<p>In this lesson, students will learn how to calculate the distance between two points in a coordinate system using both mathematical and practical situations. Students are going to:<br>- learn how to calculate the distance between two points in a coordinate system using their coordinates.<br>- learn that the formula for calculating the distance between two points is an alternative variant of the Pythagorean theorem.</p>

<p>- How may identifying regularity or recurrence help with problem-solving efficiency?&nbsp;<br>- How do spatial relationships, such as shapes and dimensions, help to create, construct, model, and portray real-world scenarios or solve problems?&nbsp;<br>- In what ways does the application of geometry shape properties aid in mathematical reasoning and problem-solving?&nbsp;<br>- How may geometric properties and theorems be utilized to describe, model, and analyze problems?&nbsp;<br>&nbsp;</p>

<p>- Pythagorean Theorem: A theorem that states the relationship between the lengths of the legs, a and b, in a right triangle and the length of the hypotenuse of the right triangle, c, is a² + b² = c².<br>- Square Root: One of two equal factors of a number.</p>

<p>- a coordinate grid from −20 to 20 on both axes for each group of 3 students&nbsp;<br>- A copy of the Civil Engineering worksheet (M-8-6-2_Civil Engineering and KEY) for each group&nbsp;<br>- The Distance Formula worksheet (M-8-6-2_Distance Formula and KEY)</p>

<p>- Assess student participation during the introduction of the taxicab and "regular" distance in order to determine comprehension level.&nbsp;<br>- Utilize the Civil Engineering worksheet to assess students' level of proficiency.&nbsp;<br>&nbsp;</p>

<p>Explicit Instruction, Modeling, and Active Engagement<br>W: In this lesson, we'll derive and use a method for calculating the distance between two locations in a coordinate system.&nbsp;<br>H: Students will apply their prior knowledge to create a useful formula that can be understood mentally rather than memorized. They'll witness practical uses for the formula. They will also learn about "taxicab" distance, which appears to be a fantastical mathematical idea.&nbsp;<br>E: Students will collaborate in groups to design a map that satisfies a variety of requirements, with each group designing their own "town." Students will be able to experiment with alternative object placements on the map, observing how shifting an object horizontally or vertically impacts the distance between it and another object.&nbsp;<br>R: Students will have the freedom to edit the placements of different points on their map as often as they like, testing out different places until they locate one that fits the specified requirements.&nbsp;<br>E: After exchanging maps with one another, each group will check that the other group's map matches all of the requirements.&nbsp;<br>T: Students can take on many tasks in this group exercise, such as creating imaginative maps or confirming distances between locations. Students can engage in multiple kinds of interaction with the activity and with each other through this activity.&nbsp;<br>O: Before students begin the self- or group-directed section of the activity, they will get guided instruction on the subject. In addition to allowing students to grasp the content, this allows them to clear up any misunderstandings during the group-guided phase.&nbsp;<br>&nbsp;</p>

<p>Display the first quadrant of a coordinate plane with the points (1, 4) and (2, 8) on the board. First, mark the point "Hotel," and then the second, "Arena."<br><br>Ask students to imagine themselves arriving at their hotel and&nbsp;needing to get a taxi to get to the arena for an event. Describe how each line on the coordinate plane represents a street in the downtown area. <strong>"What is the least number of blocks does the taxi need to go from the hotel to the arena?"</strong> <i>(5)</i><br><br>Ask students to draw or describe how a taxi may go the exactly 5 blocks from the hotel to the arena. Emphasize the two routes that have the fewest turns.<br><br><strong>"Let's call the distance between two points as long as we stay on horizontal and vertical lines as 'the taxicab distance', so we can say that the taxicab distance between (1, 4) and (2, 8) is 5."</strong><br><br>Insert a new point with the label "Restaurant" at points (6, 10). Have students calculate how far the restaurant is from the arena in a taxi. <i>(6 units)</i> As you help students realize that you can subtract the x-coordinates from one another to discover the horizontal distance and the <i>y</i>-coordinates from one another to find the vertical distance, ask them to explain their approaches.<br><br><strong>"Assume you have enough money to hire a helicopter to take you from the arena directly to the restaurant. You are no longer limited to the streets. How far is it from the arena to the restaurant in a helicopter compared to the distance you travel by taxicab?"</strong><i><strong> </strong>(It is less.)</i><br><br><strong>"To start estimating how much less, first imagine yourself traveling a very simple taxicab route."</strong> Draw a line in both the horizontal direction from (2, 10) and the vertical direction from (2, 8) to (6, 10).<br><br><strong>"Let's now draw in the possible path our helicopter may take."</strong> Draw a line connecting (2, 8) to (6, 10).<br><br><strong>"That allows us to create a right triangle. How can we find out how far the helicopter&nbsp;needs to fly?"</strong><br><br>Help students calculate the true distance by guiding them to apply the Pythagorean theorem. <i>(Actual distance: 2\(\sqrt{5}\) ≈ 4.47)</i><br><br>Have the students calculate the actual distance between the hotel and the arena. <i>(It is actually \(\sqrt{17}\) ≈ 4.12.)</i><br><br><strong>"Is there ever a situation in which the distance traveled by taxicab is the same as the actual distance between two points?"</strong> (<i>when the two points are on the same vertical or horizontal line.</i>)<br><br><strong>Activity</strong><br><br>Divide the students into groups of 3. Give a copy of the Civil Engineering worksheet (M-8-6-2_Civil Engineering and KEY) to each group, along with a coordinate plane that runs from -20 to 20 on both axes. Explain to students that they will be designing their own town on a coordinate plane using the directions provided to them.<br><br>Before you start, remind the students that the term "distance" refers to the shortest path between two sites, "as the crow (or helicopter) flies," and that "taxicab distance" refers to just following horizontal or vertical lines.<br><br>Have each group of students pass their maps to the next group once they have finished their towns. The plotted points should meet all of the specified criteria, according to the second group's verification.<br><br><strong>Extension:</strong><br><br>Introduce the distance formula to students once they have an understanding of how to use the Pythagorean theorem to find the distance between two points on a coordinate plane.</p><figure class="image"><img style="aspect-ratio:185/27;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_1.png" width="185" height="27"></figure><p>, where <i>d</i> is the distance between (\(x_1\),\(y_1\)) and (\(x_2\),\(y_2\)).</p><p>Challenge students to use the Pythagorean theorem to derive the distance formula by assigning them the Distance Formula worksheet (M-8-6-2_Distance Formula and KEY). Next, give the students practice calculating the distance between different pairs of points using the distance formula. &nbsp;</p>