<p>This lesson discusses the uses and similarities of the distance formula and the Pythagorean theorem. By the end of this lesson, students will be able to:<br>- find the shortest distance between two points on a map.<br>- relate the distance formula to the Pythagorean theorem.</p>

<p>- What is the relationship between the length of the hypotenuse of a triangle and its legs?&nbsp;<br>- What methods do we have for determining the unknown measures of a right triangle?</p>

<p>- Hypotenuse: In a right triangle, the side opposite the right angle, the longest side of the triangle.<br>- Pythagorean Theorem: A formula for finding the length of a side of a right triangle when the lengths of the other two sides are given (<i>leg</i>² + <i>leg</i>² = <i>hypotenuse</i>² or <i>a</i>² + <i>b</i>² = <i>c</i>²).&nbsp;<br>- Leg of a Right Triangle: One of the sides of the right triangle other than the hypotenuse.&nbsp;</p>

<p>- copies of Minneapolis Map (M-G-1-3_Minneapolis Map)&nbsp;<br>- copies of Distance Worksheet (M-G-1-3_Distance Worksheet.doc and M-G-1-3_Distance Worksheet KEY)&nbsp;<br>- copies of Lesson 3 Exit Ticket (M-G-1-3_Lesson 3 Exit Ticket)&nbsp;<br>- rulers</p>

<p>- Use a holistic evaluation of student presentations and explanations of their maps. Consider the relationship between written material and explained purpose, whether or not specifics support each result, and confidence in presenting.<br>- Completing the Distance Worksheet (M-G-1-3_Distance Worksheet) demonstrates how effectively students can represent ordered pairs, set up the distance formula, substitute proper data, and use appropriate and precise calculations.&nbsp;<br>- Students' Lesson 3 Exit Tickets (M-G-1-3_Lesson 3 Exit Ticket) demonstrate their ability to grasp the general concept of distance calculation without getting lost in each detail.</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: In this lesson, students will learn how to calculate the distance between two ordered pairs. They will start by recalling and applying the Pythagorean theorem and seeing it as a variation. Then they will use the Pythagorean theorem to calculate the distance between two points on a coordinate plane.<br>H: Working in pairs on a 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.&nbsp;<br>E: In group presentations, students collaborate between each pair of students to choose starting and ending points, reasoning through their route selections, and evaluating each other’s choices. The activity also demands them to plan for each individual's contributions.&nbsp;<br>R: The Distance Worksheet activity requires students to select the origin location, plot two points, compute the shortest distance, and use only horizontal and vertical directions. This allows students to visualize the base, height, and hypotenuse of each right triangle.&nbsp;<br>E: To accomplish the Lesson 3 Exit Ticket, students must write an original story explaining how to calculate the distance between two points. There are no restrictions on how the points are presented, and students are free to use ordered pairs, points on a map, or descriptive language that provides enough information to complete the assignment. This Exit Ticket assesses what students know and can perform when applying the distance formula, vertical and horizontal distances, or a correctly described method. In the extension activity, students assess their proficiency with the distance formula by recognizing triangle types based on side length.&nbsp;<br>T: This lesson explains the distance formula through real-world situations, giving students a more 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 when discussing the relationship between the Pythagorean theorem (which most students are familiar with) and the distance formula (which looks complex). In fact, the distance formula is the same as the Pythagorean theorem. Algebraically-inclined students will be happy to discover the relationship between the two, 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.&nbsp;<br>O: In this activity, students work in pairs to complete an easy task. The task becomes increasingly sophisticated when the concept of shortest distance between two points is examined. Students also have the opportunity, near the end of the lesson, to work alone on the Distance Worksheet (although this can also be completed in pairs, depending on how the first portion of the lesson went). Finally, each student works separately on the Exit Ticket.</p>

<p>This activity requires students to work in pairs. Each pair of students should be given a satellite map of a downtown area/city with generally regular square blocks, preferably in the same area as the school; see the sample Minneapolis map for an example (M-G-1-3_Minneapolis Map). Each pair of students should have a highlighter. Instruct each student to select an intersection on the map and mark it with the highlighter. If the map depicts the area surrounding the school, suggest that students select an intersection that is familiar to them (to help them visualize the situation). Otherwise, let them choose random intersections. Then, each pair should indicate the route they would take to travel from one student's point (intersection) to the point (intersection) chosen by their partner. Have each pair compute the distance (in blocks).</p><p>Next, have each group calculate the shortest distance from one point to another if they were not limited to staying on the road or going around obstacles such as buildings. Have them emphasize this route. Then, ask:</p><p><strong>"How can we find the shortest distance between your starting and ending places?"</strong></p><p><strong>"What shape have you highlighted on your map?"</strong> (<i>right triangle</i>)</p><p><strong>"What theorem do you know that relates the lengths of the sides of a right triangle?"</strong> (<i>the Pythagorean theorem</i>)</p><p>Students should use rulers to measure the distance between the two points, and then calculate how far it would take to get there by walking along the street. When students measure the sides of a triangle, they should notice that <i>a</i>² + <i>b</i>² = <i>c</i>². Show this to the class on the board. Mark a point on the board (name it A), then go up 3 inches and mark another point (call it B), then go right 4 inches (making a 90-degree angle in the process) and mark that point (call it C). Show the class that the distance between A and C is 5 inches. Use the equation <i>a</i>² + <i>b</i>² = <i>c</i>², or 3² + 4² = 5².</p><p>Now challenge students to go back to their maps and count how many right angles they can discover, as well as measure the straight-line distance versus the distance down the triangle's legs. Assist students who may require further practice as necessary.</p><p>After students have spent a few minutes exploring this exercise, bring them back together and remind them of the Pythagorean Theorem, emphasizing that a and b indicate the lengths of the legs and c represents the length of the hypotenuse.</p><p>This time, have each pair of students use the Pythagorean Theorem to calculate the shortest distance between their two points.</p><p>Choose a few groups to share their maps. Let them explain:</p><p>what points did they choose (and why, if applicable)</p><p>the street route they highlighted and how they calculated the distance</p><p>the shortest route and how they determined the distance</p><p>Ensure that by the end of the presentations, students understand how to use the Pythagorean theorem to calculate the distance between two points. Encourage the use of calculators and rounding to 1 or 2 decimal places for complicated calculations.</p><p>Share the Distance Worksheet (M-G-1-3_Distance Worksheet and KEY). Instruct students to:</p><p>1. Graph the points.</p><p>2. Draw a "street route" connecting the points.</p><p>3. Create the shortest route between the points.</p><p>4. Using the Pythagorean theorem, calculate the distance between the points.</p><p>If students are having trouble determining the lengths of the sides of the right triangles on the Distance Worksheet, instruct them to examine the two <i>x</i>-coordinates and determine how far it is from one <i>x</i>-coordinate to the other <i>x</i>-coordinate, demonstrating that the distance between the <i>x</i>-coordinates is simply the length of one leg. Repeat using the <i>y-</i>coordinates.</p><p>Once students have completed the Distance Worksheet, put the distance formula on the board.</p><figure class="image"><img style="aspect-ratio:187/58;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_163.png" width="187" height="58"></figure><p>Ask students:</p><p><strong>"How does the distance formula relate to the Pythagorean theorem?"</strong></p><p>Students should explore the differences between the two and how, algebraically, they are the same formula. The distance formula contains a square root, although the Pythagorean theorem does not. Solving the Pythagorean theorem for <i>c</i> (rather than <i>c</i>²) yields a square root. Depending on the algebra ability of the students, the algebraic relationship between the two can be thoroughly investigated.&nbsp;<br>Provide this simple example: Draw a right triangle with a right angle at the origin (0, 0) and vertices at (0, 4) and (-3, 0). Show students that the subtraction of \(x_2\) from \(x_1\) is 0 - (-3), which is 3. Then subtracting \(y_2\) from \(y_1\) yields 4 - 0, which is 4. Show that 3² + 4² is equivalent to <i>a</i>² + <i>b</i>² using the well-known equation <i>c</i>² = <i>a</i>² + <i>b</i>².</p><p>As an exit activity, students should write an explanation of how to calculate the shortest distance between two points (M-G-1-3_Lesson 3 Exit Ticket).</p><p><strong>Extension:</strong></p><p>Graph the vertices of each set of three ordered pairs on the <i>x/y</i> coordinate plane, classify each triangle based on its side lengths and angle measurements, and calculate each perimeter.</p><p>1. A (-4, -4), B (6, 0), C (3, 4)</p><p>2. A (-4, 3), B (10, 7), C (-5, 2)</p><p>3. A (-2, 2), B (6, -2), C (2, -6)</p><p>4. A (3, 6), B (7, 2), C (3, -1)</p>