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Title: The Distance Formula for the Cartesian Plane

NJ Core Curriculum Content Standards:

Problem Solving: Students will draw on their knowledge to formulate the distance formula during an interactive lecture.

Representation: Students will see graphical representations of the Pythagorean Theorem and Distance Formula.

Communication: Students will participate in an interactive lecture where they will answer questions and be asked to reason their findings in an effort to be clear and convincing to the class.

Connections: Students will form connections between The Pythagorean Theorem and The Distance Formula.

Reasoning and Proof: Students will see the proof of The Distance Formula and be able to justify it in their own words.

Resources needed: Students will need to have grid paper, a copy of the worksheet, and a calculator. Teacher will need an overhead projector.

Prior Knowledge: Students should a) Be familiar with the Cartesian Plane and be able to plot any given points b) Be able to draw lines connecting points on a plane and be familiar with properties of simple geometric figures c) Be familiar with basic concepts from algebra d) Know The Pythagorean Theorem and be able to apply it

Objectives: After completing this lesson, students should:

a) Be able to state the distance formula:

[pic]

b) Be able to apply this formula to find the distance between any two points on a graph.

c) Be able to construct a graphical representation of the Distance Formula by forming a right triangle from any given two points and using this representation to find the length between the points.

d) Understand why this formula is true, meaning they will understand the connection between the Pythagorean Theorem and The Distance Formula.

Identify important ideas in terms of the subject area: The Distance Formula is an important tool in determining the equation of a circle and will be used to show that distance is equal to the radius of a circle.

In terms of real-life applications, The Distance Formula can be used in the fields of architecture and carpentry to measure lengths between given points. In every day life, someone may use The Distance Formula to measure the size of a television by finding its diagonal.

Potential student difficulties: Students often mistake the Distance Formula for the Slope formula due to its similar notation. Time will be taken in the beginning of the lesson to refresh students understanding of calculating slope and later to note the difference between slope and distance but also their relationship.

Description of lesson:

-Students will begin the lesson by working on a “Do Now” activity that reviews their knowledge of The Pythagorean Theorem (see Do Now Worksheet). Teacher will circulate around the classroom to get a sense of students understanding.

-A class discussion will summarize the student’s results to the “Do Now” activity

-The teacher will hypothetically pose the question: We have just showed that with The Pythagorean Theorem, if we are given two sides of a right triangle we are able to find the third side. What would happen if instead, we are given two points on the Cartesian Plane, and asked to find the distance between them?

-The teacher will use the overhead projector to plot points A and B on the Cartesian Plane and ask students to identify these points as (6,4) and (-3,-4) respectively

(see Overhead 1).

-Teacher will remind students to consider their “Do Now” activity and ask for suggestions on how to find the distance between A and B.

-Teacher will proceed to draw lines and connect points A and B into a right triangle, creating a new point C at (6,-4)(see Overhead 2).

-The teacher will ask students to find the length of [pic] and [pic], labeling their lengths 8 and 9 respectively in order to find the length of [pic], i.e. the distance between A and B.

-Applying The Pythagorean Theorem, the teacher will show:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

-The teacher will note that although [pic] we must omit -12.04 because distance is positive

-The teacher will now repeat this process but with arbitrary A and B points in order to prove The Distance Formula (see Overhead 3) and below:

[pic]

[pic]

Using Pythagorean Theorem, [pic]

Take the square root of both sides to solve for distance, [pic]

-Students will be given grid paper and asked to work independently to find the distance between the following points while the teacher circulates around the room assisting students who need help

a) (1,4) and (4,0)

b) (5,1) and (5,-6)

-Class will discuss the solutions to the problems together as students volunteer to show their steps in finding the distance on the overhead projector

-Class will wrap up by answering any questions and assigning homework

(see Worksheet 1)

Time Table:

|Clock |Title |Students Doing |Teacher Doing |

|0-5 Min |Do Now |Completing |Observing student progress |

| | |“Do Now Worksheet” | |

|5-10 Min |Discussion |Participating |Explaining Do Now answers |

|10-25 Min |Lecture using Overhead|Taking Notes |Explaining relationship between The |

| | | |Pythagorean Thm. and Distance Formula. Also |

| | | |Proving The Distance Formula. |

|25-35 Min |Examples |Finding solutions to examples |Observing and providing help if needed |

|35-42 Min |Discuss |Sharing example answers |Elaborating on students understanding by |

| | | |asking open questions |

|42-45 Min |Wrap-Up |Asking questions |Answering questions and assigning Worksheet 1|

| | | |for homework |

Modifications: If students are struggling with the topic during the lecture, have all students work on Worksheet 1 in heterogeneous groups instead of individually so that students having difficulty can work with more sophisticated students. Also, provide weaker students with more organized step-by-step worksheet.

Homework: (see Worksheet 1) Students will be presented with a couple of simple examples to be worked out in a step-by-step approach similar to the lesson and also a couple of question that assesses their general understanding of the topic.

Do Now Worksheet:

1. State The Pythagorean Theorem

2. Find each missing length in the following right triangles:

Worksheet 1: Homework

1. Find the length between the following points using the Distance Formula and make a graphical representation of your findings.

a) (0,4) and (2,3)

b) (−3,17) and (15,-7)

2. Does the following formula represent the distance between points

(a,b) and (c,d)? Explain why or why not.

[pic]

3. Jack and Emily go to the same school, which is located at (0,0) on a grid. Jack walks 5 blocks east and 2 blocks south to get home from school. Emily walks 6 blocks north and 7 blocks west to get to her house from school. What is the distance between Jack and Emily’s house?

4. Given the points (-7,1) and (-1, 9), on a Cartesian Plane, construct a right triangle connecting them and find the distance between the two points using: a) The Pythagorean Theorem

b) The Distance Formula

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[pic]

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