2000 NCTM Standards Correlation

Number and Operations

Students should be able to:

• Understand numbers, ways of representing numbers, relationships among numbers, and number systems;
• Understand meanings of operations and how they relate to one another;
• Compute fluently and make reasonable estimates.

Algebra

Students should be able to:

• Understand patterns, relations, and functions;
• Represent and analyze mathematical situations and structures using algebraic symbols;
• Use mathematical models to represent and understand quantitative relationships;
• Analyze change in various contexts.

NCTM Standard	Expectations	IMP Correlation
Understand patterns, relations, and functions.	Generalize patterns using explicitly defined and recursively defined functions.	Throughout all units, all four years Regarding recursively defined functions: Year 4, Unit 1, Days 1, 7
	Understand relations and functions and select, convert flexibly among, and use various representations for them.	Year 1, Unit 3, Days 7 & 14 Year 2, Unit 1, Days 27 & 30 Year 4, Unit 4, Days 1 & 9
	Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.	Year 1, Unit 3, Days 11 & 18 Year 2, Unit 1, Days 27 & 30 Year 2, Unit 4, Days 8 & 14 Year 3, Unit 1, Days 1 & 11 Year 3, Unit 4, Days 1 & 16 Year 4, Unit 4, Days 10 & 12
	Understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions.	Year 4, Unit 4, Days 18 & 28
	Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions.	Year 1, Unit 4, Days 22 & 24 Year 2, Unit 5, Days 1– & 18 Year 3, Unit 1, Days 1 & 11 Year 4, Unit 4, Days 4 & 9, 13 & 15
	Interpret representations of functions of two variables.	Year 2, Unit 1, Day 26 Year 2, Unit 4, Days 3 & 4, 7 & 14 Year 3, Unit 3, Days 1 & 14
Represent and analyze mathematical situations and structures using algebraic symbols.	Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations.	Year 1, Unit 3, Days 9 & 10 Year 1, Unit 5, Days 13 & 14 Year 2, Unit 1, Days 6 & 7, 13 & 21 Year 2, Unit 4, Days 1 & 6
	Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases.	Year 2, Unit 1, Days 6 & 7, 13 & 22 Year 2, Unit 4, Days 1 & 6, 15 & 18 Year 3, Unit 3, Days 3 & 14, 17 & 26, 34 & 35
	Use symbolic algebra to represent and explain mathematical relationships.	Throughout all units, all four years
	Use a variety of symbolic representations, including recursive and parametric equations, for functions and relations.	Throughout all units, all four years Regarding recursive equations: Year 4, Unit 1, Days 1, 7, 20
	Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.	Throughout all units, all four years
Use mathematical models to represent and understand quantitative relationships.	Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.	Throughout all units, all four years, especially: Year 1, Unit 4, Day 26 Year 2, Unit 4, Days 1 & 3 Year 2, Unit 5, Days 1 & 2 Year 3, Unit 1, Days 1 & 3
	Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts.	Throughout all units, all four years Regarding recursive equations: Year 4, Unit 1, Days 1, 7, 20
	Draw reasonable conclusions about a situation being modeled.	Throughout all units, all four years
Analyze change in various contexts	Approximate and interpret rates of change from graphical and numerical data.	Year 3, Unit 4, Days 11 & 16

Geometry

Students should be able to:

• Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;
• Specify locations and describe spatial relationships using coordinate geometry and other representational systems;
• Apply transformations and use symmetry to analyze mathematical situations;
• Use visualization, spatial reasoning, and geometric modeling to solve problems.

NCTM Standard	Expectations	IMP Correlation
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships	Analyze properties and determine attributes of two- and three-dimensional objects.	Year 1, Unit 5, Days 6 & 16 Year 2, Unit 3, Days 2 & 27 Year 3, Unit 2, Days 2– & 15 Year 3, Unit 3, Days 8 & 14 Year 4, Unit 2, Days 8 & 33
	Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them.	Year 1, Unit 5, Days 6 & 16 Year 2, Unit 3, Days 2 & 27 Year 3, Unit 2, Days 2– & 15
	Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others.	Year 1, Unit 5, Days 9 & 16 Year 2, Unit 3, Days 7 & 8, 14, 25 Year 3, Unit 2, Days 1 & 15
	Use trigonometric relationships to determine lengths and angle measures.	Year 1, Unit 5, Days 22 & 24 Year 2, Unit 3, Days 7 & 11, 17 & 20 Year 4, Unit 1, Days 1 & 7, 11 & 13
Specify locations and describe spatial relationships using coordinate geometry and other representational systems	Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations.	Year 2, Unit 4, Days 8 & 18 Year 3, Unit 2, Days 5 & 10, 16 & 20 Year 3, Unit 3, Days 8 & 14 Year 4, Unit 1, Days 11 & 20 Year 4, Unit 2, Days 8 & 30
	Investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.	Year 2, Unit 4, Days 8 & 14 Year 3, Unit 2, Days 5 & 10, 16 & 20 Year 3, Unit 3, Days 8 & 14 Year 4, Unit 1, Days 11 & 20 Year 4, Unit 2, Days 20 & 30
Apply transformations and use symmetry to analyze mathematical situations	Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices.	Year 4, Unit 2, Days 8 & 20, 31 & 33 Year 4, Unit 4, Days 27 & 28
	Use various representations to help understand the effects of simple transformations and their compositions.	Year 4, Unit 2, Days 8 & 20, 31 & 33 Year 4, Unit 4, Days 27 & 28
Use visualization, spatial reasoning, and geometric modeling to solve problems	Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.	Year 1, Unit 5, Days 6 & 12 Year 2, Unit 3, Days 2 & 8 Year 3, Unit 2, Days 5 & 7 Year 3, Unit 3, Days 8 & 14 Year 4, Unit 2, Days 20 & 30 Year 4, Unit 3, Days 3– & 4
	Visualize three-dimensional objects from different perspectives and analyze their cross sections.	Year 2, Unit 3, Days 21 & 27 Year 3, Unit 3, Days 9 & 12 Year 4, Unit 2, Days 31 & 33
	Use vertex-edge graphs to model and solve problems.
	Use geometric models to gain insights into, and answer questions in, other areas of mathematics.	Year 2, Unit 4, throughout Year 3, Unit 3, throughout
	Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.	Year 1, Unit 5, Day 10 Year 2, Unit 3, Day 15 and supplemental problem "All About Escher"

Measurement

Students should be able to:

• Understand measurable attributes of objects and the units, systems, and processes of measurement;
• Apply appropriate techniques, tools, and formulas to determine measurements.

NCTM Standard	Expectations	IMP Correlation
Understand measurable attributes of objects and the units, systems, and processes of measurement	Make decisions about units and scales that are appropriate for problem situations involving measurement.	Year 2, Unit 3, Days 4 and Day 23 Year 3, Unit 2, Days 11 & 15
Apply appropriate techniques, tools, and formulas to determine measurements	Analyze precision, accuracy, and approximate error in measurement situations.	Throughout all units, all four years, especially: Year 2, Unit 3, Day 19 and supplemental problem "What's the Answer Worth?"
	Understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders.	Year 2, Unit 3, Days 2 & 10, 21 & 27 Year 3, Unit 2, Days 11 & 15
	Apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations.	Year 3, Unit 2, Days 11 & 15 Year 3, Unit 4, Days 12 & 14
	Use unit analysis to check measurement computations.

Data Analysis and Probability

Students should be able to:

• Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them;
• Select and use appropriate statistical methods to analyze data;
• Develop and evaluate inferences and predictions that are based on data;
• Understand and apply basic concepts of probability.

NCTM Standard	Expectations	IMP Correlation
Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them	Understand the differences among various kinds of studies and which types of inferences can legitimately be drawn from each.	Year 1, Unit 2, Days 7 & 20 Year 2, Unit 2, Days 2, 3, and throughout Year 3, Unit 5, Days 21 & 23 Year 4, Unit 5, throughout
	Know the characteristics of well-designed studies, including the role of randomization in surveys and experiments.	Year 1, Unit 4, Days 16 & 19 Year 2, Unit 2, Days 2, 3, and throughout Year 3, Unit 5, Days 21 & 23 Year 4, Unit 5, throughout
	Understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable.	Year 1, Unit 4, Days 16 & 19 Year 2, Unit 2, Days 1 & 9
	Understand histograms, parallel box plots, and scatterplots and use them to display data.	Year 1, Unit 3, Days 17 & 18 Year 4, Unit 4, Days 16 & 17 Year 4, Unit 5, Days 3-5
	Compute basic statistics and understand the distinction between a statistic and a parameter.	Year 1, Unit 4, Days 7 & 15 Year 2, Unit 2, Days 10 & 18 Year 4, Unit 5, Days 6 & 14
Select and use appropriate statistical methods to analyze data	For univariate measurement data, be able to display the distribution, describe its shape, and select and calculate summary statistics.	Year 1, Unit 4, Days 7 & 15 Year 2, Unit 2, Days 10 & 18 Year 4, Unit 5, Days 6 & 14
	For bivariate measurement data, be able to display a scatterplot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools.	Year 4, Unit 3, Days 16 & 17
	Display and discuss bivariate data where at least one variable is categorical.	Year 2, Unit 2, Days 1 & 9
	Recognize how linear transformations of univariate data affect shape, center, and spread.	Year 1, Unit 4, Day 13
	Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.	Year 1, Unit 4, Days 21 & 26
Develop and evaluate inferences and predictions that are based on data	Use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions.	Year 4, Unit 5, Days 1 & 5
	Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference.	Year 4, Unit 5, Days 11 & 22
	Evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions.	Year 1, Unit 4, Day 17 Year 2, Unit 2, Day 3
	Understand how basic statistical techniques are used to monitor process characteristics in the workplace.	Year 2, Unit 2, Days 19 & 23
Understand and apply basic concepts of probability	Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases.	Year 1, Unit 2, Days 1 & 21 Year 3, Unit 5, Days 11 & 23 Year 4, Unit 5, Days 1 & 5
	Use simulations to construct empirical probability distributions.	Year 1, Unit 2, Days 1 & 3, 12 & 17 Year 2, Unit 2, Days 10 & 18 Year 4, Unit 5, Days 2 & 5
	Compute and interpret the expected value of random variables in simple cases.	Year 1, Unit 2, Days 12 & 21 Year 4, Unit 5, Days 11 & 14
	Understand the concepts of conditional probability and independent events.	Year 1, Unit 2, Days 3 & 5 Year 3, Unit 5, Days 5 & 18 Year 4, Unit 5, Day 4
	Understand how to compute the probability of a compound event.	Year 3, Unit 5, Days 5 & 18

Problem Solving

Students should be able to:

• Build new mathematical knowledge through problem solving;
• Solve problems that arise in mathematics and in other contexts;
• Apply and adapt a variety of appropriate strategies to solve problems;
• Monitor and reflect on the process of mathematical problem solving.

NCTM Standard	IMP Correlation
Build new mathematical knowledge through problem solving	Throughout all units, all four years.
Solve problems that arise in mathematics and in other contexts	Throughout all units, all four years.
Apply and adapt a variety of appropriate strategies to solve problems	Throughout all units, all four years.
Monitor and reflect on the process of mathematical problem solving	Throughout all units, all four years.

Reasoning and Proof

Students should be able to:

• Recognize reasoning and proof as fundamental aspects of mathematics;
• Make and investigate mathematical conjectures;
• Develop and evaluate mathematical arguments and proofs;
• Select and use various types of reasoning and methods of proof.

NCTM Standard	IMP Correlation
Recognize reasoning and proof as fundamental aspects of mathematics	Throughout all units, all four years.
Make and investigate mathematical conjectures	Throughout all units, all four years.
Develop and evaluate mathematical arguments and proofs	Throughout all units, all four years.
Select and use various types of reasoning and methods of proof	Throughout all units, all four years.

Communication

Students should be able to:

• Organize and consolidate their mathematical thinking through communication;
• Communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
• Analyze and evaluate the mathematical thinking and strategies of others;
• Use the language of mathematics to express mathematical ideas precisely.

NCTM Standard	IMP Correlation
Organize and consolidate their mathematical thinking through communication	Throughout all units, all four years.
Communicate their mathematical thinking coherently and clearly to peers, teachers, and others	Throughout all units, all four years.
Analyze and evaluate the mathematical thinking and strategies of others	Throughout all units, all four years.
Use the language of mathematics to express mathematical ideas precisely	Throughout all units, all four years.

Connections

Students should be able to:

• Recognize and use connections among mathematical ideas;
• Understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• Recognize and apply mathematics in contexts outside mathematics.

Representation

Students should be able to:

• Create and use representations to organize, record, and communicate mathematical ideas;
• Select, apply, and translate among mathematical representations to solve problems;
• Use representations to model and interpret physical, social, and mathematical phenomena.