UPPER GRADES PROGRAM OVERVIEW

Transcription

1 THE ALPHABET OF MATHEMATICS Since 1975 UPPER GRADES PROGRAM OVERVIEW C O M P R E H E N S I V E S C O P E & S E Q U E N C E 10 UNITS 2,000 ACTIVITY SHEETS 60 TOPICS

2 INNOVATIVE LEARNING CONCEPTS INC. creators of TOUCHMATH TouchMath materials were first published in Innovative Learning Concepts Inc. s publications may not be reproduced or disseminated in any way, or stored in any database or retrieval system, without written permission of the publisher. However, limited permission is granted to reproduce only the blackline activity masters solely for noncommercial educational use with students in a single classroom or a single home, and for no other purpose. TouchMath, TouchNumerals and The Alphabet of Mathematics are registered trademarks of Innovative Learning Concepts Inc. Touch2Learn, TouchPoint, TouchShapes, TouchMath Tutor, TouchStar, TouchTunes, StepNumerals and Innovative Learning Concepts are trademarks of Innovative Learning Concepts Inc. of Colorado Springs, CO. Innovative Learning Concepts Inc Mark Dabling Blvd., Colorado Springs, CO Toll Free: Fax: Web: customerservice@touchmath.com Printed in the ed States of America

3 SCOPE AND SEQUENCE: UPPER GRADES Table of Contents Program Overview and Introduction... 2 Comprehensive Scope and Sequence Number Sense (Counting and Cardinality) Addition and Subtraction within Addition and Subtraction within 1, Multiplication and Division of One-Digit Numbers Multiplication and Division of Multi-Digit Numbers Mixed Operations with Whole Numbers Fractions Decimals Percents Relationships of Fractions, Decimals, and Percents Geometry Perimeter, Area, and Volume Time, Money, and Distance Measurement Data Pre-Algebra Overviews of the s Basic Scope and Sequence Notes

4 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Program Overview and Introduction Program Overview TouchMath provides this Upper Grades Program Overview and Scope and Sequence to present the story of the TouchMath approach to math standards for students with learning disabilities. The standards apply broadly to individual states, Common Core State Standards, and various versions of standards from the National Council of Teachers of Mathematics. The TouchMath Upper Grades Program, Sixth Edition, provides support for individual learning by presenting information in multiple ways and by ensuring students have the appropriate amount of meaningful repetition to master the skills. Both the content and the activities are scaffolded with small steps to ensure students understand and apply relationships to master skills. The focus is simple: Learn and apply strategies to solve problems. The use of TouchPoints as kinesthetic and/or visual cues, properties of operations, and relationships among number representations are each presented to find the match for each student. At the core of the Program is the philosophy of TouchMath: Use what comes naturally to learners to teach the skills. How To Use This Booklet Three significant components are included in this booklet: 1) A comprehensive scope and sequence 2) One-page overviews of each of the ten units that comprise the Program 3) A basic scope and sequence Comprehensive Scope and Sequence The tool is designed to show the relationships of the topics, standards, skills, and activities included in the Program. The topics appear at the top of each page. The standards or broad outcomes are emphasized in purple. The skills or objectives are highlighted in lavender. The activities that define the approach are listed in shades of blue. The delineation of the activities outlines the differences between the TouchMath Program and other published programs. On the right side of each page is a chart that indicates the broad, basic grade levels of the content. On the left side of the page in the first column is an orange speech bubble with the unit number of the Program. Everything below a speech bubble applies until a different speech bubble is included. When using this tool, it is best to read down the page to see the relationships of the skills and activities. Overviews of the s The one-page summary of each unit describes the topic presented. This introduction is related to previous learning. The modules that make up the unit are identified. The learning objectives are itemized. The overviews provide thumbnails of the content within each unit. Basic Scope and Sequence This presentation includes the topics, standards, and skills that encompass the Program. The graphic representation mirrors and identifies select sub-skills within the comprehensive scope and sequence. To access the comprehensive version online, please visit Upper Grades Scope and Sequence

5 SCOPE AND SEQUENCE: UPPER GRADES Number Sense (Counting and Cardinality) Learn Numbers to 20 1 Count 1 to 20 Count forward from any number Count backward from 10 Count backward from 12 Count backward from 15 Count backward from 18 Count backward from 20 Count backward from any number Count backward to find unknowns Read numbers in sequence Name numbers before and after a given number Name numbers after, between, and before given numbers Identify missing numbers in sequence Identify numbers out of order Read number names Read number names in a sequence Read number names in a sentence Write numbers Trace numbers Write numbers Write number names Learn Numbers to 1,000 2 Count to 100 Say and trace numbers in sequence in a hundred chart Trace and write number names for multiples of 10 Skip count to 100 by fives Skip count to 100 by tens Fill in missing numbers in a sequence Answer questions related to a sequence Identify numbers out of order

6 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Number Sense (Counting and Cardinality) continued Count to 1,000 Fill in missing numbers in a hundred chart Recognize patterns in a hundred chart Associate number names and numbers Identify numbers out of order Write a number for a number name Sequence the next two numbers from any given number Find numbers after, between, and before given numbers Write the next nine numbers after a given three-digit number Skip count to 1,000 by hundreds Represent and Compare Numbers Represent numbers with manipulatives, pictures, and TouchPoints s 1-2 Use TouchPoints to reinforce counting 0 9 Demonstrate correct Touching/Counting Patterns 1 Associate manipulatives, pictures, and TouchPoints to represent quantities Represent quantities in multiple ways Apply understanding of counting to quantities Apply one-to-one correspondence s 1-2 Use pictures of bundles of straws, beads on strings, and connecting cubes to represent numbers Demonstrate 10 as a bundle of 10 ones and 100 as a bundle of 10 tens 1 Represent numbers with pictures and TouchPoints Represent one-digit numbers with TouchPoints Transfer pictures to numerals (pictorial TouchPoints) to represent numbers Relate pictorial TouchPoints to TouchPoints Associate one-digit numbers with multiple representations Upper Grades Scope and Sequence

7 SCOPE AND SEQUENCE: UPPER GRADES Number Sense (Counting and Cardinality) continued Represent numbers to 1,000 using models and pictures Use models and pictures of base ten blocks to represent three-digit numbers Demonstrate 10 as a bundle of 10 ones and 100 as a bundle of 10 tens using models and pictures Represent numbers to 1,000 with drawings, pictures of base ten models, and place value charts Use place value charts and numerals to represent numbers Match multiple representations of numbers Relate models, pictures, and place value charts to represent numbers Represent hundreds, tens, and ones using pictures, place value charts, and numerals Represent 100 to 1,000 using place value Represent numbers with unknowns in place value charts Use expanded place value Match compact numerals to identified place value Use expanded notation to reflect place value Relate all forms of numbers: compact numerals, expanded notation, and number names Identify compact numbers from word place value names s 1-2 Compare numbers to 20 Use numerals without and with TouchPoints Use <, =, and > to express comparisons 2 Compare numbers to 100 Demonstrate comparisons using models and pictures Use place value charts and numerals to make comparisons Compare three-digit numbers using place value Compare numbers using place value and expanded notation

8 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Addition and Subtraction within Add Using Manipulatives, Pictures, and TouchPoints 1 Recognize combining parts to make a whole Use models, drawings, and TouchPoints to add Show addition as putting together groups of objects Use drawings to explain addition Demonstrate that addition is made up of parts to make a whole Relate counting strategies to addition Apply TouchPoints as tactile or visual cues Use TouchPoints on both addends to associate quantities Use TouchPoints to count on and find the sum s 1-2 Say the greater addend and use the TouchPoints on the lesser addend to count on Show addition problems using equations Demonstrate addition with TouchPoints Write addends in boxes Match expressions to sums Understand the equal sign Identify true or false for addition equations Select correct answers from two to four choices Solve addition word problems Use pictures to solve addition word problems Read one-step addition word problems Write number sentences for addition word problems Write word problems for addition number sentences Solve addition word problems using equations 1 Find sums with three to five addends Add three to five one-digit numbers Use TouchPoints to find sums of multiple addends Upper Grades Scope and Sequence

9 SCOPE AND SEQUENCE: UPPER GRADES Add Using Place Value, Properties of Operations, and Strategies Addition and Subtraction within 20 continued Add using place value Use place value charts to add Compare sums using place value Add using properties of operations Use the commutative property Use the associative property Add using strategies Use place value charts, models, TouchPoints, and visual cues to add Recognize and apply part/whole diagrams Apply patterns in addition Use Ten Frames to build tens and additional ones to add Apply fact families (relationship of three numbers, such as 5, 2, 7) Find missing addends Identify multiple addends for a given whole Compare sums Compare sums of pairs of addends Subtract Using Manipulatives, Pictures, and TouchPoints Recognize taking parts from a whole Use models, drawings, and TouchPoints to subtract Show subtraction as taking away a part from a whole Use drawings to explain subtraction Demonstrate that subtraction begins with a whole Apply counting strategies to subtraction problems Demonstrate subtraction with TouchPoints on both numbers Use pictures and TouchPoints to count to find the difference Use crossing out to show objects being taken away in pictures s 1-2 Relate subtraction to backward counting with TouchPoints Use backward counting as a subtraction strategy Apply TouchPoints only on the subtrahend

10 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Addition and Subtraction within 20 continued s 1-2 Show subtraction problems using equations Demonstrate subtraction with TouchPoints Write missing addends in boxes Match expressions to differences Identify true or false for subtraction equations Select correct answers from two to four choices Solve subtraction word problems Use pictures to solve subtraction word problems Read one-step subtraction word problems Write number sentences for subtraction word problems Write word problems for subtraction number sentences Solve subtraction word problems using equations Use pictures to show subtraction problems Apply strategies to subtract using TouchPoints Use boxes for unknowns Identify key words in word problems Subtract Using Place Value, Properties of Operations, and Strategies Subtract using place value Use place value charts, TouchPoints, and visual cues to subtract Compare differences using place value Apply patterns in subtraction Find missing addends Subtract using strategies Relate timers to backward counting and subtraction Use models, TouchPoints, and visual cues Recognize and apply part/whole diagrams Apply patterns in subtraction Use Ten Frames to build tens and ones to subtract Apply fact families (relationships of three numbers, such as 5, 2, 7) Compare differences of pairs of numbers Upper Grades Scope and Sequence

11 SCOPE AND SEQUENCE: UPPER GRADES Addition and Subtraction within 20 continued Add and Subtract Using Strategies s 1-2 Find missing addends Apply understanding of doubles to increase fluency Compose and/or decompose numbers using the associative property to create easier numbers for computation Solve for unknowns Use the relationship between addition and subtraction Compare pairs of numbers Use key words in word problems Add and subtract fluently within

12 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Addition and Subtraction within 1, Understand Place Value 2 Represent and apply understanding Represent multi-digit numbers with base ten blocks and place value charts in pictures Write compact numbers for visual representations Compare place value using models Write expanded place value using letters for places Use place value form with words for places Relate expanded place value and compact numbers Identify compact numbers for number names Relate and represent numbers in place value form to expanded notation Associate expanded notation, number name, and compact number Find mystery numbers based on place value Round numbers Apply the process for rounding two-digit numbers to the nearest ten Round three-digit numbers to the nearest ten Round three-digit numbers to the nearest hundred Apply Algorithms to Add and Subtract Add and subtract within 100 Demonstrate understanding of operations using base ten pictures and place value charts Use place value charts with visual indicators to start with the ones to perform operations Perform operations with no regrouping using TouchPoints and visual cues Calculate with two-digit and one-digit numbers Calculate with two-digit numbers Compare sums and differences Apply understanding in multi-step word problems Upper Grades Scope and Sequence

13 SCOPE AND SEQUENCE: UPPER GRADES Addition and Subtraction within 1,000 continued Add and subtract with regrouping Regroup ones to tens to add and tens to ones to subtract Use TouchPoints, base ten blocks, place value charts, and visual cues (i.e., boxes for regrouped ones to add, horizontal lines for regrouped tens to subtract, arrows above the ones as an indicator of where to start) Demonstrate algorithms with pictures Find sums with multiple two-digit addends Apply algorithms and problem solving strategies in word problems Add and subtract using strategies Use TouchPoints, place value charts, boxes and lines for regrouping, and arrows in the ones column as visual reminders Recognize and use the commutative and associative properties to add with regrouping one time Relate number families to addition and subtraction Add and subtract multiples of 10 Solve for unknowns using strategies Recognize unknown addends as subtraction Compare two numbers Make a drawing to solve word problems Identify the operation needed to solve word problems Exhibit proficiency in finding sums and differences of two-digit numbers with regrouping

14 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Addition and Subtraction within 1,000 continued Add and subtract three-digit numbers Compute with three-digit and one-digit numbers using visual cues Compute with three-digit and two-digit numbers using visual cues Regroup in the tens place when adding Regroup in the hundreds place when adding Regroup in the tens and hundreds places when adding Regroup tens to ones when subtracting Regroup hundreds to tens when subtracting Regroup tens and hundreds when subtracting Regroup when subtracting with 0 in the ones place Subtract with 0 in the tens place Subtract with 0 in both the ones and tens places Compare sums and differences using symbols Validate solutions using true/false and matching activities Find sums and differences with two three-digit numbers Add and subtract multiples of 100 Practice mixed addition and subtraction using TouchPoints, regrouping reminders, and arrows as cues Add and subtract with only TouchPoints and arrows as reminders Solve word problems with equations, drawings, algorithms, and strategies Find sums and differences with no cues Find sums and differences with unknowns in all positions Demonstrate proficiency with algorithms in computation Upper Grades Scope and Sequence

15 SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of One-Digit Numbers Show Multiplication Using Various Means 3 Multiply using equal groups of objects Relate equal sets of objects in pictures to multiplication Skip count numbers of objects Show multiplication problems as repeated addition Transfer understanding of sets of objects to numbers to skip count Use arrays of objects to skip count Skip count on a number line Fill in missing numbers in a sequence Recognize skip counting patterns in a hundred chart Use skip counting in place value to show patterns Identify numbers out of order in a sequence s 3-4 Skip count without TouchPoints as a strategy to learn multiples Skip count by 2 Skip count by 3 Skip count by 4 Skip count by 5 Skip count by 6 Skip count by 7 Skip count by 8 Skip count by

16 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of One-Digit Numbers continued Apply skip counting Interpret and create scaled bar graphs using multiples Demonstrate understanding with vertical and horizontal bar graphs Compare data in scaled bar graphs Use arrays and pictures to solve multiplication word problems Consistently apply skip counting sequences to answer word problems Find mystery numbers from clues about sequences Identify numbers after, between, and before numbers in a sequence Identify numbers that are not multiples in a given sequence Draw pictures to represent multiplication word problems Write multiplication sentences from pictures and word problems Use skip counting and multiples to answer word problems s 3-4 Demonstrate fluency with skip counting in multiple forms Multiply Using Even and Odd Numbers Find even and odd numbers to demonstrate multiplication Use pairs of objects to find even and odd numbers Use pairs in pictures to find even and odd numbers Match groups of objects in pictures to even or odd numbers Identify numbers 1 10 as even or odd to use for multiplication Find pairs and equal addends to demonstrate multiplication Write multiplication problems as equal addends from pictures Show multiplication problems as the sum of equal addends Write multiplication problems using x as the multiplication symbol Demonstrate the commutative property of multiplication Identify numbers as even or odd to find pairs of equal addends Upper Grades Scope and Sequence

17 SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of One-Digit Numbers continued Multiply Using Models and Cues 4 Show multiplication using various means Use vocabulary to express equations (factors and product) Read and write multiplication equations Demonstrate multiplication by skip counting TouchPoints Show multiplication with sets of pictures, arrays, and number lines Use sequences of multiples and TouchPoints Relate addition and multiplication equations Solve word problems with models Find products using vertical and horizontal formats Multiply by 1 5 Multiply by 1 Multiply by 2 Multiply by 3 Multiply by 4 Multiply by 5 Match products with different factors Multiply by mixed factors Evaluate and select correct products from multiple choices Find products using arrays of dots Write and solve equations to match arrays Draw arrays of dots to represent multiplication problems Relate arrays to area models Choose expressions to match word problems Solve word problems

18 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of One-Digit Numbers continued Multiply using cues Apply multiples in sequences and TouchPoints Apply patterns and place value Use the commutative property for factors Fill in missing numbers in forward-counting sequences Fill in missing numbers in backward-counting sequences Use number families Find products using sequences only (no TouchPoints) Solve word problems Multiply by 6 9 Multiply by 6 Multiply by 7 Multiply by 8 Multiply by 9 Match products with different factors Multiply by mixed factors Evaluate and select correct products from multiple choices Match models and strategies to products Demonstrate fluency with one-digit factors Identify prime numbers Identify multiples of 1 9 on a hundred chart Recognize different pairs of factors for given products Locate prime numbers Upper Grades Scope and Sequence

19 SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of One-Digit Numbers continued Divide Using Pictures, Models, and Cues 4 Show division using various means Use vocabulary to express equations (dividend, divisor, quotient) Relate multiplication and division Link dividend and divisor to factor; link quotient to product Read and write division equations Show division with sets of pictures, arrays, and area models Use sequences of multiples and tally boxes to record quantities in quotients Solve word problems with models Divide by 1 5 Divide by 1 Divide by 2 Divide by 3 Divide by 4 Divide by 5 Find quotients in vertical and horizontal formats Match quotients using different dividends and divisors Evaluate quotients for true/false Find quotients using tally boxes only Divide Using Fewer Cues Demonstrate division with divisors 6 9 Interpret, write, and solve division equations using sets of pictures, dot arrays, and area models Divide using sequences and tally boxes Find quotients in mixed review using no cues Find number families in number puzzles Match quotients to expressions Solve for the unknown x in all parts of an equation

20 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of One-Digit Numbers continued Divide by 6 9 Divide by 6 Divide by 7 Divide by 8 Divide by 9 Find quotients with no cues in various formats Match models and strategies to quotients Compare quotients using pairs of numbers Evaluate and select correct quotients from multiple choices Multiply and Divide with Strategies Find relationships Identify four equations using the three numbers in a number family Write equations for each operation Recognize facts that do not belong to number families Apply properties Use the commutative, associative, and distributive properties to multiply Use the identity and zero properties to multiply Use the identity and zero properties to divide Identify properties used to find values of unknowns Use patterns Apply given rules for in/out tables Identify rules used for given in/out tables Demonstrate multiples on a hundred chart Use a multiplication chart to find product and quotients Solve for unknowns Use relationships of given values Check results using opposite operations Apply processes to if/then statements Solve word problems Match values of unknowns in lists Find missing values in sequences Upper Grades Scope and Sequence

21 SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of One-Digit Numbers continued Use Multiples and Factors 4 Relate multiples to skip counting sequences Identify skip counting sequences (e.g., multiples of 3) Find missing multiples in sequences Identify missing multiples in the middle, at the end, and at the beginning of sequences Find missing multiples, and identify the sequences Identify the operation needed to find multiples Relate factors to multiplication and division Identify missing factors Connect missing factors to unknowns Evaluate pairs of factors for given products Recognize multiple pairs of factors for the same product Review multiples on a hundred chart or multiplication table Distinguish between multiples and factors Identify the operation needed to find factors Multiply and Divide within 100 Use symbols for numbers to practice facts Find products and quotients using no cues Compare pairs of factors to products Compare division expressions to quotients Apply multiplication and division in word problems Compare products of two pairs of factors Compare quotients of two division expressions Demonstrate fluency with mixed practice in various forms Write division expressions using both forms (i.e., and t) Validate products and quotients using true/false Use pictures and arrays to write word problems Fill in missing numbers from given number families in word problems Use strategies to solve multi-step word problems Solve for unknown factors using if/then statements Identify products and quotients from multiple choices

22 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of Multi-Digit Numbers Multiply and Divide Two-Digit Numbers by One-Digit Numbers 5 Multiply two-digit numbers by one-digit numbers Review multiplication using area models and unknowns for one-digit factors Multiply with no regrouping using TouchPoints, place value charts, and visual cues Apply understanding in one-step word problems Compare products of pairs of factors Multiply with regrouping using TouchPoints, place value charts, and visual cues Apply understanding in one-step word problems Calculate products with no regrouping and with mixed regrouping Solve two-step word problems Validate products using true/false and multiple choice Divide two-digit numbers by one-digit numbers Review division using equal sets in arrays Demonstrate division using area models Divide using sets and remainders Find quotients and remainders using area models Use tally boxes and dots to track multiples and remainders in division Divide using both formats Identify dividends and divisors without and with remainders Solve word problems Validate products using true/false and multiple choice Upper Grades Scope and Sequence

23 SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of Multi-Digit Numbers continued Use the algorithm to multiply multi-digit numbers by one-digit numbers Learn the algorithm using a visual reminder of the steps Apply the steps in the algorithm using TouchPoints, place value charts, and visual cues Multiply two-digit numbers by one-digit numbers with regrouping Find three-digit products using two-digit and one-digit factors Multiply three-digit numbers by one-digit numbers with no regrouping Apply the algorithm to find products of three-digit numbers by one-digit numbers with regrouping in all places Find products with regrouping in any place Identify places that require regrouping Multiply with 0 in any place Extend proficiency of multiplication to four-digit numbers by one-digit numbers with no regrouping Eliminate visual cues for the order of the steps to multiply Demonstrate the algorithm with regrouping Identify place values of given numbers in given products Estimate products Compare products to estimates Apply understanding in word problems

24 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of Multi-Digit Numbers continued Use the algorithm to divide multi-digit numbers by one-digit numbers Learn the algorithm using a visual reminder of the steps Apply the steps in the algorithm using tally boxes and place value indicators Apply the process using long division or short division, and check with multiplication Use the relationship of multiplication and division Divide two-digit numbers by one-digit numbers with no remainder Use the algorithm to find quotients of two-digit numbers by one-digit numbers with remainders Eliminate visual cues for the order of the steps to divide Estimate and compare quotients Identify dividends and divisors without and with remainders Divide three-digit numbers by one-digit numbers using double tally boxes for quotients and dots for remainders Divide four-digit numbers by one-digit numbers without/with remainders Eliminate tally boxes and other aids to find quotients Continue to use place value separators and arrows as indicators to bring down the next digit Demonstrate the process for long division and short division Apply understanding of multiples to determine if remainders will be present Identify ranges of quotients using greater than and less than Validate quotients by matching and multiple choice Upper Grades Scope and Sequence

25 SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of Multi-Digit Numbers continued Multiply and Divide Multi-Digit Numbers by Two-Digit Numbers 5 Multiply by two-digit numbers Find partial products using place value graphic organizers List partial products to add Multiply two-digit numbers using TouchPoints, place value separators, 0s and shaded areas, and the algorithm Find products with no regrouping Estimate products Calculate with regrouping of tens to hundreds Regroup ones and tens using arrows as indicators of starting points and boxes to record regrouping Compare products of pairs of factors Regroup tens twice using arrows as indicators of starting points Find four-digit products Estimate factors and products in word problems Compare estimated and actual products Multiply three-digit numbers with regrouping thousands only Estimate and find five-digit products Apply the algorithm to multiply three-digit numbers by two-digit numbers with regrouping in any place Compare estimates to the tens place and to the hundreds place with actual products Solve two-step word problems Find and record both partial products using TouchPoints, place value indicators, and shaded areas for the products when multiplying by 10 Validate and compare products using multiple choice

26 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of Multi-Digit Numbers continued Divide by two-digit numbers Divide by two-digit numbers using graph paper and area models Use the algorithm for long division to divide two-digit numbers without and with regrouping Round three-digit dividends and two-digit divisors to find one-digit quotients Use place value separators and spaces for recording estimates Compare quotients to numbers less than and greater than Validate quotients using true/false Find and compare quotients Find two-digit divisors using place value separators and arrows to indicate bringing down the next number Solve equations for missing factors Calculate quotients without written steps for estimating Divide with four-digit dividends and two-digit quotients Apply understanding to four-digit dividends and three-digit quotients without and with regrouping Use 0s in all places in quotients Use calculators to check equations Match problems to estimated dividends, divisors, and quotients Upper Grades Scope and Sequence

27 SCOPE AND SEQUENCE: UPPER GRADES Multiplication and Division of Multi-Digit Numbers continued Multiply and Divide with Multi-Digit Factors 5 Perform operations with no cues Solve puzzles using codes Use mixed operations on an activity sheet Identify operations to solve word problems Match problems to solutions Compare products and quotients without solving Find and compare products and quotients Evaluate solutions in multiple choice Relate acronyms for algorithms to the operations Use relationships of the two operations Verify solutions using calculators Estimate and solve word problems Solve for unknowns Demonstrate proficiency with algorithms in computation

28 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Mixed Operations with Whole Numbers Relationships of Operations 6 Add and subtract Use understanding of part/whole relationships Find unknowns in addition equations Find unknowns in subtraction equations Recognize and apply properties Compare pairs of addends to sums Compare subtraction expressions to differences Compare pairs of addends without calculating Compare subtraction expressions without calculating Solve for unknowns and match values Use relationships to solve word problems Add and multiply Demonstrate multiplication as repeated addition using sets of pictures, arrays, area models, number lines, multiples, and skip counting Recognize and apply properties Identify operations that relate to given properties Compare pairs of factors to multiple addends Solve for unknowns, and match values Identify operations to solve word problems Match operations to equations Subtract and divide Demonstrate division as repeated subtraction using backward skip counting by multiples on number lines Demonstrate division using arrays and area models Recognize and apply properties Compare expressions to solutions Solve for unknowns, and match values Select expressions to represent word problems Solve multi-step word problems Upper Grades Scope and Sequence

29 SCOPE AND SEQUENCE: UPPER GRADES Mixed Operations with Whole Numbers continued Multiply and divide Use sets of pictures, arrays, number lines, and area models to show related multiplication and division problems Write multiplication and division number sentences for word problems Recognize and apply properties Match multiplication and division equations Use if/then statements and the relationship of multiplication and division to solve for unknowns Identify and use operations to solve multi-step word problems Show multiple ways to solve word problems Compare expressions with unknowns Mixed Operations of Multi-Digit Numbers by One-Digit Numbers Two-digit numbers by one-digit numbers Apply number families and relationships of operations Identify expressions to solve word problems Identify parts and whole Apply properties and patterns Describe patterns Use pictures and equations to solve problems Find solutions with no regrouping or remainders Find solutions with regrouping and remainders Evaluate solutions to identify true or false Three-digit numbers by one-digit numbers Use 0s in all places in quotients Use 1 as addend, subtrahend, factor, and divisor Perform operations without and with regrouping and remainders Compare pairs of numbers with different operations Identify equations that require regrouping and remainders

30 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Mixed Operations with Whole Numbers continued Multi-digit numbers by one-digit numbers Evaluate and select solutions using multiple choice Find unknowns using multiple operations in expressions Solve multi-step word problems Evaluate choices, and compare solutions Four-digit numbers by one-digit numbers Apply understanding in computation Identify places that require regrouping Evaluate and select solutions using multiple choice Find unknowns in all locations Solve and compare solutions in word problems Use tables to solve word problems Mixed Operations of Multi-Digit Numbers by Two-Digit Numbers Two-digit numbers by two-digit numbers Apply properties to find solutions Use rounding to estimate solutions Identify places for regrouping Three-digit numbers by two-digit numbers Determine operations to make equations of given numbers Find missing digits in solutions Use if/then statements and relationships of operations to solve for unknowns Estimate and compare solutions without calculating Mentally compute with multiples of 10 Four-digit numbers by two-digit numbers Apply standard algorithms Solve for unknowns using properties and relationships of operations Compare solutions Upper Grades Scope and Sequence

31 SCOPE AND SEQUENCE: UPPER GRADES Mixed Operations with Whole Numbers continued Multi-digit numbers by two-digit numbers Match solutions found using different operations Create equations with the same solution but different operations Compare solutions with the same numbers but different operations Apply relationships, properties, and multiples of 10 and 100 Check solutions using calculators Create equations with unknowns to solve word problems Mixed Operations of Multi-Digit Numbers by Three-Digit Numbers Three-digit numbers by three-digit numbers Compute using standard algorithms Use calculators to check Four-digit numbers by three-digit numbers Extend computation skills Apply calculator skills Multi-digit numbers by three-digit numbers Solve for unknowns Match solutions found using different operations Compare solutions using tables and equations Compare solutions to given estimates Solve word problems Demonstrate proficiency Read solutions in all places through millions Mixed Operations with Multi-Digit Numbers by Four-Digit Numbers Four-digit numbers by four-digit numbers Perform operations Apply understanding using previous presentations of numbers Read solutions in all places through ten-millions

32 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Mixed Operations with Whole Numbers continued Mixed Practice with Mixed Operations 6 Computation Apply algorithms, properties, relationships, and strategies Use calculators efficiently and appropriately Comparisons Compare and order solutions Use number families Identify equality using true/false Unknowns Solve for unknowns in all positions and all places Evaluate and compare values of unknowns to given numbers Apply unknowns in equations using tables to solve word problems Choose the operation Identify operations that create equality of three given numbers Select operations to solve word problems Identify multiple operations to solve word problems Evaluate and select expressions with parentheses to solve word problems Proficiency with Mixed Operations Demonstrate fluency with facts in operations Apply algorithms, properties, relationships, and strategies Use tools and strategies for computation Recognize application in real world situations Analyze equations Evaluate solutions, and apply problem solving strategies Verbalize processes to explain strategies Identify strategies that contribute to achievement Upper Grades Scope and Sequence

33 SCOPE AND SEQUENCE: UPPER GRADES Fractions Represent Fractions Using Pictures, Geometric Shapes, Number Lines, and Models 7 Define and describe fractions Define fractions with pictures Demonstrate fractions on number lines Partition geometric shapes into fractional parts Relate fractions to division Recognize fractions as equal parts of a whole Describe fractions as equal parts of sets Match pictures to denominators Identify fractional parts of shapes, number lines, real world objects, and sets through sixths Read and write numerator and denominator Read and write fraction names Associate fractions, fraction names, and pictures of fractions Shade geometric shapes to represent fractions Identify, name, read, and write fractions through sixteenths Match representations of fractions to fractions and fraction names Represent fractions using sets or parts of a whole Evaluate multiple choice answers to identify fractional parts of pictures Compare fractional parts using pictures

34 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Fractions continued Understand equality and inequality Recognize equal fractions as the same amount of shading in pictures Distinguish between the number of shaded parts and the amount of shading Write equal fractions through eighths using pictures Rename familiar fractions using pictures Fill in tables with missing fractions Recognize that fractional parts decrease as denominators increase Order fractions with numerators 1 4 using shaded parts of geometric shapes, models with columns divided into parts through sixteenths, and number lines Compare fractions with the same numerator using pictures and symbols Compare fractions with different numerators and denominators using pictures and symbols Demonstrate understanding of equality and inequality using fractions, pictures, models, and symbols Extend understanding to mixed numbers Define fractions, whole numbers, and improper fractions in terms of numerators and denominators Define mixed numbers as whole numbers and fractions Identify fractions, whole numbers, improper fractions, and mixed numbers Find mixed numbers from improper fractions using pictures Use multiples to find wholes from improper fractions Show the relationships of improper fractions and mixed numbers with pictures Find improper fractions using mixed numbers with pictures Match pictures to mixed numbers Show mixed numbers as pictures, and write their improper fractions Upper Grades Scope and Sequence

35 SCOPE AND SEQUENCE: UPPER GRADES Fractions continued Relate Fractions, Whole Numbers, Improper Fractions, and Mixed Numbers 7 Apply understanding Match representations with and without pictures Compare fractions with and without models Use vocabulary accurately Apply understanding in multiple choice Fill in missing numbers in equality tables Write mixed numbers for improper fractions by dividing numerators by denominators Write improper fractions for mixed numbers using multiplication and addition Generate equivalents by multiplying by n/n Simplify fractions by dividing by n/n Compare and order fractions with different numerators, different denominators, or both Add and Subtract Using Models and Fractions 8 Demonstrate with like denominators Combine and separate fractional parts Use fractional parts to make a whole Explain sums that result in the same numerator and denominator using equivalence Demonstrate removing parts from a whole Use pictures to illustrate removing a part greater than the fraction in a mixed number Use parts of geometric shapes, number lines, and sets of objects to support operations Associate models with fractions Perform operations with no models Use relationships of addition and subtraction Fill in missing numbers in fractions

36 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Fractions continued Extend understanding to unlike denominators Identify missing addends using models Write equations to match models Combine and separate fractional parts (fractions only) Combine and separate fractional parts (mixed numbers) Use models and equivalents to solve fraction equations Solve Equations with a Step-by-Step Process Find common denominators Use steps when one denominator is a factor of the other Multiply denominators to find a product Divide the product by the greatest common factor if there is one Use equivalents to rename fractions Multiply numerators and denominators by n/n Apply understanding of least common multiples and greatest common factors Solve equations with no regrouping Use and explain processes to apply skills in word problems Compute with mixed numbers Use the above step-by-step process to solve with no regrouping Find sums or differences of fractions and then whole numbers Use these skills to solve word problems Regroup mixed numbers Add using the standard process Simplify whole numbers and improper fractions in sums Demonstrate computation Subtract using the standard process Regroup whole numbers and fractions Check subtraction by adding Match solutions to choices in a list Evaluate sums and differences using multiple choice and true/false Solve multi-step word problems Upper Grades Scope and Sequence

37 SCOPE AND SEQUENCE: UPPER GRADES Fractions continued Multiply and Divide Using Models and Processes 8 Use relationships of operations Multiply fractions by whole numbers Use models to show repeated addition of fractions Apply the process for adding fractions Simplify improper fractions in products Relate the addition process to multiplication Represent whole numbers as fractions Use the algorithm to multiply numerators by numerators and denominators by denominators Divide fractions by whole numbers Use models to explain operations Relate the division process to multiplication Rename whole numbers as fractions Invert divisors and multiply Repeat the process used for multiplication Multiply and divide fractions by fractions using algorithms Apply algorithms using relationships of operations Find common factors to rename fractions before performing operations Simplify solutions Demonstrate proficiency with algorithms in computation Compute with mixed numbers Find solutions using operations with fractions by mixed numbers Convert mixed numbers to improper fractions Repeat the process using algorithms to find solutions Compare products of pairs of factors Apply understanding to mixed numbers by mixed numbers Evaluate products and quotients using multiple choice and true/false Solve simple word problems

38 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Fractions continued Mixed Practice with Mixed Operations 8 Word Problems Fill in missing numbers in tables Use tables to solve word problems Solve multi-step word problems Associate word problems with real world applications Use money, time, and measures in computation Comparisons Compare fractions with like numerators Compare different names for the same fraction Compare improper fractions and mixed numbers Compare mixed numbers Unknowns Use relationships of operations and algorithms Solve equations with fractions, fractions and mixed numbers, and mixed numbers Evaluate solutions using multiple choice Create and solve equations for word problems Create contexts for equations Assess reasonableness of solutions Use scaling and estimation Exhibit proficiency with computation Upper Grades Scope and Sequence

39 SCOPE AND SEQUENCE: UPPER GRADES Decimals Represent Decimals Using Pictures, Models, Place Value Charts, & Numbers 7 Define decimals Relate decimals to fractions with denominators of 10, 100, or 1,000 Explain decimals using base ten blocks Associate decimals with place value and expanded notation Compare decimal values and place values Read and write decimal names Read decimals as fractions and as point Understand and use vocabulary Explain that each decimal place value is 10 times the place to the right Order decimals Demonstrate the order of decimals from the decimal point Recognize that a number gets closer to 1 as the digit in the tenths place increases Compare two-digit decimals to one-digit decimals Understand the value of 0 in decimals Compare two-digit decimals to money Compare two-digit decimals with the same numbers in different places Extend understanding Apply place value to the thousandths place Read and write decimals with place value, expanded notation, and decimal names Recognize multiple representations of the same decimal Evaluate representations of decimals using multiple choice Compare decimals of one, two, and/or three places Identify place values of given numbers in decimals

40 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Decimals continued Round decimals using vertical number lines Compare rounding ones up to tens to rounding tenths up to ones Round tenths to the nearest whole number Round hundredths to the nearest tenth Relate decimals that have the same value (e.g.,.30 and.3) Round thousandths to the nearest hundredth Round thousandths to the nearest tenth Compare decimals to whole numbers Add and Subtract Using Models, Properties, Place Value, and Relationships 8 Demonstrate the process Use base ten models to show combining or separating tenths Find solutions using place value and alignment of decimal points Demonstrate regrouping of tenths to ones Use regrouping for sums and differences of hundredths with 0 in the tenths place Add and subtract two-digit decimals with regrouping Find sums and differences with regrouping of the ones place Use skills to solve puzzles using tenths and hundredths Match expressions to solutions using one- and two-digit decimals Apply processes to the thousandths Find solutions with numbers in the thousandths place only Use symbols to compare solutions to given amounts Extend application to two- and three-digit decimals Add 0s where needed Add and subtract three-digit decimals with values in all places Upper Grades Scope and Sequence

41 SCOPE AND SEQUENCE: UPPER GRADES Decimals continued Compute with mixed decimals Find solutions with one-, two-, and three-digit decimals and whole numbers with no regrouping Apply algorithms for addition and subtraction of whole numbers Regroup as needed to find sums and differences Match solutions to choices in a list Solve multi-step word problems Identify ranges of solutions using symbols for greater than and less than Multiply and Divide Using Models, Properties, Place Value, and Relationships Demonstrate processes Use base ten models to show area (product of two factors) with tenths Use place value to identify the number of places in products Count the number of places in products, and add 0s as needed Use base ten models to show the number of times a divisor is used in the dividend Use place value to identify the number of places in the quotient Move the decimal point to change the divisor into a whole number, and move the decimal point in the dividend the same number Relate algorithms to place value Apply algorithms using tenths

42 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Decimals continued Apply algorithms Extend processes to hundredths using arrows to show the relocation of decimals in division Calculate with no cues Find sums and differences using tenths and hundredths Apply understanding to money Extend processes to thousandths Record the number of digits in a solution before computing Multiply and divide by tenths, hundredths, and thousandths Match equations using relationships of multiplication and division Evaluate products and quotients for equality Compare decimals using place value Compute with mixed decimals Solve with digits on both sides of the decimal point with no regrouping Apply understanding and algorithms to regrouping Use processes to solve multi-step word problems Solve equations, and use codes to solve puzzles Find and compare products and quotients Evaluate and select solutions using multiple choice Mixed Operations with Mixed Decimals Word problems Solve multi-step, multi-operation word problems based on real world applications Record and use information in tables to solve word problems Comparisons Compare two mixed decimals with different numbers of places, and write the symbol Unknowns Use relationships of operations and algorithms Evaluate solutions using multiple choice and true/false Use unknowns in contexts of word problems Exhibit proficiency with computation Upper Grades Scope and Sequence

43 SCOPE AND SEQUENCE: UPPER GRADES Percents Represent Percents Using Pictures, Models, Place Value, and Relationships 7 Define percents Relate percents to money and fractions using models Explain percents using base ten blocks Associate percents with place value using number lines Link percents and decimals Locate percents on grids of 100 units Compare percents less than or equal to 100% Match fraction, decimal, and percent representations Use models to identify common percents and fractions Use tables and models to compare percents and fractions Demonstrate equivalent percents and fractions using the process to find equivalent fractions Extend understanding Write percents for whole numbers Relate percents greater than 100% to mixed decimals and mixed fractions Write mixed decimals and mixed fractions for percents Compare percents Compare percents to whole numbers Compare percents to mixed decimals Locate percents within each 100% on number lines Identify percents between two given whole numbers Apply understanding using codes to locate percents on number lines Match percents to models Add and Subtract Using Models, Graphs, Place Value, and Relationships 8 Perform operations Find sums and differences of percents in pie graphs Use vertical graphs for the whole to answer related word problems Apply operations to complete tables and answer questions Compare amounts of percents

44 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Percents continued Multiply and Divide Using Relationships of Percents and Decimals 8 Use knowledge of operations with whole numbers and decimals Express percents as decimals by moving decimal points Find percents less than 100% of one- to three-digit whole numbers using the algorithm Solve real world problems (e.g., discounts, tips, sales tax, etc.) Add 0s or round where needed Multiply percents by percents Identify contexts in which percents are multiplied by percents Fill in missing percents and amounts in tables demonstrating understanding of the relationships Use percents less than, equal to, and greater than 100% in computation Mixed Operations with Percents Word problems Apply understanding using graphs and multi-operation word problems Fill in missing information in tables Comparisons Compare common fractions to percents less than 100% Compare decimals to percents Unknowns Use information in tables to solve for unknowns using multiple operations Solve for unknowns using percents greater than 100% Apply relationships of operations and algorithms in word problems Demonstrate proficiency Upper Grades Scope and Sequence

45 SCOPE AND SEQUENCE: UPPER GRADES Relationships of Fractions, Decimals, and Percents Mixed Operations with Mixed Representations 8 Compute with different representations Demonstrate proficiency with mixed operations Evaluate solutions using multiple choice Solve for unknowns in all positions and all places Solve multi-step, multi-operation word problems based on real world applications Solve equations, and use codes to solve puzzles Complete tables using word problems Make comparisons using pairs of numbers in the same representation and with the same operation Compare different representations Demonstrate proficiency

46 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Geometry Review and Extend Understanding of 2-D Shapes 9 Demonstrate knowledge of how shapes are built Review properties of shapes (e.g., number of lines) Distinguish between open and closed shapes Differentiate between 2-D shapes and polygons Use vocabulary accurately Use vocabulary related to circles Identify components of shapes Begin building the structure of relationships of components Apply knowledge of nested relationships (e.g., letters combine to make words, words combine to make phrases, etc.) to shapes Recognize and use written symbols Learn names of polygons through decagon Find fractional parts of shapes Develop strategies to divide circles and polygons Trace and then draw lines to divide shapes Divide shapes into common fractions Divide shapes into halves through sixths Divide shapes into eighths through tenths Divide shapes into twelfths and sixteenths Recognize new interior shapes formed by fractional parts Relate fractional parts to the shapes (e.g., one-sixth of a hexagon is a triangle) Upper Grades Scope and Sequence

47 SCOPE AND SEQUENCE: UPPER GRADES Geometry continued Develop greater understanding of lines and angles Recognize that lines have no endpoints, rays have one endpoint, and line segments have two endpoints Distinguish between lines and line segments Demonstrate that line segments can be measured Apply understanding of relationships of points/lines and line segments/rays/shapes Identify components of shapes by their dimensions (e.g., points are 0-D, lines are 1-D, etc.) Use points and lines to draw rays Draw and label rays Recognize the symbol for rays Combine rays to make angles Combine opposite rays to make lines Compare rays and opposite rays Write mathematical equations (e.g., AB + AC = BAC) Explain that angles are measured in degrees Recognize the relationship of degrees to circles Measure angles with protractors

48 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Geometry continued Define types of lines Describe parallel lines Draw parallel lines Use the symbol for parallel lines Identify parallel lines in polygons Draw shapes and label parallel lines Recognize intersecting lines as opposites of parallel lines Demonstrate that all perpendicular lines are intersecting lines based on their properties Explain why not all intersecting lines are perpendicular lines Draw intersecting and perpendicular lines Identify perpendicular lines in polygons Draw polygons and label the perpendicular lines Draw polygons that have intersecting lines that are not perpendicular Identify pairs of lines using multiple choice Upper Grades Scope and Sequence

49 SCOPE AND SEQUENCE: UPPER GRADES Geometry continued Define angles Describe angles using endpoint, ray, vertex, degree, arc, and arm Identify where angles are found Explain why angles are 2-D shapes Classify angles by their properties Define types of angles by their measures Use acute, right, obtuse, and straight to describe angles Draw and measure angles by their descriptors Recognize types of angles when angles are rotated Define complementary angles Identify complementary angles in shapes Describe acute angles in isosceles right triangles using pictures Define and draw supplementary angles Associate combining types of angles with addition Define types of angles by their sums when combined Compare angles by their measures Contrast angles based on their properties Identify total measures of angles in shapes Evaluate statements, and use multiple choice Classify Polygons by Their Properties Recognize lines of symmetry Identify symmetry by folding shapes and comparing the halves Define symmetry Draw lines of symmetry Generalize about symmetric figures

50 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Geometry continued Identify properties of polygons Define shapes by number of sides or lines Categorize and compare shapes based on number and kind of lines (e.g., parallel, perpendicular, or intersecting), number of equal line lengths, number and kind of angles, measure of interior angles, and sum of angles Differentiate among types of triangles based on their properties Compare and contrast five types of quadrilaterals Recognize that shapes within a given category share attributes (e.g., rectangles and squares) Describe differences among shapes within the same categories (e.g., rectangles and squares) Record properties of regular polygons, including sums of interior angles Compare pentagons, hexagons, octagons, and decagons based on their properties Create a hierarchy of shapes Classify polygons by number and kind of lines Develop strategies to estimate angles for comparisons Order polygons by type and kind of angles Record the hierarchy of shapes in an inverted triangle as a graphic organizers Match properties, pictures, and names of polygons Analyze shapes, and evaluate multiple choices Recognize that 3-D shapes are comprised of 2-D shapes Upper Grades Scope and Sequence

51 SCOPE AND SEQUENCE: UPPER GRADES Perimeter, Area, and Volume Extend Understanding of Perimeter, Area, and Volume 9 Define the measures Define the terms with identifying questions (i.e., what, where, why, and how) Apply understanding using visual representations Learn formulas for the measures Demonstrate area by tiling Show volume by inserting smaller cubes into larger ones Demonstrate understanding of formulas with unknowns Relate operations with numbers to finding measures of shapes Find areas and perimeters of polygons Practice using formulas with pictures of shapes Apply understanding in word problems using rectangles Measure other quadrilaterals Replace unknowns with numbers in formulas Compare formulas for quadrilaterals Recognize the area for a right triangle is one-half that of a rectangle Measure triangles Match perimeters and areas to right triangles Find and measure triangles within other polygons Draw triangles that have given perimeters Draw triangles that have given areas Decompose irregular polygons into rectangles to find areas and perimeters Create irregular polygons with given areas and perimeters Create pairs of polygons with the same areas Create pairs of polygons with the same perimeters Use addition and subtraction to find measures of irregular polygons Find unknown lengths or widths for given areas and perimeters Solve word problems using formulas for area and perimeter

52 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Perimeter, Area, and Volume continued Apply understanding to 3-D shapes Use pictures and formulas to find volume Find and compare volumes of cubes by filling them with smaller cubes representing one cubic unit Use formulas to measure volumes of rectangular prisms Demonstrate understanding of volume using multiple choice Identify real world applications for volume Extend Understanding of Geometry 10 Measure angles Estimate and draw three types of angles Check validity of statements about angles using true/false Describe angle measurements using circles Draw and label three types of angles inside circles Draw and label three types of angles on graphic protractors Estimate and measure given angles Estimate and sketch given angles Draw given angles using protractors Compare estimated and actual measurements Recognize angle measurement as additive Create polygons Use ordered pairs as vertices on coordinate planes Plot ordered pairs Connect vertices to form polygons Label polygons Apply relationships of sides to find line lengths Identify ordered pairs for vertices Apply understanding in real world problems with diagrams Upper Grades Scope and Sequence

53 SCOPE AND SEQUENCE: UPPER GRADES Perimeter, Area, and Volume continued Create 3-D shapes Cut out and fold nets on designated lines Represent 3-D shapes made up of rectangles and triangles Build cubes, rectangular prisms, triangular prisms, triangular pyramids, and cylinders from nets Relate 2-D shapes used to build 3-D shapes Measure 3-D shapes Identify edges, faces, and vertices Find surface areas using formulas Solve real world problems using area Recognize surface area as additive Follow directions to draw nets of cubes using models Demonstrate the surface area formula Compare volumes and surface areas

54 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Time, Money, and Distance Extend Understanding of Time 9 Tell time to the minute Review telling time to five minutes Use graphic cues on analog clocks to tell time Relate analog and digital times Draw hands on analog clocks to represent given digital times Find elapsed times Solve word problems with elapsed time within a.m. or p.m. periods Demonstrate finding elapsed time using clocks and number lines Use addition and subtraction to find elapsed time beyond a.m. or p.m. Find 12-hour times for given 24-hour times Apply understanding of elapsed time to calendars Extend Understanding of Money Use multiple operations Apply understanding of values of coins and bills to decimals Add and subtract to solve multi-step word problems Incorporate all operations in word problems Apply graphic organizers as strategies Use information in tables List amounts of money earned with different rates or different amounts of time Compare totals in lists Create drawings to solve word problems Upper Grades Scope and Sequence

55 SCOPE AND SEQUENCE: UPPER GRADES Time, Money, and Distance continued Extend Understanding of Distance 9 Use multiple operations Use addition and subtraction or multiplication and division to find total distances in word problems Find distances between international cities Record and compare distances in charts Identify ascending and descending orders of distances Apply fractions Use graphic organizers including number lines and pictures Incorporate time, money, and distance Solve word problems using multiple topics and operations Recognize and apply skills to real world activities Evaluate and select correct answers to questions using multiple choice

56 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Measurement Use U.S. Customary s of Measure 9 Understand relative measures Identify given measures on rulers Draw and measure line lengths Relate measures for inches, feet, yards, and miles using graphics Interpret relationships of measures of capacity using graphics Demonstrate measures of capacity using ounces, cups, pints, quarts, and gallons Use ounces, pounds, and tons as units of measure for mass and weight Recognize and write abbreviations for measures Convert among units of measure Relate operations to making conversions Recognize and consistently use correct operations to make conversions Convert from larger units to smaller units using tables Convert from smaller units to larger units using strategies Match different representations of units Solve real world problems using drawings and conversions Estimate and evaluate choices with conversions Use Metric s of Measure Apply vocabulary Learn the meanings of milli-, centi-, deci-, and kilo- Recognize that milli-, centi-, and deci- identify parts less than one Correlate units to place value Relate measures for length (meters) using graphic rulers Link measures for capacity (liters) using graphic measuring cups Apply measures for weight and mass (grams) using objects as guides Apply abbreviations for measures Upper Grades Scope and Sequence

57 SCOPE AND SEQUENCE: UPPER GRADES Measurement continued Convert among units of measure Apply understanding of place value and operations Use graphics and tables to convert from larger units to smaller units Convert from smaller units to larger units using strategies and operations Use drawings to represent word problems Convert units as needed to solve word problems Evaluate choices to show conversions

58 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Data Represent and Interpret Data 9 Use multiple representations for data Compare data in picture graphs Correlate and extend data in scaled picture graphs to tables Interpret data in scaled bar graphs Read and understand histograms Compare and contrast bar graphs and histograms Record data in histograms Explain data in line plots Plot data from tables Solve real world problems with data Display data in dot plots Contrast line plots and dot plots Apply understanding of data in coordinate planes Use vocabulary related to coordinate planes accurately Identify the four quadrants Locate ordered pairs in quadrant I Relate coordinate planes to maps Find coordinates for identified places Solve for unknowns in statements about locating ordered pairs Recognize appropriate representations for data Apply vocabulary to different representations Evaluate and select representations to complete statements Record data in all representations Upper Grades Scope and Sequence

59 SCOPE AND SEQUENCE: UPPER GRADES Data continued Analyze Data 9 Develop a basic understanding of statistics Define statistics as the study of numerical data See that numerical data is collected and displayed in graphs or charts Recognize distribution as how often the data is used Describe data in terms of distribution (spread, range, center, and shape) Define overall shapes of data using understanding of patterns (e.g., bell curve, clusters, symmetrical, etc.) Distinguish between statistical and non-statistical questions Apply understanding of the basics Find distribution of data in line plots and histograms Identify statistical questions Analyze information in dot plots and histograms Answer multiple choice questions about data in all representations Summarize numerical data Solve word problems

60 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Pre-Algebra Apply and Extend Understanding of Operations 10 Review properties Associate the identity and zero properties with operations Compare and contrast the identity and zero properties Use the associative and distributive properties as strategies in computation Solve and match equations using properties Identify properties in word problems using multiple choice Understand order of operations Use an acronym to remember the order Apply each step in the order Solve expressions using the order with cues Use parentheses and brackets in expressions Match expressions Practice with exponents Relate exponents to area and volume using pictures Identify base numbers and exponents Relate application of exponents to repeated multiplication Find values of numbers with exponents Compare values of numbers with exponents Relate powers of 10 to place value Demonstrate finding values of powers of 10 Explain where and why powers of 10 are used Find and compare products of exponents and factors Apply understanding by completing tables for comparisons Use expanded notation in place value charts Compare compact numerals with expanded notation Extend understanding of exponents to multiplication and division Evaluate and select correct responses using multiple choice Upper Grades Scope and Sequence

61 SCOPE AND SEQUENCE: UPPER GRADES Pre-Algebra continued Use Ratios 10 Understand ratios as comparisons of quantities Recognize formats used to express ratios Relate ratios to operations Demonstrate that numbers in ratios are addends that are compared Define rates See rates as types of ratios Compare and contrast ratios and rates Use ratios to fill in tables to compare quantities Apply operations to solve word problems with ratios Solve real world problems using ratios in tables Use tables to compare measurement conversions Learn graphic organizers to find unknowns in ratios Solve riddles using tables and codes Exhibit understanding of tape diagrams Explain real world problems with tape diagrams Recognize double number lines used to compare rates Define properties of double number lines Compare and contrast tape diagrams and double number lines Draw double number lines to solve word problems Match ratios using multiples Apply understanding in evaluating multiple choices Explore Rational Numbers Recognize relationships of numbers Use a diagram to see that natural numbers, whole numbers, and integers are rational numbers Define rational numbers Recognize rational numbers as a category of real numbers Contrast rational and irrational numbers Compare real and imaginary numbers

62 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Pre-Algebra continued Demonstrate positive and negative numbers (integers) Use number lines to demonstrate relationships of numbers Define negative numbers Find values moving left from 0 on number lines Understand that greater negative numbers have less value on number lines Read negative numbers using real world examples (e.g., thermometers, debit cards, etc.) Relate positive and negative numbers to real world examples Describe quantities using opposite directions on number lines Recognize examples of negative numbers in elevation, temperature, and debit cards Analyze questions to determine accuracy using true/false Apply operations Demonstrate addition and subtraction as opposite operations Apply understanding of number families Link operations to positive and negative numbers Represent numbers on number lines Demonstrate operations by showing movement on number lines Associate addition and subtraction symbols with values of numbers Represent numbers on coordinate planes Transfer understanding of coordinate axes to locating ordered pairs Define types of integers in each quadrant Identify quadrants on grids Use vocabulary related to locations Apply understanding to maps Upper Grades Scope and Sequence

63 SCOPE AND SEQUENCE: UPPER GRADES Pre-Algebra continued Order numbers on number lines Move right of 0 for positive numbers Compare numbers Identify comparisons in which the number on the right is less than the number on the left Write comparison symbols for pairs of positive numbers Write comparison symbols for pairs of negative numbers Reason with comparisons by asking thoughtful questions Demonstrate comparisons of multiple positive and negative numbers Apply strategies to solving word problems Relate absolute values to relationships of positive and negative numbers Define and explain absolute value Recognize the symbol Find absolute values of given numbers Conclude that absolute values are positive numbers Compare numbers and their absolute values using number lines Make a generalization about positive numbers and their absolute values and negative numbers and their absolute values Graph ordered pairs Graph ordered pairs of numbers and their absolute values Find distances between locations in all four quadrants Expand Understanding of Expressions Recognize variables Explain that variables are parts of expressions Define variables Write expressions with variables Write expressions for sentences Use variables with operations to express statements Contrast pairs of operations using the same variables

64 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Pre-Algebra continued Integrate exponents into expressions Find values of variables with exponents Use multiple operations and parentheses in expressions Follow the order of operations Compare values of expressions with different values of variables Evaluate solutions using multiple choice Identify and apply parts of expressions Define familiar vocabulary in relation to expressions (sums, factors, products, and quotients) Describe and identify terms and coefficients Label parts of expressions Simplify expressions with different operations Combine parts of expressions Apply understanding in word problems Incorporate expressions in word problems into tables to find values of variables Transfer previous learning to expressions Use properties of operations to expand expressions Apply finding greatest common factors to writing expressions with parentheses Recognize relationships of pairs of operations (e.g., addition and multiplication) Represent variables in tables with x and y Plot ordered pairs using x and y coordinates Relate values for x and y to in/out tables Compare expressions for x and y Write values of y based on values of x Upper Grades Scope and Sequence

65 SCOPE AND SEQUENCE: UPPER GRADES Pre-Algebra continued Extend Learning about Variables 10 Apply sets of values Represent variables using in/out tables Demonstrate equality of the columns Recognize that values in the Out column correspond proportionately to values in the In column Apply terms to integers Define relationships as patterns Evaluate and find values of x in equations Extend practice to include coefficients and exponents Use in/out tables to identify values of variables in expressions See relationships of operations in balancing equations with variables on one side and constants on the other Apply and identify values of variables in expressions using operations Extend the process to find values in inequalities Make generalizations based on sets of values Use variables with integers, fractions, decimals, coefficients, and exponents Use formulas Represent relationships of variables in real world formulas, [e.g., D = rt, I = P(APR), etc.] Create tables to record values for given sets of values applied to formulas Solve word problems using formulas Compare given amounts at different rates Compare different amounts at different rates Relate comparisons to real world examples

66 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Pre-Algebra continued Identify and apply different types of variables Define and relate independent and dependent variables Relate independent to cause and dependent to effect Apply understanding to equations with two variables Recognize that one variable is used to define the other Relate previously used tables to finding dependent variables Solve word problems using tables and variables Create tables with variables to plot word problems Plot two equations on the same grid Compare the slopes of the two lines Evaluate and select answers to complete statements using multiple choice Incorporate Patterns into Algebraic Thinking Understand patterns as relationships among numbers Use integers and rational numbers in patterns Perform operations when needed Find one pattern in one-operation sequences Describe patterns Identify one pattern in multi-operation sequences Apply patterns to solve word problems Extend patterns using diagrams and number lines Find the next three numbers in mixed sequences Upper Grades Scope and Sequence

67 SCOPE AND SEQUENCE: UPPER GRADES Pre-Algebra continued Compare pairs of patterns Define the input numbers when the output numbers are the same in tables Plot pairs of patterns Describe patterns on coordinate grids Explain intersections of lines on coordinate grids Use diagrams to answer real world questions Compare sequences, and explain relationships between them Compare patterns with different coins Evaluate patterns in word problems using multiple choice with tables and coordinate planes Integrate skills in pre-algebraic thinking

68 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Overview of 1 The goal of TouchMatch Upper Grades Addition and Subtraction 1 is to build the concepts of and fluency with addition and subtraction within 20. This goal includes applying various strategies to operations, manipulating whole number relationships and place value, and understanding the baseten system. The TouchMath approach focuses on the use of TouchPoints to associate numerals and quantities, which strengthens the understanding of number. The Application to Students with Disabilities in the Common Core State Standards reflects this approach: Instructional supports for learning... foster student engagement by presenting information in multiple ways and allowing for diverse avenues of action and expression. Engaging activities are presented in an intentional sequence with attention to learning style, scaffolding of vocabulary, and pacing. The skills progress from counting and using TouchPoints to constructing the concepts of addition and subtraction by putting together and taking apart sets of objects. Computation is introduced naturally with a gradual presentation of multiple strategies. Properties of operations, place value, comparisons, unknowns, and word problems integrate into the activities to ensure application of the skills. When students finish this unit, they should be able to add and subtract within 20 and demonstrate a firm foundation for work with more complex problems. Objectives 1. To count forward and backward within 20 from any number 2. To master the TouchMath Touching/Counting Patterns for each of the numerals 3. To understand the concepts of addition and subtraction 4. To use strategies to solve equations 5. To solve for unknowns in all positions 6. To apply understanding of place value 7. To compose and decompose numbers To use properties of operations 9. To apply the relationship between addition and subtraction 10. To compare sums and differences 11. To solve word problems 12. To demonstrate fluency with addition and subtraction facts within Upper Grades Scope and Sequence

69 SCOPE AND SEQUENCE: UPPER GRADES Overview of 2 The goal of TouchMatch Upper Grades Addition and Subtraction 2 is to extend the concepts of and fluency with addition and subtraction mastered in 1. This includes using various strategies with operations, strengthening the relationship between addition and subtraction, applying sequences and patterns as computational aids, and employing a wide range of visual cues. 2 continues to focus on the TouchMath approach, which uses TouchPoints to associate numerals and quantities, strengthening the understanding of number. Engaging activities are presented in an intentional sequence with attention to learning style, scaffolding of concepts and vocabulary, and pacing. The skills progress from counting to using sequences and patterns. A review of operations with one-digit numbers ensures fluency with 20. The activities and skills progress to computation with 100, first without and then with regrouping. Activities flow from the kinsethetic to the kinesthetic/pictorial to the pictorial to the pictorial/abstract and finally to the abstract developmental level. Students have opportunities to build a firm foundation with each step before progressing to the next level. The approach used throughout the unit is based on the Universal Design for Learning to present information in multiple ways. This design has been scientifically proven to assist students with disabilities in attaining higher achievement. When students finish this unit, they should be able to add and subtract within 1,000, applying processes and algorithms for computation with whole numbers. Objectives 1. To count within 1,000 from any number 2. Te read and write number names for two- and three-digit numbers 3. To skip count by 5s, 10s, and 100s, 4. To use the Touching/Counting Patterns to count 5. To add and subtract within 1,000 using TouchPoints, concrete objects, drawings, strategies, properties, and relationships 6. To solve for unknowns in all positions 7. To compare sums and differences 8. To solve one- and two-step word problems 9. To demonstrate place value of the three-digit numbers with compact numbers, place value form, and expanded notation 10. To add and subtract three-digit numbers with place value 11. To round numbers to the nearest ten and hundred 12. To add and subtract multiples of 10 and To use algorithms to add and subtract with regrouping 14. To use strategies, drawings, properties, relationships, and no visual cues to solve word problems 15. To add and subtract with zeros in all places in three-digit numbers

70 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Overview of 3 The goal of TouchMath Upper Grades 3: Skip Counting is to develop proficiency with skip counting sequences for one-digit numbers. This skill establishes the foundation for multiplication and division using the TouchMath approach. The one-digit numbers are presented in sequential order, building on processes and patterns of lesser numbers. Each of the eight modules has two clusters: (1) Pictures, Patterns, and Practice and (2) Graphs, Arrays, and Application. Basic activities within each cluster are repeated for learner familiarity and increased rate of learning. The activities move from objects to pictures to numbers and demonstrate repeated addition for each one-digit number. Other motivational activities (e.g., puzzles, mazes, codes, missing numbers, numbers out of order) provide meaningful repetition in a variety of forms. The presentation can be thought of as the three R s of skip counting: repeat, reinforce, and relate. Throughout the unit, bubble answers accustom students to more formal testing procedures. They are mixed in with written answers in the modules, but the unit review and the unit pre- and posttest require only bubble answers. The unit presents information in multiple ways, in accordance with the Universal Design for Learning. This design has been scientifically proven to assist students with disabilities in attaining higher achievement. There is a high correlation between proficiency with skip counting and success with multiplication and division. When students finish this unit, they should be able to skip count one-digit numbers within 100. Objectives 1. To skip count as repeated addition using concrete objects and pictures 2. To skip count using numbers, number lines, and hundred charts 3. To apply place value when skip counting 4. To skip count using patterns 5. To cross out wrong numbers in a skip counting sequence 6. To create and interpret graphs using skip counting sequences 7. To compare results on graphs using skip counting sequences 8. To use arrays as a strategy for skip counting 9. To apply understanding of skip counting in word problems Upper Grades Scope and Sequence

71 SCOPE AND SEQUENCE: UPPER GRADES Overview of 4 The goals of TouchMath Upper Grades 4: Multiplication and Division 1 are to build concepts of and fluency with one-digit factors. Skip counting is used to ensure learner success. The one-digit numbers are presented in sequential order, building on processes and patterns of lesser numbers. Each of the six modules builds on the one preceding it: Skip Counting, Multiplication, Division, Strategies, Multiples and Factors, and Mixed Multiplication and Division. The activities progress from concrete/ kinesthetic to realistic models (pictures, arrays, and area models) to visual cues, and finally to abstract numbers. Learners apply strategies to computation with operations. Prime and composite numbers are discovered using a hundred chart. A multiplication table is introduced as a tool for seeing relationships and finding products and quotients. Throughout the unit, bubble answers accustom students to more formal testing procedures. These bubble answers are mixed in with written answers in the modules, but the unit review and the unit pre- and posttest require only bubble answers. The unit presents information with significant changes in materials and procedures to allow learners with disabilities to achieve within the framework of the Common Core. The intentional TouchMath stepby-step approach provides meaningful repetition and maintains the fidelity of concept development and computational success. Students should be able to perform basic multiplication and division with one-digit factors when they complete this unit. Objectives 1. To demonstrate fluency with skip counting by one-digit numbers 2. To understand the concepts of multiplication and division 3. To understand multiplication as repeated addition and division as repeated subtraction 4. To apply skip counting and the TouchMath approach 5. To use multiplication and division vocabulary 6. To use the relationship of multiplication and division, patterns, and properties 7. To identify prime and composite numbers 8. To apply understanding in solving word problems, comparing products, and finding values of unknowns 9. To select and apply strategies to solve equations 10. To demonstrate fluency with one-digit factors

72 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Overview of 5 The goals of TouchMath Upper Grades 5: Multiplication and Division 2 are to strengthen concepts and build proficiency with multi-digit numbers. These skill presentations are grounded in the TouchMath approach. The factors are presented in sequential order with an increasing number of digits, working from one digit by one digit and progressing to four digits by two digits. The seven modules each build on the one preceding it: Multiplication by One Digit, Division by One Digit, Multiplication Using the Algorithm, Division Using the Algorithm, Multiplication by Two Digits, Division by Two Digits, and Mixed Practice. The activities are scaffolded from no regrouping/ no remainders to gradual inclusion of regrouping/remainders using TouchMath kinesthetic/visual cues and skip counting. Algorithms for both operations are introduced with acronyms to aid recall. Rectangular arrays, area models, and graphic organizers encourage an increased rate of learning and success. Mixed practice includes the use of calculators for checking and solving problems with multidigit numbers. Throughout the unit, bubble answers accustom students to more formal testing procedures. They are mixed in with written answers in the modules, but the unit review and the unit pre- and posttest require only bubble answers. The unit provides instructional accommodations through the TouchMath approach and materials. Research supports the use of these modifications for students with learning disabilities. When students complete this unit, they will demonstrate proficiency with multiplication and division of four digits by two digits. Objectives 1. To demonstrate understanding of multiplication and division with concrete objects, drawings, illustrations, and equations 2. To learn and consistently apply algorithms for operations with multi-digit numbers 3. To use the TouchMath approach and cues in building independence in computation 4. To apply understanding in place value and comparisons 5. To solve one- and two-step word problems 6. To apply strategies of estimation, rounding, place value, patterns, properties of operations, and the relationship of multiplication and division 7. To demonstrate the appropriate use of calculators in multi-digit operations Upper Grades Scope and Sequence

73 SCOPE AND SEQUENCE: UPPER GRADES Overview of 6 The goal of TouchMath Upper Grades 6: Mixed Operations with Whole Numbers is to ensure proficiency with the four basic operations with whole numbers. The instruction and practice begin with relationships and properties of pairs of operations. The reinforcement throughout the unit is then scaffolded sequentially using the number of digits of the numbers in the equations. Previously introduced strategies are strengthened. The six modules each provide meaningful repetition and application of the strategies and algorithms in performing computation, making comparisons (with and without solving equations), finding unknowns, using 0s in various places, and solving word problems with multi-steps. Place value is emphasized throughout the activities. The modules are Relationships of Operations, Two Four Digits by One Digit, Two Four Digits by Two Digits, Three Four Digits by Three Digits, Four Digits, and Mixed Practice with Operations. The skills advance to estimating, explaining strategies and processes, reading and applying numbers in the hundred millions, relating the lattice method to place value for multiplication, and identifying multiple ways to express equations and solutions. Throughout the unit, formal testing procedures are implemented. Multiple choice and true/false with bubble answers are incorporated. Other reviews include matching, written answers, selecting the comparison or operation symbol to be used in an expression or equation, and other engagaing activities. The unit provides instructional support for learners with disabilities by presenting and allowing multiple strategies for finding and expressing solutions. Objectives 1. To use drawings to explain operations 2. To apply strategies for solving for unknowns, including if/then statements, properties, relationships, and place value 3. To compare expressions and equations with and without performing the operations 4. To solve multi-step word problems, including choosing the operation 5. To demonsrate proficiency with algorithms 6. To understand computation as operational comparisons and distinguish between additive and multiplicative comparisons 7. To use multiples of 10 and 100 as strategies in mental math 8. To interpret patterns, tables, and graphs to solve word problems 9. To read and write write numbers through hundred millions 10. To apply and explain strategies and why they work 11. To use estimation and rounding and assess reasonableness of answers 12. To check solutions with a calculator efficiently

74 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Overview of 7 The goal of TouchMath Upper Grades 7: Fractions, Decimals, and Percents 1 is to build a strong foundation with fractions, decimals, and percents. Each of the topics is initially defined as parts of a whole. The instruction and practice begin with manipulatives and pictorial models. Place value is emphasized throughout the unit. The concepts are scaffolded to ensure learner success. Common fractions, mixed numbers, and improper fractions from wholes through sixteenths are presented. Vocabulary is introduced gradually, moving from the familiar language to the unfamiliar. Common decimals and mixed decimals from tenths through thousandths are presented sequentially, along with their fraction equivalents. Students learn to convert fractions to decimals and to round decimals. Activities introduce learners to percents less than, equal to, and greater than 100. The unit concludes with the relationships of the three representations: fractions, decimals, and percents. Each of the eight modules provides meaningful repetition and application of the strategies to develop the concepts. The approach follows the TouchMath philosophy of moving from the kinesthetic to the abstract level, including small steps to ensure the groundwork for application in computation and word problems. The modules are Fractions Concepts; Application of Concepts; Equality and Inequality; Mixed Numbers; Fractions Review; Decimals; Percents; and Relationships of Fractions, Decimals, and Percents. Throughout the unit, formal testing procedures are implemented. Multiple choice and true/false with bubble answers are incorporated. Other reviews include matching, written answers, selecting the comparison symbol to be used in an expression or equation, and other engaging activities. The unit addresses students with disabilities by providing multiple instructional supports for learning. Objectives 1. To define fractions, decimals, and percents as equal parts of a whole, using manipulatives and pictorial models 2. To understand fractions as a number of shaded parts compared to the total number of parts 3. To create and label all parts for common fractions through sixteen-sixteenths, for decimals through ten-thousandths, and for percents less than, equal to, and greater than 100% 4. To label parts of a whole on a number line using fractions, decimals, and percents 5. To see the relationships of division to fractions, decimals, and percents 6. To recognize and generate equivalent fractions using models and multiplication of the numerator and denominator by n/n 7. To simplify fractions by dividing them by n/n 8. To compare and order fractions, decimals, and percents using the relationships of the number representations 9. To apply understanding of mathematical language 10. To convert fractions, decimals, and percents to one another 11. To explain that each digit in a base ten number system is 10 times the number to the right of it 12. To relate fractions and decimals to money 13. To round decimals 14. To recognize repeating decimals 15. To use fractions, decimals, and percents interchangeably depending on the real-world situation Upper Grades Scope and Sequence

75 SCOPE AND SEQUENCE: UPPER GRADES Overview of 8 The goal of TouchMath Upper Grades 8: Fractions, Decimals, and Percents 2 is to ensure application of the concepts learned in the previous unit, with an emphasis on the relationships among fractions, decimals, and percents. The instruction and practice begin with pictorial models, with place value at the core throughout the unit. The algorithms and operations in problem solving are scaffolded to ensure learner success. Each number representation is presented individually at first. As the succeeding representation is presented, relationships are reinforced. Computation is introduced in each number representation, with comparison/contrast to whole number computation. After practice with computation, students begin problem solving: word problems, comparisons, and unknowns. Each application is scaffolded from the least complex to the most complex. Learners use visuals throughout the unit for computation, interpretation of information in graphs, and completion of graphs, charts, and tables. The unit concludes with mixed review of all of the number representations, operations, and applications that were previously practiced individually. Each of the eight modules provides meaningful repetition and application of the strategies to develop proficiency. The approach follows the TouchMath philosophy of moving from the pictorial to the abstract level, including small steps to ensure proficiency with computation and application. The modules are Addition and Subtraction of Fractions, Multiplication and Division of Fractions, Addition and Subtraction of Decimals, Multiplication and Division of Decimals, Percents, Mixed Practice (computation), Application, and Mixed Review. Formal testing procedures are included throughout the unit with multiple choice and true/false problems with bubble answers. Other reviews include matching, written answers, selecting the comparison symbol to be used in an expression or equation, and other engaging activities. The unit addresses students with disabilities by providing multiple instructional supports for learning, including step by step procedures for algortihms and visual cues with the various representations. Objectives 1. To combine and separate parts as a basis for computation with fractions, decimals, and percents 2. To match fraction, decimal, and percent representations 3. To use the concept of whole and parts as a basis for computation 4. To distinguish between whole and parts in relation to fractions and in computation 5. To use models, properties of operations, relationships of operations and representations, algorithms, and place value in computation 6. To compare fractions to fractions, decimals to decimals, percents to percents, and all representations to one another 7. To identify a benchmark fraction to use for comparisons 8. To explain the processes and/or strategies used to solve problems 9. To estimate and assess the reasonableness of solutions 10. To use visual models (e.g., graphs, charts, and tables) to find and compare information 11. To apply understanding of mathematical language and vocabulary 12. To compute with money 13. To round decimals to a given place 14. To relate solving for whole number unknowns to solving for fraction, decimal, and percent unknowns 15. To use fractions, decimals, and percents interchangeably depending on the real-world situation

76 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Overview of 9 The goal of TouchMath Upper Grades 9: Geometry, Measurement, and Data is to ensure success and familiarity with more advanced levels of previously learned concepts and with new concepts that contribute to the basis for statistics and pre-algebra. The new concepts are presented within a framework of familiar language and examples that demonstrate the relationship of the new to the previously learned skills. Geometry, measurement, and data each include relationships of the concepts and application of operations. Each of the three topics is evident in real world applications that provide relevance for the learners. Each of the eight modules provides sound presentation, guided practice, meaningful repetition, and application of the concepts to develop a strong foundation. The approach follows the TouchMath philosophy of moving from the pictorial to the abstract level, including small steps to ensure proficiency with computation and application. The modules are 2-D Shapes; Lines and Angles; Classification of 2-D Figures; Perimeter, Area, and Volume; Time, Money, and Distance; s of Measure; Data Representation; and Data Analysis. Multiple choice and true/false questions with bubble answers are incorporated. Reviews include matching, written answers, creating representations of data, and other engaging activities. The unit addresses students with disabilities by providing multiple instructional supports for learning, including step-by-step instruction and visual cues throughout the various topics in the unit. Objectives 1. To develop a strategy for dividing shapes into fractional parts 2. To identify new interior shapes of fractional parts of larger shapes 3. To understand the relationships of points, lines, rays, line segments, and angles 4. To use measurement to distinguish between lines and line segments 5. To build a hierarchy of 2-D shapes based on categories and subcategories of the types and number of lines and angles 6. To estimate angles and measure them with a protractor 7. To form and apply generalizations about symmetrical figures 8. To understand and apply the properties of shapes in finding perimeter, area, and volume 9. To create shapes with a given perimeter and/or area 10. To solve one-step through multi-step word problems for the measurement of 2-D shapes 11. To tell time to the minute 12. To implement a strategy for finding elapsed time 13. To solve word problems by finding amounts of money using multiple operations and tables 14. To use drawings and number lines to represent and calculate distance 15. To review the relative size of measurement units for the U.S. customary and metric systems 16. To use tables of measure to convert among units 17. To use strategies to solve multi-step word problems and unit conversions within the same system 18. To create, compare, and interpret data in scaled picture and bar graphs 19. To record and interpret data in histograms, line plots, and dot plots 20. To identify and locate ordered pairs on coordinate graphs 21. To develop a foundation for understanding statistics as the study of numerical data 22. To describe data in terms of distribution on histograms, line plots, dot plots, and coordinate planes 23. To find measures of variation (mean, median, and mode) 24. To analyze and summarize numerical data Upper Grades Scope and Sequence

77 SCOPE AND SEQUENCE: UPPER GRADES Overview of 10 The goal of TouchMath Upper Grades 10: Pre-Algebra is to extend understanding of number relationships so students have a firm foundation for more complex processes in math. The unit focuses on developing strategies and language to stretch students critical thinking skills through integrating and deepening previously learned concepts. A discussion of algebra what it means, who uses it, why it is used, and how it is used opens the unit. Each of the seven modules includes guided practice with concrete and visual models, following the TouchMath philosophy of moving from the pictorial to the abstract in small steps to ensure success at every level. The modules are Operations, Ratio, Rational Numbers, Expressions, Variables, Patterns, and Geometry. The unit addresses students with disabilities by providing multiple instructional supports for learning, including step by step instruction, visual cues, meaningful repetition, and clearly defined strategies throughout the various topics. When students complete this unit, they will have a greater understanding of the relationships of number concepts and their real-world importance. Objectives 1. To review properties of operations and apply the order of operations 2. To learn base numbers and exponents and understand powers of 10 and place value 3. To use tables, equivalent fractions, cross multiplication, tape diagrams, and double number lines to demonstrate ratios and rates 4. To recognize positive and negative numbers with the same value as the same distance from 0 5. To relate addition and subtraction as opposite operations to positive and negative numbers as opposites on either side of 0 6. To use number lines to show comparisons and perform operations with positive and negative numbers 7. To locate points in all four quadrants of a coordinate plane 8. To apply absolute value in identifying distances on a coordinate plane 9. To use and understand the relationships of numbers and letters in expressions 10. To relate the value of the variable to the value of the expression 11. To apply sets of rational and integer values to inequalities 12. To learn the basic principles of interest, taxes, and tips 13. To relate independent and dependent variables to cause and effect 14. To create and solve word problems with two variables 15. To build and extend progressive single-sequence patterns 16. To correlate the number of zeros at the end of numbers to place value 17. To plot two patterns on a coordinate plane and describe the resulting lines 18. To recognize angles as shapes and parts of a circle 19. To estimate and sketch angles and to use a protractor to measure and draw them 20. To identify polygons created by connecting points on a coordinate plane 21. To use and make nets for 3-D shapes 22. To find surface area and volume 23. To solve real-world problems using diagrams, relationships, equations, and strategies 24. To relate geometric principles to pre-algebra 25. To increase critical thinking and problem solving skills with the application of algebraic concepts

78 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Basic Scope and Sequence Number Sense (Counting and Cardinality) 1 Count and read numbers to 20 2 Count and skip count numbers to 1,000 Represent and compare numbers to 1,000 Addition and Subtraction within 20 1 Add using TouchPoints, cues, and strategies Add using place value and properties of operations Subtract using TouchPoints, cues, and strategies 1-2 Subtract using place value and properties of operations Add and subtract fluently Solve one-step word problems Addition and Subtraction within 1,000 2 Use place value Round numbers Apply algorithms to operations without and with regrouping Add and subtract two- and three-digit numbers with regrouping Solve two-step word problems Multiplication and Division of One-Digit Numbers 3 Represent multiplication 3-4 Skip count with TouchPoints Multiply using even and odd numbers 4 Multiply using models and cues Multiply fluently with one-digit factors Identify prime numbers Represent division Divide using models and cues Divide fluently with two-digit dividends and one-digit divisors Multiply and divide using relationships, properties, and patterns Solve for unknowns Use multiples and factors Multiply and divide fluently in mixed practice Upper Grades Scope and Sequence

79 SCOPE AND SEQUENCE: UPPER GRADES Basic Scope and Sequence continued Multiplication and Division of Multi-Digit Numbers 5 Apply algorithms and relationships of operations to multiply and divide two-digit numbers by one-digit numbers Find products/quotients for multi-digit numbers by one-digit numbers Demonstrate long division and short division Estimate products and quotients Solve equations with unknowns Apply understanding of algorithms to multi-digit factors Compare products and quotients without performing the operations Solve multi-step word problems Verify solutions with calculators Mixed Operations with Whole Numbers 6 Apply relationships of operations Compute with multi-digit numbers by one-digit numbers Estimate, compute, and compare solutions with multi-digit numbers Mentally compute with multiples of 10 and 100 Use tools and strategies for computation Find and read solutions through the ten-millions place Order solutions Recognize application in real world word problems Verbalize processes to explain strategies Demonstrate proficiency with whole numbers Fractions 7 Represent fractions Associate fractions, fraction names, and representations Understand equality and inequality Compare and order fractions through sixteenths Apply understanding to mixed numbers Generate equivalent fractions Simplify fractions

80 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Basic Scope and Sequence continued Operations with Fractions 8 Add and subtract using models Apply relationships of addition and subtraction Demonstrate computation with unlike denominators Solve equations Solve multi-step word problems Multiply and divide using models Use algorithms and relationships to compute Simplify solutions Solve simple word problems Demonstrate proficiency with mixed operations Associate word problems with real world applications Compare all forms of fractions Create contexts for equations Assess reasonableness of solutions Use scaling and estimation Exhibit proficiency with computation Decimals 7 Represent decimals Relate decimals to fractions Associate decimals with place value and expanded notation Read and write decimals Order decimals Round decimals Upper Grades Scope and Sequence

81 SCOPE AND SEQUENCE: UPPER GRADES Basic Scope and Sequence continued Operations with Decimals 8 Add and subtract with models Apply relationships of addition and subtraction Apply algorithms for whole numbers to decimals Multiply and divide using models Relate decimals to money Solve multi-step word problems Find and compare solutions Solve for unknowns in word problems Exhibit proficiency with computation Percents 7 Represent percents Relate percents to money, decimals, and fractions Associate percents with place value Compare percents Use percents less than, equal to, and greater than 100% Operations with Percents 8 Add and subtract using models Apply operations in completing tables Multiply and divide using relationships and algorithms Solve real world word problems Solve equations for unknowns using multiple operations Exhibit proficiency with computation Relationships of Fractions, Decimals, and Percents 8 Compute with mixed operations and mixed representations Solve multi-step, multi-operation word problems Exhibit proficiency with all representations and operations

82 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Basic Scope and Sequence continued Geometry 9 Identify and understand relationships of components of 2-D shapes Find fractional parts of shapes Define and identify types of lines and angles in polygons Estimate and measure angles Explain complementary and supplementary angles Classify polygons by properties Generalize about symmetric figures Differentiate among types of triangles Compare and contrast five types of quadrilaterals Recognize that shapes within a given category share attributes Create a hierarchy of shapes Recognize that 3-D shapes are comprised of 2-D shapes Perimeter, Area, and Volume 9 Define measures, and learn formulas Find areas and perimeters of polygons 10 Compare formulas for quadrilaterals Find and measure triangles within other polygons Draw triangles and rectangles for given perimeters and areas Decompose irregular polygons into rectangles Use addition and subtraction to find measures of irregular polygons Find unknown lengths for given perimeters or areas Solve word problems using formulas Use formulas to measure volumes of cubes and rectangular prisms Estimate, sketch, and draw angles with given measures Compare actual and estimated measures Create polygons on coordinate places Use nets to build cubes, rectangular prisms, triangular prisms, and cylinders Measure 3-D shapes Find surface areas using formulas Compare volumes and surface areas Solve real world word problems Upper Grades Scope and Sequence

83 SCOPE AND SEQUENCE: UPPER GRADES Basic Scope and Sequence continued Time 9 Tell time to the minute Find elapsed times Use weekly, monthly, and yearly time periods Solve word problems Money 9 Count coins and bills Compare amounts Relate money to decimals Solve word problems Apply graphic organizers as strategies to find solutions Distance 9 Find distances on number lines Compare distances Use multiple operations to solve word problems Measurement 10 Use U.S. customary units of measure Represent measurement quantities in tables and diagrams Use relative measures for length, capacity, and mass Recognize and write abbreviations Convert among U.S. units of measure Relate operations to making conversions Use metric units of measure Apply vocabulary Correlate units to place value Convert among metric units of measure Convert from smaller units to larger units and larger units to smaller units Convert units in both systems of measures to solve word problems

84 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Basic Scope and Sequence continued Data 9 Use multiple representations for data Read, understand, and compare data Interpret data in bar graphs, histograms, line plots, and dot plots Plot data from tables and word problems Understand data in coordinate planes Recognize appropriate representations of data Develop basic understanding of statistics to analyze data Define statistics in terms of data Describe distribution of data Summarize numerical data Solve real world word problems Upper Grades Scope and Sequence

85 SCOPE AND SEQUENCE: UPPER GRADES Basic Scope and Sequence continued Pre-Algebra 10 Associate properties and operations Understand and apply order of operations Use exponents Relate powers of 10 to place value Find and compare products of exponents and factors Use expanded notation in place value charts Understand ratios as comparisons of quantities Define rates as types of ratios Apply graphic organizers including tape diagrams, double number lines, and tables to find unknowns in ratios Recognize relationships of kinds of numbers (e.g., whole numbers, integers, rational numbers) Demonstrate values of integers in relationship to 0 Relate integers to real world applications Apply operations to integers Demonstrate addition and subtraction as opposite operations Link operations to positive and negative numbers Represent relationships on coordinate planes Define types of integers in each quadrant Order numbers on number lines Reason with comparisons by asking thoughtful questions Relate absolute values to relationships of integers Graph ordered pairs and their absolute values

86 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Basic Scope and Sequence continued Expressions in Pre-Algebra 10 Define and recognize variables Apply exponents Identify parts of expressions Represent variables in tables with x and y Plot ordered pairs using x and y coordinates Write the value of y based on the value of x Apply sets of values Make generalizations based on sets of values Use variables with integers, fractions, decimals, coefficients, and exponents Use formulas Solve word problems using formulas Identify different types of variables Plot two equations on the same grid, and compare their slopes Patterns in Pre-Algebra 10 Understand patterns as relationships among numbers Use integers and rational numbers in patterns Compare pairs of patterns Plot pairs of patterns on coordinate planes Integrate skills into pre-algebraic thinking Upper Grades Scope and Sequence

87 SCOPE AND SEQUENCE: UPPER GRADES Notes

88 COMPREHENSIVE SCOPE AND SEQUENCE: UPPER GRADES Notes Upper Grades Scope and Sequence

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