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About Circles

I. Section Objectives

  • Distinguish between radius, diameter, chord, tangent, and secant of a circle.
  • Find relationships between congruent and similar circles.
  • Examine inscribed and circumscribed polygons.
  • Write the equation of a circle.

II. Multiple Intelligences

  • To differentiate this lesson, break the lesson down into two sections. In the first section, cover all of the basic information about circles.
  • Have the students work on creating a diagram of a circle. Their diagram must have the following things labeled: radius, chord, diameter, secant, tangent line.
  • Encourage students to make their diagrams colorful.
  • Allow time for sharing when students are finished.
  • The second part of the lesson involves more of the operations associated with circles.
  • For this lesson, be sure that students have graph paper to work with.
  • Complete the examples on the board and walk through each of the examples and all of the steps needed to complete each one.
  • Point out where to find the radius and the ordered pair in the equation.
  • Make this section interactive so that you work through the example on the board/overhead while the students work through it in their notebooks.
  • Working through this as a whole class will help the students to follow the steps of each problem.
  • Intelligences- linguistic, visual- spatial, logical- mathematical, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Include the following notes for students.
  • Two circles are congruent if they have the same radius. Two circles are similar if they have different radii. Their similarity is shown through a ratio.
  • When writing similarity ratios be sure to simplify.
  • Remember that you can write the ratios in three ways. The text uses a colon, but you can use a fraction or the word “to”.
  • Define chord.
  • Define diameter.
  • Define secant.
  • Tangent line- touches a circle at one point. This is called the point of tangency.
  • Inscribed polygon- convex polygon inside circle.
  • Circumscribed polygon- convex polygon around circle.
  • Review convex polygons.
  • Equations with graphing- work through slowly.
  • Concentric circles- practice drawing them.

IV. Alternative Assessment

  • Assess this lesson through student drawings.
  • If the students can draw and label the parts of the circle, then they have an understanding of it.
  • If the students can graph the circles, then they have an understanding of the process.

Tangent Lines

I. Section Objectives

  • Find the relationship between a radius and a tangent to a circle.
  • Find the relationship between two tangents draw from the same point.
  • Circumscribe a circle.
  • Find equations of concentric circles.

II. Multiple Intelligences

  • Review tangents of a circle and the point of tangency.
  • Tangent to a Circle Theorem- work with this theorem by first having the students complete an exploration.
  • Have them begin by drawing a circle, the radius and a tangent line.
  • Then ask what they can observe about the relationship between the radius and the tangent line. You want them to discover the theorem on their own and then you can put a name to it for them.
  • When working with the Pythagorean Theorem and finding the hypotenuse of the triangle, ask students “How can we find the length of the hypotenuse?”
  • Once again, you want the students to come up with the Pythagorean Theorem on their own.
  • Show Pythagorean Theorem in Example 1.
  • When working with the Tangent Segments from a Common External Point Theorem- begin with an exploration.
  • Have the students draw it out and then record student observations on the board.
  • You want them to discover the theorem on their own.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal

III. Special Needs/Modifications

  • There is a lot of skills/vocabulary to review in this lesson. Do not assume that the students remember.
  • Review radius.
  • Define circumscribe.
  • Define concentric circles.
  • Review what is meant by a contradiction.
  • Example 1- rather than beginning by proving this theorem through a contradiction, use the exploration first. Then go through and show how to use a contradiction. Beginning with the contradiction can confuse many students.
  • Review the Pythagorean Theorem.
  • Review converse statements.
  • Example 5- show students where the hypotenuse is located.
  • Review how to rationalize a denominator.
  • Review HL congruence.

IV. Alternative Assessment

  • Review student thinking by observing their work during the observations.
  • Are the students able to come to conclusions about each theorem before it is actually taught?
  • Is there higher level thinking involved?
  • Can the students verbalize the steps to working through an example?

Common Tangents and Tangent Circles

I. Section Objectives

  • Solve problems involving common internal tangents of circles.
  • Solve problems involving common external tangents of circles.
  • Solve problems involving externally tangent circles.
  • Solve problems involving internally tangent circles.
  • Common tangent

II. Multiple Intelligences

III. Special Needs/Modifications

  • This lesson focuses on problem solving.
  • These notes will work for multiple intelligences and special needs students.
  • The work that we are going to do to differentiate this lesson is to be sure that the steps to working through the problems are clear and understood.
  • List out these steps as you teach the lesson. Request that the students copy these notes into their notebooks.
  • Define common internal tangent.
  • Define common external tangent.
  • Steps to common External Tangents
  • 1. Label the diagram.
  • 2. Draw a line segment that joins the centers of the two circles.
  • 3. Draw in the perpendicular segments.
  • 4. Look for any polygons – Example = rectangle.
  • Steps to Common Internal Tangents
  • 1. Look for similarity. Here in Example 2, there are similar triangles.
  • “Who can we find the length of x? Think back to our work on similar triangles.”
  • Lead students to discover working with ratios.
  • 2. Then we can find the length of the hypotenuse of the two triangles to identify the distance between the two circles.
  • Use the Pythagorean Theorem to do this.
  • Intersecting circles- define internally and externally tangent
  • Walk through each example with the students. Rely on previously learned material from this lesson.

IV. Alternative Assessment

  • Be sure that the students are making the connections between the circles, perpendicular segments and where polygons and right triangles can help them in solving each problem.
  • The students need to combine previously learned material to be successful with this lesson.
  • Review information and skills when necessary.

Arc Measures

I. Section Objectives

  • Measure central angles and arcs of circles.
  • Find relationships between adjacent arcs.
  • Find relationships between arcs and chords.

II. Multiple Intelligences

  • Define Central angle
  • Define arc
  • In each part of this lesson, work with each diagram and then have the students brainstorm different observations about each diagram. Then give them the information from each diagram.
  • For example, begin with the diagram that demonstrates the Arc Addition Postulate. It is common sense that the two arcs would add up to be the total. Give students the name of the postulate and ask them to come up with the meaning of the postulate given the diagram. Then walk them through it.
  • Do the same with the diagram on the congruent chords.
  • In Example 1- show that all angles must add to be 360^\circ.
  • Use the straight line to show the 180^\circ, then talk to the students about working through the puzzle of figuring out the measure of each angle inside the circle.
  • Keep this lesson interactive and engage students in participating in the discussion.
  • Refrain from simply presenting the material to them.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

  • Define semicircle with a diagram.
  • Define major arc with a diagram- named with three letters.
  • Define minor arc with a diagram- named with two letters (endpoints).
  • Review solving multi- step equations.
  • Present diagrams first and then engage the students in a discussion about the diagram.

IV. Alternative Assessment

  • Create a checklist of important points that you want students to discover in the lesson.
  • Then listen for these points while you discuss the information and present the material to the students.
  • During the practice exercises, pair students up to work together. This will help students to clarify the given information.

Chords

I. Section Objectives

  • Find the lengths of chords in a circle.
  • Find the measure of arcs in a circle.

II. Multiple Intelligences

  • To differentiate this lesson, have the students work on an activity in small groups.
  • The students are going to create a diagram to teach a theorem to the other students in the class.
  • Divide the students into groups of four. Each group is assigned a different theorem.
  • Group 1- The perpendicular bisector of a chord is the diameter.
  • Group 2- the perpendicular bisector of a chord bisects the arc intercepted by the chord.
  • Group 3- Congruent chords in the same circle are equidistant from the center of a circle.
  • Group 4- Two chords equidistant from the center of a circle are congruent.
  • Allow time for the students to work and then have each group teach the class about their theorem.
  • Allow time for students to ask questions.
  • From this activity, move to the longer examples in the text. The students should have an easier time working through these examples now that the theorems are very clear.
  • Intelligences- logical- mathematical, linguistic, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal

III. Special Needs/Modifications

  • Begin by defining a chord as a line segment whose endpoints are both on a circle.
  • Show students a diagram to define a chord.
  • Write out each theorem on the board. Request that students write the notes in their notebooks.
  • Review the terms diameter, bisector, perpendicular, congruent.
  • Point of the significance of the word “equidistant” in two of the theorems.

IV. Alternative Assessment

  • Use flexible grouping to engage all learners.
  • Walk around and observe students as they work on preparing their lesson.
  • Be sure that each presentation accurately teaches the content of the lesson.
  • Provide correction and feedback when necessary.

Inscribed Angles

I. Section Objectives

  • Find the measure of inscribed angles and the arcs they intercept.

II. Multiple Intelligences

  • Begin by teaching the material in the first part of this lesson. Stop before you get to the practical examples where students are actually figuring out angle measures.
  • To expand student understanding, make the corollary section interactive.
  • Have students work in pairs to draw out an example of each corollary.
  • Tell students that you will be collecting the examples at the end of the class.
  • Then move on to the actual examples in the lesson.

III. Special Needs/Modifications

  • Provide students with the following notes.
  • Inscribed angles- vertex on circles, sides are chords, intercepts an arc of the circle.
  • Review parts of an angle.
  • Review definition of a chord.
  • Measure of inscribed angle is \frac{1}{2} of the measure of the arc it intercepts.
  • Measure of center angle is 2 (measure of inscribed angle)
  • List out the inscribed angle corollaries.
  • When working on the multi-step examples, use color to help students to differentiate between which angles are being worked with and which ones aren’t being worked with. The color will help students to focus on the appropriate part of the diagram.

IV. Alternative Assessment

  • Walk around and help students as they work.
  • Collect student work from the corollaries activity.
  • Examine each example and see if it clearly demonstrates or shows the corollary.
  • Provide students with feedback/correction in the next class.

Angles of Chords, Secants and Tangents

I. Section Objectives

  • Find the measures of angles formed by chords, secants and tangents.

II. Multiple Intelligences

  • In this lesson, you are going to be working with three main theorems. The students need to learn these theorems and then prove each of the theorems.
  • Here are some notes to help students to break down each theorem.
  • Theorem- the measure of an angle formed by a chord and a tangent that intersects on the circle equals half of the measure of the intercepted arc.
  • \mathrm{mangle} = \frac{1}{2} \;\mathrm{marc}
  • Look at the first diagram and label the chord, tangent, intercepted arc and possible angles to measure.
  • Theorem- angles inside a circle
  • \mathrm{mangle} = \frac{1}{2} (\mathrm{marc}1 + \mathrm{marc}2)
  • Theorem- angles outside circle
  • \mathrm{mangle} = \frac{1}{2} (\mathrm{arc}^\circ 1 + \mathrm{arc}^\circ 2)
  • Once the students understand the three theorems, go through the example and proofs in the lesson. Ask the students to point out where the theorems are illustrated and explained in each.
  • Discuss each example and proof.
  • Allow time for questions.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

  • Review chords.
  • Review secants.
  • Review tangents.
  • Review interior angles.
  • Review exterior angles.
  • Review what an intercepted arc is and how to locate it.

IV. Alternative Assessment

  • As you work through each example, have the students work through the example in their seats.
  • When all have finished, ask the students to explain how they solved the problem.
  • Then provide feedback.
  • You can verbally check in with the students by having them raise their hands if they had the same answer. This will give you a visual cue of how many students were successful and how many were not.

Segments of Chords, Secants and Tangents

I. Section Objectives

  • Find the lengths of segments associated with circles.

II. Multiple Intelligences

  • Begin this lesson by having the students draw a circle. Then they need to use previously learned information and their text to draw in the following.
  • Tangent
  • Chord
  • Secant
  • Tangent segment
  • Chord segment
  • Secant segment
  • Have them use color to draw in each item.
  • When students are finished, explain that we are going to be using these diagrams to illustrate three different theorems.
  • When working through each theorem, give students the measurements for each section of the circle, and then have them work to figure things out.
  • Theorem- If two chords intersect, the product of segments of chord1 = product of segments of chord2.
  • Then we create two similar triangles.
  • Similar triangles- ratios
  • Add in measures and solve for the missing segment length.
  • Theorem- If two secants are drawn to a common point, a(a+b) =c(c+d)
  • The a’s = 1^{\mathrm{st}} secant
  • The c’s = 2^{\mathrm{nd}} secant
  • Draw in two triangles inside the circle.
  • Similar triangles- ratios
  • Use formula to find the length of the segment of the secant.
  • Theorem- tangent and secant- a(a+b) = c^2
  • The a’s = \mathrm{secant}
  • The c = the tangent
  • Use it to find the value of the missing tangent length.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, bodily- kinesthetic, intrapersonal.

III. Special Needs/Modifications

  • Review chords.
  • Review tangents.
  • Review secants.
  • Review squaring a number.
  • Review the distributive property.

IV. Alternative Assessment

  • Assess student learning through observation and class discussion.

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