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Let's Pitch Paper!

My college students love, love games.  I found one on Pinterest which I adapted for the new class I am teaching called Conquering College.  In this class, the students have a reading quiz over an assigned article about every three weeks, and I am always trying to think of new ways to review. I tried the game, and it was a "hit".  It is called "Pitch" and here is how you play it.

1) Divide the class into two teams and assign them two pages of the article to review.

2) Each student is to write a question about their part of the article on half a sheet of paper.

3) Select two captains to come to the front of the room.

4) Have students crumble up the paper and throw it, trying to hit the captain of the opposite team. (I have team #1 throw; then team #2. I also have four questions that I throw into the mix.)

5) The captains mix up the questions and place them on a table. They then go to board to keep score.

6) Alternating between teams, one by one, the students go and pick a question, which they must read aloud and answer correctly for their team to get a point. If the student is unable to answer, the question goes to the other team for them to answer.

7) The captains are the last students to answer a question.

The first time we played, the two teams tied so both teams received a small candy bar. The students LOVED it!!!! They were not only engaged, but they were having fun. I was surprised when they said things like, "That isn't a good question because it can be answered with a yes or a no. Or that is a well written one." I think the next time we might make airplanes and call the game "Crash Landing."

I shared the game with other faculty members, and here is what a chemistry instructor wrote me...

"I have done 'muddiest point' with my chemistry students and had them ball up and throw their papers at me...even my double section which has 62 students. I got bombarded with blue paper as the students tried really hard to hit me. It was hysterical!!!"

So, now it is your turn. Maybe you will have a "pitch" battle or just maybe, the teams will "pitch" in and help each other.

Mathematical Patterns

Since all math is based on patterns, this week, I want to target some mathematical problems in which we investigate developing patterns.

In the first example below, you will notice we begin by multiplying one by one; then 11 by 11, and so forth. Each time we multiply, the number of digits in the multiplier and the multiplicand increases. Do you see the pattern that progresses in the answer (product)? Notice how this multiplication pattern forms a triangle? Can you figure out what kind of triangle it is?


Here is another interesting pattern. In this one, instead of multiplying by 1, then 11, then 111, the answer (product) looks like the multiplier in the pattern above. Do you notice anything else significant?

Yes, we are multiplying by 9 each time. Now look at the number being added, and count the number of ones you see in each answer. Surprised? Isn’t it amazing how math is ordered, methodical and precise? Maybe that is one reason I love to teach it!

"Sum" Trick

In the book Ten Black Dots book, there are a total of 55 black dots. Normally, to find that answer, you would add the numbers together.

But did you know there is an easier way? Take 10 and divide it by 2. That equals 5. Multiply 10 x 5 and you get 50 then add in the 5 which equals 55. Too confusing? Well let's look at it in groups that equal 10.


As illustrated above, 10 is by itself so it is 10. Then if we group the numbers so that each group equals ten, we have four additional sets. All together, we have five groups of ten with five left over which equals 55.   5 x 10 = 50 + 5 = 55

This will work for every sequence of consecutive numbers which begins with one and contains an even set. In other words, sets that contain 2, 4, 6, 8, 10, 12... numbers. Merely divide the largest number by 2; multiply the largest number by the quotient, and then add the quotient.


Example:  14, 13, 12, 11, 10,  9,  8,  7,  6,  5,  4,  3,  2, 1

This will also work for an odd numbered sequence like 11 but the formula or quick trick for finding the sum is a little different. As seen below, we again divide 11 by 2, which 5.5 or rounded up equals 6. Again we group sets of two that equal 11. There are five groups plus 11 by itself so that makes a total of six groups.
Since there are no numbers left by themselves, simply multiply 11 by 6 (the rounded up quotient) to get the sum which is 66.

I love to write a series of consecutive numbers which begin with one on the board, and have the students find the answer using their calculators while I do the math in my head. Of course, they are amazed and swear that I have memorized the answer. I then ask me to give me a series (not off the wall or so large that it would take forever to use the calculator) and again I quickly give them the answer. I then teach them that math trick.

Students love "tricks" like this, but I always burst their bubble by telling them mathematicians are astute people. That's why they are always looking for faster, quicker, and smarter ways to do math!

Aliens and Trapezoids

I am always looking for ways to help my students remember things.  For example, when we learn about the properties of one, I sing (yes I do, and a little off key) One is the Loneliest Number.  Since there are so many quadrilaterals to learn (*7 in all), I create quadrilateral stories.  Here is one of my students' favorites.  (Keep in mind, these are college students.)

Once upon a time, I planted a broccoli garden in my backyard.  Since I love geometry, I placed triangle statues all around my garden.  Every morning I would go out to my garden to weed, hoe, fertilize, and water my precious broccoli plants.  One morning, I noticed several of my plants had been eaten.  I was one upset lady; so, I decided to stay up all night and watch to see which critters had the nerve to venture into my garden for a broccoli feast.

That night, I sat at my bedroom window watching the garden.  All of a sudden, out of the sky, came a UFO which landed in my backyard.  As I watched, the door of the UFO opened (I use my arms to imitate the opening door while I say, S-q-e-a-k!) and out came some little aliens.  As they approached my broccoli, they repeated, "Zoid, zoid, zoid".  (I use a high alien like voice.) Sure enough, they ate several of my plants!  They then proceeded back to their spaceship and flew away. 

The same thing happened the following night and the night after that; so, I knew something had to be done.  I went to my garage, and got out my trusty chain saw to cut off the top of each of my triangles.  (I imitate the noise of a chain saw.)  Inside each cut off triangle I placed a bunch of broccoli to entice my visitors.  I knew if those aliens got inside, they would never get out because of the slanting sides.  I went back into my house to wait.

Sure enough, like clockwork, the UFO returned.  Again, the door of the UFO opened (s-q-e-a-k!) and out came the same little aliens. They proceeded to my cut off triangles, and perched on the edge peering down at the broccoli, all the while saying, "Zoid, zoid, zoid".  One by one they leaped inside to eat the broccoli, and guess what.  I trapped-a-zoid!  Okay, you may not be laughing, but I swear this story does help my students to remember what a trapezoid is. 

Let's discuss a couple of important math things about trapezoids that you may not be aware of.   In my story, the trapezoid is an isosceles trapezoid or as sometimes called, a regular trapezoid.  Not only does it have one set of opposite sides parallel, but it also has one set of opposite sides equal (marked with the black line segments).  It also has one line of symmetry which cuts the trapezoid in half (the blue dotted line).  This special trapezoid is usually the one taught by most teachers, but it is really a special kind of trapezoid. 

   trapezoid                                   isosceles trapezoid
For a quadrilateral to be classified as a trapezoid, the shape only needs to have one set of opposite sides parallel as seen in figure one.  The first trapezoid is the one that sometimes appears on tests to "trick" our students.

In the second figure (the isosceles or regular trapezoid), the sides that are not parallel are equal in length and both angles coming from a parallel side are equal (shown on the right).  Lucky for me that I used the second trapezoid for my trap or my zoids would have been long gone, and with my entire crop of broccoli, too!

*square, rectangle, rhombus, parallelogram, trapezoid, kite, trapezium

Problem Solving Top Ten List #3


The first step in the problem solving process is to correctly identify the problem. The next is to explore, identify, and choose a problem solving strategy. The third step in the process is to correctly implement the strategy chosen. But what happens when a student swears his/her strategy isn’t working? Usually, they have a problem solving habit that I might categorize as “malfunctioning” (not effective). Let’s look at the worst problem solving habits that some of your students just might have.

  1. Trying to do it all in your head; not writing anything down.
  2. Arbitrarily choosing a strategy.
  3. Staying with a strategy when it is not working.
  4. Giving up on a strategy too early.
  5. Getting fixated on a single strategy and trying to use it for everything.
  6. Not asking yourself: “Does this make sense?”
  7. Being afraid to ask for help.
  8. Not checking your answer.
  9. Not noticing patterns.
  10. Going through the motions instead of thinking.
The student should be asking...
  1. Have I shown an adequate amount of work to demonstrate what strategy I have used?
  2. Is there more than one strategy which I could use to solve this problem?
  3. Does choosing one strategy over another make the implementation easier?
  4. Does the strategy I have chosen use any tables, charts, formulas or properties I need to review
  5. What technology or manipulatives could I use to help me solve the problem?
As mathematics teachers, what can we do in the
classroom to guide this kind of thinking?

Top Ten Reasons for Getting Stuck when Problem Solving

A good process problem uses no set algorithm to find the solution. It requires a variety of processes (problem solving strategies) to find the solution. It is a problem that is easy to understand, is interesting, perhaps even whimsical, and has numbers sufficiently small enough so that lengthy computation is unnecessary.

Standard Word Problem: Jack's family plans to rent a camping trailer for vacation. The rent is $22.50 a day. What will it cost to rent the camping trailer for one week?

A Problem that Requires Problem Solving: Drew and Addie are playing a game. At the end of each game, the loser gives the winner a chip. When they are done playing several games, Drew has won three games, but Addie has three more chips than she had when the game began. How many games did Drew and Addie play?

So what happens when your students try to do the process problem above and they have no idea what to do? In my last posting, I listed ten reasons why students get stuck when problem solving. Now let's consider why students get stuck in the first place.

Top Ten Reasons for Getting Stuck in the First Place:
  1. You tried to rush through the problem without thinking.
  2. You did not read the problem carefully.
  3. You don't know what the problem is asking for.
  4. You don't have enough information.
  5. You are looking for an answer that the problem isn't asking for.
  6. The strategy you are using doesn't work for this particular problem.
  7. You are not applying or using your strategy correctly.
  8. You failed to combine your strategy with another strategy.
  9. The problem has more than one answer.
  10. The problem cannot be solved.

Since students today tend to be more visual than anything else, a graphic organizer becomes a valuable math tool. The Triangular Graphic Organizer is generic so that it can be used to solve all kinds of formula problems such as: d=rt, A=lw, or c= a2 + b2. 

This five page handout explains in detail how to use the graphic organizer. It also contains several examples as well as a page of blank triangular graphic organizers to copy and use in your classroom.

Want the answer to the process problem? 

Check out the page above entitled: Answers to Problems.


Problem Solving - Getting Unstuck!

Math courses are not like other courses. To pass most other subjects, a student must read, understand, and recall the subject matter. However, to pass math, an extra step is required: a student must use the information they have learned to solve math problems correctly. Special math study skills are needed to help the student learn more and to get better grades.

The study of mathematics should emphasize problem solving so that students can use and apply a wide variety of strategies to investigate and understand mathematical content. In this way, they acquire confidence in using mathematics meaningfully and the assurance that they can be successful in math.

But what is a real problem that requires problem solving? A real problem solving problem presents a challenge that cannot be resolved by some routine procedure known to the student and where the student accepts the challenge! Now, there's the dilemma, students who actually accept the challenge and are persistent enough to solve the problem.

What can we, as math instructors, do when students become frustrated, exasperated and discouraged and say they are stuck? Let's look at ten ways to help them get "unstuck".

Top Ten Ways to Get Unstuck
    Free Resource
  1. Re-read the problem. 
  2. Modify your strategy. 
  3. Change your strategy. 
  4. Combine your strategy with another strategy. 
  5. Look at the problem from a new perspective. 
  6. Look at the answer. 
  7. Look at other similar problems. 
  8. Ask for help. 
  9. Wait awhile and then try again. 
  10. All of the above.
Would you like a free resource about study tips? Check out Study Tips You Won't Forget. This resource lists 20 math study tips or guidelines intended to help students succeed in math.


Problem Solving Strategies

My math classes have been looking at different problem solving strategies, trying them out, and discovering what works best individually.  We've tried:
    Poster that Hangs in My Classroom
    1. Using Models or Manipulatives
    2. Drawing a Picture
    3. Acting it Out or Role Playing
    4. Making a Chart or Table
    5. Making a List or a Graph
    6. Looking for Patterns (All math is based on patterns.)
    7. Working Backwards
    8. Guessing and Checking
    9. Making the Problem Simpler
I believe problem solving should be the central focus of any mathematics curriculum.  It is the major reason for studying math and provides a context in which concepts and skills can be learned.  It is the major vehicle for developing higher order thinking skills.  However, there is one problem solving strategy that will not work, although many students try it. It is called staring!  That is why the non-strategy poster seen above always hangs in my classroom.


Need some problem solving activities that are enjoyable, offer variety, and increase interest? Think Tank Questions is a 14 page handout that contains 46 various questions. Most subject areas are included in the questions which are appropriate for grades 2-6.