This task explores the real world topic of building light rails. Throughout the implementation of this task the students will learn about the cost of building railways and how to implement them within a budget. This task explores such mathematical concepts of using coordinates to find the distance between points, using coordinates to build polygons and find the area and length of sides, and writing equations of parallel lines.

Equation of a line with image, equations of parallel lines with image, equations of perpendicular lines with image.

This lesson unit is intended to help teachers assess how well students are able to understand the relationship between the slopes of parallel and perpendicular lines and, in particular, to help identify students who find it difficult to: find, from their equations, lines that are parallel and perpendicular; and identify and use intercepts. It also aims to encourage discussion on some common misconceptions about equations of lines.

CK-12's Basic Geometry FlexBook is designed to present students with geometric principles in a simpler, more graphics-oriented course. Students will explore geometry at a slower pace with an emphasis placed on visual aids and approachability.

Students generalize the criterion for perpendicularity of two segments that meet at a point to any two segments in the Cartesian plane.
Students apply the criterion to determine if two segments are perpendicular.

Students state the relationship between previously used formats for equations for lines and the new format a1x + a2y + c, recognizing the segments from (0,0) to (a1, a2) as a normal and - a2/a1 as a slope.
Students solve problems that are dependent upon making such interpretations.

Students recognize parallel and perpendicular lines from slope.
Students create equations for lines satisfying criteria of the kind: “Contains a given point and is parallel/perpendicular to a given line.”

The challenge of programming robot motion along segments parallel or perpendicular to a given segment leads to an analysis of slopes of parallel and perpendicular lines. Students write equations for parallel, perpendicular, and normal lines. Additionally, students will and study the proportionality of segments formed by diagonals of polygons.

In this task students make a conjecture about the slopes of perpendicular lines. This observation will be formally proved in a later task, but the representation-based argument presented in this task suggests that this is a reasonable generalization across all cases. In the previous task, students encountered the idea of perpendicularity when they were asked to rotate the lizard 90°, and they may also have noticed the idea when they considered the relationship between the line of reflection and the points on the reflected image of the lizard. Students might have used the square corner of a piece of paper or a protractor to measure these right angles. In this task they consider how the coordinate grid can be used to determine if two lines are perpendicular. This is one example of the connections that can be made between coordinate geometry and transformations. It is powerful for students to be able to draw upon two different representational systems to think about the same ideas. In this task, students also consider a definition of perpendicular that is related to reflections: Two lines are perpendicular if they meet to form congruent adjacent angles.

Image of parallel lines, image of perpendicular lines, and image of intersecting lines, each image followed by text in accordion of properties of eac.n

This is an activity. I like to call it "Making a worksheet more than a worksheet".  This is a musical review. Students will put their name on their paper, pass the paper when the music is playing and when it stops, answer the question they are on.  

This is an activity. I like to call it "Making a worksheet more than a worksheet".  This is a musical review. Students will put their name on their paper, pass the paper when the music is playing and when it stops, answer the question they are on.  

In this Khan Academy activity, students will determine if two lines are perpendicular or parallel from their equations.

Khan video developing idea of parallel and perpendicular lines using point on the line.

In this Khan Academy activity, students will determine if two lines are perpendicular or parallel from their graph.

Course presentation slides for each step of proving vertical angles are congruent, using intersecting lines and supplementary angles.

The purpose of this task is to prove that parallel lines have equal slopes and that the slopes of perpendicular lines are negative reciprocals. Students have used these theorems previously. The proofs use the ideas of slope triangles, rotations, and translations. Both proofs are preceded by a specific case that demonstrates the idea before students are asked to follow the logic using variables and thinking more generally.

In this Khan Academy activity, students will write the equations of parallel and perpendicular lines.