Lesson Plan

## Midpoints, Distance, and Slope

• Assessment Anchors
• Eligible Content
• Big Ideas
• Mathematical statements can be justified through deductive and inductive reasoning and proof.
• Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
• Objects can be transformed in an infinite number of ways. Transformations can be described and analyzed mathematically.
• Patterns exhibit relationships that can be extended, described, and generalized.
• Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
• Similarity relationships between objects are a form of proportional relationships. Congruence describes a special similarity relationship between objects and is a form of equivalence.
• Spatial reasoning and visualization are ways to orient thinking about the physical world.
• There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
• Concepts
• 2- and 3-dimensional figures
• Geometric Relations: Congruence and Similarity
• Geometric Representations
• Reasoning and Proof
• Competencies
• Define and describe types of geometrical reasoning and proof, using them to verify valid conjectures as they surface in the study of geometry; develop a counter example to refute an invalid conjecture.
• Define, describe, and analyze 2- and 3-dimensional figures, their properties and relationships, including how a change in one measurement will affect other measurements of that figure.

### Objectives

In this lesson, students will use geometric figures in the coordinate plane to find slopes of lines, distances between two points, and the midpoints between two points. Students will:

• learn how the slopes of parallel and perpendicular lines are related.
• use the distance formula to determine the length of a side in a geometric shape in the coordinate plane.
• use the midpoint formula to determine the midpoint of the segment, given its endpoints in the coordinate plane.

#### Essential Questions

• How can you use coordinates and algebraic techniques to represent, interpret, and verify geometric relationships?

### Vocabulary

• Concave: 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.  [IS.1 - Preparation]  [IS.2 - Struggling Learners and ELL Students]
• Convex: 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.
• Coordinate Plane: 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.
• Distance Formula: 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, .
• Equation: 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.
• Formula: A general answer, rule, or principle stated in mathematical language.
• Geometry: The science that treats the shape and size of things; the study of invariant properties of given elements under specified groups of transformations.
• Midpoint: The point that divides a line segment into two equal parts; the point that bisects the line.
• Parallel Lines: 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.
• Perpendicular Lines: 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.
• Polygon: 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.
• Slope: The angle of inclination; for a straight line, the tangent of the angle that the line makes with the positive x-axis.

### Duration

90–120 minutes/1–2 class periods [IS.3 - All Students]

### Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

### Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

### Formative Assessment

• View
• The Introductory Activity uses knowledge acquired from Algebra 1 on graphing linear equations to revisit the concepts of parallel and perpendicular. Students make the link between the geometry (parallel and perpendicular) and the congruent slopes and negative reciprocal associated with algebra.
• Lesson 1 Exit Ticket activity evaluates students’ understanding of correspondence between ordered pair and location on the coordinate grid, knowledge that slope is the ratio of the change in y (vertical distance) to change in x (horizontal distance), and skill in using the midpoint algorithm. The evaluation is done by requiring students to plot points, calculate slope, and calculate midpoint.

### Suggested Instructional Supports

• View
Active Engagement, Explicit Instruction

[IS.9 - All Students]

 W: This lesson begins with a pairs activity of reviewing linear functions. Students observe how the slopes of parallel lines and perpendicular lines are related. They need to know this in order to verify properties of geometric figures in the coordinate plane. They learn the distance formula to find lengths of sides of geometric figures in the coordinate plane, as well as the midpoint figure to determine the midpoint of a line segment. H: This lesson begins with an investigation that incorporates a real-world application: money. Students visualize what it means to owe money (a negative sloped line) and what it means to earn money (a positive sloped line). [IS.10 - All Students] They also design a dart board using the concepts of slope, distance, and midpoint. E: The opening investigation is a good way to gauge where students’ abilities are in terms of finding the slope of a line. This topic should be review for students, but then they will apply the slopes of lines to a geometric topic. The graphic organizer equips students with the formulas they need to solve the dart board problem. They will be interested to see how these formulas can be used in the real world and how creative they can be with math. R: Students have many opportunities to check their work and revise anything they make mistakes on. Students complete the dart board problem in pairs and then get together with another group to check work. Collaboration is a good way for students to check their thought processes and to see how other students solve problems. When some students present how they completed the dart board problem, students can refine their work before moving on to the exit ticket. E: Students can evaluate their own work when they get together with another pair. They can find other ways to solve the same problems or share with one another how they completed the task at hand. T: If necessary, the pace of the lesson can be slowed down if students need a little extra practice at finding the slope between two points. The main activity of this lesson is great for all learners. The visual learner can see where slope, distance, and midpoint can be used in a real-world application. The auditory learner is paired up with someone who can describe the process of finding slopes, distances, and midpoints. The kinesthetic learner can actually create the dart board if given the opportunity. 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 fill out a graphic organizer which they use for the main activity of the lesson: creating a dart board. Once the activity is done and the class has discussed it, students are asked to complete an exit ticket to see if they have really understood the topics of the day.

 IS.1 - Preparation In addition to the vocabulary document that follows in the materials section, consider a review of specific vocabulary terms prior to this lesson. Some of these words may need to be taught explicitly. Use a graphic organizer such as the Frayer Model to review words such as concave, convex,parallel lines, perpendicular lines etc. These may also be defined on a word wall in the classroom, or a student’s vocabulary notebook IS.2 - Struggling Learners and ELL Students A mini-graphic with key word, definition, and example/related symbols, pictures, expression may  be helpful for ELL students as well as struggling learners. IS.3 - All Students Consider pre-teaching the concepts critical to this lesson, including the use of hands-on materials . IS.4 - Struggling Learners and ELL Students Struggling students and ELL students may have difficulty with the amount of information o n the page. Consider cutting up the sheet and dissecting it into clusters of 2 or 3 problems that the students can complete IS.5 - Struggling Learners Provide physical assistance for those students who may have difficulty filling out this worksheet due to visual or physical disabilities. IS.6 - All Students Consider providing a visual of a dart board for those students who do not have the background knowledge. IS.7 - All Students Consider using this as formative assessment and using the information gained to correct any misconceptions. IS.8 - All Students Consider this a great example of minute-by-minute and/or day-by-day formative assessment that may need to be acted upon prior to students leaving the classroom. IS.9 - All Students Consider imbedding these practices throughout the lesson as frequently as possible to assure retention of new concepts. IS.10 - All Students Consider showing the students “Slope Dude” on Youtube to lock in the concept of positive and negative slope. This may help clarify the concepts for ELLs and struggling students.

### Instructional Procedures

• View

Part 1: Introductory Activity

Hand out the Intro Worksheet (M-G-5-1_Intro Worksheet.doc and M-G-5-1_Intro Worksheet KEY.doc). Have students work on it in pairs. “What can we conclude about the slopes of lines that are perpendicular?” (Their slopes multiply to negative 1.) “What can we conclude about the slopes of lines that are parallel?” (Their slopes are the same.) [IS.5 - Struggling Learners]

Hand out the Graphic Organizer (M-G-5-1_Lesson 1 Graphic Organizer.doc and M-G-5-1_Lesson 1 Graphic Organizer KEY.doc) and fill it out with students.

Part 2 (Think-Pair-Share)

“How many of you like games, such as arcade games or board games?” Allow time for students to respond. “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.” [IS.6 - All Students] Hand out copies of Mason’s Dart Board Activity (M-G-5-1_Mason's Dart Board Activity.doc and M-G-5-1_Mason's Dart Board KEY.doc) and the Dart Board Graph and Table Worksheet (M-G-5-1_Dart Board Graph and Table.doc and M-G-5-1_Dart Board Graph and Table KEY.doc).

Students should work on this in pairs using their graphic organizer. Walk around while students are working. [IS.7 - All Students] You may have to hint to them on number 2 that they need to determine if the side lengths are the same size and if the sides make a right angle (by checking their slopes). After the pairs have completed the activity, group them with another pair to check one another’s work. Groups can then present how they came up with the last two squares, as well as the slopes of their sides and lengths of side. When the dart board has been created, wrap up this activity with a discussion of how to number the different sections or how many points certain sections should be worth.

Part 3

Hand out the Lesson 1 Exit Ticket (M-G-5-1_Lesson 1 Exit Ticket.doc and M-G-5-1_Lesson 1 Exit Ticket KEY.doc) to evaluate whether students understand the concepts. [IS.8 - All Students]

Extension:

• Hand out the Lesson 1 Extension Activity (M-G-5-1_Lesson 1 Extension Activity.docx). Student responses will vary for the Extension Activity. Rectangles will vary in side length and shape. The line adjoining the midpoints of two adjacent sides will form the hypotenuse of a right triangle and the length of that hypotenuse must correspond with the distance formula calculation. Verify that students have used the correct ordered pairs in the distance formula.

### Related Instructional Videos

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DRAFT 10/13/2011