Mathematics Success Level D T819 [OBJECTIVE] The student will be able to find a missing angle measure when two non-overlapping angles form one larger angle. [PREREQUISITE SKILLS] measuring angles with a protractor, adding and subtracting whole numbers, Lesson 27 [MATERIALS] Student pages S272 S280 Transparencies T828, T830, T832, T834, and T836 String (one 6-inch piece per student) Protractor (1 per student pair) [ESSENTIAL QUESTIONS] 1. When an angle is decomposed into two angles and you know the measurement of the original angle and one of the smaller angles, explain how to find the measurement of the second angle. 2. When an angle is decomposed into two angles and you know the measurement of both smaller angles, explain how to find the measurement of the original larger angle. 3. What are non-overlapping angles? [WORDS FOR WORD WALL] non-overlapping, ray, angle, decompose, degrees [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) *For Cooperative Pairs (CP) activities, assign the roles of Partner A and Partner B to students. This allows each student to be responsible for designated tasks within the lesson. [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Graphic Organizer, Pictorial Representation, Concrete Representation, Algebraic Formula
T820 Mathematics Success Level D [WARM-UP] (5 minutes IP, WG, I) S272 (Answers are on T827.) Have students turn to S272 in their books to begin the Warm-Up. Students will classify angles as right, acute, or obtuse. They will then measure each angle with a protractor. Monitor students to see if any of them need help during the Warm- Up. Give students 3 minutes to complete the problems and then spend 2 minutes reviewing the answers as a class. {Pictorial Representation, Verbal Description} [HOMEWORK] (5 minutes) Take time to go over the homework from the previous night. [LESSON] (60 minutes M, GP, IP, CP, WG) SOLVE Problem (3 minutes WG, GP) T828, S273 (Answers on T829.) Have students turn to S273 in their books, and place T828 on the overhead. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to find a missing angle measurement when two non-overlapping angles form one larger angle. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Graphic Organizer} Additive Angle Measures - Concrete (10 minutes M, GP, WG, CP) T828, S273 (Answers on T829.) 10 minutes M, GP, WG, CP: Have students turn to S273 in their books, and place T828 on the overhead. Give each pair of students two pieces of string. Designate the roles of Partner A and Partner B. {Verbal Description, Concrete Representation, Pictorial Representation}
Mathematics Success Level D T821 MODELING Additive Angle Measures - Concrete Step 1: Have Partner A take one piece of string and place it on the desk so that it forms a straight line. Model with a piece of string on the overhead. Partner B, identify what type of angle the string represents. (straight) Record. Partner A, explain how many degrees are in a straight angle. (180 ) Record. Step 2: Have Partner B take the other piece of string and place it on the desk so that it is perpendicular to the first string. Model with a second piece of string on the overhead. Partner B, explain what the second piece of string did to the angle. (split it into two angles) Record. Partner A, identify what type of angles these are. (right) Record. How do you know? (You could put a square in the corner of each one.) Partner B, what is the measure of Angle 1? (90 ) Partner A, what is the measure of Angle 2? (90 ) Step 3: Have students add the two right angles together. What is the sum? (90 + 90 = 180 ) Record. Do the two angles take up the same amount of space as the original angle? (Yes.) What is the measurement of the angle with just one string? (180 ) What is the measurement of the two smaller angles put together? (180 )
T822 Mathematics Success Level D Additive Angle Measures - Pictorial (10 minutes M, GP, WG, CP) T830, S274 (Answers on T831.) 10 minutes M, GP, WG, CP: Have students turn to S274 in their books, and place T830 on the overhead. Pass out a protractor to each student pair. {Verbal Description, Pictorial Representation} MODELING Additive Angle Measures Pictorial Step 1: Direct students attention to Problem 1. Have students discuss how the angle is labeled. (Angles can be named with three letters, a point on one ray, the vertex, and a point on the other ray.) Trace each angle on the overhead as you say the names: Angle BAC, trace from Point B to Point A to Point C. Angle CAD, trace from Point C to Point A to Point D. Point out that these two angles share a ray and are non-overlapping. Angle BAD, trace from Point B to Point A to Point D. Point out that this angle is the same as angles CAD and BAC put together. Step 2: Model for the students how to measure angle BAC with the protractor. Be sure they line up ray AC with zero and measure where ray AB hits the protractor. Partner A, identify the measure of angle BAC. (50 ) Record. Step 3: Ask students to measure angle CAD. Have Partner A place the protractor to measure and Partner B tell how many degrees are in angle CAD. Partner B, identify the measure of angle CAD. (25 ) Record. Step 4: Partner A, determine if angle BAC and angle CAD share a common ray. (Yes.) Record. How do angles BAC and CAD relate to angle BAD? (The two angles together have the same measure as angle BAD.) Partner B, explain how you could find the measure of angle BAD without using a protractor. (Add angles BAC and CAD together.) Record. What is the measure of angle BAD? (25 + 50 = 75 ) Record.
Mathematics Success Level D T823 Step 5: Direct students attention to Problem 2. Model for the students how to measure angle LPN with the protractor. Be sure they line up ray PN with zero and measure where ray PL hits the protractor. Show students that angle LPN measures 162. Record. Step 6: Ask students to measure angle LPM. Have Partner B place the protractor to measure, and Partner A tell how many degrees are in angle LPM. How many degrees are in angle LPM? (108 ) Record. Step 7: Do the two angles share a common ray? What are the angles? (Yes, angles LPM and MPN.) Do you know the measurements of these two angles? (No, we know the largest angle LPN, and one of the smaller angles LPM.) Since we know the largest angle and only one of the smaller angles, how do you think we could find the measure of angle MPN? (Subtract the smaller angle measure from the largest angle measure.) Record. What is the measure of angle MPN? (162 108 = 54 ) Step 8: Tell students that this is called decomposing. When we decompose an angle, it means breaking the angle into two smaller angles. The two smaller parts are non-overlapping angles, which means that they share a ray but do not overlap. Since they take up the same amount of space as the original angle, the number of degrees in the two smaller angles can be added to get the number of degrees in the larger angle. Record. Step 9: Rule: When an angle is decomposed into non-overlapping parts, the angle measure of the whole equals the of the angle measures of the parts. (sum) Record.
T824 Mathematics Success Level D Additive Angle Measures - Practice (15 minutes M, GP, IP, WG, CP) T832, S275 (Answers on T833.) 3 minutes M, GP, WG, CP: Have students turn to S275 in their books, and place T832 on the overhead. {Verbal Description, Pictorial Representation, Algebraic Formula, Graphic Organizer} MODELING Additive Angle Measures Practice Step 1: Direct students attention to the graphic organizer on S275. Step 2: In Column 1, the angles are shown and the angle measures are given. Partner A, what is the measure of angle EFH? (170 ) Partner B, what is the measure of angle GFH? (38 ) Step 3: In Column 2, have students identify the two angles to add together to get the third angle. (Angle EFG and GFH equal angle EFH.) Partner B, explain how to write an equation to show that operation. ( EFG + GFH = EFH) Record. Partner B, what measurements can you fill in? (Angle EFH and Angle GFH) What do you think we could use for the measurement we don t know? (use the angle, so it looks like this: EFG + 38 = 170 ) Record. Step 4: Direct students attention to Column 3. Have students discuss what operation can be used to determine a missing addend. (the opposite of addition which is subtraction) Show the work for subtracting. Record. EFG + 38 = 170 170 38 = EFG 170 38 = 132 Step 5: Record the answer in Column 4. (132 )
Mathematics Success Level D T825 9 minutes IP, CP: Have students complete Problems 2 4 on S275. Tell students that they will be finding the total measure of the non-overlapping angles or the measure of one of the small angles when given the total measure and the measure of one angle. {Verbal Description, Pictorial Representation, Algebraic Formula, Graphic Organizer} 3 minutes WG: Use the transparency master to go over the answers. Make sure to discuss how the students knew whether they were adding or subtracting. {Verbal Description, Pictorial Representation, Algebraic Formula, Graphic Organizer} Additive Angle Measures - Word Problems (15 minutes M, GP, IP, WG, CP) T834, S276 (Answers on T835.) 5 minutes M, GP, WG, CP: Have students turn to S276 in their books, and place T834 on the overhead. {Verbal Description, Pictorial Representation, Algebraic Formula, Graphic Organizer} MODELING Additive Angle Measures Word Problems Explain to students that on page S276 they will be determining missing angle measurements in word problems. Read Problem 1 aloud. Partner A, identify the information that is given. (Angle measures 108 ; angle is split into 2 angles and one of those measures 69.) Partner B, explain how to find the missing angle. (Subtract 108-69.) Record. 7 minutes IP, CP: Have students complete the word problems on S276. Tell students that they will be finding the total measure of the non-overlapping angles or the measure of one of the small angles when given the total measure and the measure of one angle. {Verbal Description, Pictorial Representation, Algebraic Formula, Graphic Organizer} 3 minutes WG: Use the transparency master to go over the answers. Make sure to discuss how the students knew whether they were adding or subtracting. {Verbal Description, Pictorial Representation, Algebraic Formula, Graphic Organizer}
T826 Mathematics Success Level D SOLVE Problem (5 minutes GP, WG) T836, S277 (Answers on T837.) Remind students that the SOLVE problem is the same one from the beginning of the lesson. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (Students have to use decomposed angles to find the measure of an angle.) {SOLVE, Verbal Description, Graphic Organizer} If time permits (10 minutes IP, CP) S278 (Answers on T838.) Have students work in partners to complete Problems 1 and 2 on S278. [CLOSURE] (2 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. When an angle is decomposed into two angles and you know the measurements of the original angle and one of the smaller angles, explain how to find the measurement of the second angle. (Subtract the smaller angle measure from the measure of the original angle.) When an angle is decomposed into two angles and you know the measurements of both smaller angles, explain how to find the measurement of the original larger angle. (Add the two smaller angles to get the measure of the original larger angle.) What are non-overlapping angles? (two angles that share one ray) [HOMEWORK] Assign S279 and S280 for homework. (Answers on T839 and T840.) [QUIZ ANSWERS] T841 T843 The quiz can be used at anytime as extra homework or to assess how students progress on understanding the concept of additive angle measures.