principles standards

8/9/2019 Principles Standards

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A Correlation of

to the

NCTM

Principles and Standardsfor School Mathematics

Grades K-5

N/M-507


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Introduction

This document demonstrates how wellInvestigations in Number, Data, and Space aligns with

the NCTM Standards. Each mathematics standard is presented with the location of directinstruction that corresponds in theInvestigationsprogram.

Investigations in Number, Data, and Space is a Kindergarten through Grade 5 curriculum

consisting of a series of Teachers Editions that focus on major mathematical ideas, content, andpedagogy. Each book emphasizes depth of mathematical thinking over fragmented topics.

Students invent strategies and approaches to solving problems and rely less on rote learningstressed in traditional textbooks. The program blends concrete materials with appropriate

technology, including calculators in everyday mathematical lessons.

Developed by TERC under a grant from the National Science Foundation,Investigations in

Number, Data, and Space is comprehensive in its approach to students of diverse cultural,ethnic and language groups. In an effort to give mathematical lessons a broader spectrum,students are encouraged to explore working in groups, individually and as a whole class. By

incorporating these methods into everyday learning, students learn to express mathematicalthinking through talking, drawing, and writing.

Investigations in Number, Data and Space was developed after three years of nationwide

field-testing and includes teachers practical suggestions, student dialogues, and teacher notes.


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TABLE OF CONTENTS

Grades Kindergarten-2

Number and Operations Standard....1

Algebra Standard .....14

Geometry

Standard...24

Measurement Standard...34

Data Analysis and Probability Standard....40

Problem Solving Standard...47

Reasoning and Proof Standard........55Communication Standard....62

Connections

Standard.......69

Representation

Standard..........75

Grades 3-5

Number and Operations Standard......80

Algebra Standard .......103

Geometry

Standard.116

Measurement Standard.133

Data Analysis and Probability Standard......142

Problem Solving Standard.159

Reasoning and Proof Standard..169Communication Standard..179

Connections

Standard.....189

Representation

Standard.......197


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Investigations in Number, Data, and Space

to the

NCTM Standards

Grades K-2

Number and Operations Standard

In kindergarten through grade 2 all students should

Understand numbers, ways of representing numbers, relationships among numbers, and

number systems

count with understanding and recognize "how many" in sets of objects;

References:Kindergarten: Mathematical Thinking in KindergartenInvestigations 1, 2, 3

About Classroom Routines: 68-69

Pattern Trains and Hopscotch PathsAbout Classroom Routines: 92-93

Collecting, Counting, and Measuring

Investigations 1, 2, 3, 4, 5About Classroom Routines: 102-103

Counting Ourselves and Others

Investigations 1, 3, 4About Classroom Routines: 96-97

Making Shapes and Building Blocks


How Many in All?Investigations 1, 2, 3, 4


Grade 1: Mathematical Thinking at Grade 1

Investigation 2: Sessions 4-6Investigation 4: Sessions 4-6

Building Number Sense


Investigation 3: Sessions 1-9


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Investigations in Number, Data, & Space NCTM Standards

2


Quilt Squares and Block TownsInvestigation 1: Sessions 2-10


Number Games and Story ProblemsInvestigation 2: Sessions 1-8, 10-12

Bigger, Taller, Heavier, Smaller


Grade 2: Mathematical Thinking at Grade 2Investigation 2: Sessions 6, 7


Coins, Coupons, and Combinations



Putting Together and Taking Apart


Investigation 5: Sessions 2-3, 6

use multiple models to develop initial understandings of place value and the base-ten

number system;

References:

Kindergarten: Mathematical Thinking in KindergartenInvestigations 1, 2, 3








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How Many in All?

Investigations 1, 2, 3, 4About Classroom Routines: 100-101






Quilt Squares and Block Towns


Number Games and Story Problems

Investigation 2: Sessions 1-8, 10-12

Bigger, Taller, Heavier, SmallerInvestigation 2: Sessions 1-7


Investigation 2: Sessions 6, 7Investigation 4: Sessions 1-5


Coins, Coupons, and CombinationsInvestigation 1: Sessions 1-3



Investigation 2: Sessions 1-7Investigation 4: Sessions 2-4Investigation 5: Sessions 2-3, 6

develop understanding of the relative position and magnitude of whole numbers and of

ordinal and cardinal numbers and their connections;

References:


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Kindergarten: Collecting, Counting, and MeasuringInvestigations 5-6

Grade 1: Mathematical Thinking at Grade 1Investigation 2: Sessions 2-3



Number Games and Story ProblemsInvestigation 2: Sessions 6-9

Grade 2: Putting Together and Taking ApartInvestigation 2: Sessions 1-4

Investigation 5: Sessions 2-3, 6, 8

Timelines and Rhythm Patterns


develop a sense of whole numbers and represent and use them in flexible ways, including

relating, composing, and decomposing numbers;

References:

Kindergarten: Collecting, Counting, and MeasuringInvestigations 4-6

How Many in All?

Investigations 2-4

Grade 1: Mathematical Thinking in Grade 1



Building Number SenseInvestigation 1: Sessions 1-6, 9Investigation 2: Sessions 1-9

Investigation 3: Sessions 1-7, 9Investigation 4: Sessions 1-10





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Grade 2: Mathematical Thinking at Grade 2Investigation 1: Session 1

Investigation 2: Sessions 1-6, 8Investigation 4: Sessions 1, 5



Investigation 2: Session 10Investigation 3: Sessions 1-5


Putting Together and Taking ApartInvestigation 1: Sessions 1-6





connect number words and numerals to the quantities they represent, using various

physical models and representations;

References:

Kindergarten: Mathematical Thinking in KindergartenInvestigations 1, 2, 3









How Many in All?Investigations 1, 2, 3, 4


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Investigation 4: Sessions 2-4Investigation 5: Sessions 2-3, 6

understand and represent commonly used fractions, such as 1/4, 1/3, and 1/2.

Students usingInvestigations in Number, Data, and Spaceare introduced to fraction conceptsin Grade 1. Kindergarten students gain experience with preliminary concepts, which will lead

to the introduction of fractions, including the subdivision of shapes into equal parts. Grade 1students divide shapes and groups into equal parts or equal groups. They explore halves in

units on measuring capacity and length. A sample reference for Kindergarten is given below.


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References:

Kindergarten: Making Shapes and Building Blocks

Investigation 4: Choice Time: Fill the Hexagons

Grade 1: Building Number Sense

Investigation 1: Session 2


Investigation 2: Sessions 2-4Investigation 3: Session 2

Grade 2: Shapes, Halves, and Symmetry


Understand meanings of operations and how they relate to one another

understand various meanings of addition and subtraction of whole numbers and the

relationship between the two operations;

References:

Kindergarten: Collecting, Counting, and MeasuringInvestigation 4

How Many in All?Investigations 2-4




Building Number SenseInvestigation 1: Sessions 1-6, 9



Investigation 2: Sessions 1-8, 10-13Investigation 3: Sessions 1-13


Investigation 1: Session 1Investigation 2: Sessions 1-6, 8


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Investigation 2: Session 10Investigation 3: Sessions 1-5Investigation 4: Sessions 2-4





understand the effects of adding and subtracting whole numbers;

References:

Kindergarten: Collecting, Counting, and Measuring

Investigation 4

How Many in All?

Investigations 2-4

Grade 1: Mathematical Thinking in Grade 1Investigation 2: Sessions 1-6








Investigation 1: Session 1Investigation 2: Sessions 1-6, 8



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understand situations that entail multiplication and division, such as equal groupings of

objects and sharing equally.Students usingInvestigations in Number, Data, and Spaceare gradually and progressively

introduced to multiplication and division concepts in Kindergarten through Grade 2.Kindergarten students gain experience with preliminary concepts, which will lead to

understanding situations that entail multiplication and division, including the subdivision ofshapes into equal parts. Grade 1 students divide shapes and groups into equal parts and equal

groups. They also gain experience with repeated addition and skip counting. Grade 2 studentspractice skip counting by 2s, 5s, and 10s. They explore the relationship between skip

counting and grouping and apply these concepts to problem situations.References:

Kindergarten: Making Shapes and Building BlocksInvestigation 4: Choice Time: Fill the Hexagons




Investigation 2: Sessions 1-2, 4-8, 10-12






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Compute fluently and make reasonable estimates

develop and use strategies for whole-number computations, with a focus on addition and

subtraction;

References:Kindergarten: Collecting, Counting, and MeasuringInvestigation 4





Building Number SenseInvestigation 1: Sessions 1-6, 9



Investigation 1: Sessions 1-10Investigation 2: Sessions 1-8, 10-13












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develop fluency with basic number combinations for addition and subtraction;

References:









Survey Questions and Secret Rules







Investigation 2: Sessions 1-3, 6, 8Investigation 4: Sessions 1-5



Investigation 1: Sessions 1-11Investigation 2: Session 10Investigation 3: Sessions 1-5





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use a variety of methods and tools to compute, including objects, mental computation,estimation, paper and pencil, and calculators.

References:















Grade 2: Mathematical Thinking at Grade 2Investigation 1: Session 1Investigation 2: Sessions 1-3, 6, 8





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Investigation 1: Sessions 1-6Investigation 2: Sessions 1-7Investigation 3: Sessions 1-5



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Algebra Standard


Understand patterns, relations, and functions

sort, classify, and order objects by size, number, and other properties;

Kindergarten: Kindergarten students usingInvestigations in Number, Data, and Spacesort,classify, and order objects in a variety of settings. Students explore and compare the attributes

of various geometric manipulatives, including pattern blocks and geoblocks. Students classifyand sort data. They classify shapes in the environment.

Sample References:

Mathematical Thinking in Kindergarten

Investigation 1: Choice Time

Investigation 4: Choice Time: Staircase Patterns


Investigation 4Investigation 5

Investigation 6

Counting Ourselves and OthersInvestigation 1


Investigation 4

Making Shapes and Building BlocksInvestigation 1


Investigation 5

Grade 1: In addition to the following references, every text in the Grade 1Investigations in

Number, Data, and Spaceseries includes a description of sorting games in the appendix of thetext, entitled About Classroom Routines. The first game, entitled, Guess My Rule, requires

students to determine the common attribute of a set of objects. After students have becomefamiliar with this game, they can play Guess My Object, in which students ask yes-or-no

questions, based on attributes, and use the process of elimination to determine which particularobject was secretly chosen from a set of objects.

Sample References:




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Quilt Shapes and Block Towns


Grade 2:Second graders sort, classify, and order objects using a number of techniques in avariety of situations. For example, students sort manipulative materials based on their

attributes. They sort and classify information. The children sort two-dimensional geometricshapes and three-dimensional geometric solids. They use Venn diagrams to show relationships

within a group of related objects.

Sample References:

Mathematical Thinking at Grade 2Investigation 1: Sessions 2-4


Does It Walk, Crawl, or Swim?Investigation 1: Sessions 1-6



recognize, describe, and extend patterns such as sequences of sounds and shapes or simple

numeric patterns and translate from one representation to another;

References:

Kindergarten: Mathematical Thinking in KindergartenInvestigation 3

About Classroom Routines, pp. 71, 74-75

Pattern Trains and Hopscotch PathsInvestigations 1, 2, 3, 4


Collecting, Counting, and MeasuringAbout Classroom Routines, pp. 105, 108-109



Making Shapes and Building BlocksAbout Classroom Routines, pp. 109, 112-113

Appendix: Shapes Teacher Tutorial, pp. 120-144

How Many in All?About Classroom Routines, pp. 103, 106-107


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Building Number SenseInvestigation 3: Sessions 1-8Investigation 4: Session 10





Investigation 2: Sessions 2, 6-9

Grade 2: Mathematical Thinking at Grade 2Investigation 3: Sessions 1-4, 6


Investigation 2: Sessions 1-2, 4-5, 10Investigation 3: Session 1


Shapes, Halves, and SymmetryInvestigation 1: Sessions 2-8




Timelines and Rhythm PatternsInvestigation 2: Sessions 1-5

analyze how both repeating and growing patterns are generated.

References:

Kindergarten: Mathematical Thinking in Kindergarten


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Investigation 3About Classroom Routines, pp. 71, 74-75

Pattern Trains and Hopscotch Paths

Investigations 1, 2, 3, 4




Counting Ourselves and OthersAbout Classroom Routines, pp. 99, 102-103


About Classroom Routines, pp. 109, 112-113Appendix: Shapes Teacher Tutorial, pp. 120-144

How Many in All?




Building Number SenseInvestigation 3: Sessions 1-8


Survey Questions and Secret RulesInvestigation 3: Sessions 2-3



Number Games and Story ProblemsInvestigation 2: Sessions 2, 6-9

Grade 2: Mathematical Thinking at Grade 2Investigation 3: Sessions 1-4, 6



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Investigation 2: Sessions 1-2, 4-5, 10Investigation 3: Session 1


Shapes, Halves, and Symmetry

Investigation 1: Sessions 2-8Investigation 2: Sessions 1, 3Investigation 4: Sessions 1-7





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Represent and analyze mathematical situations and structures using algebraic symbols

illustrate general principles and properties of operations, such as commutativity, using

specific numbers;

References:Kindergarten: How Many in All?Investigations 2, 4

Grade 1: Mathematical Thinking at Grade 1Investigation 2: Session 4:Teacher Note




Investigation 4: Sessions 1, 6-9

Grade 2: Coins, Coupons, and CombinationsInvestigation 1: Session 1




use concrete, pictorial, and verbal representations to develop an understanding of

invented and conventional symbolic notations.

References:

Kindergarten: How Many in All?Investigations 1, 2, 3, 4

Teacher Note, p. 45

Grade 1: Building Number SenseInvestigation 1: Sessions 2, 7-9

Investigation 2: Sessions 1-2, 6-8Investigation 4: Sessions 1-2, 6-10



Investigation 3: Sessions 1-13Teacher Notes: Introducing Notation, p. 162



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Investigation 2: Sessions 1, 4-6Investigation 3: Session 5



Investigation 2: Session 3Investigation 3: Sessions 1-5Investigation 4: Sessions 2-4


Investigation 1: Sessions 5-6Investigation 3: Sessions 3-5: Teacher Note, p. 85



Use mathematical models to represent and understand quantitative relationships

model situations that involve the addition and subtraction of whole numbers, using

objects, pictures, and symbols.

References:


Investigation 4

How Many in All?

Investigations 2-4

Grade 1: Mathematical Thinking in Grade 1Investigation 2: Sessions 1-6










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Analyze change in various concepts

describe qualitative change, such as a student's growing taller;

Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, and Space

apply concepts of qualitative change to extend patterns related to geometric size and shape.Kindergarten students explore changing combinations of shapes, which can cover a given area,

and they explore changing arrangements of a given number of objects. Grade 1 studentsexplore changes in weather data the cyclical nature of time. While reading a story about weight

and capacity, first graders discuss gradual changes in the level at which a boat rests on asurface of water as animals climb into the boat, one at a time. Second graders describe

qualitative changes in rhythm patterns.

References:

Kindergarten: Pattern Trains and Hopscotch PathsInvestigations 1, 2, 3, 4

Making Shapes and Building BlocksInvestigations 2, 3, 4

How Many in All?

Investigation 2

Grade 1: Building Number SenseAbout Classroom Routines: Understanding Time and Changes




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About Classroom Routines: Understanding Time and Changes


Investigation 1: Sessions 1-2, 5-6About Classroom Routines: Understanding Time and Changes





describe quantitative change, such as a student's growing two inches in one year.

Kindergarten students describe quantitative change as they graph classroom attendance data

using an attendance stick and name cards or name pins to see how many children are presentand absent for each day of school. They estimate quantitative change as they compare how

many objects are in the Counting Jar (in About Classroom Routines) over several days. Theyillustrate quantitative change as they explore changing combinations of numbers with a given

sum.

First grade students progress from qualitative to quantitative weight comparisons as theygradually add weight units (e.g., washers, blocks) to achieve equilibrium on a balance scale.

They solve combining problems involving unknown change, where students are given initialand final amounts and are asked to come up with a quantitative representation of how the initial

value changed.

Second graders find quantitative change as they calculate differences between numbers on ahundred chart, and as they calculate the change necessary to begin at a given number and end at

one hundred. They explore and chart quantitative changes in time on timelines.


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In addition to the following references, each unit of study inInvestigations in Number,Data, and Spaceincludes a feature entitled, About Classroom Routines, which includes a

section related to changes over time. In Kindergarten this section is entitled, Calendar, andcontains activities related to the passage of time. In Grade 1 this section is entitled,

Understanding Time and Changes, and contains ideas for helping students develop an

understanding of time-related ideas such as sequencing events and understanding relationshipsamong time periods. In Grade 2 this section is entitled, Time and Time Again, and containssuggestions for activities to help students develop an understanding of time-related ideas such

as sequencing events, the passage of time, duration of time periods, and identifying importanttimes in their day.

References:


Investigations 1, 3

Counting Ourselves and OthersAbout Classroom Routines, p. 97

How Many in All?

Investigation 4

Grade 1: Survey Questions and Secret RulesInvestigation 4: Sessions 2-3




Grade 2: Coins, Coupons, and CombinationsInvestigation 4: Sessions 2-4: Dialogue Box, p. 120






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Geometry Standard


Analyze characteristics and properties of two- and three-dimensional geometric shapes anddevelop mathematical arguments about geometric relationships

recognize, name, build, draw, compare, and sort two- and three-dimensional shapes;

References:


Investigation 1: Choice Time, pp. 16-17, 22, 23

Making Shapes and Building BlocksInvestigations 1, 2, 3, 4, 5

Shapes Teacher Tutorial, pp. 117-154














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describe attributes and parts of two- and three-dimensional shapes;

References:

Kindergarten: Mathematical Thinking in KindergartenInvestigation 1: Choice Time, pp. 16-17, 22, 23


Investigations 1, 2, 3, 4, 5






Survey Questions and Secret RulesInvestigation 1: Sessions 1-2Investigation 2: Sessions 3-4








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investigate and predict the results of putting together and taking apart two- and three-

dimensional shapes.

References:



Grade 1: Quilt Squares and Block TownsInvestigation 1: Sessions 2-10, 13-15


Appendix: ShapesTutorial



Investigation 3: Sessions 3-4, 6Appendix: ShapesTutorial




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Specify locations and describe spatial relationships using coordinate geometry and other

representational systems

describe, name, and interpret relative positions in space and apply ideas about relative

position;In addition to physical manipulation of shapes and objects, students in Kindergarten throughGrade 2 usingInvestigations in Number, Data, and Spaceapply concepts of relative positions

in space through the use of Shapes. Shapesis a software program that allows students toconstruct and manipulate geometric shapes, see objects move according to rules they specify,

and explore rotation and reflection. Students in Grade 2 also use Geo-Logo software, whichenables students to extend their investigations to coordinate geometry and angles.

References:


Investigations 2, 3, 4

Grade 1: Building Number SenseInvestigation 1: Sessions 1-9



Investigation 3: Sessions 6-7Appendix: ShapesTutorial



Grade 2: Mathematical Thinking at Grade 2Appendix: ShapesTutorial

How Long? How Far?

Investigation 2: Sessions 2-8Ongoing Excursion: Geo-Logo: Shapes and Pictures


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describe, name, and interpret direction and distance in navigating space and apply ideas

about direction and distance;

In addition to physical movement and measurement of shapes and objects, students in

Kindergarten through Grade 2 usingInvestigations in Number, Data, and Spaceapply

concepts of direction and distance through the use of a software program entitled Shapes. Thisprogram allows students to construct and manipulate geometric shapes, see objects moveaccording to rules they specify, and explore rotation and reflection. Students in Grade 2 also

use Geo-Logo software, which enables students to extend their investigations to coordinategeometry and angles.

References:


Investigation 2, 3, 4

Grade 1: Quilt Squares and Block TownsInvestigation 3: Sessions 6-7

Appendix: Shapes Tutorial

Grade 2: Mathematical Thinking at Grade 2Appendix: ShapesTutorial



How Long? How Far?Investigation 1: Sessions 2-4

Investigation 2: Sessions 1-8Ongoing Excursion: Geo-Logo: Shapes and Pictures

find and name locations with simple relationships such as "near to" and in coordinate

systems such as maps.

Kindergarten students usingInvestigations in Number, Data, and Spaceuse HopscotchPaths as a means of exploring paths and relating them to patterns.

Grade 2 students use Geo-Logosoftware to explore coordinate geometry.References:

Kindergarten: Pattern Trains and Hopscotch PathsInvestigation 3

Grade 1: Quilt Squares and Block Towns


Grade 2: Putting Together and Taking ApartInvestigation 5: Session 8

How Long? How Far?


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Ongoing Excursion: Geo-Logo: Shapes and Pictures

Apply transformations and use symmetry to analyze mathematical situations

recognize and apply slides, flips, and turns;

Kindergarten students usingInvestigations in Number, Data, and Spaceapply slides and turnsas they visualize how a shape needs to be moved or turned in order to fit into a particular space

or design. In addition to physical manipulation of shapes and objects, students in Kindergartenthrough Grade 2 recognize and apply slides, flips, and turns through the use of a software

program, Shapes. This program allows students to construct and manipulate geometric shapes,see objects move according to rules they specify, and explore rotation and reflection.

References:


Investigations 2, 3

Grade 1: Quilt Squares and Block TownsInvestigation 1: Sessions 3-6, 8-10, 13-15

Investigation 3: Sessions 6-7Appendix: Shapes Tutorial





recognize and create shapes that have symmetry.

Students usingInvestigations in Number, Data, and Spaceare not formally introduced to theconcept of symmetry until Grade 2. Kindergarten and Grade 1 students explore preliminary

skills, including the manipulation of shapes through physical manipulation, drawing, andcomputer technology, as they construct murals and create designs and drawings using basic

shapes. They are exposed to symmetry as they examine and manipulate geometric shapes andsolids.

References:



Grade 1: Quilt Squares and Block TownsInvestigation 1: Sessions 1-15


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Use visualization, spatial reasoning, and geometric modeling to solve problems

create mental images of geometric shapes using spatial memory and spatial visualization;

Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, and Spacecreate mental images of geometric shapes using spatial memory and spatial visualization

throughout the course as they study geometric shapes and solids and their properties throughphysical manipulation, paper-and-pencil drawing, and the use of computer technology. For

example, Kindergarten students use Shapes software to manipulate shapes as they visualizehow a shape needs to be moved or turned in order to fit into a particular space or design. Grade

1 students wrap boxes at home and bring them to school, then classmates use spatial memoryand visualization as they attempt to identify the Mystery Boxes. Second graders create mental

images of geometric shapes using spatial memory and visualization as they follow descriptionswritten by other students to build rectangles, and as they draw rectangles from memory and

describe them in writing.References:



Investigations 1, 2, 3, 4, 5Shapes Teacher Tutorial, pp. 117-154








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Number Games and Story ProblemsInvestigation 1: Sessions 1-6Investigation 3: Sessions 6-8


Investigation 3: Sessions 1-6Appendix: ShapesTutorial




recognize and represent shapes from different perspectives;

Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, and Spacerecognize and represent shapes from different perspectives throughout the course as they study

geometric shapes and solids and their properties through physical manipulation, paper-and-pencil drawing, and the use of computer technology. For example, Kindergarten students

match a three-dimensional block to a two-dimensional outline of one of the blocks faces.Grade 1 students draw two-dimensional representations of three-dimensional objects, and

construct three-dimensional objects from two-dimensional representations. Second graderscreate two- and three-dimensional symmetrical structures with mirrors, paper folding, and

geoblocks.References:










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Putting Together and Taking ApartInvestigation 5: Session 8

How Long? How Far?



recognize geometric shapes and structures in the environment and specify their location.

References:Kindergarten: Making Shapes and Building Blocks


Grade 1: Quilt Squares and Block TownsInvestigation 1: Session 1


Grade 2: Shapes, Halves, and SymmetryInvestigation 1: Session 1


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Measurement Standard


Understand measurable attributes of objects and the units, systems, and processes of

measurement

recognize the attributes of length, volume, weight, area, and time;

In addition to the following references, each unit of study inInvestigations in Number, Data,

and Spaceincludes a feature entitled, About Classroom Routines, which includes a sectionrelated to changes over time. In Kindergarten this section is entitled Calendar and contains

activities related to the passage of time. In Grade 1 this section is entitled Understanding Timeand Changes and contains ideas for helping students develop an understanding of time-related

ideas such as sequencing events and understanding relationships among time periods. In Grade

2 this section is entitled Time and Time Again and contains suggestions for activities to helpstudents develop an understanding of time-related ideas such as sequencing events, the passageof time, duration of time periods, and identifying important times in their day.

References:


Investigation 3

Collecting, Counting, and MeasuringInvestigation 3



How Many in All?

Investigation 1











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How Long? How Far?





compare and order objects according to these attributes;

References:


How Many in All?

Investigation 1








How Long? How Far?




understand how to measure using nonstandard and standard units;

References:


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Investigation 3

How Many in All?Investigation 1








How Long? How Far?


select an appropriate unit and tool for the attribute being measured.

References:



Investigation 3





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Grade 2: Shapes, Halves, and SymmetryInvestigation 1: Sessions 2-6Investigation 2: Sessions 4-5

How Long? How Far?


Apply appropriate techniques, tools, and formulas to determine measurements

measure with multiple copies of units of the same size, such as paper clips laid end to end;References:


How Many in All?

Investigation 1





Grade 2: How Long? How Far?



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use repetition of a single unit to measure something larger than the unit, for instance,

measuring the length of a room with a single meterstick;

Kindergarten students usingInvestigations in Number, Data, and Spacequantify length

measurements using a repeating nonstandard unit. The following reference for Kindergarten is

to the estimation of a measurement of something much larger than the measuring unit, beforeactually measuring with multiple copies of units of the same size to confirm the estimate.Grade 1 students measure the capacity of a cup by repeatedly filling a spoon with sand and

emptying it into the cup. They also measure distances using their hands and feet. Secondgraders measure the length of their classroom with movable sets of linked cubes.

References:

Kindergarten: How Many in All?

Investigation 1

Grade 1: Bigger, Taller, Heavier, Smaller

Investigation 2: Session 1Investigation 3: Session 2

Grade 2: How Long? How Far?


use tools to measure;

References:


Investigation 3


How Many in All?

Investigation 1




Bigger, Taller, Heavier, SmallerInvestigation 1: Sessions 1-6Investigation 2: Sessions 1-7




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How Long? How Far?



develop common referents for measures to make comparisons and estimates.

In addition to the following references, Kindergarten students usingInvestigations in Number,

Data, and Spacedevelop common referents for units of time in throughout the course throughthe recurring feature, About Classroom Routines: Calendar: A Sense of Time.

References:


Investigation 3








Grade 2: How Long? How Far?Investigation 1: Sessions 1-8




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Data Analysis and Probability Standard


Formulate questions that can be addressed with data and collect, organize, and display

relevant data to answer them

pose questions and gather data about themselves and their surroundings;

In addition to the following references, the end of each unit ofInvestigations in Number, Data,and Spacecontains a feature entitled About Classroom Routines. In Kindergarten this includes

a section entitled, Todays Question, which consists of an activity involving students collecting,displaying, and interpreting data. Students may represent data using charts or graphs. The

Grade 1 series includes a section in the appendix entitled, Exploring Data, which includes ideasfor class surveys which involve students collecting, organizing, and displaying data. Another

section, Understanding Time and Changes, includes ideas for exploring the monthly calendarand collecting and displaying weather data. The Grade 2 series includes a section entitled,

How Many Pockets?, which describes a long-term activity in which students collect, organize,and represent data on how many pockets everyone in class is wearing on a particular day.

Students may use a Hundred Number Wall Chart and a Pocket Data Chart to interpret andrecord their data.

References:


Investigation 1

Counting Ourselves and OthersInvestigations 1-3







Coins, Coupons, and CombinationsInvestigation 1: Session 11

Investigation 2: Sessions 2, 6, 10Investigation 5: Session 1

Does It Walk, Crawl, or Swim?


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How Many Pockets? How Many Teeth?





sort and classify objects according to their attributes and organize data about the objects;

Kindergarten: Kindergarten students usingInvestigations in Number, Data, and Spacesort,classify, and order objects in a variety of settings. Students explore and compare the attributes

of various geometric manipulatives, including pattern blocks and geoblocks. Students classifyand sort data. They classify shapes in the environment.

Sample References:

Mathematical Thinking in Kindergarten

Investigation 1: Choice TimeInvestigation 4: Choice Time: Staircase Patterns



Investigation 6



Investigation 4



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Investigation 5

Grade 1: In addition to the following references, every text in the Grade 1Investigations in

Number, Data, and Spaceseries includes a description of sorting games in the appendix of thetext, entitled About Classroom Routines. The first game, entitled, Guess My Rule, requiresstudents to determine the common attribute of a set of objects. After students have become

familiar with this game, they can play Guess My Object, in which students ask yes-or-noquestions, based on attributes, and use the process of elimination to determine which particular

object was secretly chosen from a set of objects.

Sample References:



Quilt Shapes and Block Towns


Grade 2:Second graders sort, classify, and order objects using a number of techniques in avariety of situations. For example, students sort manipulative materials based on their

attributes. They sort and classify information. They sort two-dimensional geometric shapesand three-dimensional geometric solids. They use Venn diagrams to show relationships within

a group of related objects.Sample References:

Mathematical Thinking at Grade 2Investigation 1: Sessions 2-4






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represent data using concrete objects, pictures, and graphs.

In addition to the following references, the end of each unit ofInvestigations in Number, Data,and Spacecontains a feature entitled About Classroom Routines. In Kindergarten this includes

a section entitled, Todays Question, which consists of an activity involving students collecting,

representing, and interpreting data. Students may represent data using charts or graphs.Students are asked to think about what a graph represents and what it is communicating. TheGrade 1 series includes a section in the appendix entitled, Exploring Data, which includes ideas

for class surveys which involve students collecting, organizing, and displaying data. Anothersection, Understanding Time and Changes, includes ideas for exploring the monthly calendar

and collecting and displaying weather data. The Grade 2 series includes a section entitled,How Many Pockets?, which describes a long-term activity in which students collect, organize,

and represent data on how many pockets everyone in class is wearing on a particular day.Students may use a Hundred Number Wall Chart and a Pocket Data Chart to interpret and

record their data.References:



Investigations 1-3




Investigation 2: Session 1: Teacher NoteInvestigation 3: Sessions 1-2




Number Games and Story ProblemsInvestigation 1: Session 6





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Investigation 2: Sessions 2, 6





How Long? How Far?


How Many Pockets? How Many Teeth?Investigation 1: Sessions 1-5




Select and use appropriate statistical methods to analyze data

describe parts of the data and the set of data as a whole to determine what the data show.

In addition to the following references, the end of each unit ofInvestigations in Number, Data,

and Spacecontains a feature entitled About Classroom Routines. In Kindergarten this includesa section entitled, Todays Question, which consists of an activity involving students collecting,

displaying, and interpreting data. Students may use charts or graphs to describe parts of thedata and the set of data as a whole to determine what the data show. The Grade 1 series

includes a section in the appendix entitled, Exploring Data, which includes ideas for classsurveys which involve students collecting, organizing, displaying, and interpreting data.

Another section, Understanding Time and Changes, includes ideas for exploring the monthlycalendar and collecting, displaying, and interpreting weather data. The Grade 2 series includes

a section entitled, How Many Pockets?, which describes a long-term activity in which studentscollect, organize, represent, and describe data on how many pockets everyone in class is

wearing on a particular day. Students may use a Hundred Number Wall Chart and a PocketData Chart to interpret and record their data.

References:



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Investigation 1

Counting Ourselves and OthersInvestigations 1-3

Grade 1: Mathematical Thinking at Grade 1Investigation 4: Session 4Investigation 5: Sessions 1-6


Investigation 1: Sessions 5-6Investigation 2: Session 1: Teacher Note


Survey Questions and Secret RulesInvestigation 2: Sessions 1-2, 5-6








Investigation 1: Session 11Investigation 2: Sessions 2, 6, 10






How Long? How Far?



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Timelines and Rhythm PatternsInvestigation 1: Sessions 1-6Investigation 2: Sessions 2-3

Develop and evaluate inferences and predictions that are based on data

discuss events related to students' experiences as likely or unlikely.

Students using theInvestigations in Number, Data, and Spaceseries are introduced to the

concepts of probability in Grade 3. Students in Kindergarten through Grade 2 may predictfuture events based on collected data. For example, Kindergarten students predict whether or

not all of their sunflower seeds will germinate. Grade 1 students hypothesize about attendancedata on a most unusual day. Second graders make a hypothesis based on sampling and the

representation of a set of mystery data.Sample References:

Kindergarten: Counting Ourselves and OthersInvestigation 3: Dialogue Box, pp. 74-75

Grade 1: Survey Questions and Secret Rules


Grade 2: How Many Pockets? How Many Teeth?Investigation 2: Session 6

Understand and apply basic concepts of probability

Students using theInvestigations in Number, Data, and Spaceseries are introduced to theconcepts of probability in Grade 3. Students in Kindergarten through Grade 2 may predict


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future events based on collected data. For example, Kindergarten students predict whether ornot all of their sunflower seeds will germinate. Grade 1 students hypothesize about attendance

data on a most unusual day. Second graders make a hypothesis based on sampling and therepresentation of a set of mystery data.

Sample References:

Kindergarten: Counting Ourselves and OthersInvestigation 3: Dialogue Box, pp. 74-75

Grade 1: Survey Questions and Secret RulesInvestigation 4: Sessions 4-5

Grade 2: How Many Pockets? How Many Teeth?



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Problem Solving Standard

Instructional programs from kindergarten through grade 12 should enable all students

to

build new mathematical knowledge through problem solving;

Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, and Spacebuild new mathematical knowledge through problem solving throughout the course; in fact,

this is a fundamental emphasis of the series. For example, as Kindergarten studentsinvestigate properties of two-dimensional shapes, they learn to recognize these shapes in

their environment and develop vocabulary to describe and name two-dimensional shapes.Students in Grade 1 learn how to solve problems involving combining with unknown change.

Second graders are introduced to the use of Venn diagrams to organize data considering twoattributes at the same time.

Sample References:


Pattern Trains and Hopscotch PathsInvestigation 4


Investigation 1



Investigation 1






Survey Questions and Secret RulesInvestigation 1: Session 6



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Does It Walk, Crawl, or Swim?Investigation 1: Session 6




How Long? How Far?






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solve problems that arise in mathematics and in other contexts;

Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, and Spacesolve problems that arise in mathematics and in other contexts throughout the course; in fact,

this is a fundamental emphasis of the series. For example, Kindergarten students solve story

problems involving combining and separating. Students in Grade 1 plan and build a townusing geoblocks. Second graders explore coins and their values.Sample References:



Investigation 3



Investigation 2


How Many in All?

Investigation 3




Survey Questions and Secret RulesInvestigation 3: Session 1








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Timelines and Rhythm PatternsInvestigation 1: Session 3

apply and adapt a variety of appropriate strategies to solve problems;

Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, and Space

apply and adapt a variety of appropriate strategies to solve problems throughout the course;in fact, this is a fundamental emphasis of the series. For example, Kindergarten students

develop a variety of strategies for counting and keeping track of quantities. Students inGrade 1 use pictures, numbers, words, and equations to solve combining and separating story

problems. Second graders explore strategies for creating and representing symmetricalfigures.

Sample References:


Investigation 1



Investigation 2




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How Many in All?

Investigation 2


















How Long? How Far?






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monitor and reflect on the process of mathematical problem solving.

Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, and Spacemonitor and reflect on the process of mathematical problem solving throughout the course; in

fact, this is a fundamental emphasis of the series. For example, Kindergarten students reflect

on using different criteria to sort the same set of objects. Students in Grade 1 develop,describe, and justify techniques for comparing the capacities of containers. Second gradersobserve the structure and patterns on a hundred chart.

Sample References:


Investigation 4



Investigation 3



Investigation 4




Building Number SenseInvestigation 2: Session 2









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Shapes, Halves, and SymmetryInvestigation 1: Session 1






Timelines and Rhythm PatternsInvestigation 1: Session 6


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Reasoning and Proof Standard

Instructional programs from kindergarten through grade 12 should enable all students

to

recognize reasoning and proof as fundamental aspects of mathematics;

Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, and

Spacerecognize reasoning and proof as fundamental aspects of mathematics as they employmathematical reasoning in a variety of forms and settings. Students use inductive reasoning

as they generalize solution processes and draw conclusions from several trials or examples.They apply inductive reasoning as they sort and classify objects, identify similar attributes,

and identify and extend patterns.Students at all grade levels reason deductively as they use models, known facts, and

properties to draw conclusions. They reason by determining and applying relationships

between operations on numbers, between elements in patterns, between geometric shapes andsolids, and between events in time. They apply logical reasoning to solve problemsinvolving classification and sorting of objects by one or more attributes. They use spatial

reasoning as they visualize results of geometric transformations and identify shapes andsolids from different perspectives.

Sample References:


Investigation 4



Investigation 3



Investigation 3




Investigation 3: Sessions 5-7Survey Questions and Secret Rules



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How Long? How Far?





make and investigate mathematical conjectures;

Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, andSpacemake and investigate mathematical conjectures throughout the course by conducting


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the investigations into which the curriculum is organized; in fact, this is a fundamentalemphasis of the series. For example, Kindergarten students predict what comes next in a

pattern. Students in Grade 1 make and investigate conjectures regarding sorting rules.Second graders make conjectures regarding the identity of mystery photos.

Sample References:




Investigation 2



Investigation 3







Quilt Squares and Block TownsInvestigation 2: Session 7







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How Long? How Far?


How Many Pockets? How Many Teeth?Investigation 2: Session 6



develop and evaluate mathematical arguments and proofs;

Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, andSpacedevelop and evaluate mathematical arguments and informal proofs throughout the

course as they conduct and evaluate the investigations on which the curriculum is based. Forexample, Kindergarten students develop mathematical arguments for using one-to-one and

two-to-one correspondences. Students in Grade 1 develop and evaluate mathematicalarguments for solving combining and separating story problems. Second graders find the

missing part in a separating situation and record their strategies.Sample References:



Investigation 3



Investigation 1Making Shapes and Building Blocks

Investigation 4



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How Long? How Far?


How Many Pockets? How Many Teeth?Investigation 3: Session 5


select and use various types of reasoning and methods of proof.


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Students in Kindergarten through Grade 2 usingInvestigations in Number, Data, andSpacerecognize reasoning and proof as fundamental aspects of mathematics as they employ

mathematical reasoning in a variety of forms and settings. Students use inductive reasoningas they generalize solution processes and draw conclusions from several trials or examples.

They apply inductive reasoning as they sort and classify objects, identify similar attributes,

and identify and extend patterns.Students at all grade levels reason deductively as they use models, known facts, andproperties to draw conclusions. They reason by determining and applying relationships

between operations on numbers, between elements in patterns, between geometric shapes andsolids, and between events in time. They apply logical reasoning to solve problems

involving classification and sorting of objects by one or more attributes. They use spatialreasoning as they visualize results of geometric transformations and identify shapes and

solids from different perspectives.Sample References:



Investigation 4



Investigation 1


How Many in All?

Investigation 2









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Investigation 2: Sessions 2, 6-9Investigation 3: Sessions 6-8



Investigation 2: Sessions 1, 6Investigation 3: Sessions 1-4, 6


Investigation 1: Sessions 1, 4-6, 8-10Investigation 2: Sessions 1-5, 7-10




Investigation 3: Sessions 3-5Investigation 4:

principles standards

Documents