• Apply basic computational skills to problems from other subject areas and real-world situations

• Solve problems that illustrate the use of fractions and decimals

• Design and solve word problems with open sentences

• Use a variety of problem-solving strategies

• Communicate problem in own words

• Design physical representation for complex problems

      Make and Evaluate Conjectures and Arguments

• Use computation skills in investigation studies in other subject areas and games

• Conduct extended record keeping projects involving data gathering

• Communicate and verify solutions to problems, graphically and orally

     Make Conclusions based on Inductive Reasoning

• Use manipulatives to formulate and perimeter and area formulas, (e.g. circle, square, rectangle, triangle, parallelogram, irregular shapes) patterns, and relationships

     Justify Conclusions Involving Simple and Compound Statements

• Demonstrate understanding of simple and compound (e.g. and/or, not) statements (e.g. use Venn diagrams, element, intersection, union, subsets, inequalities, etc)

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• Read and write numerals to one billion

• Communicate large numbers, using powers of 10

• Understand place value concepts by using exponential notation

• Solve problems using proper and improper fractions, reducing to simplest form

• Demonstrate how to change improper fractions to mixed numbers and vice versa

• Convert common fractions to decimal form

• Convert common fractions and decimals to percent

• Understand the basic role of place value in decimal fractions

• Use the number line to model a variety of numbers (e.g. fractions, decimals, percents, whole numbers, integers)

• Use the exponential form of the bases 2, 3, 5, 10, and relate these forms to factoring (e.g. factor trees, LCM, GCF)

     Understand and Apply Ratios, Proportions and Percents

• Use circle graphs to explore the concept of percent

• Use fractional notation to express ratio and probability

• Know the relationship of fractions and ratio with shape and area

• Know how to represent a given percent and communicate orally and in writing the equivalence relationship between fractions, decimals, and percents

• Communicate how two sets of data compare, using ratios and appropriate notation (e.g. a/b; a to b; a:b)

• Know what a proportion is

     Number Theory

• Use factoring techniques to determine common denominators, common multiples, and common factors

• Communicate the concepts of prime and composite numbers

    Recognize Order Relations for Decimals, Integers, and Rational Numbers

• Use number lines to explore negative number notation for fractions

• Demonstrate how to compare decimals and common fractions, using the terms greater than, less than, between or equivalent

• Understand that zero can mean none of something or that it can represent a point on a scale and that any other number can also be depicted on the scale

• Compare size of fractions, using several methods

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• Solve multiplication problems involving three-digit numbers

• Solve division problems involving two-digit numbers

• Solve multiplication and division problems involving common fractions and mixed numbers

• Use the number line to explore adding and subtracting integers

• Solve addition and subtraction problems involving mixed numbers

• Solve addition and subtraction problems involving decimals to thousandths

• Multiply decimals to hundredths, and divide decimals to hundredths, using whole number divisors

• Solve problems in which fractions are used in everyday life

     Order of Operations

• Solve problems using the conventional rule for order of operations (1-parentheses, 2-exponents, 3-multiplication and division, left-to-right, 4-addition and subtraction, left-to-right)   

  Properties & Identity Elements

• Use distributive property to multiply mixed numbers

• Use the multiplicative inverse (reciprocal) in the division of fractions

• Know the role of additive inverse in the set of integers

    Algorithms

• Divide fractions, using a variety of approaches: factor product, partitioning, measurement, common denominator, and multiply by the reciprocal

• Communicate accurately the purpose for each step in basic calculations

• Know basic addition, subtraction, multiplication, and division facts (100 problems in 5 minutes - addition facts with sums 0-18 and subtraction with differences 0-9, Multiplication and division facts through 144)

• Develop strategies for mental math (e.g. compatible numbers, estimating)

     Ratios and Proportions

• Use scale drawings, fractions, and geometry to demonstrate ratio and proportion concepts

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• Analyze the effects of combining, subdividing, and changing basic shapes

• Use geometric ideas to solve problems

• Understand the basic characteristics of the concept of three dimensions

• Design and construct models, classify prisms, cones, cylinders, and pyramids

      Maps and Scale Drawings

• Design scale drawings like floor plans, using centimeter grids to relate scale to ratio

     Coordinate Planes

• Use geoboards and graph paper (and other hands-on methods) to explore measurement and vocabulary of geometric figures

• Know how to graph ordered pairs of numbers using x and y axes

      One and Two Dimensional Graphs

• Know how to create, read, and interpret graphs: circle, bar, histogram, line, pictograph, and stem and leaf

• Analyze histograms, line, picture, circle graphs, and stem and leaf as to what information each presents and note the advantages and disadvantages of each

     Variables to Represent Relationships

• Formulate and solve open sentences dealing with inverse operations, using letters as well as frames as placeholders

• Understand the basic characteristics of a variable

• Design a problem situation based on a given open sentence, using a single variable

      Use Concrete Models and Diagrams to Describe  Real-World Processes

• Communicate the multiplication (counting) principle as discovered through experiences with tree diagrams or lists of possible events taken in order

      Develop and Explore Models that Do and Do Not Rely on Chance

• Know what a sample space is

• Distinguish and count the elements in a sample space

• Know how to identify events with a probability equal to zero, events that are certain (probability=1), and events that happen sometimes (probability >0 and <1)

      Investigate Both Two- and Three-Dimensional Transformations

• Use concrete and artistic activities to explore the concept of symmetry

• Understand that symmetry can be analyzed by performing reflections, turns, or slides

      Use Appropriate Tools to Represent and Verify Geometric Relationships

• Know how to draw and measure plane geometric figures, using rulers, compasses, and protractors

• Use a protractor and a ruler to draw a perpendicular bisector of a line segment and an angle bisector

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• Know how to measure (including reading and recording) temperatures of familiar substances

• Demonstrate an understanding of the relationship of volume to capacity (metric and English system measure cubic centimeters, liters, cubic inch, cup, fluid ounce)

• Know how to determine which measurements (length, area, volume, mass (weight), or time) are reasonable by referring to typical values

     Select Appropriate Standard and Nonstandard Measurement Units and Tools to   Measure to a Desired Degree of Accuracy 

• Use the prefixes milli, centi, kilo and symbols g, mg, kg, mL, L, mm, km, and cm and the tools used to measure them

• Know that a protractor measures angles

• Know how to measure volume and capacity, using cubic centimeter blocks, cubic inch blocks, English system and metric measuring tools

• Use operations with metric units

• Know how to make effective use of thermometer and scale for making measurements

• Know how to estimate and then determine length, weight/mass, area, and liquid volume/capacity, using standard and nonstandard units of measure

• Understand that measurements are likely to give slightly different numbers when measured multiple times

        Develop Measure Skills and Informally Derive and Apply Formulas in direct Measurement Activities

• Know how to measure volume of prisms, using cubic units in metric and English system

• Know how to measure the area and perimeter of triangles, circles, and irregular polygons, using manipulative materials and informal methods

• Distinguish acute, obtuse, and right angles

• Use manipulatives to explore the volume of cylinders empirically (e.g. compare amount of sand used to fill various sizes of cylinders)

• Know how to approximate the area of rectangles and triangles

       Use Statistical Methods and Measures of Central Tendencies to Display, Describe, and Compare Data

• Understand difference between mode, median, and mean

• Know how to collect and organize simple data sets to answer questions

• Understand that a summary of data should include where the middle is and how much spread is around it

     Explore and Produce Graphic Representations of Data

• Read and analyze graphs that can be demonstrated by the teacher (e.g. spreadsheet, graphing calculator): bar, histogram, line

• Use pictographs and other graphic representations to model problems

• Understand the purpose of spreading data out on a number line (e.g. it helps to see what the extremes are, where they pile up, and where the gaps are located)

     Develop Critical Judgment for the Reasonableness of Measurement

• Know how to relate metric units to customary units via approximations (e.g. liters-quarts, miles-kilometers, grams-ounces)

• Make real-world comparisons of measurements

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• Know how to round numbers to the nearest hundredth and up to 10,000

• Use rounding skills to estimate

• Know how to round fractional and decimal numbers for estimates in computation

• Understand the effects of addition, subtraction, multiplication, and division

     Use Estimation to Solve Problems for Which Exact Answers are Inappropriate

• Develop an awareness of when an estimation is more appropriate than an exact answer.

     Estimate the Probability of Events

• Make predictions based on sample data

• Use arrangements and combinations (e.g. tree diagram)

• Understand that when predictions are based on what is known about the past, one must assume that the conditions stay the same from the past event to the predicted future events

     Use Simulation Techniques to Estimate Probabilities

• Conduct simulations for experiments that cannot be determined theoretically and are unwieldy to determine experimentally

     Determine Probabilities of Independent Events

• Conduct and predict outcomes of experiments with independent events.

• Understand how to express probabilities as fractions, decimals, or percents for theoretical and experimental situations such that:

• Understand how to express probabilities as fractions, decimals, or percents for theoretical and experimental situations such that: - Theoretical probability is found by:

• Understand the concept of combination or arrangement

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• Use and apply computation skills by describing and extending number patterns and sequences

• Know how to interpolate and/or extrapolate simple patterns of numbers

• Distinguish and describe simple functional relationships

     Describe and Represent Patterns and Functional Relationships, Using Tables, Charts, Graphs, and Verbal Descriptions

• Use tables and graphs to help to identify patterns

• Use a variety of representations for the same functional relationship

      Develop Methods to Solve Basic Linear Equations

• Know how to find the missing value in a proportion in which three of the numbers are known
• Distinguish between linear and quadratic relationships
• Solve one-step linear equations in one variable

     Develop an Understanding of Functions and Functional Relationships

• Develop an understanding of functions and relationships with whole numbers

• Understand that the basic function of tables and graphs is to make explicit how the values of one quantity are related to the values of another

• Use the characteristics of proportional relationships

     Apply the Concept of Similarity in Relevant Situations

• Use concrete and artistic experiences to explain similarity and congruence in plane geometric figures

     Use Properties of Polygons to Classify Them

• Develop definitions and classify polygons by properties

• Understand the basic properties of and the similarity and differences between a trapezoid, rhombus, and quadrilatera

• Know how to compare shapes in terms of parallel, perpendicular, similar, and congruent

     Explore Relationships Involving Points, Lines, Angles, and Planes

• Understand the basic characteristics of angles

• Distinguish line segments

• Formulate congruence of line segments, angles, and polygons by direct comparison given their attributes

    Develop Readiness for Basic Concepts of Right Triangle Trigonometry

• Know a right triangle contains one right angle

• Know the hypotenuse of a right triangle is opposite the right angle

• Know the hypotenuse of a triangle is greater than either of the other two legs

• Understand and investigate intuitively the concept of similarity among triangles

     Use Patterns and Functions to Represent and Solve Problems

• Use math sentences of patterns and functions to represent and solve problems

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