<p>In this lesson, students will use geometric figures in the coordinate plane to calculate line slopes, distances between points, and midpoints. The students will:&nbsp;<br>- understand how the slopes of parallel and perpendicular lines are connected.&nbsp;<br>- use the distance formula to calculate the length of a side in a geometric shape in the coordinate plane.&nbsp;<br>- use the midpoint formula to get the midpoint of the segment, given its ends in the coordinate plane.</p>

<p>- How can you use coordinates and algebraic approaches to represent, interpret, and validate geometric relationships?</p>

<p>- <strong>Concave:</strong> Curving inward; a curve is concave toward a point if it bulges away from the point; a polygon is concave if it is not convex, i.e., if at least one of its interior angles is greater than 180 degrees. &nbsp; &nbsp;<br>- <strong>Convex:</strong> Curving outward; a curve such that any straight line cutting the curve cuts it in just two points; a polygon is convex if it lies on one side of any one of its sides extended, i.e., if each interior angle is less than or equal to 180 degrees.<br>- <strong>Coordinate Plane:</strong> A surface for which any set of numbers locate a point, line, or any geometry element in space; for Cartesian coordinates, the point, can be located by its distances from two intersecting straight lines, the distance from one line being measured along a parallel to the other line.<br>- <strong>Distance Formula:</strong> The formula that represents the length of the line segment joining two 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 of the two points,&nbsp;<i>d</i> = \(\sqrt{(x_2 -x_1)^2 + (y_2 -y_1)^2}\)<br>- <strong>Equation:</strong> A statement of equality between two quantities, generally divided into two types, identities and conditional equations. A conditional equation is true only for certain values of the unknown; an identity is true for all values of the variables.<br>- <strong>Formula:</strong> A general answer, rule, or principle stated in mathematical language.</p><p><br>- <strong>Geometry:</strong> The science that treats the shape and size of things; the study of invariant properties of given elements under specified groups of transformations.<br>- <strong>Midpoint:</strong> The point that divides a line segment into two equal parts; the point that bisects the line.<br>- <strong>Parallel Lines:</strong> Equidistant, apart; if two lines are cut by a transversal, and the sum of the interior angles on one side of the transversal is less than a straight angle, the two lines will meet if produced, and will meet on that side of the transversal. Only one line can be drawn parallel to a given line through a given point not on the line.<br>- <strong>Perpendicular Lines:</strong> Two lines are perpendicular to each other if, in a plane, the slope of one of the lines is the negative reciprocal of the other; two straight lines that intersect such that they form a pair of equal adjacent angles.<br>- <strong>Polygon:</strong> A closed-plane figure consisting of points called vertices and lines called sides, which have no common point except for end points. A polygon is convex if each interior angle is less than or equal to 180 degrees. A polygon is concave if it is not convex.<br>- <strong>Slope:</strong> The angle of inclination; for a straight line, the tangent of the angle that the line makes with the positive <i>x</i>-axis.</p>

<p>- Intro Worksheet (M-G-5-1_Intro Worksheet and KEY)<br>- Lesson 1 Graphic Organizer (M-G-5-1_Lesson 1 Graphic Organizer and KEY)<br>- Mason’s Dart Board Activity (M-G-5-1_Mason's Dart Board Activity and KEY)<br>- Dart Board Graph and Table (M-G-5-1_Dart Board Graph and Table and KEY)<br>- graph paper<br>- rulers<br>- colored pencils or markers<br>- Lesson 1 Exit Ticket (M-G-5-1_Lesson 1 Exit Ticket and KEY)<br>- Lesson 1 Extension Activity (M-G-5-1_Lesson 1 Extension Activity)</p>

<p>- The Introductory Activity revisits the concepts of parallel and perpendicular that were covered in Algebra 1 through graphing linear equations. Students connect geometry (parallel and perpendicular) to algebraic concepts such as congruent slopes and negative reciprocals.&nbsp;<br>- Lesson 1 Exit Ticket activity assesses students' grasp of the correspondence between ordered pair and location on the coordinate grid, slope as the ratio of y (vertical distance) to x (horizontal distance), and proficiency with the midpoint algorithm. The evaluation is carried out by requiring students to plot points, calculate slope, and determine midpoint.<br>&nbsp;</p>

<p>Active Engagement, Explicit Instruction<br>W: The class begins with a pair activity to review linear functions. Students investigate how the slopes of parallel and perpendicular lines are related. They need to know this in order to check the attributes of geometric figures in the coordinate plane. They learn how to use the distance formula to calculate the lengths of the sides of geometric figures in the coordinate plane, as well as how to use the midpoint figure to identify the midpoint of a line.&nbsp;<br>H: This lesson begins with a real-world investigation using money as an example. Students imagine what it means to owe money (a negative sloped line) and what it means to earn money (a positive sloped line). They also create a dart board based on slope, distance, and midpoint.&nbsp;<br>E: The initial investigation assesses students' proficiency in determining the slop of a line. Students should review this topic first, but then apply slopes of lines to a geometric topic. The graphic organizer provides students with the formulas required to solve the dart board problem. They'll be curious to learn how these formulas can be applied in the real world and how creative they can be with mathematics.&nbsp;<br>R: Students can easily review and edit their work to correct any errors. Students work in pairs to finish the dart board challenge before getting together with another group to double-check their work. Collaboration is an excellent opportunity for students to test their thought processes and observe how other students solve problems. When some students demonstrate how they solved the dart board challenge, others can improve their work before going on to the exit ticket.&nbsp;<br>E: Students can evaluate their own work when collaborating with another pair. They might brainstorm alternative solutions or describe how they completed the task at hand.&nbsp;<br>T: If students require further practice determining the slope between two points, the pace of the lesson can be slowed down. The main activity in this lesson is suitable for all learners. The visual learner may see how slope, distance, and midpoint can be applied in a real-world setting. The auditory learner is matched with someone who can explain the process of calculating slopes, distances, and midpoints. If given the opportunity, the kinesthetic learner can design the dart board.&nbsp;<br>O: This lesson begins with an investigation that ties a topic students have seen before to a new concept: slopes in geometric figures. Students then complete a graphic organizer, which they will use for the main activity of the lesson: creating a dart board. Once the activity is completed and the class has discussed it, students are invited to fill out an exit ticket to check if they truly understand the topics of the day.<br>&nbsp;</p>

<p><strong>Part 1: Introductory Activity</strong></p><p>Distribute the Intro Worksheet (M-G-5-1_Intro Worksheet and KEY). Have students work in pairs. <strong>"What can we conclude about the slopes of lines that are perpendicular?" </strong>(<i>Their slopes multiply to negative one</i>.) <strong>"What can we conclude about the slopes of lines that are parallel?"</strong> (<i>The slopes are the same</i>.)</p><p>Distribute the Graphic Organizer (M-G-5-1_Lesson 1 Graphic Organizer and KEY) and have students complete it together.</p><p><strong>Part 2 (Think-Pair-Share)</strong></p><p><strong>"How many of you like games, such as arcade games or board games?" </strong>Allow students time to respond. <strong>"Today we are going to help a young man create his own dart board by using what we know about slopes of lines, squares, and finding distances and midpoints."</strong> Distribute copies of Mason's Dart Board Activity (M-G-5-1_Mason's Dart Board Activity and KEY) as well as the Dart Board Graph and Table Worksheet.</p><p>Students should work in pairs with their graphic organizers. Walk around while students are working. On number 2, you may need to remind them that they must establish whether the side lengths are the same size and whether the sides make a right angle (by examining their slopes). Once the pairs have finished the task, group them with another pair to check each other's work. Groups can then explain how they came up with the remaining two squares, including the slopes and lengths of their sides. When the dart board has been made, conclude this activity by discussing how to number the various areas or how many points certain sections should be worth.</p><p><strong>Part 3</strong></p><p>Distribute the Lesson 1 Exit Ticket (M-G-5-1_Lesson 1 Exit Ticket and KEY) to determine whether students comprehend the concepts.</p><p><strong>Extension:</strong></p><p>Distribute the Lesson 1 Extension Activity (M-G-5-1_Lesson 1 Extension Activity). Student responses to the Extension Activity will vary. Rectangles will have different side lengths and shapes. The line connecting the midpoints of two adjacent sides will create the hypotenuse of a right triangle, and its length must conform to the distance formula calculation. Check that students have used the correct ordered pairs in the distance formula.</p>