Number and Operation Concepts

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

The student produces evidence that demonstrates understanding of number and operation concepts; that is, the student:
a Consistently and accurately adds, subtracts, multiplies, and divides rational numbers using appropriate methods (e.g., the student can add + mentally or on paper but may opt to add + on a calculator) and raises rational numbers to whole number powers. (Students should have facility with the different kinds and forms of rational numbers, i.e., integers, both whole numbers and negative integers; and other positive and negative rationals, written as decimals, as percents, or as proper, improper, or mixed fractions. Irrational numbers, i.e., those that cannot be written as a ratio of two integers, are not required content but are suitable for introduction, especially since the student should be familiar with the irrational number .)

b Uses and understands the inverse relationships between addition and subtraction, multiplication and division, and exponentiation and root-extraction (e.g., squares and square roots, cubes and cube roots); uses the inverse operation to determine unknown quantities in equations.

c Consistently and accurately applies and converts the different kinds and forms of rational numbers.

d Is familiar with characteristics of numbers (e.g., divisibility, prime factorization) and with properties of operations (e.g., commutativity and associativity), short of formal statements.

e Interprets percent as part of 100 and as a means of comparing quantities of different sizes or changing sizes.

f Uses ratios and rates to express “part-to-part” and “whole-to-whole” relationships, and reasons proportionally to solve problems involving equivalent fractions, equal ratios, or constant rates, recognizing the multiplicative nature of these problems in the constant factor of change.

g Orders numbers with the > and < relationships and by location on a number line; estimates and compares rational numbers using sense of the magnitudes and relative magnitudes of numbers and of base-ten place values (e.g., recognizes relationships to “benchmark” numbers and 1 to conclude that the sum + must be between 1 and 1 (likewise, + )).

Examples of activities through which students might demonstrate understanding of number and operation concepts include:
 Sara placed one number on each side of two disks. She said that if she flipped the disks and added the two numbers facing up, the sum was always one of the following numbers: 1, .23, .87, .1. What numbers could Sara have placed on each face of the two disks? (Balanced Assessment) 1a, 5a, 5b How can you compute a 15%, 10%, or 20% tip, other than by multiplying an amount by 0.15, 0.1, or 0.2 on paper? 1a, 1c, 1e Mrs. Brown wrote a number on the chalkboard and said, “I know that to be sure I find all the factor pairs for this number I must check each number from 1 to 29.” What numbers could Mrs. Brown have written? What is the smallest number Mrs. Brown could have written? What is the largest number Mrs. Brown could have written? (Middle Grades Mathematics Project: Factors and Multiples) 1b, 1d Burger Jack’s ads claim that their -pound burger contains 50% more beef than Winnie’s -pound burger. Is their claim true? 1c, 1e The members of the school marching band wanted to arrange themselves into rows with exactly the same number of band members in each row. They tried rows of two, three, and four, but there was always one band member left over. Finally, they were able to arrange themselves into rows with exactly five in each row. What is the least number of members in the marching band? (Creative Problem Solving in Mathematics) 1d Is the sum of two consecutive integers odd or even? Always? How about the product of two consecutive numbers? Why? What can you say about three consecutive numbers? 1d Find the last two digits of . (NCTM, Mathematics Teaching in the Middle School) 1d If, in a school of 1,000 lockers, one student opens every locker, another student closes every other locker (2nd, 4th, 6th, etc.), a third student changes every third locker (opens closed lockers and closes open lockers), and so on, until the thousandth student changes the thousandth locker, which lockers are open? Why? 1d, 5a, 5b, 5c A certain clock gains one minute of time every hour. If the clock shows the correct time now, when will it show the correct time again? (Creative Problem Solving in Mathematics) 1f, 2h How much space would a million shoe boxes fill? How large would a pile of a million pennies be? How about a million sheets of notebook paper? 1g How small would a millionth of a sheet of notebook paper be? 1g

 Geometry and Measurement Concepts Middle School

The student demonstrates understanding of a mathematical concept by using it to solve problems, by representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and by 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 in the following areas; that is, the student:
a Is familiar with assorted two- and three-dimensional objects, including squares, triangles, other polygons, circles, cubes, rectangular prisms, pyramids, spheres, and cylinders.

b Identifies similar and congruent shapes and uses transformations in the coordinate plane, i.e., translations, rotations, and reflections.

c Identifies three dimensional shapes from two dimensional perspectives; draws two dimensional sketches of three dimensional objects that preserve significant features.

d Determines and understands length, area, and volume (as well as the differences among these measurements), including perimeter and surface area; uses units, square units, and cubic units of measure correctly; computes areas of rectangles, triangles, and circles; computes volumes of prisms.

e Recognizes similarity and rotational and bilateral symmetry in two- and three-dimensional figures.

f Analyzes and generalizes geometric patterns, such as tessellations and sequences of shapes.

g Measures angles, weights, capacities, times, and temperatures using appropriate units.

h Chooses appropriate units of measure and converts with ease between like units, e.g., inches and miles, within a customary or metric system. (Conversions between customary and metric are not required.)

i Reasons proportionally in situations with similar figures.

j Reasons proportionally with measurements to interpret maps and to make smaller and larger scale drawings.

k Models situations geometrically to formulate and solve problems.

Examples of activities through which students might demonstrate understanding of geometry and measurement concepts include:
 Jan says rectangles and rhombi are completely different. Joan says they are almost exactly the same. Explain and illustrate why both students are partially correct. Compare and contrast the two figures. (College Preparatory Mathematics, “Rhombus and Rectangle”) Can you think of a special shape that is both a rectangle and a rhombus? 2a Jamaal installed two sprinkler heads to water a square patch of lawn twelve feet on a side. He placed the sprinkler heads in the middle of two opposite edges of the lawn. If the spray pattern of each sprinkler is semi-circular and just reaches the two nearest corners, what percentage of the lawn is not watered by the two sprinkler heads? (College Preparatory Mathematics, “Watering the Lawn”) 2a, 2d Examine logos of businesses in the yellow pages for rotational and bilateral symmetry. 2b, 2e From square grid paper, cut “jackets” for “space food packages,” which are actually blocks of cubes (same size as the squares of the grid); each cube contains one day’s supply of food pellets, and each square of “material” costs \$1. For different space armor jackets cut from the grid paper, find the number of days the food supply will last (the volume of the package). Also find the cost of the jacket (the surface area) and the dimensions of the package. (Middle Grades Mathematics Project: Mouse and Elephant: Measuring Growth) 2c A rectangular garden has two semicircular flower beds on opposite ends. Find the area of the entire figure. (College Preparatory Mathematics, “The Garden”) 2d

 Function and Algebra Concepts Middle School

This standard describes the foundation expected of middle school students in preparation for high school mathematics. Some students will take a course in algebra before high school, and their understanding of functions and algebra should surpass what is described below.

The student demonstrates understanding of a mathematical concept by using it to solve problems, by representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and by 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 Discovers, describes, and generalizes patterns, including linear, exponential, and simple quadratic relationships, i.e., those of the form f(n)=n² or f(n)=cn², for constant c, including A=r², and represents them with variables and expressions.

b Represents relationships with tables, graphs in the coordinate plane, and verbal or symbolic rules.

c Analyzes tables, graphs, and rules to determine functional relationships.

d Finds solutions for unknown quantities in linear equations and in simple equations and inequalities.

Examples of activities through which students might demonstrate understanding of function and algebra concepts include:
 Graph and explain the growth of population over time of a colony of organisms that doubles once a day. 3a Use diagrams, tables, graphs, words, and formulas to show the relationships between the length of the sides of a square and its perimeter and area. 3a, 3b A ball is dropped from a height of sixteen feet. At its first bounce, the ball reaches a peak of eight feet. Each successive time the ball bounces, it reaches a peak height that is half that of the previous bounce. How many times will the ball bounce until it bounces to a peak height of one foot? Make a table that shows the peak bounce height of the ball for the number of bounces. Make a graph and write an algebraic expression that show the relationship between the number of bounces and the peak height of the ball. (Balanced Assessment) 3b Examine areas that can be enclosed by 24 feet of fencing and figure out the maximum area. 3b Your principal wants to hire you to work for her for ten days. She will pay you either \$6.00 each day for all ten days; or \$1.00 the first day, \$2.00 the second day, \$3.00 the third day, and so on; or \$0.10 the first day and each day thereafter twice the amount of the day before. Under which arrangement would you earn the most money? Under which arrangement would you earn the least money? (NCTM, Mathematics Teaching in the Middle School) 3c Investigate the following situation: Bricklayers use the rule N=7·L·H to determine the number N of bricks needed to build a wall L feet long and H feet high. Examine a brick wall or portion of a brick wall to see whether this seems to be true. If it works, why? If not, what would be a better formula? (UCSMP, Transition Mathematics) 3c When inquiring about the dimensions of a lot, you are told, “I can’t remember. It’s shaped like a rectangle, and I know that they needed ninety meters of fencing to enclose it. Oh, yes! The crew putting up the fence remarked that the lot is exactly twice as long as it is wide.” What are the dimensions of the lot? (Creative Problem Solving in Mathematics) 3d

 Statistics and Probability Concepts Middle School

The student demonstrates understanding of a mathematical concept by using it to solve problems, by representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and by 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; that is, the student:
a Collects data, organizes data, and displays data with tables, charts, and graphs that are appropriate, i.e, consistent with the nature of the data.

b Analyzes data with respect to characteristics of frequency and distribution, including mode and range.

c Analyzes appropriately central tendencies of data by considering mean and median.

d Makes conclusions and recommendations based on data analysis.

e Critiques the conclusions and recommendations of others’ statistics.

f Considers the effects of missing or incorrect information.

g Formulates hypotheses to answer a question and uses data to test hypotheses.

h Represents and determines probability as a fraction of a set of equally likely outcomes; recognizes equally likely outcomes and constructs sample spaces (including those described by numerical combinations and permutations).

i Makes predictions based on experimental or theoretical probabilities.

j Predicts the result of a series of trials once the probability for one trial is known.

Examples of activities through which students might demonstrate understanding of statistics and probability concepts include:
 From a sample news headline, an article, and a table of data, select and construct appropriate graphs or other visual representations of the data. Decide whether or not the headline seems appropriate. Write a letter to the editor. (Balanced Assessment) 4a, 4d, 4e, 7e In a game to see who could best guess when 30 seconds had passed, the actual times (measured by a stopwatch) of Gilligan’s guesses were 31, 25, 32, 27, and 28 seconds; Skipper’s guesses were 37, 19, 40, 36, and 22 seconds; Ginger’s guesses were 32, 38, 24, 32, and 32 seconds. Who do you think is best at estimating 30 seconds? Why? (Balanced Assessment) 4b, 4c When tossing two fair numbered cubes and finding the sum of the two numbers turned up, if the sum is seven, then Keisha gets seven points, but if the sum is not seven, then Shawna gets one point. Is the game fair or not? Explain your reasoning to Keisha and Shawna. (Balanced Assessment) 4h Decide whether it is most advantageous to use three tetrahedra, two cubes, or one dodecahedron to arrive at a specific number (like 10) when rolling polyhedral dice. 4h, 1g Lottery players often say, “Well, my numbers have to come up sometime!” Analyze a state lottery game to see how many number combinations there are and how many weeks, months, or years it will take for all of them to be drawn. 4h, 4j A poll was taken of 40 students on their favorite school lunch. The results show hamburgers and fries, 14; pizza and salad, 13; spaghetti and salad, 8; hot dogs and beans, 5; liver and spinach, 0. Assuming this is an accurate sample, if a student is chosen at random, what is the probability that he or she favors each lunch offering? If there are 400 students in the school, how many prefer hamburgers? Hot dogs? Liver? (Middle Grades Mathematics Project: Probability) 4i

 Problem Solving and Mathematical Reasoning Middle School

The student 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 The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student:
• formulates and solves a variety of meaningful problems;
• extracts pertinent information from situations and figures out what additional information is needed.

Implementation
b The student makes the basic choices involved in planning and carrying out a solution; that is, the student:
• uses and invents a variety of approaches and understands and evaluates those of others;
• invokes problem solving strategies, such as illustrating with sense-making sketches to clarify situations or organizing information in a table;
• determines, where helpful, how to break a problem into simpler parts;
• solves for unknown or undecided quantities using algebra, graphing, sound reasoning, and other strategies;
• integrates concepts and techniques from different areas of mathematics;
• works effectively in teams when the nature of the task or the allotted time makes this an appropriate strategy.

Conclusion
c The student provides closure to the solution process through summary statements and general conclusions; that is, the student:
• verifies and interprets results with respect to the original problem situation;
• generalizes solutions and strategies to new problem situations.

Mathematical reasoning
d The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student:
• formulates conjectures and argues why they must be or seem true;
• makes sensible, reasonable estimates;
• makes justified, logical statements.

Examples of activities through which students might demonstrate problem solving and mathematical reasoning include:
 Aaron’s Pizzeria will pay you \$75 if you and a partner can eat his 26" diameter pizza in one hour. To practice for this eating event, you and your friend go to Round Label Pizza. How many of Round Label’s 12" diameter pizzas would the two of you need to eat to equal the amount of pizza you hope to eat at Aaron’s? (College Preparatory Mathematics, “Pizza Problem”) 5a, 5b, 2d Design a carnival game in which paper “parachutes” are dropped onto a target of five concentric rings, for which a player scores five points for landing in the center, four points in the next ring, three in the next, two in the next, and one point for landing in the outer ring. Experiment and create rules for the game so that it is fun, makes a profit, and people feel they have a chance of winning. (Balanced Assessment) 5a, 5b, 4a, 8d Your rich uncle has just died and has left you \$1 billion. If you accept the money, you must count it for eight hours a day at the rate of \$1 per second. When you are finished counting, the \$1 billion is yours, and then you may start to spend it. Should you accept your uncle’s offer? Why or why not? How long will it take to count the money? (College Preparatory Mathematics, “Big Bucks”) 5a, 5b, 5c, 1f, 1g Starting with the numbers from 1 to 36, find out all you can about writing them as sums of consecutive whole numbers. What kinds of numbers can be written as the sum of two consecutive whole numbers; of three consecutive whole numbers; of four consecutive whole numbers? Which numbers cannot be written as the sum of consecutive whole numbers? What patterns do you notice? Why do you think they occur? (Balanced Assessment) 5b, 5c, 5d, 1d A diagonal of the 3 x 5 grid rectangle passes through seven of the squares. In the 4 x 4 grid rectangle, the diagonal passes through only four squares. Come up with at least two conjectures about grid rectangles, diagonals, and the squares they pass through. (College Preparatory Mathematics, “On Grids”) 5d, 1d

 Mathematical Skills and Tools Middle School

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, or books, perhaps working to reconstruct a method); that is, the student:

a Computes accurately with arithmetic operations on rational numbers .

b Knows and uses the correct order of operations for arithmetic computations.

c Estimates numerically and spatially.

d Measures length, area, volume, weight, time, and temperature accurately.

e Refers to geometric shapes and terms correctly.

f Uses equations, formulas, and simple algebraic notation appropriately.

g Reads and organizes data on charts and graphs, including scatter plots, bar, line, and circle graphs, and Venn diagrams; calculates mean and median.

h Uses recall, mental computations, pencil and paper, measuring devices, mathematics texts, manipulatives, calculators, computers, and advice from peers, as appropriate, to achieve solutions.

Examples of activities through which students might demonstrate facility with mathematical skills and tools include:
 How (many ways) can you use four 4s to create an expression that has a value equal to 1? (e.g., = …) (Creative Problem Solving in Mathematics) 6b, 6h Figure out how long it would take to say your name a million times; how long it would take to count to a million. 6c, 1f, 1g, 5a, 5b, 5c Accurately describe a geometric design on a 10 x 10 grid to a friend by telephone. (Balanced Assessment) 6e, 7c Use the formula A= bh for areas of triangles measured with customary and metric rulers. 6f Analyze advertisements for different music clubs and decide which offers best value for money. 6g

 Mathematical Communication Middle School

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 mathematical language and representations with appropriate accuracy, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs, and diagrams.

b Organizes work, explains facets of a solution orally and in writing, labels drawings, and uses other techniques to make meaning clear to the audience.

c Uses mathematical language to make complex situations easier to understand.

d Exhibits developing reasoning abilities by justifying statements and defending work.

e Shows understanding of concepts by explaining ideas not only to teachers and assessors but to fellow students or younger children.

f Comprehends mathematics from reading assignments and from other sources.

Examples of activities through which students might demonstrate facility with mathematical communication include:
 Use diagrams, tables, graphs, words, and formulas to show the relationship of the length of the sides of a square to its perimeter and area. 7a Use box-and-whiskers plots, stem-and-leaf plots, and bar graphs to compare characteristics of the boys and girls in the class; compare the kinds of information provided by the different displays. 7a, 7b Use symbols and a Cartesian map to explain to another student how to get from your home to school. 7c Make the following conjectures: What happens to the area of a square when you double its perimeter? What happens to the area when you triple its perimeter? Investigate to see if this is true and, if so, explain why. What does doubling the circumference of a circle do to its area? Explain. 7d, 2d, 3b Your fifth grade cousin is convinced that the probability of rolling a 12 on two numbered cubes is . Explain to your cousin why this is incorrect, and convince your cousin of the actual probability of getting 12. 7e, 4h

 Putting Mathematics to Work Middle School

The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of middle school, conducts investigations or projects drawn from three of the kinds.
A single investigation or project may draw on more than one kind.

a Data study based on civic, economic, or social issues, in which the student:
• selects an issue to investigate;
• makes a hypothesis on an expected finding, if appropriate;
• gathers data;
• analyzes the data using concepts from Standard 4, e.g., considering mean and median, and the frequency and distribution of the data;
• shows how the study’s results compare with the hypothesis;
• uses pertinent statistics to summarize;
• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.

b Mathematical model of physical phenomena, often used in science studies, in which the student:
• carries out a study of a physical system using a mathematical representation of the structure;
• uses understanding from Standard 3, particularly with respect to the determination of the function governing behavior in the model;
• generalizes about the structure with a rule, i.e., a function, that clearly applies to the phenomenon and goes beyond statistical analysis of a pattern of numbers generated by the situation;
• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.

c Design of a physical structure, in which the student:
• generates a plan to build something of value, not necessarily monetary value;
• uses mathematics from Standard 2 to make the design realistic or appropriate, e.g., areas and volumes in general and of specific geometric shapes;
• summarizes the important features of the structure;
• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.

d Management and planning, in which the student:
• determines the needs of the event to be managed or planned, e.g., cost, supply, scheduling;
• notes any constraints that will affect the plan;
• determines a plan;
• uses concepts from any of Standards 1 to 4, depending on the nature of the project;
• considers the possibility of a more efficient solution;
• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the plan.

e Pure mathematics investigation, in which the student:
• extends or “plays with,” as with mathematical puzzles, some mathematical feature, e.g., properties and patterns in numbers;
• uses concepts from any of Standards 1 to 4, e.g., an investigation of Pascal’s triangle would have roots in Standard 1 but could tie in concepts from geometry, algebra, and probability; investigations of derivations of geometric formulas would be rooted in Standard 2 but could require algebra;
• determines and expresses generalizations from patterns;
• makes conjectures on apparent properties and argues, short of formal proof, why they seem true;
• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.

Examples of investigations or projects include:
 Gather and analyze data from the neighborhood and compare the data with published statistics for the city, state, or nation. 8a, 4a, 4b, 4c, 4d 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, 3b, S2a, S3d Design and equip a recreational area on one acre with a limited budget. 8c, 1a, 2a, 2d, 2h, 2j, A1a Analyze and concoct games of chance for a school carnival. 8d, 4h, 4i, A1c Discover relationships among, and properties of, the numbers in Pascal’s triangle. Read to find more relationships and properties. 8e, 1e, 4h, 7f