Master Plane Geometry by Using Number Tiles to Solve Geometric Math Puzzles

My college students will soon start the unit on plane geometry.  I love teaching geometry because it is so visual, but there are others who despise it because of the numerous new words to learn.  In fact, our plane geometry unit alone contains over 50 terms that must be learned as well as understood.

I have found that with my students, mathematical language is either a dead language (It should be buried and never resurrected!), a foreign language (It sounds like a different language from a far away country.), a nonsense language (It makes no sense to me - ever!) or a familiar, useful language. Many times, they are unduly frustrated because mathematical language has never been formally taught or applied to real life.  For example, many primary teachers will have their children sit on the circle when in fact, the children are sitting on the circumference of the circle.  What a wonderful, concrete way to introduce children to the concept of circumference!  Yet, this teaching moment is often missed, and circumference doesn't surface again until it is time to teach the chapter on circles.

$9.75

Because I believe it is important to find different ways to introduce and practice math vocabulary, I created a new resource for Teachers Pay Teachers entitled: Geometric Math-A-Magical Puzzles.  It is a 48 page handout of puzzles that are solved like magic squares. Number tiles are positioned so that the total of the tiles on each line of the geometric shape add up to be the same sum. Most of the geometric puzzles have more than one answer; so, students are challenged to find a variety of solutions.

Before each set of activities, the geometry vocabulary used for that group of activities is listed. Most definitions include diagrams and/or illustrations. In this way, the students can learn and understand new math words without difficulty or cumbersome words. These 21 activities vary in levels of difficulty. Because the pages are not arranged in any particular order, the students are free to skip around in the book. All of these activities are especially suitable for the visual and/or kinesthetic learner.

A ten page free mini download of this item is available if you want to try it with your students. Check it out!

Problem Solving With Number Tiles in Middle School

Math Activities for Grades 5-8

I prefer using hands-on activities when teaching math. One of the most successful items I have used is number tiles. Because number tiles can be moved around without the need to erase or cross out an answer, I have discovered that students are more at ease and more willing to try challenging activities. There is something about not having a permanent answer on the page that allows the student to explore, investigate, problem solve, and yes, even guess.

I have created several number tile booklets, but the one I will feature today is for grades 5-8. It is a booklet that contains 15 different math problem solving activities that range from addition and multiplication, to primes and composites, to exponent problems, to using the divisibility rules. Since the students do not write in the book, the pages can be copied and laminated so that they can be used from year to year. These activities may be placed at a table for math practice or as a center activity. They are also a perfect resource for those students who finish an assignment or test early. Use these activities to reteach a concept to a small group as well as to introduce a new mathematical concept to the whole class.

Students solve the Number Tile Math Activities by arranging ten number tiles, numbered 0-9. Most of the number tile activities require that the students use each tile only once. The number tiles can be made from construction paper, cardboard, or square colored tiles that are purchased.  (How to make the number tiles as well as storage ideas is included in the handout.) Each problem is given on a single page, and each activity varies in difficulty which is suitable for any diverse classroom. Since the students have the freedom to move the tiles around, they are more engaged and more willing to try multiple methods to find the solution. Some of the problems will have just one solution while others have several solutions. These activities are very suitable for the visual and/or kinesthetic learner.

A free version for each of my number tile resources is also available in my TPT store.  While visiting my store, take time to check out these additional Number Tile activities.

• The A, B, C's of Number Tiles - 26 Problem Solving Math Puzzles
• Geometric Math Puzzles –Using Number Tiles Learning to Learn About Plane Geometry
• Number Tiles – Hands On Math Activities for the Primary Grades

Helping Students to Think Mathematically

What does it mean to think mathematically?  

It means using math vocabulary, language and symbols to describe or interpret mathematical concepts, procedures and to discover relationships among ideas.  Therefore when a student problem solves, they use previous knowledge, skills, and understanding of concepts to solve a problem.  This process might include formulating problems, applying a variety of strategies, or interpreting results.

What can we do to help our students become better mathematical thinkers?  

We can teach and model problem solving strategies.  We can remember and plan our lessons to involve the three stages of conceptual development: concrete, pictorial, and abstract. We can have the students talk or write about how they got an answer either with the class or with a partner.  We can use writing in the mathematics classroom (such as math journals) to allow the students to practice expository writing and show their understanding.  We can exhibit math word walls and have the students use the glossary in their book to write and define terms. 

We can also create a positive and safe classroom atmosphere for problem solving... 

By being enthusiastic and allowing the students to take risks without consequences.  By emphasizing the process as well as the answer, the students may be willing to try unconventional or different ways to solve the problem.  I always tell my students that there isn't just one right way to get an answer which surprises many of them.  In fact, this is one of the posters that hangs in my classroom.

  

As math teachers, let's continue to emphasize problem solving so that all students will acquire confidence in using mathematics meaningfully. But most of all, let's have fun while we are doing it!



$8.00 on TPT

If you are interested in having a math dictionary for your students, check out A Simple Math Dictionary. It is a 30 page student dictionary that uses easy and clear definitions as well as formulas and examples so that students can learn and understand new math words without difficulty or cumbersome language. Most definitions include diagrams and/or illustrations for the visual learner. Over 300 common math terms are organized alphabetically for quick reference.

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 c² = 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.