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The task
Students were asked to read Counting on Frank by Rod Clement and to write a letter to the author commenting on at least one example of the mathematical claims made.

Circumstances of performance
This sample of student work was produced under the following conditions:
- alone - in a group
- in class - as homework
- with teacher feedback with peer feedback
timed - opportunity for revision

Four students discussed the mathematics in depth in Counting on Frank before they wrote their analyses. The students spent class time as well as many recesses figuring out how to test the claims in the book. For example, the students discussed how to use the classroom sink to test the claim in the book about the bathroom filling up with water. The students worked together to develop strategies for testing the claims about the peas and the ball-point pen. The teacher encouraged the students’ discussions, and provided time and materials in school for them to work out their reasoning. The students met outside of class to work together on the writing. Students completed the writing individually, at home.

The students completed this activity near the end of the school year. Earlier in the year, the students had studied area and perimeter concepts in depth, as well as applications of multiplication and division in problem solving activities.

This work sample illustrates a standard-setting performance
for the following parts of the standards:

a Arithmetic and Number Concepts: Add, subtract, multiply, and divide whole numbers.
b Arithmetic and Number Concepts: Demonstrate understanding of the base ten place value system and use this knowledge to solve arithmetic tasks.
c

Arithmetic and Number Concepts: Estimate,
approximate, round off, use landmark numbers, or use exact numbers, as appropriate, in calculations.

d Arithmetic and Number Concepts: Describe and compare quantities by using concrete and real world models of simple fractions.
e Arithmetic and Number Concepts: Describe and compare quantities by using simple decimals.
f Arithmetic and Number Concepts: Describe and compare quantities by using whole numbers up to 10,000.
g Geometry and Measurement Concepts: Use basic ways of estimating and measuring the size of figures and objects in the real world.
i Geometry and Measurement Concepts: Select and use units for estimating and measuring quantities.
j Geometry and Measurement Concepts: Carry out simple unit conversions.
a Problem Solving and Reasoning: Formulation.
b Problem Solving and Reasoning: Implementation.
c Problem Solving and Reasoning: Conclusion.
a Mathematical Skills and Tools: Add, subtract,
multiply, and divide whole numbers correctly.
b Mathematical Skills and Tools: Estimate numerically and spatially.
c Mathematical Skills and Tools: Measure accurately.
d Mathematical Skills and Tools: Compute time.
f Mathematical Skills and Tools: Use +, -, x, ÷, /,…and . (decimal point) correctly in number
sentences and expressions.
h Mathematical Skills and Tools: Use recall, mental computations, pencil and paper, measuring devices, manipulatives, calculators and advice from peers, as appropriate, to achieve solutions.
a Mathematical Communication: Use appropriate mathematical terms, vocabulary, and language.
b Mathematical Communication: Show mathematical ideas in a variety of ways.
c Mathematical Communication: Explain solutions to problems clearly and logically.
d Mathematical Communication: Consider purpose and audience when communicating about mathematics.
e Mathematical Communication: Comprehend
mathematics from reading assignments and from other sources.


What the work shows
a Arithmetic and Number Concepts: The student adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is:…
• multiplies, i.e.,…uses simple rates.

The consistent and effective use of multiplication throughout the student work shows a deep understanding at the elementary level.
• divides, i.e., puts things into groups, shares equally; calculates simple rates.


• analyzes problem situations and contexts in order to figure out when to add, subtract, multiply, or divide….


The student successfully figured out how and when to use arithmetic in these multi-step problems. The analysis is particularly strong because the situations were general claims made by a character in a book, i.e., there was no hint about how to proceed to test the claims.
• solves arithmetic problems by relating addition, subtraction, multiplication, and division to one another.

The student added to complete a multiplication problem. The teacher also verified that the student divided 700 by 4 and multiplied the answer by 3 with a calculator to figure out that “three quarters of 700 equals 525.”
• computes answers mentally….

The student, as verified by the teacher, computed some of the arithmetic mentally, here and in other places in the work.

b Arithmetic and Number Concepts: The student demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is:…uses knowledge about ones, tens, hundreds, and thousands to figure out answers to multiplication and division tasks, e.g., 36 x 10, 18 x 100, 7 x 1,000, 4,000 ÷ 4.

c Arithmetic and Number Concepts: The student estimates, approximates, rounds off, uses landmark numbers, or uses exact numbers, as appropriate, in calculations.
The student used a well-developed sense of when it is appropriate to round off or estimate in these situations. Elsewhere, the student successfully used exact numbers in calculations.

d Arithmetic and Number Concepts: The student describes and compares quantities by using concrete and real world models of simple fractions; that is, finds simple parts of wholes….
The student used coffee cups and measured out 4 cups. The computation (“three quarters of 700 is 525”) is advanced for this level.

e Arithmetic and Number Concepts: The student describes and compares quantities by using simple decimals….
The student took part of the decimal answer to a division problem (13.13) and used the nearest whole number (13) to make sense of the given situation, i.e., bags of peas.


f Arithmetic and Number Concepts: The student describes and compares quantities by using whole numbers up to 10,000; that is, connects ideas of quantities to the real world….
The student called the author’s bluff by actually connecting the ideas in the book to the real world.
In each problem, the student dealt successfully with quantities larger than 10,000 (10,240 seconds; 43,680 peas; 13,500 feet). The clarity and thoroughness of communication provides evidence of understanding of these quantities and represents a strong performance for this level.

g Geometry and Measurement Concepts: The student uses basic ways of estimating and measuring the size of figures and objects in the real world, including length, width, perimeter, and area.
The student went beyond the standards by measuring the volume of the bathroom in cubic feet. The teacher verified that the students had constructed the formula for area themselves earlier in the year through conceptual activities. The students’ motivation to test the claim about the bathroom led them to a discussion of how to measure the size of the sink and the bathroom, and they constructed the idea of “layers” of length times width. The teacher then supplied them with the formula L x W x H.
The student demonstrated understanding of area by listing some of the “millions of possibilities…that are close” for 264 square feet. The student also demonstrated understanding by applying the concept of area to the size of a bathroom floor, and noticing that an area of 264 square feet would be “huge.”
The student measured the length of the pen and the length of the line and used these measurements to solve the problem.

i Geometry and Measurement Concepts: The student selects and uses units, both formal and informal as appropriate, for estimating and measuring quantities such as…length, area, volume, and time.
The student used centimeters to measure the pen, and feet to measure the line.
The student used a coffee cup to measure volume.
The student used seconds to measure time.


j Geometry and Measurement Concepts: The student carries out simple unit conversions, such as between cm and m, and between hours and minutes.
The student converted peas per day to peas per year, seconds to minutes, minutes to hours, and feet to yards.

a Problem Solving and Reasoning: Formulation. Given the basic statement of a problem situation, the student:
• makes the important decisions about the approach, materials, and strategies to use, i.e., does not merely fill in a given chart, use a pre-specified manipulative, or go through a predetermined set of steps.

The student made the decisions to use the sink, the peas, the coffee cup, the ball-point pen, and the 100 feet long piece of paper.
• uses previously learned strategies, skills, knowledge, and concepts to make decisions.

Throughout the three problems, the student drew upon knowledge and skills related to multiplication, division, and measurement (i.e., length, area, and volume).
• uses strategies, such as using manipulatives or drawing sketches, to model problems.

The student devised effective strategies, such as timing how long the water took to fill the sink, using the coffee cup to measure the peas, and measuring the decrease in length of the ink in the pen.


b Problem Solving and Reasoning: Implementation. The student makes the basic choices involved in planning and carrying out a solution; that is, the student:
• makes up and uses a variety of strategies and approaches to solving problems….


• makes connections among concepts in order to solve problems.

There are many connections between arithmetic and measurement throughout the work.
• solves problems in ways that make sense and explains why these ways make sense, e.g., defends the reasoning, explains the solution.

Statements like, “His whole kitchen would be small like a grocery bag or the peas are as big as softballs,” show that the student made sense of the problem.
Throughout the work, the student impressively, consistently, and accurately explicated the reasoning used.


c Problem Solving and Reasoning: Conclusion. The student moves beyond a particular problem by making connections, extensions, and/or generalizations….
The assignment asked the student to focus on one claim made in the book. The student chose to extend the assignment by analyzing and writing about more than one claim. This is a reasonable extension for elementary school level.

a Mathematical Skills and Tools: The student adds, subtracts, multiplies, and divides whole numbers correctly; that is:
• knows single digit addition, subtraction, multiplication, and division facts.

The student showed a command of basic facts, which was needed to carry out these and other calculations.
• adds…numbers with several digits.


• multiplies and divides numbers with one or two digits.


• multiplies and divides three digit numbers by one digit numbers.

b Mathematical Skills and Tools: The student estimates numerically and spatially.

c Mathematical Skills and Tools: The student measures length, area,…height,…and volume accurately in both the customary and metric systems.
The teacher verified the accuracy of the measurements.

d Mathematical Skills and Tools: The student computes time (in hours and minutes)….

f Mathematical Skills and Tools: The student uses +, -, x, ÷, /,…and . (decimal point) correctly in number sentences and expressions.

h Mathematical Skills and Tools: The student uses recall, mental computations, pencil and paper, measuring devices,…manipulatives, calculators…and advice from peers, as appropriate, to achieve solutions….
The student drew upon many resources to complete the assignment.

a Mathematical Communication: The student uses appropriate mathematical terms, vocabulary, and language, based on prior conceptual work.
The student used many mathematical terms appropriately, including “estimate,” “measure,” “average,” “multiply,” “divide,” “seconds,” “minutes,” “area,” “long,” “high,” “equal,” “cm,” “ft.”
The use of the term “cubic feet” and the formula for volume go beyond what is required at this level.
The student wrote “yds” after 13,500, but this mislabeling did not interfere with the correct mathematics in the work.

b Mathematical Communication: The student shows mathematical ideas in a variety of ways, including words, numbers, and symbols….

c Mathematical Communication: The student explains solutions to problems clearly and logically, and supports solutions with evidence, in both oral and written work.

d Mathematical Communication: The student considers purpose and audience when communicating about mathematics.
The student focused consistently on the purpose of disputing the claims in the book and wrote directly to the author.

e Mathematical Communication: The student comprehends mathematics from reading assignments and from other sources.
The student even comprehended mathematical information from a contractor about the average size of a bathroom.
The student misspelled a few words (e.g., “definetely”) and made some grammatical mistakes (e.g., capitalizing “book” in the middle of the sentence). The teacher did not require the students to fix the errors.