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The task
These tasks all focus on the same arithmetic and number concepts and skills, although they were drawn from four different classrooms.

In Sample 1, the teacher included the following instructions on a classroom test:
Using what you know about multiplication, find the answer to the equation below. Show all your work and clearly explain how you got your answer.

63 x 46 =

In Sample 2, the teacher gave the following written instructions as part of a class exercise:
Solve each problem two ways.

522 - 367 =
87 x 9 =

In Sample 3, the teacher gave the following oral instructions at the end of a long unit of work:

Make a catalogue of all the ways the class has come up with to multiply large numbers.

Make up a two digit by two digit multiplication problem and use these ways to find the answer.

In Samples 4 and 5, the teacher gave the following written instructions to the students:

There were 3 bags of M & M candies. Four children decided to open all three bags and share the M & M’s fairly. Each bag had 52 M & M’s in it. How many M & M’s did each child get?

In Sample 4, the teacher orally added the instruction, “Show all your work and clearly explain how you got your answer.” In Sample 5, the teacher included this direction with the written instructions.

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

Sample 1 was part of a classroom test. For Samples 2 and 3, students were permitted to talk as they worked. Samples 4 and 5 were produced as class work.

Sample 1 was produced before the class had received any instruction about two digit by two digit multiplication, although they had experience in developing strategies for simpler computation.


Sample 1

Sample 2 was produced in a classroom where students’ computation strategies were regularly shared, and several were posted on the wall.

Sample 3 was produced after a long unit of work in which students created, developed, and learned a range of strategies for single and double digit multiplication.

Samples 4 and 5 were produced after minimal instruction about division. The students did have some experience with developing and explaining various strategies for other computation problems.
These work samples illustrate standard-setting
performances 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 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 in calculations.
d Arithmetic and Number Concepts: Describe and compare quantities by using simple fractions.
a Mathematical Skills and Tools: Add, subtract,
multiply, and divide whole numbers correctly.
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, and pencil and paper 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.
What the work shows
a Arithmetic and Number Concepts: The student adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is:

adds, i.e., joins things together, increases.
All students added as they solved larger problems.
subtracts, i.e., takes away, compares, finds the difference.
multiplies, i.e., uses repeated addition, counts by multiples, combines things that come in groups, makes arrays….
The student combined doubling with repeated addition.
The student used repeated addition of 85.
The student showed how the problem could be solved by making 62 circles with 85 stars in each circle.
The student drew a base ten block array.
divides, i.e., puts things into groups, shares equally….
The student “split” 80 into 40 and 40 in order to multiply.
The student divided 156 by 4 although the place value position of the quotient is incorrect.
The student divided 100 by 4, 50 by 4, and 6 by 4.
analyzes problem situations and contexts in order to figure out when to add, subtract, multiply, or divide.

In Samples 4 and 5, the instructions give no indication about how arithmetic can be used to solve the problem: the students used their own strategies.
• solves arithmetic problems by relating addition, subtraction, multiplication, and division to one another.
The students related addition to multiplication or division
These parts of the work demonstrate using addition to solve a subtraction problem.
• computes answers mentally, e.g., 27 + 45, 30 x 4.
All students used mental computation effectively throughout all of their 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:


Sample 2


Sample 3

• counts 1, 10, 100, or 1,000 more than or less than, e.g., 1 less than 10,000, 10 more than 380, 1,000 more than 23,000, 100 less than 9,000.

The student counted 3, 30, and 100 more.

• 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.

The students broke the numbers apart, e.g., 40 x 9, 8 x 9.
The student broke 156 into 100, 50, and 6 and divided each part by 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 landmark numbers to go from 367 to 370 to 400 to 500.
The student appropriately rounded off at first, then used exact numbers to finish the task.

d Arithmetic and Number Concepts: The student describes and compares quantities by using concrete and real world models of simple fractions; that is:


Sample 3

finds simple parts of wholes.
The student applied concrete knowledge to split “80 in half.”
The student demonstrated knowledge of adding like fractions.
The student recognized equivalent fractions and simple fractions as parts of whole numbers.

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.
Here, as in other places, the students’ knowledge of basic facts is demonstrated in correct solutions to larger problems.
adds and subtracts numbers with several digits.
Here, as in other places, the samples demonstrate correct work.
multiplies and divides numbers with one or two digits.
Here, as in other places, the samples demonstrate correct work.
multiplies and divides three digit numbers by one digit numbers.


f Mathematical Skills and Tools: The student uses +, -, x…correctly in number sentences and expressions.


Sample 4


Sample 5

These parts of the samples demonstrate correct use of symbols. This is also evident throughout the samples.
h Mathematical Skills and Tools: The student uses recall, mental computations, pencil and paper…to achieve solutions.

a Mathematical Communication: The student uses appropriate mathematical terms, vocabulary, and language, based on prior conceptual work.
Each sample includes evidence that demonstrates this kind of understanding.
b Mathematical Communication: The student shows mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.

c Mathematical Communication: The student explains solutions to problems clearly and logically, and supports solutions with evidence, in both oral and written work.
Each sample includes explanations for the solutions given.

There are some errors of spelling, grammar, and usage in these samples of student work. For example, in Sample 3, the student spelled a few words inconsistently (e.g., “multiplied” is correct while “multiplyed” is not) and misspelled others (e.g., “Catalouge” instead of “Catalogue”). In Sample 4, there are some spelling errors (e.g., “divied” instead of “divided”) and some slips in tenses. All of the samples were taken from work done in class which were not edited or revised.