G R A D E
IMPLEMENTATION
GUIDE
Everything you need to know to get the most
out of the TouchMath Second Grade Program
ad st!
Re s fir
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th
S E C O N D
INNOVATIVE LEARNING CONCEPTS INC.
creators of TOUCHMATH®
TouchMath materials were first published in 1975. 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™, TouchStar™, TouchMath Tunes™, TouchMath Tutor™, StepNumerals™ and Innovative Learning Concepts™ are trademarks of Innovative Learning
Concepts Inc. of Colorado Springs, CO.
Innovative Learning Concepts Inc. • 5445 Mark Dabling Blvd., Colorado Springs, CO 80918-3800
Toll Free: 1-800-888-9191 • Fax: 1-719-593-2446 • Web: www.TouchMath.com • E-mail: [email protected]
Printed in the United States of America
TouchMath Table of Contents
Navigating the Guide
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–2
• Philosophy of TouchMath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
• Overview of the TouchMath Second Grade Program and Vocabulary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Second Grade Sequence of Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–10
Second Grade Unit Box Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Second Grade Program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–15
• Unit 1: Adding and Subtracting Within 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
• Unit 2: Adding and Subtracting Within 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
• Unit 3: Adding and Subtracting Within 1,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
• Unit 4: Time, Money, Measurement, Data, and Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Second Grade Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–28
• Module Descriptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–21
• How to Use the Module Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
• Instructional Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
• Reproducible Activity Sheets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
• Bibliography for Literature Connections included in Second Grade Module Guides . . . . . . . . . . . . . . . 25–26
• Parent/Guardian Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
• Assessment and Progress Monitoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
TouchMath PLUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Teaching Aids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30–31
Classroom Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32–33
Common Core State Standards
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34–35
TouchMath Introduction
Philosophy
The philosophy of TouchMath.
The TouchMath philosophy may be stated in five easy words: REACH AND TEACH ALL
LEARNERS. The educators who have developed and expanded the TouchMath Program
throughout the last 40 years have worked constantly to meet this goal. Every TouchMath unit,
module, and worksheet reflects their dedication.
Janet Bullock
President/CEO
The success TouchMath has experienced is largely due to its multisensory approach. It is unique
in that it combines the use of the four major senses of seeing, saying, hearing, and touching with
hands-on manipulatives and paper/pencil written activities. As students engage all of their
senses, each individual within the group has the opportunity to assimilate information whether
they are visual, auditory, or tactile/kinesthetic learners.
TouchMath activities are meticulously scaffolded and provide all learners with ample
opportunities for success. As each skill is mastered, the next simple step is introduced. The TouchMath worksheets are
clean and uncluttered with age-appropriate artwork. The variety, quantity, and quality of the activities provided make
reinforcement immediately available on an as-needed basis. Testing and tracking materials provide instant feedback.
Teacher materials are straightforward and easy to use. Skills on each page are clearly defined, connected to the
Common Core State Standards, and presented using specific methodology. Instructional strategies include real world
connections, books to read, group and individual games, coloring, cutting-and-pasting, dot-to-dots, and a host of
other inviting motivators.
The TouchMath materials in this unit may be used as a complete core program or as a supplementary approach for
struggling students. The materials are effective in large group situations, in small groups, or for individual learners.
Students master each concept as they see it, say it, hear it, touch it, learn it.
1
© 2012
Second Grade Implementation Guide
TouchMath Introduction CONTINUED
Overview
Overview of the
TouchMath Second Grade Program.
The TouchMath Second Grade Program is designed to ensure learner success with four critical concepts: (1) extending
understanding of base 10 notation, (2) building fluency with addition and subtraction, (3) using standard units of
measure, and (4) describing and analyzing shapes. The content was built around the Common Core State Standards,
and each module includes a correlation to these standards. For easy reference, a complete list of the second grade
standards can be found on pages 34–35.
The TouchMath approach scaffolds the learning from using concrete objects, to using pictures, to using numbers.
Multiple representations are included on the pages so that learners at all developmental levels are successful.
Students will move naturally from the concrete level to the abstract level as they are ready.
Three of the four Second Grade Units relate to numbers and operations. These first three units are organized to present
small clusters of numbers within addition and subtraction, continually relating the two operations to each other. Unit
1 focuses on number concepts and operations within 50, including regrouping. Unit 2 extends the learning within
100 and introduces multiplication. Unit 3 applies the understanding within 1,000 and extends the understanding of
multiplication. The units build on the foundation of each previous unit, providing students with varied experiences
with numbers through 1,000. The goal of these first three units is for students to add and subtract fluently within 100 by
the end of second grade. (Students should be able to add and subtract fluently within 20 using mental strategies and
know the sums and differences of all two one-digit numbers.) Meaningful repetition using multiple representations,
coupled with the tactile, visual approach of TouchPoints, provides all learners with the support they need to meet
and/or exceed the goal of applying addition and subtraction within 100 consistently.
Unit 4 presents experiences with telling time, learning about money, representing and interpreting data (including
graphs and line plots), reasoning with length (including measurement, comparison, estimation, and relationship to
addition and subtraction), reasoning with shapes (including application of students’ understanding of the attributes
of shapes), partitioning rectangles into same-size squares as foundational work for area, and partitioning shapes
into equal shares for foundational work in understanding fractions. Skill development moves naturally from using
concrete objects to cutting and pasting pictures, to matching, and finally to drawing pictures.
Word problems are included and reinforced throughout each unit. The content is designed to relate to topics of
interest to children. The readability may be too high for some learners, so it is recommended that the teacher go over
the word problems with the students.
Cross-curricular examples are included throughout all four units. The pictures on student pages cross all curriculum
areas, and interesting information about science-related pictures is included. (Example: Horse or pony height is
measured by a unit of measurement known as a hand. One hand is equal to four inches.)
Vocabulary for each unit is included in each unit overview. Within the Instructional Strategies, these vocabulary
words are identified in boldface. The vocabulary builds on each previous unit, continually strengthening oral fluency,
and is intended to be used by the teacher and reinforced in language with the students. The words are presented in
print to the students.
© 2012
Second Grade Implementation Guide
2
TouchMath Sequence of Skills
Second Grade
Sequence of skills.
The Sequence of Skills is provided for an easy reference to all of the skills in the Second Grade Program and is
organized to mirror the order in which the skills are presented within each of the Common Core State Standards.
Included skills are scaffolded from the concrete to the more abstract, ensuring the learners’ understanding, and are
correlated to the Second Grade Common Core State Standards. (Please access TouchMath PLUS for a correlation
by standard.)
Use the Sequence of Skills to track the skills you have taught, recording dates to provide evidence of instruction. Or,
make copies of it to record individual student performance for each skill. Used in conjunction with the Progress
Monitoring Record, this can be useful for IEPs, parent conferences, and differentiated instruction.

OPERATIONS AND ALGEBRAIC THINKING
Add and Subtract Using Manipulatives, Pictures, and TouchPoints
2.OA.1, 2
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Show addition as putting together groups of objects
Show subtraction as taking away a part from a whole
Count groups of objects to get the sum or difference
Use drawings to explain addition and subtraction
Identify parts and wholes
Represent problems in multiple ways
Count the quantities and use TouchPoints to find sums and differences
Relate addition and subtraction to counting using TouchPoints
Tell word problems using pictures
Read one-step word problems
Write number sentences for word problems
Write word problems for number sentences
Use equations
Compare sums and differences using >, =, or <
Apply properties of operations
Apply the relationship between addition and subtraction
Match expressions to sums and differences
Identify true or false
Select correct answers from two to four choices
Find sums and differences within 13
 Apply strategies including TouchPoints
 Add and subtract fluently within 10
 Use Ten Frames to build tens and additional ones
 Count on from the larger addend and count backward from the minuend
 Cross out objects in pictures to demonstration subtraction
 Use boxes for unknowns
 Use other symbols for unknowns
Add and Subtract Using Pictures and TouchPoints
2.OA.1, 2
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Identify the correct operation and operation sign
Apply operations in vertical and horizontal formats
Identify and say the larger addend and minuend
Use TouchPoints on the smaller addend and subtrahend to count
and find sums and differences
• Compare sums and differences using >, =, or <
• Find sums and differences within 20
 Apply understanding of TouchPoints as tactile or visual cues
 Extend adding and subtracting fluently to within 20
 Recognize and apply patterns in addition and subtractions
(e.g., +4 or –4 to a sequence of numbers)
© 2012
Second Grade Implementation Guide
TouchMath Sequence of Skills CONTINUED
Second Grade
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Use doubles and doubles +/– 1 to build fluency
Identify multiple addends for a given whole
Compose 10 using the associative property
Decompose numbers to make easier sums
Apply understanding of number families
Add and Subtract Using TouchPoints and Visual Cues
2.OA.1, 2
2.NBT.6
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Use place value and indicators of where to begin
Compute with a one-digit number and a two-digit number
Solve problems with two two-digit numbers
Read and solve one- and two-digit numbers
Compare sums and differences using >, =, or <
Find sums and differences within 50 (no regrouping)
 Find sums and differences on a fifty chart
 Add up to four addends
 Select operation and solve problems
 Solve with unknowns in various positions
• Find sums and differences within 100
Skills listed under Number & Operations in Base Ten
Multiply Using Equal Groups of Objects
2.OA.3, 4
© 2012
• Sort concrete objects into equal groups
 Correlate the objects to TouchPoints on the numbers
 Relate the objects to pictures
 Use the pictures as TouchPoints on the numbers
 Show the problems as repeated addition of the same number
 Show multiplication as groups of 2, 3, 4, and 5
 Apply understanding in solving word problems
 Draw pictures to represent word problems
• Transfer learning from objects and pictures to arrays of dots
 Transfer from pictorial TouchPoints to TouchPoints
 Ring equal groups of dots in arrays
• Transition to skip counting without TouchPoints
 Tell word problems from pictures
 Find missing numbers in sequences
 Relate skip counting to equal groups of objects
 Solve word problems and draw pictures to represent them
 Find odd and even numbers using pairs
º Use objects
º Use pictures
º Match groups of pictures of objects to even (or odd) numbers
 Identify numbers 1–10 as even or odd
 Find pairs of two equal addends for numbers 11–25 using pictures
 Write the problem as the sum of the two equal addends
 Write the problem using x as the multiplication symbol
 Demonstrate the commutative property of multiplication
 Identify numbers 11–25 as even or odd
 Multiply by 2, 3, 4, and 5 (up to 5 x 5) using pictures, equal addends, arrays of dots,
and multiplication equations
 Draw arrays of dots to represent problems
 Use pictures to solve word problems
 Use drawings and equations to solve word problems
Second Grade Implementation Guide
4
TouchMath Sequence of Skills CONTINUED
Second Grade

NUMBER AND OPERATIONS IN BASE TEN
Understand Place Value Using Manipulatives, Pictures, Charts, and Numbers
2.NBT.1, 4
• Represent hundreds, tens, and ones using base 10 blocks
 Demonstrate that 10 is a bundle of 10 ones and a 100 is a bundle of 10 tens
 Relate pictures of base 10 blocks and place value charts
 Represent each multiple of 100 using models, pictures, and charts
• Find mystery numbers based on place value
• Represent 100–1,000 using expanded place value
 Represent numbers with unknowns in place value charts
 Match compact numerals to identified place value
• Use expanded notation to reflect place value
• Compare numbers using place value and expanded notation
• Find unknowns in compact numerals, place value, and expanded notation
• Identify compact numbers from written word place values
• Find mystery numbers based on written clues
• Match representations of numbers using words, mystery numbers, compact numerals,
place value, and expanded notation
Count, Read, and Write Numbers to 1,000
2.NBT.2–3
• Sequence count and read numbers for each hundred using a hundred chart
 Sequence count by fives and tens within hundreds
 Practice odd and even numbers within each hundred
• Find mystery numbers based on understanding the sequence of numbers
 Identify numbers that come immediately after a given number
 Identify numbers that come between two numbers
 Identify numbers that come immediately before a given number
• Use meaningful repetitions to sequence numbers in a variety of ways
• Compare numbers
• Find unknowns based on comparisons and place value
• Apply understanding of sequence and place value in word problems
• Read number words and write numerals in sequence
 Skip count and write numbers in sequence by fives to 100
 Use number words in flow charts to skip count by tens to 100
 Read, write, and skip count by hundreds to 1,000
 Use understanding of skip counting by fives, tens, and hundreds to find unknown numbers
• Write base 10 numerals in place value charts and as number names
 Match base 10 numerals and number names
 Write numbers from expanded forms
• Relate and write all forms of numbers: compact numerals, expanded forms,
and number names
• Apply understanding in writing numerals from number names in flow charts
• Integrate writing number names into finding missing numbers in sequence
Add and Subtract with Models and Visual Cues
2.NBT.4–7, 9 •
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Use base 10 blocks and Ten Frames to demonstrate regrouping
Use place value and indicators of where to begin
Associate models, pictures, place value chart, and numerals
Use TouchPoints as tactile or visual cues
Apply understanding of operations with one-digit and two-digit numbers
Extend learning to computing with two-digit numbers
Read and solve two-step word problems
Compare sums and differences
Find sums and differences within 50 (with regrouping)
 Use visual cues to support regrouping (boxes for addition and lines for subtraction of the tens)
 Add up to three addends
 Solve with unknowns in various positions
© 2012
Second Grade Implementation Guide
TouchMath Sequence of Skills CONTINUED
Second Grade
 Match models, pictures, place value charts, and numerals with/without TouchPoints to
represent problems
 Use input/output tables for meaningful repetitions
 Represent problems with drawings
 Select operations and solve problems
 Demonstrate problems with expanded place value
 Apply understanding of the relationship of addition and subtraction by using number families
 Match expressions to sums and differences
 Use the associative property to provide multiple solutions
 Provide written explanations or drawings of problems
 Extend application of finding 10 as a strategy
 Apply strategies in solving problems
• Find sums and differences with 100 (with and without regrouping)
 Determine if regrouping is needed
 Use if/then statements
 Use a hundred chart to demonstrate problems and their answers
 Work equations in both vertical and horizontal formats
 Confirm answers by matching
 Add up to four addends
 Perform operations without TouchPoints
 Solve increasingly complex word problems
 Demonstrate fluency using strategies
Add and Subtract with Strategies
2.NBT.8
• Find sums and differences using multiples of 10
 Add and subtract multiples of 10 with multiples of 100
 Use the relationship of addition and subtraction
 Find unknowns in all positions
 Apply understanding of if/then statements
 Solve word problems
 Add and subtract multiples of 10 with multiples of 100 and multiples of 10 (e.g., 250 + 40)
 Add and subtract multiples of 10 with a three-digit number (e.g., 957 – 50)
• Find sums and differences using multiples of 100
 Add and subtract multiples of 100 with multiples of 100
 Add and subtract multiples of 100 with multiples of 100 and multiples of 10
 Add and subtract multiples of 100 with a three-digit number
 Find and apply patterns in sequences of numbers
 Apply understanding in flow charts
 Compare sums and differences
 Use input/output tables for meaningful repetitions
 Match sums and differences
• Demonstrate adding and subtracting multiples of 10 and multiples of 100 mentally
• Use place value to find easier sums and differences
• Use number families
• Use properties of operations
• Use multiples of 10 and 10 +/– 1
• Select expressions that to not make a given sum or difference
• Use problem solving strategies
 Draw a picture
 Find a pattern
 Make a table
 Find unnecessary information
• Demonstrate adding and subtracting fluently within 100
Add and Subtract Three-Digit Numbers (within 1,000)
2.NBT.7
© 2012
• Extend understanding of regrouping with models
• Apply using visual cues for finding sums and differences
 Use place value charts and arrows for indicators of where to start
Second Grade Implementation Guide
6
TouchMath Sequence of Skills CONTINUED
Second Grade
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 Use boxes for regrouping in addition and lines for regrouping in subtraction
Order sums and differences from least to greatest and greatest to least
Compare sums and differences
Apply understanding in word problems using problem solving strategies
Transfer learning to computing with no visual cues
Solve for unknowns in all positions
Use multiple addends
Use new problem solving strategies
 Guess and check
 Write a number sentence
 Use logic
 Estimate
Apply multiple problem solving strategies to solve word problems
Select multiple expressions for a given sum or difference
Demonstrate using mixed addition and subtraction in flow charts
Use new problem solving strategies
 Work backward
 Choose an operation (calculation)
Apply problem solving strategies in complex word problems
Explain why addition and subtraction strategies work
Demonstrate understanding of adding and subtracting with 1,000

MEASUREMENT & DATA
Reason with Length
2.MD.1, 2
2.MD.4
2.MD.3
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• Learn about the ruler—standard (customary) measurement
 Find that a ruler equals 12 inches or one foot
 Demonstrate how to measure with and read the measurement using a ruler
 Measure line lengths up to 12 inches to the nearest number of whole-length units
 Identify objects that can be measured with a ruler (up to 12 inches)
 Use the customary abbreviation for inches and feet
• Learn about the yardstick
 Find that a yardstick equals three feet
 Demonstrate how to measure with and read the measurement using a yardstick to the nearest
number of whole-length feet
 Identify objects that can be measured with a yardstick
 Use the customary abbreviation for yards
• Learn about the tape measure
 Find that the tape measure shows markings for inches and feet
 Demonstrate how to measure with and read the measurement using a tape measure for
lengths greater than three feet
 Identify objects that can be measured with a tape measure
• Identify the tool that should probably be used to measure an object
 Measure and record the measurement
 Understand that the greater the length of an object, the larger the tool that should be used
 Explain that a tool that is too large to use to measure length of an object is cumbersome
 Select the appropriate tool and measure the lengths of common objects
 Measure objects twice with different tools
 Demonstrate that the larger the unit (tool) used, the fewer number of units in
the measurement
 Show that the smaller the unit used, the more accurate the measurement
• Compare lengths of two or more objects using <, =, or >
 Estimate standard length, comparing inches and feet
 Estimate standard length, comparing inches to inches and feet to feet
 Find the difference in length of two lines
• Learn about the metric ruler (metric measurement)
 Find that a metric ruler equals ~30 centimeters
 Measure line lengths up to 30 centimeters to the nearest number of whole-length units
© 2012
Second Grade Implementation Guide
TouchMath Sequence of Skills CONTINUED
Second Grade
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2.MD.4
2.MD.3
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2.MD.5
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2.MD.6
 Use the customary abbreviation for centimeters
 Identify objects that can be measured with a metric ruler
Learn about the meter stick
 Find that a meter stick equals 100 centimeters
 Demonstrate how to measure with and read the measurement using a meter stick to the
nearest number of whole-length meters
 Demonstrate how to read the measurement using a meter stick for centimeters
 Identify objects that can be measured with a meter stick
 Use the customary abbreviation for meters
Identify the tool that should probably be used to measure an object
 Measure and record the measurement
 Understand that the greater the length of an object, the larger the tool that should be used
 Explain that a tool that is too large to use to measure the length of an object is cumbersome
 Select the appropriate tool and measure the lengths of common objects
 Measure objects twice with different tools
 Demonstrate that the larger the unit (tool) used, the fewer number of units in
the measurement
 Show that the smaller the unit used, the more accurate the measurement
Compare lengths of two or more objects using <, =, or >
 Estimate metric length, comparing centimeters and meters
 Estimate metric length, comparing centimeters to centimeters and meters to meters
 Find the difference in length of two lines
Relate addition and subtraction to length
 Use the drawing of a ruler with a centimeter markings
 Measure line lengths
 Cut out line lengths and lay them end-to-end to add or with the shorter one on top of the
other to subtract
 Record the sums and differences of line lengths in equations
 Measure the line lengths of the sides of shapes (readiness for perimeter)
 Find the sum and difference of the line lengths in shapes
 Apply understanding in word problems with diagrams and pictures
 Solve for an unknown in word problems
 Add and subtract differences in diagrams
 Relate distances to lengths
 Apply understanding in word problems with diagrams and pictures including unknowns
 Write equations to solve problems with addition and subtraction of length/distance
 Represent line lengths on number line diagrams
 Demonstrate finding sums and differences of line lengths on number line diagrams
Tell Time
2.MD.7
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© 2012
Understand a 24-hour day
Trace hands on analog clock and corresponding time on a digital clock
Identify a.m. and p.m. activities
Relate a.m. and p.m. activities to personal experiences
Tell time
 Tell time to the hour
 Tell time to the half hour
 Skip count by fives to 60 using the star indicators on the analog clock
 Identify each five-minute interval between 11:30 a.m. and 12:30 p.m.
 Tell time to the nearest five minutes
Associate time on analog and digital clocks with personal experiences
Select the analog clock that reflects the time associated with an activity
Demonstrate telling time to the nearest five minutes with a.m. and p.m.
Find elapsed time
 Identify time one to two hours later
 Identify time one to two hours earlier
Second Grade Implementation Guide
8
TouchMath Sequence of Skills CONTINUED
Second Grade
Learn about Money
2.MD.8
• Identify coins
 Recognize the coin front and back
 Read and write the coin name
 Demonstrate value of coin using ¢ and $
• Identify one-dollar bill
 Read and write the bill name
 Demonstrate the value of the bill
 Know the purpose of the decimal point in the representation with $
• Count values of multiple coins
 Count the number of same-type coins
º Name and write the value of the multiple same-type coins using skip counting
º Compare values of multiple same-type coins (e.g., 8 nickels compared to 8 dimes)
º Skip count by 25 to count value of quarters
º Find value of one dollar using same-type coins
º Match multiple same-type coins to their values using $
 Count values of mixed coins
º Count values with the coin values arranged from greatest to least
º Represent values using ¢ and $
º Count values with the coin values arranged randomly
º Identify the coins for given values
º Use a problem solving strategy to find the value using the fewest number of coins
º Find one dollar using coins of multiple values
º Match values of mixed coins and dollars
 Compare values of mixed coins and dollars
 Apply understanding in word problems
 Use patterns and/or missing addends and/or subtraction to solve word problems
Represent and Interpret Data
2.MD.9, 10
• Create and interpret pictorial graphs
 Create graphs from given data
 Record data on both vertical and horizontal graphs
 Create graphs with up to four categories and 10 data points in each category
 Compare data from the graphs
• Construct and interpret bar graphs
 Create graphs from given data
 Select answers from four choices about the graphs
 Select the graph that represents given data
• Generate measurement data using the lengths of objects
 Use non-standard, customary, and metric measurements
 Record the data in a table
 Order the measurements from least to greatest
• Transfer the data to line plots
 Interpret the data in the line plots
 Find differences in line lengths represented in the line plots
• Measure objects with a picture of a centimeter ruler (up to 15 cm)
 Record lengths as whole-number units
 Select answers from four choices to questions about the line plots

GEOMETRY
Reason with Shapes
2.MD.1–3
9
•
•
•
•
•
Recognize and draw shapes
Identify the number and kind of sides and angles (e.g., equal, different)
Trace and write the name
Select a shape in different orientations and sizes from other shapes
Connect dots using a ruler to draw the shapes
© 2012
Second Grade Implementation Guide
TouchMath Sequence of Skills CONTINUED
Second Grade
• Draw shapes with no guides
• Distinguish between shapes with the same number of sides and angles using defining attributes
 Shapes with three sides and three angles: triangles (equilateral, right, isosceles, and irregular)
 Shapes with four sides and four angles: squares, rhombuses, rectangles, parallelograms,
and trapezoids
 Shapes with six sides and six angles: hexagons
 3-D shapes: cubes
 Match shapes, shape names, and definitions
• Partition rectangles into same-size squares (readiness for area)
• Follow dotted lines, then gray lines, then dots to be connected
• Use vertical and horizontal orientations
 Two, four, and six same-size squares
 Eight and nine same-size squares
 Ten and twelve same-size squares
• Identify true or false for statements about the number of squares in
given numbers of rows and columns
• Partition rectangles and circles into two, three, and four equal shares
(readiness for fractions)
• Shade the identified equal share, trace the fraction word name, and associate the fraction
• Connect dots using a ruler to divide the shapes into equal parts
• Identify the fraction for the shaded part
• Match partitioned circles and rectangles to the fraction and the fraction word names
(e.g., one third, a third of, three thirds, one whole)
• Draw the identified fractional part of rectangles
• Recognize that equal shares of identical wholes need not have the same shape
© 2012
Second Grade Implementation Guide
10
TouchMath Second Grade Contents
What is in the box?
Examining the Second Grade box contents.
Four unit boxes make up the Second Grade Program. These boxes contain the basic TouchMath materials needed in each
unit and are designed for effortless organization, ease of use, and clutter-free storage. When using a particular unit, turn
the inner box containing the module folders around 180 degrees and slide it back into the box backward, which allows for
easy examination. Once you have completed the unit, turn the inner box back around for space-efficient storage.
The titles of the Second Grade Units are as follows:
Unit 1
Unit 2
Unit 3
Unit 4
Adding and Subtracting Within 50
Adding and Subtracting Within 100
Adding and Subtracting Within 1,000
Time, Money, Measurement, Data, and Geometry
In an upright position, the Second
Grade Unit box allows for easy
access to materials while stored
on a bookshelf or desktop; from
a horizontal position, materials
are within reach when stored in a
drawer or filing cabinet.
Included are six modules inside
each of the Second Grade Unit
boxes (24 modules if you have all
four units). Each module title is
printed on the module spine.
Each module contains 25
reproducible student activities (left),
plus a complete Module Guide with
Instructional Strategies and answer
key (right).
There are also three sets of FlipCards in
each unit. These cards are designed to be
used for demonstration, additional practice
by small groups of learners, reinforcement
and/or extensions of skills, or another way of
presenting the skill.
The code included on the outside of every
Second Grade Unit box will allow you to
access TouchMath PLUS, the online portion
of the second grade materials. These
materials include the same pages available
in this Second Grade Unit. They enable
you to have computer fingertip access to
all activities and Module Guides anytime,
anywhere, for planning and preparation.
Online Digital Activation
n
Second Grade
de 2011 Unit 01
E
L
P
M
A
S
TouchMath FlipCards [Included] Order
extra sets! TM1160 
11
© 2012
Second Grade Implementation Guide
TouchMath Second Grade Program
UNIT 1: Adding, and Subtracting Within 50
Overview
Unit 1:
Adding and Subtracting
Within 50
This unit reviews basic addition and subtraction, scaffolding the review by increments
that support learners in their mastery of finding sums and differences within 13, then
20, and then 50. TouchPoints, whole/part diagrams, Ten Frames, place value charts,
indicators of where to begin, and fifty charts are used to provide visual, kinesthetic
cues. Students identify and ring the operation on the pages with mixed skills. Matching
activities, true/false statements, word problems, comparisons, and unknowns are
included throughout the activity pages.
This unit then extends addition and subtraction to include regrouping. Instruction
begins with base 10 blocks and Ten Frames to model regrouping, then moves to pictures
of the models with the manipulatives, and finally ends with the pictures themselves. The
learning is transferred to TouchPoints and place value charts. A variety of activities is
used to ensure student engagement as they progress through the developmental understanding of regrouping in
addition and subtraction. Strategies are introduced to aid in learning. Examples include expanded notation forms,
number families, properties of operations, find ten, find doubles, and find doubles +/– 1. Application of the skills in
solving one- and two-step word problems is inherent throughout the unit. Multiple testing formats are included, and
the relationship between addition and subtraction is thoroughly reinforced.
Vocabulary for Unit 1:
Number Concepts
Number Operations
Fifty chart
Part
Place value form
Ten Frame
Whole
© 2012
Addend
Associative property
Commutative property
Diagonal
Doubles
Doubles + 1
Doubles – 1
Equation
Expression
If/then statement
In/out table
Number family
Regroup
Rule
Strategy
Subtrahend
Symbol
Unknown
Second Grade Implementation Guide
12
TouchMath Second Grade Program CONTINUED
UNIT 2: Adding and Subtracting Within 100
Overview
Unit 2:
Adding and Subtracting
Within 100
This unit begins by extending understanding of base 10, using concrete models,
pictorial representations, and place value charts to establish the foundation for
understanding three-digit numbers. The instruction is then scaffolded to transfer
learning from place value charts and models to place value form (e.g., 3 hundreds +
1 ten + 9 ones = 319) and then to expanded notation (300 + 10 + 9 = 319). Frequent
reviews that include identification of the numbers in a given place value are built into
the program. As students demonstrate familiarity with place value, the learning is
extended to counting, reading, and writing numbers to 1,000. Hundred charts, mystery
numbers, color-by-number pictures, dots-to dots, and more are included to provide
meaningful repetition of the skills. Students integrate their understanding of place
value as they read and write numbers, including number names, numbers missing in
sequence, and numbers that are five, ten, or one hundred more than a given number.
Multiplication is introduced using objects, pictures, TouchPoints, arrays of dots, and skip counting to build the
concept of repeated addition. Arrays with up to five rows and five columns are used. The multiple presentations
provide learners with kinesthetic, visual, and abstract approaches to the development of understanding. The
learning is applied in word problems, and students are asked to draw pictures representing the problems.
Adding and subtracting within 100 are extended from Unit 1, where students worked with sums and differences
within 50. Regrouping is reviewed with base 10 blocks, Ten Frames, and place value charts. The learning is
scaffolded as it was in Unit 1, now to include two-digit numbers. Addition and subtraction are presented separately
in this unit to provide adequate time to reinforce the learning. Students identify if regrouping is needed to find the
answer. The activities provide many experiences that contribute to adding and subtracting fluently within 100.
Vocabulary for Unit 2:
All vocabulary words used in Unit 1, as well as those listed below:
Number Concepts
After
Array
Before
Between
Compact number
Even number
Expanded notation
Hundreds
Odd number
Ones
Place value
Place value chart
Skip count
Standard form
Tens
Value
13
Number Operations
Multiplication
Multiply
Regroup
Repeated addition
© 2012
Second Grade Implementation Guide
TouchMath Second Grade Program CONTINUED
UNIT 3: Adding and Subtracting Within 1,000
Overview
Unit 3:
Adding and Subtracting
Within 1,000
This unit builds on the foundation of the base 10 system developed in the two previous
units. Students begin by adding and subtracting multiples of 10 and 100 with two- and
three-digit numbers. The learning is scaffolded to add the multiples first to the hundreds,
then to the hundreds and tens, and finally to all three digits. Adding and subtracting with
100 is then reviewed, with mixed operations on the pages. Application in word problems,
comparisons, unknowns, and computation strategies are incorporated. Additionally,
twelve problem solving strategies are introduced individually with visuals and guided
support in learning to apply methods to solving word problems.
The solving of equations with two-digit numbers is transferred to three-digit numbers,
again with addition and subtraction presented individually and then as mixed operations.
The presentation mirrors those of the introductions in the two previous units. Students
use diagrams and draw pictures to represent the word problems as they work systematically to find solutions. Students
have many opportunities to work with three-digit numbers throughout their experiences with this unit.
The unit concludes with a review of multiplication. These activities broaden and extend the introduction in the previous
unit. The presentation begins with finding odd and even numbers using pairs, with pictures representing objects
typically in pairs (e.g., shoes). The learning is then scaffolded to apply the concept of odd and even in finding two equal
addends (through sums of 25). Again, arrays and skip counting are presented for finding products up to 25. Repeated
addition continues to be reinforced, and the multiplication symbol (x) is introduced. The activities conclude with
application of the concept in word problems. Students ring groups and/or draw pictures to represent the problems. A
firm foundation for multiplication is evident once students have the experiences provided in Units 2 and 3.
Vocabulary for Unit 3:
All vocabulary words used in Units 1 and 2, as well as those listed below:
Number Concepts
Digit
Expanded notation
© 2012
Number Operations
Array
Calculate
Calculation
Choose a calculation
Draw a picture
Even number
Estimate
Find a pattern
Find extra information
Guess and check
Make a table
Mental math
Mentally
Multiple
Multiplication symbol
Multiply
Odd number
Product
Problem solving strategy
Use logic
Work backward
Write a number sentence
Second Grade Implementation Guide
14
TouchMath Second Grade Program CONTINUED
UNIT 4: Time, Money, Measurement, Data, and Geometry
Overview
Unit 4:
Time, Money, Measurement, Data,
and Geometry
This unit provides hands-on activities for time, money, measurement, data, and
geometry. The time in hours, half hours, and five-minute intervals is to be traced,
matched, or identified. Learning about a 24-hour day, a.m. and p.m., and elapsed
time are all presented. The activities with money include coin and value identification,
counting money, identifying coins/dollar bills to match given values, and comparing
values.
The data section presents picture and bar graphs that are used for students to record
and interpret data in both vertical and horizontal formats. The learning is extended to
collecting, recording, and organizing the data, and transferring the data to line plots. The
activities conclude with interpreting line plots.
The final sections in measurement and data include measuring and estimating lengths in standard units, using both
customary and metric measurement. Students learn to use the most appropriate standard tools to measure, record,
and compare lengths. The learning is scaffolded to using addition and subtraction with line lengths. Students use
diagrams and drawings of rulers to measure and record the distance around given objects (as a soft introduction to
perimeter) and between cities. The information is then transferred to number lines.
Module 6 relates to geometry, introducing shapes based on the number of sides and angles: three-sided shapes
(four kinds of triangles), four-sided shapes (square, rhombus, rectangle, parallelogram, and trapezoid), other
polygons (including the hexagon), and a 3-D shape (cube). Activities include tracing and writing names, comparing
and contrasting shapes (e.g., square and rhombus), and drawing shapes. The activities conclude with partitioning
rectangles into same-size squares (as a soft introduction to area) and dividing rectangles and circles into halves,
thirds, and fourths. Instruction includes the fraction word names and number representations.
Vocabulary for Unit 4:
Time
Money
a.m.
Analog clock
Digital clock
Elapsed time
Five minutes
p.m.
15
Cent
Coin
Decimal point
Dime
Dollar
Dollar bill
Nickel
Penny
Quarter
Symbols ¢ and $
Value
Measurement
Data
Centimeter (cm)
Foot (ft.)
Inch (in.)
Inch ruler
Measuring tape
Measuring tool
Meter (m)
Meter stick
Metric measurement
Metric ruler
Mile (mi.)
Standard measurement
Tape measure
Yard (yd.)
Yardstick
© 2012
Bar graph
Category
Data
Interpret
Line plot
Picture graph
Represent
Geometry
Angle
Attribute
Corner
Edge
Equilateral triangle
Face
Fourth
Half (halves)
Hexagon
Irregular triangle
Isosceles triangle
Parallelogram
Partition
Rhombus
Right triangle
Third
Trapezoid
Second Grade Implementation Guide
TouchMath Second Grade Modules
Module Descriptions
Four units. 24 modules. One complete program.
UNIT 1:
Adding, and Subtracting within 50
Module 1: Adding and Subtracting Within 13
Module 1 presents strategies and drawings to help
students internalize sums and differences within 13. The
presentation is scaffolded to work with one-digit numbers
within 9, then 10, and then 13. Drawings for part/whole
relationships are used, incorporating missing numbers
(unknowns) to reinforce the relationships. Tracing,
followed by filling in missing information, is required
as students demonstrate their increased understanding
of the concept. Ten Frames and use of dots for
representation are used to build a firm foundation of the
number 10. Number families are used to ensure mastery
of the relationship of addition and subtraction within 13.
TouchPoints provide kinesthetic and/or visual support
for the learners. If/then statements, puzzles, mazes, and
matching activities are presented to maintain student
interest. One-step word problems and comparisons are
reinforced throughout the module. Problem solving is
addressed through specific identification and practice
of strategies. When students complete this module,
they will have had many experiences that contribute to
developing fluency within 13.
Module 2: Adding and Subtracting Within 20
Module 2 extends the activities used in Module
1, eliminating the models and concrete activities
previously incorporated. The activities are further
scaffolded by adding the same second addend to each
number, and then increasing the second addend by one
and continuing the same pattern. A section on using
strategies has been added to focus on specific properties
and/or functions within operations to enhance learning.
Problem solving and/or comparisons are included
throughout the module. New testing formats, a new
symbol for unknowns, and two-step word problems are
introduced. The Problem Solving section is continued
from the previous module, and the part/whole and
addition/subtraction relationships are still reinforced
throughout. When students finish this module, they will
have had many experiences that build fluency within
20. If students have not demonstrated 80% accuracy,
remediation may be necessary.
Module 3: Adding and Subtracting Within 50
In Module 3, the development of the concepts of addition
and subtraction is extended to include increasingly
larger numbers; however, no regrouping is introduced.
Visual/kinesthetic cues—TouchPoints, place value
© 2012
Second Grade Implementation Guide
indicators, and arrows above the ones column to
remind students of the starting point—are included
to provide support within the developmental level of
the learner. The activities scaffold from two-digit and
one-digit numbers, to two-digit numbers, to multiple
two-digit addends. A fifty chart is introduced as another
strategy for finding sums and differences. Drawings and
equations are modeled and encouraged throughout the
Problem Solving section, and reinforcement of strategies
and properties of operations is continually presented.
Two-step word problems are further developed. When
students have finished this module, they will have
had many experiences of transferring the concepts of
adding and subtracting within 20 to working with larger
numbers. The complexity continues to be gradual and
step-by-step.
Module 4: Adding with Regrouping
Module 4 introduces regrouping since students now
have a firm foundation of working with two-digit
numbers. This concept is presented slowly with the use
of concrete materials and pictorial representations/
drawings to demonstrate the concept. Base 10 blocks
and Ten Frames are used to demonstrated trading in 10
ones for 1 ten, with the instruction presented in place
value charts to reinforce the understanding. These
models are eliminated as students progress through the
activities in this module; however, the visual cues used
in Module 3 continue to provide support. TouchPoints
on the smaller addend and symbols for unknowns
in all positions are incorporated. The learning is
reinforced and applied in the Problem Solving section.
When students finish this module, they will have had
experiences at all developmental levels in building a
foundation for the concept of regrouping in addition.
Module 5: Subtracting with Regrouping
The activities in Module 5 replicate the instruction
provided in Module 4. Students have extensive support
at all developmental levels in building the concept of
regrouping in subtraction. They begin by seeing the
association of the written numerals with the problems,
then progress to providing some of the numerals, and
finally advance to filling in the numerals that correspond
with the models. Students demonstrate increasing
amounts of understanding throughout the activities.
In the Problem Solving section, the four related facts
of addition and subtraction numbers are emphasized.
When students finish this module, they will have had
many experiences at all developmental levels in building
a foundation for the concept of regrouping in subtraction.
16
TouchMath Second Grade Modules CONTINUED
Module Descriptions
Module 6: Adding and Subtracting with Regrouping
Module 6 is a review and practice module to put
together the concepts of adding and subtracting within
50. Strategies are emphasized, including place value
notation, tally marks as an alternative drawing approach,
and properties of operations as application for continued
practice. To continue the integration of strategies, building
10 is used, and doubles and doubles +/– 1 are reinforced.
Additionally, the Problem Solving section provides
ample opportunities to create and use drawings, solve
for unknowns, and apply learning in word problems.
When students finish this module, they will have had
many experiences that contribute to a firm foundation for
increased complexity with two-digit numbers.
are presented. Odd and even numbers, multiples of
five, and numbers out of order are included. Puzzles,
mazes, dot-to-dots, hidden pictures, and mystery
numbers provide a variety of motivating experiences.
Sequencing from any number is also presented,
scaffolded to include numbers after, then numbers
between, and finally numbers before given numbers.
The Problem Solving section continues to be a focus
with comparing numbers, finding unknowns, and
solving word problems. Application of place value form
is again presented to reinforce understanding of the
relationship between counting/reading and place value.
When students finish this module, they will demonstrate
sequencing of numbers within 1,000.
UNIT 2:
Module 3: Reading and Writing to 1,000
Module 3 provides activities that contribute to building
the skill development of reading numbers, which was
initiated in the previous module. Writing numbers up to
1,000, both as numerals and as words, is incorporated,
and place value is reinforced. The structure of the
module originates with filling in missing numbers on
a hundred chart that has the even numbers presented.
Reading number words and sequencing them by fives,
tens, and hundreds is also included. Student interest is
enhanced through dot-to dots with number words, flow
charts, and questions about numbers 5, 10, and 100.
Matching, mazes, multiple choice questions, and in/
out tables are designed to provide practice within each
hundred sequence to ensure mastery of reading and
writing number names and numerals. When students
finish this module, they will have had many experiences
that contribute to a firm grasp of representing numbers
up to 1,000.
Adding and Subtracting within 100
Module 1: Understanding Place Value: Hundreds,
Tens, and Ones
In Module 1, various models and approaches are
presented to extend the foundation of place value
to include three-digit numbers. The strategies begin
with representing numbers using base 10 blocks and
then incorporating the blocks with pictures and place
value charts. An in-depth presentation of the value of
numbers is developed. The numbers are scaffolded
gradually, working within increments of 200 to reduce
learner anxiety. Frequent reviews are included to
provide formative assessment opportunities. The
concept of number value is further developed to include
expanded notation and compact/standard numerals,
and students are encouraged throughout the process
to explain the presentation, using drawings or written
details. Activities using place value charts conclude the
instruction related to the understanding of place value.
Comparisons, unknowns, puzzles, mystery numbers,
and riddles are woven throughout the activities to ensure
regular application of the skills. The Problem Solving
section is provided as a focus with specific approaches
to making comparisons, finding unknowns, and solving
word problems structured for student learning. The
number words for ones, tens, and hundreds are used
so that students read the words as they say them. When
students finish this module, they will have the basis
for understanding a base 10 system and manipulating
numbers within it.
Module 2: Counting and Reading to 1,000
Activities in Module 2 are structured to extend counting
and reading numbers to 1,000. Counting to a four-digit
number is scaffolded through practice within each of
the hundreds. Associating the written numeral with its
verbal name and reinforcing the sequencing of numbers
17
Module 4: Understanding Multiplication
Module 4 introduces the concept of multiplication,
presented as repeated addition, up to five times five.
The operation sign for multiplication is not formally
introduced. (The operation sign is introduced in Unit
3, Module 6.) Students use pictures as TouchPoints on
the numerals, trace the numerals, and ring the quantity
being multiplied in sequential order (e.g., 2 x 2, 3 x 2,
4 x 2, 5 x 2). The activities progress to using arrays of
dots to represent the quantities, and TouchPoints
are removed in the final activity on each page. Word
problems are used for review, directing students to
draw a picture of the problem in order to solve it. The
review also includes ringing groups of dots in arrays to
represent the problem. The next step in scaffolding the
concept is removing the pictures and using TouchPoints
and skip counting to demonstrate repeated addition.
The module concludes with skip counting with no
TouchPoints. The sequence is twos, fives, threes, and
© 2012
Second Grade Implementation Guide
TouchMath Second Grade Modules CONTINUED
Module Descriptions
then fours since most learners will already be proficient
in counting by twos and fives. This order allows students
to move from the known/familiar to the unknown/
unfamiliar. When students finish this module, they will
have had many experiences with repeated addition as
the model for multiplication. NOTE: Some students may
be ready to transfer understanding of repeated addition
to multiplication. An option is included to have students
write multiplication equations. Directions for both
approaches are included in the Instructional Strategies.
Module 5: Adding Within 100
The activities in Module 4 extend learning from adding
within 50 to adding within 100. Students have continued
support at all development levels in building this
concept. Base 10 blocks and Ten Frames are continued,
and visual cues (modified place value charts, arrows,
and boxes above the tens to remind students to regroup)
replace these models in the next level of scaffolding.
One- or two-step word problems and comparisons
are included throughout the section on regrouping.
Students then progress to mixed practice, without
the regrouping box. Students apply the concept of
regrouping through in/out tables, hundred charts, and
tables with the problems and unknowns in all positions.
Frequent reviews are included to ensure understanding,
and the Problem Solving section emphasizes making
comparisons, finding unknowns, and solving word
problems. When students finish this module, they will
have had many experiences at all developmental levels
in applying regrouping in addition within 100.
Module 6: Subtracting Within 100
In Module 6, the activities spiral back to Unit 1, Module
5, which uses base 10 blocks and pictures to show
regrouping in subtraction. The instruction mirrors that
of the previous module in the current unit, Adding
within 100. This transference of learning from addition
to subtraction further develops the relationship between
the two operations. When students finish this module,
they will have had many experiences at all developmental
levels in applying regrouping in subtraction within 100.
UNIT 3:
Adding and Subtracting within 1,000
Module 1: Adding and Subtracting Multiples of 10 and 100
The activities in Module 1 are scaffolded from adding
and subtracting multiples of 10 to/from 100 and
then to/from each multiple of 100 in sequence. The
relationship of addition and subtraction is reinforced by
presenting the addition problem and its complementary
subtraction problem. The skills are extended to adding
and subtracting multiples of 10 within a three-digit
© 2012
Second Grade Implementation Guide
number that is also a multiple of 10 (e.g., 540 + 40 and
580 – 40). Boxes for unknowns are included in the
problems, if/then statements are incorporated to
develop deeper understanding of the process used,
and word problems provide opportunities to apply
the learning. The instruction then progresses to
adding multiples of 10 to all three-digit numbers (e.g.,
649 + 30 and 679 – 30) without regrouping. Students
are challenged to compare and create drawings from
the problems. The same process is used for adding and
subtracting multiples of 100, moving gradually from
adding and subtracting to/from multiples of 100 to
multiples of 10 within the 100s–900s and finally to any
number within the 100s–900s. Computation strategies
(including doubles and number families) are reinforced,
and in/out tables provide practice in a variety of formats.
The activities are structured to support the learner’s
mental mastery of the skill. FlipCards Adding and
Subtracting Multiples of 10 and FlipCards Adding and
Subtracting Multiples of 100 are included with the unit
to provide additional reinforcement through a variety
of activities. When students complete this module,
they will have a basis for understanding adding and
subtracting multiples of 10 and 100.
Module 2: Adding and Subtracting Within 100
In Module 2, the instruction focuses on using strategies
both in operations and in problem solving. All strategies
are presented with activities and visuals that have been
previously presented to ensure that learners begin with
what is familiar as they move into the more unfamiliar.
The activities include practice with and without
regrouping. For the computation strategies, instruction
is presented to begin using the more familiar form
of expanded notation through building 10. Number
families, properties of operations, multiples of 10 (from
the previous module), and multiples of 10 and 10 +/– 1
are implemented. Matching, finding pairs of addends
that make a given sum, identifying addends that do and
do not make a given sum, and flow charts are used to
encourage student engagement. The problem solving
strategies include draw a picture, find a pattern, make a
table, and extra information. Students are encouraged
to use the strategies that provide the most meaning to
them as they solve increasingly more complex word
problems. When students complete this module, they
will have had many experiences adding and subtracting
within 100 to solve one- and two-step word problems.
Module 3: Adding 3-Digit Numbers
Learning in Module 3 is transferred from working within
100 to working within 1,000. Base 10 blocks are used
to provide the concrete experience of demonstrating
regrouping hundreds. Place value models and charts
18
TouchMath Second Grade Modules CONTINUED
Module Descriptions
are used with the base 10 materials and then presented
independently for visual support. Modified place value
charts, indicators above the ones column, and boxes
for the numbers regrouped are used to ensure learner
success. The complexity of the problems is scaffolded,
beginning with regrouping only ones, then tens, then
hundreds, and finally all digits. Students begin by
adding a one-digit number, then a two-digit number,
and finally a three-digit number to a three-digit number.
Multiple addends are incorporated. The visual cues
are eliminated as students gain more experience and
success. Word problems are included throughout the
module, and they are structured around the problem
solving strategies presented in the previous module.
Two new strategies are introduced: guess and check and
write a number sentence. Students are encouraged to
use both strategies of computation and problem solving
as they continue in their learning of adding larger
numbers.
Module 4: Subtracting 3-Digit Numbers
Module 4 activities mirror those for addition in the
previous module. Regrouping is extended to include
zero. Applying understanding of place value is reinforced
throughout the module. The visual cues are removed so
that students internalize the process of subtracting. The
use of unknowns in various positions continues to be
used. Word problems are used extensively, and students
continue to demonstrate the strategies presented in the
previous module. Two new problem solving strategies
are introduced: use logic and estimate. The learning is
presented slowly with visuals to ensure learner success.
When students finish this module, they will have
had many experiences that contribute to mastery of
subtracting three-digit numbers.
Module 5: Adding and Subtracting 3-Digit Numbers
Module 5 encourages the use of the strategies presented
in the two previous modules as students begin mixed
practice. The activities are scaffolded to increase
in the number of digits and the complexity of the
problems. Students begin with no regrouping and move
systematically to regrouping all digits. Unknowns in all
positions and comparisons are continued. Identifying
part/whole relationships, drawing pictures, and
applying the relationship of addition and subtraction
are reinforced. Word problems continue to foster the
application of the strategies previously implemented
in this unit. Two new strategies are presented: work
backward and choose a calculation. As students
move through this module, they will continue to gain
confidence in manipulating numbers up to 1,000.
19
Module 6: Reviewing Multiplication
Module 6 focuses on reviewing multiplication, extending
and expanding the introduction to multiplication
presented in Unit 2, Module 4. The learning is initiated
with identifying odd and even numbers by finding
pairs. Students use pictures and numerals together as
they begin the process before progressing to using the
numerals independently. The activities then move to
finding two equal addends. All of these skills are related
to odd and even numbers. The symbol for multiplication
is presented, and a soft introduction to the commutative
property is included (2 x 5 and 5 x 2). Creating arrays
up to five rows and five columns using pictures and
then dots to represent problems is then scaffolded. The
problems are presented as addition of multiple addends
and as multiplication. To conclude the module, students
apply the learning in word problems, demonstrating
understanding through drawing pictures and explaining
the operation. FlipCards Representing Multiplication
with Objects are included for remediation and
reinforcement. When students finish this module, they
will have had many experiences that relate to building a
firm foundation for multiplication.
UNIT 4:
Time, Money, Measurement, Data,
and Geometry
Module 1: Telling Time
Module 1 begins with an introduction to a 24-hour day
with a.m. and p.m., using both analog and digital clocks.
The learning is scaffolded from tracing the hands on
analog clocks to tracing the hands on analog clocks and
writing the digital time. The analog clocks have fivepoint stars as visual cues for the five-minute increments.
A 60-minute hour is introduced, and its representation is
shown on an analog clock. Telling time is then gradually
introduced, beginning with the hour, then the half
hour, and then five minutes. With five-minute intervals,
students are encouraged to transfer their understanding
of counting by fives. The activities conclude with reading,
writing, and selecting appropriate times for certain
activities, and finding elapsed time is briefly introduced.
When students finish this module, they will have had
many experiences with telling time.
Module 2: Learning about Money
Module 2 is built on identifying coins and the one-dollar
bill, counting the values of multiple coins of the same
type (e.g., nickels), counting the values of mixed coins,
finding the coins to represent given values, and finally
problem solving with money. To ensure learner success,
one coin is presented with multiple activities before
© 2012
Second Grade Implementation Guide
TouchMath Second Grade Modules CONTINUED
Module Descriptions
introducing a new coin. Instruction is encouraged using
actual or plastic coins, demonstrating the fronts and
backs of the coins, writing the coin names, and matching
the coin values. Both cent and dollar values (e.g., 25¢
and $0.25) are represented. Students then count and
record the number of multiple coins of the same type
and write the value. The learning is then scaffolded to
include counting multiple mixed coins. Students learn
by starting with the largest coin, finding its value, and
counting on the remaining coin values. The coins are
then mixed, and students order them by value and
repeat the process learned for counting the total value.
The activities are extended to include the one-dollar
bill. Comparisons and word problems conclude the
instruction. When students finish this module, they will
have had many experiences learning about money.
Module 3: Representing and Interpreting Data
Module 3 covers three skills: collecting, organizing,
representing, and interpreting data. The graphic
organizers used are vertical and horizontal graphs
and line plots. Activities begin with recording and
interpreting data (provided on the page) in picture
graphs before moving on to bar graphs. The learning is
scaffolded from three to four categories with up to 10
data points per category. After experiences with reading
and constructing graphs, collecting and organizing data
are presented. Students measure common classroom
objects and lengths provided on the page (e.g., animal
tails) using both non-standard and standard tools, as
well as drawings of rulers. The data is recorded and
then ordered from least to greatest to be transferred
later to line plots. The activities are organized to support
implementation in a variety of approaches, depending
on the needs of the learners. Each skill can be introduced
independently, or all the skills can be taught at one time.
When students finish this module, they will have had
many experiences generating data from measurements,
representing that data in line plots, picture graphs, and
bar graphs, and solving problems with graphs.
Module 4: Measuring, Comparing, and
Estimating Length
In Module 4, initial instruction is presented using
standard tools: rulers, yardsticks, and tape measures.
Metric tools include the metric ruler and meter stick.
The learning begins with these guidelines: A ruler is
used to measure lengths up to one foot (12 inches), a
yardstick is used for lengths greater than a foot and up to
a yard, and the tape measure is used for lengths greater
than a yard. The concept is expanded to include placing
a tool end-to-end for multiple feet, meters, etc. Students
use their skill of skip counting by threes to determine
yards on the tape measure. Each tool is introduced,
© 2012
Second Grade Implementation Guide
and students are guided through measuring various
lengths and identifying objects in their environments
that would probably be measured using the specific
tool. The activities are scaffolded to identify the tool for
a list of objects, to measure the objects, and to record
the measurements. Students then measure a given
list of objects with two different tools and record the
measurements, building the ideas that the smaller the
unit, the greater the number of them and the smaller
the unit, the more accurate the measurement. After the
multiple experiences described above, students estimate
lengths of objects by selecting from a pair of choices
(e.g., 2 inches or 2 feet). The process is presented first
with standard (customary) units and then with metric
units. No conversion between the systems is presented.
When students finish this module, they will have a firm
foundation for the concepts of measurement.
Module 5: Relating Addition and Subtraction to Length
Module 5 extends the activities from the previous
module. The process of adding and subtracting length is
presented through measuring line lengths with a ruler,
recording the lengths, cutting out the lines, and laying
them end-to-end for addition and with the shorter one
on top of the longer one for subtraction. The learning is
extended from measuring line lengths to measuring how
many given units are needed to frame a given object, a soft
introduction to perimeter. Students identify how much
of a given material is needed for each of two objects and
then find the sum or difference of the material needed.
Unknowns in equations are introduced gradually and
within the context of real-world situations. The learning
is then scaffolded to include distances between cities.
Number lines are introduced to demonstrate relative
distance, giving students a visual reference for addition
and subtraction of distances. The concept of number
lines is identical to that of adding and subtracting line
lengths mentioned above. When students complete this
module, they will have manipulated measurements on
number lines and in equations to understand addition
and subtraction of length.
Module 6: Reasoning with Shapes
Module 6 presents three sets of skills/activities. The
first set introduces shapes according to the number
of sides: triangles (equilateral, right, isosceles, and
irregular), quadrilaterals (square, rhombus, rectangle,
parallelogram, and trapezoid), other polygons
(pentagon, hexagon, heptagon, octagon, nonagon, and
decagon), and one 3-D shape (cube). The learning has
been structured to identify the critical attributes of a
shape, trace its name, select it from other shapes, and
then draw it. The process for drawing the shapes begins
with dots at vertices that can be connected using a ruler
20
TouchMath Second Grade Modules CONTINUED
Module Descriptions
and progresses to free-hand drawing. Comparisons and
contrasts are used to provide deeper understanding
through analysis of the shapes (e.g., Is every square a
rectangle? Is every rectangle a square?). The second
set of skills/activities is partitioning rectangles into
same-size squares, which is a soft introduction to area.
Students begin by tracing dotted lines, then use dots
and light gray lines as guides, use dots as references for
the process, use light gray lines with no dots, and finally
partition rectangles into a given number of squares with
no support. Students are directed to use their learning
of tools (rulers) for the process. The third set of skills/
activities presents partitioning rectangles and circles
21
into fractional parts: halves, thirds, and fourths. The
same process for dividing shapes into equal shares
mentioned above is used to scaffold the learning. Writing
the fractions and their various names (e.g., one-fourth
or a fourth of) are presented through tracing. Matching
activities and partitioning shapes independently
conclude the presentation. When students finish this
module, they will have had many experiences that guide
them into learning to reason with shapes.
 Hexagon is the only other polygon presented in the
activity pages. The additional shapes are included in
TouchMath PLUS.
© 2012
Second Grade Implementation Guide
TouchMath Second Grade Modules CONTINUED
How to Use the Module Guides
A look at the Module Guide.
Each Second Grade Unit has six modules, and each module includes a Module Guide. Each Module Guide
summarizes the content for that module, provides Instructional Strategies paired with the student activity worksheets
(including an answer key), contains Parent/Guardian Communication Letters, and includes a Progress Monitoring
Record. These essential Module Guides are organized for quick reference and easy use in a busy classroom. They
contain the following:
•
•
•
•
•
•
•
•
•
•
An overview that provides the scope of content
Correlation of the Second Grade Common Core State Standards to the content
Objectives that define a focus for the skills presented
Prerequisites that identify experiences needed prior to the lessons
Vocabulary that emphasizes teacher language to be used in concept building
Materials that will be necessary for the lessons
Instructional Strategies matched with student activity sheets
Tests as formative assessment tools
Parent/Guardian Communication letters and suggestions for parents/guardians
A Progress Monitoring Record to track student progress and to plan differentiated instruction
Page-by-page instructions
with answer keys
Suggestions for
instruction
Overview with
vocabulary and
resources
Module closure
instructions
Progress
Monitoring Record
Listing of all activity
sheet topics and
module contents
© 2012
Content review with
posttest instructions
Parent
communication
Second Grade Implementation Guide
22
TouchMath Second Grade Modules CONTINUED
Instructional Strategies
Support right at your fingertips.
Each Module Guide includes Instructional Strategies to be used with the student activity worksheets. These are designed
to provide suggestions for the teacher to help implement the TouchMath Program. The strategies are based on a modified
direct instruction model, featuring effective principles of teaching and learning. Meaningful repetition incorporates a
variety of approaches to ensure multiple experiences for the learners. Suggested directions for pages with apple icons are
included in the Instructional Strategies instead of being printed directly on the corresponding student activity worksheets.
This eliminates interference by words that most students cannot read and makes it easy for the teacher to modify the
directions. However, simple directions are included on the independent practice and take home pages. Comments for
the teacher (not necessarily to be shared with the students) are indicated with brackets in the Instructional Strategies. As
frequently as possible, lesson presentations relate math to the child’s world and are cross-curricular, using a variety of
classroom settings (e.g., large group, small group, pairs, etc.) for practical use in a second grade classroom.
• Anticipatory Sets—activities to engage students in the lesson and show them the focus of the skill being
presented
• Teacher Input/Modeling—information about the skill shared with the students and demonstrated by the
teacher
• Checking for Understanding—active participation techniques used to quickly ensure that students have the
necessary understanding before beginning practice
• Review and Closure—a capsule of the learning and restatement of the skill
• Test Directions—modifiable script for test administration
• Remediate, Reinforce, and Challenge—differentiation activities that provide for individual student needs
• Real World Connections—application of the skills in the learners’ world
• Literature Connections—sample books that are relevant to the skill taught
Special symbols used on the activity sheets:
Guided Practice
Suggested directions for and implementation
of the activity sheets, to be monitored by
the teacher in a whole-class setting or by a
paraprofessional or parent volunteer in a small
group
Independent Practice
Pages for student application of the skill with
little supervision
Anticipatory sets help
introduce students to
lesson concepts.
: Activity Sheets 143–149
ctional Strategies
TouchMath Instru
Objective
To solve addition
problems
and subtraction word
with regrouping
lem Solving
Prob
plates
Jane places three
Think about this:
plate has 7
a word problem.
ns, and the third
ral ways to solve
. How
plate has 17 raisi
es there are seve
ns, the second
on the second plate
ns
raisi
6
raisi
has
the
Say, “Sometim
plate
s, and Lara eats
table. The first
first and third plate
of raisins on the
the raisins on the
this question,
raisins. Janice eats eat altogether?”
t ways to solve
eren
they
do
diff
ns
ral
raisi
seve
y
man
know there are
lesson, you will
finish with this
Say, “When you
same.”
will always be the
but the answer
Anticipatory Set
TouchMath Instru
ctional Strategies
–149
Activity Sheets 143
same, regardless
er remains the
out that the answ
6 + 7 = 13 on the
lem, and point
plates, so write
adding the
o solve the prob
first and third
theule
property to
Mod
ether. Now show
raisins on
iative
s altog
6 raisinity
assoc
the
CONTINUED
d. Janice eats the
Shee
they eat 30Activ
ts 143–144 Prob
Show how to use h the numbers are addeed.
.
13 + 17 = 30. So
add
same
so
,
the
whic
is
lem Solving
plate
in
30,
d
nd
secon
answer,
of the order
the raisins on the
to show that the
7,
eats
+
17
+
Lara
.
6
e
ple
45 cars in it. 6 more
whiteboard
, for exam
Guid
e parking lot has
Th
ed
in another order
solving
it, “Th
of
Prac
bers
ng
solve
tice:
num
standi
them
Activity under
three
em, and have
ts do not demonstrateSheet 143
• Tellstudenclass
n the flowing probl
children
that
ng lot?” [ If some the
RSTANDING: Give
regrouping.] they will compare sums
are in the parki
stude
m withnts
CHECK FOR UNDE
as they use logic
and
. How many cars
two-step word proble
and various strate differences on this page. Guid
en 11 cars leave
expression make
ling using another
16+16
the modeeling
cars drive in. Th
e
gies to help them
s the compariso
31-16
using strategies, repeat
18+17 18+18
decide which
n true.
two-step problems
• Direct children’s atten
19+19
40-24 38-19
45-17
18+19
tion to the first
of 18 + 17. Enco
32-15
prob
lem.
First have stude
urage them to
nts find the sum
use doubles-m
18 + 17 = 35.
40-24
Now explain that
inus-one: If 18
19+19
33-16 34-17
+ 18 = 36, then
they must find
35. By now, many
15+15
21- 4
16+1
an
7
expre
stude
ssion that is less
nts may be on
18+18
40-22
the five two-d
their way to deve
than
igit doubles with
18+15
loping fluenc
sums less than
of which are in
y with
50 that require
24+24
this problem.
regro
So one way to
38-19
that since 16
24+18 23+23
answer this prob uping, three
+ 16 = 32 and
21+21
40-21 26- 8
32 is less than
lem is to know
students to sugg
41-17
22+22
35, choice A is
est other ways
correct. Encourage
45-26
knowing that
to
fi
nd
the corre
choices B–D cann
15+15
ot be correct beca ct answer choice, such as
than the adde
45-16
nds in the first
42-14
use the addends
38- 9
expression. Alter
18+18
choice A is less
are greater
16+17
44- 8
natively, they
than 35 because
38-19
50-21
might sugg
first expression
its addends are
17+17
.
less than the adde est that
nds in the
• Continue in the same
manner for
143
share the vario
us strategies they the rest of the problems. Invit
Suggest that stude
e students to
could
nts ring each comp use to make the correct
comparisons.
for the correct
arison symbol
comparison.
to be sure they
are looking
27+ 8 = 17+1
8
Independ
acher Modeling
Teacher Input/Te
Name
A
A
B
B
C
C
D
D
A
A
B
B
C
C
D
D
A
Take Home
Pages for reinforcement of the skill at home
Special symbols used in the Module Guides:
Stop Signs
Visual reminders not to proceed past a review
page unless students demonstrate learning of the
skill
Did You Know?
Interesting facts related to the pictures
B
C
D
D
Strategies for each
activity sheet are
displayed on the page.
A
© 2012
nit 1
Unit
Second Grade
A
B
B
C
C
D
D
© 2012
SGASW50
Compare Sums
Directions: Fill
ent Practice:
• Tell students that they Activity Sheet 144
6
Module Guide
and Differences
143
Name
in the unknowns
in the equations.
blue
red
blue
31- 15 = 42-2
6
27+ 8 = 17+18
will find unkn
working indep
owns for each
ende
problem on the
class. Again, enco ntly, you may want to work
page. Before
one or more prob
urage children
lems with the
unknowns in
to use various
each equa
strategies to help
blue 12 them find the
• In the first row, say tion.
toget
her,
“27 plus what
children ring
number equa
the equal sign
ls
in
17
red.
find the unkn
+ 18?” Have
Tell the class
own number
that this indic
that makes both
a blue crayon
expressions equa ates they must
to ring 27 in the
l. Have them use
first expression
Elicit that 27
is 10 more than
and 17 in the
other expression
17,
make the two
.
expressions equa so the unknown must be
10 less than 18
l. Ask, “What num
that 8 is 10 less
to
than 18, so 8 is
ber is 10 less than
the unknown.
oval. Then say
18?” Elicit
Instruct stude
together,
nts to write 8 in
• Continue in the same “27 plus 8 equals 17 plus 18.”
the
manner for
to share strate
gies they can use each of the problems, enco
uragi
to help them fi
nd the unknowns ng students
.
red
13 +28 = 24+1
7
43 -17 = 348
blue
29+16 = 27
+18
31- 15 = 42-26
23-19 = 33
-29
44-27 = 3518
© 2012
SGASW50
Find Unknowns
144
144
13 +28 = 24+17
Activity sheet thumbnails
provide answer keys.
Draw, Write, Share
Opportunities for students to apply the learning
and share it with a partner
23
A
B
C
43 -17 = 34- 8
29+16 = 27 +18
13
23-19 = 33 -29
© 2012
© 2012
Second Grade
Unit 1
Module Guide
6
Second Grade Implementation Guide
TouchMath Second Grade Modules CONTINUED
Reproducible Activity Sheets
Build a foundation of key concepts.
Each unit in the TouchMath Second Grade Program contains 150 reproducible student activity sheets, which are
organized by skill into six modules of 25 easy-to-use worksheets each. These pages are aligned with the Common
Core State Standards and are designed to prepare children for the skills that they will need to complete second grade.
All skills are transparent and are organized from the concrete to the abstract—from simple to more complex.
The icon in the upper left corner of each activity sheet identifies the intended use. An apple icon ( ) emphasizes that
the page is to be done with the guidance and support of the teacher. A pencil icon ( ) indicates that some students
may be able to complete the page independently, with simple directions and limited support of the teacher. The house
icon ( ) references a page that can be taken home for student reinforcement of the skills being taught. Answer keys to
all worksheets can be found in the appropriate Module Guide alongside the Instructional Strategies.
The worksheets are designed to incorporate classroom counters as well as the manipulatives provided in each unit.
They feature opportunities to discuss and build mathematical language naturally. The clean, uncluttered pages also
allow students the necessary space to develop tactile skills as they master new concepts. As children see, say, hear, and
touch while they practice math, they learn and remember.
Icons denote individual practice ( ),
teacher-guided ( ), and take home
( ) worksheets.
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© 2012
Second Grade Implementation Guide
24
TouchMath Second Grade Modules CONTINUED
Bibliography
Literature Connections included
in the Second Grade Module Guides.
UNIT 1
UNIT 2
Atwood, Megan. Sparrows Singing. Minneapolis: Magic
Wagon, 2012.
Aker, Suzann. What Comes in 2’s, 3’s, & 4’s? New York: Simon
and Schuster Books for Young Readers, 1990.
Barchers, Suzanne. Harvest Time Subtraction. Mankato, MN:
Capstone Press, 2011.
Appelt, Kathi. The Bat Jamboree. New York: Morrow Junior
Books, 1996.
Bowen, Anne. The Great Math Tattle Battle. Morton Grove, IL:
A. Whitman, 2006.
Cleary, Brian P. The Action of Subtraction. Minneapolis:
Millbrook Press, 2006.
Boynton, Sandra. Hippos Go Berserk! New York: Simon &
Schuster Books for Young Readers, 1996.
Clement, Rod. Counting On Frank. Milwaukee: G. Stevens
Children’s Books, 1991.
Bradbury, Judy. One Carton of Oops! New York: Learning
Triangle Press, 1997.
Driscoll, Laura. The Blast Off Kid! New York: Kane Press, 2003.
Burningham, John. The Shopping Basket. New York: Crowell,
1980.
Friedman, Aileen. The King’s Commissioners. New York:
Scholastic, 1994.
Cato, Sheila. Subtraction. Minneapolis: Carolrhoda Books,
1999.
Goldstone, Bruce. Great Estimations. New York: Henry Holt,
2006.
Cleary, Brian P. The Mission of Addition. Minneapolis:
Millbrook Press, 2005.
Goldstone, Bruce. Greater Estimations. New York, NY: Henry
Holt, 2008.
DeRubertis, Barbara. A Collection for Kate. New York: Kane
Press, 1999.
Kaye, Marilyn. A Day with No Math. Orlando: Harcourt Brace
Jovanovich, 1992.
Duke, Kate. Twenty Is Too Many. New York: Dutton Children’s
Books, 2000.
LaRochelle, David. 1+1=5. New York: Sterling Publishing,
2010.
Giganti, Paul. Each Orange Had 8 Slices. New York:
Greenwillow Books, 1992.
LoPresti, Angeline Sparagna. A Place for Zero. Watertown,
MA: Charlesbridge, 2003.
Kroll, Virginia. Equal
Charlesbridge. 2005.
Shmequal.
Watertown,
MA:
Leedy, Loreen. Mission Addition. New York: Holiday House,
1999.
Leedy, Loreen. Subtraction Action. New York: Holiday House,
2000.
Day—Hooray! New York:
Murphy, Stuart J. Mall Mania. New York: HarperCollins,
2006.
Murphy, Stuart J. Safari Park. New York: HarperCollins, 2002.
May, Eleanor. Mac & Cheese, Pleeeeze! New York: Kane Press,
2008.
Murphy, Stuart J. Too Many Kangaroo Things to Do! New
York: HarperCollins, 1996.
Merriam, Eve. 12 Ways to Get to 11. New York: Simon &
Schuster Books for Young Readers, 1993.
Neuschwander, Cindy. Amanda Bean’s Amazing Dream.
New York: Scholastic, 1998.
Nagda, Ann Whitehead. Panda Math. New York: Henry Holt,
2005.
Nolan, Helen. How Much, How Many, How Far, How Heavy,
How Long, How Tall Is 1000? Toronto: Kids Can Press, 1995.
Neitzel, Shirley. The Bag I’m Taking to Grandma’s. New York:
Greenwillow Books, 1995.
Packard, Edward. Big Numbers. Brookfield, CT: Millbrook
Press, 2000.
Ramsay, Mark. The Good Neighbors’ Cheese Feast. East
Aurora, NY: Strategic Educational Tools, 2009.
Ross, Tony. Centipede’s 100 Shoes. New York: Henry Holt, 2003.
Ramsay, Mark. The Good Neighbors Store an Award. East
Aurora, NY: Strategic Educational Tools, 2009.
Rocklin, Joanne. Just Add Fun! New York: Scholastic, 1999.
Shaskan, Trisha Speed. If You Were a Minus Sign. Minneapolis:
Picture Window Books, 2009.
Shaskan, Trisha Speed. If You Were a Plus Sign. Minneapolis:
Picture Window Books, 2009.
Slade, Suzanne. What’s New at the Zoo? Mount Pleasant, SC:
Sylvan Dell Publishing, 2009.
25
Murphy, Stuart J. Earth
HarperCollins, 2004.
Schwartz, David M. How Much Is a Million? New York:
Lothrop, Lee & Shepard Books, 1985.
Schwartz, David M. If You Hopped Like a Frog. New York:
Scholastic, 1999.
Slade, Suzanne. What’s the Difference? Mount Pleasant, SC:
Sylvan Dell Publishing, 2010.
Stamper, Judith Bauer. Breakfast at Danny’s Diner. New York:
Grosset & Dunlap, 2003.
Tang, Greg. Math Potatoes. New York: Scholastic, 2005.
© 2012
Second Grade Implementation Guide
TouchMath Second Grade Modules CONTINUED
Bibliography
UNIT 3
UNIT 4
Adler, David A. Mystery Math. New York: Holiday House, 2011.
Aber, Linda Williams. Carrie Measures Up! New York: Kane
Press, 2001.
Adler, David A. You Can, Toucan, Math. New York: Holiday
House, 2007.
Adler, David A. 3D, 2D, 1D. New York: Crowell, 1975.
Anno, Masaichiro and Mitumasa Anno. Anno’s Mysterious
Multiplying Jar. New York: Penguin Putnam Books for Young
Readers, 1999.
Axelrod, Amy. Pigs Will Be Pigs. New York: Aladdin/Simon &
Schuster Children’s Publishing, 1997.
Buckless, Andrea. Too Many Cooks! New York: Scholastic,
2000.
Clements, Andrew. A Million Dots. New York: Simon &
Schuster Books for Young Readers, 2006.
Chalmers, Mary. Six Dogs, Twenty-Three Cats, Forty-Five Mice,
and One Hundred Sixteen Spiders. New York: Harper & Row,
1986.
Connelly, Luella. Let’s Measure It! Cypress, CA: Creative
Teaching Press, 1995.
Cobb, Annie. The Long Wait. New York: Kane Press, 2000.
Dobson, Christina. Pizza Counting. Watertown, MA:
Charlesbridge, 2003.
Franco, Betsy. Double Play. Berkeley: Tricycle Press, 2011.
Helakoski, Leslie. The Smushy Bus. Brookfield, CT: Millbrook
Press, 2002.
Losi, Carol A. 512 Ants on Sullivan Street. New York: Scholastic,
1997.
McKissack, Pat. A Million Fish—More or Less. New York: Alfred
A. Knopf, 1996.
Murphy, Stuart J. Coyotes All Around. New York: HarperCollins,
2003.
Bader, Bonnie. Graphs. New York: Grosset & Dunlap, 2003.
Fisher, Leonard Everett. Look Around: A Book About Shapes.
New York: Viking Kestrel, 1987.
Friedman, Aileen. A Cloak for the Dreamer. New York:
Scholastic, 1994.
Friedman, Mel. Kitten Castle. New York: Kane Press, 2001.
Harris, Trudy. The Clock Struck One. Minneapolis: Millbrook
Press, 2009.
Hulme, Joy N. Sea Squares. New York: Hyperion Books for
Children, 1991.
Hutchins, Pat. Clocks and More Clocks. New York: Macmillan,
1970.
Leedy, Loreen. The Great Graph Contest. New York: Holiday
House, 2005.
Leedy, Loreen. Measuring Penny. New York: Henry Holt, 1997.
Murphy, Stuart J. Shark Swimathon. New York: HarperCollins,
2001.
Lionni, Leo. Inch by Inch. New York: Knopf Books for Young
Readers, 2010.
Murphy, Stuart J. Sluggers’ Car Wash. New York: HarperCollins,
2002.
Murphy, Stuart J. Lemonade for Sale. New York: HarperCollins, 1998.
Napoli, Donna Jo. Corkscrew Counts. New York: Henry Holt,
2008.
Myller, Rolf. How Big Is a Foot? New York: Dell, 1991.
Neuschwander, Cindy. Sir Circumference and All the King’s
Tens. Watertown, MA: Charlesbridge, 2009.
Pappas, Theoni. The Adventures of Penrose the Mathematical
Cat. San Carlos, CA: Wide World, 1997.
Pilegard, Virginia. The Warlord’s Beads. Gretna, LA: Pelican,
2001.
Murphy, Stuart J. The Penny Pot. New York: HarperCollins, 1998.
Nagda, Ann Whitehead. Tiger Math. New York: Henry Holt, 2000.
Nathan, Cheryl. The Long and Short of It. Mahwah, NJ:
BridgeWater Books, 1998.
Older, Jules. Telling Time. Watertown, MA: Charlesbridge, 2000.
Pluckrose, Henry. Length. Chicago: Childrens Press, 1995.
Pluckrose, Henry. Time. Chicago: Childrens Press, 1995.
Sayre, April Pulley. One Is a Snail, Ten Is a Crab. Cambridge:
Candlewick Press, 2003.
Rocklin, Joanne, and Marilyn Burns. The Case of the Backyard
Treasure. New York: Scholastic, 1998.
Schwartz, David M. On Beyond a Million. New York: Random
House, 1999.
Rocklin, Joanne, and Marilyn Burns. The Case of the Missing
Birthday Party. New York: Scholastic, 1997.
Tang, Greg. Math Appeal. New York: Scholastic, 2003.
Viorst, Judith. Alexander, Who Used to Be Rich Last Sunday.
New York: Atheneum, 1978.
Tang, Greg. Math for All Seasons. New York: Scholastic, 2002.
Tang, Greg. Math-terpieces. New York: Scholastic, 2003.
Wells, Robert E. Is a Blue Whale the Biggest Thing There Is?
Morton Grove, IL: A. Whitman, 1993.
Tibo, Gilles. Simon and the Snowflakes. Montreal: Tundra
Books, 1988.
Williams, Rozanne Lanczak. The Coin Counting Book.
Watertown, MA: Charlesbridge, 2001.
© 2012
Second Grade Implementation Guide
26
TouchMath Second Grade Modules CONTINUED
Parent/Guardian Communication
Connect school to home.
The TouchMath Second Grade Program contains 28 Parent/Guardian Communication Letters, with seven letters in
each unit. The first module of each unit has two letters: (1) an introductory letter to be sent out at the beginning of the
module to inform parents that TouchMath will be used with students during the school year and (2) a letter to be sent
out at the end of the module informing parents/guardians that their child has completed the necessary requirements
to move on to the skill set in the next module. Modules 2, 3, 4, 5, and 6 each have one letter at the end of the module to
keep parents/guardians informed of their child’s progress. The letters are designed to be printed on school letterhead
and signed by the teacher. Spanish versions are available online at TouchMath PLUS.
All letters define the skill focus, describe activities that will be used in the classroom, identify vocabulary that will be
used in the classroom, provide a statement of proficiency, suggest ways parents can help at home, and offer a Parent/
Guardian Tip with ideas for helping parents/guardians structure supporting activities.
Formatting designed to copy or
print on your school’s letterhead
Dear Parent/G
uardian,
We will be usi
ng the TouchM
the test, and
ed
ster
ian,
ath program thr
ma
,
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m,
signed to provid
ath program
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t 1, Module 1
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with
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many objects wer
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determine how
many remain out
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with and how
were to begin
uency with addit
dent practice the
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thin
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take away som
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at home.
me. Have your
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help your child
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child find
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it! Add it and sub
suggesting way
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le and
ncing Ball
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Tip:
subtraction facts
parts or the who
Parent/Guardian help your child practice addition and
y
either one of the
problem, with
the ball that man
nce
cise and
subtraction
nswer and bou
Get some exer
the answ
an addition or
r player must say
player calls out
lay again!
together. One
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– 5 = ___ or 11 –
ect. Switch role
missing (e.g., 11
the answer is corr
player confirms
times. The first
w w w. t o u c h
math.com
Parent/Guardian
Tip:
Area for your signature
Raid your recycli
ng bin, and pla
y Knock Them
plastic bottles.
Down with you
Set up the can
r child. Use a bal
s like bowling
how many we
pins. Take turns
l and 10 to 13 em
re knocked dow
rolling the ball
pty cans or
n and how ma
addition senten
at the cans/bottl
ny are left. Th
ce.
es and countin
en use the num
g
bers to write a
subtraction or
w w w. t o u c h
math.com
Parent/Guardian Tip for at-home
continued learning
27
© 2012
Second Grade Implementation Guide
TouchMath Second Grade Modules CONTINUED
Assessment/Progress Monitoring
Review. Assess. Record.
Regular assessment of specific skills is essential to ensure student success. At the second grade level, formal
assessments must be combined with ongoing classroom teacher observations and should measure what has been
taught. These assessments should occur upon completion of work on particular skill, and the results should be used
for re-teaching and remediation.
Each module includes a pretest and a posttest, which are to be used for student placement and as an indication of skill
proficiency. They may also be used as a tool to direct future planning by enhancing data-driven decision making and
identifying needs for differentiated instruction.
A Progress Monitoring Record is also included in each Module Guide. The top of the sheet identifies each skill and
the worksheet on which it is taught in the module, and space is provided for teacher notations. Teachers may use any
appropriate record keeping method they wish (e.g., check marks, different colors for completed or needs help, letter
grades, etc.). When used effectively, these monitoring records provide an ongoing snapshot of each student’s progress
and are very helpful for conferences with the parent/guardian.
Topics for each activity sheet are detailed
for easy reference and progress monitoring.
Skills for each module are clearly defined.
TouchMath Progress Monitoring Record
Second Grade Unit 1, Module 1
Adding and Subtracting within 13
Skills
Subtract within 9
Review +/– within 9
Review +/– within 9
Find Sums of 10
Subtract from 10
Add and Subtract within 10
Add and Subtract within 10
Review +/– within 10
Review +/– within 10
Add within 13
Subtract within 13
Add and Subtract within 13
Add and Subtract within 13
Review +/– within 13
Review +/– within 13
Compare Sums
Compare Differences
Find Unknowns in Addition
Find Unknowns in Subtraction
Solve Word Problems
Solve Word Problems
Review Problem Solving
Posttest: Adding and Subtracting within 13
Adding and Subtracting
within 13
Add within 9
Adding and Subtracting
within 10
Pretest: Adding and Subtracting within 13
Activity Sheets
Adding and
Subtracting
within 9
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Student Name
Problem Solving
Comments
The record includes room to record
individual student names, track lesson
completion, and make notes on progress.
© 2012
Second Grade Implementation Guide
28
TouchMath PLUS
Overview
Access your Second Grade Unit from
anywhere ... plus a whole lot more!
Every Second Grade Unit comes complete with a one-year subscription to the online lesson management tool
TouchMath PLUS. Your subscription gives you exclusive access to all of the printed material contained in the unit. The
user-friendly interface allows you to print the student worksheets you need, when you need them—an excellent tool
for lesson planning on the go. By taking advantage of your free subscription to TouchMath PLUS, you can:
• Organize classroom clutter and plan lessons on the go based on individual student need
• Keep track of student progress with testing and recording pages, reteach when necessary, and be prepared for
parent/teacher conferences
• Create folders to plan lessons by time frame (daily, weekly, monthly class lessons), topic, or individual student
• Save time by finding what you need, when you need it
• Reduce stress by letting TouchMath PLUS do the heavy lifting, helping you create and manage only what is
relevant for your students
• Access more than 50 extension pages per unit, organized by module, which can be used for remediation or
reinforcement in the classroom. These pages may also be sent home for application, remediation, and/or
extensions on the skill
To access/activate your subscription:
1. Go directly to www.TouchMathPlus.com or visit www.TouchMath.com and click the TouchMath PLUS icon on
the homepage. If you are not logged in to your online TouchMath account, you will be prompted to do so; if you
do not already have an online account with TouchMath, you will first need to register.
2. When registered/logged in, click on the Register License button. If you have already activated your license
code(s), proceed to step 4. If this is your first time accessing TouchMath PLUS, continue to step 3.
3. Before accessing TouchMath PLUS for the first time, you will be prompted to enter your activation license
code(s). This is the 20-letter sequence of letters seen on the outside of the unit packaging, under the bar code.
NOTE: You will need to activate each unit license number before gaining access to the worksheets contained in
that unit.
4. You can now click the TouchMath PLUS link and begin sorting, customizing, and printing your lessons!
5. Bookmark the page, or return to www.TouchMathPlus.com or www.TouchMath.com, to access your account in
the future.
E
L
P
SAM
Online Digital Activation
Activ tio
ond Grade 2012 Unit 01
Second
The code included on the outside of the Second Grade Unit box
will allow you to access TouchMath PLUS, the online portion of
the Second Grade materials.
29
© 2012
Second Grade Implementation Guide
TouchMath Teaching Aids
FlipCards
FlipCards add meaningful practice
to ensure skill acquisition.
Learning is expedited and recall is enhanced when students have varied, meaningful repetition in the learning process.
FlipCards provide the opportunity to see, say, and hear in order to master concepts. The skills are scaffolded to provide
visuals at all developmental learning levels. As the title suggests, the cards are dual-sided for visual reinforcement of
addition and subtraction, counting and skip counting, multiplication using objects, reasoning with shapes, learning
about money, measurement, and much more. The cards are flipped for the paired relationship.
Each unit in the TouchMath Second Grade Program includes three sets of FlipCards. Each set contains 55+ full-size
cards (or 110+ half-size cards), as well as teaching suggestions. References are incorporated throughout the Instructional
Strategies for remediating, reinforcing, and/or extending the learning objective with the FlipCards. The cards are
designed to be used for demonstration with a large group, for activities at centers for small groups, or for re-teaching
for individuals. They support development of short-term and long-term memory and fluency with concepts. Activities
include sequencing, counting, matching, visualizing, verbalizing, comparing, using horizontal and vertical formats,
recalling, relating, representing, measuring, and more. Their versatile applications bring extended value to your learners.
Unit 1 FlipCards
1. Addition and Subtraction Facts 10–20
2. Addition with Unknowns 1–20
3. Subtraction with Unknowns 1–20
Unit 2 FlipCards
1. Number Cards 1–100
2. Number Cards 10s to 1,000
3. Skip Counting Cards by 5s, 10s, and 100s
© 2012
Second Grade Implementation Guide
Unit 3 FlipCards
1. Adding and Subtracting Multiples of 10
2. Adding and Subtracting Multiples of 100
3. Representing Multiplication with Objects
Unit 4 FlipCards
1. Learning about Money
2. Selecting Tools for Measurement
3. Reasoning with Shapes
30
TouchMath Teaching Aids CONTINUED
Optional Teaching Aids
Additional tools for the classroom.
Also available but not included are classroom posters and other TouchMath manipulatives mentioned in the Module
Guides. These TouchMath products are available individually or in the Second Grade Complete Program.
The TouchMath Second Grade Program also uses an assortment of manipulatives typically found in second grade
classrooms. These include (but are not limited to) the following examples: various counters, attribute objects, counting
discs, connecting cubes, pennies, beans, spinners, etc.
Student Number Cards
Desktop TouchLines
TouchPoint Posters
TM285
TM212
TM215
Skip
Sk
ip C
ip
Counting
ounting P
Poster
osst
ster & C
CD
D Se
Sett
TM1127
TM11
27
TouchMath Math Fans
TM1107
TouchMath TouchNumerals
TM1015
TouchMath Tutor Second Grade
Software
TM1010
Purchase these teaching aids individually or save big when you order them all as part of the
TouchMath Second Grade Differentiated Instruction Set! (TM1260)
31
© 2012
Second Grade Implementation Guide
TouchMath Classroom Guide
Setting up the Classroom
Make it easy on yourself!
Having the TouchMath Program in the classroom is one part of a successful overall strategy to implement quality
math instruction. To ensure user fidelity and augment the effectiveness of the program, suggestions are included for
classroom use.
Using this Classroom Guide will help maximize instruction time and individual learning time for each specific task.
The Classroom Guide is divided into three sections: Classroom Setup, Program Implementation, and Suggested
Program Implementation Tips. A checklist is included for easy reference.
TouchMath Classroom Setup
•
•
•
•
•
•
•
Display all TouchMath classroom aids during math instruction, and use them with the students.
Provide centers featuring FlipCards, counters, and other manipulatives to reinforce learning.
Have books identified in Literature Connections available to read.
Place the Progress Monitoring Record in an easily accessible place.
Store the program box on a shelf for later use.
Schedule a paraprofessional to meet with students needing individual instruction.
Establish small group areas for use with a paraprofessional or parent volunteer to differentiate instruction and
provide individualized guidance.
TouchMath Program Implementation
• Incorporate the objectives listed in the Module Guide into your lesson plans and/or IEPs.
• Include in your lessons the visual, auditory, and tactile/kinesthetic components mentioned in the Instructional
Strategies.
• Model using TouchMath materials, manipulatives, and counters during whole group instruction.
• Use the vocabulary words included for each objective in your instruction with the children.
• Assess students at the end of each module to monitor progress.
• Send home the completed worksheets to keep parents/guardians informed of the program material.
• Send home the Parent/Guardian Communication Letter at the beginning of the unit and upon completion of
each module.
• Use opportunities outside of math instructional time (art, music, p.e., transitions) to reinforce TouchMath
strategies and provide meaningful repetitions.
• Incorporate the skills in the TouchMath Program throughout the day.
TouchMath Program Implementation Tips
• TouchPoints: Provide explicit instruction to master the Touching/Counting Patterns.
• Instructional Strategies: Modify the Teacher Input/Modeling, Checking for Understanding, and directions for
Guided Practice, implementing those that work for you and your learners.
• Student activity worksheets: Use only the worksheets that are needed to advance the learning of individual
students.
• Practice: Use enough pages to provide meaningful repetition of the skill for the developmental level of the
student.
• Extra support: Schedule a parent volunteer or paraprofessional to work with small groups or individuals who
need more experience with the skill.
• Classroom setup: Locate details and examples in the Implementation Guide.
• Amount of time: In general, structure activities within the two to two-and-a-half times the learner’s
developmental age (e.g., If the student is 7 years old developmentally, activities will be most effective if they are
14 to 18 minutes long.). Independent time provides opportunities for additional experiences. Since 150 pages
of activities are included in each unit, using one to four pages per day will provide ample opportunities for
experiential learning throughout the year.
© 2012
Second Grade Implementation Guide
32
TouchMath Classroom Guide CONTINUED
Checklist
Classroom guide checklist.
Use all TouchMath classroom aids during math instruction.
Set up small group areas.
Feature FlipCards and other counters in centers.
Keep the Progress Monitoring Record in an accessible place.
Store the program box.
Schedule paraprofessionals or parent volunteers.
Include objectives from the Module Guide in lesson plans and IEPs.
Incorporate visual, auditory, and kinesthetic components in lesson plans.
Modify Instructional Strategies to meet the needs of the learners.
Identify and use student activity worksheets for meaningful repetition.
Allocate time based on the developmental level of the learner.
Model using TouchMath materials during large group instruction.
Provide explicit instruction with TouchPoints.
Include vocabulary words in instruction.
Read books listed in Literature Connections.
Reinforce TouchMath strategies throughout the day.
Assess students at the end of each module.
Send home Parent/Guardian Communication Letters at the end of each module.
Regularly send home completed student activity worksheets.
33
© 2012
Second Grade Implementation Guide
TouchMath Standards
Common Core State Standards
Built with Common Core Standards in mind.
The Common Core State Standards (CCSS) were developed based
on two decades of research into how students learn mathematics.
The research included comparison of the United States to other
countries, focusing on both student performance and instructional
programs. As a result, two themes have emerged as paramount
in building effective programs: (1) standards must have greater
focus and coherence, and (2) standards must guide the learners in
understanding mathematics.
www.corestandards.org
Standards for Practice and Standards for Content are parts of the Common Core State Standards. The critical areas
for emphasis in second grade are on base 10 notation, addition and subtraction, linear measurement, and reasoning
with geometric shapes. Through clearly defined instruction that guides learners in the acquisition of mathematical
language and concept development, the expected outcome is higher achievement in analysis and problem solving.
The Standards for Mathematical Practice and the Standards for Content are listed below.
Standards for Mathematical Practice
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning.
Second Grade Common Core State Standards
Operations and Algebraic Thinking 2.OA
Represent and solve problems involving addition
and subtraction.
1. Use addition and subtraction within 100 to solve
one- and two-step word problems involving
situations of adding to, taking from, putting
together, taking apart, and comparing, with
unknowns in all positions, e.g., by using drawings
and equations with a symbol for the unknown
number to represent the problem.
Add and subtract within 20.
2. Fluently add and subtract within 20 using mental
strategies. By end of Grade 2, know from memory
all sums of two one-digit numbers.
Work with equal groups of objects to gain
foundations for multiplication.
3. Determine whether a group of objects (up to 20)
has an odd or even number of members, e.g., by
pairing objects or counting them by 2s; write an
equation to express an even number as a sum of
two equal addends.
© 2012
Second Grade Implementation Guide
4. Use addition to find the total number of objects
arranged in rectangular arrays with up to 5 rows
and up to 5 columns; write an equation to express
the total as a sum of equal addends.
Number and Operations in Base Ten 2.NBT
Understand place value.
1. Understand that the three digits of a three-digit
number represent amounts of hundreds, tens,
and ones; e.g., 706 equals 7 hundreds, 0 tens, and
6 ones. Understand the following as special cases:
• 100 can be thought of as a bundle of ten tens—
called a “hundred.”
• The numbers 100, 200, 300, 400, 500, 600, 700,
800, 900 refer to one, two, three, four, five, six,
seven, eight, or nine hundreds (and 0 tens and 0
ones).
2. Count within 1000; skip-count by 5s, 10s, and 100s.
3. Read and write numbers to 1000 using base-ten
numerals, number names, and expanded form.
4. Compare two three-digit numbers based on
meanings of the hundreds, tens, and ones digits,
34
TouchMath Standards CONTINUED
Common Core State Standards
using >, =, and < symbols to record the results of
comparisons.
Use place value understanding and properties of
operations to add and subtract.
5. Fluently add and subtract within 100 using
strategies based on place value, properties of
operations, and/or the relationship between
addition and subtraction.
6. Add up to four two-digit numbers using strategies
based on place value and properties of operations.
7. Add and subtract within 1000, using concrete
models or drawings and strategies based on
place value, properties of operations, and/or the
relationship between addition and subtraction;
relate the strategy to a written method. Understand
that in adding or subtracting three-digit numbers,
one adds or subtracts hundreds and hundreds,
tens and tens, ones and ones; and sometimes it
is necessary to compose or decompose tens or
hundreds.
8. Mentally add 10 or 100 to a given number 100–
900, and mentally subtract 10 or 100 from a given
number 100–900.
9. Explain why addition and subtraction strategies
work, using place value and the properties of
operations.
Measurement & Data 2.MD
Measure and estimate lengths in standard units.
1. Measure the length of an object by selecting and
using appropriate tools such as rulers, yardsticks,
meter sticks, and measuring tapes.
2. Measure the length of an object twice, using length
units of different lengths for the two measurements;
describe how the two measurements relate to the
size of the unit chosen.
3. Estimate lengths using units of inches, feet,
centimeters, and meters.
4. Measure to determine how much longer one
object is than another, expressing the length
difference in terms of a standard length unit.
Relate addition and subtraction to length.
5. Use addition and subtraction within 100 to solve
word problems involving lengths that are given
35
in the same units, e.g., by using drawings (such as
drawings of rulers) and equations with a symbol for
the unknown number to represent the problem.
6. Represent whole numbers as lengths from 0 on a
number line diagram with equally spaced points
corresponding to the numbers 0, 1, 2, ..., and
represent whole-number sums and differences
within 100 on a number line diagram.
Work with time and money.
7. Tell and write time from analog and digital clocks
to the nearest five minutes, using a.m. and p.m.
8. Solve word problems involving dollar bills,
quarters, dimes, nickels, and pennies, using $ and
¢ symbols appropriately. Example: If you have
2 dimes and 3 pennies, how many cents do you
have?
Represent and interpret data.
9. Generate measurement data by measuring
lengths of several objects to the nearest whole unit,
or by making repeated measurements of the same
object. Show the measurements by making a line
plot, where the horizontal scale is marked off in
whole-number units.
10. Draw a picture graph and a bar graph (with
single-unit scale) to represent a data set with up
to four categories. Solve simple put-together, takeapart, and compare problems using information
presented in a bar graph.
Geometry 2.G
Reason with shapes and their attributes.
1. Recognize and draw shapes having specified
attributes, such as a given number of angles or a
given number of equal faces. Identify triangles,
quadrilaterals, pentagons, hexagons, and cubes.
2. Partition a rectangle into rows and columns of
same-size squares and count to find the total
number of them.
3. Partition circles and rectangles into two, three, or
four equal shares, describe the shares using the
words halves, thirds, half of, a third of, etc., and
describe the whole as two halves, three thirds, four
fourths. Recognize that equal shares of identical
wholes need not have the same shape.
© 2012
Second Grade Implementation Guide