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?