Grade 3: Place Value Unit
Assessment Math Task
One Hundred Miles
A team of five bikers are training to ride a combined total
of one hundred miles in a race. The bikers practice every
Sunday from seven o’clock in the morning to ten thirty
in the morning. During practice a biker who rides 20 or
more miles when rounded to the nearest ten is ready for
the race. This past Sunday the first biker rode 28 miles.
The second biker rode 24 miles. The third biker rode
27 miles. The fourth biker rode 12 miles. The fifth biker
rode 8 miles. Did the team of five bikers meet the goal of
riding a combined total of one hundred miles? How many
bikers are ready for the race? Show all your mathematical
thinking.
© 2013
exemplars.com
1
One Hundred Miles
Place Value Unit
Mathematical Processes: 3.1A, 3.1B, 3.1E, 3.1G
Task
A team of five bikers are training to ride a combined total of one hundred miles in a race. The
bikers practice every Sunday from seven o’clock in the morning to ten thirty in the morning.
During practice a biker who rides 20 or more miles when rounded to the nearest ten is ready
for the race. This past Sunday the first biker rode 28 miles. The second biker rode 24 miles.
The third biker rode 27 miles. The fourth biker rode 12 miles. The fifth biker rode 8 miles. Did
the team of five bikers meet the goal of riding a combined total of one hundred miles? How
many bikers are ready for the race? Show all your mathematical thinking.
TEKS Unit of Study and Evidence
Place Value Unit
The Place Value Unit involves understanding the relative position, magnitude and
relationships within the numeration system in order to answer questions such as:
• How could you use base-10 blocks to show what the numerals in this number mean?
• How can you use the additive property of place value to decompose this number?
• What other way(s) can you use thousands, hundreds, tens, and ones to show this
number without changing its value?
Exemplars Task-Specific Evidence
This task requires students to use place value to round whole numbers to the nearest 10. The
students are also expected to add numbers to determine a combined total.
Underlying Mathematical Concepts
• Rounding whole numbers to the nearest 10
• Adding or combining whole numbers
• Comparison
Possible Problem-Solving Strategies
•
•
•
•
2
Model (manipulatives)
Diagram/Key
Table
Number line
exemplars.com
800-450-4050
Possible Mathematical Vocabulary/Symbolic Representation
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Model
Diagram/Key
Table
Number line
Greather than (>)/Less than (<)
Estimation
Odd/Even
Dozen
Time notation: 9:00 AM
Hours, minutes
Average
Total/Sum
Per
Fractions
1/2, 1/4
Percents
25%, 50%
Ordinal numbers
1st, first, 2nd, second, 3rd, third ...
Miles
5,280’ (ft.)
Ones, tens
Place value
Possible Solutions
3 bikers are ready to race: the first, second and third bikers. The determination of whether the
team has ridden 100 miles will depend on whether the student adds the actual miles ridden
or the rounded numbers.
Biker
Miles
Tens
Ones
4 5
Rounded
Number
1
28
2
8
30
2
24
2
4
20
3
27
2
7
30
4
12
1
2
10
5
8
0
8
10
28 + 24 + 27 + 12 + 8 = 99
30 + 20 + 30 + 10 + 10 = 100
(continued on next page)
3
exemplars.com
800-450-4050
Possible Solutions (cont.)
Biker 1
Biker 2
Biker 3
Biker 4
Biker 5
0
5
10
15
Miles
20
Biker
Number
Miles
Biked
Mile Rounded to
the Nearest Ten
Total
Miles
1
28
30
28
2
24
20
52
3
27
30
79
4
12
10
91
5
8
10
99
25
30
Possible Connections
Below are some examples of mathematical connections. Your students may discover some
that are not on this list.
•
•
•
•
•
•
•
•
•
•
4
Compare the difference in estimated total miles ridden to exact total miles ridden.
The 5th biker rode 1/3 as far as the 2nd biker.
Bikers 1 and 2 ride a total of 52 miles. That is more than 1/2 of the goal.
Biker 4 rides 1/2 the distance of biker 2.
If each biker rides 20 miles for a total of 100 miles, it is an equal share of miles per
biker.
1 mile is 5,280 feet.
Biker 3 rides an odd number of miles. All other bikers ride an even number of miles.
12 miles is a dozen.
Relate to a similar task and state a math link.
Solve more than one way to verify the answer.
exemplars.com
800-450-4050
Novice Scoring Rationales
5
Criteria and
Performance Level
Assessment Rationales
Problem Solving
Novice
The student’s strategy of adding the numbers 20, 28, 24, 12,
and 8 would not work to solve the task. The student’s answer,
“no,” is based on an incorrect strategy so is not considered
correct.
Reasoning Proof
Novice
The student does not demonstrate correct reasoning of the
underlying concepts in the task. The student finds the total
of numbers that five bikers ride on Sunday and does not
address the concept of rounding that is needed to solve the
second part of the task.
Communication
Novice
The student does not use any mathematical language to
communicate her/his reasoning and proof.
Connections
Novice
The student does not make a mathematically relevant
observation about her/his solution.
Representation
Novice
The student does not use a representation to support her/his
reasoning.
exemplars.com
800-450-4050
Novice
P/S R/P Com Con Rep A/Level
N
6
exemplars.com
N
N
N
N
N
800-450-4050
Apprentice Scoring Rationales
7
Criteria and
Performance Level
Assessment Rationales
Problem Solving
Apprentice
The student’s strategy of using a table to indicate the number of racers, miles rode, and total miles would work to solve
the first part of the task but the student indicates 18 miles
instead of eight miles for the fifth biker. This leads to a total
of 109 miles. The student’s answer, “YES-109 miles,” is not
correct. The student rounds 18 instead of eight, which results
in an incorrect answer, “4 can race,” for the second part of
the task.
Reasoning Proof
Practitioner
The student demonstrates correct reasoning of the underlying concepts in the task. The student finds the total miles
that five bikers ride but makes the error of adding 18 miles
instead of eight. The student demonstrates understanding
of rounding numbers to the nearest 10. Using 18 instead of
eight is a careless error and is not considered a lack in reasoning and proof.
Communication
Practitioner
The student correctly uses the mathematical terms miles,
total from the task. The student also correctly uses the terms
table, dozen.
Connections
Practitioner
The student makes the mathematically relevant observation,
“12 miles is a dozen miles.”
Representation
Apprentice
The student’s table is appropriate but not accurate. The
student states 18 “miles rode” by biker five instead of eight
miles. The 109 total miles for the five bikers is not correct.
exemplars.com
800-450-4050
Apprentice
P/S R/P Com Con Rep A/Level
A
8
exemplars.com
P
P
P
A
A
800-450-4050
Practitioner Scoring Rationales, Student 1
9
Criteria and
Performance Level
Assessment Rationales
Problem Solving
Practitioner
The student’s strategy of using a table to indicate the number of racers, miles ridden, and total miles works to solve the
first part of the task. The student’s answer, “did not ride 100
miles,” is correct. The student’s strategy of making a table to
indicate the miles each biker rides, the rounding of the miles,
and if each biker is ready to race is correct. The student’s
second answer, “Bikers 1 2 and 3 can race,” is correct.
Reasoning Proof
Practitioner
The student demonstrates correct reasoning of the underlying concepts in the task. The student finds the total miles
that five bikers ride and correctly rounds the miles that each
biker rides.
Communication
Practitioner
The student correctly uses the mathematical terms miles,
total from the task. The student also correctly uses the terms
table, “estimatation” (estimation), AM, Sunday, Saturday.
The words round and add are not considered mathematical
language as the rubric does not accept the use of verbs as
communication.
Connections
Practitioner
The student makes the mathematically relevant observations,
“biker 1-fastest racer,” “biker 8-slowest racer,” and, “add
the estimatation-it is 100 miles but you can’t use that just 99
miles. 3 round up 28, 27, 8. 2 round down 24, 12,” “They
only ride in the AM,” and, “Biker 4 and 5 should practice
longer on Sunday or on Saturday.”
Representation
Practitioner
The student’s first table is appropriate and accurate. The
labels and entered data is accurate. The student’s second
table is also appropriate and accurate. The labels and
entered data is accurate.
exemplars.com
800-450-4050
Practitioner, Student 1
P/S R/P Com Con Rep A/Level
P
10
exemplars.com
P
P
P
P
P
800-450-4050
Practitioner Scoring Rationales, Student 2
11
Criteria and
Performance Level
Assessment Rationales
Problem Solving
Practitioner
The student’s strategy of using a diagram to indicate the
rounded number of racers, miles ridden, and then adding
the miles for a total of 100 miles works to solve the first part
of the task. The student’s answer, “YES They did ride 100
miles,” is correct. The student’s strategy of using her/his key
again to indicate the miles each biker rides, the rounding of
the miles, and if each biker is ready to race is correct. The
student’s second answer, “onle 3 can race,” is correct.
Reasoning Proof
Practitioner
The student demonstrates correct reasoning of the
underlying concepts in the problem. The student finds the
total miles that five bikers ride and correctly rounds the miles
that each biker rides to determine that three bikers are ready
to race.
Communication
Practitioner
The student correctly uses the mathematical term miles from
the task. The student also correctly uses the terms diagram,
key, more.
Connections
Practitioner
The student makes the mathematically relevant observation,
“ 28 m – 8 m = 20 m between B1 and B5. The fastest and
slowest bikers.”
Representation
Practitioner
The student’s diagram is appropriate and accurate. The key
and entered data is accurate.
exemplars.com
800-450-4050
Practitioner, Student 2
P/S R/P Com Con Rep A/Level
P
12
exemplars.com
P
P
P
P
P
800-450-4050
Practitioner Scoring Rationales, Student 3
Criteria and
Performance Level
Assessment Rationales
Problem Solving
Practitioner
The student’s strategy of using a table to indicate the number of racers, total miles, and then adding the miles for a
total of 99 miles works to solve the first part of the task. The
student’s answer, “NO,” is correct. The student’s strategy of
using a number line to indicate the miles each biker rides,
the rounding to nearest 10, and circling the miles that would
round to or past 20 and putting an X by the number of miles
that would not, works to solve the second part of the task.
The student’s answer, “3 are ready,” is correct.
Reasoning Proof
Practitioner
The student demonstrates correct reasoning of the underlying concepts in the task. The student finds the total miles
that five bikers ride and correctly determines that three bikers are ready to race.
Communication
Practitioner
The student correctly uses the mathematical terms miles,
total, first from the task. The student also correctly uses the
term table, number line.
Connections
Practitioner
The student makes the mathematically relevant observation,
“1st Biker is the fastest.”
Representation
Practitioner
The student’s table is appropriate and accurate. All labels
are included and the entered data is correct. The student’s
number line is appropriate and accurate. All labels are
included and the jump for miles and the line for the rounding
results are correct.
NOTE: Students may choose printed number lines to use
as a representation if they independently select one from a
variety of number lines left on a math shelf, table, etc.
13
exemplars.com
800-450-4050
Practitioner, Student 3
P/S R/P Com Con Rep A/Level
P
14
exemplars.com
P
P
P
P
P
800-450-4050
Practitioner, Student 3 (cont.)
15
exemplars.com
800-450-4050
Expert Scoring Rationales
16
Criteria and
Performance Level
Assessment Rationales
Problem Solving
Expert
The student’s strategy of using a table to indicate the number of bikers, number of miles, total miles, and ready to race
works to solve the task. The student’s answer, “NO, 3 bikers-1, 2, and 3,” is correct. The student brings prior knowledge of fractions, elapsed time, and average miles per hour
to her/his solution.
Reasoning Proof
Expert
The student demonstrates correct reasoning of the underlying concepts in the task. The student finds the total miles
that five bikers ride and correctly determines that three
bikers are ready to race. The student applies correct reasoning of fractions, elapsed time and miles per hour in her/his
solution.
Communication
Expert
The student correctly uses the mathematical terms miles,
total, 5th, 1st from the task. The student also correctly uses
the terms more, number, AM, hours/hrs, min., “averge”
(average), per. The student correctly uses the mathematical
notation 1/4, 3 1/2, 7:00, 8:00, 9:00, 10:00, 10:30.
Connections
Expert
The student makes the mathematically relevant Connection
observations, “They need 1 more mile for 100 miles,” “The
5th biker is slowest,” and, “The 1st biker is the fastest.” The
student makes the Expert connections, “Biker 2 rode almost
1/4 of 100. 1 more miles is 25-1/4 of 100,” “7:00 AM-8:00
AM-1 hour,” “8:00 AM-9:00 AM- 2 hours, 9:00 AM-10:00
AM-3 hours, 10:00 AM-10:30 AM-3 1/2 hours total time
riding.” The student finds “210 min total.” The student also
determines, “I think the averge is about 6 miles per hour,” by
dividing 20 miles by 3 hours and then by 4 hours. It appears
that the student is not able to divide the 20 miles by 3 1/2
hours so rounds both down and up and selects six miles per
hour.
Representation
Expert
The student’s table is appropriate and accurate. All labels
are included and the entered data is correct. The student
uses the data presented on the table to support work with
fractions, elapsed time, and miles per hour.
exemplars.com
800-450-4050
Expert
P/S R/P Com Con Rep A/Level
E
17
exemplars.com
E
E
E
E
E
800-450-4050
Expert (cont.)
18
exemplars.com
800-450-4050