Games - An Important Part of Learning Math

I currently teach remedial math students on the college level. These are the students who fail to pass the math placement test to enroll in College Algebra - that dreaded class that everyone must pass to graduate. The math curriculum at our community college starts with Basic Math, moves to Fractions, Decimals and Percents, and then to Basic Algebra Concepts. Most of my students are smart and want to learn, but they are deeply afraid of math. I refer to them as mathphobics.

We all have this type of student in our classrooms, whether it is middle school, high school, or college. When working with this type of student, it is important to bear in mind how all students learn. I always refer back to the Conceptual Development Model which states that a student must first learn at the concrete stage (use manipulatives) prior to moving to the pictorial stage, and well in advance of the abstract level (the book). This means that lessons must include the use of different manipulatives.

I use games a great deal because it is an easy way to introduce and use manipulatives without making the students feel like “little kids.” I can also control the level of mathematical difficulty by varying the rules; thus, customizing the game to meet the instructional objectives my students are learning. However, as with any classroom activity, teachers should monitor and assess the effectiveness of the games. 

When using games, other issues to think about are:

1. Excessive competition. The game is to be enjoyable, not a “fight to the death”.
2. Mastery of the mathematical concepts necessary for successful play. Mastery should be at an above average level unless teacher assistance is readily available when needed. A game should not be played if a concept has just been introduced.
3. Difficulty of the rules. If necessary, the rules should be modified or altered in order that the students will do well.
4. Physical requirements (students with special needs). These should be taken into account so that every player has an opportunity to win.

In addition to strengthening content knowledge, math games encourage students to develop such skills as staying on task, cooperating with others, and organization. Games also allow students to review mathematical concepts without the risk of being called “stupid”. Furthermore, students benefit from observing others solve and explain math problems using different strategies.

Games can also….

1. Pique student interest and participation in math practice and review. 
2. Provide immediate feedback for the teacher. (i.e. Who is still having difficulty with a concept? Who needs verbal assurance? Why is a student continually getting the wrong answer?)
3. Encourage and engage even the most reluctant student.
4. Enhance opportunities to respond correctly.
5. Reinforce or support a positive attitude or viewpoint of mathematics.
6. Let students test new problem solving strategies without the fear of failing.
7. Stimulate logical reasoning.
8. Require critical thinking skills.
9. Allow the student to use trial and error strategies. 

Mathematical games give the learner numerous opportunities to reinforce current knowledge and to try out strategies or techniques without the worry of getting the “wrong” answer. Games provide students of any age with a non-threatening environment for seeing incorrect solutions, not as mistakes, but as steps towards finding the correct mathematical solution. 

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If you want a challenging but fun and engaging math game, try Contact. It is a fun and attention-grabbing way for students to review basic math facts and to use critical thinking without doing another “drill and kill” activity.

Fractions - Identifying Equivalent Fractions, Reducing Fractions to Lowest Terms

Here is a Halloween riddle: Which building does Dracula like to visit in New York City? Give up? It's the Vampire State Building!! (Ha! Ha!) Here is another riddle. What do ghosts eat for breakfast? Scream of Wheat and Ghost Toasties!

Okay, so what do these riddles have to do with teaching math? I have been attempting to come up with ways for my students to recognize fractional parts in lowest terms. As you know from this blog, I have used Pattern Sticks, the Divisibility Rules, and finding Digital Root. These are all strategies my students like and use, but to be a good mathematician requires practice - something most of my students dread doing. I can find many "drill and kill" activities, but they tend to do just that, drill those who don't need it and kill those who already know how to do it. So to drill and "thrill", I created fractional word puzzles for specific times of the year.

The one for October is Halloween Fraction Riddles. It contains ten riddles that the students must discover by correctly identifying fractional parts of words. For instance, my first clue might be:

The first 2/3's of WILLOW. The word WILLOW contains six letters. It takes two letters to make 1/3; therefore, the first 2/3's would be the word WILL. This causes the students to group the letters (in this case 4/6), and then to reduce the fraction to lowest terms. The letters are a visual aid for those students who are still having difficulty, and I observe many actually drawing lines between the letters to create groups of two. 

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At first, I thought my students would breeze through the activities, but to my surprise, they proved to be challenging as well as somewhat tricky - just perfect for a Trick or Treat holiday. Maybe this is an activity you would like to try with your intermediate or middle school students. Just click on this link: Halloween Fraction Riddles.

How to Have Successful Parent/Teacher Conferences

If you are like most teachers, you are preparing for your first round of parent/teacher conferences. Now that I teach on the college level, this is one activity I currently don't have to do, but when I did, I really did enjoy them. Why? Because I was prepared with more than just the student's grades. Here are some of the ways I got ready.

First, in preparing for parent/teacher conferences, what can you do on a daily basis? Is the conference based on simply talking about grades or are there additional items that need discussing? How can an observation be specific without offending the parent or guardian? How is it possible to remember everything?

I kept a clipboard in my classroom on which were taped five 6” x 8” file cards so they overlapped - something like you see in the two pictures above. Each week, I tired to evaluate five students, writing at least two observations for each child on the cards. At the end of the week, the file cards were removed and placed into the children's folders. The next week, four different students were chosen to be evaluated. In this way, I did not feel overwhelmed, and had time to really concentrate on a small group of children. By the end of 4-5 weeks, each child in the class had been observed at least twice. By the end of the year, every child had been observed at least eight different times.

Below are sample observations which might appear on the cards.

Student	Date	Observation	IEP	ESL
Mary Kay	8/20   8/28	Likes to work alone; shy and withdrawn;  wears a great deal of make-up. She has a good self concept and is friendly. Her preferred learning style is  visual based on the modality survey.	X
Donald	9/19   9/21	Leader, at times domineering, likes to  play games where money is involved. His preferred learning style is auditory  (from the modality survey). He can be a  “bully,” especially in competitive games. He tends to use aggressive language with  those who are not considered athletic.

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By the time the first parent/teacher conferences rolled around, I had at least two observations for each child. This allowed me to share specific things (besides grades) with the parents/guardians. As the year progressed, more observations were added; so, that a parent/guardian as well as myself could readily see progress in not only grades, but in a student's behavior and social skills. The cards were also an easy reference for filling out the paperwork for a 504 plan or an IEP (Individual Education Plan). As a result of utilizing the cards, I learned pertinent and important facts related to the whole child which in turn created an effective and relevant parent/teacher conference.

To keep the conference on the right track, I also created a checklist to use during parent/teacher conferences.  It features nine characteristics listed in a brief, succinct checklist form. During conferences, this guide allows me to have specific items to talk about besides grades. Some of the characteristics included are study skills and organization, response to assignments, class attitude, inquiry skills, etc. Since other teachers at my school were always asking to use it, I rewrote it and placed it in my TPT store. It is available for only $2.00, and I guarantee it will keep your conferences flowing and your parents focused! When you have time, check it out!

What Is A Palindrome?

I was getting ready to pay for my meal at a buffet when I noticed the cashier's name tag.  It read "Anna" to which I replied, "Your name is a palindrome!"  The cashier just stared at me in disbelief.  I explained that a palindrome was letters that read the same backwards as forwards. Because you could read her name forwards and backwards, it qualified as a palindrome. She replied that she remembered a math teacher talking about those because of patterns (I love that math teacher), and she remembered the phrase "race car" was a palindrome.  We then began sharing palindromes that we knew such as radar, level and madam while my family waited impatiently in line. (Sometimes they have little patience with my math conversations.)

The word palindrome is derived from the Greek word palíndromos, which means "running back again". A palindrome can be a word, phrase or sentence which reads the same in both directions such as: "Eva, can I stab bats in a cave?" or "Was it a car or a cat I saw?" or "Rats live on no evil star."

But did you know there are also palindromic numbers?  A palindromic number is a number whose digits are the same if read in both directions (as seen on your left).   Whereas "1234" is not a palindromic number, because backwards it is "4321" which is not the same. 

Suppose a person starts with the number one and lists the palindromic numbers in order: 11, 22, 33, 44, 55...etc.  Can you continue the list? 

Did you notice that palindromic numbers are symmetrical?  Look carefully at the 17371 shown above.  It is symmetrical (when a figure can be folded along a line so the two halves match perfectly) on either side of the three whether read left to right or vice versa.

Palindromic numbers are very simple to generate from other numbers with the help of addition.

Try this:

1. Write down any number that has more than one digit. I will use 47.
2. Write down that number in reverse beneath the first number. (See illustration below.)
3. Add the two numbers together. (121)
4. 4. The sum of 121 is undeniably a palindrome.

Try an easy number first, such as 18.  At times you will need to use the first addition answer and repeat the process of reversing and adding. You will almost always get a palindrome answer within six steps.  Try one of these numbers 68 or 79.  Be careful because if you pick a number greater than 89, arriving at the palindromic answer will take more steps, but it will still work.  (See the two examples below.)

But don't try 196!  In fact, avoid it like the plague!   A computer has already gone through several thousand stages, and it still hasn't come up with a palindromic answer!

Example #1:

• Start with 75.
• Reverse 75 which makes 57.
• Add 75 and 57 and you get 132.  The answer 132 is not a palindrome.
• SO reverse 132, and it becomes 231.
• Add 132 and 231, and the answer is 363. 
• Since 363 is a palindrome, we are done!

Example #2:

• Begin with 255.
• Reverse 255 to get 552.
• Add 255 and 552. The answer is 807 which is not a palindrome.
• SO reverse 807 to get 708.
• Add 807 and 708. The answer of 5151 is not a palindrome.
• SO reverse 1515 to get 5151.
• Add 1515 and 5151 which is 6666.
• This is a palindrome; so, we are done!

Do We Say "Fall" or "Autumn"? Doing Science Investigations Using Leaves

October is just around the corner.  October means football (Ohio State, of course), cooler weather, and gorgeous leaves. (It is also the month my husband and I were married.) In October, we see the leaves turning colors, and the deciduous trees shedding their leaves.

Another name for fall is autumn, a rather strange name to me. Through research, I discovered that the word autumn is from the Old French autumpne, automne, which came from the Latin autumnus. Autumn has been in general use since the 1960's and means the season that follows summer and comes before winter.

Fall is the most common usage among those in the United States; however, the word autumn is often interchanged with fall in many countries including the U.S.A. It marks the transition from summer into winter, in September if you live in the Northern Hemisphere or in March if you live in the Southern Hemisphere.  It also denotes when the days are noticeably shorter and the temperatures finally start to cool off. In North America, autumn is considered to officially start with the September equinox. This year it was on September 22nd.

With all of that said, the leaves in our neighbor's yard have already begun to fall into ours which aggravates my husband because he is the one who gets to rake them. Maybe focusing on some activities using leaves will divert his attention away from the thought of raking leaves to science investigations.  

Remember ironing leaves between wax paper?  We did that in school when I was a little girl (eons and eons ago).  Here is how to do it.

1. Find different sizes and colors of leaves.
2. Tear off two sheets about the same size of waxed paper.
3. Set the iron on "dry".  No water or steam here!
4. The heat level of the iron should be medium.
5. Place leaves on one piece of the waxed paper.
6. Lay the other piece on top.
7. Iron away!

You can also use this activity to identify leaves.  According to my husband who knows trees, leaves and birds from his college studies, we "waxed" a maple leaf, sweet gum leaf, elm leaf, cottonwood leaf (the state tree of Kansas - they are everywhere), and two he doesn't recognize because they come from some unknown ornamental shrubs.

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Maybe you would like to use leaves as a science investigation in your classroom. I have one in my Teacher Pay Teachers store that is a six lesson science performance demonstration for grades K-2. The inquiry guides the primary student through the scientific method and includes: 

1. Exploration time
2. Writing a good investigative question
3. Making a prediction
4. Designing a plan
5. Gathering the data
6. Writing a conclusion based on the data. 

Be"leaf" me, your students will have fun!

Does A Circle Have Sides?

Believe it or not, this was a question asked by a primary teacher.  I guess I shouldn't be surprised, but in retrospect, I was stunned. Therefore, I decided this topic would make a great blog post.

The answer is not as easy as it may seem. A circle could have one curved side depending on the definition of "side!"  It could have two sides - inside and outside; however this is mathematically irrelevant. Could a circle have infinite sides? Yes, if each side were very tiny. Finally, a circle could have no sides if a side is defined as a straight line. So which definition should a teacher use?

By definition a circle is a perfectly round 2-dimensional shape that has all of its points the same distance from the center. If asked then how many sides does it have, the question itself simply does not apply if "sides" has the same meaning as in a rectangle or square.

I believe the word "side" should be restricted to polygons (two dimensional shapes). A good but straight forward definition of a polygon is a many sided shape.  A side is formed when two lines meet at a polygon vertex. Using this definition then allows us to say:

1) A circle is not a polygon.

2) A circle has no sides.

One way a primary teacher can help students learn some of the correct terminology of a circle is to use concrete ways.  For instance,  the perimeter of a circle is called the circumference.  It is the line that forms the outside edge of a circle or any closed curve. If you have a circle rug in your classroom, ask the students is to come and sit on the circumference of the circle. If you use this often, they will know, but better yet understand circumference.

For older students, you might want to try drawing a circle by putting a pin in a board. Then put a loop of string around the pin, and insert a pencil into the loop. Keeping the string stretched, the students can draw a circle!

And just because I knew you wanted to know, when we divide the circumference by the diameter we get 3.141592654... which is the number π (Pi)!  How cool is that?

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If you are studying circles in your classroom, you might like this resource. It is a set of two circle crossword puzzles that feature 18 terms associated with circles. The words showcased in both puzzles are arc, area, chord, circle, circumference, degrees, diameter, equidistant, perimeter, pi, radii, radius, secant, semicircle, tangent and two. The purpose of these puzzles is to have students practice, review, recognize and use correct geometric vocabulary in a fun, non-threatening way.

I'm Pro-Tractor! Correctly Teaching and Using Protractors

Using a protractor is supposed to make measuring angles easy, but somehow some students still get the wrong answer when they measure. Here are a few teacher tips that might help.

1)  Make sure that each student has the SAME protractor.  (To avoid having many sizes and types, I purchase a classroom set in the fall when they are on sale.)  If each student's protractor is the same, you can teach using the overhead or an Elmo, and everyone can follow along without someone raising their hand to declare that their protractor doesn't look like that!  (Since the protractor is clear it works perfectly on the overhead. No special overhead protractor is necessary.)

2) Show how the protractor represents 1/2 of a circle.  When two are placed together with the holes aligned, they actually form a circle.

3) Talk about the two scales on the protractor, how they are different, and where they are located.  It's important that the students realize that when measuring to start at zero degrees and not at the bottom of the tool.  They need to understand that the bottom is actually a ruler. 

I use a couple of word abbreviations to help my students remember which scale to use.

4)  When the base ray of an angle is pointing to the right, I tell the students to remember RB which stands for Right Below.  This means they will use the bottom scale to measure. 

5) When the base ray of an angle is pointing to the left, I tell the students to remember LT which are the beginning and ending letters of LefT. This means they will use the top scale to measure the angle.

6) Of course the protractor has to be on the correct side.  It's amazing how many students try to measure when the protractor is backwards.  All the information is in reverse!

7)  Make sure the students line up the hole with the vertex point of the angle, aligning the line on the protractor that extends from the hole, with the base ray.  Even if they choose the correct scale, if the protractor is misaligned, the answer will be wrong.

8)  Realize that the tools the students use are massed produced, and to expect students to measure to the nearest degree is impossible.  To purchase accurate tools such as engineer uses would cost more than any of us are willing to spend!

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If you would like supplementary materials for angles, check out these two products: Angles: Hands On Activities  or  Geometry Vocabulary Crossword Puzzle.

Using a Math Survey to Determine A Student's Attitude Towards Math

Math is really important in our daily lives and can help us be successful, but many studies show that students aren't doing well in math. That's why we, as teachers, need to pay attention to things that can make us better at teaching it. One big thing that affects students’ ability to learn math is their attitude towards it. This means how they feel about math, whether they like it or not. If students have a positive attitude, they will think math is important and try harder to do well in it. Their attitude towards math also affects their choices for the future. If they don't like math, they might avoid taking math classes in college or picking careers that use math.

So how do math teachers get some insight into a student’s math attitude? Math attitude surveys can be beneficial. Just like we pre-assess our students to determine their understanding of math concepts, such as place value or multiplication, so we know the best entry point for new instruction, it’s equally important that we uncover the attitudes our students have about learning math.

To start, look for a survey that measures what you think is important. You can easily find them by searching "math surveys for students" on Google. I looked at a lot of surveys, but none were right for me, so I made my own. I wanted a math survey that was simple to give, beneficial for students in the upper grades (I teach remedial math at the college level), and would give me a comparison from the beginning to the end of the semester.

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It is important to understand your students' strengths and weaknesses in the subject, and that's where this math survey comes in. It consists of ten statements and four thought-provoking questions, specifically designed to reveal insights into your students' math abilities. The statements are easy to complete - students simply check a box that reflects their beliefs, with options ranging from strongly disagree to strongly agree. But the real gems lie in the four short answer questions, where students can share their thoughts and ideas in their own words. After the survey is complete, the students’ responses are compiled and placed in their personalized profile folder, which they receive at the end of the semester. This allows you to not only gauge their progress but also tailor your teaching to their individual needs. You'll have a better understanding of your students, and they'll have a clearer picture of their own strengths and areas for improvement. It's a win-win situation for everyone.

The Pros and Cons of Testing, Testing, Testing!

Tests are here to stay whether we like it or not. As I read various blogs, I am finding more and more teachers who are frustrated over tests and their implications. I am seeing many of my former student teachers leave the teaching profession after only two or three years because of days structured around testing.

High stakes tests have become the “Big Brother” of education, always there watching, waiting, and demanding our time. As preparing for tests, taking pre-tests, reliably filling in bubbles, and then taking the actual assessments skulk into our classroom, something else of value is replaced since there are only so many hours in a day. In my opinion, tests are replacing high quality teaching and much needed programs such as music and art. I have mulled this over for the last few months, and the result is a list of pros and cons regarding tests.

Testing Pros

1. They help teachers understand what students have learned and what they need to learn.
2. They give teachers information to use in planning instruction. 
3. Tests help schools evaluate the effectiveness of their programs. 
4. They help districts see how their students perform in relation to other students who take the same test. 
5. The results help administrators and teachers make decisions regarding the curriculum. 
6. Tests help parents/guardians monitor and understand their child's progress. 
7. They can help in diagnosing a student's strengths and weaknesses. 
8. They keep the testing companies in business and the test writers extremely busy. 
9. Tests give armchair educators and politicians fodder for making laws on something they know little about.  

                                           **The last two are on the sarcastic side.**

Testing Cons

1. They sort and label very young students, and those labels are nearly impossible to change.
2. Some tests are biased which, of course, skew the data. 
3. They are used to assess teachers in inappropriate ways. (high scores = pay incentives?) 
4. They are used to rank schools and communities. (Those rankings help real estate agents, but it is unclear how they assist teachers or students.) 
5. They may be regarded as high stakes for teachers and schools, but many parents and students are indifferent or apathetic. 
6. They dictate or drive the curriculum without regard to the individual children we teach. 
7. Often, raising the test scores becomes the single most important indicator of overall school improvement. 
8. Due to the changing landscape of the testing environment, money needed for teachers and the classroom often goes to purchasing updated testing materials. 
9. Under Federal direction, national testing standards usurp the authority of the state and local school boards. 
10. Often they are not aligned with the curriculum a district is using; so, curriculum is often changed or narrowed to match the tests. 

Questions That Need to Be Asked

1. What is the purpose of the test?
2. How will the results be communicated and used by the district? 
3. Is the test a reflection of the curriculum that is taught? 
4. Will the results help teachers be better teachers and give students ways to be better learners?
5. Does it measure both a student's understanding of concepts as well as the process of getting the answer? 
6. Is it principally made up of multiple choice questions or does it does it contain any performance based assessment? 
7. What other means of evaluation does the school use to measure a child's progress? 
8. Is it worth the time and money?

How to Overcome Mathphobia (a hatred of Math) and Be a Success in Math

I HATE Math!

I teach Mathphobics on the college level who aren't always thrilled to be in my math class. Last week, as the students were entering and finding seats, I was greeted with, “Math is my worst enemy!” I guess this particular student was waiting for an impending Math Attack. But then I began thinking, “Should this student wait to be attacked or learn how to approach and conquer the enemy?” Since winning any battle requires forethought and planning, here is a three step battle plan for Mathphobics.

1) Determine why math is your enemy. Did you have a bad experience? Were you ever made to feel stupid, foolish, or brainless? Did your parents say they didn’t like math, and it was a family heredity issue? (One of the curious characteristics about our society is that it is now socially acceptable to take pride in hating mathematics. It’s like wearing a badge of honor or is that dishonor? Who would ever admit to not being able to read or write?) Math is an essential subject and without math, not much is possible...not even telling time!

2) Be optimistic. Suffering from pessimism when thinking of or doing math problems makes it impossible to enjoy math. Come to class ready to learn. At the end of class, write down one thing you learned or thought was fun. I realize math teachers are a big part of how a student views math. In fact, one of the most important factors in a student’s attitude toward mathematics is the teacher and the classroom environment. Just using lecture, discussion, and seat work does not create much interest in mathematics. You've been in that class. Go over the homework; do samples of the new homework; start the new homework. Hands-on activities, songs, visuals, graphic organizers, and connecting math to real life engage students, create forums for discussion, and make math meaningful and useful.

3) Prove Yourself. Take baby steps, but be consistent. Faithfully do the homework and have someone check it. Don’t miss one math class! You can’t learn if you aren't there. Join in the discussions. Think about and write down your questions and share them with your teacher or with the class. Study for an upcoming test by reviewing 15 minutes each night a week before the test. Get help through tutoring, asking your instructor, or becoming a part of a study group. Keep in mind, no one is destined for defeat!

So don’t just sit there and wait for the dreaded Math Attack. Meet it head on with a three step battle plan in hand!
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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. Toprchase 20 beneficial math study tips, just download this resource.

You are invited to the Inlinkz link party!

Click here to enter

What To Do With Those Annoying Cell Phones in the Classroom

Most of us can't live without our cell phones.  Unfortunately, neither can our students.  I teach on the college level, and my syllabus states that all cell phones are to be put on "silent", "vibrate", or turned off when class is in session.  Sounds good, doesn't it?  Yet, one of the most common sounds in today's classrooms is the ringing of a cell phone, often accompanied by some ridiculous tune or sound effect that broadcasts to everyone a call is coming in.  It’s like “technological terror" has entered the classroom uninvited.  Inevitably, this happens during an important part of a lesson or discussion, just when a significant point is being made, and suddenly that "teachable moment" is gone forever.

What are teachers to do?  Some instructors stare at the offender while others try to use humor to diffuse the tension. Some collect the phone, returning it to the student later.  A few have gone so far as to ask the student to leave class.

In my opinion the use of cell phones during class time is rude and a serious interruption to the learning environment. What is worse is the use of the cell phone as a cheating device.  The college where I teach has seen students take a picture of the test to send to their friends, use the Internet on their phone to look up answers, or have answers on the phone just-in-case.  At our college, this is cause for immediate expulsion without a second chance.  To avoid this problem, I used to have my students turn their cell phones off and place them in a specific spot in the classroom before the test was passed out.  Unfortunately, the students’ major concern during the test was that someone would walk off with their phone.  Not exactly what I had planned!

It's a CUTE sock and
perfect for a cell phone!

A couple of years ago, a few of us in our department tried something new.  Each of us has purchased those long, brightly colored socks that seem to be the current fashion statement.  (I purchased mine at the Dollar Tree for $1.25 a pair.)  Before the test, each student had to turn off their cell phone, place it in the sock, tie the sock into a knot and place the sock in front of them. This way, the student still had control over their cell phone and could concentrate on doing well on the test, and I did not have to constantly monitor for cheating.

At the end of the semester, we compared notes.  Overall, we found that the students LOVED this idea.  Many said their students were laughing and comparing their stylish sock with their neighbor's.  I was surprised that a few of the students even wanted to take their sock home with the matching one – of course.  So here is a possible side benefit....maybe socking that cell phone away caused my students to TOE the line and study!

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Need more ideas for helping with those annoying classroom irritations? Here is  resource that offers a number of practical and realistic ideas about classroom management and how to eliminate those day-after-day aggravating and annoying student problems that keep resurfacing in your classroom. It is perfect for novice teachers, beginning teachers or for student teachers. It is also a good review for those who have been teaching for a number of years.

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.

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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

Using the Book, Anno's Mysterious Multiplying Jar, to Learn About Factorials

Let's look at one more book in this series of books that link literature and math. This book is more for those taking algebra as it as to do with factorials.  Factorial is a word that mathematicians use to describe a special kind of numerical relationship. Factorials are very simple things. They are just products, indicated by the symbol of an exclamation mark. The factorial function (symbol: !) means to multiply a series of descending natural numbers. For instance, "five factorial" is written as "5!" (a shorthand method) and means 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in determining the numbers of combinations and permutations and in finding probability.

A Few Are Available
on Amazon

Now all of that may seem above your mathematical head, but let me introduce you to the book Anno's Mysterious Multiplying Jar by Masaichir and Mitsumasa Anno.  It is a story about one jar and what is inside it. Anno begins with the jar, which contains one island, that has two countries, each of which has three mountains. The story continues like this until 10 is reached.  The colorful pictures are arranged within borders on the page as many times as the number of objects being discussed. For instance when four walled kingdoms are introduced, four kingdoms are on the page.

The explanation of 10! in the back of the book is also very helpful. Even if children do not understand the concept being taught, they will certainly appreciate the detailed colored drawings and imaginative story! The book is best for kids who have been introduced to at least basic multiplication facts, but younger kids will enjoy counting and looking at the pictures even if the rest of it is over their heads; so, this book helps with multiplying skills as well as the mathematical concept of factorials.

You might give the students a worksheet to keep track of how many islands, rooms, etc. there are. The final question is how many jars are there. Hopefully there will some students who catch on to the factorial concept, find the pattern and discover the answer! 

Here is an example of how you might use factorials in solving a word problem.  How many different arrangements can be made with the letters from the word MOVE?  Because there are four different letters and four different spaces, this is how you would solve the problem.

____   ____   ____   ____ 

Four Possible Spaces

All four letters could be placed in the first space. Once the first space is filled, only three letters remain to fit in the second space. Once the second space is filled with a letter, two letters remain to write in the third space. Finally, only one letter is left to take the fourth and final space. Hence, the answer is a factorial (4!) = 4 × 3 × 2 × 1 = 24 arrangements.

Try some problems in your classroom. Start with an imaginary character, Cal Q. Late, who is working at an Ice Cream Store called Flavors. A hungry customer orders a triple scoop ice cream cone with Berry, Vanilla, and Bubble Gum ice cream. How many different ways could Cal Q. Late stack the ice cream flavors on top of each other?

You could answer the question by listing all of the possible orders of the three ice cream flavors from top to bottom. (Students could have colored circles of construction paper to physically rearrange.)

• 
  Bubble Gum - Berry - Vanilla
• Bubble Gum - Vanilla - Berry

• Berry - Vanilla - Bubble Gum
• Berry - Bubble Gum - Vanilla

• Vanilla - Berry - Bubble Gum
• Vanilla - Bubble Gum - Strawberry

Or, if we use factorials, we arrive at the answer much faster: 3! = 3 × 2 × 1 = 6

Learning about patterns and the use of factorials will stretch a students' mathematical mind. Why not try a few problems in your classroom? And by all means, check out Anno's Mysterious Multiplying Jar.

A Day With No Math - Another Book that Links Math and Literature

Who wants to read about math? Who even likes it? Many, many times I have heard a parent of one of my students say, "I understand why my child cannot do math. I was never very good at math, either." Right! So you weren't good at reading; so, your child should be illiterate? So you don't like to play sports; so, PE should be optional? I don't think so.
 

My goal in life is to make people, students, adults, children, comfortable with math; to see its value; to learn to at least like it. After all, there isn't a day that goes by that you don't use math in some form. Did you read a clock today? Did you buy something with money? Did you go to the home improvement store to buy paint? Did you cook or keep score while you played a game? That is all math. Useful - right?


Ask yourself or your students, "What would happen if suddenly there were no numbers?" To find out, read A Day with No Math by Marilyn Kaye, published by Harcourt Brace Jaovanovich, Inc. in 1992. It is a great read aloud book. It's one I have used in workshops and in my own classroom with children, college students and adults. The book demonstrates how mathematics plays an important role in our daily lives and shows the reader how time, measurement, money and other mathematics are used everyday. The story helps kids to understand that math is a part of all aspects of our every day life and without it, our life would be such a mess. Try reading this if you hate math or even if you love it, and you will be surprised at how much math you really know. It will give you a different appreciation for math 

This is a book teachers will treasure to have in your classroom library. Currently, it's difficult to find, but Amazon seems to have a few copies 

Ten Black Dots - Another Book that Links Math and Literature

I am an avid reader, and I love books that integrate math and literature. Lately, my blog has featured books that link the two.  

Available on Amazon
for $7.60

Today's featured book is Ten Black Dots by Donald Crews (Greenwillow Books, 1986).  This picture book is for grades PreK-2 and deals with numbers and operations. 

The book asks the question, What can you do with ten black dots?  Then the question is answered throughout the book by using  illustrations of everyday objects beginning with one dot and continuing up to ten. Simple rhymes accompany the pictures such as:

"Two dots can make the eyes of a fox, Or the eyes of keys that open locks."

Materials Needed: 

• Unifix cubes or Snap Cubes (multi-link cubes) as seen on the right
• Black circles cut from construction paper or black circle stickers
• Crayons
• Pencils
• Story paper
• Calculators -simple ones like you purchase for $1.00 at Walmart

Activities:

1)  Read the book a number of times to your class.  Let the students count the dots in each picture. On about the third reading, have the children use the snap cubes to build towers that equal the number of dots in each picture.

2)  Have the children think of different ways to make combinations, such as: How could we arrange four black dots?  (e.g. 1 and 3, 4 and 0, 2 and 2)  Have the children use black dots or snap cubes to make various combinations for each numeral from 2-10.

3)  This is a perfect time to work on rhyming words since the book is written in whimsical verse. Make lists of words so that the students will have a Word Wall of Rhyming Words for activity #4.

• How many words can we make that rhyme with:  sun?  fox?  face?  grow?  coat?  old?  rake?  rain?  rank?  tree?
• Except for the first letter, rhyming words do not have to be spelled the same.  Give some examples (fox - locks or see - me)

4)  Have the children make their own Black Dot books  (Black circle stickers work the best although you can use black circles cut from construction paper. I'm not a big fan of glue!)  Each child makes one page at a time.  Don't try to do this all in one day.  Use story paper so that the children can illustrate how they used the dots as well as write a rhyme about what they made.  Collate each book, having each child create a cover.

5)  Have the children figure out how many black dots are needed to make each book. (The answer is 55.)  This is a good time to introduce calculators and how to add numbers using the calculator.

If you can't find Ten Black Dots in your library, it is available on Amazon.