Math
Spring Operational 2015
Algebra 2
PBA Item #12
Cost of Store Delivery
VF884971
Prompt
Rubric
Task is worth a total of 6 points.
VH884971 Rubric Part A
Score Description
4
Student response includes the following 4 elements.
• Modeling component = 4 points
o Model for Van: C = 8.9p
o Model for Truck: C = 6.7p + 12
 C represents the total cost, and p represents the fuel
cost per gallon.
o Justification for the model for the van
o Justification for the model for the truck
Sample Student Response:
Van: C = 8.9p
Truck: C = 6.7p + 12
C represents the total cost, and p represents the fuel cost per
gallon.
First determine the total distance that would be traveled by each
vehicle type.
For the truck, use the sum of the segment lengths connecting a
circuit of the given points, starting and ending at the distribution
center. For the van, determine the distance from the distribution
center to each point and multiply the sum by 2 (to give the
round‐trip distances). Divide these values by the respective fuel
efficiency values to determine the total number of gallons of fuel
used. Finally, write linear equations based on these values and
the loading dock fee in the case of the truck.
3
2
1
0
Student
Student
Student
Student
response
response
response
response
includes 3 of the 4 elements.
includes 2 of the 4 elements.
includes 1 of the 4 elements.
is incorrect or irrelevant.
VH884971 Rubric Part B
Score Description
2
Student response includes the following 2 elements.
• Modeling component = 1 point
o Justification for the computation
•
Computation component = 1 point
o Reporting the findings
1
0
Sample Student Response:
It is more cost efficient to use the van when this inequality is
true:
6.7p + 12 > 8.9p
12 > 2.2p
5.45 > p
When the cost of fuel is less than $5.45 per gallon, it is more cost
efficient to drive the van.
Student response includes 1 of the 2 elements.
Student response is incorrect or irrelevant.
Anchor Set
A1 – A13
A1
Part A: Score Point 4
Part B: Score Point 2
A1b
Annotations
Anchor Paper 1
Part A: Score Point 4
This response receives full credit. The student includes each of the four required elements:
•
A correct model for the large truck is included (The cost for the large truck is
20
3
x + 12
dollars).
•
A correct model for the small van is included (The cost for the small van is
80
9
x
dollars).
•
Correct justification for the large truck model is included and is calculated correctly
(60 miles for 9 miles per gallon costs).
•
Correct justification for the small van model is included and is calculated correctly
(160 miles for 18 miles per gallon).
Note that the variable x used in both equations is defined correctly—both models use the
term “dollars.”
Part B: Score Point 2
This response receives full credit. The student includes both of the two required elements:
•
x + 12 >
Correct justification for the computation is provided ( 20
3
80
9
x) with correct
work to reach 5.4 > x.
•
Correct findings are provided (Therefore the small van is more efficient when the price
of gas is less than $5.40). The number is lower than $5.45 because the student used
fractions instead of rounded decimals, which is acceptable since they are equivalent.
A2a
Part A: Score Point 4
Part B: Score Point 2
A2b
Annotations
Anchor Paper 2
Part A: Score Point 4
This response receives full credit. The student includes each of the four required elements:
•
A correct model for the large truck is included (Total cost for a trip for Large Truck:
x + 12).
$ 20
3
•
A correct model for the small van is included (Total cost for a trip for small van:
$ 80
x).
9
•
Correct justification for the large truck model is included (Both modes calculate the
total amount of gas used by the vehicle, then multiply that by the cost of gas. Then
any additional cost is added).
•
Correct justification for the small van is included (18miles/1 = 1/160 miles=9/80gal).
Note that the variable x used in both equations is defined correctly (Both modes calculate the
total amount of gas used by the vehicle, then multiply that by the cost of gas. Then any
additional cost is added).
Part B: Score Point 2
This response receives full credit. The student includes both of the two required elements:
•
Correct justification for the computation is provided (
work to reach x = 5.4.
•
20
3
x + 12 =
80
9
x) with correct
Correct findings are provided (Therefore it is more cost efficient to use the van when
the gas price is under $5.40). The number is lower than $5.45 because the student
used fractions instead of rounded decimals, which is acceptable since they are
equivalent.
A3a
Part A: Score Point 4
Part B: Score Point 1
A3b
Annotations
Anchor Paper 3
Part A: Score Point 4
This response receives full credit. The student includes each of the four required elements:
•
A correct model for the large truck is included (6.67c +12 = x).
•
A correct model for the small van is included (8.89c =x).
•
Correct justification for the large truck is included and is calculated correctly (60 miles
for 9 miles per gallon costs).
•
Correct justification for the small van is included and is calculated correctly (160 miles
for 18 miles per gallon).
Note that the variable x used in both equations is defined correctly (x = total cost of
delivery). The extra variable of c as cost of gas is seen as a minor flaw and does not detract
from complete understanding.
Part B: Score Point 1
This response receives partial credit. The student includes one of the two required elements:
•
Correct findings are provided (It is more cost efficient for a van when you arent
traveling so many miles or the gas prices arent over $5.41 per gallon). The number is
lower than $5.45 because the student used repeating decimals from Part A instead of
rounded decimals.
Correct justification for the computation is missing.
A4
Part A: Score Point 4
Part B: Score Point 0
Annotations
Anchor Paper 4
Part A: Score Point 4
This response receives full credit. The student includes each of the four required elements.
•
A correct model for the large truck is included (y = 6.7x + (3 × 4)).
•
A correct model for the small van is included (y = 8.8x).
•
Correct justification for the large truck is included (The semi will travel 60 miles at 9
mpg so it uses 6.7 gal of fuel).
•
Correct justification for the small van is included (the van travels 160 miles at 18 mpg,
so it uses 8.9 gal of fuel).
Note that the variable x used in both equations is defined correctly (where x = cost of fuel).
Part B: Score Point 0
This response receives no credit. The student includes neither of the two required elements.
The justification for the computation is incorrect (y = 6.7(3) + (3 × 4), y = 20.1 + 12, y =
32.1; y = 8.9(3), y = 26.7); instead of placing the two models into an inequality and solving
for x, the student substitutes a fuel price of $3 for each and solves, comparing the two
results. While this method will determine which vehicle is cheaper to drive, it will not
determine the point at which it becomes more cost efficient to use the van for deliveries.
The findings are incorrect (y = 32.1; y = 26.7); the point at which it becomes more cost
efficient to drive the van is not found.
A5
Part A: Score Point 2
Part B: Score Point 2
Annotations
Anchor Paper 5
Part A: Score Point 2
This response receives partial credit. The student includes two of the four required elements.
•
A correct model for the large truck is included (y =
•
A correct model for the small van is included (y =
The justification for the large truck is missing.
20𝑥
3
80𝑥
9
+ 12).
).
The justification for the small van is missing.
Note that the variables used in both equations are defined correctly (y is the total cost and x
is the cost of one gallon of gas).
Part B: Score Point 2
This response receives full credit. The student includes each of the two required elements.
•
•
Correct justification for the computation is included (It is more cost efficient to use the
80𝑥
20𝑥
van for deliveries than the truck . . . it makes the inequality
<
+ 12 true).
9
3
A correct final answer for the question is included (when the cost of gas is less than
$5.40); this answer is correct for the inequality given.
A6
Part A: Score Point 1
Part B: Score Point 2
Annotations
Anchor Paper 6
Part A: Score Point 1
This response receives partial credit. The student includes one of the four required elements.
•
A correct model for the large truck is included (cost = 6.7x + 12).
The model for the small van is incorrect (cost = 17.8x); dividing the number of miles
traveled, 160 miles, by the fuel efficiency, 18 miles per gallon, yields a coefficient of 8.9
times the cost of gas, not 17.8.
The justification for the large truck model is incomplete (60mi in all, so 6.7 gallons for trip,
$12 for store fees); no mention is made of the fuel efficiency for the truck so the number of
gallons of gas, 6.7, is not explained.
The justification for the small van model is incomplete (160mi in all, so 17.8 gallons for trip);
no mention is made of the fuel efficiency for the van so the number of gallons of gas, 17.8, is
not explained.
Note that the variable x used in both equations is defined correctly (price of gas per gallon =
x; price of gas = x).
Part B: Score Point 2
This response receives full credit. The student includes both of the two required elements.
•
Correct justification for the computation is included (6.7x + 12 > 17.8x; 12 > 11.1x;
1.08 > x); the justification is correct given the incorrect model for the small van in
Part A.
•
A correct final answer for the question is included (When gas costs less than $1.08);
this answer is correct given the incorrect model for the small van in Part A.
Note that a scoring decision was made that if an error is made in one part of the response
but the incorrect answer is used correctly in the next part, the response can receive full credit
for correctly using the incorrect answer from the first part.
A7
Part A: Score Point 3
Part B: Score Point 0
Annotations
Anchor Paper 7
Part A: Score Point 3
This response receives partial credit. The student includes three of the four required
elements.
60
•
A correct model for the large truck is included (y =
•
Correct justification for the large truck model is included (60 miles is the shortest
distance and it’s divided by the miles per gallon, 9. the constant, 12, is the total fees
that are paid).
•
Correct justification for the small van model is included (60 miles is the shortest
distance and it’s divided by the miles per gallon 18).
The model for the small van is incorrect (y =
60
18
9
x +12).
x); the equation y =
160
18
x would be correct.
Note that the variables used in both equations are defined correctly (y is the total cost and x
is the cost of gas per gallon).
Part B: Score Point 0
This response receives no credit. The student includes neither of the two required elements.
The justification for the computation is missing.
The final answer to the equation is missing.
A8
Part A: Score Point 2
Part B: Score Point 0
Annotations
Anchor Paper 8
Part A: Score Point 2
This response receives partial credit. The student includes two of the four required elements.
•
A correct model for the large truck is included (large truck = 12 + 6.67 × x).
•
A correct model for the small van is included (small van = 8.89 × x).
The justification for the large truck is missing.
The justification for the small van is missing.
Note that the variable x used in both equations is defined correctly (x is how much gas
costs).
Part B: Score Point 0
This response receives no credit. The student includes neither of the two required elements.
The justification for the computation is missing.
The final answer to the equation is incorrect (when gas prices are 5 dollars or lower); it is
more cost efficient to use the van for deliveries when gas prices are less than $5.45.
A9
Part A: Score Point 1
Part B: Score Point 1
Annotations
Anchor Paper 9
Part A: Score Point 1
This response receives partial credit. The student includes one of the four required elements.
•
A correct model for the small van is included (C =
The model for the large truck is incorrect (C =
be $12, not $3.
60
9
160
18
x).
x + 3); the total fees for using the dock will
The justification for the large truck model is incomplete (60 is the miles for 1 trip; 9 is the
miles per gallon; 3 is the fee); the justification does not explain the relationship between the
miles per trip and the miles per gallon.
The justification for the small van model is missing.
Note that the variables used in both equations are defined correctly (x is the cost of gas; C is
the cost of 1 trip).
Part B: Score Point 1
This response receives partial credit. The student includes one of the two required elements.
•
Correct justification for the computation is included (
160
18
x=
60
9
x + 3); the response
sets the two equations given in Part A equal to each other, which is correct
justification.
The final answer for the question is incorrect (x =
27
20
); while this fraction is correct for the
incorrect equation given, it is not reported as the dollar amount where it would become more
cost efficient to use the van. Furthermore, the response indicates that it becomes more cost
efficient after a specific number of trips, which is incorrect.
A10
Part A: Score Point 0
Part B: Score Point 1
Annotations
Anchor Paper 10
Part A: Score Point 0
This response receives no credit. The student includes none of the four required elements.
The model for the large truck is incorrect (c = 7x = 12); 7 gallons of gas is not the same as
2
6 gallons of gas. Note that rounding the coefficient of the variable x to the nearest whole
3
number is not considered acceptable.
The model for the small van is included (c = 8x); 8 gallons of gas is not the same as 8
gallons of gas. Note that truncating the coefficient of the variable x is not considered
acceptable.
9
10
The justification for the large truck model is missing.
The justification for the small van model is missing.
Note that the variables used in both equations are correctly defined (c = cost; x = the cost of
a gallon of fuel).
Part B: Score Point 1
This response receives partial credit. The student includes one of the two required elements.
•
Correct justification for the computation is included (when the cost of fuel multiplied
by 8 is less than the cost of fuel multiplied by 7 and added to 12); this justification is
correct for the incorrect models given in Part A.
The final answer for the question is missing.
Note that a scoring decision was made that if an error is made in one part of the response
but the incorrect answer is used correctly in the next part, the response can receive full credit
for correctly using the incorrect answer from the first part.
A11
Part A: Score Point 1
Part B: Score Point 0
Annotations
Anchor Paper 11
Part A: Score Point 1
This response receives partial credit. The student includes one of the four required elements.
•
A correct model for the small van is included (c = 160 ÷ 18g).
The model for the large truck is incorrect (c = 60 ÷ 9g); the total fee for using the loading
docks is not applied.
The justification for the large truck model is incomplete (60 = total miles); the response does
not explain the fuel efficiency of the truck or the relationship between the miles and the fuel
efficiency.
The justification for the small van model is incomplete (160 = total miles); the response does
not explain the fuel efficiency of the van or the relationship between the miles and the fuel
efficiency.
Note that the variables used in both equations are defined correctly (c = cost; g = gallons of
gas).
Part B: Score Point 0
This response receives no credit. The student includes neither of the two required elements.
The justification for the computation is missing.
The final answer to the equation is missing (it is more cost efficient to use the van when
dilivering to store A and store D); the cost of gas at the point where delivering with the small
van is more cost efficient is not provided.
A12
Part A: Score Point 0
Part B: Score Point 0
Annotations
Anchor Paper 12
Part A: Score Point 0
This response receives no credit. The student includes none of the four required elements.
60
The model for the large truck is incomplete ( ); the total docking fee is not added, and the
9
variable representing the cost per gallon of gas is missing.
The model for the small van is incomplete (8.89); the variable representing the cost per
gallon of gas is missing.
The justification for the large truck model is incomplete (large truck miles: 60; total of 12
60
2
…used to find how much gas is used. 6 gallons); the
dollars for using loading dock….
9
3
significance of the denominator 9 in finding how much gas is used is not explained here.
The justification for the small van model is incomplete (small van miles: 160;
160
18
gallons
used.8.89 gallons used); the significance of the denominator 18 in finding how many gallons
is used is not explained here.
No variables are used or defined.
Part B: Score Point 0
This response receives no credit. The student includes neither of the two required elements.
The justification for the computation is incorrect (It has better gas mileage and since the trip
is short you would save on gas); no attempt is made to calculate the point when the cost of
gas makes delivering with the small van more cost efficient.
The final answer to the equation is missing (For short deliveries the small van would be more
cost efficient); the cost of gas at the point where delivering with the small van is more cost
efficient is not provided.
A13
Part A: Score Point 0
Part B: Score Point 0
Annotations
Anchor Paper 13
Part A: Score Point 0
This response receives no credit. The student includes none of the four required elements.
The model for the large truck is incorrect (50 ÷9 = 5.5; 4 × 3 = 12; 12 + 5.5 = 17.5); the
model does not take into account the cost of gas and incorrectly calculates the total number
of miles driven.
The model for the small van is incorrect (310 ÷ 18 = 17.2); the model does not take into
account the cost of gas and incorrectly calculates the total number of miles driven.
The justification for the large truck model is missing.
The justification for the small van model is missing.
No variables are used or defined.
Part B: Score Point 0
This response receives no credit. The student includes neither of the two required elements.
The justification for the computation is missing.
The final answer to the equation is missing (it is more effective for shorter trips not longer
trips); the cost of gas at the point where delivering with the small van is more cost efficient is
not provided.
Practice Set
P101 - P105
P101
P102
P103
P104
P105
Practice Set
Paper
Score
P101
3,0
P102
1,0
P103
0,0
P104
2,0
P105
1,0