6th to 8th
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
Calculation Nation®: DiRT Dash
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
Dirt Dash Activity Sheet
Dirt Dash Answer Key
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
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.
discussion should engage students and access their prior knowledge and experience,
because the problem requires students to make mathematical decisions about
traversing different terrains.
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.
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?
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
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?