1. All students understand place value of whole numbers and decimals to two decimal places, how these relate to simple fractions, and use concepts of negative numbers

• read and write whole numbers in the millions
  
• order and compare whole numbers and decimals to two decimal places
  
• round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand or hundred thousand
  
• decide when a rounded solution is called for, and explain why this is the case
  
• interpret different meanings for fractions including parts of a whole, parts of a set, indicated division of whole numbers and quantities (and measures) between whole numbers on a number line; and relate to simple decimals on a number line
  
• write tenths and hundredths in decimal and fraction notation and know fraction/decimal equivalents for halves and fourths (e.g., = 0.5 or .50; 7/4 = 1 3/4 = 1.75)
  
• write the fraction represented by a drawing of parts of a figure; represent a given fraction using drawings
  
• use concepts of negative numbers (e.g., on a number line, in counting, in temperature, "owing")
  
• identify the relative position of fractions, mixed numbers, and decimals to two decimal places on the number line.
  

2. All students extend their use and understanding of whole numbers to addition and subtraction of simple decimals

• estimate and compute the sum or difference of whole numbers and positive decimals to two places
  
• round two place decimals to one decimal or the nearest whole number, and use rounding to judge the reasonableness of an answer.
  

3. All students solve problems involving addition, subtraction, multiplication and division of whole numbers, including the addition and subtraction of negative numbers, and understand the relationships among the operations

• demonstrate understanding of, and the ability to use standard algorithms for addition and subtraction of multi-digit numbers
  
• demonstrate understanding of, and ability to use standard algorithms for multiplying a multi-digit number by a two digit number and long division for dividing a multi-digit number by a one digit number; use relationships between them to simplify computations and to check results
  
• solve problems involving multiplication of multi-digit numbers by two-digit numbers
  
• solve problems involving division of multi-digit numbers by one-digit numbers.
  

4. All students know how to factor small whole numbers

• understand that many whole numbers decompose in different ways (e.g., 12 = 4 x 3 = 2 x 6 = 2 x 2 x 3)
  
• 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 prime numbers.

1. All students use and interpret variables, mathematical symbols and properties to write and simplify expressions and sentences

• use letters, boxes, or other symbols to stand for any number in simple expressions or equations (e.g., demonstrate understanding and use of a concept of a variable)
  
• interpret and evaluate mathematical expressions that use parentheses
  
• use parentheses to indicate which operation to perform first when writing expressions containing more than two terms and different operations
  
• use and interpret formulas (e.g., Area = length times width or A = lw) to answer questions about quantities and their relationships
  
• understand that an equation such as y = 3x + 5 is a prescription for determining a second number when a first number is given.
  

2. All students know how to manipulate equations

• know and understand that equal added to equals are equal
  
• know and understand that equals multiplied by equals are equal.

1. All students understand perimeter and area

• measure the area of rectangular shapes, using appropriate units square centimeter2,, square meter2, square kilometer2,, square inches2, square yard2, square mile2
  
• recognize that the rectangles having the same area can have different perimeters
  
• understand that the same number can be the perimeter of different rectangles, each having a different area
  
• understand and use formulas to solve problems involving perimeters and areas of rectangles and squares. Use these formulas to find the areas of more complex figures by dividing them into parts with these basic shapes.
  

2. All students use two-dimensional coordinate grids to represent points and graph lines and simple figures

• draw the points corresponding to linear relationships on graph paper (e.g., draw the first ten points for the equation y = 3x and connect them using a straight line)
  
• understand that the length of a horizontal line segment equals the difference of the x-coordinates
  
• understand that the length of a vertical line segment equals the difference of the y-coordinates.
  

3. All students demonstrate understanding of plane and solid geometric objects. They use this knowledge to show relationships and solve problems

• identify lines that are parallel and perpendicular
  
• identify the radius and diameter of a circle
  
• identify congruent figures
  
• identify figures that have bilateral and rotational symmetry
  
• know the definitions of right angle, acute angle and obtuse angle. They understand that 90, 180, 270, and 360 degrees are, respectively, associated with 1/4, 1/2, 3/4 and full turns.
  
• visualize, describe and represent 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; and draw patterns (of faces) for a solid that when folded will make a model of the solid
  
• know the definitions of different triangles (e.g., equilateral, isosceles, scalene) and identify their features
  
• know the definition of different quadrilaterals (e.g., rhombus, square, rectangle, parallelogram, trapezoid).

1. All students organize, represent and interpret numerical and categorical data, and clearly communicate their findings

• formulate survey questions, systematically collect and represent data on a number line, and coordinate graphs, tables and charts
  
• identify the mode(s) for sets of categorical data, and the mode(s), median, and any apparent outliers for numerical data sets
  
• interpret one- and two-variable data graphs to answer questions about a situation.
  

2. All students make predictions for simple probability situations

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

1. All students make decisions about how to approach problems

• analyze problems by identifying relationships, discriminating relevant from irrelevant information, sequencing and prioritizing information, and observing patterns
  
• determine when and how to break a problem into simpler parts.
  

2. All students use strategies, skills and concepts in finding solutions

• use estimation to verify the reasonableness of calculated results
  
• apply strategies and results from simpler problems to more complex problems
  
• use a variety of methods such as words, numbers, symbols, charts, graphs, tables, diagrams and models to explain mathematical reasoning
  
• 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
  
• indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy
  
• make precise calculations and check the validity of the results from the context of the problem.
  

3. All students move beyond a particular problem by generalizing to other situations

• evaluate the reasonableness of the solution in the context of the original situation
  
• note method of deriving the solution and demonstrate conceptual understanding of the derivation by solving similar problems
  
• develop generalizations of the results obtained and extend them to other circumstances. H:\whoppe\STNDRDS\Math\Math4th.wpd.frm 2/99