During What could you do to the angles to reliably compare their measures?

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1 Measuring Angles LAUNCH (9 MIN) Before What does the measure of an angle tell you? Can you compare the angles just by looking at them? During What could you do to the angles to reliably compare their measures? After Were you surprised by the results? Why or why not? How do you think the size of an airplane s nose angle affects how the airplane flies? PART 1 (8 MIN) During the Intro Why can the ABC shown also be named CBA, but not BCA? While solving the Example How can you position the protractor to measure an angle? How do you decide which of the two scales on the protractor to use? After solving the Example What is the advantage of a protractor that has two scales instead of one? Dana Says (Screen 2) Use the Dana Says button to discuss how measuring angles is useful in everyday situations. PART 2 (7 MIN) During the Intro Which types of angles can have only one measure? Which types can have more than one possible measure? Explain. Why is a right angle a good gauge for measuring other angles? While solving the problem What sorts of everyday objects have right angles that can help you gauge the measure of other angles? Dana Says (Screen 2) Use the Dana Says button to spark a discussion about how a line is also an angle. How can two rays form a line and an angle? PART 3 (8 MIN) Before solving the problem What does the diagram tell you about angle 1? While solving the problem How can you use what you know about angle 1 to find x? CLOSE AND CHECK (8 MIN) How can you compare the sizes of two angles? How can you classify angles by their measures?

2 Measuring Angles LESSON OBJECTIVES 1. Name parts of a geometric figure using appropriate letters and symbols 2. Measure parts of geometric figures using the appropriate tools and units of measure. 3. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations to solve problems by reasoning about the quantities. FOCUS QUESTION All angles are formed by two rays. What makes angles different from each other? MATH BACKGROUND In previous grades, students were exposed to vocabulary about angles and angle relationships. They are familiar with terms such as ray, angle, right angle, and perpendicular, and they should be able to identify examples of each term in a diagram. In this lesson, students learn that the size of an angle can be measured in degrees. They visually estimate which of two angles has a greater measure, measure angles precisely with a protractor, and classify angles by their measures. Students also solve algebraic equations involving angle measures. This lesson builds a foundation for talking about angle relationships later in the topic. For example, in upcoming lessons students will see that angle measures are the basis for classifying two angles as complementary or supplementary. LAUNCH (9 MIN) Objective: Compare angle measures without a protractor. Author Intent This Launch helps students think creatively about how to compare angles without tools such as a protractor to assist them. Students must decide whether the measure of an angle should be based on the length of its sides or the distance between the sides. Questions for Understanding Before What does the measure of an angle tell you? [It tells you the size of the opening between the angle s two sides.] Can you compare the angles just by looking at them? [No; you can make a guess about their sizes, but each angle is facing a different direction, which makes it hard to check your guess.] During What could you do to the angles to reliably compare their measures? [Sample answer: You could trace 1 on a piece of paper. Then you could overlay the copy of 1 on 2 so that the angles have the same vertex and one matching side. This will show which angle has a greater distance between its sides.] After Were you surprised by the results? Why or why not? [Sample answer: Yes; I was surprised because the measures of the angles looked about the same, although 1 seemed smaller because its sides are shorter.]

3 How do you think the size of an airplane s nose angle affects how the airplane flies? [Sample answer: A smaller angle creates a sharper point. This probably helps the plane cut through the atmosphere better so that it flies faster.] Solution Notes It is not necessary to trace both sides of an angle to make a comparison. Students could also use an edge of a piece of paper to represent one side and then trace the second side starting from a point along the paper s edge. As long as students align that point and edge with the vertex and side of the other angle, they should be able to make a good comparison. Students could also fold the piece of paper into a paper airplane whose nose is the same size as one of the angles, and then place the nose inside the other angle to make a comparison. Connect Your Learning Move to the Connect Your Learning screen. Start a conversation about what makes two angles different from each other. Even though the rays forming one angle may have a different length than the rays forming the other angle, discourage students from identifying this as a difference. Emphasize that the distance between an angle s rays (that is, how wide an angle opens) gives a more meaningful way to compare angles. PART 1 (8 MIN) Objective: Use a protractor to measure degrees in an angle. Author Intent Students use a protractor to find the measures of two angles in degrees. In the Launch, students had to rely on a visual comparison of the sizes of two angles to determine which had the greater measure. Now they see that angle measures can be described quantitatively, so that the task of determining which angle is bigger can be reduced to comparing numbers. Instructional Design Use the Intro to define important vocabulary and notation related to angles and to show how to measure an angle with a protractor. Make sure students understand that the sides of an angle are the same as the two rays that form the angle. Move to the Example. The Example shows two angles as well as a protractor that you can drag and rotate. Call two students to the whiteboard. Have one student use the protractor to measure 1, and have the other student use it to measure 2. Questions for Understanding During the Intro Why can the ABC shown also be named CBA, but not BCA? [The first letter must be a point on one side of the angle, the second letter must be the vertex, and the third letter must be a point on the other side of the angle. The name CBA follows this pattern, but BCA does not because the vertex B is not the second letter.] While solving the Example How can you position the protractor to measure an angle? [Sample answer: Line up the lower side of the angle with the protractor s base so the side passes through 0 and the vertex is at the center of the protractor.]

4 How do you decide which of the two scales on the protractor to use? [Sample answer: One side of the angle will pass through 0 on one of the two scales. Use the same scale to read off the angle measure.] After solving the Example What is the advantage of a protractor that has two scales instead of one? [Sample answer: It allows you to measure an angle no matter how it is positioned.] Dana Says (Screen 2) Use the Dana Says button to discuss how measuring angles is useful in everyday situations. Solution Notes If students are confused about how to use a protractor, the animated solution will model for them how to position and read the protractor to ensure an accurate measurement. It may be difficult for students to read off angle measures if the sides of the angles are not long enough to intersect the correct scale. You can remind students that a side can be of any length without changing the final angle measure. Show students how they can use a straightedge to extend a side through the protractor scale. Although it is usually easier to measure an angle by positioning the protractor s base along the side that is more nearly horizontal, you can show students how starting from the other side will give the same result. Error Prevention Some students may be confused about which scale to use to measure a given angle. Encourage students to first estimate whether the angle is greater than, less than, or equal to 90, since right angles are easy to identify. This allows students to check the reasonableness of their measurement and decide if they used the correct scale. Got It Notes If you show answer choices, consider the following possible student errors: Students who choose A are using the wrong scale on the protractor. They should be using the bottom scale but are using the top scale instead. Students who choose B are using the wrong scale; they are also reading that scale incorrectly, as though the numbers were increasing from right to left rather than from left to right. Students who choose D are making a similar mistake with the other scale. PART 2 (7 MIN) Objective: Name angles based on measure. ELL Support Beginning Prepare a set of cards with pictures of right, acute, obtuse, and straight angles. Say an angle type aloud (right, acute, obtuse, or straight) and have students choose the matching cards. Intermediate Have students create and play their own sorting game with a partner, where one student says an angle type and the other student draws that type of angle on a card. Then have students switch roles. After several drawings are made, partners can then work together to classify each drawing and explain the definition of that type of angle.

5 Advanced Have students create their own mnemonics for recalling each term and then explain their strategies to a small group. Each group should discuss the various mnemonics and propose the best ones to the class. [Sample: Obtuse sounds like AAH when you open your mouth wide so it refers to an angle larger than 90 degrees but less than 180 degrees.] Author Intent Students classify angles as acute, right, obtuse, or straight. The measure of each angle is not actually given, so students have to visually estimate whether the measure is less than, equal to, or greater than 90 (or equal to 180 ). The angles have different orientations to reinforce the fact that an angle s measure does not depend on its orientation. Instructional Design Use the Intro to show students how angles can be classified based on their measures. You can invite students to make a conjecture about the measure of each type of angle shown in the diagram, and then click on the angle type to see if they are correct. Move to the Example. There are six angles that students need to classify. Have a different student classify each angle by dragging the appropriate angle type to the space below the angle. Ask the students to explain their reasoning. Questions for Understanding During the Intro Which types of angles can have only one measure? Which types can have more than one possible measure? Explain. [A right angle and a straight angle can have only one measure. A right angle must measure 90 and a straight angle must measure 180. An acute angle and an obtuse angle can have many possible measures. An acute angle can have any measure between 0 and 90, and an obtuse angle can have any measure between 90 and 180.] Why is a right angle a good gauge for measuring other angles? [Sample answer: A right angle has a very recognizable form; when one side is horizontal, the other side is vertical. An angle that opens less wide than a right angle is acute, and an angle that opens wider than a right angle is obtuse or straight.] While solving the problem What sorts of everyday objects have right angles that can help you gauge the measure of other angles? [Sample answer: A piece of paper has four corners that all form right angles. I can compare a corner of a piece of paper to other angles and decide whether the angles are smaller or larger than 90º.] Dana Says (Screen 2) Use the Dana Says button to spark a discussion about how a line is also an angle. How can two rays form a line and an angle? [If the rays share the same endpoint but point in opposite directions, together they form a line. As long as the rays share an endpoint but do not point in the same direction, they form an angle.] Solution Notes Students may struggle to estimate the measure of the angle that appears to be upside down. Talk about ways you could change the orientation of the angle to better compare it to a right angle. Have a student trace the angle onto a piece of paper and then adjust the position of the angle based on others suggestions. If students need proof that the angle has the same measure no matter how it is positioned, use a protractor to measure the angle in each orientation.

6 Differentiated Instruction For struggling students: Some students may need help remembering the names for the angle types. You could use props or arm gestures to visually illustrate the types of angles. For example, acute angles can look like a capital A, and they have a cute little size that is smaller than 90º. For advanced students: Have students draw circles and divide them into halves, thirds, quarters, fifths, etc. they should name the type of angle formed and measure the measure of the angles in each circle. Although students may not yet have learned that there are 360 degrees in a circle, they can make a conjecture from their results. Error Prevention Although two of the angles in the Example look like right angles, students should realize, based on the differences in notation and the comparisons they make to right angles in everyday objects, that one of the angles measures slightly less than 90º. Use this situation to reinforce the importance of relying on notation, and not assumptions, to logically deduce characteristics of geometric figures. Got It Notes This Got It approaches angle identification in a reverse manner from the Example students must determine which angle type is missing. This extends students practice identifying angle types and prepares them to break down adjacent and overlapping angles in later lessons. If you show answer choices, you may want students to name each angle they can identify and record the name next to the angle type that describes it. By process of elimination, students can determine which angle type is not represented. To help students avoid errors naming and identifying the angles, encourage them to use their hand or a piece of paper to cover the part of the diagram that is not included in a given angle. This will help students visualize the individual angles better. Got It 2 Notes Students make a conjecture and then construct an argument to support their conjecture. Although most students will probably use words to explain their reasoning, they can also use symbols: If x and y are the measures in degrees of the two acute angles, then x 90 and y 90. So x y , and therefore the sum of the measures of the acute angles is less than the measure of a straight angle. PART 3 (8 MIN) Objective: Construct simple equations to solve problems about angles. ELL Support On the Student Companion page for the Part 3 Got It, there are two tasks for students to complete and discuss: How did you apply what you know about solving equations to solve this problem? Describe the steps you took to set up the equation to solve. Beginning Have students write and compare the equations they wrote for the problem. Pay particular attention to the level of detailed language that students are using when explaining how they applied their understanding of the situation to writing the equation.

7 Intermediate Have students use the Know-Need-Plan organizer to break down the problem. Then have them write and compare the equations, checking their work against what was written in the graphic organizer. Advanced Have students identify 2 to 3 reasoning errors that might be made when writing an equation to describe the problem situation. Then have them collect data on how many classmates made those types of reasoning errors when they first wrote their equations to the problem. Author Intent Students use what they know about the measure of a right angle to solve an equation for an unknown value. This helps students make a connection between algebra and geometry, which prepares them to integrate these disciplines in later lessons and math courses. Questions for Understanding Before solving the problem What does the diagram tell you about angle 1? [The diagram shows that angle 1 is a right angle, which means that its measure is 90.] While solving the problem How can you use what you know about angle 1 to find x? [Since you know that the measure of 1 is both 3x 15 and 90, you can solve the equation 3x to find x.] Solution Notes Setting up the equation is likely the biggest challenge students face in finding x. Make sure students understand that two different expressions can describe the measure of angle 1. Since the measure is both 3x 15 and 90, it must be the case that 3x Students may wonder what x means in this situation. You can explain that this problem helps them practice using algebra to solve problems in geometry; however, in this instance, x does not have a specific meaning. Got It Notes The Got It is similar to the Example, but this time the algebraic expression is part of the diagram of the angle. This helps familiarize students with other ways they might see algebra incorporated into geometry problems. If you show answer choices, consider the following possible student errors: Students who choose A may have subtracted 30 from the right side of the equation 2x instead of adding 30. Students who select C may have set up the equation 2x instead of 2x 30 60, perhaps mistakenly assuming that they must use information from the Example. Students who choose D may have correctly gotten to the step 2x 90 of the solution, but then may have multiplied the right side by 2 instead of dividing by 2.

8 CLOSE AND CHECK (8 MIN) Focus Question Sample Answer Angle measures make angles different from each other. Angles between 0 and 90 are acute angles. Angles that measure exactly 90 are right angles. Angles between 90 and 180 are obtuse angles. Straight angles have a measure of exactly 180. Focus Question Notes As you discuss how angles differ from each other, make sure students understand that even if appearance seems to be a differentiator, the only way to know that angles are different is by their measure. Discuss how students can use right angles as a gauge for whether an angle measures greater than or less than 90. Essential Question Connection The Essential Question asks how you can best describe relationships between angles, and which relationships are more useful than others. Use the following questions to turn the discussion toward the Essential Question. How can you compare the sizes of two angles? [Sample answer: You can compare the measures of the angles in degrees. The angle with the greater measure can be considered bigger. ] How can you classify angles by their measures? [Sample answer: You can classify an angle as acute, right, obtuse, or straight based on how the angle s measure is related to 90 and 180.]