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Problem Solving Strand

Students will build new mathematical knowledge through problem solving

Know the difference between relevant and irrelevant information when solving problems

Understand that some ways of representing a problem are more efficient than others

Interpret information correctly, identify the problem, and generate possible strategies and solutions

Students will solve problems that arise in mathematics and in other contexts

Act out or model with manipulatives activities involving mathematical content from literature

Formulate problems and solutions from everyday situations

Translate from a picture/diagram to a numeric expression

Represent problem situations verbally, numerically, algebraically, and/or graphically

Select an appropriate representation of a problem

Understand the basic language of logic in mathematical situations (and, or, and not)

Students will apply and adapt a variety of appropriate strategies to solve problems

Work in collaboration with others to solve problems

Translate from a picture/diagram to a number or symbolic expression

Use trial and error and the process of elimination to solve problems

Model problems with pictures/diagrams or physical objects

Analyze problems by observing patterns

Make organized lists or charts to solve numerical problems

Students will monitor and reflect on the process of mathematical problem solving

Discuss with peers to understand a problem situation

Determine what information is needed to solve problem

Determine the efficiency of different representations of a problem

Differentiate between valid and invalid approaches

Understand valid counterexamples

Explain the methods and reasoning behind the problem solving strategies used

Discuss whether a solution is reasonable in the context of the original problem

Verify results of a problem

Reasoning and Proof Strand

Students will recognize reasoning and proof as fundamental aspects of mathematics

Recognize that mathematical ideas can be supported using a variety of strategies

Understand that mathematical statements can be supported, using models, facts, and relationships to explain their thinking

Students will make and investigate mathematical conjectures

Investigate conjectures, using arguments and appropriate mathematical terms

Make and evaluate conjectures, using a variety of strategies

Students will develop and evaluate mathematical arguments and proofs

Justify general claims or conjectures, using manipulatives, models, expressions, and mathematical relationships

Develop and explain an argument verbally, numerically, algebraically, and/or graphically

Verify claims other students make, using examples and counterexamples when appropriate

Students will select and use various types of reasoning and methods of proof

Support an argument through examples/counterexamples and special cases

Devise ways to verify results

Communication Strand

Students will organize and consolidate their mathematical thinking through communication

Provide an organized thought process that is correct, complete, coherent, and clear

Explain a rationale for strategy selection

Organize and accurately label work

Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others

Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models, and symbols in written and verbal form

Answer clarifying questions from others

Students will analyze and evaluate the mathematical thinking and strategies of others

Understand mathematical solutions shared by other students

Raise questions that elicit, extend, or challenge others' thinking

Consider strategies used and solutions found by others in relation to their own work

Students will use the language of mathematics to express mathematical ideas precisely

Increase their use of mathematical vocabulary and language when communicating with others

Use appropriate vocabulary when describing objects, relationships, mathematical solutions, and rationale

Decode and comprehend mathematical visuals and symbols to construct meaning

Connections Strand

Students will recognize and use connections among mathematical ideas

Understand and make connections and conjectures in their everyday experiences to mathematical ideas

Explore and explain the relationship between mathematical ideas

Connect and apply mathematical information to solve problems

Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole

Understand multiple representations and how they are related

Model situations with objects and representations and be able to draw conclusions

Students will recognize and apply mathematics in contexts outside of mathematics

Recognize and provide examples of the presence of mathematics in their daily lives

Apply mathematics to problem situations that develop outside of mathematics

Investigate the presence of mathematics in careers and areas of interest

Recognize and apply mathematics to other disciplines and areas of interest

Representation Strand

Students will create and use representations to organize, record, and communicate mathematical ideas

Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations

Explain, describe, and defend mathematical ideas using representations

Read, interpret, and extend external models

Use standard and nonstandard representations with accuracy and detail

Students will select, apply, and translate among mathematical representations to solve problems

Use representations to explore problem situations

Investigate relationships between different representations and their impact on a given problem

Students will use representations to model and interpret physical, social, and mathematical phenomena

Use mathematics to show and understand physical phenomena (e.g., determine the perimeter of a bulletin board)

Use mathematics to show and understand social phenomena (e.g., construct tables to organize data showing book sales)

Use mathematics to show and understand mathematical phenomena (e.g., Find the missing value: (3 + 4) + 5 = 3 + (4 + ___ )

Number Sense and Operations Strand

Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems

Read and write whole numbers to trillions

Define and identify the commutative and associative properties of addition and multiplication

Define and identify the distributive property of multiplication over addition

Define and identify the identity and inverse properties of addition and multiplication

Define and identify the zero property of multiplication

Understand the concept of rate

Express equivalent ratios as a proportion

Distinguish the difference between rate and ratio

Solve proportions using equivalent fractions

Verify the proportionality using the product of the means equals the product of the extremes

Read, write, and identify percents of a whole (0% to 100%)

Solve percent problems involving percent, rate, and base

Define absolute value and determine the absolute value of rational numbers (including positive and negative)

Locate rational numbers on a number line (including positive and negative)

Order rational numbers (including positive and negative)

Students will understand meanings of operations and procedures, and how they relate to one another

Add and subtract fractions with unlike denominators

Multiply and divide fractions with unlike denominators

Add, subtract, multiply, and divide mixed numbers with unlike denominators

Identify the multiplicative inverse (reciprocal) of a number

Represent fractions as terminating or repeating decimals

Find multiple representations of rational numbers (fractions, decimals, and percents 0 to 100)

Evaluate numerical expressions using order of operations (may include exponents of two and three)

Represent repeated multiplication in exponential form

Represent exponential form as repeated multiplication

Evaluate expressions having exponents where the power is an exponent of one, two, or three

Students will compute accurately and make reasonable estimates

Estimate a percent of quantity (0% to 100%)

Justify the reasonableness of answers using estimation (including rounding)

Algebra Strand

Students will represent and analyze algebraically a wide variety of problem solving situations

Translate two-step verbal expressions into algebraic expressions

Students will perform algebraic procedures accurately

Use substitution to evaluate algebraic expressions (may include exponents of one, two and three)

Translate two-step verbal sentences into algebraic equations

Solve and explain two-step equations involving whole numbers using inverse operations

Solve simple proportions within context

Evaluate formulas for given input values (circumference, area, volume, distance, temperature, interest, etc.)

Geometry Strand

Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes

Calculate the length of corresponding sides of similar triangles, using proportional reasoning

Determine the area of triangles and quadrilaterals (squares, rectangles, rhombi, and trapezoids) and develop formulas

Use a variety of strategies to find the area of regular and irregular polygons

Determine the volume of rectangular prisms by counting cubes and develop the formula

Identify radius, diameter, chords and central angles of a circle

Understand the relationship between the diameter and radius of a circle

Determine the area and circumference of a circle, using the appropriate formula

Calculate the area of a sector of a circle, given the measure of a central angle and the radius of the circle

Understand the relationship between the circumference and the diameter of a circle

Students will apply coordinate geometry to analyze problem solving situations

Identify and plot points in all four quadrants

Calculate the area of basic polygons drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths)

Measurement Strand

Students will determine what can be measured and how, using appropriate methods and formulas

Measure capacity and calculate volume of a rectangular prism

Identify customary units of capacity (cups, pints, quarts, and gallons)

Identify equivalent customary units of capacity (cups to pints, pints to quarts, and quarts to gallons)

Identify metric units of capacity (liter and milliliter)

Identify equivalent metric units of capacity (milliliter to liter and liter to milliliter)

Determine the tool and technique to measure with an appropriate level of precision: capacity

Students will develop strategies for estimating measurements

Estimate volume, area, and circumference (see figures identified in geometry strand)

Justify the reasonableness of estimates

Determine personal references for capacity

Statistics and Probability Strand

Students will collect, organize, display, and analyze data

Develop the concept of sampling when collecting data from a population and decide the best method to collect data for a particular question

Record data in a frequency table

Construct Venn diagrams to sort data

Determine and justify the most appropriate graph to display a given set of data (pictograph, bar graph, line graph, histogram, or circle graph)

Determine the mean, mode and median for a given set of data

Determine the range for a given set of data

Read and interpret graphs

Students will make predictions that are based upon data analysis

Justify predictions made from data

List possible outcomes for compound events

Determine the probability of dependent events

Determine the number of possible outcomes for a compound event by using the fundamental counting principle and use this to determine the probabilities of events when the outcomes have equal probability

Information Source: NYSED.GOV CS&IT (standards pdf)