Math 7

Unit 4 Circles and Area

Lesson 4.2  Circumference of a Circle

Focus

The FOCUS section introduces the main concept that will be covered in the lesson.

Investigate the relationship between the circumference and diameter of a circle.

There isn’t a Focus activity in this lesson.

Explore

Explore an idea or problem.
The exploration provides the background for the ideas presented in the lesson.

There are MANY ways you can explore the ideas of this lesson.

Three different approaches are given below,  your teacher may tell you to follow a specific approach.
If not, you are welcome to do one (or more!) of the sections below to help explore these important ideas about circles.

TEXT Approach Page 133

Read/Work through the  Explore  section on Page 133 in your textbook.
It’s super valuable to actually DO the activity!
You learn a lot by doing things “hands on”.

This video goes through the steps of the exploration for three objects.
LINK

A fantastic video that shows you exactly what this exploration is all about.
LINK

Computer  approach A

Video Tutorial  (ExploreA)

Computer  approach B

A slightly different approach, looking at “rolling circles” instead of a “fixed” circle.

Demonstration

You use the following demonstration to do a similar exploration  to the one given in the textbook.
Follow along by working through the sections below

The following is a "do it yourself" way to come up with an approximate value of π .

Not totally required, but I think that "hands on" can help you to remember better....

In the animation, you control three circles, slowly rolling over to the right.    
(and you can control the speed, by showing the animation controls)

Each circle rolls at the same speed over to the right, so the larger circle will cover more ground.

The colored numbers at the right give the horizontal location of the colored dots.

Video Tutorial   (ExploreB)

Collect data I

At their starting positions, it is easy to find the diameter of each circle.

Write those numbers down  (and then check your answers before moving on)

Answers

Diameter of small circle  =   20
Diameter of medium circle = 40
Diameter of large circle = 60

Collect data II

You can tell when the circles have made a complete revolution by using the colored dot on each one.

When a circle has made a complete revolution, it has marked off a length EQUAL TO ITS' CIRCUMFERENCE along the horizontal axis

Watch carefully and find the circumference of each circle and write those numbers down

Answers

These are estimates, but your answers shouldn't be too different from mine.  I took these measurements when t=10   in the animation.

Small diameter = 63
Medium diameter = 126
Large diameter = 188

Calculate

The question mathematicians wondered about was,  how is the circumference C  related to the diameter d  of a circle?

You might calculate  C +  d    or  C - d    or  C × d   and so on, looking for patterns or something "significant".

I'll tell you right out that what we are looking for  is   (C/d)

Find the value of   (C/d)   for each circle  using the numbers you wrote down.  Check your answers below.

Answer

Here are the answers I got for (C/d)     using the numbers I typed above

Interpret

All of these numbers give us the approximate value of   π.

Two of my answers are a little too big, and one is a little too small,  but all close to the value of π which is, to 2 decimals,   3.14

The circumference of a circle, divided by the diameter of a circle , is equal to π

Assuming we could measure perfectly, which we can't!

Connect

Connect summarizes the math in the lesson.
A presentation of ideas connected to the lesson focus.

TEXT Pages 134 - 135

Read/Work through the  Connect  section on Pages 134 - 135  in your textbook.

Even if you KNOW you are going to watch the video, READING this section is still important.

Getting the same information in multiple ways can help with understanding.

Watch

Video Tutorial

Connect  Alternate approach  (optional)

Connect summarizes the math in the lesson.
A presentation of ideas connected to the lesson focus.

The information that follows is optional, but it goes over the ideas of the lesson, in yet another way, to hopefully help with understanding.

Working with the diameter

The circumference of a circle, divided by the diameter of a circle , is equal to π

So, using an equation, that says that  ((C )/d)= π

If we rewrite that,  we get  C = π d    (same equation in a different form)

How can we interpret that equation?

Check out the demonstration below.

I have marked the diameter of the circle.  (the line through the center of the circle)

I have also marked off lengths 	EQUAL to the diameter along the horizontal line.

Each section of the line, marked by dots, is equal to the length of the diameter.

I've also "painted" the circumference of the circle in red.

Imagine, as the circle rolls, it's going to leave behind a paint trail.

When the circle makes ONE COMPLETE REVOLUTION,  it has marked off a bit more than three diameters.

One complete revolution of a circle is exactly  π d   (about  3.14  times the diameter)

The circumference of a circle,  C ,   is equal  to  π d                 C = π d

Working with the radius

Recall that the diameter of a circle is TWICE the length of the radius

This gives us a similar presentation, except the dots on the line now mark off lengths equal to the radius of the circle

When the circle makes ONE COMPLETE REVOLUTION,  it has marked off a bit more than six radii  
(the plural of radius, pronounced ray dee eye)

One complete revolution of a circle is exactly  2 π r    (about 6.28 times the radius)

The circumference of a circle,  C ,   is equal  to  2 π r                 C = 2 π r

Example

The Example will give you TOOLS for solving problems, and also show you that there may be different ways to approach a question.

Example

TEXT Pages 135 - 136

Work through the Example on Pages 135 - 136  in your textbook.

Watch

Video Tutorial

Practice Problems

Doing Practice Problems

Practice problems reinforce the math presented in this lesson.

•	Work through the practice problems for this lesson.
	These problems are for practice, you don’t need to hand them in!
	(your teacher may assign specific questions for you to complete)

•	Use the answers in the back of the book to CHECK YOUR WORK!

•	If you make a mistake, TRY TO FIX IT!

•	If you need help or want more information, view the video solutions provided.

•	If you still need more help,  CONTACT YOUR TEACHER!

Types of Practice Problems

Assessment Focus a question that gives a good example of the kind of question you would be expected to answer after completing the lesson

Take It Further questions offer enrichment and extension.

Video Solution Links

Check your work for every problem you do!
If you make a mistake,  try to fix it!
Use the video solutions for more information.
If a problem still doesn’t make sense, ASK YOUR TEACHER FOR HELP!

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