Rational Numbers and the Coordinate Plane LAUNCH (8 MIN) Before How can you use the numbers placed on the grid to figure out the scale that is used? Can you tell what the signs of the x- and y-coordinates for points A and B will be just by seeing where they lie on the coordinate plane? Explain. During Is either of the coordinates of point A an integer? Explain. After Does it matter whether you use decimals or fractions for the coordinates? PART 1 (8 MIN) How can you find the coordinates of a point if the point is not on the grid lines? What is the scale of each axis? How will it help you find the coordinates of the points? Lisa Says (Screen 2) Use the Lisa Says button to connect what students know about plotting ordered pairs of whole numbers to plotting ordered pairs of rational numbers. Why do you think a point on the grid is called an ordered pair? Would a scale of 1 -unit rather than 1 -unit on both the x- and y-axes be more helpful? 2 What do you notice about the signs of the points in each of the four quadrants? PART 2 (8 MIN) How can you use the signs of the coordinates to plot the points on the coordinate grid? The scales for both axes are in integers. The coordinates of some of the points are rational numbers. How can you plot the points? After completing the solution Could you have determined which quadrant a point will lie in without graphing it? Explain. PART 3 (8 MIN) Why does the sign of the x-coordinate not change when you reflect a point across the x-axis? Before completing the first statement How can you use the information you learned in the Intro to find the coordinates of the reflected point without graphing? Lisa Says (Screen 2) Use the Lisa Says button to connect reflections in the coordinate plane to mirror images in real life, and to the concept of opposites. What shape do the four points make if you connect them? CLOSE AND CHECK (8 MIN) In which quadrant would you find ordered pairs that have negative x- and y-coordinates? Suppose you have a coordinate plane with axes whose scales only show integer values. How can you use it to graph an ordered pair whose coordinates are rational numbers?
Rational Numbers and the Coordinate Plane LESSON OBJECTIVES 1. Find and position pairs of integers and other rational numbers on a coordinate plane. 2. Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both coordinate axes. FOCUS QUESTION How is a number line related to the coordinate plane? How are rational numbers used within the coordinate plane? MATH BACKGROUND In the previous topic, students extended their knowledge of the coordinate plane to all four quadrants and plotted ordered pairs with integer coordinates. Now that students have worked with rational numbers and used number lines to compare and order rational numbers, they work with ordered pairs with rational-number coordinates. Students plot points and match coordinates to points in the coordinate plane. Review quadrant numbers from the previous topic as needed. They also review how various reflections over the axes affect the coordinates of a point. In the next lesson, students learn about polygons and use polygons in the coordinate plane to find side lengths or horizontal and vertical lines. LAUNCH (8 MIN) Objective: Identify rational-number coordinates of points in the coordinate plane. Students use points on the axes of the coordinate grid to determine the scale for both axes and to find the coordinates of two ordered pairs with rational-number coordinates. Before How can you use the numbers that are placed on the grid to figure out the scale that is used? [Sample answer: When you see that two boxes represent 1 unit, you know that 1 box represents 0.5 unit.] Can you tell what the signs of the x- and y-coordinates for points A and B will be just by seeing where they lie on the coordinate plane? Explain. [Sample answer: Yes; points in Quadrant II, where point A lies, have negative x-coordinates and positive y-coordinates. Points in Quadrant IV, where point B lies, have positive x-coordinates and negative y-coordinates.] During Is either of the coordinates of point A an integer? Explain. [No; the x-coordinate is between 1 and 2, and the y-coordinate is between 1 and 2. Neither coordinate of Point A can be an integer.] After Does it matter whether you use decimals or fractions for the coordinates? [No; you should use the form that you find most useful in the situation.] After students have recognized that the scale on the axes is 0.5 unit, have students write in the coordinates that are missing on the axes to help them identify the coordinates of points A and B.
Connect Your Learning Move to the Connect Your Learning screen. Begin a discussion about number lines and the coordinate plane. Ask students how the coordinate plane in the Launch made it easy to plot rational numbers. Look for students to describe the choices you make when you label axes, which some students may identify as the scale. PART 1 (8 MIN) Objective: Find pairs of rational numbers on a coordinate plane. Students write the ordered pair that describes each point on the coordinate grid and identify the quadrant. There is one point in each quadrant. Instructional Design Use the Intro to show that some coordinates do not fall on the grid lines. On Screen 2, the scale is 0.5 unit for both axes, and 3 out of points lie between grid lines. How can you find the coordinates of a point if the point is not on the grid lines? [You can approximate the coordinates using the two closest grid lines.] What is the scale of each axis? How will the scale help you find the coordinates of the points in this problem? [Sample answer: Both axes have a scale of 1 2 unit. If a coordinate is halfway between two integers, the point still falls on a grid line.] Lisa Says (Screen 2) Use the Lisa Says button to connect what students know about plotting ordered pairs of whole numbers to plotting ordered pairs of rational numbers. Why do you think a point on the grid is called an ordered pair? [Sample answer: It is called an ordered pair because the two numbers that are needed to plot the point are in a specific order: the first number is the x-coordinate, and the second number is the y-coordinate.] Would a scale of 1 -unit rather than 1 -unit on both the x- and y-axes be more 2 helpful? Explain. [Sample answer: Yes; a scale of 1 -unit on both axes would be easier because three of the four points lie in between the grid lines. If the scale were in 1 -unit, all of the points would lie on the grid lines.] What do you notice about the signs of the points in each of the four quadrants? [Sample answer: They are different depending on the quadrant. For example, any ordered pair in Quadrant II has coordinates (negative, positive).] Use the Full Plane mode of the Coordinate Grapher Tool to make a scale of 1 -unit on both axes. This way, students will be able to name the coordinates of the four ordered pairs more easily because they will lie on the grid lines. After students have found the
coordinates of the ordered pairs with this 1 -unit grid, allow them to use the grid given in the problem to see if they can name the coordinates using the 1 -unit grid. 2 Got It Notes This problem only asks for one of the four points plotted, but you can easily ask students to label all four points. Students may reverse the coordinates of point V without realizing it because they are equal. Ask students to explain the difference between the coordinates to check understanding. If you show answer choices, consider the following possible student errors: Students may choose A, B, or D if they find the coordinates for points T, S, or U, respectively. PART 2 (8 MIN) Objective: Position pairs of rational numbers on a coordinate plane. ELL Support On the Student Companion page for the Part 2 Got It, there are two tasks for students to complete and discuss: Circle four words in the problem that are important for understanding the problem. Take turns reading one of the circled words, and explain what the word means. Beginning Provide students with a list of the words plot, point, identify, and quadrant. Talk first about the words that describe an action plot and identify. Then discuss the words point and quadrant. Intermediate Provide a list of the words students should have included as circled words. Discuss any word that they did not circle, and why it is important to understand words like identify and plot as well as the words point and quadrant. Advanced Have students write 1 or 2 more sentences that use related math words to describe the graph. These would include use of the words x-coordinate, y-coordinate, x-axis, and y-axis. In this problem, students plot ordered pairs with rational-number coordinates on the coordinate grid and identify the quadrant in which each point lies. Instructional Design Students use Full Plane mode of the Coordinate Grapher tool to plot ordered pairs. They must use approximation to figure out where to place each point in relation to the scale given on the grid. How can you use the signs of the coordinates to plot the points on the coordinate grid? [The signs tell you what quadrant the point is in.]
The scales for both axes are in integers. The coordinates of some of the points are rational numbers. How can you plot the points? [Sample answer: You can find the two integers on either side of the coordinate and then estimate the location.] Could you have determined which quadrant a point will lie in without graphing it? Explain. [Sample answer: Yes; you can tell by the signs of the coordinates in which quadrant a point will lie.] How do you remember the numbers of the quadrants? [Sample answer: I remember that Quadrant I has both positive coordinates and that the quadrants are numbered in a counter-clockwise direction.] Students may graph all four points and then identify the quadrants, or they may graph one point at a time and then identify the quadrant that it falls in. Students may be able to identify the quadrant for a point without graphing it, based on the signs of its coordinates. Encourage students to articulate the patterns that they use to determine in which quadrant a point lies. Students may not realize that a point such as point E does not belong in any quadrant because it lies on an axis. Differentiated Instruction For struggling students: To help students keep the coordinates straight, have one student trace along the x-axis and another along the y-axis. Explain that traditionally, you use the x-coordinate first because it is listed first. Students can color code the x- and y-coordinates with different colors and then color the x- and y-axes with the corresponding colors. For advanced students: Turn off the Data Panel feature of the tool and encourage students to use what they know about locating rational numbers on a number line. Emphasize that approximation is the only way to position the points correctly. Turn on the Data Panel feature after students finish solving to check the accuracy of their approximations. Got It Notes Unlike the Example, the points in this problem are already plotted, and the problem statement asks students to identify which point has the given coordinates. Students should pay attention to the signs of the coordinates: a point with a positive x-coordinate and a negative y-coordinate lies in Quadrant IV. Only points A and C have positive x-coordinates and negative y-coordinates, so answer choices B and D can be eliminated. If you show answer choices, consider the following possible student errors: Students may choose B if they mix up the signs of the coordinates. Students may choose D if they switch the order of the coordinates. PART 3 (8 MIN) Objective: Recognize that when two ordered pairs differ only by signs, the locations of the points are related by a reflection across the x- or y-axis. Students recall what they learned about reflecting points over axes in the previous topic and extend it to work with points whose coordinates are rational numbers. They
use the sign changes they are shown in the Intro to write the coordinates of points reflected over the x-axis, the y-axis, and both axes in the Example. Instructional Design Call on students to click the radio buttons and launch animations that explain each type of reflection to the class. Use the examples shown in the coordinate grid to help students realize that reflecting across the x-axis changes the sign of the y-coordinate, and vice versa. Move to Screen 2. Discuss how the coordinates change for each reflection and have students drag the correct ordered pair to complete each statement. Why does the sign of the x-coordinate not change when you reflect a point across the x-axis? [Sample answer: When a point is reflected across the x-axis, the point s horizontal position does not change, so the x-coordinate does not change.] Before completing the first statement How can you use the information you learned in the Intro to find the coordinates of the reflected point without graphing? [The Intro shows that in a reflection across the x-axis, only the y-coordinate changes sign.] How can you use the Coordinate Grapher Tool to check your answers? [Sample answer: You can plot each point and its reflection to see whether the two points are actually reflections of each other over the given axis.] Lisa Says (Screen 2) Use the Lisa Says button to connect reflections in the coordinate plane to mirror images in real life, and to the concept of opposites. Why does Lisa use the word opposite? [Changing the sign of a coordinate is the same as using its opposite.] What shape do the four points make if you connect them? [Sample answer: a square] For a challenge, ask students to find a point that is its own reflection across the y-axis. Students may cite examples, such as 0, 2, before they recognize that every point on the y-axis reflects to itself. Error Prevention Students may benefit from the Coordinate Grid in the Grids and Organizers menu to refer to when solving the problem. Make sure the axes are labeled clearly so that students remember which axis is which. Got It Notes If you show answer choices, consider the following possible student errors: Students may choose A if they reflect the point across the y-axis. They may select B if they choose the point itself. If they reflect the point across both axes, they may choose D.
CLOSE AND CHECK (8 MIN) Focus Question Sample Answer The coordinate plane is formed by two number lines, one horizontal and one vertical, that intersect at the origin. You can use pairs of rational numbers, called coordinates, to describe the location of a point in the coordinate plane. Focus Question Notes Look for students to connect the coordinate plane to the horizontal number line with which they have worked for years. This horizontal number line is called the x-axis, where positive numbers lie to the right of 0 and negative numbers lie to the left of 0. On the vertical number line, which is called the y-axis of the coordinate plane, positive numbers lie above 0 and negative numbers lie below 0. Students should know that to plot a point on the coordinate plane, they need both an x- and a y-coordinate. Rational numbers are plotted on the coordinate plane just as integers are. But sometimes you must figure out where to place a point if it lies between the grid lines. You can also use the signs of the coordinates to identify the quadrant in which a point is located. You may also wish to extend the discussion to how reflections across an axis affect signs as well. 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 recognize how positive and negative rational numbers can be shown on a coordinate plane. In which quadrant would you find ordered pairs that have negative x- and y-coordinates? [Sample answer: Ordered pairs that have negative x- and y-coordinates are found in the third quadrant.] Suppose you have a coordinate plane with axes whose scales only show integer values. How can you use the coordinate plane to graph an ordered pair whose coordinates are rational numbers? [Sample answer: You can compare the integers on the axes to the coordinates of the ordered pair. You can approximate where to plot the points in relation to the integers that are shown on the scale. The points will fall between grid lines.]