Journey
This problem gives you the chance to:
• draw and interpret a graph of speed, distance and time
Here is a description of a car journey.
“I left home at 2:00 hours. I traveled for half an hour at forty miles an hour,
then for an hour at fifty miles an hour. I had a half hour stop for lunch, then I
travelled for two hours at fifty-five miles an hour.”
1. Complete this table showing the distances traveled by the end of each stage of my journey.
Time in hours
Distance from home in miles
2.
2:00
0
2:30
3:30
4:00
6:00
Draw a distance-time graph for this journey on the grid below.
240
Distance from home in miles
220
200
180
160
140
120
100
80
60
40
20
0
2:00
2:30
3:00
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3:30
4:00
Time in hours
Page 40
4:30
5:00
5:30
6:00
Journey Test 7
3. What is the average speed for the whole journey?
Explain how you figured it out.
4. Use your graph to find:
a. How far from home I had traveled by 5:15.
miles
b. At what time I had traveled 60 miles from home.
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Journey Test 7
Journey
Rubric
The core elements of performance required by this task are:
• draw and interpret a graph of speed, distance and time
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Table correctly completed:
2ft
Partial credit
2.
1 error
(1)
Graph correctly drawn
2ft
2
Partial credit
1 or 2 errors
3.
(1)
Gives correct answer: 45 mph
and shows 180 ÷ 4
1ft
2
1
Gives correct answers:
4.a
b
About 140 miles
1ft
About 3:20. In the correct interval on graph.
1ft
Total Points
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2
7
Journey Test 7
Journey
Work the task and look at the rubric. What are the big mathematical ideas being assessed in
the task? ______________________________________________________________
What experiences have students had with finding rates and working with the formula d=rt?
Do you think students understand the idea of cumulative distance? Look at student work on
the table. How many of your students thought the distance traveled for the first half hour
was:
20
40
25
22.5
What was the confusing thing about finding this distance? How are the misconceptions
slightly different? Now look to see values for 4:00. How many students:
Same value as 3:30 Went back to 0 Increased the distance Put a rate (55) instead of
a distance
Now look at their graphs. Did students:
• Make a step graph?
• Make a bar graph?
• Forget to start at 0?
• Make all their points follow a straight line?
What other graphing errors did you notice?
Finding average speed was difficult for students. How many of your students put:
45mph Divided Divided Divided by 4
Put a
50
Estimated
Other
by 3
by 5
(but not using
total
or “by
d/t)
looking”
What are the different misunderstandings about average speed? What were students really
averaging? How can you help students to relate this average speed to the distance formula?
To the graph? How is this “average” conceptually different from finding average salary?
How are these “averages” the same?
Were your students able to read points of the graph in part 4? What types of experiences
have your students had in reading this type of graph? In making their own graphs? How
does the mathematics change when students make their own graph? How are the cognitive
demands different?
7th grade – 2007
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Looking at Student Work on Journey
Student A is able to do all the parts of the task. There is an implied use of d=rt in the
multiplication work under the table. Notice the clear explanation in part 3.
Student A
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Student A, part 2
Student B is able to complete the table and graph correctly. But look at the work of Student
B in part 3. Why doesn’t this strategy work? How is the student thinking about finding
average? What is the confusion for the student? How could the student make a small change
to make this strategy work?
Student B
Why does the strategy for student C work? Will it always work? How is it different from
Student A and how is it the same?
Student C
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Student D does not understand distance from home in the table. The student forgets to
divide the initial rate in half for half hour of the journey. Notice that the student makes a
step graph, which is inappropriate for the context. The car would be in multiple locations at
the same time. In part 5 the student seems to think about total distance, but then multiplies
by 4. The student does not have the academic language to describe the rates and labels them
miles.
Student D
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Student E completes the table correctly, showing total distance and graphs all the points
correctly. However the student forgets to connect the distance at 2:30 back to the origin.
The student makes a common error for finding average speed. What is the student confused
about or why doesn’t this strategy work? What questions might you ask during a class
discussion to press students’ thinking about the mathematics of average speed?
Student E
Student F understands the total distance and is able to complete the table. However the
student makes a bar graph. Why is this graph inappropriate for this situation? What is
incorrect about the information conveyed by the bars?
Student F
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Student G is struggling with the ideas of total distance. The graph shows continual progress
with a correct break in progress for lunch. However the student starts for home after lunch
rather than continuing the journey. However the table does not show any of the ideas about
combining distances. What might be the next move to help this child build on her knowledge
or clarify some of the issues about distance traveled?
Student G
Student H and I have difficulty with the action of the story. They both want to return to zero
distance, maybe confusing speed with distance. Notice that student I actually plots a correct
point for 2:30, but draws the line missing that point in order to make a smooth straight line.
This eliminates the change of speed at 2:30.
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Student H
Student I
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Student J is only able to graph points. The student doesn’t know to halve the rate for 2:30 to
find distance. The table just is a record of starting speeds. Notice the lack of multiplicative
thinking in part 3. The student does not relate the questions in part 4 to the graph.
Student J
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7th Grade
Student Task
Core Idea 3
Algebra and
Functions
Task 3
Journey
Draw and interpret a graph of speed, distance and time.
Understand relations and functions, analyze mathematical
situations, and use models to solve problems involving quantity and
change.
• Use graphs to analyze the nature of changes on quantities in
linear relationships.
Mathematics in this task:
• Ability to use rates and time to find distance traveled
• Make a line graph to show total distance traveled over time
• Use graph or d=rt to find average speed for a journey
• Read and estimate points on a time and distance graph
Based on teacher observations, this is what seventh graders knew and were able to do:
• Use data from a table to plot graphs
• Read data from a line graph
Areas of difficulty for seventh graders:
• Finding total distance
• Separating rates from distance
• Finding average speed for a journey
7th grade – 2007
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The maximum score available for this task is 7 points.
The minimum score for a level 3 response, meeting standards, is 3 points.
Many students, about 79%, could graph from their table with only one or two errors. About
half the students could make a graph from their table with no errors and read one of the
points off the graph or make a graph with an error and read both points off the graph in part
4. About 25% could make a table of total distance traveled given information about rate and
time, make a correct graph of data in the table, and read points from the graph. About 4%
could meet all the demands of the task including calculating average speed for a journey.
21% scored no points on this task. 88% of the students with this score attempted the task.
Journey
7th grade – 2007
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Points
0
Understandings
Misunderstandings
88% of the students attempted
the task.
Students had trouble graphing the data in
their table. Almost 7% of the students
made bar graphs. 5% did not start the line
graph at 0. 7% were off by one point.
Students had trouble reading data on the
graph. 11% of the students didn’t have
graphs that ever reached a height of 60
miles. 8% did not attempt this part of the
task.
Students had difficulty converting from
rates to total distance. 16% of the students
forgot to halve the first rate. 6% show a
gain of distance during the lunch break.
7% return to 0 at lunch. 16% do not
double the 55 when finding the final
distance in the table.
Students had difficulty finding average
speed of the journey. 40% divided the
three different rates by three. 9% divided
the five numbers in the boxes by 5. 10%
estimated or said they just looked at the
graph. 8% divided the wrong quantity by 4
1
Students met with partial
success on graphing their data.
3
Student had partial success on
the graph and could read both
points on the graph in part 4 or
they could make the graph
correctly and give the distance
traveled for 5:15.
5
Students could complete the
table of distance traveled,
graph the data from the table,
and read points on a graph.
7
Students could complete the
table of distance traveled,
graph the data from the table,
and read points on a graph.
Students could also find the
average speed.
7th grade – 2007
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Implications for Instruction
Students need more experience working with time and distance problems in context.
Students did not seem to understand the idea of total or cumulative distance traveled, but
thought about each part separately. Many students did not consider the amount of time at a
given rate, e.g. if you travel 40 mph for half an hour, you actually only go 20 miles. If you
stop for a rest, you don’t go back to 0 miles traveled. Rate is a big idea mathematically, but
is often not given extended thought in textbooks. Students need to work with the distance
formula for solving problems and also for graphing.
Students had a great deal of difficulty graphing the situation. The cognitive demands for
making a graph are very different from that of reading information from a graph that is
already made. Many students confuse the action of the story with the shape of the graph.
For example, they want the graph to return to zero when the car is not moving. They don’t
think that this would imply that the car instantly returns home when the car stops for gas, a
light, or lunch. Some students did not realize that this is not a situation for using a bar graph.
Students at this grade level should be comfortable with line graphs. Students should move
beyond how to make a graph and have discussions about what types of graphs could best
suit a particular context.
As with Suzi’s Company, students had difficulty thinking about mean in a context or from a
graph or table. Students wanted to average speeds even thought they represented different
amounts of time. They tried to divide by 5 because their was 5 times on the table instead of
dividing by total distance by the number of hours. Students need to have frequent
experiences dealing with mathematical ideas in context and making their own
representations to help them think about the situation.
Ideas for Action Research - Making Graphs for Stories
Having students make their own graphs about situations helps them to understand the logic
of the graph and see how the lines do not represent the action of the story. Consider giving
your students some story situations and have them make a graph of the general situation, not
necessarily dealing with the issues of an exact scale. For example:
A factory cafeteria contains a vending machine selling drinks.
On a typical day:
- The machine starts half full.
- No drinks are sold before 9 a.m. or after 5 p.m.
- Drinks are sold at a slow rate throughout the day, except during the morning and lunch
breads (10:30-11 am and 102 pm) when there is a greater demand.
- The machine is filled up just before the lunch break. (It takes about 10 minutes to fill).
Make a sketch to the graph to show how the number of drinks in the machine might vary
from 8 am to 6 pm.
What does the student have to understand about graphing to do this task? What do you think
the graph might look like? What errors do you anticipate students might make? How does
this help you think about how to process this activity? What are the mathematics you want
to bring out or highlight as students discuss their work?
7th grade – 2007
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Now consider another task from the Shell Centre book, The Language of Functions and
Graphs.
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How did this activity add to student understanding of line graphs? What did the student
have to think about to be successful? What evidence of understanding did you see in student
work?
7th grade – 2007
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