Tiles.  The empty packages represented the variable “x,” each orange candy represented “+ 1” and each brown (or yellow) candy represented “-1.”  Shon and Serena also helped the students understand how to use the manipulatives.  I thought the lesson would only last about five minutes, but it lasted the remainder of the class period.  The students did well with the activity, but it would not have gone so well if my fellow students were not there assisting mostly because of English language issues.

Mark asked me to teach the lesson again to his third-period class.  He even bought more candy so I wouldn’t have to.  His third-period class had six or seven students in attendance.  Because Mark bought M&M’s which come in many different colors, I labeled the diagram I drew on the board with +’s and –‘s instead of O’s and B’s when I explained how their mats should look as we did each step.   Doing that made my explanation to third period clearer than my explanation was to the first period.

When we were almost finished, Mark asked me to write more problems so the students could continue practicing balancing equations.  Instead, I asked a student to write a problem for the class.  She quickly wrote one on paper and then on the whiteboard.  After most of the class members had solved her problem, I had her explain the steps to the class.  She did a terrific job and we all clapped.  I asked another student to write a problem.  She shared a problem, and we cheered for her after she explained the steps.  Eventually, every class member wrote a problem for the class to solve, and we cheered after they explained the steps to solve it with their newly acquired English skills.  Mark also wrote a problem, one that I had thought to be too simple to put on the board:
x – 2 = -2.  It turned out not to be too trivial.  Some students needed to manipulate what happens in that case as well.  Shon and Serena assisted some of the students, but clearly before the class period was over most of the students did not need much help.

When class was finished, Mark met with the three of us.  He told us he really liked the activity and that any time you mix candy and learning together, it’s going to be a hit.  He said when I introduce an activity, I need to slow down.  I need to make sure everyone understands what they are supposed to do.  He thanked me and started his next class.  Shon and Serena both enjoyed helping students with the activity.  Serena said slowing down when giving directions seems to be one of the most common suggestions she hears given to pre-service teachers.  She said it might be helpful to have an equal sign between the two rectangles on the mat so students would know that the two sides are supposed to be equal.  Shon mentioned that when I explained what to do, I didn’t stress that we were solving for x so students might not understand what they need to do when they have similar problems to solve but no candy to use as a manipulative.  All of these are good suggestions that will improve my presentation to help students learn better.

As this paper illustrated, reflection when teaching is very important.

Now for a little about the number 933:

All of its digits are divisible by 3, so 933 is divisible by 3.

  • 933 is a composite number.
  • Prime factorization: 933 = 3 × 311
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 933 has exactly 4 factors.
  • Factors of 933: 1, 3, 311, 933
  • Factor pairs: 933 = 1 × 933 or 3 × 311
  • 933 has no square factors that allow its square root to be simplified. √933 ≈ 30.545049

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Tag Cloud