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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.

NCTM Standard

Expectations

IMP Correlation

(Years, Units, and Days listed are examples of places where the topic is discussed.)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Develop a deeper understanding of very large and very small numbers and of various representations of them.

Year 2, Unit 5, Days 13 & 17

 

Compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions.

Year 1, Unit 1, Days 13 & 14

Year 2, Unit 1, Days 13 & 16

Year 4, Unit 3, Days 6 & 7

 

Understand vectors and matrices as systems that have some of the properties of the real-number system.

Year 3, Unit 3, Days 27 & 35

 

Use number-theory arguments to justify relationships involving whole numbers.

Year 1, Unit 1, Days 3, 10 & 12

Year 2, Unit 1, Day 17

Understand meanings of operations and how they relate to one another.

Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities.

Year 2, Unit 3, Days 18, 19

Year 2, Unit 5, Days 1 & 12

 

Develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices.

Year 3, Unit 3, Days 27 & 35

 

Develop an understanding of permutations and combinations as counting techniques.

Year 3, Unit 5, Days 11 & 18

Year 4, Unit 5, Days 2 & 5

Compute fluently and make reasonable estimates

Develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases.

Regarding real numbers:

Throughout all units, all four years

Regarding vectors and matrices:

Year 3, Unit 3, Days 27 & 35

Year 4, Unit 2, Days 9, 16, 32

 

Judge the reasonableness of numerical computations and their results.

Throughout all units, all four years

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.

NCTM Standard

IMP Correlation

Recognize and use connections among mathematical ideas

Throughout all units, all four years.

Understand how mathematical ideas interconnect and build on one another to produce a coherent whole

Throughout all units, all four years.

Recognize and apply mathematics in contexts outside of mathematics

Throughout all units, all four years.

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.

NCTM Standard

IMP Correlation

Create and use representations to organize, record, and communicate mathematical ideas

Throughout all units, all four years.

Select, apply, and translate among mathematical representations to solve problems

Throughout all units, all four years.

Use representations to model and interpret physical, social, and mathematical phenomena

Throughout all units, all four years.


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