Lesson Plan

## The Distance Formula and the Pythagorean Theorem

• 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.
• Some geometric relationships can be described and explored as functional relationships.
• 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
• Analytic Geometry
• Geometric Relations: Congruence and Similarity
• Geometric Representations
• Trigonometric Ratios
• Competencies
• 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.
• Use concepts of congruence and similarity to relate and compare 2- and 3-dimensional figures, including trigonometric ratios.
• Use coordinates and algebraic techniques to interpret, represent, and verify geometric relationships.

### Objectives

This lesson covers the distance formula and the Pythagorean theorem, their uses, and their similarities. By the end of this lesson, students will be able to:

• find the shortest distance between two points on a map.

• relate the distance formula to the Pythagorean theorem.

#### Essential Questions

• What is the relationship between the length of the hypotenuse of a triangle and its legs?

• What methods do we have to determine the unknown measures of a right triangle?

### Vocabulary

• Hypotenuse: In a right triangle, the side opposite the right angle, the longest side of the triangle.

• Pythagorean Theorem: [IS.1 - All Students] A formula for finding the length of a side of a right triangle when the lengths of the other two sides are given (leg2 + leg2 = hypotenuse2 or  a2 + b2 = c2).  [IS.2 - Struggling Learners]

• Leg of a Right Triangle: One of the sides of the right triangle other than the hypotenuse. [IS.3 - All Students]

### Duration

45–60 minutes [IS.4 - All Students]

### Prerequisite Skills

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

### Related Materials & Resources

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### Formative Assessment

• View
• Use a holistic evaluation of student presentations and explanations of their maps. Consider correspondence between written material and explained purpose, whether or not details support each result, and confidence in presenting.

• Completion of the Distance Worksheet (M-G-1-3_Distance Worksheet.doc) shows how well students can represent ordered pairs, set up the distance formula and substitute correct data, and use appropriate and accurate calculations.

• Students’ Lesson 3 Exit Tickets (M-G-1-3_Lesson 3 Exit Ticket.doc) show the degree to which they can make general sense of how to calculate distance without getting lost in each detail.

### Suggested Instructional Supports

• View
Active Engagement, Modeling, Explicit Instruction W: In this lesson, students will learn how to find the distance between two points described as ordered pairs. They will begin by recalling and using the Pythagorean theorem and seeing it as a variation. Then they will apply the Pythagorean theorem to find the distance between two points on a coordinate plane. H: Working in pairs on the city map activity provides students a real-world application of routes and distances in a familiar location and takes advantage of their assessments of one other’s work. E: Group presentations engage collaboration between each pair of students in choosing starting and ending points, reasoning through their route selections, and evaluating each other’s choices. The activity also requires them to plan for the contributions of each individual. R: The Distance Worksheet is an activity for which students choose the location of the origin, plot the two given points, calculate the shortest distance, and the distance using only horizontal and vertical directions. In this way, students can visualize base, height, and hypotenuse of each right triangle. E: To complete the Lesson 3 Exit Ticket students must create an original narrative that describes how to find the distance between two points. There are no specifications as to how the points are described, and students are free to choose ordered pairs, points on a map, or descriptive language that contains enough information to complete the task. This Exit Ticket evaluates what students know and are able to do in using the distance formula, vertical and horizontal distances, or an appropriately described method. In the extension activity, students are evaluating their competence with the distance formula by identifying triangle types according to side length. T: This lesson provides an explanation of the distance formula in terms of real-world examples, providing students with concrete (rather than abstract) ways to think about the distance formula (both in terms of its derivation and its applicability). It also appeals to more abstract students in the discussion of the connection between the Pythagorean theorem (a formula that most students are familiar with) and the distance formula (a formula that looks complicated). In fact, the distance formula and the Pythagorean theorem are one and the same. Algebraically-inclined students will be pleased to find the relationship between the two, and those students who are not as comfortable with algebra will be comfortable with being able to use the Pythagorean theorem as a substitute for the distance formula. O: Students begin working in pairs for this activity on a straightforward, concrete task. The task becomes more abstract as the concept of the shortest distance between two points begins to be explored. Students also get a chance, near the end of the lesson, to work individually on the Distance Worksheet (although this can also be completed in pairs, depending on how the first portion of the lesson went). Lastly, students work individually on the Exit Ticket.

 IS.1 - All Students Consider explaining where this term comes from IS.2 - Struggling Learners Be sure struggling students have prerequisite skills to understand this concept IS.3 - Preparation Consider using graphic organizers (e.g., Frayer Model, Verbal Visual Word Association, Concept Circles) to review key vocabulary prior to the lesson. Some additional vocabulary to review in this lesson may include: ratio, quotient, variables, numerator, denominator and factor/factoring. IS.4 - All Students Consider preteaching the concepts critical to this lesson, including the use of hands-on materials. Throughout the lesson (based on the results of formative assessment), consider the pacing to be flexible to the needs of the students. Also consider the need for reteaching and/or review both during and after the lesson as necessary. IS.5 - All Students Consider having these materials available in alternate formats such as large print, digital, etc. IS.6 - All Students Ensure that students understand the proper implementation of cooperative learning.  Encourage them to consistently use the language of mathematics, specifically Pythagorean theorem, hypotenuse, etc.  Show many visuals and models of the Pythagorean theorem. IS.7 - All Students Consider modeling this for students. IS.8 - All Students Model this as a think-aloud IS.9 - All Students Monitor student pairs and discuss the activity with them. IS.10 - All Students Review the distance formula with the students.  Model how it should be used. IS.11 - All Students Allow for multiple means of presentation: oral, visual, written, etc. IS.12 - All Students Reteach if necessary. IS.13 - All Students Allow for alternate means of responding such as oral or digital.

### Instructional Procedures

• View

Students should work in pairs for this activity. Each pair of students should be given a satellite map of a downtown area/city, [IS.6 - All Students] somewhere with mostly regular square blocks and preferably in the same area as the school; see sample Minneapolis map for an example (M-G-1-3_Minneapolis Map.doc). Each pair of students should also have a highlighter. Instruct each student to choose an intersection on the map and mark it with the highlighter. [IS.7 - All Students] If the map is of an area surrounding the school, recommend that students choose an intersection familiar to them (so they can better visualize the situation). Otherwise, let them choose random intersections. Then, each pair should highlight the path they would drive to get from one student’s point (intersection) to the point (intersection) their partner chooses. Have each pair calculate this distance (in blocks).

Next, have each group consider the shortest distance from one point to the other if they were not restricted to staying on the road or going around obstacles such as buildings. [IS.8 - All Students] Have them highlight this route. Then ask:

How can we find the shortest distance between your starting and ending places?”

What shape have you highlighted on your map?” (right triangle)

What theorem do you know that relates the lengths of the sides of a right triangle?” (Pythagorean theorem)

Have students measure the distance between the two points using rulers, and then measure the distance it would take to get to that point by going down the streets. If students measure the sides of a triangle, have them compare and see that for their measurements, a2 + b2 = c2. Demonstrate this to the class on the board. Mark a point on the board (call it A), and then go up 3 inches and mark another point (call it B); then go right (forming a 90 degree angle in the process) 4 inches and mark that point (call it C). Measure the distance between A and C, and show the class that is it 5 inches; show them that for this example, a2 + b2 = c2, or 32 + 42 = 52.

Now challenge students to go back to their maps and see how many right angles they can find, and have them measure the straight-line distance versus the distance going down the legs of the triangle. Assist students who might need additional practice as necessary. [IS.9 - All Students]

After giving students a few minutes exploring this exercise, have students gather back together, and remind students of the Pythagorean Theorem, taking care to stress that a and b represent the lengths of the legs and c represents the length of the hypotenuse.

Have each pair of students find the shortest distance between their two points using the Pythagorean Theorem this time. [IS.10 - All Students]

Select a few groups to present their maps. [IS.11 - All Students] Have them explain:

• what points they selected (and why, if applicable)

• the street route they highlighted and how they found that distance

• the shortest route and how they found that distance

Ensure that, by the end of the presentations, it is clear to students how to use the Pythagorean theorem to find the distance between two points. [IS.12 - All Students] Encourage the use of calculators and rounding to 1 or 2 decimal places for complicated calculations.

Distribute the Distance Worksheet (M-G-1-3_Distance Worksheet.doc and M-G-1-3_Distance Worksheet KEY.doc). Instruct students to:

1. Graph the points.

2. Draw the “street route” between the points.

3. Draw the shortest route between the points.

4. Use the Pythagorean theorem to find the distance between the points.

If students have difficulty determining the lengths of the sides of the right triangles on the Distance Worksheet, ask students to examine the two x-coordinates and determine how far it is from one x-coordinate to the other x-coordinate and illustrate that the distance between the x-coordinates is simply the length of one of the legs. Repeat this with the y-coordinates.

After students have finished the Distance Worksheet, write the distance formula on the board:

How does the distance formula relate to the Pythagorean theorem?”

Students should note the differences between the two and discuss how the two are, algebraically, the same formula. For example, the distance formula has a square root in it, and the Pythagorean theorem does not; however, solving the Pythagorean theorem for c (rather than c2) results in a square root. Depending on the class’s algebra skills, the algebraic relationship between the two can be explored in depth.
Show this simple example: Draw a right triangle with the right angle on the origin (0, 0) and two vertices at (0, 4) and (-3, 0). Show students that the subtraction of x2 from x1 is the same as 0 – (-3), which is 3. Then the subtraction of y2 from y1 is the same as 4 – 0, which is 4. Now show that 32 + 42 is the same as a2 + b2 from the more familiar c2 = a2 + b2.

As an exit activity, have students write an explanation of how to find the shortest distance between two points (M-G-1-3_Lesson 3 Exit Ticket.doc). [IS.13 - All Students]

Extension:

• For each set of three ordered pairs, graph the vertices on the x/y-coordinate plane, classify each triangle according to its side lengths and angle measures, and find each perimeter.

1. A (-4, -4), B (6, 0), C (3, 4)

2. A (-4, 3), B (10, 7), C (-5, 2)

3. A (-2, 2), B (6, -2), C (2, -6)

4. A (3, 6), B (7, 2), C (3, -1)

### Related Instructional Videos

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DRAFT 08/31/2011