Thank you for your interest in NCTM’s Illuminations. Beginning in mid-April, all Illuminations content will be moving to nctm.org/illuminations. Interactives will remain openly available and NCTM members will have access to all Illuminations lessons with new filtering and search options. We hope you will continue to utilize and enjoy these resources on nctm.org.

## DiRT Dash: Planning the Fastest Route across Various Terrains

• Lesson
6-8
1

The shortest distance between two points is a line. But what is the shortest time to travel between two points on different terrains? In this lesson, students will predict, estimate and then calculate the path that results in the fastest time to travel between two points when different terrains affect the fastest path. This lesson is designed as an introduction to the Calculation Nation® game DiRT Dash and prepares students to apply mathematics to improve their performance in the game.

A few days before teaching this lesson, have students log onto Calculation Nation® and play the DiRT Dash game at home, the school library, or other available computers.

Tell students that in a few days they will be learning about mathematics that will be useful for playing DiRT Dash well.

To prepare for this lesson, make enough copies of the Dirt Dash Activity Sheet for each student. This activity sheet is broken up into three sections to (1) introduce the lesson, (2) supplement the main lesson, and (3) provide reflection and assessment.

To introduce the lesson, build on students’ experiences with playing Dirt Dash over the past few days. As an in-class demonstration, you might attach sandpaper to one side of a board, then elevate the board and observe a toy car as it rolls down across the surface. (Sandpaper can be substituted with any material that will cause more friction.) Afterwards, flip the board over to the smooth side, elevate the board to the same level and observe the toy car roll down this surface. Which surface was faster? [The surface without the sand paper.] How did the terrain affect the speed of the car? [Answers will vary. Sample: The sand paper provided more friction.]

Using this demonstration as a starting point, lead a discussion about various terrains that are faster or slower to traverse. You might also discuss local terrains that are familiar to students such as walking through thick snow, running on cinder versus paved or synthetic tracks, walking on ice, skating on ice, running in sand, and swimming.

The problem that serves as the basis of this lesson features the context of a beach where students run on sand and swim in the ocean. You might also review why distance is the product of rate and time by providing a sample calculation such as: if you drive 70 mph for 3 hours, how much distance will you cover? [210 miles=(70mph)(3h)]. Show students how the hours cancel while the miles remain.

The class discussion should engage students and access their prior knowledge and experience, because the problem requires students to make mathematical decisions about traversing different terrains.

Once students have discussed different terrains and understand the effect of various terrains on speed, point to the "Intro Problems" in the activity sheet and play the motivational video. Pause the video twice, each time before the two segment paths appear in the sand and in the water; allow students time to think about, predict, and calculate the time to travel to their Brother. (These images also appear in Questions 1 and 2 on the activity sheet, if you do not wish to use the video.)

To estimate the distances, students might mark the edge of a sheet of paper then align the marks along the x-axis; use the width of one or more of their fingers to establish a unit scale along the x-axis and then convert the number of finger widths to the scaled distance; or employ some other estimation technique. Asking students to make and reflect on predictions establishes buy‑in and helps students develop number sense as they try to evaluate the accuracy of their predictions throughout the lesson.

Note that the distance between the two points in the sand in Question 1 correspond to the endpoints of the hypotenuse of a 5-12-13 right triangle, but the sides of the right triangle whose hypotenuse is between the two points in the water does not form a Pythagorean triple. Optionally, you might question students about why they obtained a terminating decimal for the time to travel between the points in the sand but did not obtain a terminating decimal for the time between the points in the water.

Point to the "Dirt Dash- Two Terrains" portion of the activity sheet. You might want to give students time to formulate questions 3 and 4 on their own, based on their observations from the motivational video, before asking them to read the questions. Questions 3 and 4 are guided questions about the direct line path when the two points lie in different terrains. Invite student observations from the motivational video, and discuss the goal of reaching their brother as fast as possible. If necessary, you might offer some guidance, such as, “In the video it took a really long time for you to swim a long distance to your brother. Wouldn’t you like to spend less time in the water?”  To emphasize this point, Question 4 asks students to identify two points on the shoreline whose corresponding paths would take more time and less time to travel than the direct line path that students found in question 3. Recognizing that traveling a shorter distance in the water generally corresponds to a faster overall time is an important idea for students to understand as they continue working through the Two Terrains" portion of the activity sheet.

In Question 6, students use a table to organize the distances and times for traversing the sand and ocean based on the point where they enter the ocean. They then use the table to estimate the entry point corresponding to the fastest path to their Brother. The table intentionally includes x = 15, the point that minimizes the distance swum in the water, as this point does not produce the fastest path. Watch for students who doubt their calculations or who otherwise think they did something wrong, since their intuition thus far may have been to minimize their distance in the water, the slower terrain to traverse. In question 7 students are prompted to use the results in the table to estimate the entry point corresponding to the fastest path. Question 7 also provides a differentiation opportunity for advanced students, so monitor students who finish early and prompt them for ways to improve their estimates including adding rows to the table for additional entry points, or making a scatter plot then sketching a function to fit the points and estimating the function’s minimum point. See Extension 3 for additional ideas about improving estimates.

The last page of the activity sheet, “Dirt Dash- Reflection and Extension,” contains reflection and assessment questions. The applet allows users to change the speed for each terrain and to move the starting and ending points. There are many variations of this problem, and particularly interesting is analyzing how the fastest path changes when the speed changes over only one of the terrains. You can use the applet yourself in class to present and investigate some of these variations for students to consider. Alternatively, you might let students use the applet themselves to investigate questions 9 and 10 in a computer lab or for homework; student use of the applet can be an effective use of technology that enables students to explore and make sense of these variations on their own.

Assessment Options

1. Have students write a detailed journal entry to demonstrate their understanding of this problem using the prompt “What would you tell a friend who missed today’s class about the fastest path to travel across two terrains? Be sure to address questions 8, 9, and 10 from the activity sheet.
2. Asks students to play a one-player game of Dirt Dash on Calculation Nation®, and have them bring up a course and select a car. Then, have students calculate the amount of time it will take them to reach FINISH. Before they pres Start, have them show you their calculations and estimate, and then watch as they traverse the course. How close was their estimate? What would account for any difference? They can write their answers to these questions in a journal entry, or they could just have a brief conversation with you about the results.
Note that the speeds shown in the game are expressed in miles per hour (mph), but distances within the game are shown in feet. It will therefore be necessary to convert the speeds from mph to feet per second (fps). Before students prepare their estimates, you may want to review how to make these conversions. The process involves multiplying the speed in mph by 5280 feet per mile and dividing by 3600 sec per hour, which will cause units to cancel and convert the speed to fps. (As a shortcut, you could just suggest that kids multiply the speed in mph by 1.5, as that gives a slightly high but still reasonable estimate of the speed in fps; the error is only 2.2%. The benefit of this shortcut is that students can continue to focus on the main content of the lesson, which is not conversions but the relationship between distance, rate, and time.) It should also be noted that the horizontal distance from START to FINISH is about 750 feet, and the vertical distance is about 400 feet, so the straight line distance between the two points is about 850 feet.
Alternatively, you could project a route from Dirt Dash (either directly from the game or as a screen capture) and have all students compute an estimate. This method allows all students to consider a common scenario, which then promotes discussion about the solution.
3. Students can apply their learning from this lesson as they play Dirt Dash on Calculation Nation. Students should analyze each map, plan the fastest route, and select their car accordingly from the Options tab before starting each race. After completing a best-of-five match, students should click on the Race Stats tab and revisit each of the maps to analyze their car selection and critique the path that they took. Students can also click the Personal Stats tab to see the distances and times they spent on each terrain, calculate the ideal path based on their selection of car, and then compare their performance against the ideal path. Have students write a paragraph answering either or both of these prompts:
Explain how completing this lesson has changed your ability to implement the optimal race strategy when playing Dirt Dash.
Give at least one specific example of something you learned from analyzing your race statistics that will help you improve your race strategy in the future.

Extensions

1. Using the activity Two Terrains, students can experiment with variations on this problem. Some variations include considering how the fastest path changes when the off-road speed is greater than or equal to the on-road speed [always travel along the straight line path to the finish], and generalizing the fastest path given a relationship between on-road and off-road speeds.
Two Terrains
2. Students should brainstorm and write about applications of this problem in their own lives. A common application is determining when to take the back roads to drive somewhere because the distance is shorter, versus traveling a longer distance on the highway to take advantage of faster speeds. Students might also use websites like Google Maps to investigate various routes around their own cities, given actual speed limits and real-time traffic conditions.
3. Challenge students to write an equation for calculating the total time to reach their brother as a function of the x-coordinate of their entry point into the ocean. Obtaining the equation might best be structured by revisiting the table on the activity sheet.
1. Have students make a graph by plotting Entry Point vs. Total Time
2. Sketch a function through these points.
3. Write in words the procedures they used to complete each row.
4. Repeat those procedures for the variable x instead of a known value.
5. Their graphical sketch should fit closely to the graph of the equation. Students can use a graphing calculator to find the minimum point of this function, which gives the optimal entry point and therefore the fastest path to reach their brother. Utilizing multiple representations (pictorial, graphical, tabular, and verbal) helps students make sense of this problem and can provide effective support as students work to model this problem symbolically.

Questions for Students

1. We know that the shortest distance and the shortest time between two points is along the line connecting them. But why is it beneficial to travel a longer distance when the points are on different terrains with different speeds of travel?

[It is advantageous to travel a greater distance on the faster terrain and reduce the distance traveled on the slower terrain.]

2. Why is the fastest path NOT the path through (13, 0) that minimizes the distance traveled through the water, the slower terrain?

[More time is spent running the extra distance to (13,0) through the sand than is saved by swimming a slightly shorter distance in the water. Moving to the left of x = 13 noticeably shortens the distance on the sand while increasing the distance in the water by only a small amount].

Teacher Reflection

• Was the motivation video effective in prompting students to ask the question you wanted them to ask? Is there a better way to stimulate students’ questioning for this problem?
• Were you able to make effective use of technology through the motivational video and the applet to engage students more deeply and interactively with this problem?
• Which strategies or questions did you use to differentiate this problem for low or high achievers?
• Were there aspects of this problem that were too advanced for your students? Did the opening discussion about traversing different terrains, reviewing d = r × t or accessing other prior knowledge adequately support students to engage with this problem?
• How well were students able to model this problem through multiple representations? Was the use of pictorial, tabular, verbal, graphical, and/or symbolic representations effective in supporting students to gain entry to this problem and to develop a deeper understanding of the underlying mathematics?
• How do you know whether students were engaged meaningfully with this problem throughout the lesson? How might you sustain high levels of engagement through portions of the lesson when students may have become less engaged?
• What evidence did you observe that shows your students met the learning objectives for this lesson? How were you able to communicate your learning objectives to students throughout the lesson?

### Learning Objectives

Students will:

• Estimate and calculate distances between two points.
• Estimate and calculate time to travel between two points given speed.
• Predict, estimate, and calculate the fastest path between two points located on two different terrains.
• Describe using precise mathematical language how changing travel speeds across different terrains affects the fastest path between two points.

### NCTM Standards and Expectations

• Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
• Use geometric models to represent and explain numerical and algebraic relationships.
• Solve simple problems involving rates and derived measurements for such attributes as velocity and density.

### Common Core State Standards – Mathematics

• CCSS.Math.Content.6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, ''This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.'' ''We paid $75 for 15 hamburgers, which is a rate of$5 per hamburger.''