7
th
Grade Math
(Mon/Tues) What are we doing today?
• Begin Unit 3 – We will be
covering information in the first few lessons but not using the book directly today
• Unit 2 Exams have been graded and are in PowerSchool. We will discuss this briefly today.
• Reminder: MATHia is again due on Saturday this week.
Unit 2 Exam
• If you have not taken the exam due to being absent last week, you should e-mail me immediately. If I do not get a message by 3:30pm on Tuesday, December 1, then you will have to take the makeup
version of the exam.
• On Friday, we will go over the Unit 2 exam in class.
• If you are not satisfied with your score, you will have the opportunity to change it in the future. However, that will not occur this week. I suspect it will occur at the end of next week at the earliest. I will
announce it in class when we get closer to that time. The makeup exam will occur on your own time and will not happen during a class meeting.
Learning Targets
• I can understand the terms radius, diameter, and circumference which are related to the measurements of circles.
• I understand that 𝜋 represents an irrational number which is
approximately equal to 3.14 and is the constant of proportionality relating the measurement of the diameter to the circumference.
• Given the radius or diameter of a circle, I can find the circumference.
• Given the circumference of a circle, I can find the radius or diameter.
• I understand that the formulas 𝐶 = 𝜋𝑑 and 𝐶 = 2𝜋𝑟 are proportional equations relating the diameter and radius to the circumference.
What is a circle?
• Obviously, I would assume
everyone knows what a circle is and could distinguish a circle from an oval or a square.
• What is the mathematical definition of a circle?
• A circle is all the points which are the same distance from a given point (the center).
Size of a Circle
• If I asked, “How big is this circle?”, how would you
respond?
• What would you need to be able to measure to be able to
Measurements of a Circle
• The radius of a circle runs from the center to the edge of the circle.
• The diameter of a circle runs
from one edge to the other and passes through the center.
• The diameter of a circle is twice the length of the radius. Of
course, that also means the radius is half the length of the diameter.
Circumference
• For polygons, the distance
around them is referred to as
perimeter. In the past, you have likely found the perimeter of
polygons such as squares or triangles by adding the lengths of all sides.
• The distance around a circle is called its circumference.
Essentially, it is the perimeter of a circle.
Problem #1 (Not in Book)
• In this table, I have measured the diameter and circumference of various circles. In each
instance, I rounded to the nearest tenth of an inch.
• Is this a proportional
relationship? Hint: Find the constant of proportionality if possible.
Problem #1 (ANSWER)
• For each row, I have divided
circumference by diameter to look for the constant of proportionality. On this table, I have added this
measurement “C/D”.
• In each row, the value of C/D is different though the numbers are very similar.
• Could there be a reason why this number is similar but not exactly the same number?
Problem #1 (ANSWER)
• Since I was measuring the
circumference and rounding off each value, that would introduce some potential errors to my values.
• Thus, the differences in the value of C/D are due to the rounding and/or measurement errors. The relationship between diameter and circumference is proportional with a constant of proportionality
Pi
• The constant of proportionality between the diameter and circumference of any circle is the mathematical constant 𝜋. The symbol 𝜋
is a Greek letter.
• Generally, we will estimate 𝜋 to be 3.14. However, 𝜋 is an irrational number which
means it cannot be written as a fraction. This also means it is a decimal which never ends and never repeats.
• In Desmos, you can type “pi” to get 𝜋 to
appear in the calculation box. Most scientific or graphing calculators have a 𝜋 button which will estimate the value of 𝜋 to a certain
Calculating Circumference
• Since 𝜋 is the constant of proportionality relating the
diameter to the circumference, we can use the formula 𝐶 = 𝜋𝑑
to calculate the circumference of any circle given the diameter.
• Similarly, we can use 𝐶 = 2𝜋𝑟 to calculate the circumference
given the radius since the
diameter is twice the size of the radius.
𝐶 = 𝜋𝑑
𝐶 = 2𝜋𝑟
Example
• Find the circumference of this circle.
Example (ANSWER)
• Find the circumference of this circle.
• 𝐶 = 𝜋𝑑
• We know that d = 4. Thus, 𝐶 = 𝜋 ∙ 4.
• Since 𝜋 ≈ 3.14, we have 3.14 ∙ 4 which is approximately 12.56. So, 12.56 mi is our solution.
• Note: It is also common, especially in later math classes, to not estimate 𝜋 and leave it in the answer. In this case the answer would just be 4𝜋. As with
variables, the 𝜋 symbol is written after the number it is multiplied by. I may ask for the answer to be estimated or in exact form. If neither is specified, you can
Problem #2 (Not in Book)
• Find the circumference of this circle. Give both the exact
answer and an estimated answer rounded to 2 decimal places.
Problem #2 (ANSWER)
• The exact answer is 6𝜋 which is approximately 18.84 cm.
• 𝐶 = 𝜋𝑑 → 𝜋 ∙ 6 = 6 ∙ 3.14 ≈ 18.84
• Note: If you use the 𝜋 button on a calculator, you may get a slightly different answer. For example, my calculator would give
18.8495559215 which would round to 18.85. I will always consider
that when you give estimated answers.
Problem #3
• Find the circumference of this circle. Give both the exact
answer and an estimated answer rounded to 2 decimal places.
Problem #3 (ANSWER)
• The exact answer is 14𝜋 which is approximately 43.96.
• 𝐶 = 2𝜋𝑟 → 2 ∙ 𝜋 ∙ 7 = 14𝜋
• Alternatively, we could just double the radius to get a
diameter of 14 and work this problem just like the previous problem.
Classwork
• The link on the right will let you practice finding circumference.
• This is not being graded to see if you get them all correct. It is
practice only. However, it will show the answers after you finish.
• I will post the link in the chatbox during class. https://forms.office.com/Pages/R esponsePage.aspx?id=fSGa33NnF 021d4XclgeCIslaDQMWMuNBpze NhT_XGjpUMENRT1BKTUZJWUk4 NDlTTEVIMTNXM0NVTS4u
(Wed/Thurs) What are we doing today?
• Unit 3 – Continue Discussion of Circumference
• Unit 2 Exams have been graded and are in PowerSchool. We will review them on Friday. If you
did not take it, you should e-mail me immediately.
• Reminder: MATHia is again due on Saturday this week.
Learning Targets
• I can understand the terms radius, diameter, and circumference which are related to the measurements of circles.
• I understand that 𝜋 represents an irrational number which is
approximately equal to 3.14 and is the constant of proportionality relating the measurement of the diameter to the circumference.
• Given the radius or diameter of a circle, I can find the circumference.
• Given the circumference of a circle, I can find the radius or diameter.
• I understand that the formulas 𝐶 = 𝜋𝑑 and 𝐶 = 2𝜋𝑟 are proportional equations relating the diameter and radius to the circumference.
Classwork
• Reminder: If you have not completed the classwork assignment from Monday/Tuesday, you should do so. It is linked in the General
channel of our team. Only 31 out of 88 students had completed it as of Wednesday morning.
Finding Circumference
If given diameter…
• The formula is 𝐶 = 𝜋𝑑.
• You can simply multiply by 𝜋.
• If asked to find the exact solution, then you can leave 𝜋 in the
solution.
• If asked to approximate the
solution, you can use 3.14 for 𝜋 or use the 𝜋 button on a calculator.
If given radius…
• The formula is 𝐶 = 2𝜋𝑟.
• You will need to multiply by 2 which would give diameter then you can multiply by 𝜋.
• If asked to find the exact solution, then you can leave 𝜋 in the
solution.
• If asked to approximate the
solution, you can use 3.14 for 𝜋 or use the 𝜋 button on a calculator.
Problem #1 and #2
Problem #1
• Find the circumference of a circle with a diameter of 3.5. Write your answer both as an exact and approximate answer.
Problem #2
• Find the circumference of a
circle with a radius of 8. Write your answer both as an exact and approximate answer.
Problem #1 (ANSWER)
SOLUTION
• First, you should always ask
yourself, “Do I have the radius or diameter?” Here you have the
diameter. So, you do not need to multiply by 2.
• 𝐶 = 𝜋𝑑 → 𝜋 ∙ 3.5 → 3.14 ∙ 3.5
• The circumference is 3.5𝜋
(exact) or approximately 10.99.
Problem #1
• Find the circumference of a circle with a diameter of 3.5. Write your answer both as an exact and approximate answer.
Problem #2 (ANSWER)
SOLUTION
• First, you should always ask yourself,
“Do I have the radius or diameter?”
Here you have the radius. So, you do need to multiply by 2.
• 𝐶 = 2𝜋𝑟 → 2 ∙ 𝜋 ∙ 8 → 3.14 ∙ 8
• The circumference is 16𝜋 (exact) or approximately 50.24.
• Note: On either of these problems, you might have a slightly different, but still correct, answer if you used the 𝜋 button on a calculator.
Problem #2
• Find the circumference of a circle with a radius of 8. Write your answer both as an exact and approximate answer.
Problem #3 (Finding Diameter/Radius)
• What if I have circumference and need to find the diameter or radius?
Problem #3 (ANSWER)
• The answer is approximately 5.25.• Keep in mind that 𝐶 = 𝜋𝑑.
• We know the circumference is 32.97. So, we can divide by 𝜋 to find the diameter. 32.97
3.14 =
10.5
• In the problem, we are asked to find the
radius though. So we need to divide this by 2.
10.5
2 = 5.25
• Always make sure you have answered the question. Here if we got 10.5 we would have the diameter and not be answering what is asked. We can also use the formula to check our answer. 𝐶 = 2 ∙ 3.14 ∙ 5.25 = 32.97
Problem #4
• If the circumference of a circle is approximately 53.38, what is the diameter and radius of this circle?
Problem #4 (ANSWER)
• The diameter is 17 and the radius is 8.5.
• Since we have the
circumference, we can divide by
𝜋 to get the diameter.
53.38
3.14 = 17
• Then, we can divide by 2 to find the radius. 17 2 = 8.5 • 𝐶 = 53.38 • 𝑑 = ? • 𝑟 = ?
Practice Problem (ANSWER)
• The flying disc has an inaccurate measurement.
• Notice that we do not need to exactly multiply by 𝜋 to see this. I simply multiplied by 3 to see which was not close to being correct.
• Since 28 is not close to 69, we would know something is wrong about the measurements.
Practice Problem #1 (ANSWER)
• The distance the rider travels is approximately 251.2 or 80𝜋
exactly.
• Since we have the diameter, we just need to multiply by 𝜋.
Practice Problem #2 (ANSWER- Identify Parts)
• PART A: This is the radius.
• PART B: This is the diameter.
• PART C: This is the diameter.
• PART D: This is the circumference.
• PART E: This is the radius.
• PART F: This is the diameter.
• PART G: This is the circumference.
Practice Problem #2 (ANSWER
–
Measurements)
• Part A: 𝑟 = 5, 𝑑 = 10, 𝐶 = 31.4 • Part B: 𝑟 = 1.9, 𝑑 = 3.8, 𝐶 = 11.9 • Part C: 𝑟 = 7, 𝑑 = 14, 𝐶 = 44.0 • Part D: 𝑟 = 11.9, 𝑑 = 23.9, 𝐶 = 75 • Part E: 𝑟 = 7, 𝑑 = 14, 𝐶 = 44.0 • Part F: 𝑟 = 5, 𝑑 = 10, 𝐶 = 31.4 • Part G: 𝑟 = 30.3, 𝑑 = 60.5, 𝐶 = 190 • Part H: 𝑟 = 60, 𝑑 = 120, 𝐶 = 376.8• Note: All answers have been rounded to 1 decimal place using 3.14 for 𝜋.
Practice Problem #3 (ANSWER)
• The perimeter is approximately 42.84 or exactly 6𝜋 + 24.
• Notice how this is half of a circle. So we will find the circumference and then cut it in half. We have the diameter. So 𝐶 = 12𝜋 ≈ 37.68. Half of this is 18.84.
• Then we just need to add the other exterior sides (not the one inside). So 18.84 + 12 + 12 = 42.84.
(Fri) What are we doing today?
• Review Unit 2 Exam
• Reminder: Classwork from Mon/Tues is posted in the
General channel. It will close today at 4pm. As of this
morning, less than half of you have completed it.
• Reminder: MATHia is again due on Saturday this week.
