<p>In this unit, students will learn how to measure the steepness of a line on the coordinate plane. Students will:&nbsp;<br>- learn about the slope of a line and how to find it.<br>- use the Distance Formula to compute the distance between two points.&nbsp;<br>- to get the middle of two points, apply the midpoint formula.</p>

<p>- How would you explain the relationship between congruence and similarity in two and three dimensions?&nbsp;<br>- How are coordinates algebraically transformed to express, interpret, and validate geometric relationships?</p>

<p>- <strong>Distance:</strong> Between any two points, the length of the line segment joining the points. In analytic geometry it is found by taking the square root of the sum of the squares of the differences of the corresponding rectangular Cartesian coordinates (ordered pairs) of the two points. In the plane, this is d = \(\sqrt{(x_2 -x_1)^2 + (y_2 - y_1)^2}\).<br>- <strong>Midpoint:</strong> The point that divides the given line segment into two equal parts; the point that bisects the line. If the two endpoints of the line have the Cartesian coordinates (\(x_1\)), \(y_1\)) and (\(x_2\), \(y_2\)), the coordinates of the midpoint are x = \(x_1 + x_2 \over 2\), y = \(y_1 + y_2 \over 2\).<br>- <strong>Slope:</strong> Of a line, the tangent of the angle that the line makes with the positive x-axis; in rectangular Cartesian coordinates, slope = \(y_2 - y_1 \over x_2 - x_1\), where (\(x_1\), \(y_1\)) and (\(x_2\), \(y_2\)) are points on the line, and designated by m; also Δ\(Δy \over Δx\) ; also known as \(rise\over sun\).<br>- <strong>Slope-intercept form:</strong> <i>y = mx + b</i>.</p>

<p>- copies of Lesson 3 Graphic Organizer (M-G-7-3_Lesson 3 Graphic Organizer and KEY)<br>- copies of Ball Activity sheet (M-G-7-3_Ball Activity)<br>- copies of Dot Activity sheet (M-G-7-3_Dot Activity)<br>- 16 eight-foot ropes marked off with tape at each foot<br>- eight athletic balls (soccer balls, basketballs, volleyballs, etc.)<br>- poster-sized graph paper<br>- sticky dots<br>- copies of Lesson 3 Exit Ticket (M-G-7-3_Lesson 3 Exit Ticket and KEY)</p>

<p>- Observations during the ball and dot exercises must include an assessment of students' knowledge of slope (positive or negative), distance (Pythagorean Theorem or Distance Formula), and midpoint.&nbsp;<br>- The Lesson 3 Exit Ticket assignment must be evaluated using students' estimates of the ordered pairs that correspond to their selected cities.<br>&nbsp;</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: This lesson starts with a discussion about measurement, which students use daily. They will learn new formulas for estimating distance, especially when usual measuring equipment are not around. After this lesson, students will be able to compute the distance, midpoint, and slope of the line formed by the two points.&nbsp;<br>H: Students are engaged throughout the lesson. They begin by filling out a graphic organizer, but the instruction then becomes active, visual, and relates the new material to real-world applications.&nbsp;<br>E: After completing a graphic organizer, students can use it for the following activities. The ball and dot activities let students to "do" and "see" the mathematics underlying the new formulas. They play two roles: passing, bouncing, and rolling a ball, and then solving for distance, midpoint, and slope. Because the coordinates are not provided, students are responsible for checking each other's work. If one of them computed a different solution, they must work together to solve the problem. Walk around and check student work.&nbsp;<br>R: Students first see distance, midpoint, and slope problems in the graphic organizer. They try them on their own, then collaborate with a partner to &nbsp;correct mistakes before moving on to the ball activity. Because there is more than one activity, students have time to modify and rethink their work before going on to the next one. The dot activity comes before the exit ticket, so they have another chance to improve their work before the lesson ends.&nbsp;<br>E: This lesson requires extensive participation with peers. They assess each other's performance in the ball and dot activities before moving on. If there is a disagreement, clarify the mistakes.&nbsp;<br>T: This lesson is designed for kinesthetic and visual learners. The ball activity takes students outside of the classroom and into the human coordinate plane. They are acting as coordinates and calculating their own distance. Being the subject and focus of the class activity is enjoyable for them. The dot activity begins similarly to the Pin the Tail on the Donkey game. Students make dots on paper with their eyes closed. This keeps them from simply choosing positive numbers. It is a good visual activity because the paper and dots are larger than the usual materials used. This lesson is flexible for both students who need more practice and students who go beyond the standards. Students that need additional practice should be given the four coordinates on their exit ticket.&nbsp;<br>O: The lesson progresses from measurement to activity. It begins with a discussion of measurement, followed by note-taking and measurement-related activities. Students spend the most of the class in groups, but the exit ticket is individual. This aids in the evaluation of student learning.</p>

<p>Use a blackboard, whiteboard, or overhead projector to display the following information to the class.</p><p><strong>Distance</strong></p><p><strong>"The distance between two points in a plane is measured by the length of the straight line that joins the points."</strong> Refer to the Vocabulary section for a definition of the Distance Formula. <strong>"If the points are defined as Cartesian coordinates, the distance between them is expressed by the following formula, generally known as the Distance Formula:</strong></p><figure class="image"><img style="aspect-ratio:206/82;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_78.png" width="206" height="82"></figure><p>where the two points are represented by the ordered pairs (\(x_1\), \(y_1\)) and (\(x_2\), \(y_2\)).</p><p><strong>"If we look at the Pythagorean Theorem, which represents the length of the hypotenuse of a right triangle as \(c^2 = a^2 + b^2\), we can see the similarity between the two formulas. Taking the square root of both sides of the equation (c = \(\sqrt{a^2 + b^2}\)). If we replace with a and b, they look exactly the same."</strong></p><p><strong>"Let's use the Distance Formula to calculate the length of a straight line connecting two ordered pairs (−8, −3) and (7, 5). Note that for these two ordered pairs, \(x_1\) = −8, \(y_1\) = −3, \(x_2\) = 7, and \(y_2\) = 5, substituting these values of </strong><i><strong>x</strong></i><strong> and </strong><i><strong>y</strong></i><strong> into the Distance Formula,”</strong></p><figure class="image"><img style="aspect-ratio:208/183;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_79.png" width="208" height="183"></figure><figure class="image"><img style="aspect-ratio:380/343;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_80.png" width="380" height="343"></figure><p><strong>The graph of the two ordered pairs (−8, −3) and (7, 5) shows a right triangle with vertices at (−8, −3), (7, −3), and (7, 5). The line connecting (−8, −3) and (7, 5) serves as the hypotenuse. The graph shows that the base of the right triangle is 15 [(−8, −3) to (7, −3)], whereas the other leg is the altitude from (7, −3) to (7, 5), which is 8. From the Pythagorean theorem,</strong></p><p><strong>\(c^2 = 15^2 + 8^2\)</strong></p><p><i><strong>c</strong></i><strong> = \(\sqrt{289}\), which is 17, which yields the same result as the Distance Formula."</strong></p><p><strong>Midpoint</strong></p><p><strong>"From the graph of the hypotenuse, the line joining the two ordered pairs (−10, 0) and (10, 2), it looks like the midpoint is close to the origin."</strong></p><p><strong>"Is the midpoint of the base of the triangle near the </strong><i><strong>y</strong></i><strong>-axis?" (</strong><i><strong>yes</strong></i><strong>)</strong></p><p><strong>Let's figure out where the midpoint is. From our definition of midpoint, the coordinates are </strong><i><strong>x</strong></i><strong> = \(x_1+x_2 \over 2\), </strong><i><strong>y</strong></i><strong> = \(y_1+y_2 \over 2\)."</strong></p><p><strong>Substitute \(x_1\), \(y_1\), \(x_2\), and \(y_2\).</strong></p><figure class="image"><img style="aspect-ratio:198/103;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_81.png" width="198" height="103"></figure><p><strong>The midway is represented by the ordered pair (-\(1 \over 2\), 1).</strong></p><p><strong>Slope</strong></p><p><strong>"If the line determined by the points we have been working with, (−8, −3) and (7, 5), were the graph of a linear equation, where </strong><i><strong>y</strong></i><strong> is the dependent variable and </strong><i><strong>x</strong></i><strong> is the independent variable, how could we identify that equation?"</strong></p><p><strong>"Remember the slope-intercept form of a linear equation, </strong><i><strong>y = mx + b</strong></i><strong>; the </strong><i><strong>y</strong></i><strong>-intercept is </strong><i><strong>b</strong></i><strong> and the slope is </strong><i><strong>m</strong></i><strong>."</strong></p><p><strong>"Refer to the right triangle above. What is the </strong><i><strong>y</strong></i><strong>-intercept?"</strong> (1\(4 \over 15\) because y - \(y_1\) = m(x - \(x_1\)) + b, where <i>m</i> is the slope (altitude of the right triangle divided by its height), and <i>b</i> is the <i>y</i>-intercept).</p><figure class="image"><img style="aspect-ratio:199/101;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_82.png" width="199" height="101"></figure><p><strong>"To represent the slope, go back to the definition of slope in the Vocabulary section, 'the tangent of the angle the line makes with the positive </strong><i><strong>x</strong></i><strong>-axis.'"</strong></p><p><strong>In a right triangle, such as the one depicted above, the tangent to the positive </strong><i><strong>x</strong></i><strong>-axis of the angle whose vertex is at (−8, −3) is the ratio of the opposite side of the right triangle (distance from (7, −3) to (7, 5)) to the adjacent side (distance from (−8, −3) to (7, −3)). That ratio is \(8 \over 15\)."</strong></p><p><strong>"Return to the point-slope form of the linear equation: </strong><i><strong>y = mx + b</strong></i><strong>; </strong><i><strong>m</strong></i><strong> = \(8 \over 15\) and </strong><i><strong>b</strong></i><strong> = \(19 \over 15\), therefore the equation is </strong><i><strong>y</strong></i><strong> = \(8 \over 15\)</strong><i><strong>x</strong></i><strong> + \(19 \over 15\). The slope in this example may also be seen along the path of the line whose distance we first calculated. Consider the point at (−8, −3) moving to the right and rising above the </strong><i><strong>x</strong></i><strong>-axis. It moves 15 units to the right horizontally and 8 units up vertically. The change in the </strong><i><strong>y</strong></i><strong>-direction is positive 8, whereas the change in the </strong><i><strong>x</strong></i><strong>-direction is positive 15. This is another way to characterize the slope on a coordinate plane: change in </strong><i><strong>y</strong></i><strong> divided by change in </strong><i><strong>x</strong></i><strong>, commonly known as slope = \(Δy \over Δx\)."</strong></p><p><strong>Part 1</strong></p><p>Begin the next section of the lesson with a discussion of measurements. Ask students what they measure on a daily basis and how they derive that measurement. Examples include telling time using a clock or a watch, measuring ingredients with spoons or cups, weighing with a scale, and calculating distance with an odometer. <strong>"What do we do if we don't have our usual measuring tools?"</strong> Allow them a few minutes to think about it.</p><p>Ask, <strong>"How many of you estimate your measurements? Alternatively, use a formula instead of a measurement tool. Today, we'll discover a method for estimating distance without a measuring tape or yard stick."</strong></p><p>To complete this task, use the Graphic Organizer (M-G-7-3_Lesson 3 Graphic Organizer and KEY). Have students complete both the organizer and the examples. They can complete the tasks independently before collaborating with a partner to discuss them.</p><p><strong>Part 2: Ball Activity</strong></p><p>Go to a large place, like the cafeteria or gym. Lay out two 8-foot ropes as if they were the coordinate plane's <i>x</i> and <i>y</i> axes. As a demonstration, arrange two students anywhere on the "plane" and hand them one of the athletic balls. <strong>"If </strong>[name of first student] <strong>passed the ball to</strong> [name of second student], <strong>how could we determine the distance of the pass? Keep in mind that I did not bring a measuring tape or yard stick"</strong>. Give students a few minutes to think about it and answer the question. (Answer:<i> select the points and name the ordered pairs of coordinates</i>.)</p><p><strong>"Let's designate the locations they're standing with coordinates. Assume</strong> [name of first student] <strong>is standing at (−2, −1) and</strong> [name of second student]<strong> at (3, 4). How could I measure the pass now?" </strong>(<i>Use the Distance Formula</i>.)</p><p>Place students in groups of four. Lay out the remaining 8-foot ropes in the same manner as the first set. Distribute the Ball Activity sheet (M-G-7-3_Ball Activity). Students calculate the distance of the pass, the midpoint between the two locations, and the slope of the line the ball rolls along from one spot to the next. The instructions are on the Ball Activity sheet.</p><p>If there aren't enough ropes or balls, divide the class in half. Half of the class does the Ball Activity, while the other half does the Dot Activity; then they exchange.</p><p><strong>Part 3: Dot Activity</strong></p><p>This activity can be completed in pairs. Tape a sheet of poster-size graph paper to the wall. Have students sketch the <i>x</i>- and <i>y</i>-axes. Hand out the Dot Activity sheet (M-G-7-3_Dot Activity) and a sheet of sticky dots.</p><p><strong>Part 4</strong></p><p>Distribute the Lesson 3 Exit Ticket (M-G-7-3_Lesson 3 Exit Ticket and KEY) to see whether students comprehend the concepts.</p><p><strong>Extension:</strong></p><p>Students can make a map of their town for someone new to the area. They should include schools, police and fire stations, petrol stations, hospitals, and restaurants. The map should be produced on a coordinate plane, with each building identified with its coordinates. Below the map, students should make a "Index" of the distances and midpoints between each establishment.</p>