File Name: Algebraic Reasoning_Geometry_Physics_Distance vs Displacement_5-27-2011.docx Lesson Title: Distance vs. Displacement Targeted Grade Band: 4 5 6 7 X 8 9 10 11 12 Targeted Mathematics Strand: X Algebra Reasoning Targeted Science Strand: Earth Science Agriculture X Geometry Life Science Technology Measurement Probability Quantitative Reasoning Space Science X Physics Chemistry Statistics Science Scientific Investigation and Reasoning Total Estimated Time for this Lesson: 2 hours Prerequisites: Students should be able to evaluate and algebraic expression. Attachments: Title File Name Worksheet: Calculating Distance and Calculating Distance and Displacement.docx Displacement KEY: Calculating Distance and Displacement Calculating Distance and Displacement- KEY.pdf Reference: Real Number System Real Number System.docx Example #1 Example#1.pdf Example #2 Example#2.pdf Worksheet: Density Activity: Data Collection Density Activity: Data Collection.docx KEY: Density Activity: Data Collection Density Activity: Data Collection KEY.pdf 1
I. Learning Objectives and Standards Alignment At the end of this lesson, students will be able to Describe the difference between additive reasoning and multiplicative reasoning Use the Pythagorean Theorem to determine the shortest distance between two points. Find the distance traveled between two points Mathematics Learning Objectives National TEKS Texas CCRS Mathematics Standards A 8.2A II.C.1. G,M 8.9 III.A.3.a. A,G,M 8.9 III.A.3.b. Science Learning Objectives At the end of this lesson, National Science TEKS Texas CCRS students will be able to Standards Find the displacement A 6.6B VIII.C.1.a.b.c. between two points B Show that B 6.2E II.A.5.6. displacement is commutative Calculate density A 6.6B VIII.C.1.a.b.c. At the end of this lesson, students will be able to N/A At the end of this lesson, students will be able to N/A Technology Standards National Education TEKS Technology Standards Agriculture Standards National Agriculture, Food and Natural TEKS Resources Career Cluster Content Standards Texas CCRS Texas CCRS 2
II. Materials Activity #1: Derivation of the distance formula Item n/a Reference: Real Number System Example #1 Example #2 Quantity (per student or per group) Student Materials Instructor Materials 1 per instructor 1 per instructor 1 per instructor Activity #2: Distance vs. Displacement Item Quantity (per student or per group) Student Materials Worksheet #1: Calculating Distance and 1 per student Displacement Instructor Materials KEY -Calculating Distance and Displacement 1 per instructor Activity #3: Density Activity: Data Collection Item Quantity (per student or per group) Student Materials Worksheet: Density Activity: Data Collection 1 per student 150 ml of water per graduated cylinder Ruler 1 per group 250 ml Graduated Cylinder 1 per group Triple Beam Balance 1 per group Aluminum ( short rectangular prism) 1 per group Aluminum ( long rectangular prism) 1 per group Copper (short rectangular prism) 1 per group Copper (short rectangular prism) 1 per group Instructor Materials KEY: Density Activity: Data Collection 1 per instructor III. References Reference (from Glenbrook South Physics): www.glenbrook.k12.il.us/gbssci/phys/class/1dkin/u1l1c.html The Math Worksheet Site: On-line Math Worksheet Generator http://themathworksheetsite.com/coordinate_plane.html Merriam-Webster Word Central http://www.wordcentral.com/ 3
IV. Terms and Definitions Math Term Definition Source Distance the space or amount of space between two points, lines, surfaces, or objects Merriam-Webster Word Central Commutative relating to, having, or being the property of giving the same mathematical result no matter in which order two numbers are used with an operation Merriam-Webster Word Central Science Term Definition Source Displacement the difference between the first position of an object and any later position. Merriam-Webster Word Central Density the mass of a substance per unit volume <density expressed in grams per cubic centimeter> Merriam-Webster Word Central 4
V. Language Issues A. Same words that have different meanings Dense - Scientist may describe elements as being dense. Saying that a real number is dense does not mean that we calculate the reaction of its mass to its volume as in physics. B. Different words that have the same meanings Displacement and Distance Displacement in physics is defined as an object s overall change in position; it is a vector quantity. The distance formula used to calculate the value of displacement. C. Supposedly interchangeable words D. Homonyms E. Words with variety of meanings in different contexts VI. Parallel Concepts Density of real numbers parallels the general physics idea of density in the notion of compactness. VII. Misconnections In mathematics, we often use the shortest distance between two points to describe the displacement. The shortest distance may not be the path traveled. In mathematics, it is important to pay attention to the context. If you are discussing the distance between two points in coordinate system, then mathematicians use the distance formula to calculate this distance. However, if the problem describes the path traveled parallel to the y-axis and then turn and travel the distance parallel to the x-axis, the distance traveled is not defined by the distance between the two points. 5
VIII. Sequencing of the Lesson Procedures Introduction/Set: Activity #1: Derivation of the distance formula Notes Name some subsets of the Reals? Integers, Counting Numbers, Zero, Let s talk about the real numbers. Recap (see diagram) If we wanted to draw a picture of the real numbers it would look like this. Draw real number line. One thing we want to talk about is the relationship of some numbers relative to its distance from zero. If I want to know how far away the number 2 is away from 0, you would say what? 2 units Could we say how far away is 0 from 2 and get the same answer? Yes Could I do this with any number in the Reals? Yes Relative to its distance from zero? Yes Mathematicians have established a notation for the idea of distance and that notation is absolute value. Write this on the board. 2 0 2 0 2 2 It s also okay to think about the relationship between two other numbers not just 0. Write this on the board. 5 4 1 Let s put this into words. How would I say this? How far away is 5 from 4? How far away is 4 from 5? And the answer is? 1 unit Let s generalize. Write this on the board. 0 = Let s put this into words. How would I say this? How far away is a number away from O? And this? Write this on the board. The distance between two numbers. 6
What if you have two number lines? What is this called? Coordinate Plane So now I have two number lines. If I asked you to plot two points on the x-axis and tell me the distance between the two you could do this right? Give an example. (See Example #1) Now plot (3,3) and (6,9) ask students to find the distance between the two points. How would I do this? Make a cross section of both points and explore what can be extracted from the graph. In particular the distance of the two points in respect to the x and y axis. Once this is illustrated students can use the Pythagorean Theorem to find the length of the hypotenuse of the triangle. 6 3 ² 9 5 ² ² What does the square mean for the value of this? Always positive. It could also be zero. So you can now write it like this. 6 3 ² 9 5 ² ² 3 ² 4 ² ² 9 + 16 = c² 25=c² c = 5 Now can we generalize the process? (see Example #2 page) ² ² ² ² ² ² c= ² ² First, find the fathom point. Then find the horizontal distance. Then use the Pythagorean Theorem. Activity #2: Calculating Distance and Displacement So how does this help us in science? Have each student complete #1 and #2 on Worksheet #1: Calculating Distance and Displacement or use this sheet as a class example. Discussion: How far did the teacher travel? 12 meters How did you solve this 7
problem? 4 2 4 2 12 2 2 Displacement is the difference between the first position of an object and any later position. How far is the teacher from her original point? 0 units Correct. She was displaced 0 units. Discussion points: The physics teacher has walked a total distance of 12 meters, but her displacement is 0 meters. During the course of her motion, she has covered 12 meters of ground (distance = 12 m). Activity #3: Density Activity: Data Collection When she is finished walking, she is not out of place i.e., there is no displacement for her motion (displacement = 0 m). Displacement, being a vector quantity, must give attention to direction. Use the same discussion points for the remaining sample problems. For this activity students should be placed in small groups. Pass out materials and worksheet for this lab. The conclusion section can be done in student groups or in a whole class setting. Summary: An extension to this lesson could be to calculate percent error. Is displacement commutative? Yes Which method for finding the displacement is more accurate? calculating 8
IX. Assessments 1. 2.
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Key: 1. Mathematics, Objective 1, A 2. Mathematics, Objective 4, D 3. Mathematics, Objective 4, B 4. Mathematics, Objective 4, A 5. Science, Objective 4, A 6. Science, Objective 4, A 13