By the end of kindergarten, students understand the consistency of small numbers, quantities, and simple shapes in their everyday environment. They count, compare, describe and sort objects, and develop a sense about properties and patterns.

NUMBER SENSE

FUNCTIONS/ALGEBRA

1. Students sort and classify objects.
1. identify, sort and classify objects by attribute and identify objects that do not belong to a particular grouping

MEASUREMENT

1. Students understand that there are properties such as length, weight, capacity and time and that comparisons can be made by using these properties.
   1. compare the length, weight and capacity of objects by making direct comparisons or using reference objects (e.g., shorter/longer/taller, lighter/heavier, which holds more?)
   2. demonstrate understanding of concepts of time (e.g., morning, afternoon, evening, day, yesterday, tomorrow, week, year) and the purpose of tools that measure time (e.g., clock, calendar)
   3. name the days of the week

GEOMETRY

1. Students identify common geometric objects in their environment and describe their features.
2. identify and describe common geometric objects (e.g., circle, triangle, square, rectangle, cube, sphere, cone, cylinder)
3. compare familiar plane and solid objects by common attributes (e.g., position, shape, size, roundness, number of corners)

STATISTICS AND PROBABILITY

1. Students collect information about objects and events in their environment.
1. collect data and record the results using objects, pictures and picture graphs
2. identify, describe and extend simple patterns involving shape, size, or color such as circle, triangle, or red, blue

MATHEMATICAL REASONING

1. Students make decisions about how to set up a problem.
1. decide about the approach, materials and strategies to use
2. use tools, manipulatives or sketches to model problems
2. Students solve problems in reasonable ways
1. explain the reasoning used with concrete objects and pictorial representations

By the end of first grade, students understand and use the concept of "tens" and "ones" in the place value number system. They add and subtract small numbers with ease. They measure with simple units and locate objects in space. They describe data and analyze and solve simple problem situations.

NUMBER SENSE

1. Students understand and use numbers up to 100.
   1. count, read and (using numerals) write whole numbers to 100
   2. compare and order whole numbers to 100 using the symbols for less than, equal to, or greater than (<,=,>)
   3. represent equivalent forms of the same number through the use of physical models, diagrams and number expressions (to 20) (e.g., 8: 4+4, 5+3, 2+2+2+2, 10-2, 11-3)
   4. identify and know the value of coins
2. Students demonstrate the meaning of addition and subtraction and use these operations to solve problems.
   1. know addition facts (sums to 10) and the corresponding subtraction facts
   2. identify one more than, one less than, a given number
   3. count by 2s, 5s and 10s with numbers to 100
   4. show the meaning of addition (combining, increasing) and subtraction (taking away, comparing, finding the difference)
   5. solve addition and subtraction to problem with one- and two-digit numbers with no regrouping (e.g., 5+61=__)
   6. find the sum of three one-digit numbers
3. Students use estimation strategies in computation and problem solving that involve numbers that use the one, tens.

FUNCTIONS/ALGEBRA

1. Students use number sentences to solve problems.
   1. write and solve number sentences from problem situations that express relationships involving addition and subtraction
   2. understand the meaning of the symbols +, -, =
   3. create problem situations that could lead to given number sentences involving addition and subtraction

MEASUREMENT

1. Students use direct comparison and non-standard units to describe the measurements of objects.
   1. compare the length, weight and volume of two or more objects using direct comparison or a non-standard unit
   2. tell time to the nearest hour and half hour, and compare time related to events (e.g., before/after, shorter/longer)

GEOMETRY

1. Students identify common geometric figures, classify them by common attributes
   1. identify rectangles, squares and circles, and the faces of three-dimensional objects
   2. classify familiar plane and solid objects by common attributes, such as color, position, shape, size, roundness, number of corners
   3. follow directions about location
   4. describe and arrange objects in space in terms of proximity, position and direction (e.g., near, far, below, above, up, down, behind, in front of, next to, left/right)

STATISTICS AND PROBABILITY

1. Students organize, represent and compare categorical data on simple graphs and charts.
   1. sort objects and data by common attributes and describe the groups formed using categorical labels
   2. represent and compare data (largest, smallest, most often, least often) using pictures, bar graphs, tally charts and picture graphs
2. Students sort objects, and create and describe patterns involving numbers, shape, size, rhythm, or color.
   1. describe, extend and explain how to get to the next element in simple repeating patterns (rhythmic, numeric, color and shape patterns)

MATHEMATICAL REASONING

1. Students make decisions about how to set up a problem.
   1. decide about the approach, materials and strategies to use
   2. use tools, such as manipulatives or to model problems
2. Students solve problems and justify their reasoning.
   1. use estimation to verify the reasonableness of calculated results
   2. explain the reasoning used and justify the procedures selected
   3. make precise calculations and check the validity of the results from the context of the problem

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By the end of second grade, students understand place value and number relationships as they add and subtract and use simple concepts of multiplication. They measure quantities with appropriate units. They classify and see relationships among shapes by paying attention to the elements that compose them. They collect and analyze data and verify answers.

NUMBER SENSE

1. Students understand the relationship among numbers, quantities and place value in whole numbers up to 1000
   1. count, read, write in numerals whole numbers to 1,000 and identify the place value for each digit
   2. use words, models and expanded form to represent numbers (to 1000)
   3. order and compare whole numbers up to 1,000 using the symbols <,=,>
   4. count and group objects into tens and ones and use words, models and symbols to represent the number (e.g., 3 groups of ten and 4 more is 34 or 30+4)
   5. show different combinations of coins that equal the same value
2. Students estimate, calculate and solve problems involving addition and subtraction of two- and three-digit numbers.
   1. understand and use the inverse relationship between addition and subtraction (e.g., an opposite number sentence for 8+6=14 is 14-6=8) to solve problems and check solutions
   2. find the sum or difference of two whole numbers up to three digits long
   3. use the inverse relationship between addition and subtraction to solve problems and commit them to memory
3. Students understand that fractions can refer to parts of a set and parts of a whole.
   1. recognize, name and compare unit fractions up to 1/10
   2. recognize fractions of a whole and parts of a group (e.g., 1/4th of a pie, 2/3'rds of 15 balls)
   3. know that when all fractional parts are included, the result is equal to the whole
4. Students model and solve problems by representing, adding and subtracting amounts of money.
   1. solve problems using combinations of coins and bills
   2. know and use the decimal notation and the dollar and cents symbols for money

FUNCTIONS/ALGEBRA

1. Students model, represent and interpret number relationships to create and solve problems involving addition and subtraction.
   1. use the commutative and associative properties to simplify and check results
   2. match mathematical problems and number sequences involving addition and subtraction
   3. solve addition and subtraction problems using data from simple charts, picture graphs and number sentences

MEASUREMENT

1. Students understand that measurement is accomplished by identifying a unit of measure, repeating that unit and comparing it to the item to be measured.
   1. measure the length of objects by repeating a non-standard or standard unit
   2. use different units to measure the same object and predict whether the measure will be greater or smaller when a different unit is used
   3. measure the length of an object to the nearest inch and/or centimeter
   4. tell time to the nearest quarter hour and know time relationships (minutes in an hour, days in a month, weeks in a year)

GEOMETRY

1. Students identify and describe the elements that compose common figures in the plane and common objects in space.
   1. describe and classify plane and solid geometric shapes (e.g., circle, triangle, square, rectangle, sphere, pyramid, cube, rectangular prism) according to the number and shape of faces, edges and vertices
   2. put shapes together and take them apart to form other shapes (e.g., two congruent right triangles can form a rectangle)

STATISTICS AND PROBABILITY

1. Students collect, record, organize, display and interpret numerical data on bar graphs and other representations.
   1. record numerical data in systematic ways, keeping track of what/who has been counted
   2. represent the same data set in more than one way (e.g., charts with tallies, and bar graphs)
   3. identify features of data sets (range and mode)
   4. ask and answer simple questions related to data representations
2. Students demonstrate an understanding of patterns and how they repeat and describe them in general ways.
   1. recognize, describe, extend and explain how to get the next term in linear patterns
   2. solve problems involving simple number patterns

MATHEMATICAL REASONING

1. Students make decisions about how to set up a problem.
   1. decide about the approach, materials and strategies to use
   2. use tools, such as manipulatives or sketches to model problems
2. Students solve problems and justify their reasoning.
   1. use estimation to verify the reasonableness of calculated results
   2. explain the reasoning used and justify the procedures selected
   3. make precise calculations and check the validity of the results from the context of the problem
3. Students note connections between one problem and another.

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By the end of third grade, students deepen their understanding of place value and their understanding of and skill with addition, subtraction, multiplication and division of whole numbers. They estimate, measure and describe objects in space. They use patterns to solve problems. They represent number relationships and conduct simple probability experiments.

NUMBER SENSE

1. Students understand place value of whole numbers.
   1. count, read, write in numerals whole numbers to 100,000
   2. compare and order whole numbers to 100,000
   3. identify the place value for each digit in numbers to 10,000
   4. round off numbers to 100,000 to the nearest ten, hundred and thousand
   5. use expanded notation to represent numbers (e.g., 3200 = 3000 + 200 + 6)
2. Students calculate and solve problems involving addition, subtraction, multiplication and division
1. find the sum or difference of two whole numbers between 0 and 10,000
2. memorize to automaticity the multiplication table for numbers between 1 and 12
3. use the inverse relationship of multiplication and division to compute and check results
4. solve simple problems involving multiplication of two-digit numbers by one-digit numbers
5. solve division problems in which a two-digit number is evenly divided by a one-digit number
6. understand the special properties of 0 and 1 in multiplication and division
7. determine the unit cost when given the total cost and number of units (two-digit by one-digit division)
8. solve problems which combine two or more of the skills above
3. Students understand the relationship between whole numbers, simple fractions and decimals.
1. compare fractions represented by drawings or concrete materials to show equivalency, and to add and subtract simple fractions in context (e.g., _ of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is more than 1/8)
2. add and subtract simple fractions with common denominators (e.g., determine that 1/8 + 3/8 ???
3. solve problems involving the addition and subtraction of money amounts in decimal notation
4. solve problems involving the multiplication and division of money amounts in decimal notation using whole number multipliers and divisors ($.90 x 6)
5. know and understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents is _ of a dollar, 75 cents is 3/4 of a dollar)

FUNCTIONS/ALGEBRA

1. Students select appropriate symbols, operations and properties to represent, describe, simplify and solve simple number relationships.
   1. represent relationships of quantities in the form of mathematical expressions, equations, or inequalities (3 < 4; 3 + 1 5)
   2. solve problems involving numeric equations or inequalities
   3. select appropriate operational and relational symbols to make an expression true (e.g., 4 __ 3 = 12, what operation symbol goes in the blank?)
   4. express simple unit conversions in symbolic form (e.g., #inches = #feet x 12)
   5. recognize the commutative and associative properties of multiplication (if 5 x 7 = 35, then what is 7 x 5? If 7 x 15 = 105, then what is 7 x 3 x 5?)
2. Students represent simple functional relationships.
   1. solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the per unit cost)
   2. extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses can be calculated by counting by 4's or by multiplying the number or horses by 4)

MEASUREMENT AND GEOMETRY

1. Students choose and use appropriate units and measurement tools to quantify the properties of objects.
   1. choose appropriate units (metric and U.S. customary) and tools, and estimate and measure length, liquid volume and weight
   2. estimate or determine the area and volume of plane and solid figures, respectively, by covering them with squares or counting the number of cubes that fill them
   3. find the perimeter of any polygon using standard measurements
   4. carry out simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes)
2. Students describe and compare the attributes of plane and solid geometric figures and use their understanding to show relationships and solve problems.
   1. identify and describe and classify polygons (including pentagons, hexagons and octagons)
   2. identify different types of triangles (e.g., two equal sides for the isosceles triangle, three equal sides for the equilateral triangle, right angle for the right triangle)
   3. identify different types of quadrilaterals (e.g., parallel sides for the parrelogram, right angles for the rectangle, equal sides and right angles for the square)
   4. identify right angles in geometric figures or in appropriate real objects and determine whether other angles are greater or less than a right angle
   5. identify and describe common three-dimensional geometric objects (cube, rectangular solid, sphere, prism, pyramid, cone, cylinder)
   6. identify the common solid objects that are the component parts needed to make a more complex solid object

STATISTICS AND PROBABILITY

1. Students conduct simple probability experiments by determining the number of possible outcomes, and make simple predictions.
   1. identify whether common events are certain, likely, or unlikely
   2. record the possible outcomes for a simple event (e.g., tossing a coin) and systematically keep track of the outcomes when the event is repeated many times
   3. summarize and display the results of probability experiments in a clear and organized way (e.g., use a bar graph or a line plot)
   4. use the results of probability experiments to predict future events ( e.g., a line plot to predict the temperature forecast for the next day)

MATHEMATICAL REASONING

1. Students make decisions about how to approach problems.
   1. analyze problems by identifying relationships, discriminating relevant from irrelevant information, sequencing and prioritizing information, and observing patterns
   2. determine when and how to break a problem into simpler parts
2. Students use strategies, skills and concepts in finding solutions.
   1. use estimation to verify the reasonableness of calculated results
   2. apply strategies and results from simpler problems to more complex problems
   3. use a variety of methods such as words, numbers, symbols, charts, graphs, tables, diagrams and models to explain mathematical reasoning
   4. express solutions clearly and logically using appropriate mathematical notation, terms and clear language, and support solutions with evidence, in both verbal and symbolic work
   5. make precise calculations and check the validity of the results from the context of the problem
3. Students move beyond a particular problem by generalizing to other situations.
   1. evaluate the reasonableness of the solution in the context of the original situation
   2. develop generalizations of the results obtained and extend them to other circumstances

Return to Table of Contents

By the end of fourth grade, students understand large numbers and addition, subtraction, multiplication and division of whole numbers. They describe and compare simple fractions and decimals. They understand the relationship between area and perimeter and the properties of and the relationships between plane geometric figures. They collect, represent and analyze data to answer questions.

NUMBER SENSE

1. Students understand place value of whole numbers and decimals to two decimal places, and how these relate to simple fractions.
   1. read and write whole numbers in the millions
   2. use mental arithmetic to find the sum or difference of two 2-digit numbers
   3. order and compare whole numbers and decimals to two decimal places
   4. round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand or hundred thousand
   5. decide when a rounded solution is appropriate and justify it
   6. write and recognize simple fractions and decimals as equivalents
   7. write the fraction represented by a drawing; represent a given fraction using drawings
2. Students extend their use and understanding of whole numbers to addition and subtraction of simple decimals.
   1. compute the sum or difference of whole numbers and positive decimals
   2. round two place decimals to one decimal place or the nearest whole number
3. Students solve problems involving addition, subtraction, multiplication and division of whole numbers and understand the relationships among the operations.
   1. demonstrate their use of multi-step addition and subtraction
   2. estimate and solve problems involving multiplication of multi-digit numbers by two-digit numbers
   3. estimate and solve problems involving division of multi-digit numbers by one-digit numbers
4. Students know how to factor small whole numbers.
   1. whether a number is a factor or multiple of a given whole number, they understand that many whole numbers can be factored in different ways
   2. they know that numbers such as 2,3,5,7,11 do not have any factors except 1 and themselves, and that such numbers are called primes and the others are composite numbers

FUNCTIONS/ALGEBRA

1. Students use and interpret variables, mathematical symbols and properties to write and simplify expressions and sentences.
   1. demonstrate understanding and use of concept of a variable
   2. apply the order of operations
   3. build an understanding for basic geometric shapes and formulas

MEASUREMENT AND GEOMETRY

1. Students understand perimeter and area
   1. measure the area of rectangular shapes, using appropriate units
   2. understand that the rectangles having the same area can have different perimeters and rectangles having the same perimeter can have different areas
2. Students demonstrate understanding plane and solid geometric objects. They use this knowledge to show relationships and solve problems.
   1. identify lines that are parallel and perpendicular
   2. identify the radius and diameters of a circle
   3. identify congruent figures
   4. identify figures that have bilateral and rotational symmetry
   5. know the definitions of right angle, acute angle and obtuse angle
   6. visualize, describe geometric solids (e.g., prisms, pyramids, etc.) in terms of the number and shape of faces, edges and vertices; interpret two-dimensional representations of three-dimensional objects
   7. know the definition of different triangles (equilateral, isosceles, scalene)
   8. know the definition of different quadrilaterals (rhombus, square, rectangle, parallelogram, trapezoid)

STATISTICS AND PROBABILITY

1. Students make predictions for simple probability situations.
   1. represent all possible outcomes for a simple probability situation in an organized way (e.g., tables, grids, tree diagrams)
   2. express outcomes of experimental probability situations verbally and numerically (e.g., 3 out of 4; 3/4)

MATHEMATICAL REASONING

1. Students make decisions about how to approach problems.
   1. analyze problems by identifying relationships, discriminating relevant from irrelevant information, sequencing and prioritizing information, and observing patterns
   2. determine when and how to break a problem into simpler parts
2. Students use strategies, skills and concepts in finding solutions.
   1. predict outcomes and make reasonable estimates
   2. apply strategies and results from simpler problems to more complex problems
   3. use a variety of methods such as words, numbers, symbols, charts, graphs, tables, diagrams and models to explain mathematical reasoning
   4. express the solution clearly and logically using appropriate mathematical notation and terms and clear language, and support solutions with evidence, in both verbal and symbolic work
   5. indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy
3. Students move beyond a particular problem by generalizing to other situations
   1. evaluate the reasonableness of the solution in the context of the original situation
   2. note method of deriving the solution and demonstrate conceptual understanding of the derivation by solving similar problems
   3. develop generalizations of the results obtained and extend them to other circumstances

Return to Table of Contents

By the end of fifth grade, students understand addition, subtraction, multiplication and division of whole numbers, fractions and decimals. They understand the relationship between area and perimeter and the properties of, and the relationships between, basic geometric figures. They collect, represent and analyze data to answer questions.

NUMBER SENSE

1. Students compute numbers, positive and negative numbers, decimals and fractions and understand the relationship between decimals, fractions and percents.
   1. estimate, round and manipulative very large (e.g., millions) and very small (e.g., thousandths) numbers
   2. determine the prime factors of all numbers through 50 and write numbers as the product of their prime factors
   3. use simple concepts of negative numbers (e.g., on a number line, in counting, in temperature, and in "owing")
   4. identify the relative position of fractions, mixed numbers, and decimals to two decimal places on the number line

FUNCTIONS/ALGEBRA

1. Students perform calculations and solve problems involving addition, subtraction and simple multiplication and division of fractions and decimals.
   1. add, subtract, multiply and divide with whole numbers, decimals and fractions
   2. become proficient with division, including division with positive decimals and long division with multiple digit divisors
   3. solve simple problems involving the addition and subtraction of fractions and mixed numbers and express answers in simplest form
   4. understand the concept of multiplication and fractions
2. Students use variables in simple expressions, compute the value of the expression for specific values of the variable.
   1. interpret information on a graph or in an equation
   2. use a letter to represent an unknown number; write and solve simple algebraic expressions
   3. know and use the distributive property in equations and expressions with variables

MEASUREMENT AND GEOMETRY

1. Students understand and compute volumes and areas of simple objects and use this understanding to solve problems.
   1. use and interpret formulas for basic geometric shapes
   2. understand and use formulas to solve problems involving perimeters and areas of rectangles and squares
   3. construct cubes and rectangular boxes from two-dimensional patterns and use this to compute the surface area for these objects
   4. understand the concept of volume and use the appropriate units
   5. differentiate between and use appropriate units of measures for, two- and three- dimensional objects (perimeter, area, and volume)
   6. measure, identify and draw angles, perpendicular and parallel lines, rectangles and triangles, using appropriate tools (e.g., straight edge, ruler, compass, protractor and drawing software)
   7. draw two-dimensional views of three-dimensional objects made from rectangular solids
2. Students use two-dimensional coordinate grids to find locations and represent points and graph lines and simple figures.
   1. understand that the length of a horizontal line equals the difference of the x- coordinates and that the length of a vertical line equals the difference of the y-coordinates
3. Students will become familiar with units of time.
   1. convert basic units of time (i.e., 60 sec. = 1 min., 60 min. = 1 hr., 24 hr. = 1 day)
   2. will be able to determine elapsed time to the quarter hours

STATISTICS AND PROBABILITY

1. Students display, analyze, compare and interpret different data sets.
   1. understand the concept of average
   2. organize and display single-variable data in simple graphs and representations
   3. represent data on a number line, coordinate graphs, tables and charts
   4. interpret one- and two-variable data graphs to answer questions about a situation

MATHEMATICAL REASONING

1. Students make decisions about how to approach problems.
   1. analyze problems by identifying relationships, discriminating relevant from irrelevant information, sequencing and prioritizing and observing patterns
   2. determine when and how to break a problem into simpler parts
2. Students use strategies, skills, and concepts in finding solutions.
   1. predict outcomes and make reasonable estimates
   2. apply strategies and results from simpler problems to more complex problems
   3. use a variety of methods such as words, numbers, symbols, pictures, charts, graphs, tables, diagrams and models to explain mathematical reasoning
   4. express the solution clearly and logically using appropriate mathematical notation and terms and clear language, and support solutions with evidence, in both verbal and symbolic work
   5. make precise calculations and check the validity of the results from the context of the problem
3. Discrete - Students use strategies to find and understand combinations.
   1. use estimation to find the possible number of combinations
   2. use a variety of methods to show the possible combinations

By the end of sixth grade, students will understand and accurately compute and solve problems with whole numbers, fractions, and decimals. They begin using letters for numbers in formulas. They solve 1-step linear equations.

NUMBER SENSE

1. Students compare and order fractions, decimals, and mixed numbers.
   1. compare and order fractions, decimals, and mixed numbers, place them on a number line
2. Students calculate and solve problems involving addition, subtraction, multiplication and division of whole numbers, fractions and decmals.
   1. solve problems involving addition, subtraction, multiplication and division of  fractions and explain why a particular operation was used for a given situation
   2. explain the meaning of multiplication and division of fractions and perform the calculations
   3. determine the least common multiple and greatest common factor of whole numbers. Use them to solve problems.
   4. raise whole numbers to whole number powers to cubic powers.
   5. determine the prime factors of all numbers and write numbers as the product of their prime factors, using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3)
   6. become proficient with division, including long division with multiple digit divisors
   7. become proficient with multiplication (multi-digit by multi-digit)
   8. solve simple problems including ones arising in concrete situations involving the addition and subtraction of fractions and mixed numbers and express answers in simplest form
   9. understand the concept of multiplication and division of fractions

FUNCTIONS/ALGEBRA

1. Students use variables in simple expressions; compute the value of the expression for specific values of the variable.
   1. use information taken from a graph or equation to answer questions about a problem situation
2. Students will solve simple linear equations.
   1. write and solve one-step linear equations in one variable
   2. apply order of operations and the commutative, associative and distributive properties to evaluate expressions and justify each step in the process
   3. solve problems using correct order of operations manually and by using a calculator

MEASUREMENT AND GEOMETRY

1. Students identify, describe, draw and classify properties of, and relationships between, plane and solid geometric figures.
   1. measure, identify and draw angles, perpendicular and parallel lines, rectangles and triangles, using appropriate tools (e.g., straight edge, ruler, compass, protractor and drawing software)
   2. know that the sum of angles of any triangle is 180 and the sum of the angles of any quadrilateral is 360 and use this information to solve problems
   3. visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids
2. Students compute the perimeter and area of simple geometric figures.
   1. know the formulas for perimeter and area for rectangle and square
   2. use formulas to compute perimeter and area for rectangle and square
3. Students understand and compute length conversions given in different units.
   1. know the commonly used units of length and distance (e.g., inches, feet, yards, miles)
   2. length and distance conversions are given in different units (e.g., between inches and feet, between feet and yards)

STATISTICS AND PROBABILITY

1. Students will identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph.
   1. know how to write ordered pairs correctly
2. Students display, analyze, compare and interpret different data sets.
   1. know the concepts of average (mean, median, and mode)
   2. organize and display single-variable data in appropriate graphs and representations, and explain which types of graphs are appropriate for different kinds of data sets
   3. identify the mode and median for given sets of data
   4. compute the mean for given sets of data

MATHEMATICAL REASONING

1. Students make decisions about how to approach problems.
   1. analyze problems by identifying relationships, discriminating relevant from irrelevant information, identifying missing information, sequencing and prioritizing information and observing patterns
   2. determine when and how to break a problem into simpler parts
2. Students use strategies, skills and concepts in finding solutions.
   1. predict outcomes and make reasonable estimates
   2. apply strategies and results from simpler problems to more complex problems
   3. solve for unknown or undecided quantities using algebra, graphing, sound reasoning and other strategies
   4. use a variety of methods such as words, numbers, symbols, pictures, charts, graphs, tables, diagrams and models to explain mathematical reasoning
   5. express the solution clearly and logically using appropriate mathematical notation and terms and clear language, and support solutions with evidence, in both verbal and symbolic work
   6. make precise calculations and check the validity of the results from the context of the problem
3. Students move beyond a particular problem by generalizing to other situations.
   1. evaluate the reasonableness of the solution in the context of the original situation
   2. note patterns in the solution and use these patterns to extend the solution to similar problems
   3. develop generalizations of the results obtained and the strategies used and extend them to new problem situations

Return to Table of Contents

In seventh grade, students are preparing for pre-algebra, getting considerable practice at manipulating numbers and equations with an emphasis on the general principles at work. Fractions are added, subtracted, multiplied, divided, and simplified using factoring of numerator and denominators. Students change from one representation of rational numbers (fractions, decimals, and percent) to another. Students will compute with the four arithmetic operations with positive and negative numbers. Students learn the concepts of and how to calculate the range, mean, median and mode of data. They relate statistics to daily life, analyzing data and sampling processes for possible bias and misleading conclusions. The other area of emphasis is the concept and computation of ratios and proportions, including computing percentages (e.g., tax, tips, interest). Students learn about pi and the formulas for the circumference and area of a circle.

NUMBER SENSE

1. Students will understand the relationship between decimals, fractions, and percents.
   1. interpret percents as part of a hundred; find decimal and percent equivalents for common fractions; explain why they represent the same value
   2. identify and represent decimals, fractions, and mixed numbers on a number line
2. Students compute problems involving fractions, decimals, ratios, proportions, and percentages.
   1. interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities using appropriate notations (a:b, a to b, a/b)
   2. use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon)
   3. use cross-multiplication as a method for solving problems (understanding if as as multiplication of both sides of an equation by a multiplicative inverse)
   4. add and subtract fraction using factoring to find common denominators
   5. multiply and divide fractions and simplify fractions by using factoring
   6. master the concept of multiplying and dividing with decimals
   7. compute a given percent of a whole number
3. Students will understand and compute problems with positive and negative integers in all four operations.
   1. identify and represent positive and negative integers on a number line
   2. compare and order positive and negative integers
   3. interpret and apply the rules of operation for positive and negative integers
4. Students will understand and compute squares and cubes of non-negative whole numbers.
   1. interpret and compute examples as repeated multiplication
   2. read and write exponential expressions

FUNCTIONS/ALGEBRA

1. Students use variables in simple expressions, compute the value of the expression for specific values of the variable, and plot and interpret the results.
   1. use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable for a given situation
   2. identify and graph ordered pairs in the four quadrants of the coordinate plane
2. Students solve problems involving rates and proportions.
   1. convert from one unit of measurement to another (e.g., from inches to feet, from centimeters to meters)
   2. demonstrate understanding that rate is a measure of one quantity per unit value of another quantity (e.g., scale on a map)
   3. solve simple problems involving rates, average speed, distance, and time
3. Students will write verbal expressions and sentences as algebraic expressions and equations; they will evaluate algebraic expressions and solve linear equations.
   1. write and solve linear equations with one variable
   2. write and evaluate an algebraic expression for a given situation using one variable
   3. apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions and justify each step in the process
   4. solve problems using correct order of operations manually and by using a calculator

MEASUREMENT AND GEOMETRY

1. Students compute the perimeter, circumference, and area of common geometric objects.
   1. routinely use formulas for finding the perimeter, circumferences, and areas of basic two-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles and circles
2. Students deepen their understanding of measurement of plane and solid shapes and use this understanding to solve problems.
   1. understand the formula for the circumference and area of a circle
   2. know common estimates of pi (3.14 or 22/7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements
3. Students understand and compute time conversions given in different units.
   1. know the commonly used units of time (e.g., minutes, hours, days, year)
   2. time conversions are given in different units (e.g., between minutes and hours, between hours and days)
4. Students develop an understanding of two and three dimensional geometric figures.
   1. identify and characterize various geometric figures
   2. construct geometric figures
   3. measure figures with a straightedge to 1/4" and/or 5 mm
   4. become familiar with a compass in constructing figures

MATHEMATICAL REASONING

1. Students make decisions about how to approach problems.
   1. analyze problems by identifying relationships, discriminating relevant from irrelevant information, identifying missing information, sequencing and prioritizing information and observing patterns
   2. determine when and how to break a problem into simpler parts
2. Students use strategies, skills and concepts in finding solutions.
   1. predict outcomes and make reasonable estimates
   2. apply strategies and results from simpler problems to more complex problems
   3. solve for unknown or undecided quantities using algebra, graphing, sound reasoning and other strategies
   4. use a variety of methods such as words, numbers, symbols, pictures, charts, graphs, tables, diagrams and models to explain mathematical reasoning
   5. express the solution clearly and logically using appropriate mathematical notation and terms and clear language, and support solutions with evidence, in both verbal and symbolic work
   6. make precise calculations and check the validity of the results from the context of the problem
3. Students determine solution is complete and move beyond a particular problem by generalizing to other situations.
   1. evaluate the reasonableness of the solution in the context of the original situation
   2. note patterns in the solution and use these patterns to extend the solution to similar problems
   3. develop generalizations of the results obtained and the strategies used and extend them to new problem situations

STATISTICS AND PROBABILITY

1. Students display, analyze, compare, and interpret different data sets, including data sets that are not the same size.
   1. know the concepts of mean, median, and mode; compute and compare them in simple examples and notice that they can differ
   2. organize and display single-variable data in appropriate graphs and representations (e.g., histogram, line graphs, bar graphs), and explain which types of graphs are appropriate for different kinds of data sets
   3. understand how additional data added to data sets can effect these computations of measures of central tendency
   4. understand how the inclusion or exclusion of outliers affect measure of central tendency
   5. know why a specific measure of central tendency (mean, median, mode) provides the most useful information in a given context
2. Students use data samples of a population and describe the characteristics and limitations of the samples
   1. compare different samples from a population with the data from the entire population, and identify when it makes sense to use a sample
   2. identify different ways of selecting a sample (e.g., random sampling, convenience sampling, response to a survey) and which makes a sample more representative for a population
   3. analyze data displays and explain how the way the question was asked might have influenced the results obtained, and/or how the way the results were displayed might have influenced the conclusions reached
   4. identify data that represent sampling and explain why the sample (and the display) may be biased
   5. identify claims based on statistical data and, in simple cases, evaluate the validity of the claims

GRADE 8

By the end of eighth grade, students understand, use and connect a variety of techniques for solving equations. They understand the meaning of variables, expressions and equations. Students evaluate, graph and interpret the graphs of a variety of functions. Students apply proportional reasoning to solve problems involving scale drawings and similar figures, and connect geometric situations to algebraic and numerical situations.

NUMBER SENSE

1. Students know the properties of and compute with rational numbers expressed in a variety of forms.
   1. read, write and compare rational numbers in scientific notation (positive and negative powers of 10), approximate numbers using scientific notation
   2. add, subtract, multiply and divide rational numbers, integers, fractions and decimals and take rational numbers to whole number powers
   3. convert fractions to decimals and percents and use these representations in estimation, computation and applications
   4. differentiate between rational and irrational numbers
   5. know that every fraction is either a terminating or repeating decimal and be able to convert terminating decimals into reduced fractions
   6. calculate percent of increases and decreases of a quantity
   7. solve problems that involve discounts, markups, commissions, profit and simple compound interest
2. Students use exponents, powers, and roots and use exponents in working with fractions
   1. understand negative whole number exponents; add, subtract, multiply and divide expressions involving exponents with a common base
   2. add and subtract fractions using factoring to find common denominators
   3. multiply and divide fractions, and simplify fractions using exponent rules
   4. use the inverse relationship between raising to a power and root extraction for perfect square integers; which are not square, determine the two integers between which its square root lies, and explain why
   5. understand the meaning of the absolute value number, interpret it as the distance of the number from zero on a number line and determine the absolute value of real numbers

FUNCTIONS/ALGEBRA

1. Students express quantitative relationships using algebraic terminology, expressions, equations, inequalities and their graphs.
   1. use variables and appropriate operations to write an expression, equation, or system of equations which represent a verbal description (e.g., three less that a number, half as large as area A)
   2. use order of operations correctly to evaluate algebraic expressions such as 3(2x+5)^
   3. simplify numerical expressions by applying properties of rational numbers (identity, inverse, distributive, associative, commutative)
   4. use algebraic terminology correctly (e.g., variable, equation, term, expression)
   5. represent quantitative relationships graphically ( e.g., determine an appropriate scale, graph a line from a set of points, and interpret the meaning of a specific part of a graph in terms of the situation represented by the graph)
2. Students interpret and evaluate expressions involving integer powers and simple roots.
   1. interpret positive whole number powers as repeated multiplication and negative whole numbers as repeated division or multiplication by the multiplicative inverse; simplify and evaluate expressions that include exponents
3. Students graph and interpret linear and some non-linear functions
   1. graph functions of the form y=n x^2 and y=n x^3 and use in solving problems
   2. plot the values from the volumes of a 3-d shape for various values of its edge lengths (e.g., cubes with varying edge lengths or a triangle prism with a fixed height and a varying length equilateral triangle base)
   3. graph linear functions, noting that the vertical change (change in y-value) per unit horizontal change (change in x-value) is always the same and know that the ratio ("rise over run") is called the slope of a graph
   4. plot values of the quantities whose ratio is always the same (cost vs. number of an item, feet vs. inches, circumference vs. diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities.
4. Students solve simple linear equations and inequalities over the rational numbers.
   1. solve two-step linear equations in one variable over the rational numbers, interpret the solution(s) in terms of the context from which they arose, and verify the reasonableness of the results
   2. solve multi-step problems involving rate, average speed, distance and time

MEASUREMENT AND GEOMETRY

1. Students deepen their understanding of measurement of plane and solid shapes and use this understanding to solve problems.
   1. know and use the formulas for the volume of triangular prisms and cylinders (area of base x height); compare and explain the similarity between these formulas and the formula for the volume of a rectangular solid
2. Students identify and describe the properties of, and relationships among, two- and three-dimensional figures.
   1. identify angles as complementary and/or supplementary and provide descriptions of these terms
   2. use the properties of complimentary and supplementary angles and of the angles of a triangle to solve problems involving an unknown angle
   3. draw quadrilaterals and triangles given information about them (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle)
3. Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems.
   1. compare weights, capacities, geometric measures, times and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters
   2. construct and read scale drawings and models
   3. use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems, checking units of the solutions; and use dimensional analysis to check the reasonableness of the answer
4. Students compute the perimeter, area and volume of common geometric objects and use these to find measures of less common objects; they know how perimeter, area, and volume are affected under changes of scale.
   1. routinely use formulas for finding the perimeter and areas of basic two-dimensional figures and for the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, cones and circular cylinders
   2. estimate and compute the area of more complex or irregular two- and three- dimensional figures by breaking them up into more basic geometric objects
   3. compute the length of the perimeter, the surface areas of the faces, and the volume of a 3-d object built from rectangular solids. They understand that when the length of all dimensions are doubled or tripled, the unit measures are increased by the same factor
   4. relate the changes in measurement under change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = (12)^2 square inches, 1 cubic inch = (2.6)^3 cubic centimeters)
5. Students know the Pythagorean Theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures.
   1. identify and construct basic elements of geometric figures, (e.g., altitudes, midpoints, diagonals, angle bisectors and perpendicular bisectors; and central angles, radii, diameters and chords of circles) using compass and straight-edge
   2. understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections
   3. know and understand the Pythagorean Theorem and use it to find the length of the missing side of a right triangle and lengths of other line segments
   4. empirically verify the Pythagorean Theorem by direct measurement
   5. demonstrate an understanding of when two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures
   6. construct two-dimensional patterns for three-dimensional models such as cylinders, prisms and cones
   7. identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and how two or more objects are related in space (e.g., skew lines, the possible ways three planes could intersect)

STATISTICS AND PROBABILITY

1. Students determine theoretical and experimental probabilities and use these to make predictions about events.
   1. represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome
   2. use data to estimate the probability for future events (e.g., batting averages or number of accidents per mile driven)
   3. represent probabilities as ratios, proportions, decimals and percents between 0 and 1 and check that probabilities computed are reasonable; know how this is related to the probability of an event not occurring
   4. understand that the probability of either of two events occurring is the sum of the two individual probabilities and that the probability of one event following another, in independent trials, is the product of the two probabilities
   5. understand the difference between independent and dependent events and how this affects the results for specific probability situations
2. Students collect, organize and represent data sets that have one or more variables and identify relationships among variables within a data set both manually and by using an electronic spreadsheet program. (ECTL recommendations)
   1. know various forms of display for data sets, including a stem-and-leaf plot or box- and-whisker plot; use them to display a single set of data or compare two sets of data
   2. represent two numerical variables on a scatter plot and informally describe how the data points are distributed and whether there is an apparent relationship between the two variables (e.g., time spent on homework and grade level)
   3. understand the meaning of and be able to compute the minimum, the lower quartile, the median, the upper quartile and the maximum of a data set

MATHEMATICAL REASONING

1. Students make decisions about how to approach problems.
   1. analyze problems by identifying relationships, discriminating relevant from irrelevant information, identifying missing information, sequencing and prioritizing information and observing patterns
   2. formulate and justify mathematical conjectures based upon a general description of the mathematical question or problem posed
   3. determine when and how to break a problem into simpler parts
2. Students use strategies, skills and concepts in finding solutions.
   1. predict outcomes and make reasonable estimates
   2. apply strategies and results from simpler problems to more complex problems
   3. solve for unknown or undecided quantities using algebra, graphing, sound reasoning and other strategies
   4. make and test conjectures using both inductive and deductive reasoning
   5. use a variety of methods such as words, numbers, symbols, charts, graphs, tables, diagrams and models to explain mathematical reasoning
   6. express the solution clearly and logically using appropriate mathematical notation and terms and clear language, and support solutions with evidence, in both verbal and symbolic work
   7. indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy
   8. make precise calculations and check the validity of the results from the context of the problem
3. Students determine solution is complete and move beyond a particular problem by generalizing to other situations.
   1. evaluate the reasonableness of the solution in the context of the original situation
   2. note patterns in the solution and use these patterns to extend the solution to similar problems
   3. develop generalizations of the results obtained and the strategies used and extend them to new problem situations

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