# READ ME FIRST. Investigations 2012 for the Common Core State Standards A focused, comprehensive, and cohesive program for grades K-5

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2 Number and Operations Whole Number Operations Understanding and working with an array model of multiplication Unit 1 Math Focus Points using arrays to model multiplication situations breaking an array into parts to find the product represented by the array using arrays to find factors of 2-digit numbers identifying features of numbers, including prime, square, and composite numbers (1.3, 2.3, 2.4, 2.5) Whole Number Operations Reasoning about numbers and their factors Unit 1 Math Focus Points finding the multiples of a number by skip counting determining whether one number is a factor or multiple of another identifying the factors of a given number identifying all the factors of 100 using knowledge of the factors of 100 to find factors of multiples of 100 using known multiplication combinations to find related multiplication combinations for a given product (e.g., if 4 x 50 = 200, then 8 x 25 = 200) using representations to show that a factor of a number is also a factor of its multiples (e.g., if 25 is a factor of 100, then 25 is also a factor of 300) Unit 3 Math Focus Points understanding the effect of multiplying by a multiple of 10 (e.g., describing the relationship between 3 x 4 and 3 x 40) finding multiples of 2-digit numbers describing a sequence of multiples in order to predict other multiples determining the effect on the product when a factor is doubled or halved Whole Number Operations Understanding and using the relationship between multiplication and division to solve Unit 3 Math Focus Points solving division story using and interpreting division notation solving division by making groups of the divisor using known multiplication combinations to solve division Unit 8 Math Focus Points representing a multiplication or division problem with pictures or diagrams, including arrays and pictures of groups using a story problem represented by a multiplication expression to keep track of parts of the problem Whole Number Operations Describing, analyzing, and comparing strategies for adding and subtracting whole numbers Unit 5 Math Focus Points representing addition and subtraction on a number line identifying, describing, and comparing addition and subtraction strategies by focusing on how each strategy starts developing arguments about why two addition expressions are equivalent (e.g., = ) using story context and representations to support explanations about equivalent addition expressions understanding the meaning of the steps and notation of the U.S. algorithms for addition and subtraction (4.4A) identifying, describing, and comparing subtraction strategies by focusing on how each strategy starts 1

3 developing arguments about how the differences represented by two subtraction expressions are related (e.g., and ) using story contexts and representations to support explanations about related subtraction expressions Whole Number Operations Understanding different types of subtraction Unit 5 Math Focus Points understanding the action of subtraction representing subtraction situations Whole Number Operations Representing the meaning of multiplication and division Unit 3 Math Focus Points representing a multiplication or division problem with pictures, diagrams, or models using arrays to model multiplication making sense of remainders in terms of the problem context creating a story problem to represent a division expression comparing visual representations of multiplication situations Whole Number Operations Understanding division as making groups of the divisor Unit 8 Math Focus Points solving division by breaking the problem into parts using multiples of 10 to solve division using the relationship between multiplication and division to solve division Whole Number Operations Developing strategies to multiply/divide 4-digit numbers Unit 8 Math Focus Points multiplying a 4-digit number by a 1-digit number (2.4A) dividing a 4-digit number by a 1-digit number (3.5A) The Base-Ten System Extending knowledge of the number system to 10,000 Unit 5 Math Focus Points reading, writing, and sequencing numbers to 1,000 and 10,000 using >, =, and < to compare numbers to 1,000 (1.2, 1.5A) to 1,000,000 (3.6A) writing numbers to 1,000 in expanded form (i.e., 435 = ) (1.3, 1.5A) to 1,000,000 (3.6A) understanding the structure of 10,000 equivalence to one thousand 10s, one hundred 100s, and ten 1,000s understanding the place value concepts to 1,000,000 (3.6A) recognizing the place value of digits in large numbers rounding numbers to the nearest ten and the nearest hundred (1.5A) to 1,000,000 (3.6A) Computational Fluency Fluency with multiplication combinations to 12 x 12 Unit 1 Math Focus Points identifying and learning multiplication combinations not yet known fluently using known multiplication combinations to determine the products of more difficult combinations solving word that involve multiplicative comparison (1.6A) 2

4 Computational Fluency Solving multiplication with 2-digit numbers Unit 3 Math Focus Points developing strategies for multiplying that involve breaking apart numbers reviewing multiplication combinations to 12 x 12 multiplying multiples of 10 Unit 8 Math Focus Points estimating solutions to 2-digit multiplication multiplying multiples of 10 solving 2-digit multiplication by breaking the problem into smaller parts and combining the subproducts solving 2-digit multiplication by changing one factor to create an easier problem Computational Fluency Adding and subtracting accurately and efficiently Unit 5 Math Focus Points adding and subtracting multiples of 10, 100, and 1,000 using multiples of 10 and 100 to find the difference between any 3-digit number and 1,000 adding 3- and 4-digit numbers using clear and concise notation for recording addition and subtraction strategies (4.4A) finding combinations of 3-digit numbers that add to 1,000 solving subtraction by breaking numbers apart solving multistep addition and subtraction combining positive and negative numbers Rational Numbers Understanding the meaning of fractions and decimal fractions Unit 6 Math Focus Points finding fractional parts of a rectangular area finding fractional parts of a group (of objects, people, etc.) interpreting the meaning of the numerator and the denominator of a fraction writing, reading, and applying fraction notation representing fractions greater than 1 identifying everyday uses of fractions and decimals reading and writing tenths and hundredths representing tenths and hundredths as parts of an area Rational Numbers Comparing the values of fractions and decimal fractions Unit 6 Math Focus Points identifying relationships between unit fractions when one denominator is a multiple of the other (e.g., halves and fourths, thirds and sixths) comparing the same fractional parts of different-sized wholes identifying equivalent fractions ordering fractions in justifying their order through reasoning about fraction equivalencies and relationships representing fractions using a number line comparing fractions to the landmarks 0, ½, 1, and 2 ordering decimals and justifying their order through reasoning about representations and meaning of the numbers identifying decimal and fraction equivalents 3

5 Computation with Rational Numbers Using representations to add and subtract rational numbers Unit 6 Math Focus Points using representations to add fractions that sum to 1 estimating sums of fractions adding fractions with the same and related denominators (e.g., halves, fourths, and eighths; thirds and sixths) using visual representations to subtract fractions with like denominators (1.8A) subtracting fractions with like denominators (1.8A) adding and subtracting mixed numbers with like denominators using representations and reasoning about fractions and the operations (2.7A) estimating sums of decimal numbers adding decimal numbers that are multiples of 0.1 and 0.25 (e.g., ) using representations to combine tenths and hundredths Computation with Rational Numbers Using representations to multiply fractions Unit 6 Math Focus Points multiplying a whole number and a fraction (3A.1, 3A.2, 3A.3) using visual models to solve word involving multiplication of a whole number and a fraction (3A.1, 3A.2, 3A.3) Computation with Measurements Solving with measurement Unit 7 Math Focus Point using the four operations to solve word involving measurements (3.5B) Patterns, Functions, and Change Using Tables and Graphs Using graphs to represent change Unit 9 Math Focus Points interpreting the points and shape of a graph in terms of the situation the graph represents finding the difference between two values on a line graph discriminating between features of the graph that represents quantity and those that represent changes in quantity plotting points on a coordinate grid to represent a situation in which one quantity is changing in relation to another identifying points in a graph with corresponding values in a table and interpreting the numerical information in terms of the situation the graph represents comparing situations by describing differences in their graphs describing the relative steepness of graphs or parts of graphs in terms of different rates of change comparing tables, graphs, and situations of constant change with those of non-constant change Using Tables and Graphs Using tables to represent change Unit 9 Math Focus Points interpreting numbers in the table in terms of the situation they represent using tables to represent the relationship between two quantities in a situation of constant change 4

6 Linear Change Describing and representing a constant rate of change Unit 9 Math Focus Points describing the relationship between two quantities in a situation of constant change, taking into account the beginning amount and a constant increase creating a representation for a situation of constant change finding the value of one quantity in a situation of constant change, given the value of the other writing an arithmetic expression for finding the value of one quantity in terms of the other in a situation of constant change making rules that relate one variable to another in situations of constant change using symbolic letter notation to represent the value of one variable in terms of another Data and Probability Data Analysis Representing data organizing ordered numerical data to describe a data set using a line plot to represent ordered numerical data representing two sets of data in order to compare them Unit 6 Math Focus Point making a line plot to display a data set of measurements involving fractions (2.7A) Data Analysis Describing, summarizing, and comparing data describing the shape of a data set: where the data are spread out or concentrated, what the highest and lowest values are, what the range is, and what the outliers are determining the range of a data set describing and interpreting data that compare to groups describing what values are typical or atypical in a data set comparing two sets of data using the shape and spread of the data finding the median of a data set using medians to compare groups considering what information a median does and does not provide Data Analysis Analyzing and interpreting data drawing conclusions based on data developing arguments based on the data Data Analysis Designing and carrying out a data investigation recording and keeping track of data considering how well a data representation communicates to an audience developing and revising a survey question Probability Describing the probability of an event associating the word probability with the likelihood of an event (3.1) arranging events along a line representing the range of certain to impossible (3.1, 3.2, 3.3, 3.5) associating verbal descriptions of probability with numeric descriptions (3.2) using numbers from 0 to 1 as measures of probability (3.2, 3.3, 3.5) 5

7 comparing the expected probability of an event with the actual results of repeated trials of that event (3.3, 3.4) Geometry Features of Shape Describing and classifying two-dimensional figures Unit 4 Math Focus Points defining polygons as closed figures with line segments as sides, and vertices classifying polygons by attribute, including number of sides, length of sides, and size of angles combining polygons to make new polygons recognizing number of sides as a descriptor of various polygons identifying parallel lines and perpendicular lines (2.3A, 2.4, 2.5) identifying right angles, acute angles, and obtuse angles (2.3A, 2.4, 2.5) identifying right triangles (2.3A, 2.5) developing vocabulary to describe the attributes and properties of quadrilaterals understanding the relationship between squares and rectangles making designs with mirror symmetry Features of Shape Describing and measuring angles Unit 4 Math Focus Points recognizing angle measure as additive (e.g., = 90, = 60, = 120 ) (3.1, 3.2, 3.3) identifying a right angle as 90 degrees measuring acute angles by relating them to 90 degrees using known angles to find a measure of other angles drawing lines, parts of lines, and angles (3.4A) understanding the relationship between the degree measure of an angle and circular arcs (3.4A) measuring angles using a protractor (3.4A) Features of Shape Describing properties of three-dimensional shapes Unit 7 Math Focus Points describing attributes of geometric solids (1.1) naming geometric solids (1.1) Features of Shape Translating between twodimensional and three-dimensional shapes Unit 7 Math Focus Points understanding how 3-D solids project silhouettes with 2-D shapes (for example, how a cone can produce both triangular and circular silhouettes) (1.2, 1.3, 1.4) decomposing images of 3-D shapes and then recombining them to make a given structure (1.3) visualizing what 3-D figures look like from different perspectives (1.3, 1.4, 2.5) recognizing how components of 3-D cube buildings come together to form the whole building (2.1) drawing silhouettes of 3-D cube buildings from different perspectives (2.2, 2.3, 2.5) integrating different silhouettes of an object, both to form a mental model and to build the whole object (2.4) Measurement Linear Measurement Measuring with standard units Unit 2 Math Focus Point using U.S. standard units to measure lengths longer than the measuring tool 6

8 Unit 4 Math Focus Points reviewing the lengths of units of measure (inches, feet, yards, centimeters, meters) using U.S. standard and metric units to accurately measure length estimating lengths based on common units (centimeter, inch, foot, yard, meter) determining when estimates or exact measurements are needed finding perimeter using standard units recognizing in explaining possible sources of measurement error comparing different paths that have the same length Unit 7 Math Focus Points converting measurements in larger units to smaller units (3.5A) making tables of equivalent measurements (3.5A) Unit 9 Math Focus Points measuring in centimeters Area Measurement Finding and understanding area Unit 4 Math Focus Points finding the area of symmetrical designs understanding that the larger the unit of area, the smaller the number of units needed to measure the area dividing irregular polygons into two shapes that have equal area finding the area of polygons by decomposing shapes finding the area of polygons using square units finding the area of rectangles finding the area of triangles in relation to the area of rectangles Volume Structuring rectangular prisms and determining their volume Unit 7 Math Focus Points seeing that cubes filling a rectangular prism can be decomposed into congruent layers finding the volume of cube buildings designing patterns for boxes that hold a given number of cubes (volume) developing a strategy for determining the volume of rectangular prisms finding the number of cubes (volume) that will fit into a box made by a given pattern doubling the number of cubes for a given box and considering how that changes the dimensions of the original box Ten-Minute Math Closest Estimate Units 8 and 9 Math Focus Points approximating numbers to nearby landmark numbers, e.g., multiples of 10 or 100 calculating mentally comparing answer choices to find the one closest to the actual answer Counting Around the Class Units 1, 3, 6, and 8 Math Focus Points finding the multiples of numbers through skip counting becoming familiar with multiplication patterns understanding the relationship between skip counting and multiplication skip counting by fractions (U6 2.7A, 3A.1, 3A.2, 3A.3) Practicing Place Value Units 5, 6, and 7 Math Focus Points recognizing and interpreting the value of each digit in three- and four-digit numbers reading and writing numbers up to 10,000 7

9 writing numbers in expanded form (i.e., 435 = ) (U5 1.3, 1.4, 2.1, 2.4, 2.5, 3.4, 3.5, 3.6A, 4.3, 4.4A, 4.4, 4.5; U6 1.8A) adding multiples of 10 to, and subtracting multiples of 10 from, three- and four-digit numbers reading and writing decimal fractions and decimal numbers adding tenths and hundredths to, and subtracting them from, decimal fractions and decimal numbers comparing two 3-, 4-, or 5-digit numbers using <, >, or = (U7 3.5A, 3.5B) Quick Images Units 1, 4, and 7 Math Focus Points organizing and analyzing visual images developing language and concepts needed to communicate about spatial relationships writing equations to describe shape patterns (Unit 1) decomposing images of 2-D shapes and then recombining them to make a given design (Unit 4) identifying parallel lines, perpendicular lines, and right triangles in images (U4 4.1, 4.2, 4.3, 4.4) decomposing images of 3-D shapes and then recombining them to make a given structure (Unit 7) Quick Survey Units 2, 6, and 9 Math Focus Points describing features of the data interpreting a set of data Today's Number Units 1, 2, 4, and 5 Math Focus Points generating equivalent expressions for a number using particular constraints practicing computation skills using notation to record expressions 8

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