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Relationship Between Circumference, Radius and Diameter - What Is Pi?

BIlly Edward Bush B.A, M.Ed., Ed.D
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Students will be able to apply the formulas for area and circumference of circles to solve real-world and mathematical problems.

Class experience

US Grade 7 - 10
  • Common Core Standards CCSS.MATH.CONTENT.7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Addresses all 8 Standards for Mathematical Practice Day #1 Elements of a Circle Lesson Objective Students will understand the defining elements of a circle and be able to find the radius and diameter of a circle. Big Idea The students will be able to identify and draw the parts of a circle. The students will be able to identify and/or calculate the measure of the diameter and radius of a circle. The students will be able to identify the total number of degrees in a circle. Vocabulary Circle- The set of all points in a plane that are the same distance from a point called the center. Diameter- The distance across a circle through the center. Radius- The distance from the center of a circle to any point on the circle. Chord- A line segment that has both endpoints on the circle. Arc- A curved line that is part of the circumference of a circle. Central Angle- An angle whose vertex is the center of the circle. Center Point- The point inside a circle that is the same distance from all points on the circle. Circumference- The distance around the circle. Explore Ask students to draw a large circle on a sheet of paper. From this position, ask students how they formed the circle and what a circle is. As students share out, have a class scribe (write down) their ideas on their notes. To deepen the understanding of what a circle is, I will visually model what a circle is. Continue the class discussion on what a circle is until I have had 1-3 students to contribute their ideas. Students may need a bit of support. The main ideas we are trying to get are as follows: A circle is a series of points all equal distance away from a defined point (the center). A circle is a continuous curve. The circle is technically the space confined within the points. Review students’ notes. Create a foldable on the parts of a circle. Each student needs to obtain two small square sheets of paper in two different colors. Have the students trace the same small circular item (bottom of mug, jar lid, etc.) on each sheet of paper. Cut out both circles. You should now have two congruent circles. Fold both circles horizontally and vertically. Cut a slit to the center on each. Overlap the two circles. (see attachment for example) Each student will proceed to label the parts of the circle to better understand the parts/concept of the parts of a circle. Summarize Exit Ticket students will participate in an online interactive activity testing their understanding/concept of what was reviewed today. Day #2 Relationship Between Circumference and Diameter - What is pi? Objective Students will understand the relationship between circumference and diameter. Big Idea Did you say pie? or pi? Bring in a circular item for class, and it if happens to be food - AWESOME! Students will understand the relationship between circumference and diameter. Explore Pi Lab: As an introduction to the unit, students will along with me to explore the ratio between the circumference of a circle and its diameter. I ask students in advance to bring in one circular item. While the students are completing the discovery activity, I will monitor progress and providing assistance when necessary. It is important that this be a student discovery, MP 1, however, so my help will be limited to procedures of wrapping string around the item, and finding the diameter - attention to precision MP 6 will be important as this will help with developing the formula later on. As students begin to finish the activity, we will come back together as a class and discuss their findings. Hopefully, students will correctly identify the formula and pi. Analyzing their findings and looking at developing formulas brings in mathematical practices 7 and 8 - looking for repeated reasoning. This lesson will not work for all students, so I have included some Scaffolding Ideas. Summarize Exit Ticket: After all classwork, problems have been worked out on the board, I will wrap up the lesson by asking two questions: “How is the circumference of a circle related to its diameter?” and “How do we calculate the circumference of a circle given the diameter or the radius?” I will have students write their answers on the back of their opener and turn in as an exit ticket. Day #3 Circumference and Area of Circles Objective Students will be able to apply the formulas for area and circumference of circles to solve real-world and mathematical problems. Big Idea Ever play a song back in your head over and over? You will be singing about the area and circumference all day after this lesson! Launch Students will watch the following https://www.flocabulary.com/circles/ This song is a catchy way for students to remember the area and circumference of circles. I will play the song while I get the notes passed out. Area and Circumference Notes: After the fun song, I will move into some formal notes on area, and then a few whole class practice problems. I will work a few problems out with the kids and then set the off to complete the remaining problems on their own. I am curious to see how kids do with the bicycle question - I have found in the past that many students struggle to realize this is a circumference question. Instructional Strategy - How do table challenges work?: Students will respond to 7 area and circumference problems. This is a good way for me to assess who gets it and who doesn't :) Additionally, their are word problems included so students will have to determine whether they are solving for area or circumference. Summarize + Homework Class Discussion: I will wrap up the lesson by asking “Who can give us a real world example of when we would apply the area of a circle problem?” (MP 4) taking volunteers for responses. The purpose of this activity will be to make sure that students understand when it would be appropriate to calculate area, versus circumference. Normally, lots of hands go up for this question – which is always a good sign! Homework: After discussion of the question, Students will have homework, and allow students the last few minutes of class to look it over and ask any questions needed for clarification. Philosophy on Homework Day #4 Virtual review time Students will be given an online activity with performance task to test for mastery. Outcome 1: Students will utilize operations with rational numbers and use the order of operations to evaluate numerical and algebraic expressions. 1.1 Add, subtract, multiply, and divide positive rational numbers (integers, fractions, decimals). 1.2 Add, subtract, multiply, and divide positive and negative rational numbers. 1.3 Evaluate numerical and algebraic expressions using the order of operations. 1.4 Evaluate exponential and radical expressions. Outcome 2: Students will solve simple one-variable equations. 2.1 Solve one-step, one-variable equations. Outcome 3: Students will calculate perimeter, circumference, area, surface area, and volume of two and three-dimensional shapes. 3.1 Define basic concepts associated with geometry. 3.2 Calculate the perimeter/circumference of a two-dimensional shape. Outcome 4: Students will use various problem-solving strategies to solve real-life applications. 4.1 Translate a verbal model into an algebraic expression or equation. 4.2 Solve problems involving percents, ratios, and proportions. 4.3 Convert units of measurement within a system (U.S. Customary or Metric). Outcome 5: Students will analyze and interpret a given set of data. 5.2 Graph data and interpret graphs of data. 5.3 Find the overall mean given the means of two or more sets of data. Outcome 6: Students will use the basic properties of the number system. 6.1 Restate the equivalent form of numbers using exponents, radicals, scientific notation, absolute values, fractions, decimals, and percents. 6.2 Use the number line to order numbers, add and subtract signed numbers, and graph one-variable inequalities. 6.3 Use place value to order and round numbers. 6.4 Factor whole numbers and find the least common multiple and greatest common factors of two or more numbers. 6.5 Classify a number as prime or composite using division rules.
Algebra is critically important because it is often viewed as a gatekeeper to higher-level mathematics and it's a required course for virtually every postsecondary school program.  I have 6 basic reasons why I think offering this class is so important.  

1)Algebra is Faster And Better Than “Basic” Math
Just as multiplying two by twelve is faster than counting to 24 or adding 2 twelve times, algebra helps us solve problems more quickly and easily than we could otherwise. Algebra also opens up whole new areas of life problems, such as graphing curves that cannot be solved with only foundational math skills.

2) Algebra is Necessary to Master Statistics and Calculus

While learning one kind of math to learn more kinds of math may not be an immediately satisfying concept, statistics and calculus are used by many people in their jobs. For example, on my other side job as a research person for a local non-profit organization, I use statistics every day. I help departments identify ways to measure their success. In general, statistics are used in certain jobs within businesses, the media, health and wellness, politics, social sciences, and many other fields. Understanding statistics makes us wiser consumers of information and better employees and citizens.

Calculus helps us describe many complex processes, such as how the speed of an object changes over time. Scientists and engineers use calculus in research and in designing new technology, medical treatments, and consumer products. Learning calculus is a must for anyone interested in pursuing a career in science, medicine, computer modeling, or engineering.

3) Algebra May Be a Job Skill Later

A student may be confident they are not going into any career needing statistics or calculus, but many people change jobs and entire careers multiple times in their working life. Possessing a firm knowledge and understanding of algebra will make career-related changes smoother.

4) Algebra Can Be Useful in Life Outside of the Workplace

I have found algebra helpful in making financial decisions. For example, I use algebra every year to pick a health care plan for my family using two-variable equations to find the break-even point for each option. I have used it in choosing cell phone plans. I even used it when custom-ordering bookshelves for our home. 

5) Algebra Reinforces Logical Thinking

I would not use algebra as the only means of teaching logic. There are more direct and effective means of doing so, but it is a nice side-benefit that the two subject areas reinforce one another.

6) Algebra is Beautiful

The beauty of algebra is an optional benefit because one has to truly choose to enjoy it, but algebra provides us with a basic language to describe so many types of real-world phenomena from gravity to the population growth of rabbits. That five letters can be used to describe how an entire category of matter, namely ideal gases, behaves is amazing and beautiful in its simplicity.

There is also a beauty when we start with a complex-looking problem and combine and simplify over and over until we have one value for each variable. The process can be enjoyable and the result immensely satisfying.

Algebra is an important life skill worth understanding well. It moves us beyond basic math and prepares us for statistics and calculus. It is useful for many jobs some of which a student may enter as a second career. Algebra is useful around the house and in analyzing information in the news. It also reinforces logical thinking and is beautiful.

So, keep an open mind about why we learn algebra and look for ways to share its applications with students. Dispel the stigma that it is a boring list of rules and procedures to memorize. Instead, consider algebra as a gateway to exploring the world around us. Those are our top reasons why we learn algebra, and there are plenty more. 
Homework Offered
Practice Work Practice work will be graded weekly day. Once an assignment has been graded you cannot receive full credit on that homework. I will utilize an exponential decay for makeup or to complete missing work. School policy says that if you do not complete assigned homework you will need to come in and complete the assignment before the due date. Once the term ends homework must be turned in before I submit grades. Practice will be in the form of on line work that will be due on the day you take the chapter test. There should be plenty of time for you to complete your practice work during time period of the class.
1 - 2 hours per week outside of class
Assessments Offered
Test and Quizzes Tests and quizzes will be given on daily basses.
Grades Offered
Yarn, Calculators, One to two circular items
In addition to the Outschool classroom, this class uses:
Joined March, 2020
Teacher expertise and credentials
Master's Degree in Education
California Teaching Certificate - Elementary Education
California Teaching Certificate - Mathematics
I am currently a 7th 8th and 9th grade Math Middle School Teacher and truly love what I am doing. When it comes to teaching, I believe it is about developing strong relationships that tap into students' passions and provide relevancy to students... 
Group Class


for 3 classes
3x per week, 1 week
55 min

Completed by 39 learners
Live video meetings
Ages: 12-16
1-12 learners per class

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