Polygons in the Coordinate Plane

Transcription

1 Polygons in the Coordinate Plane LAUNCH (8 MIN) Before How can you find the perimeter of the sandbox that the park worker made? During How will you determine whether the park worker s plan for the sandbox is correct? How can you use the given length of board to find the length of each side of the sandbox? After How can you plot the side lengths of the square if they are not whole numbers? KEY CONCEPT (3 MIN) Use the animation to define polygon. The animation is helpful because it shows both examples of shapes that are polygons and examples of shapes that are not polygons. Could a circle be considered a polygon? Explain. PART 1 (7 MIN) Does it matter in what order you plot the vertices of the polygon? Explain. How can you prove that the shape you have made is a polygon? Could you have determined the name of the polygon without graphing the points? PART (7 MIN) During the Intro How can you add two numbers to find the length of AC? Do you have to add them to find the distance? Lisa Says (Screen ) Use the Lisa Says button to connect the length of a segment to the distance between its endpoints. Which numbers in the ordered pairs will you need to focus on in order to find the lengths of AB and CD? Explain. After finding the horizontal segment lengths How was the process for finding the lengths of AB and CD different? When finding the segment lengths, why didn t you add or subtract the y-coordinates? PART 3 (7 MIN) Jay Says (Screen 1) Use the Lisa Says button to discuss how coordinate planes are used in computer graphics to make lifelike images. How will you use the scale on the axes to guide you in plotting the points? Examine the given coordinates of home plate. Do any of the points show reflections of each other? How do you know? If you could have created your own scale on the grid, what scale would you have used? CLOSE AND CHECK (8 MIN) Think of the example of the graph of home plate on the coordinate plane. What lengths could you determine from the graph and what lengths could you not find? Imagine you want to find a vertical distance on a coordinate plane. One point is above the x-axis and another point is below the x-axis. How will you find the distance between them?

2 Polygons in the Coordinate Plane LESSON OBJECTIVES 1. Draw polygons in the coordinate plane given coordinates for the vertices.. Use coordinates to find the length of a side of a polygon joining points with the same first coordinate or the same second coordinate. FOCUS QUESTION What makes a figure a polygon? Why might you want to draw a polygon in the coordinate plane? MATH BACKGROUND In the previous lesson, students extended their work in the coordinate plane to include ordered pairs with rational-number coordinates. They plotted and identified points in all four quadrants and used the quadrant to identify the signs of the coordinates of a point. In this lesson, students use the vocabulary of polygons to draw figures in the coordinate plane and find side lengths as horizontal and vertical distances. Although distance in the coordinate plane is typically found using subtraction, students may not know how to subtract a negative rational number from a positive rational number. Therefore, the solutions in this lesson add the absolute values of the two coordinates to find the total distance. In the final lesson of the topic, students solve more problems involving rational numbers, absolute value, and points and polygons in the coordinate plane. Students will next work with polygons in the topic Area. In Grade 7, students work more with rational numbers and the four basic operations. In Grade 8, they learn about one more type of real number: irrational numbers. LAUNCH (8 MIN) Objective: Draw a polygon in the coordinate plane and find distance (perimeter) using coordinates. In this problem, students analyze a real-world problem that was solved incorrectly on a coordinate grid. Students need to find the side length of a square given its perimeter. To solve the problem correctly, students must be able to interpret the problem and then plot rational numbers on the coordinate grid. Instructional Design Students must know the basic properties of a square and how to find its perimeter. Review these concepts if needed. Once students find the coordinates of a square sandbox, call on students to draw on the provided grid using a different color from the park worker s diagram. Compare the two drawings. Before How can you find the perimeter of the sandbox that the park worker made? [Sample answer: You can use the coordinates given to find the lengths of each of the four sides and add them together.]

3 During How will you determine whether the park worker s plan for the sandbox is correct? [You need to check that all four sides are the same length and that the perimeter is.] How can you use the given length of board to find the length that each side of the square sandbox should be? [You can divide by 4 to find the length of each side.] After The coordinate grid shown uses a scale of 1 unit and only shows whole numbers. How can you plot the side lengths of the square if they are not whole numbers? [The length of each side of the square is 5.5 units. You can find the point halfway between 5 and 6.] Solution Notes Students may prefer to use a grid with a 1 -unit scale because the coordinates they will plot for the sandbox are either 0 or 5.5. You can repeat this activity for any perimeter that is not a multiple of 4, such as 30 feet. Connect Your Learning Move to the Connect Your Learning screen. Begin a discussion about whether the coordinate plane helped students complete the Launch more efficiently. Ask if it was easier drawing the square on the plane than it would have been on unlabeled paper. Look for students to talk about needing to correctly draw the corner of the square, etc. Also, make sure students understand the animation in the Launch in which the boss complains that he wanted a square rectangle. Make sure students understand that all squares are rectangles, but not vice versa. KEY CONCEPT (3 MIN) Teaching Tips for the Key Concept Use the animation to define polygon. The animation is helpful because it shows both examples of shapes that are polygons and examples of shapes that are not polygons. Students are also able to see how to create a polygon by plotting points on the coordinate grid and connecting them. Point out that the plural form of vertex is vertices. Could a circle be considered a polygon? Explain. [Sample answer: No; even though a circle is a closed figure, it is not made up of line segments.] PART 1 (7 MIN) Objective: Draw polygons in the coordinate plane given rational number coordinates for the vertices. Students plot ordered pairs with rational-number coordinates on the coordinate plane. They connect the points to make a polygon and then name the polygon.

4 Instructional Design Call on students to plot each of the five points. Ask them how they found each point using the coordinates. Does it matter in what order you plot the vertices of the polygon? Explain. [No; the order does not matter. Once all five points are plotted, you will connect them to form a polygon.] How can you use the grid on the axes to assist you in plotting the points that are not integer values? [Sample answer: The grid goes up by 1 integer for each line. Therefore, for the points whose values are not integers, you can estimate where they lie in relation to the integers on the grid.] How can you prove that the shape you have made is a polygon? [Sample answer: The shape is a closed shape with five line segments that do not cross.] Could you have determined the name of the polygon without graphing the points? [Yes; there are five total points, so you know immediately that connecting the points to make a polygon will result in a pentagon.] Is it possible to draw a figure that is not a polygon? [Yes; you can make the connecting lines cross or use curves instead of line segments.] Solution Notes Students may plot the points of the pentagon in any order. Once the five points are plotted, students can connect the five vertices to make the pentagon. Consider using the Coordinate Grapher Tool to help students plot the five points and make the pentagon. Differentiated Instruction Because some of the coordinates include a fractional part, work with a grid with a 1 -unit scale. While the shape of the figure does not change, all points will lie on grid lines, which can help students understand how to plot points that do not fall on grid lines. Do additional examples with grids that are 1 unit, 11 Got It Notes units, and units. Students should recall that a vertex of a polygon is the point where two line segments meet. Each of the five points in the graph is a vertex of the polygon. Students may find all five vertices and then check whether any of the ordered pairs in the list match one of the vertices. If you show answer choices, consider the following possible student errors: Students may choose A, B, or C if they do not consider all three points. Make sure students realize that more than one ordered pair can correspond to a vertex of the polygon.

5 PART (7 MIN) Objective: Apply techniques for drawing polygons in the coordinate plane given rational number coordinates for the vertices or for using coordinates and distance to find the length of a side of a polygon joining points with the same first coordinate or the same second coordinate in the context of solving real-world problems. Students find horizontal and vertical distances using the coordinates of the vertices of a polygon. They either add or subtract the distances based on whether the line segment crosses an axis. Instructional Design Have students click on each radio button to show that you can find lengths of horizontal and vertical line segments using the coordinates of the ordered pairs. Note that you cannot do the same for the third side of the triangle because it is a diagonal line segment. Stress that the process for finding the length of a line segment changes based on whether the line segment crosses an axis (meaning the points are on either side of the axis). On Screen, present students with a polygon with two horizontal sides and two diagonal sides. They will solve a problem involving a vertical line in the Got It. During the Intro How can you add two numbers to find the length of AC? Do you have to add them to find the distance? [Sample answer: Segment AC stretches across Quadrants I and II. There is one horizontal length on each side of 0. To find the length of the segment, you add the length to the left of 0 and the segment to the right of 0. You could instead subtract one number from the other.] Lisa Says (Screen ) Use the Lisa Says button to connect the length of a segment to the distance between its endpoints. Students will need to count grid marks, or subtract coordinates, in order to determine length. You may wish to discuss absolute value to prepare students to find the length of a segment that spans two quadrants. Which numbers in the ordered pairs will you need to focus on in order to find the lengths of AB and CD? Explain. [Sample answer: You need to look at the x-coordinates because you are trying to find two different horizontal distances (AB and CD).] How can you tell what operation you need to use to find the length of segment CD? [Sample answer: You know that you have to use subtraction because the points are on the same side of the y-axis.] After finding the horizontal segment lengths How was the process for finding the lengths of AB and CD different? [Sample answer: To find the length of AB, you add the absolute values of the x-coordinates of A and B. To find the length of CD, you subtract the x-coordinates of C and D.] When finding the segment lengths, why didn t you add or subtract the y-coordinates? [Sample answer: You are finding two horizontal lengths. The distance between the x-coordinates gives you a horizontal distance. The difference between the y-coordinates is 0.]

6 Solution Notes Some students may use the grid and count the squares to find the length of segments AB and CD. It is important for students to recognize how they can find the length of the two segments using addition or subtraction when the segments cannot be measured by counting grid squares. Error Prevention Students may correctly find the x-coordinates, which are needed to find the length of AB, but add the values of the x-coordinates rather than the absolute values. To help students avoid making this error, ask how many units point A is from the y-axis and how many units point B is from the y-axis. Help students realize that they can add these two distances to find the total length. Provide another example for student to do in pairs and have each student explain their work to a partner. Got It Notes This problem now asks students to find a vertical distance that involves adding the absolute values of the y-coordinates. If you show answer choices, consider the following possible student errors: If students add the y-coordinates, they may choose A. Students may choose D if they add the absolute values of the x-coordinates rather than the y-coordinates. PART 3 (7 MIN) Objective: Use coordinates and distance to find the length of a side of a polygon joining points with the same first coordinate or the same second coordinate. Students plot the given coordinates to make home plate in the coordinate plane and find one side length. This problem applies students knowledge of finding a horizontal distance using rational-number coordinates of vertices. Lisa Says (Screen 1) Use the Lisa Says button to discuss how coordinate planes are used in computer graphics to make lifelike images. How will you use the scale on the axes to guide you in plotting the points? [Sample answer: Each square on the grid represents units. For example, when you plot 8 1, you go 1 4 of the distance from 8 to 10.] Examine the given coordinates of home plate. Do any of the points show reflections of each other? How do you know? [Sample answer: Points A and E are reflections of each other. Also, Points B and D are reflections of each other. You can tell because their x-coordinates are opposites and their y-coordinates are the same.] If you could have created your own scale on the grid, what scale would you have used and why? [Sample answer: You could use a scale of 1 the points have either 8 1 or!8 1 in them.] unit because all of

7 Differentiated Instruction For struggling students: Students may benefit from a coordinate grid with a scale of 1 rather than a scale of. Students may have an easier time plotting the points that have 8 1 or 8 1 as coordinates because the point will lie halfway between the two squares rather than one-fourth of the way. For advanced students: Challenge students to find the coordinates of a home plate that is 1 the size of the one given. Students will have to take 1 of each x, y pair to find the coordinates of a home plate that is 1 the size of the one given. This would show a dilation, which students will learn about when they learn the various transformations in the coordinate plane. Got It Notes Students apply their knowledge of how to find the length of a horizontal segment representing the side of a polygon using its coordinates. In this example, adding the absolute values of the x-coordinates will yield length AE. CLOSE AND CHECK (8 MIN) Focus Question Sample Answer A figure is a polygon if it is closed and formed by three or more line segments that do not cross. Drawing a polygon in the coordinate plane can help you determine distances, such as the lengths of the polygon s sides. Focus Question Notes Look for students to discuss how drawing a polygon on plain paper is different from drawing a polygon using a coordinate plane. Students may find it is easier to draw a polygon on a coordinate plane, and more efficient when they need to determine the lengths of the sides of the polygon. Essential Question Connection The Essential Question asks, Why do we need positive and negative numbers? How do you know when to use positive numbers and when to use negative numbers? Use the questions that follow to help students explore polygons drawn on a coordinate plane. Think of the example of the graph of home plate on the coordinate plane. What lengths could you determine from the graph and what lengths could you not find? [Sample answer: When a polygon is represented on a coordinate plane, it is easy to find the lengths of the sides that are horizontal or vertical. You do not yet know how to determine the lengths of sides that are slanted.] Imagine you want to find a vertical distance on a coordinate plane. One point is above the x-axis and the other point is below the x-axis. How will you find the distance between the two points? [Sample answer: You can find the y-coordinates of both points. Then you can add the absolute values of the y-coordinates to find the vertical distance between the two points.]