Chocolates Problem Solving Task Tom is buying his mother her favorite chocolate for Valentines Day If 14 of a pound of the candy sells for $139 what will it cost Tom to purchase his gift that weighs 2 12 pounds

1

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

Scores and Commentary Paper 1 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

6 6 6 6 6

Making Sense of the Task The student interprets the task as a rate problem and states they

are going to use a proportion to solve it Heshe shows an understanding of rate using the

diagram and also shows an understanding of how to connect the rate to a proportion The

student is also able to flexibly interchange frac14 with 025 All of this combined with the

connection to another context (I usually use proportions to solve rate problems for examplehellip)

makes this paper thoroughly developed and enhanced

Representing and Solving the Task The strategy of drawing the visual quarter portions to

show that there are 10 portions of frac14 in 25 pounds is effective and complete It is enhanced

when the student generalizes the problem into a ldquoraterdquo problem and states that they ldquousually

use proportionsrdquo to solve this type of problem The student is able to generalize how to solve

a proportion by giving the second example with the pushups

Communicating and Reasoning The student uses precise mathematical language (rate and

proportion) The communication is enhanced by showing the labels for the numbers in the

proportion the clearly labeled diagram and all the places where the student takes time to

explain their steps (cross multiply simplify etchellip)

Accuracy The solution is correct and clearly stated It is enhanced by the connection

between the diagram and the proportion as well as the studentrsquos ability to generalize the

problem as a rate problem The additional context also helps to enhance the solution

Reflecting and Evaluating The solution was stated in the context of the problem The

student worked the problem using two methods (the diagram and the proportion) and

commented on hisher general approach for these types of problems (rate problems)

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

2

1

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

Scores and Commentary Paper 1 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

6 6 6 6 6

Making Sense of the Task The student interprets the task as a rate problem and states they

are going to use a proportion to solve it Heshe shows an understanding of rate using the

diagram and also shows an understanding of how to connect the rate to a proportion The

student is also able to flexibly interchange frac14 with 025 All of this combined with the

connection to another context (I usually use proportions to solve rate problems for examplehellip)

makes this paper thoroughly developed and enhanced

Representing and Solving the Task The strategy of drawing the visual quarter portions to

show that there are 10 portions of frac14 in 25 pounds is effective and complete It is enhanced

when the student generalizes the problem into a ldquoraterdquo problem and states that they ldquousually

use proportionsrdquo to solve this type of problem The student is able to generalize how to solve

a proportion by giving the second example with the pushups

Communicating and Reasoning The student uses precise mathematical language (rate and

proportion) The communication is enhanced by showing the labels for the numbers in the

proportion the clearly labeled diagram and all the places where the student takes time to

explain their steps (cross multiply simplify etchellip)

Accuracy The solution is correct and clearly stated It is enhanced by the connection

between the diagram and the proportion as well as the studentrsquos ability to generalize the

problem as a rate problem The additional context also helps to enhance the solution

Reflecting and Evaluating The solution was stated in the context of the problem The

student worked the problem using two methods (the diagram and the proportion) and

commented on hisher general approach for these types of problems (rate problems)

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

2

Scores and Commentary Paper 1 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

6 6 6 6 6

Making Sense of the Task The student interprets the task as a rate problem and states they

are going to use a proportion to solve it Heshe shows an understanding of rate using the

diagram and also shows an understanding of how to connect the rate to a proportion The

student is also able to flexibly interchange frac14 with 025 All of this combined with the

connection to another context (I usually use proportions to solve rate problems for examplehellip)

makes this paper thoroughly developed and enhanced

Representing and Solving the Task The strategy of drawing the visual quarter portions to

show that there are 10 portions of frac14 in 25 pounds is effective and complete It is enhanced

when the student generalizes the problem into a ldquoraterdquo problem and states that they ldquousually

use proportionsrdquo to solve this type of problem The student is able to generalize how to solve

a proportion by giving the second example with the pushups

Communicating and Reasoning The student uses precise mathematical language (rate and

proportion) The communication is enhanced by showing the labels for the numbers in the

proportion the clearly labeled diagram and all the places where the student takes time to

explain their steps (cross multiply simplify etchellip)

Accuracy The solution is correct and clearly stated It is enhanced by the connection

between the diagram and the proportion as well as the studentrsquos ability to generalize the

problem as a rate problem The additional context also helps to enhance the solution

Reflecting and Evaluating The solution was stated in the context of the problem The

student worked the problem using two methods (the diagram and the proportion) and

commented on hisher general approach for these types of problems (rate problems)

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

2

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

2

Scores and Commentary Paper 2 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

6 6 6 5 6

Making Sense of the Task The interpretation is thoroughly developed by stating a plan and

describing the solution method and carrying it out It is enhanced through the extension to

another context reinforcing the first strategy

Representing and Solving the Task The representation is insightful by converting it to

decimals to simplify calculations and to more readily see the multiple of 10 It is enhanced by

the comparison to another context All steps in the process are clearly recorded

Communicating and Reasoning The student uses the language of mathematics both

verbally and symbolically providing a tutorial to solving proportions It is enhanced by the

example that allows the reader to move easily from one thought to another

Accuracy The correct solution is enhanced by showing the different representations of

decimals and fractions to solve the problem but because the second context doesnrsquot really

extend the solution we donrsquot feel this is a 6

Reflecting and Evaluating The student reworked the task using a different method (fractions)

thus applying proportional reasoning in an analytical way

Even though not every bullet is evident the tutorial nature of the paper warrants a 6

Note It is not necessary for a student to complete every bullet to earn a 6

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

3

Scores and Commentary Paper 3 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

4 4 4 4 4

Making Sense of the Task the problem was adequately developed and displayed showing

there are 10 groups of frac14 pounds of chocolate which indicates proportional thinking

Representing and Solving the Task Effective and complete strategies were applied as

shown by multiplying the 10 parts by $139 to arrive at the answer of $1390

Communicating and Reasoning A clear and coherent path is shown throughout the work

sample leading to the clearly identified solution ldquoIt will cost Tom $1390rdquo

Accuracy The solution $1390 is a mathematically justifiable solution to the task and is

supported by the work

Reflecting and Evaluating The student reviews and reflects on the reasonableness of the

strategies throughout the work sample by checking the original mental calculations (frac14 x 101 =

2 frac12) and by working backwards ldquoI will check by working backwardsrdquo

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

4

Scores and Commentary Paper 4 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

4 3 3 4 2

Making Sense of the Task The concept of knowing that frac14 x 10 = 2 frac12 is adequately

developed The student knew to take 10 x $139 (the 10 representing how many frac14 lbs)

Representing and Solving the Task The representation is only partially complete The use of

mathematical symbols does not show where the 10 comes from If 2 24 = 104 had been

shown it could have been a 4

Communicating and Reasoning There is evidence of only partial communication of the

process with significant gaps There is a gap in that there is no clear explanation of how 10 was

derived

Accuracy $1390 a mathematically justifiable solution to the task The work supports it

Reflecting and Evaluating The checking of one calculation is a partial attempt to work

backwards but does not justify how 10 connects to the problem and solution process

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

3

Scores and Commentary Paper 3 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

4 4 4 4 4

Making Sense of the Task the problem was adequately developed and displayed showing

there are 10 groups of frac14 pounds of chocolate which indicates proportional thinking

Representing and Solving the Task Effective and complete strategies were applied as

shown by multiplying the 10 parts by $139 to arrive at the answer of $1390

Communicating and Reasoning A clear and coherent path is shown throughout the work

sample leading to the clearly identified solution ldquoIt will cost Tom $1390rdquo

Accuracy The solution $1390 is a mathematically justifiable solution to the task and is

supported by the work

Reflecting and Evaluating The student reviews and reflects on the reasonableness of the

strategies throughout the work sample by checking the original mental calculations (frac14 x 101 =

2 frac12) and by working backwards ldquoI will check by working backwardsrdquo

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

4

Scores and Commentary Paper 4 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

4 3 3 4 2

Making Sense of the Task The concept of knowing that frac14 x 10 = 2 frac12 is adequately

developed The student knew to take 10 x $139 (the 10 representing how many frac14 lbs)

Representing and Solving the Task The representation is only partially complete The use of

mathematical symbols does not show where the 10 comes from If 2 24 = 104 had been

shown it could have been a 4

Communicating and Reasoning There is evidence of only partial communication of the

process with significant gaps There is a gap in that there is no clear explanation of how 10 was

derived

Accuracy $1390 a mathematically justifiable solution to the task The work supports it

Reflecting and Evaluating The checking of one calculation is a partial attempt to work

backwards but does not justify how 10 connects to the problem and solution process

Scores and Commentary Paper 3 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

4 4 4 4 4

Making Sense of the Task the problem was adequately developed and displayed showing

there are 10 groups of frac14 pounds of chocolate which indicates proportional thinking

Representing and Solving the Task Effective and complete strategies were applied as

shown by multiplying the 10 parts by $139 to arrive at the answer of $1390

Communicating and Reasoning A clear and coherent path is shown throughout the work

sample leading to the clearly identified solution ldquoIt will cost Tom $1390rdquo

Accuracy The solution $1390 is a mathematically justifiable solution to the task and is

supported by the work

Reflecting and Evaluating The student reviews and reflects on the reasonableness of the

strategies throughout the work sample by checking the original mental calculations (frac14 x 101 =

2 frac12) and by working backwards ldquoI will check by working backwardsrdquo

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

4

Scores and Commentary Paper 4 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

4 3 3 4 2

Making Sense of the Task The concept of knowing that frac14 x 10 = 2 frac12 is adequately

developed The student knew to take 10 x $139 (the 10 representing how many frac14 lbs)

Representing and Solving the Task The representation is only partially complete The use of

mathematical symbols does not show where the 10 comes from If 2 24 = 104 had been

shown it could have been a 4

Communicating and Reasoning There is evidence of only partial communication of the

process with significant gaps There is a gap in that there is no clear explanation of how 10 was

derived

Accuracy $1390 a mathematically justifiable solution to the task The work supports it

Reflecting and Evaluating The checking of one calculation is a partial attempt to work

backwards but does not justify how 10 connects to the problem and solution process

Standard 722 Apply ratio and proportionality to solve problems including percent and simple probability

4

Scores and Commentary Paper 4 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

4 3 3 4 2

Making Sense of the Task The concept of knowing that frac14 x 10 = 2 frac12 is adequately

developed The student knew to take 10 x $139 (the 10 representing how many frac14 lbs)

Representing and Solving the Task The representation is only partially complete The use of

mathematical symbols does not show where the 10 comes from If 2 24 = 104 had been

shown it could have been a 4

Communicating and Reasoning There is evidence of only partial communication of the

process with significant gaps There is a gap in that there is no clear explanation of how 10 was

derived

Accuracy $1390 a mathematically justifiable solution to the task The work supports it

Reflecting and Evaluating The checking of one calculation is a partial attempt to work

backwards but does not justify how 10 connects to the problem and solution process

Scores and Commentary Paper 4 Chocolates

Making Sense of

the Task

Representing

and Solving the

Task

Communicating

and Reasoning

Accuracy Reflecting and

Evaluating

4 3 3 4 2

Making Sense of the Task The concept of knowing that frac14 x 10 = 2 frac12 is adequately

developed The student knew to take 10 x $139 (the 10 representing how many frac14 lbs)

Representing and Solving the Task The representation is only partially complete The use of

mathematical symbols does not show where the 10 comes from If 2 24 = 104 had been

shown it could have been a 4

Communicating and Reasoning There is evidence of only partial communication of the

process with significant gaps There is a gap in that there is no clear explanation of how 10 was

derived

Accuracy $1390 a mathematically justifiable solution to the task The work supports it

Reflecting and Evaluating The checking of one calculation is a partial attempt to work

backwards but does not justify how 10 connects to the problem and solution process