Learning Targets: pply the Segment ddition Postulate to find lengths of segments. Use the definition of midpoint to find lengths of segments. SUESTED LERNIN STRTEIES: Close Reading, Look for a Pattern, Think-Pair-Share, Vocabulary Organizer, Interactive Word Wall, Create Representations, Marking the Text, Visualization, Identify a Subtask, Discussion roups In geometry, axioms, or postulates, are statements that are accepted as true without proof in order to provide a starting point for proving rules. Like point, line, and plane, distance along a line is an undefined term in geometry used to define other geometric terms. For example, the length of a line segment is the distance between its endpoints. If two points are no more than 1 foot apart, you can find the distance between them by using an ordinary ruler. (The inch rulers below have been reduced to fit on the page.) inches 1 2 3 4 5 6 7 In the figure, the distance between point and point is 5 inches. Of course, there is no need to place the zero of the ruler on point. In the figure below, the 2-inch mark is on point. In this case,, measured in inches, is 7-2 = 2-7 = 5, as before. 2 3 4 5 6 7 8 9 The number obtained as a measure of distance depends on the unit of length. For example, the distance between two points in inches will be a different number than the distance between the two points in centimeters. 1. Determine the length of each segment in centimeters. D E F MTH TERMS To prove a rule, at least one other rule must be used. So in order to develop geometry, some rules, called postulates, are accepted without proof. REDIN MTH denotes the distance between points and. If and are the endpoints of a segment (), then denotes the length of. MTH TERMS The Ruler Postulate a. To every pair of points there corresponds a unique positive number called the distance between the points. Continued PLN Pacing: 1 class period Chunking the Lesson #1 4 #5 #6 #11 #12 Example #14 Lesson Practice TECH ell-ringer ctivity sk students how they can measure the length of the classroom using only a 12-inch ruler. They will probably describe a process in which each 12-inch segment is marked and then the segments are added. Congratulate them and tell them that in this lesson the process they described will be formalized. 1 4 Close Reading, Look for a Pattern, Vocabulary Organizer, Marking the Text, Visualization efore students begin Items 1 4, read the text together, reminding them that not all terms can be defined and not all statements can be proven. Some terms can only be described and not defined, and some statements, called postulates, are intuitively obvious and cannot be proven. Confirm that students are measuring the line segments in Item 1 correctly. a. DE = 2.5 b. EF = 5.7 c. DF = 8.2 2. ttend to precision. Determine the length of each segment in centimeters (to the nearest tenth). a. KH = 4.9 b. H = 5.7 c. K = 10.6 H K b. The points on a line can be matched with the real numbers so that the distance between any two points is the absolute value of the difference of their associated numbers. TECHER to TECHER COMMON MISCONCEPTIONS e sure that students understand the difference between segment,, and, or the distance from to. Developing Math Language Monitor group discussions to ensure that all members of the group are participating and that each member understands the language and terms used in the discussion. ctivity 12 eometric Figures 219
Continued 1 4 () Have students read Items 2 and 3 and predict the pattern they expect to see. In Item 4, they will apply the pattern. Universal ccess s students respond to questions or discuss possible solutions to problems, monitor their use of new terms and descriptions of applying math concepts to ensure their understanding and ability to use language correctly and precisely. 5 Create Representations Encourage students to represent each situation by drawing and labeling the segment being described and writing an equation to find the missing measures. Discuss what it means for a point to be between two other points. (In particular, if is between and C, you can assume that,, and C are collinear and that + C = C.) Emphasize that the Segment ddition Postulate applies only if the points are collinear. CONNECT TO P Students will need to find distances in the coordinate plane in P Calculus. s you model the solutions to Item 5, make sure you draw some of the segments vertically or diagonally and not just horizontally so students are used to finding measures from those perspectives as well. You could also ask students to represent vertical and horizontal segments in the coordinate plane and find their measure. Finally, work with students to understand that if the measure of an entire segment is x and one part is some number b less than x, then the other part s measure would be x b. ELL Support s you guide students through their learning of these new essential mathematical terms, explain meanings in ways that are accessible for your students. s possible, provide concrete examples to help students gain understanding. Encourage students to use a graphic organizer such as Unknown Word Solver to make notes about new terms and their understanding of what they mean and how to use them to describe precise mathematical concepts and processes. MTH TERMS Item 4 and your answer together form a statement of the Segment ddition Postulate. MTH TIP For each part of Item 5, make a sketch so that you can identify the parts of the segment. CONNECT TO P You will frequently be asked to find the lengths of horizontal, vertical, and diagonal segments in the coordinate plane in P Calculus. 3. Using your results from Items 1 and 2, describe any patterns that you notice. Sample answer: When you add the lengths of the two shorter segments, you get the length of the longest segment. 4. iven that N is a point between endpoints M and P of line segment MP, describe how to determine the length of MP, without measuring, if you are given the lengths of MN and NP. MN + NP = MP 5. Use the Segment ddition Postulate and the given information to complete each statement. a. If is between C and D, C = 10 in., and D = 3 in., then CD = 13 in.. b. If Q is between R and T, RT = 24 cm, and QR = 6 cm, then QT = 18 cm. c. If P is between L and, PL = x + 4, P = 2x - 1, and L = 5x - 3, then x = 3 and L = 12. The midpoint of a segment is the point on the segment that divides it into two congruent segments. For example, if is the midpoint of C, then C. 3 3 6. iven: M is the midpoint of RS. Complete each statement. a. If RS = 10, then SM = 5. b. If RM = 12, then MS = 12, and RS = 24. C 220 Springoard Integrated Mathematics I, Unit 3 Lines, Segments, and ngles
7. Points D and E are aligned with a ruler. Point D is at the mark for 4.5 cm, and the distance between points D and E is 3.4 cm. t which two marks on the ruler could point E be located? 8. Point N is the midpoint of F. If FN = 2x, what expression represents F? 9. Reason abstractly. Does a ray have a midpoint? Explain. 10. ive an example that illustrates the Segment ddition Postulate. Include a sketch with your example. Continued 6 Vocabulary Organizer, Interactive Word Wall In addition to midpoint of a segment, you may also introduce the concept of segment bisector. Encourage students to illustrate given information before completing missing information. TECHER to TECHER CLSSROOM-TESTED TIP You may wish to mention that a number line can be thought of as a one-dimensional coordinate system and that students will see how these ideas work on a two-dimensional coordinate system in the next activity. You can also use a number line to find the distance between two points. 11. What is LM? L M 5 4 3 2 1 0 1 2 3 4 5 Point L is at -3, and point M is at 4. LM = -3-4 = -7 = 7 The midpoint of a segment is halfway between its endpoints. So, if you know the coordinates of the endpoints, you can average them to find the coordinate of the midpoint. 12. What is the coordinate of the midpoint M of PQ? Explain your reasoning. P 0 4 8 12 16 20 24 28 32 36 40 44 Point P is at 8, and point Q is at 40. 8 + 40 = 48 = 24 2 2 The coordinate of the midpoint M of PQ is 24. P 16 M 16 Q Q to ensure that they understand the concepts associated with distance between two points, and how to use the coordinates on a number line to find the distance between two points and the coordinate of the midpoint of the segment. nswers 7. the mark for 1.1 cm or the mark for 7.9 cm 8. 4x 9. No. ray extends infinitely in 1 direction, so it cannot be divided into 2 equal parts. 10. Sample answer: Point is between points and C. = 5, C = 4, and C = 9, so + C = C. 5 4 C 0 4 8 12 16 20 24 28 32 36 40 44 9 11 12 Vocabulary Organizer, Interactive Word Wall, Create Representations Items 11 and 12 relate the concept of distance on a line to the number line. e sure that students do not equate distance on a number line with the coordinates of the endpoints. Emphasize that distance is the absolute value of the difference of the coordinates. lso, make sure that students distinguish between the coordinate of midpoint M and its distance from either endpoint of the segment. ctivity 12 eometric Figures 221
Continued to ensure that they understand how the concepts of distance, coordinates, and midpoint relate to each other on a number line. Students who struggle with these concepts may benefit from counting units from one endpoint to the other. nswers 13. a. 24 b. 3 c. 1 d. 11 Example Discussion roups, Critique Reasoning Have students work in small groups and explain how the reason given for each step justifies the statement on the left. Monitor students group discussions to ensure that complex mathematical concepts are being verbalized precisely and that all group members are actively participating in discussions through sharing ideas and through asking and answering questions appropriately. You may want to have students complete the graphic organizer Collaborative Dialogue as they provide explanations for the steps in the example. Point out that the question prompt for each step is: How does the reason justify the step? MTH TIP number line represents a one-dimensional coordinate system. You will explore the concepts of distance and midpoint using a two-dimensional coordinate system in ctivity 14. WRITIN MTH Use when you talk about segment. Use when you talk about the measure, or length, of. 13. Use the number line to solve each problem. a. What is KL? K 12 9 6 3 0 3 6 9 12 15 18 b. What is the coordinate of the midpoint of KL? c. Point C lies between points K and L. The distance between points K and C is 1 of KL. What is the coordinate of point C? 3 d. Point N lies between points C and L. The distance between points C and N is 3 of CL. What is the coordinate of point N? 4 You can use the definition of midpoint and properties of algebra to determine the length of a segment. Example If Q is the midpoint of PR, PQ = 4x - 5, and QR = 11 + 2x, determine the length of PQ. ecause Q is the midpoint of PR, you know that PQ QR and PQ = QR. PQ = QR 4x - 5 = 11 + 2x Substitution Property 2x - 5 = 11 Subtraction Property of Equality 2x = 16 ddition Property of Equality x = 8 Division Property of Equality Now substitute 8 for x in the expression for PQ. PQ = 4x - 5 = 4(8) - 5 = 27 Make a sketch of PR and its midpoint Q. Label the lengths of PR, PQ, and QR. P 27 Q 27 R 54 L 222 Springoard Integrated Mathematics I, Unit 3 Lines, Segments, and ngles
Try These a. If Y is the midpoint of WZ, YZ = x + 3, and WZ = 3x - 4, determine the length of WZ. WZ = 26 iven: M is the midpoint of RS. Use the given information to find the missing values. b. RM = x + 3 and MS = 2x - 1 c. RM = x + 6 and RS = 5x + 3 x = 4 and RM = 7 x = 3 and SM = 9 When you bisect a geometric figure, you divide it into two equal or congruent parts. 14. Reason abstractly. Line segment WZ bisects XY at point Z. What are two conclusions you can draw from this information? Sample answer: XZ ZY and Z is the midpoint of XY. 15. Explain how to find the distance between two points on a number line. 16. Mekhi knows that H = 7 and HJ = 7. ased on this information, he claims that point H is the midpoint of J. Is Mekhi s claim necessarily true? Make a sketch that supports your answer. 17. iven: T is the midpoint of JK, JK = 5x - 3, and JT = 2x + 1. Determine the length of JK. 18. Reason quantitatively. lies on a number line. The coordinate of point is -6. iven that = 20, what are the two possible coordinates for point? LESSON 12-3 PRCTICE 19. iven: Point K is between points H and J, HK = x - 5, KJ = 5x - 12, and HJ = 25. Find the value of x. 20. If is the midpoint of C, = x + 6, and C = 5x - 6, then what is C? 21. Point P is between points F and. The distance between points F and P is 1 of F. What is the coordinate of point P? 4 F MTH TERMS geometric figure that bisects another figure divides it into two equal or congruent parts. Continued 14 Create Representations Draw a line segment on the board and bisect it into two equal parts to help students grasp the concept of bisect. to ensure that they understand how to find the distance between two points on a number line and that they do not confuse the coordinates of the endpoints with the distance. The distance between two points on a number line is the absolute value of the difference of the coordinates, but it is also acceptable for students to count units on the number line. nswers 15. Subtract the coordinates of the points. Then find the absolute value of the difference. SSESS Students answers to the Lesson Practice items will provide a formative assessment of their understanding of using the Segment ddition Postulate and the definition of midpoint to find lengths of segments, and of students ability to apply their learning. Short-cycle formative assessment items for are also available in the ssessment section on Springoard Digital. Refer back to the graphic organizer the class created when unpacking Embedded ssessment 1. sk students to use the graphic organizer to identify the concepts or skills they learned in this lesson. 0 2 4 6 8 10 12 14 16 18 20 22 22. Use appropriate tools strategically. nne has a broken ruler. It starts at the 3-inch mark and ends at the 12-inch mark. Explain how nne could use the ruler to measure the length of a line segment in inches. 23. If P is the midpoint of ST, SP = x + 4, and ST = 4x, determine the length of ST. 24. Some geometric rules are postulates. Compare and contrast postulates and other geometric rules. 16. No. The claim is not true if points, H, and J are not collinear. Sample sketch: 17. JK = 22 18. 26 and 14 7 7 J H 223 LESSON 12-3 PRCTICE 19. x = 7 20. C = 12 21. 8 22. Sample answer: lign one endpoint with the mark for 3 inches. lign the other endpoint with the ruler, and note the mark closest to this endpoint. Then subtract 3 from this mark. 23. ST = 16 24. Postulates are similar to other geometric rules because they can be used to prove other statements. Postulates are different because they are accepted without proof, but other rules must be proved. DPT Check students answers to the Lesson Practice to ensure that they have mastered the content of this lesson. Students should be able to relate coordinates on a number line with the distance between two points on the line and find the midpoint of a segment. If some students struggle with these concepts, you may want to have students use grid paper to construct their number lines and to count spaces between coordinates. See the ctivity Practice on page 231 and the dditional Unit Practice in the Teacher Resources on Springoard Digital for additional problems for this lesson. You may wish to use the Teacher ssessment uilder on Springoard Digital to create custom assessments or additional practice. ctivity 12 eometric Figures 223