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Grade Standard # Standard Description Sub Point # Sub Point Description Book/ Unit Investigation Book/ Unit (2) Investigation (2) Book/ Unit (3) Investigation (3) Book/ Unit (4) Investigation (4) Book/ Unit (5) Investigation (5) Count = 3 Kindergarten Standard 1:  Count objects through 30 and use numbers to express quantity. K.1a    Count up to 30 objects and tell how many there are in the counted group of objects. Counting Ourselves and Others How Many Are We? Collecting Counting and Measuring Counting Books Collecting Counting and Measuring Taking Inventory Kindergarten Standard 1:  Count objects through 30 and use numbers to express quantity. K.1b    Count up to 30 objects in groups of 2, 5 and 10. Number Games and Story Problems Twos, Fives, and Tens Coins, Coupons, and Combinations Groupings by Twos, Fives, and Tens Kindergarten Standard 1:  Count objects through 30 and use numbers to express quantity. K.1c    Read aloud and write the numerals 1 to 30 and match numerals with the numbers used in counting. Counting Ourselves and Others Mathematical Teaching In Kindergarten Counting Jar Mathematical Thinking In Kindergarten Attendance Building Number Sense Counting Kindergarten Standard 1:  Count objects through 30 and use numbers to express quantity. K.1d    Recognize and use correctly numerical adjectives (e.g. first, fourth, last). Kindergarten Standard 2:  Use number notation and place value up to 25. K.2a    Understand that numbers 1 through 9 represent “ones.” Kindergarten Standard 2:  Use number notation and place value up to 25. K.2b    Understand that numbers 11 through 25 consist of one or two “tens” and some “ones.” Kindergarten Standard 2:  Use number notation and place value up to 25. K.2c    Count to 100 by tens. Number Games and Story Problems Twos, Fives, and Tens Coins, Coupons, and Combinations Groupings by Twos, Fives, and Tens Kindergarten Standard 3: Compare numbers up to 30 and sets of up to 10 objects. K.3a    Compare sets of ten or fewer objects and identify which are equal to, more than, or less than others. Collecting Counting and Measuring Comparing Towers Collecting Counting and Measuring Counting and Comparing Collecting Counting and Measuring Least To Most Exploring Numbers Building Number Sense Kindergarten Standard 3: Compare numbers up to 30 and sets of up to 10 objects. K.3b    Place numbers 1 through 10 in their correct sequence. Kindergarten Standard 3: Compare numbers up to 30 and sets of up to 10 objects. K.3c     Compare numbers up to 30 (e.g. which is/has more?  Which is/has fewer?  Which is/has less?) Collecting Counting and Measuring Comparing Towers Collecting Counting and Measuring Counting and Comparing Collecting Counting and Measuring Least To Most Mathematical Thinking at Grade 1 Exploring Numbers Kindergarten Standard 4: Understand addition as putting together and subtraction as breaking apart for totals equal to or less than 10. K.4a    Understand add as “put together” and solve addition problems with numbers less than 10. How many In All? Six Tiles How many In All? Story Problems How many In All? Blue and Red Crayons Mathematical Thinking at Grade 1 Exploring Numbers Building Number Sense Addition and Subtraction Kindergarten Standard 4: Understand addition as putting together and subtraction as breaking apart for totals equal to or less than 10. K.4b    Understand subtract as “break apart” or “take away” and solve subtraction problems using numbers between 1 and 10. Building Number Sense Addition and Subtraction Kindergarten Standard 4: Understand addition as putting together and subtraction as breaking apart for totals equal to or less than 10. K.4c    Add and subtract single digit numbers who total (or difference) is between 1 and 10. Mathematical Thinking at Grade 1 Exploring Numbers Building Number Sense Building Numbers in Different Ways Building Number Sense Addition and Subtraction Kindergarten Standard 4: Understand addition as putting together and subtraction as breaking apart for totals equal to or less than 10. K.4d    Solve problems, make drawings, and create illustrative stories involving addition and subtraction of numbers between 1 and 10. How many In All? Story Problems How many In All? Story Problems How many In All? Blue and Red Crayons Mathematical Thinking at Grade 1 Exploring Numbers Building Number Sense Addition and Subtraction Kindergarten Standard 4: Understand addition as putting together and subtraction as breaking apart for totals equal to or less than 10. K.4e    Be able to write expressions (e.g. 5 + 2 or 7 – 3) to represent situations involving addition or subtraction of numbers less than 10. Building Number Sense Addition and Subtraction Kindergarten Standard 4: Understand addition as putting together and subtraction as breaking apart for totals equal to or less than 10. K.4f     Use the commutative property to reduce the number of addition facts that must be learned. Kindergarten Standard 4: Understand addition as putting together and subtraction as breaking apart for totals equal to or less than 10. K.4g    Use addition facts to solve subtraction problems. Kindergarten Standard 5: Compose and decompose numbers 2 through 10. K.5a Understand that numbers greater than 2 can be decomposed in several different ways. Collecting Counting and Measuring Six Tiles Kindergarten Standard 5: Compose and decompose numbers 2 through 10. K.5b Recognize 6 through 10 as a five and some ones. Kindergarten Standard 6: Compare the length, weight and capacity of objects. K.6a Make direct comparisons between objects (e.g. recognize which is shorter, longer, taller, lighter, heavier, or holds more). Collecting Counting and Measuring Comparing Towers Collecting Counting and Measuring Least To Most Bigger Taller Heavier Smaller Weighing and Balancing Bigger Taller Heavier Smaller Filling Bigger Taller Heavier Smaller Measuring Length Kindergarten Standard 6: Compare the length, weight and capacity of objects. K.6b Recognize that length, weight, and capacity are preserved when objects are decomposed and recomposed. Kindergarten Standard 6: Compare the length, weight and capacity of objects. K.6c Recognize tools used to measure length, weight and capacity. Bigger Taller Heavier Smaller Weighing and Balancing Kindergarten Standard 7: Recognize and use words that represent periods of time. K.7a Recognize day, night, morning, afternoon, evening, yesterday, today, tomorrow. Mathematical Thinking In Kindergarten Calendar Kindergarten Standard 7: Recognize and use words that represent periods of time. K.7b Recognize the role of clocks or calendars in measuring and keeping track of time. Mathematical Thinking at Grade 1 Calendar Mathematical Thinking In Kindergarten Daily Schedule Kindergarten Standard 7: Recognize and use words that represent periods of time. K.7c Identify daily landmark times such as bedtime or lunch time. Mathematical Thinking In Kindergarten Daily Schedule Kindergarten Standard 8: Create, explore and describe shapes K.8a Identify common shapes such as rectangles, circles and triangles. Making Shapes and Building Blocks 2D shapes Around Us Kindergarten Standard 8: Create, explore and describe shapes K.8b Make compound pictures by putting together simple geometric shapes. Making Shapes and Building Blocks Exploring Shapes With the Computer Making Shapes and Building Blocks Making Shapes and Building Blocks Kindergarten Standard 9: Identify, sort and classify objects. K.9a Sort by attribute and identify objects that do not belong in a particular group. Counting Ourselves and Others What Did You Eat for Lunch? Counting Ourselves and Others Collecting Data About Our Class Kindergarten Standard 10: Create, describe and extend simple patterns. K.10a Use repeating, growing and shrinking patterns. Pattern Trains and Hopscotch Paths Exploring Patterns Pattern Trains and Hopscotch Paths What Comes Next? Pattern Trains and Hopscotch Paths Pattern Borders 1st Grade Standard 1: Understand and use number notation and place value up to 100. 1.1a Count to 100 by ones and tens, and relate these number words to groups of tens and ones. Number Games and Story Problems Investigation 2, sessions 6-8 and 10-12 1st Grade Standard 1: Understand and use number notation and place value up to 100. 1.1b Read and write numbers up to 100 in numerals and in words. Building Number Sense Investigation 3, sessions 1 and 2 Number Games and Story Problems Investigation 2, sessions 6-8 1st Grade Standard 1: Understand and use number notation and place value up to 100. 1.1c Recognize the place value of numbers (hundreds, tens & ones). Coins, Coupons and Combinations(2nd Grade) Investigation 4, sessions 2-4 and daily class routine with # of the day Putting Together and Taking Apart(2nd Grade) All of Investigation 1, specifically sessions 1 and 2 emphasize the strategy of manipulating numbers. Place Value is taught as a daily routine of manipulating numbers with addition and subtraction. 1st Grade Standard 2: Compare numbers up to 100 and arrange them in numerical order. 1.2a Arrange numbers in increasing and decreasing order. Building Number Sense Investigation 3, sessions 1 and 2 Note that students do not anything in decreasing order. 1st Grade Standard 2: Compare numbers up to 100 and arrange them in numerical order. 1.2b Locate numbers up to 100 on the number line. Building Number Sense Investigation 3, sessions 1 and 2, and 5-7 Number Games and Story Problems Investigation 2, sessions 6-8 In both units using the 100's chart instead of number line. 1st Grade Standard 2: Compare numbers up to 100 and arrange them in numerical order. 1.2c Compare two or more sets of objects in terms of the difference in the number of elements. Mathematical Thinking at Grade 1 Investigation 2, sessions 1-3 1st Grade Standard 3: Understand how to add, subtract, compose and decompose numbers up to 100. 1.3a Use efficient methods to add and subtract numbers to 100, and to solve associated word problems. Building Number Sense Investigation 1, sessions 1,2, 7-9; all of Investigation 2; Investigation 3, sessions 5-7; all of Investigation 4 Number Games and Story Problems All of Investigation 1; Investigation 2, sessions 1-8 and 10-12; all of Investigation 3 Methods are used for quantities up to 40. Putting Together and Taking Apart(2nd Grade) All of Investigation 2; All of Investigation 4; Investigation 5, sessions 1, 4, 5, and 7 Coins, Coupons and Combinations (2nd Grade) All of Investigation 3 1st Grade Standard 3: Understand how to add, subtract, compose and decompose numbers up to 100. 1.3b Understand how to compose and decompose numbers up to 100. Building Number Sense Investigation 1, sessions 1,2 and 5-9; all of Investigation 2; Investigation 3, sessions 1,2, 5-7; all of investigation 4 Number Games and Story Problems All of Investigation 1; Investigation 2, sessions 1-8 and 10-12; all of Investigation 3 Putting Together and Taking Apart(2nd Grade) All of Investigation 3; All of Investigation 4; Investigation 5, sessions 2 and 3 1st Grade Standard 3: Understand how to add, subtract, compose and decompose numbers up to 100. 1.3c Recognize that adding and subtracting are inverse operations. Number Games and Story Problems Investigation 1, sessions 2-9; all of Investigation 3 Inverse operations not explicitly taught, but briefly covered and encouraged. Putting Together and Taking Apart(2nd Grade) All of Investigation 3 The entire unit emphasizes that students can use both addition and subtraction to solve the same problem. 1st Grade Standard 3: Understand how to add, subtract, compose and decompose numbers up to 100. 1.3d Create drawings, stories and equations to illustrate addition and subtraction of numbers smaller than 100. Building Number Sense Investigation 2, sessions 1, 2 and 6-8; all of Investigation 4 Number Games and Story Problems All of Investigation 1; Investigation 2, sessions 1, 2, 4-8, and 10-12; all of Investigation 3 1st Grade Standard 3: Understand how to add, subtract, compose and decompose numbers up to 100. 1.3e Understand how to use groups of tens and ones to add numbers greater than ten. Number Games and Story Problems Investigation 1, sessions 2-9; Investigation 2, sessions 10-12; all of Investigation 3 Strategy not explicitly taught, but often encouraged in the Spring to find friendly 10 combinations. 1st Grade Standard 3: Understand how to add, subtract, compose and decompose numbers up to 100. 1.3f Solve verbal problems by using diagrams, numbers and symbolic expressions. Building Number Sense Investigation 4, sessions 1 and 2 Number Games and Story Problems Investigation 3, sessions 1 and 2 Students express the verbal idea while the teacher records responses. 1st Grade Standard 4: Measure length, weight, capacity and time. 1.4a Use rulers, scales and containers to measure and compare the dimensions, weight and capacity (volume) of classroom objects. Bigger, Taller, Heavier, Smaller All of Investigations 1, 2 and 3 Note that students do not use rulers rather, non-standard units of measure (pencils, cubes, hands, feet). From Paces to Feet (2nd Grade) All of Investigation 2; All of Investigation 3; All of Investigation 4 1st Grade Standard 4: Measure length, weight, capacity and time. 1.4b Compare lengths, weights and capacities of pairs of objects. Bigger, Taller, Heavier, Smaller All of Investigation 1; Investigation 2, sessions 2-7; all of Investigation 3 1st Grade Standard 4: Measure length, weight, capacity and time. 1.4c Tell time from analog (round) clocks in half-hour intervals. Not covered in 1st Grade Investigations 1st Grade Standard 5: Count, add and subtract money up to $100.00. 1.5a Tell the different denominations of coins and notes. Number Games and Story Problems Investigation 2, session 3 Money is only counted up to one dollar and only penny, nickel, dime and quarter and used. 1st Grade Standard 5: Count, add and subtract money up to $100.00. 1.5b Tell the amount of money in cents up to $1 and in dollars up to $100. Number Games and Story Problems Investigation 2, session 3 The unit works with coins only. Putting Together and Taking Apart(2nd Grade) Investigation 2, sessions 5 and 6 1st Grade Standard 5: Count, add and subtract money up to $100.00. 1.5c Add and subtract money; in cents only; in dollars only (within the scope of Standard 1.3). Number Games and Story Problems Investigation 2, session 3 Money is only added. Mathematical Thinking at Grade 3 Investigation 2, sessions 5-7 1st Grade Standard 5: Count, add and subtract money up to $100.00. 1.5d Solve word problems using addition and subtraction of money; in cents only; in dollars only. Number Games and Story Problems Investigation 2, sheet 14 Cents are added up to 10 cents. Mathematical Thinking at Grade 3 Investigation 2, sessions 5-7 1st Grade Standard 6: Make picture graphs of measurements and use them to pose and solve problems. 1.6a Explain the meaning of picture graphs. Survey Questions and Secret Rules Investigation 2, sessions 1-6; Investigation 3, sessions 1 and 2; Investigation 4, sessions 2 and 3 Mathematical Thinking at Grade 1 Investigation 5, sessions 3-6 1st Grade Standard 6: Make picture graphs of measurements and use them to pose and solve problems. 1.6b Make picture graphs from collected or given data. Survey Questions and Secret Rules Investigation 2, sessions 1, 2 and 5-6; Investigation 3, session 1; Investigation 4, sessions 2-5 Mathematical Thinking at Grade 1 Investigation 5, sessions 3 and 4 Students use manipulatives and then transfer to a picture graph. 1st Grade Standard 7: Recognize, describe, draw and manipulate two-dimensional figures. 1.7a Identify circle, triangle, square and rectangle. Quilt Squares and Block Towns Investigation 1, session 1, 3-6, and 11-15; Investigation 2, sessions 1, 2, 4-6 and 7; Investigation 3, sessions 1-5 1st Grade Standard 7: Recognize, describe, draw and manipulate two-dimensional figures. 1.7b Sort objects by shape and size. Quilt Squares and Block Towns Investigation 1, sessions 11 and 12; all of Investigation 2; Investigation 3, sessions 1-5 1st Grade Standard 7: Recognize, describe, draw and manipulate two-dimensional figures. 1.7c Describe and draw shapes resulting from rotation and flips of simple two-dimensional figures. Quilt Squares and Block Towns Investigation 1, sessions 3-15 Students draw 3-D shapes and flipping takes place on the computer program. 1st Grade Standard 8: Recognize and extend simple patterns. 1.8a Sort objects and create and describe patterns by number, shapes, sizes, rhythms or colors. Survey Questions and Secret Rules All of Investigation 1; Investigation 2, sessions 3-6; Investigation 3, sessions 1 and 2; Investigation 4, sessions 2 and 3 Mathematical Thinking at Grade 1 Investigation 1, sessions 2-4; Investigation 2, sessions 2 and 3; all of Investigation 3; Investigation 4, session 5; all of Investigation 5 Continues throughout all 1st grade investigation units. Number Games and Story Problems-number patterns.  Quilt Squares and Block Towns-shapes and colors.  Bigger, Taller, Heavier, Smaller-sizes.  Mathematical Thinking at Grade 1-rhythms and colors. 1st Grade Standard 8: Recognize and extend simple patterns. 1.8b Distinguish between repeating and growing patterns. Mathematical Thinking at Grade 1 Investigation 3, sessions 1 and 2; Investigation 4, session 5 Number Games and Story Problems Investigation 2, sessions 2 and 9 Mathematical Thinking at Grade 1-repeating patterns.  Number Games and Story Problems-growing patterns 1st Grade Standard 8: Recognize and extend simple patterns. 1.8c Identify and describe the next element in simple repeating patterns. Mathematical Thinking at Grade 1 Investigation 3, sessions 1 and 2; Investigation 4, session 5 Number Games and Story Problems Investigation 2, sessions 2 and 9 2nd Grade Standard 1: Understand and use number notation and place value up to 1000. 2.1a Count to 1000 by ones, tens and hundreds. Putting Together and Taking Apart Investigation 2 Coins, Coupons, and Combinations Investigation 2 Mathematical Thinking at Grade 3 Investigation 1 Only to 100 2nd Grade Standard 1: Understand and use number notation and place value up to 1000. 2.1b Read and write numbers up to 1000 in numerals and in words. Coins, Coupons, and Combinations Investigation 4 Mathematical Thinking at Grade 3 Investigation 1 Only to 100 2nd Grade Standard 1: Understand and use number notation and place value up to 1000. 2.1c Recognize the place value of numbers (hundreds, tens & ones). Coins, Coupons, and Combinations Classroom Routines: Today's Number Putting Together and Taking Apart Investigation 5 and Classroom Routines: Today's Number Up to 200 2nd Grade Standard 1: Understand and use number notation and place value up to 1000. 2.1d Understand and utilize the relative values of the different number places. Classroom Routines: How Many Pockets covers Estimating     MTG3:Teacher Note  page 46 2nd Grade Standard 1: Understand and use number notation and place value up to 1000. 2.1e Compare numbers up to 1000. Coins, Coupons, and Combinations Investigation 4 Putting Together and Taking Apart Investigation 2 Mathematical Thinking at Grade 3 Investigation 1 Up to 100 2nd Grade Standard 2: Know how to locate numbers on the number line. 2.2a Know that on the number line larger numbers are located to the right and smaller numbers are located to the left. Lynn Kuske Lesson 2nd Grade Standard 2: Know how to locate numbers on the number line. 2.2b Know how to locate zero on the number line. Lynn Kuske Lesson 2nd Grade Standard 2: Know how to locate numbers on the number line. 2.2c Use number line pictures and manipulatives to illustrate the actions of adding and subtracting. Lynn Kuske Lesson 2nd Grade Standard 2: Know how to locate numbers on the number line. 2.2d Know how to place -1, -2, -3, … on the number line as mirror images of 1, 2, 3 … with respect to zero. Lynn Kuske Lesson 2nd Grade Standard 3: Understand how to add and subtract numbers up to 1000 and solve related word problems. 2.3a Understand how to efficiently add and subtract two- and three-digit numbers. Coins, Coupons, and Combinations Investigation 3  Investigation 4 Putting Together and Taking Apart Investigation 1, 2, 3, 4, and 5 Mathematical Thinking at Grade 3 Investigation 2      Investigation 4 Up to 100 2nd Grade Standard 3: Understand how to add and subtract numbers up to 1000 and solve related word problems. 2.3b Understand that subtracting is the inverse of adding, and use this inverse to solve problems. Putting Together and Taking Apart Investigation 3 Mathematical Thinking at Grade 3 Investigation 3 Coins, Coupons, and Combinations Investigation 3 a+b=c needs to be explicitly taught; Traditional Algorithm needs to be taught when appropriate; See CCC page 102 teacher note;  PTTA pg 32-34 Teacher Note  2nd Grade Standard 3: Understand how to add and subtract numbers up to 1000 and solve related word problems. 2.3c Solve word problems of more than one step that use numbers with three or fewer digits. Putting Together and Taking Apart Investigation 3   Investigation 5 Coins, Coupons, and Combinations Investigation 3   Investigation 4 Does not include 3 digit problems; Some problems with irrelevant information but don't have problems with insufficient information 2nd Grade Standard 3: Understand how to add and subtract numbers up to 1000 and solve related word problems. 2.3d Add and subtract mentally with ones, tens and hundreds. Putting Together and Taking Apart Investigation 5 Mathematical Thinking at Grade 3 Investigation 1 Only 2 digit numbers, no 3 digit numbers; Mental Math in the Primary (several lessons), Classroom Routines from 3rd Grade; 2nd Grade Standard 3: Understand how to add and subtract numbers up to 1000 and solve related word problems. 2.3e Create stories, drawings, and equations to illustrate arithmetic problems with unknowns of various types. Putting Together and Taking Apart Investigation 1, 2, 4, and 5 Coins, Coupons, and Combinations Investigation 1, 3, 2nd Grade Standard 4: Multiply small whole numbers by repeated addition and understand division as the inverse of multiplication. 2.4a Count by steps of 2, 3, 4, 5, and 10 and relate patterns in these counts to multiplication. Things That Come in Groups Investigation 2 Coins, Coupons, and Combinations Investigation 2 2nd Grade Standard 4: Multiply small whole numbers by repeated addition and understand division as the inverse of multiplication. 2.4b Know how to multiply and divide numbers within the 2, 3, 4, 5, and 10 multiplication table. Things That Come in Groups Investigation 2    Investigation 3 2nd Grade Standard 4: Multiply small whole numbers by repeated addition and understand division as the inverse of multiplication. 2.4c Know the multiplication table up to 5 x 5. Emphasis on Recall Stage       Supplement with Multiplication Made Easy 2nd Grade Standard 4: Multiply small whole numbers by repeated addition and understand division as the inverse of multiplication. 2.4d Recognize different interpretations of multiplication and explain why they are equivalent. Things That Come in Groups Investigation 3   Investigation 1   Investigation 2 2nd Grade Standard 4: Multiply small whole numbers by repeated addition and understand division as the inverse of multiplication. 2.4e Recognize different interpretations of division and explain why they are equivalent. Things That Come in Groups Investigation 4 more emphasis on the relationship of multiplication and division 2nd Grade Standard 4: Multiply small whole numbers by repeated addition and understand division as the inverse of multiplication. 2.4f Solve multiplication and division problems involving repeated groups and arrays of small whole numbers. Things That Come in Groups Investigation 1, 2, 3, and 4 more emphasis on the relationship of multiplication and division; no number line 2nd Grade Standard 5: Understand and use unit fractions. 2.5a Recognize that unit fractions such as ½, ⅓, ¼, …, 1/12 represent one part in the division of a whole into equal parts. Fair Shares Investigation 1 ,2 Shapes, Halves, and Symmetry Investigation 3 FS doesn't cover 1/7, 1/9, 1/11;  SHS only covers 1/2, 1/3, and 1/4 2nd Grade Standard 5: Understand and use unit fractions. 2.5b Recognize, name and compare unit fractions with denominators up to 12. Fair Shares Investigation 1, 3 2nd Grade Standard 5: Understand and use unit fractions. 2.5c Know how to place halves (e.g. ½, 3/2 or 1½, 5/2 or 2½, etc.) on the number line. Fair Shares Investigation 2 Does not place fractions on a Number Line 2nd Grade Standard 6: Add, subtract, compare and estimate measurements. 2.6a Estimate, measure, and calculate length in meters, decimeters, centimeters, yards, inches and feet. 3rd Grade Foss except for decimeters; Not very much with in. and feet 2nd Grade Standard 6: Add, subtract, compare and estimate measurements. 2.6b Measure the lengths of sides and diagonals of common two-dimensional figures such as triangles, squares, rectangles and other polygons. 3rd Grade Foss except for the Language of Perimeter and Circumference 2nd Grade Standard 6: Add, subtract, compare and estimate measurements. 2.6c Solve simple word problems involving length. 3rd Grade Foss 2nd Grade Standard 7: Understand how to tell time. 2.7a Tell, write and use time measurements from (round) clock faces and from digital clocks and translate between these two. Possible Website lessons:  www.time-for-time.com   2nd Grade Standard 7: Understand how to tell time. 2.7b Add and subtract times both mentally and on paper. Possible Website lessons:  www.time-for-time.com   2nd Grade Standard 7: Understand how to tell time. 2.7c Understand and use comparative phrases such as “in fifteen minutes”, “half an hour for now”, “ten minutes late.” Possible Website lessons:  www.time-for-time.com   2nd Grade Standard 8: Understand how to count, add and subtract money. 2.8a Read, write add and subtract money using standard decimal notation. Coins, Coupons, and Combinations Investigation 2 Need to emphasize standard decimal notation 2nd Grade Standard 8: Understand how to count, add and subtract money. 2.8b Solve simple word problems solving money. Putting Together and Taking Apart Investigation 1, 2, 3, 4, and 5 Mathematical Thinking at Grade 3 Investigation 2 2nd Grade Standard 9: Represent measurements by means of bar graphs. 2.9a Select appropriate scales and make them explicit. How Many Pockets? How Many Teeth? Investigation 1    Investigation 2 Investigation 3 is valuable because students need to come up with their own data and representation. 2nd Grade Standard 9: Represent measurements by means of bar graphs. 2.9b Use bar graphs as a context for addition and subtraction problems. How Many Pockets? How Many Teeth? Investigation 1  Investigation 2   Need to do more of these problems 2nd Grade Standard 9: Represent measurements by means of bar graphs. 2.9c Solve word problems where information is contained in a bar or picture graphs. How Many Pockets? How Many Teeth? Investigation 1  Investigation 2 Need to do more of these problems      None for Vertical representations 2nd Grade Standard 10: Recognize, classify and transform geometric figures in two and three dimensions. 2.10a Identify, describe and compare common geometric shapes in two and three dimensions. Shapes, Halves, and Symmetry Investigatin 1    Investigation 2 Circumference is missing; Three dimensional element of spheres, cones, and cylinders is missing 2nd Grade Standard 10: Recognize, classify and transform geometric figures in two and three dimensions. 2.10b Classify familiar plane and solid objects by common geometric attributes and explain which attributes are being used. Shapes, Halves, and Symmetry Investigation 1  Investigation 2 Weight is missing; The beginning of these concepts are taught in First Grade Foss: Solids and Liquids unit.  We still need more. 2nd Grade Standard 10: Recognize, classify and transform geometric figures in two and three dimensions. 2.10c Rotate, flip, and fold shapes to explore the effect of transformations. Shapes, Halves, and Symmetry Investigation 2        Investigation 4 Flips, Turns, and Area Investigation 1 2nd Grade Standard 11: Make, identify and extend patterns. 2.11a Fill in tables based on stated rules to reveal patterns.  Illluminations Algebra strand lessons; Patterns that Grow;  Counting Around the Class 3rd Grade Standard 1: Understand how to read, write, add and subtract numbers up to 10,000. 3.1a Read and write numbers up to 10,000 in numerals and in words. Landmarks in the Thousands Investigation 4 4th Grade Unit Ten-minute Math Calendar Math/Counting Around the Class (with variations) 3rd Grade Standard 1: Understand how to read, write, add and subtract numbers up to 10,000. 3.1b Recognize the place values in numbers (thousands, hundreds, tens, ones) and understand what quantities each digit represents. Landmarks in the Hundreds (ones, tens, hundreds) Mathematical Thinking in Grade 5 (thousands) Landmarks in the Thousands 3 and 4 3rd Grade Standard 1: Understand how to read, write, add and subtract numbers up to 10,000. 3.1c Compare and order numbers up to 10,000. Landmarks in the Thousands 3rd Grade Standard 1: Understand how to read, write, add and subtract numbers up to 10,000. 3.1d Understand and use grouping for addition and ungrouping for subtraction. Landmarks in the Thousands 2 and 3 3rd Grade Standard 1: Understand how to read, write, add and subtract numbers up to 10,000. 3.1e Add and subtract two-digit numbers mentally. Mathematical Thinking at Grade 3 Combining and Comparing Investigation 1, 2, &3 MT@G3 is taught in 2nd grade Mental Math in the Middle Grades 3rd Grade Standard 1: Understand how to read, write, add and subtract numbers up to 10,000. 3.1f Judge the reasonableness of estimated answers. Comes up in Ten-Minute Math activities, but is never explicitly taught. 3rd Grade Standard 1: Understand how to read, write, add and subtract numbers up to 10,000. 3.1g Solve a range of addition and subtraction problems with the unknown in all possible positions. Combining and Comparing All Investigations Mathematical Thinking in Grade 4 Investigation 3 Landmarks in the Thousands Investigation 2 and 3 Money, Miles, and Large Numbers Investigation 1 2nd Grade - How Long? How Far? 3rd Grade Standard 2: Understand how to multiply and divide with simple numbers. 3.2a Express a multiplication statements in terms of division and vice-versa. Things that Come in Groups Investigation 4 Arrays and Shares Packages and Groups Does not use the terms product, quotient, or remainder In the 3rd and 4th grade units there is never a standard expectation of when basic multiplication facts should be memorized and to what extent (10x10? 12x12?) 3rd Grade Standard 2: Understand how to multiply and divide with simple numbers. 3.2b Know the multiplication table up to 10 x 10. Work with mutliplication facts up to 10x10, but are not expected to be memorized. Nimble with Numbers (3/4 or 4/5) Multiplication Facts Made Easy, anything from this book. 3rd Grade Standard 2: Understand how to multiply and divide with simple numbers. 3.2c Understand and solve multiplication and division problems that involve groups, area and arrays. Things that Come in Groups Investigation 1, 3 Arrays and Shares Packages and Groups Investigation 3 3rd Grade Standard 2: Understand how to multiply and divide with simple numbers. 3.2d Describe contexts for multiplication and division facts. Things that Come in Groups Investigation 1, 3, & 5 Arrays and Shares 3rd Grade Standard 2: Understand how to multiply and divide with simple numbers. 3.2e Multiply numbers up to 100 by numbers up to 10. Things that Come in Groups Investigation 5 Arrays and Shares Investigation 3 Packages and Groups Investigations 1,2, and 3 3rd Grade Standard 2: Understand how to multiply and divide with simple numbers. 3.2f Understand the inverse relationship between addition and subtraction and between multiplication and division, and the commutative and associative properties of multiplication and addition. Packages and Groups Investigation 3 Constant exposure but never explicitly taught. 3rd Grade Standard 2: Understand how to multiply and divide with simple numbers. 3.2g Make and test conjectures that probe understanding of the operations of addition, subtraction, multiplication and division. Nimble with Numbers Possible Equations Strike 3rd Grade Standard 3: Solve contextual, experiential, and word problems requiring more than one arithmetic operation. 3.3a Represent problems mathematically using diagrams, numbers and symbolic expressions.  Express answers in appropriate verbal or numerical form. Combining and Comparing Investigation 4 Money, Miles, and Large Numbers Investigation 1 Packages and Groups (4th grade) Arrays and Shares (4th grade) Nimble with Numbers (4/5, 3/4) Detect the Digits/Detecting Digits 3rd Grade Standard 3: Solve contextual, experiential, and word problems requiring more than one arithmetic operation. 3.3b To avoid excess reliance on verbal skills, use real contexts as well as written “word problems” as prompts for tasks. Things that Come in Groups Investigation 1 Landmarks in the Hundreds Investigation 2 Fair Shares Investigation 3 The types of word problems in this standard are only brought up once in each unit.  Teachers should be aware of their need to modify wording of word problems to meet this standard.  3rd Grade Standard 4: Understand and use fractions. 3.4a Understand that a fraction is defined in relation to a whole.  Specifically, if some who is divided into d equal parts, the fraction n/d represents n of these parts. Fair Shares Investigation 1 (sub-point concept #2) Flips, Turns, and Area sub-point concept #1 Different Shapes, Equal Pieces Sub-point concept # 1, 2, 4, and exposed to 5 Terms numerator and denominator are never used. 3rd Grade Standard 4: Understand and use fractions. 3.4b Recognize from the definition of fraction that a fraction n/d is equal to 1 when n = d, is less than 1 when n < d, and is greater than 1 when n > d. Fair Shares Investigation 2 3rd Grade Standard 4: Understand and use fractions. 3.4c Be able to compare two fractions with the same denominator. Different Shapes, Equal Pieces 3rd Grade Standard 4: Understand and use fractions. 3.4d Locate fractions with denominators 2, 4, 8 and 10 on the number line. Fair Shares Different Shapes, Equal Pieces Never use 10 in 3rd grade 3rd Grade Standard 5: Understand what it means to add and subtract fractions with equal denominators. 3.5a Recognize how adding and subtracting fractions with equal denominators can be thought of as the joining and taking away, respectively, of contiguous segments of the number line. Fair Shares All Investigations Different Shapes, Equal Pieces Nimble with Numbers (4/5) Fraction Tic-Tac-Toe 3rd Grade Standard 5: Understand what it means to add and subtract fractions with equal denominators. 3.5b Understand that a fraction n/d is the sum of n unit fractions of denominator d. Fair Shares All Investigations Different Shapes, Equal Pieces 3rd Grade Standard 5: Understand what it means to add and subtract fractions with equal denominators. 3.5c Add and subtract fractions with equal denominators flexibly, efficiently and accurately. Fair Shares All Investigations Different Shapes, Equal Pieces 3rd Grade Standard 6: Understand and use the idea of equivalent fractions. 3.6a Know the meaning of equivalent fractions, namely that two fractions are equivalent if they refer to the same amount with respect to some whole. Fair Shares Investigation 1 & 2 Different Shapes, Equal Pieces Nimble with Numbers (5/6) Ten Tallies Win, If…Then… *Equilivant fractions is covered in similar ways in both 3rd and 4th grades.  Never covered deeply: students are not necessarily seeing that 9/6 is the same as 1 1/2. 3rd Grade Standard 6: Understand and use the idea of equivalent fractions. 3.6b Understand, for example, why the fact that 3 x 3 = 9 and 3 x 4 = 12 makes ¾ equivalent to 9/12. Fair Shares Investigation 3 Different Shapes, Equal Pieces Nimble with Numbers (4/5) Finding Fractions Alluded to, but not directly taught 3rd Grade Standard 6: Understand and use the idea of equivalent fractions. 3.6c Use equivalent fractions to compare fractions with denominators 2, 4, 8 and 10. Fair Shares Investigation 1 & 2 Different Shapes, Equal Pieces Never 10 3rd Grade Standard 7: Recognize why measurements need units, and know how to use common units. 3.7a Know the names, abbreviations, and approximate magnitudes of common units: Measurement (science) Paces to Feet (2nd grade) Combining and Comparing (money) Investigation 1 & 2 Science is metric based, Paces to Feet talks about length.  Combining and Comparing uses money. 3rd Grade Standard 7: Recognize why measurements need units, and know how to use common units. 3.7b Know common within-system equivalencies: Measurement (science) Metric system - 1 meter = 100 cm, 1 kg = 1000 g, 1 l = 1000ml Paces to Feet (2nd grade) Investigation 2 - 12 inches=1 foot Combining and Comparing (money) Measurement Benchmarks (5th grade) - Investigation 3 - It's About Time - might be used in 3rd to cover time measurement standards) 3rd Grade Standard 7: Recognize why measurements need units, and know how to use common units. 3.7c Chose appropriate units of measure, estimate common measurements, and use tools to make measurements. Measurement (science) meter stick w/ cm Paces to Feet (2nd grade) Investigation 2 - inch stick = 1 ft Nimble with Numbers (4/5) How Many? How Much? 3rd Grade Standard 7: Recognize why measurements need units, and know how to use common units. 3.7d Know that the standard unit of volume is a unit cube (a cube of side one unit) and that the capacity (volume) of a container is the number of unit cubes that fill the inside. Landmarks in the Thousands (4th grade) Containers and Cubes (5th grade) Investigation 1 Exploring Solids and Boxes (grade 3 - not taught) Investigation 4 Might be best to draw from ES&B instead of 5th 3rd Grade Standard 7: Recognize why measurements need units, and know how to use common units. 3.7e Use decimal notation to express, add and subtract amounts of money. Combining and Comparing Investigation 3 - money Landmarks in the Hundreds Investigation 2 - money Measurement (science) metric system Coins, Cupons, and Combinations (2nd grade - money - investigation 2) (Mathematical Thinking at Grade 2 goes over coin values - Investigation 4) 3rd Grade Standard 7: Recognize why measurements need units, and know how to use common units. 3.7f Solve problems requiring the addition and subtraction of lengths, weights, capacities, times and money. Combining and Comparing Investigation 3 - money Landmarks in the Hundreds Investigation 2 - money Measurement (science) metric system Money, Miles, and Large Numbers (4th grade) Math Thinking at Grade 4 does a lot of work with coin values. 3rd Grade Standard 8: Recognize basic elements of geometric figures and use them to describe shapes. 3.8a Identify points, rays, line segments, lines, planes, and right angles in everyday settings. Sunken Ships and Grid Patterns (4th grade) Turtle Paths (3rd - not taught) Picturing Pollygons (5th grade) Sunken Ships and Grid Patterns should be taught in 3rd grade after Up and Down the Number Line. 3rd Grade Standard 9: Identify and draw perpendicular, parallel, and non-parallel lines and planes. 3.9a Know that the edges of a polygon are called parallel or perpendicular if they lie on parallel or perpendicular lines, respectively.  Similarly, know that the faces of a three dimensional solid are called parallel or perpendicular if they lie in parallel o Picturing Pollygons (5th) Investigation 2 - parallel and types of angles Nope. 3rd Grade Standard 9: Identify and draw perpendicular, parallel, and non-parallel lines and planes. 3.9b Recognize parallel and perpendicular line segments and planes in everyday settings. Picturing Pollygons (5th) Investigation 2 - parallel and types of angles Nope. 3rd Grade Standard 9: Identify and draw perpendicular, parallel, and non-parallel lines and planes. 3.9c Be able to draw perpendicular, parallel, and non-parallel line segments using rulers and squares. Picturing Pollygons (5th) Investigation 2 - parallel and types of angles Nope. 3rd Grade Standard 9: Identify and draw perpendicular, parallel, and non-parallel lines and planes. 3.9d Be able to identify when two planes are perpendicular or parallel. Picturing Pollygons (5th) Nope. 3rd Grade Standard 10: Explore and identify familiar two- and three-dimensional shapes. 3.10a Describe and classify plane figures (e.g. circle, triangle, square, rectangle, polygon) and solid shapes (e.g. sphere, pyramid, cube, rectangular prism) according to the number and shape of faces, edges and vertices. Exploring Solids and Boxes Investigation 1-3 Seeing Solids and Silhouettes (4th grade) Sunken Ships and Grid Patterns Investigation 2 We would like Exploring Solids and Boxes added to the 3rd grade and have Flips, Turns, and Area moved to 2nd. 3rd Grade Standard 10: Explore and identify familiar two- and three-dimensional shapes. 3.10b Know how to put shapes together and take them apart to form other shapes. Picturing Pollygons (5th) Investigation 2 - parallel and types of angles 3rd Grade Standard 10: Explore and identify familiar two- and three-dimensional shapes. 3.10c Identify angles, edges, perpendicular and parallel edges and vertices in two-dimensional shapes. Picturing Pollygons (5th) Investigation 2 - parallel and types of angles 3rd Grade Standard 10: Explore and identify familiar two- and three-dimensional shapes. 3.10d Identify right angles, edges, vertices, perpendicular and parallel planes in three- dimensional shapes. Picturing Pollygons (5th) Investigation 2 - parallel and types of angles 3rd Grade Standard 11: Make and test conjectures about whole number arithmetic. 3.11a Understand basic properties of odd and even numbers. Nope, never taught (any grade) 3rd Grade Standard 11: Make and test conjectures about whole number arithmetic. 3.11b Recognize and extend patterns given in tables of data. Up and Down the Number Line Extending patterns (not in tables), but as change over time Changes Over Time Investigation 1 Ten-minute Math Counting around the Class Nimble with Numbers 4th Grade Standard 1: Understand how to read, write, add and subtract numbers up to 1,000,000. 4.1a Read and write numbers up to 1,000,000 in numerals and in words. Data Around Us Investigation 3 Sessions 1 - 4 4th Grade Standard 1: Understand how to read, write, add and subtract numbers up to 1,000,000. 4.1b Recognize the place values in numbers (hundred thousands, ten thousands, thousands, hundreds, tens, ones) and understand what quantities each digit represents. Landmarks in the Thousands Investigation 2 Sessions2, 3, 4 4th Grade Standard 1: Understand how to read, write, add and subtract numbers up to 1,000,000. 4.1c Compare and order numbers up to 1,000,000. Data Around Us (no longer used by 7th grade) Investigation 4 Sessions 1 - 3 4th Grade Standard 2: Understand the need to approximate or estimate and know how to do it. 4.2a Round off numbers to the nearest 10, 100 or 1,000. Money, Miles, and Large Numbers Investigation 1 Sessions 4, 5, 7, 8 (note: only rounding to landmark $) No formula rounding curricula found 4th Grade Standard 2: Understand the need to approximate or estimate and know how to do it. 4.2b Mentally estimate answers to problems involving addition, subtractions, and multiplication. Mathematical Thinking at Grade 4 Investigation 1 Sessions 2, 3, 4; Investigation 2 Sessions 1, 2 4th Grade Standard 2: Understand the need to approximate or estimate and know how to do it. 4.2c Judge the accuracy appropriate to given problems or situations. Landmarks in the Thousands Investigation 1 Session 1 4th Grade Standard 3: Understand and identify small prime and composite numbers. 4.3a List all factors of whole numbers up to 50. Arrays and Shares Investigation 2 Sessions 5, 6 4th Grade Standard 3: Understand and identify small prime and composite numbers. 4.3b List the first 10 multiples of a given 1 digit number. Arrays and Shares Investigation 1 Session 1 4th Grade Standard 3: Understand and identify small prime and composite numbers. 4.3c Determine if a 1-digit number is a factor of a given whole number and whether a given number is a multiple of a given 1-digit number. Arrays and Shares Investigation 1 Session 1, 2  Packages and Groups Investigation 1 Sessions 1, 2, 4th Grade Standard 3: Understand and identify small prime and composite numbers. 4.3d Recognize that some whole numbers can be expressed as a product of factors in more than one way (e.g. 12 = 1 x 12 = 3 x 4 = 2 x 6 = 2 x 2 x 3). Arrays and Shares Investigation 2 Session 2, 3 Landmarks in the Thousands Investigation 1 Session 2 4th Grade Standard 4: Understand how to multiply small multi-digit numbers and to divide by a single digit. 4.4a Understand how to multiply any multi-digit number by a 1-digit number and how to multiply any 3-digit number by a 2-digit number flexibly, efficiently and accurately (without a calculator). Arrays and Shares Investigation 3 Session 1 - 5 Packages and Groups Investigation 2 Session 1, 2, 3 4th Grade Standard 4: Understand how to multiply small multi-digit numbers and to divide by a single digit. 4.4b Understand how to divide numbers up to 1,000 by a single digit number and by 10 flexibly, efficiently and accurately (without a calculator). Arrays and Shares Investigation 2 Sessions 7, 8 (dividend in the tens) Packages and Groups Investigation Session 1, 2, 3, 7, 8, 10 (dividend in the hundreds) Mathematical Thinking at Grade 5 Investigation 3 Sessions 2, 3, 4 4th Grade Standard 4: Understand how to multiply small multi-digit numbers and to divide by a single digit. 4.4c Recognize and be able to explain and correct common computational errors. 4th Grade Standard 5: Solve multi-step problems using whole numbers and combinations of the four arithmetic operations. 4.5a Solve problems of various types (mathematical tasks, word problems, contextual questions, and setting from the world outside the classroom) that require more than one of the four arithmetic operations. Arrays and Shares Investigation 2 Sessions 7, 8; Investigaiton 3 Sessions 1 - 5 Packages and Groups Inv. 1 Sessions 4, 5; Inv. 2 Sessions 1 - 3; Inv. 3 Sessions 1, 2, 3, 7, 8, 10 4th Grade Standard 5: Solve multi-step problems using whole numbers and combinations of the four arithmetic operations. 4.5b Understand and use parentheses:  know why they are needed, when and how to use them, and how to evaluate expressions containing them. Say it with Symbols Investigation 1 4th Grade Standard 5: Solve multi-step problems using whole numbers and combinations of the four arithmetic operations. 4.5c Understand the role and function of remainders in division. Arrays and Shares Investigation 2 Session 7, 8; Investigation 3 Session 2, 3, 4 Packages and Groups Investigation 3 Sessions 1, 2 4th Grade Standard 5: Solve multi-step problems using whole numbers and combinations of the four arithmetic operations. 4.5d Use the inverse relation between multiplication and division to check results when solving problems. Note: No specific instruction / no worksheet requirement (4 - 7th) 4th Grade Standard 5: Solve multi-step problems using whole numbers and combinations of the four arithmetic operations. 4.5e Translate a problem’s verbal statements or contextual details into appropriate diagrams and numerical expressions, and express answers in appropriate verbal or numerical form. Changes Over Time Investigation 3 Sessions 1- 8 Patterns of Change Investigation 2 Sessions 1 - 5 4th Grade Standard 5: Solve multi-step problems using whole numbers and combinations of the four arithmetic operations. 4.5f Provide units in answers and use estimation to judge reasonableness. Measurement Benchmarks (no longer used in 5th grade) Investigation 1 Sessions 2, 3; Investigation 3 Sessions 1 4th Grade Standard 5: Solve multi-step problems using whole numbers and combinations of the four arithmetic operations. 4.5g Write word problems representing a given number sentence and use the four operations to write number sentences for given situations. Arrays and Shares Investigation 2 Session 7, 8 Packages and Groups Investigation 3 Session 1, 10 4th Grade Standard 6: Understand equivalent fractions. 4.6a Understand that equivalent fractions represent the same point on the number line. Different Shapes, Equal Pieces Investigation 3 - Sessions 3 - 5 4th Grade Standard 6: Understand equivalent fractions. 4.6b Understand that any two fractions can be written as two equivalent fractions with equal denominators. Different Shapes, Equal Pieces Investigation 3 - Session 3 (note: lesson does not specify yet skill applied) 4th Grade Standard 6: Understand equivalent fractions. 4.6c Place fractions m/n on the number line. Different Shapes, Equal Pieces Investigation 3 - Session 4 and 5 (followed by assessment SS15) 4th Grade Standard 6: Understand equivalent fractions. 4.6d Use equivalent fractions to compare any two fractions. Different Shapes, Equal Pieces Investigation 2  Sessions 1, 2; Investigation 3 Sessions 1,2 4th Grade Standard 7: Understand how to add and subtract fractions and use these skills to solve problems. 4.7a Add fractions by rewriting them as equivalent fractions with a common denominator. Name That Portion Investigation 2 Session 3, 7 and 8 4th Grade Standard 7: Understand how to add and subtract fractions and use these skills to solve problems. 4.7b Understand mixed numbers as an alternate notation for fractions greater than 1. Name That Portion Investigation 2 Session 4; Investigation 3 Session 3, 4 and 5 4th Grade Standard 7: Understand how to add and subtract fractions and use these skills to solve problems. 4.7c Solve multi-step problems involving addition and subtraction of fractions and whole numbers. Name That Portion Investigation 2 Sessions 1, 2, 3, 7, 8 4th Grade Standard 8: Understand and use decimal numbers with up to two decimal places. 4.8a Understand decimal digits in the context of place value for determinating decimals up to two decimal places. Money, Miles, and Large Numbers Investigation 1 Sessions 1 - 8 4th Grade Standard 8: Understand and use decimal numbers with up to two decimal places. 4.8b Use decimal notation to convert grams to kilograms and meters to kilometers, and cents to dollars. Money, Miles, and Large Numbers ($ only) Investigation 1 Session 3 Note: See standard 4.9d 4th Grade Standard 8: Understand and use decimal numbers with up to two decimal places. 4.8c Add and subtract decimals with up to two decimal places. Money, Miles, and Large Numbers Investigation 1 Sessions 4, 5 (adding only) 7, 8 (adding up from decimal to whole #) 4th Grade Standard 8: Understand and use decimal numbers with up to two decimal places. 4.8d Write tenths and hundredths in decimal and fraction notation and know the fraction and decimal equivalents for halves, fourths and fifths. Between Never and Always Investigation 1 Session 1, 2 (note: skill only covered at start - not reinforced) Name That Portion Investigation 1 Sessions 2, 3, 4 4th Grade Standard 8: Understand and use decimal numbers with up to two decimal places. 4.8e Solve multi-step problems involving addition and subtraction of decimals, fractions and whole numbers. Money, Miles and Large Numbers Investigation 1 Sessions 4 - 8 (counting up decimals) Different Shapes, Equal Pieces Investigation 1 Session 5; Investigation 2 Session 3 (fraction equations adding to 1) Name That Portion Note: Assessment for Investigation 2 and 3 requires multi-step problem solving (per 5th grade, there is no multi-step practice) 4th Grade Standard 9: Understand how to measure perimeter, area and volume and how to solve problems using these concepts. 4.9a Understand how measurement of the length of lines, the area of rectangles and the volume of rectangular prisms are based on standard units. Containers and Cubes Investigation 4 Sessions 4 and 5 (standard units for volume only) Picturing Polygons Investigation 3 Sessions 5 and 6 (length of sides and area) 4th Grade Standard 9: Understand how to measure perimeter, area and volume and how to solve problems using these concepts. 4.9b Know how to calculate the perimeter and area of a rectangle, and the volume and surface area of a rectangular prism. Covering and Surrounding Investigation 3 Session 1 (perimiter and area) Filling and Wrapping Investigation 2 Sessions 1, 2 4th Grade Standard 9: Understand how to measure perimeter, area and volume and how to solve problems using these concepts. 4.9c Know and use common units of measure of length, area and volume in both metric and English units. Containers and Cubes Investigation 1 Sessions 1, 3; Investigation 2 Session 1, Paces to Feet Inv. 1 Session 1 (standard); Inv. 2 Session 5, 6, 7 (metric) 4th Grade Standard 9: Understand how to measure perimeter, area and volume and how to solve problems using these concepts. 4.9d Know common within-system equivalencies. checked grades 2 - 8 - not in units but knowledge is assumed and applied in assignments 4th Grade Standard 9: Understand how to measure perimeter, area and volume and how to solve problems using these concepts. 4.9e Visualize, describe and draw the relative sizes of length, area and volume units in the different measurement systems. Covering and Surrounding Investigation 1 Sessions 1 - 4 (length, area) Filling and Wrapping Investigation 2 Session 1; Investigation 3 Session 1 - 3 4th Grade Standard 9: Understand how to measure perimeter, area and volume and how to solve problems using these concepts. 4.9f Solve problems involving area, perimeter, surface area or volume of rectangular figures. Covering and Surrounding Investigation 3 Session 1 (perimiter and area) Filling and Wrapping Investigation 2 Sessions 1, 2 4th Grade Standard 10: Record, arrange, present and interpret data using tables and bar graphs. 4.10a Recognize that most measurements are approximations. Containers and Cubes Investigation 1 Session 3 4th Grade Standard 10: Record, arrange, present and interpret data using tables and bar graphs. 4.10b Create and interpret bar graphs and associated tables of data. The Shape of the Data Inv. 1 Sessions 1, 2, 3; Inv. 2 Sessions 1 - 7; Inv. 3 Session 1 - 5 4th Grade Standard 11: Know and use the definitions of angle, polygon and circle. 4.11a Know that an angle in a plane is a region between two rays. Picturing Polygons Investigation 2 Session 8; Investigation 3 session 3 4th Grade Standard 11: Know and use the definitions of angle, polygon and circle. 4.11b Know that if angle A is contained in another angle B, then angle B is said to be bigger than angle A. Picturing Polygons Investigation 2 Session 8 4th Grade Standard 11: Know and use the definitions of angle, polygon and circle. 4.11c Know and use the basic properties of square, rectangle, and isosceles, equilateral and right triangles. Picturing Polygons Investigation 1 Sessions 1, 2; Investigation 2 Sessions 1, - 4; Inv. 3 Sessions 1, 2, 4th Grade Standard 11: Know and use the definitions of angle, polygon and circle. 4.11d Know what a polygon is and be able to identify and draw some examples. Picturing Polygons Investigation 1 Sessions 1 - 4 4th Grade Standard 11: Know and use the definitions of angle, polygon and circle. 4.11e Know and use the basic properties of a circle. Covering and Surrounding Investigation 7 Session1 4th Grade Standard 12: Use properties of arithmetic to solve simple problems. 4.12a Know and use the commutative and associative properties. Say it with Symbols Investigation 3 (commutative only) Note: teachers state associative property not specifically taught 4th Grade Standard 12: Use properties of arithmetic to solve simple problems. 4.12b Know and use the distributive property. Say it with Symbols Investigation 3 4th Grade Standard 12: Use properties of arithmetic to solve simple problems. 4.12c Find the unknown in a simple linear equation. Moving Straight Ahead Investigation 4 and 6 4th Grade Standard 13: Evaluate simple expressions. 4.13a Find the value of expressions such as na + b and na – b where a, b, and n are whole numbers and na > b. 4th Grade Standard 13: Evaluate simple expressions. 4.13b Evaluate expressions such as             ,where a, b, c, and n are whole numbers. Name That Portion Investigation 2 Sessions 1 and 2 Bits and Pieces 2 Investigation 4 4th Grade Standard 13: Evaluate simple expressions. 4.13c Evaluate expressions such as              where a and b are single digit whole numbers. Name That Portion Investigation 2 Sessions 1 and 2 Bits and Pieces 2 Investigation 4 5th Grade Standard 1: Understand and use the relation between prime and composite numbers. 5.1a Know the definitions of prime and composite numbers. Mathematical Thinking At Grade 5 1 Prime Time: Factors and Multiples (6) 5 Term composite is  mentioned in text, but not as part of the lesson. 5th Grade Standard 1: Understand and use the relation between prime and composite numbers. 5.1b Know that every composite number greater than 1 can be written as a product of two or more prime numbers in just one way (except for order). Prime Time: Factors and Multiples (6) 5 Add in GCM and LCF 5th Grade Standard 1: Understand and use the relation between prime and composite numbers. 5.1c Write composite numbers up to 50 as a product of prime numbers. Prime Time: Factors and Multiples (6) 5 5th Grade Standard 2: Understand and use division of whole numbers. 5.2a Know and use the long division algorithm. Nimble with Numbers 4/5 Not taught prior to 6th, but expected to be known. 5th Grade Standard 2: Understand and use division of whole numbers. 5.2b Divide numbers up to 1,000 by numbers up to 100 using long division. Skill Power 5 Not taught prior to 6th, but expected to be known. 5th Grade Standard 2: Understand and use division of whole numbers. 5.2c Know and use simple mental methods where appropriate. Building On Numbers You Know 3,5 Mental Math in the Middle Grades Division pg 99-106.  Mental Math is practice 5th Grade Standard 3: Understand how to add and subtract fractions. 5.3a Add fractions with unequal denominators by rewriting them as equivalent fractions with equal denominators. Name That Portion 2,3 Bits and Pieces II (6) 4 Nimble with Numbers 4/5 and 5/6 5th Grade Standard 3: Understand how to add and subtract fractions. 5.3b Add and subtract mixed numbers. Name That Portion 2 Bits and Pieces II (6) 4 Nimble with Numbers 4/5 and 5/6 Only addressed in the extensions. Adding but not necessarily subtracting.  5th Grade Standard 3: Understand how to add and subtract fractions. 5.3c Find the unknown in simple linear equations involving fractions. Bits and Pieces II (6) 4 Nimble with Numbers  5/6 5th Grade Standard 4: Understand what it means to multiply fractions and know how to do it. 5.4a Understand that the multiplication of a fraction by a whole number is the repeated addition of the fraction. Bits and Pieces II (6) 5 5th Grade Standard 4: Understand what it means to multiply fractions and know how to do it. 5.4b Understand the formula                 for unit fractions. Bits and Pieces II (6) 5 5th Grade Standard 4: Understand what it means to multiply fractions and know how to do it. 5.4c Know the meaning of the product of two fractions in terms of an area model. Bits and Pieces I (6) 3 5th Grade Standard 4: Understand what it means to multiply fractions and know how to do it. 5.4d Be able to derive and explain the formula. Bits and Pieces II (6) 5 concept, but can't explain the formula 5th Grade Standard 4: Understand what it means to multiply fractions and know how to do it. 5.4e Understand why the fraction     of c is               the same as     . Name That Portion 2 Bits and Pieces 1(6) 1 5th Grade Standard 4: Understand what it means to multiply fractions and know how to do it. 5.4f Understand in terms of the area model that if a, b, c and d are all > 0, and        < 1,              then            <  . Bits and Pieces II (6) 5 not explicitly taught 5th Grade Standard 4: Understand what it means to multiply fractions and know how to do it. 5.4g Recognize that the formula                         proves that the multiplication of fractions obeys the commutative, associative and distributive laws. 5th Grade Standard 4: Understand what it means to multiply fractions and know how to do it. 5.4h Solve problems using the product of fractions. Bits and Pieces II (6) 5 5th Grade Standard 5: Be able to explain and use the interpretation of a fraction as division. 5.5a Understand why the fraction         is the same as Building On Numbers You Know 3 Name That Portion 3 5th Grade Standard 5: Be able to explain and use the interpretation of a fraction as division. 5.5b Understand and be able to divide a fraction by a fraction and to solve related problems. in updated teacher's addition 5th Grade Standard 5: Be able to explain and use the interpretation of a fraction as division. 5.5c Relate division with remainder to the conversion of fractions greater than 1 to mixed numbers. Name That Portion 2 5th Grade Standard 5: Be able to explain and use the interpretation of a fraction as division. 5.5d Understand division as the inverse of multiplication. 5th Grade Standard 6: Understand how to multiply terminating decimals by whole numbers. 5.6a Multiply and divide decimals up to two decimal places by whole number multiples of 10, 100, 1000. Bits and Pieces II (6) 6 only taught with calculators 5th Grade Standard 6: Understand how to multiply terminating decimals by whole numbers. 5.6b Multiply one- and two-digit whole numbers by decimals up to two decimal places. Bits and Pieces II (6) 6 Nimble with Numbers 4/5  and 5/6 5th Grade Standard 6: Understand how to multiply terminating decimals by whole numbers. 5.6c Understand the effect on decimal places when a number is multiplied by a power of ten. Bits and Pieces II (6) 6 5th Grade Standard 6: Understand how to multiply terminating decimals by whole numbers. 5.6d Be able to explain that multiplication of a number by a decimal may result in a smaller number. 5th Grade Standard 7: Understand and use ratios and percentages. 5.7a Recognize ratio as a comparison of two quantities by division. 5th Grade Standard 7: Understand and use ratios and percentages. 5.7b Understand that percentage is a standard ratio with denominator 100. Name That Portion 1,3,4 Not as a ratio, but as a fraction. 5th Grade Standard 7: Understand and use ratios and percentages. 5.7c Know by heart common percentages and ratios based on fractions whose denominators are 2, 3, 4, 5, or 10. Name That Portion 1,2,3 5th Grade Standard 7: Understand and use ratios and percentages. 5.7d Express the ratio between two quantities as a percent, and a percent as a ratio. Name That Portion 1,3 Percents, but not ratios are addressed.  5th Grade Standard 7: Understand and use ratios and percentages. 5.7e Create and solve world problems involving ratio and percentage. Name That Portion 1,2 Building on Numbers You Know Student Sheets 18, 22, 25, 36, 40 Percents, but not ratios are addressed. (Students solve, but do not create their own.) No "profit" and "loss" addressed either.  5th Grade Standard 8: Solve multi-step problems using multi-digit positive numbers, fractions and decimals. 5.8a Solve problems of various types – mathematical tasks, word problems, contextual questions and questions from settings outside the classroom. Name That Portion 2,3 Bits and Pieces II (6) 2 5th Grade Standard 8: Solve multi-step problems using multi-digit positive numbers, fractions and decimals. 5.8b Use a mixture of arithmetic operations, parentheses, and arithmetic laws (commutative, associative, distributive). Skill Power 5 Practice only, not taught.  5th Grade Standard 8: Solve multi-step problems using multi-digit positive numbers, fractions and decimals. 5.8c Translate verbal statements of a problem into appropriate diagrams and numerical expressions, and express answers in verbal or numerical form. Bits and Pieces II (6) 2 not explicitly taught, but throughout CMP 5th Grade Standard 8: Solve multi-step problems using multi-digit positive numbers, fractions and decimals. 5.8d Use mental arithmetic with simple multiplication and division of whole numbers, fractions and decimals. Building On Numbers You Know 1 Mental Math in the Middle Grades Multiplication with decimals pg 90-92 10 Minute Math pg. 31-33 Mental Math is practice 5th Grade Standard 9: Convert measurements of length, weight, area, volume and time within a single system. 5.9a Emphasize conversions that are common in daily life. Picturing Polygons 1 Bits and Pieces II (6) 3 Skill Power 5 Chapter 6 Volume in Containers and Cubes that most don't get to.  5th Grade Standard 10: Understand, derive and use the formula for the area of a triangle. 5.10a Know how to derive the formula A = ½ bh for the area of a triangle. Covering and Surrounding (6) 6 Identify triangles by angles, but not area. 5th Grade Standard 11: Know how to find the area of a polygon by decomposing it into triangles. 5.11a Be able to derive a formula for the area of a parallelogram from that of a triangle. Covering and Surrounding (6) 6 taught in reverse..take apart parallelograms first 5th Grade Standard 11: Know how to find the area of a polygon by decomposing it into triangles. 5.11b Find the area of a convex polygon by decomposing it into triangles. 5th Grade Standard 11: Know how to find the area of a polygon by decomposing it into triangles. 5.11c Find the area of other polygons that can be paved by triangles. Covering and Surrounding (6) 6 5th Grade Standard 12: Find, interpret and use the average (mean) of a set of data. 5.12a Given a set of data including whole numbers, fractions and decimals, be able to calculate the mean. Data About Us (6) 5 5th Grade Standard 12: Find, interpret and use the average (mean) of a set of data. 5.12b Given the mean and other incomplete information about a set of data, be able to infer other characteristics of the data set. Data About Us (6) 5 5th Grade Standard 13: Understand how to measure angles in degrees and solve problems involving angles. 5.13a Understand what is meant by degree and be able to measure angles in degrees. Picturing Polygons 2 Shapes and Designs (6) 3 5th Grade Standard 13: Understand how to measure angles in degrees and solve problems involving angles. 5.13b Know and be able to use the definitions of right angle, acute angle, and obtuse angle. Picturing Polygons 2,3 Skill Power 5 Chapter 8 Do not use perpendicular 5th Grade Standard 13: Understand how to measure angles in degrees and solve problems involving angles. 5.13c Recognize that angles on a straight line add up to 180° and that angles around a point add up to 360°. Picturing Polygons 2,3 5th Grade Standard 14: Understand how to interpret and plot points on the coordinate plane. 5.14a Be fluent in associating an ordered pair with a point and a point with an ordered pair. Picturing Polygons 1 5th Grade Standard 14: Understand how to interpret and plot points on the coordinate plane. 5.14b Know that positions on the coordinate plane are determined in relation to the coordinate axes, a pair of numbered lines that are placed perpendicularly to each other so that the zero point of each line coincides. Picturing Polygons 1 5th Grade Standard 14: Understand how to interpret and plot points on the coordinate plane. 5.14c Identify characteristics of the set of points that define vertical and horizontal line segments and find the length of vertical or horizontal line segments identified by the ordered pairs at their endpoints. (no title) Greatest Common Factor Nimble with Numbers 5/6 6th Grade Standard 1: Demonstrate thorough understanding of negative numbers. 6.1a Know the definitions of a negative number and of integers. Accentuate the Negative Investigation 1 6th Grade Standard 1: Demonstrate thorough understanding of negative numbers. 6.1b Understand why –(–a) = a for any integer a. Accentuate the Negative Investigation 3 6th Grade Standard 1: Demonstrate thorough understanding of negative numbers. 6.1c Be able to explain why subtracting a smaller number from a larger number is the same as adding the negative of the smaller number to the larger.  That is if a > b, then a – b = a + (–b). Accentuate the Negative Investigation 3 6th Grade Standard 1: Demonstrate thorough understanding of negative numbers. 6.1d Recognize that a – b can be defined for all numbers to be a + (–b), even when b > a. Accentuate the Negative Investigation 2 6th Grade Standard 1: Demonstrate thorough understanding of negative numbers. 6.1e Know how to locate negative numbers on the line and be able to explain why, for example, –a is the mirror image of a with respect to 0. Accentuate the Negative Investigation 1 6th Grade Standard 1: Demonstrate thorough understanding of negative numbers. 6.1f Know that the sum of two integers satisfies the same commutative and associative laws as does the sum of two positive whole numbers. Accentuate the Negative Investigation 2 Vocabulary is not explicitly taught 6th Grade Standard 1: Demonstrate thorough understanding of negative numbers. 6.1g Know that the product of two integers satisfies the same commutative, associative, and distributive laws as does the product of two positive whole numbers. Accentuate the Negative Investigation 4 Vocabulary is not explicitly taught 6th Grade Standard 1: Demonstrate thorough understanding of negative numbers. 6.1h Understand why the definition of negative numbers require that the product of two negative numbers must be positive. Accentuate the Negative Investigation 4 6th Grade Standard 2: Understand division of positive fractions and mixed numbers, and use division to solve a variety of problems. 6.2a Understand that division of fractions has the same meaning as does division of whole numbers. Bits and Pieces II Investigation 7 In the updated version of Bits and Pieces II (2002 teachers edition), there is an investigation 7 which covers division of fractions 6th Grade Standard 2: Understand division of positive fractions and mixed numbers, and use division to solve a variety of problems. 6.2b Be able to use and explain the “invert and multiply” rule stated above. Bits and Pieces II Investigation 7 6th Grade Standard 2: Understand division of positive fractions and mixed numbers, and use division to solve a variety of problems. 6.2c Find unknowns in division and multiplication problems of the form a/x = b using both whole and mixed numbers. 6th Grade Standard 3: Understand the notation and calculation of positive whole number powers. 6.3a Recognize the definition and notation of powers of numbers. Prime Time Investigation 5 6th Grade Standard 3: Understand the notation and calculation of positive whole number powers. 6.3b Emphasize two special cases:  powers of 2 and powers of 10. Growing, Growing, Growing Investigation 1 6th Grade Standard 4: Calculate with and solve problems using rational numbers and percentages. 6.4a Calculate with integer exponents using the law of exponents. HIAG 2 Unit 4, Lesson 4 or 5 It is very briefly in Growing, Growing, Growing, page 56, ACE #7 6th Grade Standard 4: Calculate with and solve problems using rational numbers and percentages. 6.4b Express numbers and solve problems in scientific notation. Data Around Us Investigation 4 6th Grade Standard 4: Calculate with and solve problems using rational numbers and percentages. 6.4c Solve multi-step problems involving more than one arithmetic operation using rational numbers. Bits and Pieces II Investigation 6 6th Grade Standard 4: Calculate with and solve problems using rational numbers and percentages. 6.4d Solve word problems involving sales tax, tips, simple interest and discount. Bits and Pieces II Investigation 1 and 2 6th Grade Standard 5: Understand the meaning of probability and how it is expressed and used. 6.5a Understand and correctly interpret different ways of expressing probabilities (e.g. as decimals, percents, odds, etc.) How Likely Is It? Investigation 1, 2 and 4 It is only addressed in the ACE Problems 6th Grade Standard 5: Understand the meaning of probability and how it is expressed and used. 6.5b Know that the probability of an event is a number between zero and one, and that if p is the probability that an event will occur, then 1 – p is the probability that it will not occur. How Likely Is It? Investigation 2 and 4 page 18, ACE Problems 5-13; page 35, ACE Problems 1-2 (needs teacher enhancement) 6th Grade Standard 5: Understand the meaning of probability and how it is expressed and used. 6.5c Understand and calculate both theoretical and empirical probabilities for simple experiments with equally probable outcomes (e.g. tossing dice, flipping coins, spinning spinners). How Likely Is It? Investigation 2 through 6 6th Grade Standard 6: Understand and use basic properties of triangles and quadrilaterals. 6.6a Understand and use the facts that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360°. Shapes and Designs Investigation 4 6th Grade Standard 6: Understand and use basic properties of triangles and quadrilaterals. 6.6b Verify the basic properties of parallelograms (i.e. opposite sides and opposite angles are equal) by direct measurement. Shapes and Designs Investigation 2 6th Grade Standard 6: Understand and use basic properties of triangles and quadrilaterals. 6.6c Use a protractor and ruler to draw triangles and quadrilaterals (squares, rectangles, parallelograms, and trapezoids) from data given in either numerical or geometric form. Shapes and Designs Investigation 3 The ruler work is absent 6th Grade Standard 6: Understand and use basic properties of triangles and quadrilaterals. 6.6d Recognize, understand and use trapezoids. Picturing Polygons Investigation 2 and 3 6th Grade Standard 7: Know basic definitions and facts about angles, lines and triangles in two dimensions and use these facts to solve problems. 6.7a Know the triangle inequality, verify it through measurement, and use it to solve problems. Shapes and Designs Investigation 2 6th Grade Standard 7: Know basic definitions and facts about angles, lines and triangles in two dimensions and use these facts to solve problems. 6.7b Know the meaning of adjacent, complementary and supplementary angles, and use them to describe geometric figures. 6th Grade Standard 7: Know basic definitions and facts about angles, lines and triangles in two dimensions and use these facts to solve problems. 6.7c Understand that when a line intersects two parallel lines, the corresponding angles and alternate interior angles are equal. HIAT 3 Unit 4 6th Grade Standard 7: Know basic definitions and facts about angles, lines and triangles in two dimensions and use these facts to solve problems. 6.7d Know the meanings of interior and exterior angles and be able to explain why each exterior angle of a triangle is equal to the sum of the opposite interior angles. HIAT 3 Unit 4 6th Grade Standard 7: Know basic definitions and facts about angles, lines and triangles in two dimensions and use these facts to solve problems. 6.7e Understand why, if a line intersects two other lines in such a way as to make the corresponding angles and alternate interior angles equal, then these two lines must be parallel. HIAT 3 Unit 4 6th Grade Standard 7: Know basic definitions and facts about angles, lines and triangles in two dimensions and use these facts to solve problems. 6.7f Understand why the sum of the interior angles of an n-sided convex polygon is (n – 2) x 180°. Shapes and Designs Investigation 4 Page 48, ACE Problem #6, but much more is needed 6th Grade Standard 7: Know basic definitions and facts about angles, lines and triangles in two dimensions and use these facts to solve problems. 6.7g Understand why the sum of exterior angles of a convex polygon is 360°. 6th Grade Standard 7: Know basic definitions and facts about angles, lines and triangles in two dimensions and use these facts to solve problems. 6.7h Use these properties to solve problems involving angles and triangles. HIAT 3 Unit 4 6th Grade Standard 8: Understand and use the definition of congruence for simple polygons. 6.8a Understand the definition of congruence for polygons. Picturing Polygons Investigation 3 6th Grade Standard 8: Understand and use the definition of congruence for simple polygons. 6.8b Identify congruent polygonal figures. HIAT 3 Unit 4 6th Grade Standard 9: Know and use the definitions and basic properties of rotations, reflections and translations in the plane. 6.9a Understand the meaning of rigid motion in the plane. Shapes and Designs Investigation 5 Translations aren't covered 6th Grade Standard 9: Know and use the definitions and basic properties of rotations, reflections and translations in the plane. 6.9b Know how to define congruence by use of rotations, reflections and translations. Shapes and Designs Investigation 5 The term 'congruent' is not used 6th Grade Standard 10: Understand and make use of different kinds of symmetry in the plane. 6.10a Explain the symmetry of geometric figures from the standpoint of rotations, reflections and translations. Shapes and Designs Investigation 5 very basic, and translations aren't covered 6th Grade Standard 10: Understand and make use of different kinds of symmetry in the plane. 6.10b Identify and utilize bilateral symmetry and rotational symmetry in regular polygons. Shapes and Designs Investigation 5 6th Grade Standard 10: Understand and make use of different kinds of symmetry in the plane. 6.10c Identify and utilize translational symmetry in tessellations of the plane. Shapes and Designs Investigation 1 6th Grade Standard 10: Understand and make use of different kinds of symmetry in the plane. 6.10d Use reflections to see why the base angles of an isosceles triangle are equal, and why the bisector of the angle opposite the base is the perpendicular bisector of the base. Shapes and Designs Investigation 5 concept is barely covered 6th Grade Standard 11: Translate verbal expressions into algebraic expressions and use these expressions to solve problems. 6.11a Observe the notational conventions regarding the use of letters. Variables and Patterns Investigation 4 6th Grade Standard 11: Translate verbal expressions into algebraic expressions and use these expressions to solve problems. 6.11b Evaluate expressions, including those with exponents Say It With Symbols Investigation 1 6th Grade Standard 11: Translate verbal expressions into algebraic expressions and use these expressions to solve problems. 6.11c Understand that relationships between quantities can be represented using formulas or verbal descriptions, and can be suggested by graphs and tables. Variables and Patterns Investigation 1 through 6 6th Grade Standard 11: Translate verbal expressions into algebraic expressions and use these expressions to solve problems. 6.11d Solve problems involving linear functions whose coefficients and domain are integers. Moving Straight Ahead Investigations 1 through 6 7th Grade Standard 1: Know how to interpret and convert to and from the decimal forms of rational numbers. 7.1a Understand that every rational number is the division of two integers. 8th Looking for Pythagoras Investigation 5 7th Grade Standard 1: Know how to interpret and convert to and from the decimal forms of rational numbers. 7.1b Understand that the decimal form of a rational number either terminates or eventually repeats.  8th Looking for Pythagoras Investigation 5 7th Grade Standard 1: Know how to interpret and convert to and from the decimal forms of rational numbers. 7.1c Know how to use long division to express a fraction as a (possibly repeating) decimal and how to express repeating decimals in fractional form. 8th Looking for Pythagoras Investigation 5 7th Grade Standard 1: Know how to interpret and convert to and from the decimal forms of rational numbers. 7.1d Know divisibility rules and use them to help factor numbers. 5th grade fifth grade 7th Grade Standard 2: Add, subtract, multiply and divide rational numbers and use these skills to solve mathematical, everyday and word problems. 7.2a Understand why the result of dividing two negative integers is positive. Accentuate the Negative Investigation 4 7th Grade Standard 2: Add, subtract, multiply and divide rational numbers and use these skills to solve mathematical, everyday and word problems. 7.2b Transform numbers in different forms (fractions, decimals, mixed numbers) so as to permit efficient calculation. 6th Bits and Pieces I and II 7th Grade Standard 2: Add, subtract, multiply and divide rational numbers and use these skills to solve mathematical, everyday and word problems. 7.2c Understand why the procedures used for calculation and transformation of numbers produce correct answers. 6th Bits and Pieces I and II 7th Grade Standard 2: Add, subtract, multiply and divide rational numbers and use these skills to solve mathematical, everyday and word problems. 7.2d Make judicious use of calculators or computers that recognizes and compensates for the dangers of calculator roundoff. Comparing and Scaling Investigation 5 large numbers on calculators, contained in Data around Us, no longer used 7th Grade Standard 2: Add, subtract, multiply and divide rational numbers and use these skills to solve mathematical, everyday and word problems. 7.2e Check answers both by estimation and by appropriate independent calculations. Comparing and Scaling Investigation 5 7th Grade Standard 3: Understand and use the properties and structure of the real number line. 7.3a Locate rational numbers on the number line. Accentuate the Negative 7th Grade Standard 3: Understand and use the properties and structure of the real number line. 7.3b Recognize that the definitions and properties of integers extend to rational numbers. 8th Looking for Pythagoras Investigation 5 7th Grade Standard 3: Understand and use the properties and structure of the real number line. 7.3c Know and use standard rules for inequalities when comparing rational numbers. HIAT 3 Unit 3 7th Grade Standard 3: Understand and use the properties and structure of the real number line. 7.3d Know by heart the squares of numbers from 1 to 12 (thus the square roots of square numbers between 1 and 144) and the cubes of numbers from 1 to 5. 7th Grade Standard 3: Understand and use the properties and structure of the real number line. 7.3e Recognize that unless they are integers, square, cube and nth roots of integers are not rational. 7th Grade Standard 3: Understand and use the properties and structure of the real number line. 7.3f Understand why  . 7th Grade Standard 3: Understand and use the properties and structure of the real number line. 7.3g Be able to estimate square roots and cube roots and use calculators to find good approximations. 8th Looking for Pythagoras Investigation 2 7th Grade Standard 4: Know how to collect, organize an analyze both single variable and two-variable data. 7.4a Represent data faithfully using a variety of graphs and charts including bar graphs, line graphs, pie charts, box-and-whisker plots, stem-and-leaf plots and scatter plots.  Interpret and solve problems using information presented in these same forms. 8th Samples and Populations Investigation 1, 2 6th Data about Us Investigation 3 7th Grade Standard 4: Know how to collect, organize an analyze both single variable and two-variable data. 7.4b Compute, graph and interpret relative and cumulative frequencies. Comparing and Scaling Investigation 5 terms "relative" and "cumulative" not used 7th Grade Standard 4: Know how to collect, organize an analyze both single variable and two-variable data. 7.4c Find and interpret the mean (average), mode, median, upper quartile, lower quartile and inner-quartile range of a set of data. 8th Samples and Populations Investigation 1 7th Grade Standard 5: Know how to do basic constructions using a straightedge and compass. 7.5a  Basic constructions include (a) a perpendicular from a point to a line, (b) bisection of an angle and of a line segment (by construction of its perpendicular bisector), and (c) construction of a hexagon on a circle. 7th Grade Standard 5: Know how to do basic constructions using a straightedge and compass. 7.5b Other constructions (e.g. of an equilateral triangle given one side; of a square inscribed inside a circle) build on these basic constructions. 7th Grade Standard 6: Demonstrate thorough knowledge of similarity of triangles, including definitions, criteria and applications. 7.6a Know the definition of similarity for triangles. Stretching and Shrinking Investigation 4 7th Grade Standard 6: Demonstrate thorough knowledge of similarity of triangles, including definitions, criteria and applications. 7.6b Know that two triangles are similar if the ratios of the lengths of corresponding sides are equal (SSS criterion). Stretching and Shrinking Investigation 3 7th Grade Standard 6: Demonstrate thorough knowledge of similarity of triangles, including definitions, criteria and applications. 7.6c Know that two triangles are similar if the ratios of the lengths of two pairs of corresponding sides are equal and the corresponding angles between them are equal (SAS criterion). Stretching and Shrinking Investigation 5 7th Grade Standard 6: Demonstrate thorough knowledge of similarity of triangles, including definitions, criteria and applications. 7.6d Know that two triangles are similar if the measures of two pairs of angles are equal (AA criterion). Stretching and Shrinking Investigation 3 Only found in ACE4 # 28 teacher Notes, p. 58h 7th Grade Standard 6: Demonstrate thorough knowledge of similarity of triangles, including definitions, criteria and applications. 7.6e Be able to identify and use equality of ratios of different line segments when a line inside a triangle is drawn parallel to one side. Stretching and Shrinking Investigation 5 7th Grade Standard 6: Demonstrate thorough knowledge of similarity of triangles, including definitions, criteria and applications. 7.6f Know how to use similar triangles to find the lengths of unknown line segments in a triangle.  In particular, be able to employ the notion of similarity to solve geometric problems where direct measurement is difficult or impossible. Stretching and Shrinking Investigation 5 7th Grade Standard 7: Demonstrate understanding of similarity of other two-dimensional figures. 7.7a Recognize that the notion of similarity has two different definitions that turn out to be equivalent. Stretching and Shrinking Investigation 3 7th Grade Standard 7: Demonstrate understanding of similarity of other two-dimensional figures. 7.7b Be able to explain why congruent figures are similar. Stretching and Shrinking Investigation 3 7th Grade Standard 7: Demonstrate understanding of similarity of other two-dimensional figures. 7.7c Be able to show why analogues of the SSS, SAS and AA criteria for similarity of triangles do not work for polygons with more than 3 sides. 7th Grade Standard 7: Demonstrate understanding of similarity of other two-dimensional figures. 7.7d Understand and make use of the scale factor through which a figure can be transformed into a different figure that is precisely similar to the original. Stretching and Shrinking Investigation 1 7th Grade Standard 8: Demonstrate thorough understanding of slope -- what it means, how to calculate it, and how to use it in problems. 7.8a Be able to calculate the slope of a line from its definition as the ratio of the lengths of two line segments – the “rise” and “run” – that together with the line form a right triangle. Moving Straight Ahead Investigation 5 7th Grade Standard 8: Demonstrate thorough understanding of slope -- what it means, how to calculate it, and how to use it in problems. 7.8b Know how to find the slope of a line in a coordinate plane. Moving Straight Ahead Investigation 5 7th Grade Standard 8: Demonstrate thorough understanding of slope -- what it means, how to calculate it, and how to use it in problems. 7.8c Use similar triangles to show that the calculated slope of a line is the same no matter which two points one uses to perform the calculation. Moving Straight Ahead Investigation 5 7th Grade Standard 8: Demonstrate thorough understanding of slope -- what it means, how to calculate it, and how to use it in problems. 7.8d Understand that two lines in a coordinate plane are parallel exactly when they have the same slope and are perpendicular when and only when the product of their slopes is –1. 8th Looking for Pythagoras Investigation 4 7th Grade Standard 8: Demonstrate thorough understanding of slope -- what it means, how to calculate it, and how to use it in problems. 7.8e Know how to find the slopes of physical objects (roads, roofs, ramps, stairs) and express the answer as a decimal, ratio or percent. Moving Straight Ahead Investigation 5 7th Grade Standard 9: Use the arithmetic properties of numbers to simplify expressions and solve equations. 7.9a Add, subtract and multiply simple polynomial expressions. Say it with Symbols Invetigation 2 7th Grade Standard 9: Use the arithmetic properties of numbers to simplify expressions and solve equations. 7.9b Apply algebraic order of operations and the commutative, associative and distributive properties to evaluate expressions; and justify each step in the process. Say it with Symbols Investigation 3 7th Grade Standard 9: Use the arithmetic properties of numbers to simplify expressions and solve equations. 7.9c Recognize and use the fact equals added to equals are equal and that equals multiplied by equals are equal. Moving Straight Ahead Investigation 4 7th Grade Standard 9: Use the arithmetic properties of numbers to simplify expressions and solve equations. 7.9d Keep equations of the form ax + b = c and ax + b = cx + d “balanced” by carrying out the same operations on each side. Say it with Symbols Investigation 4 7th Grade Standard 10: Know what is meant by the graph of an equation. 7.10a Create tables of coordinate points for simple linear and quadratic equations. Frogs, Fleas, and Painted Cubes Investigation 2(Quadratic) Variables and Patterns Investigation 4 (Linear) 7th Grade Standard 10: Know what is meant by the graph of an equation. 7.10b Plot points of the graph of simple equations in the coordinate plane by hand Variables and Patterns Investigation 4 7th Grade Standard 10: Know what is meant by the graph of an equation. 7.10c Be able to match graphs to equations in simple linear and quadratic cases. Frogs, Fleas, and Painted Cubes Investigation 2 7th Grade Standard 11: Understand fully the properties of linear functions y = kx + b and use these properties in a variety of applications. 7.11a Understand that a linear function is characterized by a constant rate of change, m. Variables and Patterns Investigation 3 7th Grade Standard 11: Understand fully the properties of linear functions y = kx + b and use these properties in a variety of applications. 7.11b Know that a direct proportional relationship between y and x can be represented by a linear function y = mx where b = 0 and m is called the constant of proportionality. Moving Straight Ahead Investigation 2 7th Grade Standard 11: Understand fully the properties of linear functions y = kx + b and use these properties in a variety of applications. 7.11c Understand why the graph of a linear function y = mx + b is a straight line. Moving Straight Ahead Investigation 2 7th Grade Standard 11: Understand fully the properties of linear functions y = kx + b and use these properties in a variety of applications. 7.11d Know that in the graph of y = mx + b, the slope is m and b is the y-coordinate of the point where the graph crosses the y-axis (i.e. the value of y when x = 0) Moving Straight Ahead Investigation 3 7th Grade Standard 11: Understand fully the properties of linear functions y = kx + b and use these properties in a variety of applications. 7.11e Understand that a graph of y = mx is a line with slope m that passes through the point (0, 0). Moving Straight Ahead Investigation 3 7th Grade Standard 11: Understand fully the properties of linear functions y = kx + b and use these properties in a variety of applications. 7.11f Translate common linear phenomena into symbolic statements and interpret m and b in y = mx + b in terms of the original situation. Moving Straight Ahead Investigation 4 7th Grade Standard 11: Understand fully the properties of linear functions y = kx + b and use these properties in a variety of applications. 7.11g Work fluently with directly proportional relationships and linear functions; solve word problems involving linear phenomena and recognize contextual situations in which linear models are appropriate. Moving Straight Ahead Investigation 4 7th Grade Standard 12: Recognize that many functions, including inverse relationships, are not linear. 7.12a Know that an inversely proportional relationship between y and x is represented by y = k/x where k is some non-zero number. Thinking with Math Models Investigation 2 7th Grade Standard 12: Recognize that many functions, including inverse relationships, are not linear. 7.12b Understand that the graph of y = k/x is not a line, and that it crosses neither the x- nor y-axis (i.e. x cannot be 0 for obvious reasons so it cannot cross the y-axis and y cannot be 0 since k is a non-zero number so it cannot cross the x-axis). 7th Grade Standard 12: Recognize that many functions, including inverse relationships, are not linear. 7.12c Recognize relationships between quantities that are inversely proportional, for example, the relationship between the lengths of the base and side of a rectangle of fixed area. Frogs, Fleas, and Painted Cubes Investigation 1 7th Grade Standard 12: Recognize that many functions, including inverse relationships, are not linear. 7.12d Recognize whether information given in a table, graph or formula suggests a direct proportional, linear or inversely proportional, or other non-linear relationship. Thinking with Math Models Investigation 4 7th Grade Standard 12: Recognize that many functions, including inverse relationships, are not linear. 7.12e Solve word problems involving direct proportion, linear equations and inverse proportion. Comparing and Scaling (direct proportion); Moving Straight Ahead (linear equations);Thinking with Math Models (inverse proportion) Investigaiton 3; Investigation 4, Investigation 2 (respectively) 8th Grade Standard 1: Understand the definition of irrational numbers and know some examples. 8.1a By definition numbers that cannot be expressed as the ratio of two integers are called irrational numbers. Looking For Pythagoras Investigation 5 8th Grade Standard 1: Understand the definition of irrational numbers and know some examples. 8.1b Name some common examples of irrational numbers and locate them approximately on the number line. Looking For Pythagoras Investigation 5 8th Grade Standard 1: Understand the definition of irrational numbers and know some examples. 8.1c Be able to show that any non-zero rational multiple of an irrational number is irrational. 8th Grade Standard 2: Understand and use derived quantities and indirect measurements. 8.2a Understand and use units in ratio and multiplicative quantities such as velocity, density, area, foot-pound, watt-hour and person-days. 8th Grade Standard 2: Understand and use derived quantities and indirect measurements. 8.2b Know how to plan and carry out indirect measurements. Stretching and Shrinking (7th) Investigation 5 8th Grade Standard 2: Understand and use derived quantities and indirect measurements. 8.2c Understand how the precision of measurement influences accuracy of quantities derived from these measurements. 8th Grade Standard 2: Understand and use derived quantities and indirect measurements. 8.2d Understand the behavior of derived measurements such as weighted averages and percentage change. 8th Grade Standard 2: Understand and use derived quantities and indirect measurements. 8.2e Make within-system conversions of derived quantities (e.g. feet per second to miles per hour). 8th Grade Standard 3: Demonstrate thorough mastery of ratios, rates and percentages. 8.3a Solve problems involving simple and compound interest. (Simple Interest)       Bits and Pieces II Investigation 1 & 2 Thinking With Math Models Investigation 3 Growing, Growing, Growing Investigation 3 8th Grade Standard 3: Demonstrate thorough mastery of ratios, rates and percentages. 8.3b Solve simple proportion problems. Comparing and Scaling (7th) Investigation 3, 4, & 5 8th Grade Standard 4: Understand probability and relative frequency and use these concepts to solve problems. 8.4a Understand the meaning of probability and solve simple probability problems. What Do you Expect (7th) 8th Grade Standard 4: Understand probability and relative frequency and use these concepts to solve problems. 8.4b Know the definition of relative frequency and its relation to probability. How Likely Is It (6th) Investigation 1,2,3,4 What Do You Do Expect (7th) Investigation 1 8th Grade Standard 5: Demonstrate thorough understanding of the Pythagorean theorem. 8.5a Understand how to prove the Pythagorean Theorem and its converse. Looking For Pythagoras Investigation 3 8th Grade Standard 5: Demonstrate thorough understanding of the Pythagorean theorem. 8.5b Use the Pythagorean theorem and its converse to solve perimeter, area, and volume problems and to find distances between points in the Cartesian coordinate system. Looking For Pythagoras Investigation 3 & 4 8th Grade Standard 6: Demonstrate thorough understanding of the area and circumference of a circle. 8.6a Understand why the area of a circle equals its radius times one-half of its circumference. Covering and Surrounding (6th) Investigation 7 8th Grade Standard 6: Demonstrate thorough understanding of the area and circumference of a circle. 8.6b Understand that the circumference of a circle is directly proportional to its diameter and that the area of a circle is directly proportional to the square of its radius. Covering and Surrounding (6th) Investigation 7 8th Grade Standard 6: Demonstrate thorough understanding of the area and circumference of a circle. 8.6c Understand that the constant of proportionality is the same in both cases and is what we call π. Covering and Surrounding (6th) Investigation 7 8th Grade Standard 6: Demonstrate thorough understanding of the area and circumference of a circle. 8.6d Know and use formulas for the circumference and area of a circle, semi-circle, and quarter-circle. Covering and Surrounding (6th) Investigation 7 8th Grade Standard 7: Solve problems involving perimeter, area, volume and surface area. 8.7a Find the perimeter or area of any region in the plane bounded by line segments and circular arcs. Filing and Wrapping (7th) Investigation 5 8th Grade Standard 7: Solve problems involving perimeter, area, volume and surface area. 8.7b Know how to use formulas for volumes of common solids. Filing and Wrapping (7th) Investigation 3, 4, & 5 8th Grade Standard 7: Solve problems involving perimeter, area, volume and surface area. 8.7c Solve problems involving the surface area of prisms, pyramids and cylinders. Filing and Wrapping (7th) Investigation 3, 4 & 5 8th Grade Standard 8: Understand properties of scaling, dilation and representation of three-dimensional objects in two dimensions. 8.8a Understand that similar polygons with scale factor r have areas related by a factor of r². Stretching and Shrinking (7th) Investigation 4 8th Grade Standard 8: Understand properties of scaling, dilation and representation of three-dimensional objects in two dimensions. 8.8b Understand the definition and basic properties of dilation. Stretching and Shrinking (7th) Investigation 1 8th Grade Standard 8: Understand properties of scaling, dilation and representation of three-dimensional objects in two dimensions. 8.8c Understand that a dilation moves a line to a line with the same slope and a triangle to a similar triangle. Stretching and Shrinking (7th) Investigation 1 & 2 8th Grade Standard 8: Understand properties of scaling, dilation and representation of three-dimensional objects in two dimensions. 8.8d Understand that a dilation can be used to define the similarity of arbitrary plane figures (in addition to polygons). Stretching and Shrinking (7th) Investigation 1 & 2 8th Grade Standard 8: Understand properties of scaling, dilation and representation of three-dimensional objects in two dimensions. 8.8e Represent and work with solids by means of orthogonal views, picture views and nets. Filing and Wrapping (7th) Investigation 1 8th Grade Standard 9: Understand and solve problems involving simultaneous linear equations or inequalities in two variables. 8.9a Understand why the coordinates of the point of intersection of the lines represented by two linear equations are the solutions to the system. Moving Straight Ahead (7th) Investigation 4 Say It With Symbols Investigation 2 & 4 8th Grade Standard 9: Understand and solve problems involving simultaneous linear equations or inequalities in two variables. 8.9b Solve simultaneous linear equations in two variables using the method of substitution. HIAG 2 Unit 1 8th Grade Standard 9: Understand and solve problems involving simultaneous linear equations or inequalities in two variables. 8.9c Understand why the graph of a linear inequality is a half-plane. HIAT 3 Unit 1 8th Grade Standard 9: Understand and solve problems involving simultaneous linear equations or inequalities in two variables. 8.9d Understand that when both sides of an inequality are multiplied or divided by a negative number, the inequality is reversed, but that all other basic operations when applied to both sides preserve the inequality. HIAT 3 Unit 1 8th Grade Standard 9: Understand and solve problems involving simultaneous linear equations or inequalities in two variables. 8.9e Solve linear inequalities in one and two variables. HIAT 3 Unit 1 8th Grade Standard 10: Be able to use the mathematical notation for an arbitrary function. 8.10a Understand a function as a rule that assigns one number (the output)  to another number (the input). HIAT 3 Unit 3 8th Grade Standard 10: Be able to use the mathematical notation for an arbitrary function. 8.10b Use the notation f(x) to represent the output that the function f produces when given the input x. HIAT 3 Unit 3 8th Grade Standard 10: Be able to use the mathematical notation for an arbitrary function. 8.10c Understand that to solve the linear equation f(x) = g(x) means to find all the values of x for which the equation is true. HIAT 3 Unit 3 8th Grade Standard 10: Be able to use the mathematical notation for an arbitrary function. 8.10d Be able to explain why the solution to the linear equation f(x) = g(x) is the x-value of the intersection of the graphs of f(x) and g(x). HIAT 3 Unit 3 8th Grade Standard 10: Be able to use the mathematical notation for an arbitrary function. 8.10e  Be able to explain why the solution to the linear equation h(x) = 0 is the x-value of the intersection of the graphs of h(x) and the x-axis. HIAT 3 Unit 3 8th Grade Standard 11: Know, understand and use common nonlinear functions, identities and equations. 8.11a Understand and use the following identities:   HIAT 3 Unit 3 8th Grade Standard 11: Know, understand and use common nonlinear functions, identities and equations. 8.11b Recognize common nonlinear functions (including inverse, polynomial, and exponential) and their graphs. Thinking With Mathematical Models Investigation 2 Frog, Fleas, & Painted Cubes Investigation 1,2 & 4 Growing, Growing, Growing Investigation 1-4 8th Grade Standard 11: Know, understand and use common nonlinear functions, identities and equations. 8.11c Work fluently with nonlinear functions in geometric contexts of length, area and volume. HIAG 2 Unit 4 8th Grade Standard 11: Know, understand and use common nonlinear functions, identities and equations. 8.11d Graph simple quadratic functions. Frog, Fleas, and Painted Cubes Investigation 1 & 2 8th Grade Standard 11: Know, understand and use common nonlinear functions, identities and equations. 8.11e Solve factorable quadratic equations. Say It With Symbols Investigation 4