Performance Descriptions: Elementary School Mathematics
 To see how these performance descriptions compare with the expectations for middle school and high school, click here. The examples that follow the performance descriptions for each standard are examples of the work students might do to demonstrate their achievement. The examples also indicate the nature and complexity of activities that are appropriate to expect of students at the elementary level. Depending on the nature of the task, the work might be done in class, for homework, or over an extended period. The cross-references that follow the examples highlight examples for which the same activity, and possibly even the same piece of work, may enable students to demonstrate their achievement in relation to more than one standard. In some cases, the cross-references highlight examples of activities through which students might demonstrate their achievement in relation to standards for more than one subject matter. The cross-references after the examples that begin “E,” “S,” and “A” refer to the performance standards for English Language Arts, Science, and Applied Learning respectively. See, for example, the cross-references after the examples of activities for .
Arithmetic and Number Concepts

The student demonstrates understanding of a mathematical concept by using it to solve problems, representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and explaining it to someone else. All three ways of demonstrating understanding—use, represent, and explain—are required to meet this standard.

The student produces evidence that demonstrates understanding of arithmetic and number concepts; that is, the student:

 a Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: • adds, i.e., joins things together, increases; • 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, uses area models, computes simple scales, uses simple rates; • 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; • solves arithmetic problems by relating addition, subtraction, multiplication, and division to one another; • computes answers mentally, e.g., 27 + 45, 30 x 4; • uses simple concepts of negative numbers, e.g., on a number line, in counting, in temperature, “owing.”

 b Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is: • 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; • 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 Estimates, approximates, rounds off, uses landmark numbers, or uses exact numbers, as appropriate, in calculations. d Describes and compares quantities by using concrete and real world models of simple fractions; that is: • finds simple parts of wholes; • recognizes simple fractions as instructions to divide, e.g., 1/4 of something is the same as dividing something by 4; • recognizes the place of fractions on number lines, e.g., in measurement; • uses drawings, diagrams, or models to show what the numerator and denominator mean, including when adding like fractions, e.g., 1/8 + 5/8, or when showing that 3/4 is more than 3/8; • uses beginning proportional reasoning and simple ratios, e.g., “about half of the people.”

 e Describes and compares quantities by using simple decimals; that is: • adds, subtracts, multiplies, and divides money amounts; • recognizes relationships among simple fractions, decimals, and percents, i.e., that 1/2 is the same as 0.5, and 1/2 is the same as 50%, with concrete materials, diagrams, and in real world situations, e.g., when discovering the chance of a coin landing on heads or tails.

 f Describes and compares quantities by using whole numbers up to 10,000; that is: • connects ideas of quantities to the real world, e.g., how many people fit in the school’s cafeteria; how far away is a kilometer; • finds, identifies, and sorts numbers by their properties, e.g., odd, even, multiple, square.
 Examples of activities through which students might demonstrate conceptual understanding of arithmetic and number include: Use examples and drawings to show a third grader who is having trouble understanding multiplication why 3 x 6 = 6 x 3. 1a Use base ten blocks and numerals or other models and representations to solve 43 x 38, and show how the numbers get taken apart in the process of solving such a problem, e.g., 43 can become 40 + 3, or 38 can become 40 - 2. 1a, 1b Figure out how many Valentine’s Day cards would be exchanged in total, if there are 30 students in the class and if everyone gave everyone else one card. 1a, 1b, 1c Draw and explain many different ways to make 263, using tens, hundreds and ones. 1a, 1b, 1c Organize a budget for a project. 1a, 1b, 1c, 1d, 1e, 1f, A1a, A1b, A1c Solve the following problem: “You have a book to read that is 100 pages long. You must read it in five days. How many pages should you read each day?” 1a, 1b, 1c, 1f Determine the possible number of objects in a group given this information: when put in groups of three, none is left over; when put in groups of two, one is left over; when put in groups of five, one is left over; there are fewer than 40 objects altogether. 1a, 1b, 1c, 1f Make reasonable estimates and then calculate accurately the number of beans in a cup, seeds in a pumpkin, ceiling tiles in a room, and wheels on 37 tricycles. 1a, 1b, 1c, 1f Show and explain to a classmate the different coin combinations that make 75¢, excluding pennies. 1a, 1b, 1c, 1f Find one possible answer for addends that equal the sum of 8,829, when the digits 1, 2, 3, 4, 5, 6, 7, and 8 are used only once in the solution. 1a, 1b, 1c, 5a, 5b, 5c Analyze seasonal variations in temperature, including negative values. 1a, 1c, 1f, S3c, A1a Put together fraction pieces to make a whole in different ways, e.g., 1/8 + 1/8 + 3/4. 1c, 1d Draw diagrams to explain how three pizzas can be shared equally by four people. 1c, 1d Make approximate counts of different kinds of animals in different zones of a small tide pool area. 1c, 1f, S6b Draw and label diagrams to show the fractional value of each piece of a set of tangram pieces, if all seven pieces (i.e., the whole set) are equal to one whole. 1d Pretend you see a highway sign that reads, “Exit A, 1 mile. Exit B, 2¼ miles,” and shortly thereafter you see a sign that reads, “Exit A, ¼ mile.” Figure out how far away exit B must now be. (Balanced Assessment) 1d, 1f Make up stories that go with number sentences, e.g., 5 x 9 = 45; 27 x 6 = ?. 1f

Geometry and Measurements Concepts

The student demonstrates understanding of a mathematical concept by using it to solve problems, representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and explaining it to someone else. All three ways of demonstrating understanding—use, represent, and explain—are required to meet this standard.

 The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student: a Gives and responds to directions about location, e.g., by using words such as “in front of,” “right,” and “above.” b Visualizes and represents two dimensional views of simple rectangular three dimensional shapes, e.g., by showing the front view and side view of a building made of cubes. c Uses simple two dimensional coordinate systems to find locations on a map, and represent points and simple figures. d Uses many types of figures (angles, triangles, squares, rectangles, rhombi, parallelograms, quadrilaterals, polygons, prisms, pyramids, cubes, circles, and spheres) and identifies the figures by their properties, e.g., symmetry, number of faces, two- or three- dimensionality, no right angles. e Solves problems by showing relationships between and among figures, e.g., using congruence and similarity, and using transformations including flips, slides, and rotations. f Extends and creates geometric patterns using concrete and pictorial models. g Uses basic ways of estimating and measuring the size of figures and objects in the real world, including length, width, perimeter, and area. h Uses models to reason about the relationship between the perimeter and area of rectangles in simple situations. i Selects and uses units, both formal and informal as appropriate, for estimating and measuring quantities such as weight, length, area, volume, and time. j Carries out simple unit conversions, such as between cm and m, and between hours and minutes. k Uses scales in maps, and uses, measures, and creates scales for rectangular scale drawings based on work with concrete models and graph paper.

 Examples of activities through which students might demonstrate conceptual understanding of geometry and measurement include: Locate a point on a map using its distance away from another given point, the map’s scale, and compass point directions, e.g., N, NE, E, SE, etc. 2a, 2c, 2k Given views of three faces of a three dimensional figure made of stacked cubes, represent the views of the other faces and figure out the total number of cubes used to construct the figure. 2b Explain and show why a square is not a cube. 2d Describe a shape from among the following, using as few attributes as possible to distinguish it from the others: right triangle, equilateral triangle, trapezoid, and parallelogram. 2d Design and label a quilt square which includes two lines of symmetry, four congruent shapes, and two similar shapes; then make a final version at twice the original size, i.e., a “two to one” scale, and add color while maintaining symmetry. 2d, 2e, 2f, 2k Draw at each of three consecutive 90° rotations in a clockwise direction. 2e Find all the shapes that can be made with five squares if the sides touch completely. 2e Figure out the approximate area and perimeter of the bottom of a shoe. 2g Solve the following problem, using concrete materials as appropriate: “Given a diagram of a fenced-in, rectangular garden plot with dimensions three meters by eight meters, find its area and perimeter; design a second garden plot using less fencing, but providing greater area; design a third plot using more fencing, but providing less area. Of all possible rectangular designs using the original amount of fencing, which provides the greatest area?” 2g, 2h, 2i Put five objects, such as books, rocks, or pumpkins, in rank order by weight, first by estimating and then by measuring exactly. 2i Figure out exactly how much time would you gain for the completion of a project by waking up at 6:15 rather than 7:00 for five school days. 2i, 2j

Function and Algebra Concepts

The student demonstrates understanding of a mathematical concept by using it to solve problems, representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and explaining it to someone else. All three ways of demonstrating understanding—use, represent, and explain—are required to meet this standard.

 The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student: a Uses linear patterns to solve problems; that is: • shows how one quantity determines another in a linear (“repeating”) pattern, i.e., describes, extends, and recognizes the linear pattern by its rule, such as, the total number of legs on a given number of horses can be calculated by counting by fours; • shows how one quantity determines another quantity in a functional relationship based on a linear pattern, e.g., for the “number of people and total number of eyes,” figure out how many eyes 100 people have all together. b Builds iterations of simple non-linear patterns, including multiplicative and squaring patterns (e.g., “growing” patterns) with concrete materials, and recognizes that these patterns are not linear. c Uses the understanding that an equality relationship between two quantities remains the same as long as the same change is made to both quantities. d Uses letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations with concrete materials, i.e., demonstrates understanding and use of a beginning concept of a variable.

 Examples of activities through which students might demonstrate conceptual understanding of functions and algebra include: Find, make, and describe linear patterns on the 99-chart, e.g., 4, 14, 24, 34. 3a Given the situation described in the Christmas carol, “The Twelve Days of Christmas,” determine how many gifts would have been given in total; describe any pattern you notice that helps in solving the problem. 3a, 3b Show how the letters “aab, aab,...,” can represent the pattern “metal, metal, plastic,...,” “leaf, leaf, rock,...,” or many other patterns. 3a, 3d Solve the following problem: When building a staircase out of cubes, one step uses a total of one cube, two steps a total of three cubes, three steps a total of six cubes, etc. Find how many cubes are used for six steps, nine steps, n steps. Describe the pattern. 3b Observe and record, in a two-column table, multiplicative patterns with concrete materials, e.g., how many regions are produced by increasing the numbers of folds of paper; and recognize that this type of pattern does not proceed in a linear way, i.e., not 2, 4, 6, 8, etc., but 2, 4, 8, 16, etc. 3b Build the fourth, fifth, and sixth iterations in the following sequence: ; use any given number to make a square number; and recognize that the number of little squares in the pattern does not proceed in a linear fashion. 3b Given x + y = 10 (or, say,+ __=10), figure out all the whole numbers that will make the equation true, i.e., use the variables to show all the ways to make ten by adding two whole numbers together. 3c, 3d Plot points on a coordinate graph according to the convention that (x,y) refers to the intersection of a given vertical line and a given horizontal line. 3b

Statistics and Probability Concepts

The student demonstrates understanding of a mathematical concept by using it to solve problems, representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words), and explaining it to someone else. All three ways of demonstrating understanding—use, represent, and explain—are required to meet this standard.

 The student produces evidence that demonstrates understanding of statistics and probability concepts in the following areas; that is, the student: a Collects and organizes data to answer a question or test a hypothesis by comparing sets of data. b Displays data in line plots, graphs, tables, and charts. c Makes statements and draws simple conclusions based on data; that is: • reads data in line plots, graphs, tables, and charts; • compares data in order to make true statements, e.g., “seven plants grew at least 5 cm”; • identifies and uses the mode necessary for making true statements, e.g., “more people chose red”; • makes true statements based on a simple concept of average (median and mean), for a small sample size and where the situation is made evident with concrete materials or clear representations; • interprets data to determine the reasonableness of statements about the data, e.g., “twice as often,” “three times faster”; • uses data, including statements about the data, to make a simple concluding statement about a situation, e.g., “This kind of plant grows better near sunlight because the seven plants that were near the window grew at least 5 cm.” d Gathers data about an entire group or by sampling group members to understand the concept of sample, i.e., that a large sample leads to more reliable information, e.g., when flipping coins. e Predicts results, analyzes data, and finds out why some results are more likely, less likely, or equally likely. f Finds all possible combinations and arrangements within certain constraints involving a limited number of variables.

 Examples of activities through which students might demonstrate conceptual understanding of statistics and probability include: Generate survey questions, such as: How many people are in your family? What is your hair color? What do you do at recess? Predict the survey results, carry out the survey, graph the data, and write true statements about the data. 4a, 4b, 4c, 4d Figure out a need, e.g., the librarian’s purchase of new library books; and collect data in order to make a recommendation. 4a, 4b, 4c, 4d Make conjectures, after sampling the particular situation, about how many raisins are in a given box taken from a set of boxes or what colors of tiles are in a bag. 4a, 4b, 4c, 4d Investigate the temperature over the entire school year. 4a, 4b, 4c, 4d, S3c, A1a Investigate the possible and likely or unlikely outcomes when rolling two number cubes and recording the sums. 4d, 4e, 4f Design spinners with regions red, blue and green, according to the following instructions: Spinner #1: red will certainly win; Spinner #2: red cannot win; Spinner #3: red is likely to win; Spinner #4: red, blue and green are equally likely to win; Spinner #5: red or blue will probably win. 4e Find all of the different combinations possible using three ice cream flavors, two sauces, and one topping; then discover which would allow the most new combinations—an additional ice cream flavor or a new topping. (Balanced Assessment) 4f

Problem Solving and Reasoning

The student demonstrates logical reasoning throughout work in mathematics, i.e., concepts and skills, problem solving, and projects; demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow; and solves problems that make demands on all three aspects of the solution process—formulation, implementation, and conclusion.

 Formulation a 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; • uses previously learned strategies, skills, knowledge, and concepts to make decisions; • uses strategies, such as using manipulatives or drawing sketches, to model problems.

 Implementation b 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 and uses or learns approaches that other people use, as appropriate; • makes connections among concepts in order to solve problems; • solves problems in ways that make sense and explains why these ways make sense, e.g., defends the reasoning, explains the solution. Conclusion c The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: • explains a pattern that can be used in similar situations; • explains how the problem is similar to other problems he or she has solved; • explains how the mathematics used in the problem is like other concepts in mathematics; • explains how the problem solution can be applied to other school subjects and in real world situations; • makes the solution into a general rule that applies to other circumstances.

 Examples of activities through which students might demonstrate facility with problem solving and reasoning include: Suppose you are given a string that is sixteen inches long. If you cut or fold it in any two places, will it always make a triangle? 5a, 5b, 5c Figure out how many handshakes there will be altogether if five people in a room shake each other’s hand just once. 5a, 5b, 5c Prove or disprove a classmate’s claim that 3/4 and 5/6 are really the same because both have one piece missing. 5a, 5b, 5c Given an unopened bag of chocolate chip cookies, with cookies arranged in rows, show and explain whether the manufacturer’s claim of “More than 1,000 chips in every bag!” is reasonable. 5a, 5b, 5c Figure out the value of each tangram piece if the small tangram triangle is worth one cent and the value is proportional to the area. 5a, 5b, 5c For the sum of 8,829, find one possible answer for addends when the digits 1, 2, 3, 4, 5, 6, 7, and 8 are used only once in the solution. 5a, 5b, 5c

Mathematical Skills and Tools

The student demonstrates fluency with basic and important skills by using these skills accurately and automatically, and demonstrates practical competence and persistence with other skills by using them effectively to accomplish a task, perhaps referring to notes, books, or other students, perhaps working to reconstruct a method; that is, the student:

 a Adds, subtracts, multiplies, and divides whole numbers correctly; that is: • knows single digit addition, subtraction, multiplication, and division facts; • adds and subtracts numbers with several digits; • multiplies and divides numbers with one or two digits; • multiplies and divides three digit numbers by one digit numbers.
 b Estimates numerically and spatially. c Measures length, area, perimeter, circumference, diameter, height, weight, and volume accurately in both the customary and metric systems. d Computes time (in hours and minutes) and money (in dollars and cents). e Refers to geometric shapes and terms correctly with concrete objects or drawings, including triangle, square, rectangle, side, edge, face, cube, point, line, perimeter, area, and circle; and refers with assistance to rhombus, parallelogram, quadrilateral, polygon, polyhedron, angle, vertex, volume, diameter, circumference, sphere, prism, and pyramid. f Uses +, -, x, ÷, /, , \$, ¢, %, and . (decimal point) correctly in number sentences and expressions. g Reads, creates, and represents data on line plots, charts, tables, diagrams, bar graphs, simple circle graphs, and coordinate graphs. h Uses recall, mental computations, pencil and paper, measuring devices, mathematics texts, manipulatives, calculators, computers, and advice from peers, as appropriate, to achieve solutions; that is, uses measuring devices, graded appropriately for given situations, such as rulers (customary to the 1/8 inch; metric to the millimeter), graph paper (customary to the inch or half-inch; metric to the centimeter), measuring cups (customary to the ounce; metric to the milliliter), and scales (customary to the pound or ounce; metric to the kilogram or gram). Examples of activities through which students might demonstrate facility with mathematical skills and tools include: Know that 6 x 7 = 42. 6a Use an efficient mental approach to estimate the total cost of items costing \$2.94, \$1.28 and \$0.74. 6a, 6b, 6d, 6h Mentally add two digit numbers correctly during problem solving. 6a, 6h Decide whether to use a calculator, paper and pencil or mental arithmetic to figure out 6 x 6,000. 6a, 6h Figure out rectangular areas correctly when designing a floor plan for a “dream house.” 6b, 6c Measure the circumference of a pumpkin accurately by using a piece of yarn, and then laying it next to a ruler. 6c, 6e Use a calculator to check the arithmetic in a project. Use a table to record functions such as how many chairs fit at how many tables. 6g Use a Venn diagram to record students who wore a sweater to school and students who walked to school. 6g Make a bar graph or simple circle graph to show how many students like different kinds of vegetables. 6g, 6f

Mathematical Communication

The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:

 a Uses appropriate mathematical terms, vocabulary, and language, based on prior conceptual work. b Shows mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models. c Explains solutions to problems clearly and logically, and supports solutions with evidence, in both oral and written work. d Considers purpose and audience when communicating about mathematics. e Comprehends mathematics from reading assignments and from other sources. Examples of activities through which students might demonstrate facility with mathematical communication include: Use words, numbers, or diagrams to explain how to take numbers apart in order to solve problems using mental math, e.g., 25 x 6. One way is “20 x 6 is 120, and 5 x 6 is 30; 120 + 30 = 150.” Or, “25 x 4 is 100, and 25 x 2 is 50, and so the answer is 150 because 100 + 50 = 150.” 7a, 7b, 7c Explain why 34 + 17 3417 to a first grader or to a visitor from outer space. 7a, 7b, 7c, 7d Show 1/2 + 3/4 in pictures and diagrams so a younger student could understand the sum. 7a, 7b, 7c, 7d Use clear, correct mathematical language to describe a shape composed of several geometric solids so that it could be reproduced exactly by a person who cannot see the shape. 7a, 7b, 7c, 7d Represent survey data about student school lunch preferences in graphical, written, and numerical form in order to make a clear and effective recommendation to kitchen staff about menu changes. 7a, 7b, 7c, 7d Give an oral presentation of a preliminary investigation of classification of shapes, in order to get peer feedback; then revise the classification scheme to make it clearer. 7a, 7b, 7c, 7d, E3c Prepare a report, including graphs, charts, and diagrams, on the optimal number and location of recycling containers, based on data from the classroom and the entire school. 7a, 7b, 7c, 7d, 7e, E2a, S4b, S6d, S7a, A1b

Putting Mathematics to Work

The student conducts at least one large scale project each year, beginning in fourth grade, drawn from the following kinds and, over the course of elementary school, conducts projects drawn from at least two of the kinds.

 A single project may draw on more than one kind. a Data study, in which the student: • develops a question and a hypothesis in a situation where data could help make a decision or recommendation; • decides on a group or groups to be sampled and makes predictions of the results, with specific percents, fractions, or numbers; • collects, represents, and displays data in order to help make the decision or recommendation; compares the results with the predictions; • writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.

 b Science study, in which the student: • decides on a specific science question to study and identifies the mathematics that will be used, e.g., measurement; • develops a prediction (a hypothesis) and develops procedures to test the hypothesis; • collects and records data, represents and displays data, and compares results with predictions; • writes a report that compares the results with the hypothesis; supports the results with diagrams, charts, and graphs; acknowledges assistance received from parents, peers, and teachers.

 c Design of a physical structure, in which the student: • decides on a structure to design, the size and budget constraints, and the scale of design; • makes a first draft of the design, and revises and improves the design in response to input from peers and teachers; • makes a final draft and report of the design, drawn and written so that another person could make the structure; acknowledges assistance received from parents, peers, and teachers.

 d Management and planning, in which the student: • decides on what to manage or plan, and the criteria to be used to see if the plan worked; • identifies unexpected events that could disrupt the plan and further plans for such contingencies; • identifies resources needed, e.g., materials, money, time, space, and other people; • writes a detailed plan and revises and improves the plan in response to feedback from peers and teachers; • carries out the plan (optional); • writes a report on the plan that includes resources, budget, and schedule, and acknowledges assistance received from parents, peers, and teachers.

 e Pure mathematics investigation, in which the student: • decides on the area of mathematics to investigate, e.g., numbers, shapes, patterns; • describes a question or concept to investigate; • decides on representations that will be used, e.g., numbers, symbols, diagrams, shapes, or physical models; • carries out the investigation; • writes a report that includes any generalizations drawn from the investigation, and acknowledges assistance received from parents, peers, and teachers.

 Examples of projects include: Develop questions and a hypothesis for a study of students’ diets; collect, organize, display, and analyze the data; and make recommendations to the school community based on the data. 8a, S2a, S4c, A1b Compare the growth of a set of plants under a variety of conditions, e.g., amount of water, fertilizer, duration and exposure to sunlight. 8b, S2a Make a design for a tree house that accounts for physical and financial constraints. 8c, A1a Plan a class camping trip, including making a schedule, researching costs and facilities, developing a budget. 8d, S2a, S2c, S4b, S4c, A1c Plan and conduct a probability study that compares the results from three different spinners, e.g., . 8e