Plotting Ordered Pairs on a Four Quadrant Grid Grade Five

Transcription

1 Ohio Standards Connection Geometry and Spatial Sense Benchmark C Specify locations and plot ordered pairs on a coordinate plane. Indicator 6 Extend understanding of coordinate system to include points whose x or y values may be negative numbers. Mathematical Processes Benchmark H Use representations to organize and communicate mathematical thinking and problem solutions. Lesson Summary: In this lesson, students label and represent ordered pairs on a four-quadrant grid. Students demonstrate prior knowledge of plotting ordered pairs in the first quadrant. Introduction to the four-quadrant grid begins with students creating and discussing number lines that include negative numbers. Practice in creating the axes on a coordinate grid is a component of the lesson. Students discuss the importance of direction in plotting ordered pairs and explain how to plot ordered pairs. Estimated Duration: One-and-a-half hours Commentary: As students ideas about the number system expand to include negative numbers, they can work in all four quadrants of the Cartesian plane. Provide opportunities for students to describe each quadrant to build understanding. Common errors occur when students switch the order when plotting the pair. Strategies, such as, thinking x comes before y in the alphabet will help students remember the order. Have students compare points such as (3, -4) and (-3, 4). Also, mistakes occur when the coordinate pair includes a zero. Pre-Assessment: Informally assess students abilities to plot ordered pairs in Quadrant I. Distribute Plotting Ordered Pairs on a Grid Pre- Assessment, Attachment A, to individual students. Circulate around the room to observe students plotting the ordered pairs in the first quadrant. Make note of students who reverse the order when plotting the ordered pairs. Select different students to plot the ordered pairs on an overhead transparency and explain how they arrive at the location. Scoring Guidelines: Informally assess as students plot the ordered pairs. Write anecdotal notes of misconceptions students have. Discuss any misconceptions as students explain plotting the ordered pairs. 1

2 Post-Assessment: Students plot ordered pairs on a blank, four-quadrant grid. Distribute Plotting Ordered Pairs on a Four Quadrant Grid Post-Assessment Attachment B to each individual student. Collect the assessment and evaluate for accuracy and understanding. Scoring Guidelines: Score the assessment by evaluating the students responses for accuracy. Instructional Procedures: Part One 1. Read, The Rene Descartes Story, Attachment E, to the class to begin the lesson. Ask, Descartes used coordinate grids to describe the position of the fly on the ceiling. How does this discovery apply to you? (Students can describe the position of their seats in the rooms, such as row 4, seat 2. ) 2. Draw a horizontal number line on a transparency sheet on the overhead projector. Say, I have $5. Where is this on the number line? I owe Mary $8. Where is this on the number line? Listen to the responses. 3. Draw a vertical number line. Ask students to describe this line. (It is the same as the horizontal line, but the orientation has changed.) Have students plot integers on the vertical line. 4. Overlay the horizontal number line and the vertical number line on the overhead projector. Have students describe what they see. 5. Distribute grid paper to each student. Discuss the connection between the uses of axes in social studies contexts (such as maps or globes) and the axes on the grid. Have students draw the x- and y-axes on the grid paper. Observe students creating their grids, ask and answer questions to clarify misconceptions. Select students to go to the overhead projector to draw their grids, if you notice differences in the axes. 6. Plot some points in each of the four quadrants. Ask students what they see in each quadrant. Since students are familiar with plotting points in the first quadrant, have them discuss what they notice. (All of the numbers are positive. The numbers get larger on the horizontal line as they move to the right.) Do this for each quadrant. Have students write notes about the points in each quadrant in their journals. 7. Have students plot ordered pairs once they understand what is in each quadrant. Have them explain the direction they use to plot the points. Do this for each quadrant before giving them points for all quadrants. 8. Present several ordered pairs and have students plot and explain their thinking. Sample exercises include: (-3, 4), (0, -5), (4, -2) and (-6, -5). Have pairs share their work with the class. Have pairs use the overhead projector in their demonstration. Continue to provide ordered pairs and check for students accuracy. Ensure that students start from the origin with each ordered pair. Ask students what ordered pairs mean. Focus on the word ordered as the dialogue takes place. 9. Write ordered pairs, such as (3, 3), (-3, 3), (-3,-3) and (3,-3) on the overhead projector. Have students label the coordinates to reveal understanding and progress. Because the same 2

3 integers are used in each of the four pairs, it is easy to reveal the students who understand the four quadrants and those who do not. 10. Have students create geometric figures on the coordinate grid, such as rectangles or hexagons. a. Model the process by plotting points on the grid (using all four quadrants) and connecting the points to make a shape. b. Have students draw a simple shape or picture on a four-quadrant grid and give the ordered pairs for the simple shape or picture. c. Pair students and have them create the drawing for each other using the ordered pairs. Allow the pairs to compare the simple shapes and discuss any errors in the drawing. 11. Provide students with a blank grid. Instruct them to put the axes on the grid and in each quadrant, describe the coordinates of the types of points in each quadrant. Also, have them write a description in their journals on how to plot ordered pairs, using examples of ordered pairs for all four quadrants. Collect the journal entry and assess for understanding and progress toward expectations. Differentiated Instructional Support: Instruction is differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). Provide grid paper with larger squares. Provide grids with axes already given and labeled. Use self-adhesive colored dots to plot points. Extension: Have the students identify the paths between points on a grid using the grid lines and give determine the distance between the points. Use Paths on a Grid, Attachment F. Home Connections: Have students plan a trip or vacation utilizing the coordinate graphing skills to determine the route. Provide an outline map of Ohio, Florida or any other state and have students plot various cities or have students use an atlas and plot tourist attractions. Materials and Resources: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not endorse any particular resource. The Web addresses listed are for a given site s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. 3

4 For the teacher: grid paper, overhead projector, overhead transparency coordinate grids For the students: pencils, grid paper, journals Vocabulary: coordinates ordered pairs origin quadrant x-axis y-axis Research Connections: Marzano, Robert J., Jane E. Pollock and Debra Pickering. Classroom Instruction that Works: Research-Based Strategies for Increasing Student Achievement, Alexandria, Va.: Association for Supervision and Curriculum Development, Pask, Gordon. Conversation, Cognition and Learning. New York: Elsevier, Reimer, Wilbert and Luetta Reimer. Historical Connections in Mathematics, Volume III. Fresno, Calif. AIMS Education Foundation, Wickett, Marilyn, Katherine Kharas and Marilyn Burns. Lessons for Algebraic Thinking. Math Solutions Publications Attachments: Attachment A, Plotting Ordered Pairs on a Grid, Pre-Assessment Attachment B, Plotting Ordered Pairs on a Grid, Pre-Assessment Answer Key Attachment C, Numerology Island, Post-Assessment Attachment D, Numerology Island, Post-Assessment Answer Key Attachment E, The Rene Descartes Story Attachment F, Paths on a Grid 4

5 Attachment A Plotting Ordered Pairs on a Grid, Pre-Assessment Name Date Directions: Plot each ordered pair on the grid. Label the points as give. 1. Point A (1, 1) 2. Point B (5, 1) 3. Point C (7, 3) 4. Point D (4, 0) 5. Point E (0, 6) 6. Explain how to plot the ordered pair (4, 2). 5

6 Attachment B Plotting Ordered Pairs on a Grid, Pre-Assessment Answer Key Name Date Directions: Plot each ordered pair on the grid. Label the points as give. 1. Point A (1, 1) 2. Point B (5, 1) 3. Point C (7, 3) 4. Point D (4, 0) 5. Point E (0, 6) 6. Explain how to plot the ordered pair (4, 2). Begin at (0, 0) and go to the right 4 spaces or go to 4 on the x-axis. Then go up 2 places and place a point. 6

7 Attachment C Numerology Island, Post-Assessment Directions: Plot the following ordered pairs on the Island of Numerology and label with the given letter. A. City Hall (0, 0) B. Tourist Center (-5, -5) C. Sanders Elementary School (3, -7) D. Library (-4, 2) E. Police Department (6, 6) Directions: Give the ordered pair for each location. 1. Post Office 2. Hospital 3. Fire Department 4. Community Swimming Pool 5. Movie Theater 7

8 Attachment D Numerology Island, Post-Assessment Answer Key 1. Post Office (-1, 9) 2. Hospital (2, -3) 3. Fire Department (-8, -4) 4. Community Swimming Pool (0, -10) 5. Movie Theater (5, 7) 8

9 Attachment E The Rene Descartes Story Rene Descartes (day-kart) was a French mathematician, philosopher and scientist who is often called the father of modern mathematics. His studies of mathematics opened up many doors in the areas of geometry and algebra. His studies of geometry lead to the development of analytic geometry, a way of connecting algebra and geometry that enables mathematicians to display an equation as a set of points on a graph. One late morning Descartes was lying in bed. He noticed a fly crawling on the ceiling. As the fly was crawling to a corner, Descartes thought of how he could mathematically describe the fly s position. He realized he could express the fly s position in terms of its distance from the adjacent wall. The story goes that this was the birth of coordinate geometry, also known as the Cartesian coordinate system (the position of a point in a plane may be determined relative to intersecting lines known as axes). 9

10 Attachment F Paths on a Grid The school is located at (-1, 0) on the grid. Plot the following points on the grid. Amy s house (-1, 6) Kim s house (2, -3) Chris s house (-5, -5) Kevin s house (4, 3) The four students walk to school. Answer the questions by identifying the shortest paths on grid lines for each student and comparing the distance of the paths. Who walks the farthest distance to school? Which two students walk the same distance to school? Who walks 8 units to the school? You walk 12 units to school. Plot a point to show where your house could be. 10