<p>In this lesson, students investigate another attribute of circles: area. Students will:&nbsp;<br>- create their own approximation of the area of a circle using the area of a circumscribed square.&nbsp;<br>- learn the formula for calculating the area of a circle given its radius or diameter.&nbsp;<br>- use the area formula to calculate the radius or diameter when given the area.&nbsp;<br>&nbsp;</p>

<p>- How do spatial relationships, such as shape and dimension, help to create, construct, model, and represent real-world scenarios or solve problems?&nbsp;<br>- How may geometric properties and theorems used to describe, model, and analyze problems?&nbsp;<br>- How may patterns be used to describe mathematical relationships?&nbsp;<br>- How may detecting repetition or regularity assist in solving problems more efficiently?&nbsp;<br>- How may using geometric shape features help with mathematical reasoning and problem solving?&nbsp;<br>&nbsp;</p>

<p>- Circumference: The distance around a circle. (C = 2π<i>r</i>)</p>

<p>- one copy of the Area Worksheet (M-7-4-2_Area Worksheet and KEY) for each pair of students&nbsp;<br>- one copy of Optional Circles in Context sheet (M-7-4-2_Optional Circles in Context and KEY) for each student (<i>optional</i>)&nbsp;<br>- one copy of the Lesson 2 Exit Ticket (M-7-4-2_Lesson 2 Exit Ticket and KEY) for each student</p>

<p>- Use the Lesson 2 Exit Ticket (M-7-4-2_Lesson 2 Exit Ticket and KEY) to assess students' grasp of the relationship between the area and circumference of a circle and its radius and diameter.<br>- Assess students' comprehension using their completed Area Worksheet (M-7-4-2_Area Worksheet and KEY).&nbsp;<br>- The Optional Circles in Context sheet (M-7-4-2_Optional Circles in Context and KEY) can be used to assess student proficiency.<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Metacognition&nbsp;<br>W: Students will learn how to calculate the area of a circle. They will also be able to calculate the radius and diameter of a circle or circular object when they have known its area.&nbsp;<br>H: Students will use existing knowledge about the area of a square to make estimates about the area of a circle.&nbsp;<br>E: Students will explore ideas about finding area by making estimates, then refining those estimates and methods before making educated guesses about the formula for the area of a circle.&nbsp;<br>R: Students will improve their area formula based on additional problems, and then rehearse their knowledge through teacher-led questioning and a partner-completed worksheet.&nbsp;<br>E: In Activity 2, students' grasp of the material can be assessed through both teacher-led discussions and interactions with their partners. Students may also be evaluated based on their completed worksheet from Activity 2.&nbsp;<br>T: Use the Extension section to personalize the lesson to students' specific requirements. The Routine section provides opportunities to review lesson concepts throughout the year. The Small Group section is intended for students who might benefit from more learning or practice opportunities. The Expansion section offers suggestions for students who are willing to go above and beyond the requirements of the standard.&nbsp;<br>O: The lesson starts with a teacher-led investigation of the new topic. Students are free and encouraged to make predictions and estimates and back them up with mathematical reasoning. Students should enjoy the exploratory aspect of Activity 1 (rather than simply being told the formula and being expected to apply it). After determining the formula, students will work with a partner to investigate how radius, diameter, area, and circumference are all related.&nbsp;</p>

<p><strong>Activity 1</strong><br><br>Draw a square on the board with a circle circumscribed inside it as a "template" for the first activity:</p><figure class="image"><img style="aspect-ratio:179/176;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_41.png" width="179" height="176"></figure><p><strong>"Here, we have a circle drawn inside a square. The radius of the circle is labeled as </strong><i><strong>r</strong></i><strong>. What is the diameter of the circle?"</strong> (<i>2r</i>) If students struggle to use a variable such as <i>r</i> right away, start the activity with a specified value for <i>r</i>. <strong>"How long is one side of the square?"</strong> (<i>2r</i>) <strong>"And how long is the other side of the square?"</strong> (<i>2r</i>)&nbsp;<br><br>Make sure students understand how to calculate the area of a square; it is not necessary to determine that the area is 4<i>r²</i>.<br><br><strong>"What is the area of the square if the radius of the circle is 1 unit?"</strong> (<i>4 square units</i>) <strong>"What do we know about the area of the circle compared to the area of the square?"</strong> (<i>It has a lesser area than the area of the square.</i>) Ask students to estimate the area of the circle.&nbsp;<br><br>It may be helpful to divide the square into quadrants.</p><figure class="image"><img style="aspect-ratio:191/184;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_42.png" width="191" height="184"></figure><p><strong>"What is the area of each quadrant?"</strong> Shade in one quadrant so that students understand which part of the diagram you are referring to. Students should know that it is \(1 \over 4\) square unit. Ask students to estimate the area of the part of the circle in that quadrant. To help, ask, <strong>"Does the circle take up more than half the area in the quadrant?"</strong> (<i>Yes</i>) <strong>"Does it take up more than \(3 \over 4\) of the area of the quadrant?"</strong> Responses to this question will be mixed - it really takes up \(π \over 4\), or around 78.5% of the area. Use \(3 \over 4\) as a guess.&nbsp;<br><br><strong>"So, if the area of the entire square is 4 square units, and the area of the circle is the area of the square, approximately what is the area of the circle?"</strong> (<i>3 square units</i>)<br><br><strong>"That's a pretty good estimate. An estimate, to the nearest hundredth, is actually 3.14 square units. So an estimate that the circle's size is about \(3 \over 4\) the area of the square is a pretty good one."</strong> Write the following information on the board:&nbsp;<br><br><i>r</i> = 1 unit, <i>A</i> ≈ 3.14 square units&nbsp;<br><strong>"Is there anything special or unique about the number we got for the area of our circle with a radius of 1 unit?"</strong> Students should understand that it is (approximately) π.&nbsp;<br><br><strong>"So based on the radius of the circle, which was 1 unit, and the fact that the area is π, what do you think we have to do with the radius to determine the area?"</strong> Students should recommend that we multiply the two of them together; in other words, that A = π<i>r</i>.<br><br><strong>"To put our assumption to the test, let's pick a different value for the radius. Suppose the radius was 10 units. What would the area of the square be?"</strong> Make sure students understand that each side of the square is 2<i>r</i> units (in this case, 20 units). The area measures 400 square units. <strong>"So, if we think our circle has an area of about \(3 \over 4\) that of the square, what's the approximate area of the circle?"</strong> (<i>300 square units</i>)<br><br><strong>"Now, if we use our guess at a formula to find the area and multiply the radius of 10 units by π, what do we get for an area?"</strong> (<i>31.4 square units</i>) <strong>"Can this guess be correct, based on the fact that we know the area of the square is 400 square units?"</strong> (<i>No</i>)<br><br><strong>"So, we need a little more information to develop a better hypothesis about how to find the area. I'll tell you that the actual area of the circle rounded to the nearest whole number, is 314."</strong> Write the following information on the board:&nbsp;<br><br><i>r</i> = 10 units, <i>A</i> ≈ 314 square units.&nbsp;<br><strong>"Clearly, 314 is not equivalent to 10 times π. How many times greater than π is 314?"</strong> (<i>100</i>) <strong>"So, based on the fact that our area here is 100π, can anyone come up with a formula that uses the radius to get 100π?"</strong> Help students recognize that 100π = <i>r</i>²π.&nbsp;<br><br>Write A = π<i>r</i>² on the board.<br><br><strong>"Let's test out one more example. Suppose the radius of our circle is 8 units. What is the area of the square?"</strong> (<i>256 square units</i>) <strong>"And, based on our estimate that the circle has about \(3 \over 4\) the area of the square, what is the area of the circle?"</strong> (<i>192</i>)&nbsp;<br><br><strong>"Let's test our formula, A = π</strong><i><strong>r</strong></i><strong>², with a radius of 8 units and see if we can get close to 192. What does the formula indicate our area should be?"</strong> (<i>approximately 201</i>) <strong>"Pretty close to our estimate—I think our formula works!"</strong><br><br>Explain to students that when they said the area was 201 square units, they were approximating. Even if they read all the digits on their calculator, it was still an approximation because the digits of π continue indefinitely. <strong>"So, it is more accurate to just write the answer to this problem, for example, as 64π. In other words, <u>not</u> multiplying 64 by 3.14 yields an accurate result."</strong>&nbsp;<br><br>Using the formula, give students different values for radii of circles and ask them to calculate the area, as well as give them with different diameters and ask for the area. Move to Activity 2 once students have gained confidence in using the formula.<br><br><strong>Activity 2</strong><br><br><strong>"Now, given the radius or diameter, you can calculate the area of a circle. If you know the area of a circle, can you work backwards to calculate its radius? Suppose there is a tarp that covers a circular swimming pool. You know the tarp covers an area of 144π square feet. How can you calculate the radius?"</strong>&nbsp;<br><br>Help students realize that they must take the square root of 144. If students have trouble with square roots, asking them, <strong>"What number squared equals 144?"</strong> or <strong>"What number times itself equals 144?"</strong> should help.&nbsp;<br><br>Explain to students that since the formula for area is π<i>r</i>², we can write π<i>r</i>² = 144π and cancel out the πs on each side.&nbsp;<br><br><strong>"What if the area of the tarp is 615.44 square feet, and we need to calculate the radius? What makes this situation different from the previous one?"</strong> Students should notice that the previous area contained π (i.e., was stated in terms of π), whereas this one does not.<br><br><strong>"So, we can write π</strong><i><strong>r</strong></i><strong>² = 615.44, but now what?"</strong>&nbsp;<br><br>Students should realize that they still need to remove the π, and that when it was "canceled" in the previous problem, we were actually dividing both sides of the equation by π. <strong>"So, we still need to divide both sides of the equation by π, although we'll divide both sides by 3.14, our approximation for π instead. How does our equation look after dividing both sides by π?"</strong> (<i>r² = 196</i>) <strong>"So what is the radius of our tarp?"</strong> (<i>14 feet</i>)&nbsp;<br><br>Explain that the general method is to get <i>r</i>² by itself by dividing both sides of the equation by π (or 3.14). Then simply take the square root of both sides.<br><br><strong>"Now, what if the area of our tarp is 706.5 square feet and we want to know its </strong><i><strong>diameter</strong></i><strong>? What steps do we have to do differently, or are there any additional steps do we need to perform?"</strong> Guide students to the idea that the first half of the problem is the same: find the radius. Then, to find the diameter, double the radius.&nbsp;<br><br>Students should work in pairs to complete the Area Worksheet (M-7-4-2_Area Worksheet and KEY). If students have already mastered circumference, include the last column on the worksheet; otherwise, they should ignore it. Instruct students to express their areas and circumferences in terms of π wherever possible. If they are given an area that is not expressed in terms of π, they should use π = 3.14.<br><br>Ask students to turn in their completed worksheets at the end of class.&nbsp;<br><br>This lesson explained the area of a circle using existing knowledge, incorporating a diagram, and estimating abilities to appeal to visual learners and those who prefer an intuitive approach. Students then used the actual formula, which is slightly more abstract, to appeal to students who like computation and algebraic concepts.&nbsp;<br><br><strong>Extension:</strong>&nbsp;<br><br>Use the strategies listed below to adjust the lesson to your students' needs throughout the year.<br><br><strong>Routine:</strong> Finding the area of circles can be integrated into a wide range of other geometric topics throughout the year, and with simple numbers, it can easily be used in future quizzes, entrance or exit tickets, etc. Students can also study and develop their understanding of (estimating) square roots in the context of the area of circles. Students can practice circular calculations by playing the game below.&nbsp;<br><a href="http://www.quia.com/cb/10522.html">http://www.quia.com/cb/10522.html</a>&nbsp;<br><br><strong>Small Group:</strong> Students who require further practice can be divided into small groups and given the Area Worksheet (M-7-4-2_Area Worksheet and KEY). Students can also practice calculating the areas of circles with the following game. <a href="http://www.aaamath.com/exp612x2.htm">http://www.aaamath.com/exp612x2.htm</a>&nbsp;<br><br>Students who could benefit from supplementary teaching will find the following website useful.&nbsp;<br><a href="http://www.mathsisfun.com/geometry/circle-area.html">http://www.mathsisfun.com/geometry/circle-area.html</a>&nbsp;<br><br><strong>Expansion:</strong> For students who are ready to take on a challenge that goes above and beyond the standard. This lesson can be extended to include ellipses, which have the formula πab, where a and b are the lengths of the "long" and "short" radii of the ellipse. Students will discover that the formula for the area of a circle is simply a special example of this, where <i>a = b = r</i>, and therefore a circle is essentially a type of ellipse. Use the following websites as resources. Have students write a brief explanation of how to calculate the area of an ellipse. They can provide some instances using calculations. <a href="http://www.math.hmc.edu/funfacts/ffiles/10006.3.shtml">http://www.math.hmc.edu/funfacts/ffiles/10006.3.shtml</a>&nbsp;<br><br>The following website allows you to check your answers. <a href="http://www.csgnetwork.com/areaellipse.html">http://www.csgnetwork.com/areaellipse.html</a>&nbsp;<br><br>The lesson can also be expanded to incorporate three-dimensional shapes with circles (mainly cylinders and cones, but spheres could also be included due to similar formulas). Finding the surface area and volume of these 3D objects requires calculating the area and circumference of circles. (for determining the surface area of a cylinder).</p>