6 On the Line Copyright © 2026 Pearson Education, Inc. One the Line (50 – 60 minutes) Learning Objective(s): Students will find the midpoint and weighted average between two points. Students will understand the difference between a midpoint and a weighted average. Students will interpret midpoints and weighted averages in a real-world context. Material needed: Student pages: On the Line Calculator Graphing software (optional) Colored pencils (optional) Lesson Procedure: Warm–Up 10 minutes Prompt: Suppose that two points on a number line represent two distinct locations. One location is used more than the other. How would you find a point between them that reflects this difference? How is this different from just finding the midpoint between the two points? Discuss: distance, midpoint, weighted average Guided Instruction 15 minutes Present: scenario for On the Line. Example: On the number line, points A and B are located at 4 and 12. What is the midpoint of A and B? 8 What is the weighted average point dividing segment AB in the ratio 1 : 3, closer to point A? 6 How does finding the midpoint and weighted average of two points on a number line compare to finding the midpoint and weighted average of a segment on the coordinate plane? On the coordinate plane, finding the midpoint and weighted average of a segment involves separately averaging (weighted or unweighted) the x– and y–coordinates to determine a single point. Encourage students to explore and recognize patterns in ratios and coordinates to apply them to new models. Review: key terms – midpoint, weighted average, ratio midpoint: a point on a line segment that divides the segment into two equal parts, making it the exact middle of the segment weighted average: calculation used to find a point that is closer to the points with higher "weights" ratio: a comparison of two numbers or quantities that shows how many times one value contains the other Independent Practice 20 minutes Distribute: student activity On the Line Allow students to work individually or in pairs. Closure 10–15 minutes Review Answers: 1. Check students’ graphs. 2. (5, 9) 3. (4, 8) 4. (4, 8); The distance from Entrance A to (4, 8) is about 2.8 units and the distance from Entrance A to (5, 9) is about 4.2 units. 5. Sample answer: The weighted average at (4, 8) is better because it puts the kiosk closer to where most people enter. 6. a. (6, 10); b. Entrance B; the distance from Entrance A to (6, 10) is about 5.7 units and the distance from Entrance B to (6, 10) is about 2.8 units. 7. a. (3.2, 7.2); b. closer to Entrance A; it has four times the weight; c. The kiosk is not ideal for those using Entrance B, but it is more practical overall for most visitors who use Entrance A. 8. Sample answer: (4, 8) is more efficient because it better reflects the real traffic pattern. Discuss: How can understanding the difference between a midpoint and a weighted average encourage better decisions when choosing locations in real-life situations? Can you think of examples when applying these concepts might change the outcome?
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