Intensive Support - samresources.scholastic.com

| Intensive Support

Background and Overview ........................................................................................... 145

Math Foundations: Understanding Student Needs .............................. 146

Math Foundations Diagnostic Assessment ..................................................149

Math Foundations Diagnostic Assessment: Quantity Concepts ...............................152

Math Foundations Diagnostic Assessment: Counting System ................................. 154

Quantity Concepts Intervention ............................................................................ 160

Lesson 1 Quantities 1–10 .......................................................................................................162

Lesson 2 Quantities 1–20 ..................................................................................................... 164

Lesson 3 Plus 10, Minus 10 at the Decades .....................................................................166

Lesson 4 Plus 5, Minus 5 Within the 5-Scale ................................................................ 168

Lesson 5 Plus and Minus 1 and 2 From Any Number ..................................................170

Lesson 6 Plus 10, Minus 10 From Any Number ...............................................................172

Lesson 7 Plus and Minus 1–10 .............................................................................................. 174

Counting System Intervention ................................................................................. 176

Lesson 1 Counting by 1s .......................................................................................................178

Lesson 2 Counting by 10s .................................................................................................. 180

Lesson 3 Counting by 5s ..................................................................................................... 182

Lesson 4 Counting by 10s From Any Number ............................................................... 184

Lesson 5 Counting by 2s (Even Numbers) ..................................................................... 186

Lesson 6 Counting by 2s (Odd Numbers) ....................................................................... 188

Lesson 7 Counting by 100s ................................................................................................ 190

Lesson 8 Counting by 25s ...................................................................................................192

Technical Support ................................................................................................................ 194

Common Core State Standards © copyright 2010.

Copyright © 2012 by Houghton Mifflin Harcourt Publishing Company

All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying or recording, or by any information storage and retrieval system, without the prior written permission of the copyright owner unless such copying is expressly permitted by federal copyright law.

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INTENSIVE SUPPORT

145Background and Overview

Background and OverviewMost students develop foundational understanding of quantities and counting that is necessary to understand and learn basic facts. Others, for a variety of reasons, lack some or much of this conceptual or skill foundation. These Intensive Support interventions serve as a safety net for those students. Students with partial understanding or skills may give the impression that they know the concepts, but they may not have developed the mental fl exibility, for example, to increase 14 by 10 without counting. Lack of strong basic understanding and agile use of a mental number line are evidence of an insuffi cient mathematical foundation. This section includes diagnostic assessments, research-based strategies, and targeted lessons to help students develop robust and fl exible number sense.

Conceptual and Procedural InterventionThe interventions are targeted instructional strategies developed for students who need additional support in the following areas:

Quantity ConceptsProfi ciency in this area means that students possess a robust understanding of quantities and quantity relationships, for example, the “threeness” of 3, the “fi veness” of 5, and the quantity/size/distance relationship of 3 and 5.

Counting SystemProfi ciency in this area means that students possess nimble counting skills for navigating the base-ten number system, for example, knowing how to count forward by 10s and 5s or count back by 5s from 50 on a mental number line.

Using the InterventionsYou can use the intervention lessons in any of these ways:

• to replace grade-level math curriculum for a period of time forstudents who need intensively focused foundational support

• to supplement the ongoing grade-level curriculum

• to serve as a precursor to the FASTT Math Next Generation software

• to accompany the FASTT Math Next Generation software

Determining Student NeedsStudents whose fact fl uency is less than 50% are good candidates for these interventions. The FASTT Math Placement Assessment will alert you to students whose fact fl uency is less than 50%. Use the following Math Foundations Diagnostic Assessment to determine which intervention—Quantity Concepts or Counting System—is appropriate for each student.

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146 FASTT Math Next Generation

Math Foundations: Understanding Student NeedsFor students to be successful in math, they need a foundation that includes spatial, linguistic, and strategic understanding of numbers and their relationships. Students need a mental picture of how much each number represents. They also need mastery of the language of numbers and counting. A sense of quantity provides the fundamental mental images students need to understand numbers; counting follows, providing language for those quantities and helping students to abstract from physical representations.

Quantity Concepts InterventionUnderstanding Quantities and Quantity RelationshipsThe Quantity Concepts intervention focuses on developing a student’s mental representation of quantity (the “fi veness” of 5) and the relationships between quantities (increases and decreases). In large measure, number is a perceived aspect of objects in space, so our mental representation of quantity and the increasing and decreasing of quantity is grounded in our processing of spatial information.

For some students, this spatial underpinning for math does not fully develop. Such students need to go back to physical representations and rebuild this foundation, harnessing their relatively better-developed language skills. The key to their securing a math foothold is to return to physical number representations (objects) and to fi rmly connect these to verbal representations (words), to written representations (numerals), and to actions on the number line (our system of numbers).

Determining Students Who Need Intervention From late second grade and beyond, students needing this degree of conceptual intervention in quantity and quantity relationships are uncommon. They generally fall into one of three categories:

• students with discernible neurological challenges (for example,traumatic brain injury, Turner’s syndrome, cerebral palsy)

• students with an overall slow-learning profi le

• students who are deemed dyscalculic, meaning they have aprimary math disability presumed to relate to underlying nonverbalor spatial defi cits

It is clear that math ability depends on one’s grasp of the most fundamental concepts: numerical values and their relationships.

—Krasa and Shunkwiler, 2009, page 23

Note: Students who need this level of intervention are not yet ready for the FASTT Math software.

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INTENSIVE SUPPORT

Math Foundations: Understanding Student Needs

How the Intervention WorksIn the Quantity Concepts intervention, teachers model how to use the manipulatives to represent quantities and they verbalize their thought processes. They offer students many opportunities for repetitive practice until the students can readily show and tell on their own. The work proceeds to securing the connections between actions and words, using math manipulatives and written numerals, written algorithms, and the written number system.

Counting System InterventionDeveloping Skills for Navigating the Number SystemWell-developed counting skills are critical to solid foundation for computation. Students need to be facile at navigating a mental number line, progressing forward or backward from any spot, counting by 2s, 5s, and so on. Agility with counting provides a base for understanding arithmetic operations and developing crucial mental math skills.

Determining Students Who Need InterventionMany students in late second grade and beyond who perform slowly or inconsistently in basic arithmetic lack facility in navigating our system of numbers. Counting, inherent in our number system, manifests in repeating patterns that are both audible and visible. Students who do not “catch” these repeating patterns do not yet fully grasp counting and do not develop math agility.

How the Intervention WorksWhere students break down in counting is often where their understanding of the number system leaves off. The Counting System intervention engages students in counting up, counting on, counting back, counting by, counting on by, and counting back by. These are skills that comprise the foundations for mental math.

Students With Dyscalculia Because students with dyscalculia sometimes do not display obvious characteristics, the nature of their math learning diffi culties may be misinterpreted or even overlooked. These students often display other talents and strengths, commonly in verbal areas. Their strengths can divert the unskilled eye from recognizing the seriousness of their math learning needs. It is important to provide in-depth assessment for students who may have a signifi cant mathematics disability but are linguistically bright or gifted.

1 20 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Note: Students whose understanding of quantity is robust but who need work on counting may combine the Counting System lessons with FASTT Math.

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Motivating Students in Need of InterventionStudents who need either of the math foundations interventions, Quantity Concepts or the Counting System, are not performing well in comparison with their peers. As they lose their mathematical footing in late second grade and beyond, not only do they lag behind classmates, they also internalize negative beliefs that are often hard to unlearn.

At the start of any intervention, as you focus on what students really need in the way of basics, it is common to elicit signs of underlying I can’t or I won’t scenarios. Students beginning intervention may be hyperreactive to peer put-downs and teacher irony—or even teacher praise—as these can be viewed as the continuation of previous frustration or embarrassment. Students may raise barriers to learning, such as derailing instruction, denigrating activities as boring or babyish, investing minimal attention, or giving up. Such self-protective moves require skilled handling so as not to further undercut an already eroded sense of self-effi cacy. There are several keys to providing a safe and encouraging arena for interventions.

Progress ChartingReal progress, even in small increments, is a potent motivator. Being able to experience distinct steps forward is key, which is why progress charting is so effective—it offers clear, visual evidence that a student’s efforts are working. Successfully navigating a number line, completing number grids and worksheets, and progressively fi lling in the Fact Grid in the FASTT Math software all provide students with measurable signs of progress that feedself-confi dence and further effort.

Reinforcement and FeedbackRegular descriptive acknowledgment from a teacher provides the most profound sort of support by focusing encouragement. Such descriptive support requires knowing that what the student has done is actually a step forward, is heading in the right direction, or is what you want more of. Be specifi c and timely. You paused and listened to yourself—good strategy! Being explicit helps students recognize what they have learned and use it later. And clearly describing what the learner has just done is benefi cial because it invites the student to congratulate himself or herself—yes, I did do that . . . and you noticed!

Supportive EnvironmentA safe social arena is key to taking the risks of learning. Peer put-downs—blatant or subtle, verbal or just a look—make risk-taking many times more diffi cult. Put-downs are a major hindrance to classroom learning. If allowed to infect small-group intervention, they can halt progress. Eliminating put-downs can be as easy as being clear, fi rm, and alert to every instance. And by teaching students to praise and encourage one another, you can create a supportive learning environment for all students.

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149

INTENSIVE SUPPORT

Math Foundations Diagnostic Assessment


PurposeThe purpose of this assessment is to determine which students might benefi t from the Quantity Concepts and/or Counting System intervention lessons. When students do not master basic number facts in a timely fashion, we tend to serve up more fact practice, neglecting to look underneath at more fundamental understandings and skills. With this assessment, you can probe students’ underlying foundation for the learning of math facts in addition, subtraction, multiplication, and division.

When to Use the Diagnostic InterviewUse this diagnostic interview when you have

• observed a student and have concerns that he or she has poor fact fl uency or hasyet to establish basic number sense or counting strategies.

• received a notifi cation within the Teacher Dashboard or SAM of a student’s lowperformance on the initial Placement Assessment.

How the Assessment Is OrganizedThis diagnostic math interview has two parts corresponding to the interventions in this guide.

Quantity ConceptsUnderstanding quantity and quantity relationships Students needing intervention at this level are rare in comparison with those needing to strengthen their counting foundation. These students require highly concrete, language-intensifi ed instruction, tailored to their particular needs. This section of the assessment looks at students’ sense of quantity and their ability to relate number and quantity.

Counting SystemDeveloping skills for navigating the number system Students requiring this intervention need to strengthen their underlying counting skills and connect those skills to moves on an extended number line. These are the stepping-stones to mental math. Students needing to strengthen their counting and number line skills are more common than most teachers realize. This section of the assessment examines student nimbleness with counting, forward and backward, by different increments and from different starting points.

Note: Assessments and reproducibles are available as resources through SAM. (See Getting Started in the Program Overview section of this guide for more details.)

• Digital timer or stopwatch

• Math FoundationDiagnostic Assessment(pages 152–157)

• 10–15 loose paper clips

• 3 paper clip chains(10 paper clips per chain)

• Number Line 0–50reproducible (page 158)

• Reading Numeralsreproducible (page 159)

MATERIALS

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Administering the AssessmentBefore administering the assessment, familiarize yourself with the assessment format, questions, activities, and recording sheets. Organize the materials so that they will be handy when you need them. Find a place where you can work with a student quietly and without interruption for up to 20˛minutes.

• Start at the Beginning Assessment of each student should begin with the fi rst item ofthe Quantity Concepts Assessment and proceed in order. The student who experiences nodiffi culty with Quantity Concepts should speed along and complete the section in less thantwo minutes, proceeding immediately to the Counting section.

• Administer Individually This math diagnostic interview should be administered by ateacher or qualifi ed aide to one student at a time.

• Administer Orally This assessment is a probing diagnostic interview, aimed atdetermining an individual student’s math needs, not a paper-and-pencil test. Administerthe assessment orally, accepting oral responses from the students. This verbal interactionprovides a signifi cant window into what the student really knows and is able to do.

• Do Not Let It Drag Take care not to let the assessment go on too long. When you observethat a student can only complete a task via tedious counting or laborious strategies, do notprolong that item. Cut a task short when it becomes clear that the student doesn’t know it,doesn’t understand it, or can only accomplish it laboriously.

• Note Breakdown Points It is important to note what the student can and cannot do and toidentify where skills break down (for example, the student can count like lightning to 100,but then leaps to 200). It is also important to note laborious strategies the student uses (forexample, fi ngers-tapping-mouth to count each addend).

• Note Other Performance Characteristics It is important to note accompanying aspectsof the student’s performance—qualitative indicators related to self-regulation, language,and even emotional response. The emotional signals comprise a legitimate focus ofintervention unto themselves. For example, we can teach students to be more purposeful,more willing, and less reactive. As you observe the student’s interactions with the tasks,note anything that might relate to how you target follow-up intervention activities.

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151

INTENSIVE SUPPORT


Some Performance Indicators to Note

Evidence of problems related to self-regulation

• Lack of awareness of errors

• Impulsive errors

• Diffi culty concentrating or staying focused

• Problems returning to focus

• Rigid adherence to ineffi cient procedure

• Diffi culty shifting between items

Evidence of language-based difficulties

• No or limited use of self-talk

• Confusion about teacher’s language use

• Misuse of math terminology

• Confused explanations

• Word and name retrieval diffi culties

Evidence of emotional responsiveness and frustration

• Self-anger or anger at others

• Denigrating the task

• Sidetracking

• Somatizing (perceived pain, illness)

• Sadness/hopelessness

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FASTT Math Next Generation


Quantity Concepts Notes on Performance

Show on your fi ngers:

a. 2 plus 3 . . . Then, How much is that?

b. 4 plus 5 . . . Then, How much is that?

c. 6 plus 4 . . . Then, How much is that?

Profi cient Not Profi cient

Profi ciency means doing the following for items a, b, and c:

• showing total number of fi ngers without countingthem individually

• answering “how much” without counting thefi ngers again

Is 6 closer to 12 or to 10? Profi cient Not Profi cient

Profi ciency means answering “10” with assurance after a brief thinking pause.

Put 9 paper clips on the table.

a. Point to the 9 paper clips.Count these.

b. Show me 7 paper clips.

c. With the paper clips, show me3 plus 4.


Profi ciency means the student

a. counts accurately and easily.

b. does not recount, but shows all but two.

c. splits the 7 paper clips easily into 3 and 4without counting each paper clip in each group.



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Quantity Concepts

Teacher Name:

Student Name:

Student Age: Class/Grade:

Interviewer: Date:

Page 1 of 2

Directions: Say the words in italics to the student.

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Proficient on items 1–6, continue Assessment

If the student has shown profi ciency with items 1–6, it is likely that this student does not need the Quantity Concepts Intervention. If the student only experiences diffi culty with the number line tasks (item 7), then proceed to the Counting System Assessment Items (pages 154–157).

Not proficient on items 1–6, go to Quantity Concepts Intervention

For students who demonstrate diffi culty with the concepts sampled above, provide instruction offered in the Quantity Concepts Intervention. See pages 160–175 of this guide. Since the Quantity Concepts

portion of the assessment only briefl y samples performance, you may want to further probe aspects of these students’ math knowledge (for example, reading and writing numerals and number sentences, math vocabulary, counting skills) to be able to include these in your Quantity Concepts Intervention.

You may also choose to further investigate these students’ skills by continuing to administer the Counting System Assessment. If you do so, take care not to press the student to demonstrate skills not yet developed. Once you discern a breakdown point, either proceed to the next type of item or conclude the assessment.

Quantity Concepts (continued) Notes on Performance (continued)

Prepare 3 strings of 10 paper clips each.

a. Provide 1 string of paper clips.Here is a string of paper clips.How many paper clips are there?

b. Provide 2 more 10-Strings.How many paper clips are there now?

c. Take back one 10-String.How many paper clips are there now?


Profi ciency means the student

a. counts paper clips accurately and easily.

b. answers “30” without counting individualpaper clips.

c. answers “20” without counting individualpaper clips.



Lay out the Number Line 0–50 with only 10, 20, 30, 40, and 50 showing (see reproducible on page 158).

a. Can you show me where we could put thenumber 13 on this number line?

b. Where could we put 15?

c. Where could we put 25?

d. Where could we put 36?


Profi ciency means doing the following for items a, b, c, and d:

performing accurately and easily without counting up from one

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Counting System Notes on Performance

NUMERALS

Show student one line of numerals at a time on the Reading Numerals reproducible (see page 159). For each line, say: Read these numerals:

a. 5, 8, 12, 17

b. 29, 68, 85

c. 103, 152, 210, 574

d. 1,021; 1,604


Note errors, confusion, or breakdowns in performance.

COUNTING

Count for me starting at 1. Have student start with 1 and stop at 115, if accurate. Stop earlier if inaccurate.



For all items 3–5, have student count for only a short stretch of numbers.

Count, starting with 7 (to about 15)





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Teacher Name:

Student Name:

Student Age: Class/Grade:

Interviewer: Date:

Counting System

Page 1 of 4

Directions: Say the words in italics to the student.

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Counting System (continued) Notes on Performance (continued)

COUNTING




Count, starting with 996 (to about 1,005)



Count by 2s. Have student start with 2 and stop at 40, if accurate. Stop earlier if inaccurate.



Count by 10s. Have student start with 10 and stop at 110, if accurate. Stop earlier if inaccurate.



Count by 5s Have student start with 5 and stop at 75, if accurate. Stop earlier if inaccurate.



Count by 25sHave student start with 25 and stop at 325, if accurate. Stop earlier if inaccurate.



Counting backward from 15 (go all the way to 1)

Counting backward from 31 (to about 18)

Counting backward from 63 (to about 48)



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10

11

12

13

14

15

16

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MENTAL MATH

What is:

6 – 1

12 – 1

70 – 1

150 – 1

Accurate and rapid

Accurate with counting

Inaccurate/confused

N ote specifi c errors of –1 performance

What is:

6 + 2

2 + 14

26 – 2

35 + 2

Accurate and rapid


Inaccurate/confused

Note specifi c errors of +/–2 performance

What is:

10 + 7

25 + 10

33 – 10

10 + 41

Accurate and rapid


Inaccurate/confused


What is:

5 + 20

45 + 5

35 – 5

85 + 5

Accurate and rapid


Inaccurate/confused

Note specifi c errors of +/– 5 performance

What is:

25 + 25

125 – 25

Accurate and rapid


Inaccurate/confused

17

21

25

29

18

22

26

30

19

23

27

31

20

24

28

32

33

34

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MENTAL MATH

25 + 50

75 + 25


What is:

11 + 11

20 + 20

12 + 12

Accurate and rapid


Inaccurate/confused

Note specifi c errors of + 11, + 20, + 12 performance

37

35

38

36

39

For students working on addition and subtraction

Proficient (few difficulties) on items 1–39, continue with FASTT Math Instructional Software

If the students are profi cient overall with counting (items 1–16) and mental math (items 17–39), they may proceed with the FASTT Math software.

Proficient (few difficulties) on items 1–16 but labored on items 17–39, go to the Counting System Intervention

If students are slow and labored with addition and subtraction facts, then they may benefi t from the Counting System Intervention. See pages 176–193.

Not Proficient (many difficulties) with items 1–39, go to the Counting System Intervention

If students stumble particularly with items 1–16, they likely need the entire instructional sequence offered in the Counting System Intervention. Even students with select diffi culties among the counting system items can strengthen their skills by proceeding through all the Counting System Intervention lessons. See pages 176–193 of this guide.

For students ready for multiplication and division

Proficient (few difficulties) on items 1–39, continue with FASTT Math Instructional Software

Proficient (few difficulties) on items 1–16 but labored on items 17–39, go to the Counting System Intervention

If students are still slow and labored with additionand subtraction facts, then they may benefi t fromthe Counting System Intervention lessons. See pages 176–193. You may also consider assigning these students to the addition component of the FASTT Math Instructional Software and having them progress through subtraction before beginning multiplication.

Not Proficient (many difficulties) with items 1–39, go to the Counting System Intervention

If students stumble particularly with items 1–16, they likely need the entire instructional sequence offered in the Counting System Intervention. Even students with select diffi culties among the counting system items can strengthen their skills by proceeding through all the Counting System Intervention lessons. See pages 176–193 of this guide. You should consider assigning these students to the addition component of the FASTT Math software and having them progress through subtraction before beginning multiplication.

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0–50 Number LineFor use with Quantity Concepts Assessment item 7.

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Reading NumeralsFor use with Counting System Assessment item 1. Fold or cover parts of the sheet to display each line for students to read.

a. 5 8 12 17

b. 29 68 85

c. 103 152 210 574

d. 1,021 1,604


Quantity Concepts Intervention

Understanding Quantity and Quantity RelationshipsQuantity concept development can be slow going for students with weak foundational understanding. It requires an attentive focus on seeing and feeling the “fourness” of 4 and the “eightness” of 8, for example, and on calibrating the quantities and the distance between them. Students must learn to verbalize what they are doing as they compare quantities, order them from lesser to greater, and match them to a number line repeatedly over time. By modeling actions and related words and then having students repeat, teachers direct and focus learners’ attention. Combining actions, concrete materials, and clear verbal descriptions develops more robust mental representations of quantity and the relationships among quantities.

Using ManipulativesIt is important to use one type of structured concrete manipulative (such as linking cubes or colored number rods) in order to establish all the basic concepts. Color is a powerful conceptual support when used strategically, so be sure to maintain consistency of color usage when using linking cubes. For example, all 2-Stacks should be the same color (say, green) and all 3-Stacks should be another color (say, maroon). It is also critical to keep the linking cubes attached to establish a stable mental representation. The manipulatives should map to the number line, with one unit of manipulative fi tting into a unit space on the number line.

Once students readily demonstrate concepts with one type of manipulative across the range of actions (combining, separating, comparing, ordering, grouping, regrouping), have them use other materials, for example, base-ten blocks or paper clips.

Using the Quantity Concepts InterventionFrequency of InterventionPlan for brief frequent sessions, ideally fi ve times a week for intensive 15-minute sessions or three times a week for 20- to 30-minute sessions. The approximate time expected for each part of the session is shown for a 20-minute session. This particular intervention is likely to be carried out one-to-one or in a very small group, so pacing, repetition, and emphasis should be adjusted to the˛individual learner and the time available.

“Students use numbers … to represent quantities and to solve quantitative problems, such as counting objects in a set … comparing sets or numerals … [and] quickly recognizing the cardinalities of small sets of objects.”

— Common Core State Standards, 2010, page 9

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161

INTENSIVE SUPPORT

Quantity Concepts Intervention

Adapting the LessonsIt is diffi cult to predict which methods of instruction will spark understanding for the students who require this intervention. In the fi rst lesson, look for what “clicks”—the mix of language, verbalizations, and manipulatives—to help each student develop a sense of quantity. Then adapt what you are teaching, how you model the concept, and the amount of time you spend to meet each student’s needs.

Progression of ConceptsThe lessons should be taught in sequence, as each concept builds upon the previous one. Practice each concept until students show a secure, consistent understanding, which may take more than one session.

Teacher PreparationMuch of this intervention involves using a consistent set of manipulatives that match a growing number line. It is best to prepare these materials ahead of time. For your convenience, we have included templates for making your own number lines, manipulatives, and number cards. Templates are available through SAM.

Create• Sets of color-consistent cube stacks

1–10 (ten of each stack)

To create color-consistent stacks,snap and tape together linking cubes,or print the PDF Stack templates oncolored-paper stock and cut out.

Note: The colors are not important, butchoosing a particular color to correspondto a particular number is crucial. Here isa sample color scheme:

1–light blue2–green 3–maroon4–dark blue 5–red

6–orange 7–yellow8–brown 9–white10–black

• Sets of cube stacks 11–20 (made bycombining black 10-Stacks with colorconsistent cube stacks 1–10)

Print and Assemble• Number Line 0–20

(Print PDF Number Line 0–60 andassemble to create your own.)

• Number Line 0–100(Print PDF Number Line 0–60 andNumber Line 61–120 and assemble tocreate your own.)

• Sets of Number Cards 1–100(Print and cut up PDF Number Cards1–100, or create with index cards.)

Note: The Number Line PDFs aredesigned to match basic linking cubes.

Accessing the PDF filesThe PDF fi les for the Quantity Concepts Intervention are accessible through SAM. (See Getting Started in the Program Overview section for more details.)

MATERIALS (per student)

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LESSON 1 Quantities 1—10


OBJECTIVES

• Develop a mental representation ofand understand how to compareand order quantities 1–10.

• Map quantities 1–10 to thenumber line and connect physicalquantities with written numerals.

• Connect physical quantities,the number line, and thecorresponding 11, 21 writtennumber sentences.

SUMMARY

Students work with cube stacks to gain a multisensory feel for quantities 1–10. They compare and order the stacks, align them with the number line, and add or subtract 1.

MATERIALS

For each student:

• 1 set of cube stacks 1–10• Number Line 0–20• 1 set Number Cards 1–10

MATHEMATICAL LANGUAGE

• compare• shorter, longer• order• number line• one more, one less, the same as• plus one, minus one, equals• number sentence

Model & ExploreCompare Quantities (3 minutes)

This activity develops a multisensory feel for quantities. Students feel, see, and verbalize the relationships between two quantities.

• Select a red 5-Stack, show it, count it, and say the number. Modelvisualizing the quantity. I am going to hold this red 5-Stack and makean image of it in my mind. Repeat with a green 2-Stack.

• Model comparing the stacks. Place the 5-Stack below the 2-Stack.Feel how much longer one is than another and model describing thedifferences in length: The red 5-Stack is 3 cubes longer than thegreen 2-Stack. The green 2-Stack is 3 cubes shorter than the red5-Stack.

• Have students practice by selecting two stacks, showing each stack,and naming the quantities. Prompt students with: Show me. Tell me.

• To develop stable mental representations of quantity, tell students: Holdeach stack. Feel how much longer one is than the other. Make animage in your mind. Then have students place the stacks horizontallyand compare them, using these frames to describe the relationship:

The -Stack is cubes longer than the -Stack.

The -Stack is cubes shorter than the -Stack.

Order Quantity and Number (3 minutes)

• Model ordering the stacks vertically from least to greatest, like anascending staircase. Begin by placing the 1-Stack next to the 2-Stack.

• Model talking about the differences by pointing and using the phrase“one more.” The green stack is one more than the light blue stack.

• Place the maroon 3-Stack. Why is the maroon 3-Stack next? (Themaroon 3-Stack is one more than the green 2-Stack).

• Model ordering from greatest to least, using the phrase “one less.”

• Ask students to show and describe one-more and one-lessrelationships as they order the remaining stacks. Prompt students with:First show me. Then tell me. ©

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Lesson 1: Quantities 1—10

Map to the Number Line (5 minutes)

• Display the Number Line 0–20. Model visualizing the number line. I see the number linestarts at 0, ends at 20, and goes up by ones. I will make a number line image in my mind.

• Model mapping the 1-Stack and 2-Stack to the number line beginning at 0. Point to eachspace (not tick mark) on the number line and verbalize what you are mapping. When I putthe red 5-Stack on the number line, it starts at 0 and ends at 5. It takes up fi ve spaces.

• Invite students to place the remaining cube stacks on the number line in order and verbalizethe quantities, ending numbers, and number of spaces on the number line.

• Next, place the cube stacks on the number line beginning at different starting numbersWhen I place the red 5-Stack at 10, it ends at 15. The red 5-Stack still takes up 5 spaces.

1 20 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Connect to Numerals and Number Sentences (5 minutes)

• Match a Number Card 1–10 to the corresponding stack. Here is a 6 card, and here is anorange 6-Stack. Show how the stack maps to the number line.

• Connect addition to the corresponding number expression. I want to add one more. This isan orange 6-Stack plus a light-blue 1-Stack. I write it with numbers: 6 1 1.

• Connect the physical quantities to the corresponding written notation. I can show the6+1-Stack on the number line. It is the same as a yellow 7-Stack. I write: 6 1 1 5 7.

• Write other 11 and 21 number sentences on the board. Read the number sentences aloud.Prompt students to: Show me with cubes. Now show it to me on the number line.

1 20 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Practice (3 minutes)

Have students work in pairs, alternating roles, to complete this activity.

Invite students to place Number Cards 2–9 facedown in a pile. Student A selects a number card and uses the number to write a +1 number sentence. Student B checks the number sentence by building the corresponding +1 stack and fi nding the total cube stack.

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LESSON 2 Quantities 1—20


OBJECTIVES

• Develop a mental representationof and compare and orderquantities 1–20.

• Map quantities 1–20 to thenumber line and connect physicalquantities with written numerals.

• Connect physical quantities,the number line, and thecorresponding 11, 21 writtennumber sentences.

SUMMARY

Students work with cube stacks to gain a multisensory feel for quantities 1–20. They compare and order the stacks, align them with the number line, and add or subtract 1.

MATERIALS

For each student:

• 1 set of cube stacks 1–10• 1 set of cube stacks 11–20• Number Line 0–20• 1 set of Number Cards 1–20


• compare• shorter, longer• order• number line• one more, one less, the same as• plus one, minus one, equals• number sentence


• Select a black 10-Stack, show it, and say the number. To help studentsdevelop a solid mental representation, model visualizing the quantity. Iam going to hold this black 10-Stack and make a picture of it in mymind. Repeat with a 12-Stack.

• Model comparing the stacks. Place the black 10-Stack above the12-Stack and feel how much longer one is than the other. Describe thedifferences in length: The 12-Stack is 2 cubes longer than the black10-Stack. The black 10-Stack is 2 cubes shorter than the 12-Stack.

• Have students select two stacks between 11 and 20, show them, andname the quantities. Prompt students with: Show me. Now tell me.

• To develop secure foundational concepts of quantity, tell students: Holdeach stack. Feel how much longer one is than the other. Make apicture in your mind. Then have students compare the horizontal cubestacks using these sentence frames to respond:




• Guide students to order the cube stacks from 1 to 20.

• Model talking about the differences by pointing and using the phrases“one more” and “one less.” The 11-Stack is next to the 10-Stackbecause the 11-Stack has one more cube. The 10-Stack is next tothe 11-Stack because the 10-Stack has one less cube.

• Guide students to answer questions like these. For each, promptstudents with: Show me. Now tell me.

Why is the 13-St ack next to the 12-Stack?(The 13-Stack is one more than the 12-Stack.)

Why is the 13-Stack next to the 14-Stack?(The 13-Stack is one less than the14-Stack.)

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• Show students the Number Line 0–20. Model mapping stacks, in increasing order, to thenumber line. Begin with the 11-Stack. When I place the 11-Stack on the number line, Isee that it starts at 0 and ends at 11. It takes up 11 spaces. Point to each space on thenumber line as you state the quantity. Repeat, mapping the 12-Stack to the number line.

• Have students place increasing stacks on the number line, stating the quantity andthe ending number. Develop verbal connections: Show me on the number line. Tellme in words.

• Use different starting numbers, continuing to verbalize the quantities and ending numbers.When I place the 11-Stack at 3, it ends at 14. The 11-Stack still takes up 11 spaces.

• Ask questions to help to solidify students’ concepts of quantity.

If I place the 11-Stack at 3, where does it end? (14)

How many spaces does it take up? (11)

1 20 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


• Match a Number Card to the corresponding stack. Here is an 11 card, and here is an11-Stack. Show how the stack maps to the number line.

• Connect addition and subtraction to the corresponding expression. I want to show oneless. I can show it and tell it in words: This is an 11-Stack minus a light-blue 1-Stack. Ican write it with numbers: 11 2 1.

• Connect the physical quantities to the corresponding written notation. I can show the11-minus-1-Stack on the number line. When I take away the light-blue 1-Stack, I havea black 10-Stack. So, I write: 11 2 1 5 10.

• Write other +1 and –1 number sentences on the board. Invite students to demonstrate andverbalize the written problem using cubes on the number line.


Display a 11 cube stack between 10 and 20. Prompt students to write the corresponding 11˛and 21 number sentence. Continue with at least three other +1 cube stacks.

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LESSON 3 Plus 10, Minus 10 at the Decades


OBJECTIVES

• Develop a mental representationof “tenness” and understand howto compare and order multiples of10 up to 100.

• Connect physical quantities ofmultiples of 10 to the decadeson the number line and to thecorresponding 110, 210 writtennumber sentences.

SUMMARY

Students combine 10-Stacks to gain a multisensory feel for multiples of 10 through 100. They compare and order the stacks, align them with the number line, and add or subtract 10.

MATERIALS

For each student:

• 10 black 10-cube stacks• Number Line 0–100• Number Cards (10, 20, 30, 40, 50,

60, 70, 80, 90, 100)


• compare• shorter, longer• order• number line• ten more, ten less, the same as• plus ten, minus ten, equals• number sentence


• Select a 10-Stack, show it, and say the number. Model visualizing the10-Stack to help students develop a solid mental representation of thequantity. I am going to hold and feel this 10-Stack. Now I close myeyes and make an image of it in my mind.

• Pick up two more 10-Stacks. Explain and demonstrate how to combinethem to make a 20-Stack.

• Place the 10-Stack above the 20-Stack to compare the stacks.Describe the relationship: The 20-Stack is 10 cubes longer than the10-Stack. The 10-Stack is 10 cubes shorter than the 20-Stack.

• Have students combine 10-Stacks to make quantities 20 and 30. Tellstudents: Hold both stacks. Feel the difference between stacks. Makean image in your mind. Then have students compare the cube stacksusing these sentence frames:



• Repeat with other multiples of 10 up to 50.


• Have students order the cube stacks from 10 to 40 in ascending orderto develop an understanding of “10 more” and “10 less.”

• Model describing the differences between stacks using the phrases“10 more” and “10 less.” The 30-Stack is next to the 20-Stack—30 is10 more than 20, and 20 is 10 less than 30.

• Guide students to answer comparison questions, fi rst promptingstudents with: Show me. Now tell me.

Why is the 20-Stack next to the 10-Stack? (20 is 10 more than 10;10 is 10 less than 20.)

Why is the 20-Stack next to the 30-Stack? (30 is 10 more than 20;20 is 10 less than 30.)

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167Lesson 3: Plus 10, Minus 10 at the Decades


• To develop a feel for “tenness,” map consecutive 10-Stacks (10, 20, 30, . . . , 100) to theNumber Line 0–100. Discuss the quantities as you map the stacks to the number line:I place the 10-Stack at 0 on the number line. It ends at 10. It takes up 10 spaces. WhenI place the 20-Stack on the number line, it starts at 0 and ends at 20. It takes up 20spaces.

• Compare the quantities using the phrase “10 more.” The 20-Stack takes up 10 morespaces on the number line than the 10-Stack. So, I can say that 20 is 10 more than 10.

• Invite students to place additional 10-Stacks in consecutive order on the number line andverbalize the quantities and comparisons. Prompt students with: Show me on the numberline. Now tell it to me in words.

• Use different starting numbers, continuing to demonstrate and verbalize the quantitiesand relationships. I place the 10-Stack at 20 and it ends at 30. The 10-Stack takes up10 spaces. So, 30 is 10 more than 20.

16 1715 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

• Model mapping consecutive 10-Stacks to the number line from greatest to least (100, 90,. . . , 10). Describe the relationships using the phrase “10 less.”


• Select a Number Card, name it, and use 10-Stacks to show the quantity. Place the stackon the number line.

• Connect addition to the corresponding number expression. I want to add 10 more. Ican show it and tell it in words: This is a 20-Stack plus a 10-Stack. I can write it withnumbers: 20 1 10.

• Connect the physical quantities to the corresponding written notation. I can show the20+10-Stack on the number line. It is the same as a three 10-Stacks, a 30-Stack. So Iwrite: 20 1 10 5 30. I can say it in words: 20 plus 10 is equal to 30.

• Repeat the process, modeling how to show, say, and write –10 number sentences.


Show 110 or 210 stacks and have students write 110 or 210 number sentences. Use prompts: Show me. Now tell me. Now write it with numbers.

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LESSON 4 Plus 5, Minus 5 Within the 5-Scale


OBJECTIVES

• Develop a mental representationof “fi veness” and understand howto compare and order multiples of5 to 100.

• Connect physical quantities ofmultiples of 5 to multiples of 5on the number line and to thecorresponding 15, 25 writtennumber sentences.

SUMMARY

Students combine 5-Stacks to gain a multisensory feel for multiples of 5 through 100. They compare and order the stacks, align them with the number line, and add or subtract 5.

MATERIALS

For each student:

• 20 red 5-cube stacks• Number Line 0–100• Number Cards (multiples of 5)


• compare• shorter, longer• order• number line• fi ve more, fi ve less, the same as• plus fi ve, minus fi ve, equals• number sentence


• Show a 5-Stack, say the number, and model visualizing the quantity.I am going to hold and feel this red 5-Stack. I close my eyes andmake an image of it in my mind.

• Explain and demonstrate how to combine two more 5-Stacks to make a10-Stack. Model visualizing the stack.

• Place the 5-Stack above the 10-Stack to compare the stacks. Feel thedifferences in length and model how to describe the relationship:The 10-Stack is 5 cubes longer than the 5-Stack. The 5-Stack is 5cubes shorter than the 10-Stack.

• Have students combine 5-Stacks to make quantities 10 and 15. Tellstudents: Hold each stack. Feel the difference in lengths. Make animage in your mind. Then have students compare the cube stacksusing these sentence frames:

The -Stack is 5 cubes longer thanthe -Stack.

The -Stack is 5 cubes shorter thanthe -Stack

• Repeat with other multiples of 5 to 50.


• Guide students to order the cube stacks from 5 to 30 in ascendingorder to develop a feel for “5 more” and “5 less.”

• Model describing the differences between stacks using the phrases“5 more” and “5 less.” The 30-Stack is next to the 25-Stack becausethe 30-Stack has 5 more cubes than the 25-Stack—30 is 5 morethan 25, and 25 is 5 less than 30.

• Guide students to answer comparison questions, fi rst promptingstudents with: Show me. Now tell me.

How much longer is the 15-Stack than the 10-Stack? (5 cubeslonger)

How much more is 15 than 10? (5 more)

How much less is 10 than 15? (5 less)

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169Lesson 4: Plus 5, Minus 5 Within the 5-Scale


• Build an understanding of “fi veness” by mapping consecutive 5-Stacks (5, 10, . . . , 100) tothe Number Line 0–100. Begin with a 5-Stack. I place the 5-Stack at 0 on the number line.It ends at 5. It takes up 5 spaces. Repeat with a 10-Stack.

• Model comparing the quantities and verbalizing the increases using the phrase “5 more.”The 10-Stack takes up 5 more spaces than the 5-Stack. So, I can say 10 is 5 more than 5.

• Invite students to place consecutive 5-Stacks on the number line and verbalize the quantityand increases. Prompt students with: Show me on the number line. Now tell it to me inwords.

• Use different starting numbers, continuing to demonstrate and verbalize the quantities andrelationships. I can move the 5-Stack to start at 15; it ends at 20. The 5-Stack still takesup 5 spaces. So, I can see that 20 is 5 more than 15.

11 1210 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

• Repeat, mapping consecutive “–5” stacks (100, 95, . . . , 5) to the number line anddiscussing the decreases using the phrase “5 less.”


• Show a Number Card, name it, and use 5-Stacks to build the quantity. Then place the stackon the number line.

• Model connecting addition to the corresponding number expression. I want to add 5 more.I can show it and tell it in words: This is a 15-Stack plus a 5-Stack. I can write it withnumbers: 15 1 5.

• Connect the physical quantities to the corresponding written notation. The 15+5-Stack isthe same as a 20-Stack. So I write: 15 1 5 5 20.

• Repeat, modeling how to show, say, and write –5 number sentences.


Have students work in pairs, alternating roles, to complete this activity.

• Students place their cube stacks and Number Cards (multiples of 5) facedown in a pile.

• Student A selects a number card and writes a 15 number sentence.

• Student B checks the number sentence by building the corresponding 15 stack and fi ndingthe total cube stack.

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LESSON 5 Plus and Minus 1 and 2 From Any Number


OBJECTIVES

• Develop a mental representationof and understand how tocompare and order increases anddecreases of 1 and 2.

• Connect physical quantities,the number line, and thecorresponding 11, 12, and 21,22 written number sentences.

SUMMARY

Students combine 1-Stacks or 2-Stacks with stacks through 100to gain a multisensory feel forincreases and decreases of 1 and 2.They compare and order the stacks,align them with the number line,and add or subtract 1 and 2.

MATERIALS

For each student:

• 2 sets of cube stacks 1–10• 1 set of cube stacks 11–20• Number Line 0–100• Number Cards 1–100


• compare• order• more, less, the same as• number line• plus, minus, equals• number sentence


Begin this activity using the 1–10 Stacks and progress to using larger, combination stacks (11 and greater).

• Demonstrate combining 1-Stacks with other stacks to representquantities that are “one more” or “one less.” I am holding a red5-Stack. I can show “one more” by adding a light blue 1-Stack.

• Place the combined stack below a 6-Stack. Model comparing quantitiesusing the phrases “one more” and “one less.” This stack is the samelength as an orange 6-Stack. So, the orange 6-Stack is one morethan the red 5-Stack. The red 5-Stack is one less than the orange6-Stack.

• Have students build and compare “one more” stacks. Invite studentsto show and tell about their stacks using the phrases “one more” and“one less.”

• Repeat, combining 2-Stacks with other stacks to build and comparestacks that are “two more” and “two less.”


• Show students the green 2-Stack. Guide students to use 1-Stacks tobuild a stack that shows “one more” and a stack that shows “one less.”

• Have students place the three stacks in ascending order. Discuss thedifferences by pointing to each stack and modeling the phrases “onemore” and “one less.” The 3-Stack is next to the 2-Stack because3 is one more than 2. The 1-Stack is next to the 2-Stack because1 is one less than 2.

• Repeat by showing progressively greater stacks from 4 to 20 andhaving students use the 1-Stack to build stacks that show “one more”and “one less.” Encourage students to verbalize the differencesbetween stacks. Prompt them with: Show me. Now tell me.

• Repeat, having students use the green 2-Stack to build, order, anddescribe stacks that are “two more” and “two less.”

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171Lesson 5: Plus and Minus 1 and 2 From Any Number


• Map a 5-Stack to the Number Line 0–100. Tell the starting and ending points and quantity.

• Combine a 1-Stack with other stacks to model quantities that are “one more.” I can show 1more. That is a red 5-Stack plus a light blue 1-Stack. I’ll call this a 5+1 stack. Place the511 stack on the number line and match it to the stack of the same length. My 5+1 stack isthe same as a 6-Stack. They both end at 6 on the number line. So, 6 is 1 more than 5.

• Then model “2 more” on the number line by combining a 5-Stack with a 2-Stack, placing the5+2 stack on the number line, and matching it to the yellow 7-Stack.

• Have students show “1 more” and “2 more” on the number line. Guide them: Show me the[number]-Stack. Now, use your stacks to show me 1 more. Show me the [number]+1stack on the number line. Tell it to me in words. Show and tell me the stack that is thesame length. Now show me 2 more. . . .

• Use different starting numbers, showing and verbalizing the quantities and relationships.I place my 5-Stack at 12. It ends at 17. When I place my 5+1 stack at 12, it ends at 18. 18is 1 more than 17. When I place my 5+2 stack at 12, it ends at 19. 19 is 2 more than 17.

• When students can confi dently show 1 more and 2 more from different starting numbers,model “1 less” and “2 less” using –1 and –2 stacks on the number line.

1 20 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


• Match the Number Card 3 to the corresponding stack. Map the 3-Stack to the number line.

• Connect addition and subtraction to the corresponding expression. I want to showone more. I can show it and tell it in words: This is a 3-Stack plus a light blue 1-Stack.I can write it with numbers: 3 1 1.

• Connect the physical quantities to the corresponding written notation. I can show the311 stack on the number line. It is the same as a dark blue 4-Stack. I write: 3 1 1 5 4.

• Repeat, connecting stacks to 12, 21, and 22 number sentences.


Show several [number]11 and [number]12 cube stacks and prompt students to write corresponding 11, 21, 12, and 22 number sentences.

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LESSON 6 Plus 10, Minus 10 From Any Number


OBJECTIVES

Connect the physical quantity of 10 to the number line and to the corresponding +10, –10 written number sentences.

SUMMARY

Students slide a 10-Stack up and down the number line to determine the [number]+10 and [number]–10 quantity from any starting point.

MATERIALS

For each student:

• one 10-cube stack• Number Line 0–100• Number Cards 1–100


• number line• ten more, ten less, the same as• plus ten, minus ten, equals• number sentence

Model & ExploreCompare and Order Quantities (3 minutes)

Beginning with this lesson, students no longer compare and order quantities as the fi rst step of the lesson. By this point, students have graduated to knowing that the physical quantity of, say, 8, is represented by the 8 on the number line. They should understand without manipulatives that starting at a number on the number line is the same as placing the corresponding stack at 0 on the number line.


• Map a 10-Stack to the Number Line 0–100 at 0. To help studentsdevelop a secure mental representation, model visualizing the quantity.I am going to close my eyes and make a number line picture in mymind. I see that the 10-Stack takes up 10 spaces. It ends at 10.

• Slide the 10-Stack to 20 on the number line. I move the 10-Stack tobegin at 20; it ends at 30. So, 30 is 10 more than 20. Next, slide the10-Stack to begin at 30 and discuss the increase. Continue sliding the10-Stack along the number line at the decades and discussing10 more.

• Use different starting numbers that are not decade numbers, continuingto demonstrate and verbalize the quantities and relationships. When Iplace the 110-Stack at 8, it ends at 18. So, 18 is 10 more than 8.

• Have students continue to slide a 10-Stack along the number line,prompting them with: Show me on the number line. Now tell me inwords. Guide students to recognize and talk about “10 more.”

When you place the 10-Stack at 18, where does it end? (28)

How does 28 compare to 18? (28 is 10 more than 18.)

Where does the 10-Stack end when you slide it to 28? (38)

• Model mapping 10-Stacks to the number line to demonstrate “10 less.”Place a 10-Stack between 100 and 90. I can also show 10 less on thenumber line. I place a 10-Stack so that it begins at 100 and ends at90. So, 90 is 10 less than 100.

11 1210 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

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173Lesson 6: Plus 10, Minus 10 From Any Number

• Slide the –10-Stack so that it extends from 90 to 80. When I move the 10-Stack to beginat 90, it ends at 80. So, 80 is 10 less than 90. Continue sliding the 10-Stack along thenumber line and discussing the decreases.

• Invite students to slide a 10-Stack along the number line, using starting numbers that arenot decade numbers, for example, 74, 64, 54. Prompt students with: Show me on thenumber line. Now tell it to me in words.


• Choose a Number Card, name it, point to the number on the number line. Here is a 46card. I am going to fi nd 46 on the number line.

• Connect addition to the corresponding number expression. I want to add 10 more. I canshow it and tell it in words: I place a 10-Stack starting at 46. That is 46 plus 10. I canwrite it with numbers: 46 1 10.

• Make the connection between the physical quantities and the corresponding writtennotation. I see that the 10-Stack ends at 56. So 56 is ten more than 46. I write that as:46 1 10 5 56.

• Repeat, modeling how to show, say, and write –10 number sentences. Here is an 82 card.I am going to fi nd 82 on the number line. I want to show 10 less. I place a –10-Stackstarting at 82. That is 82 minus 10. The –10-Stack ends at 72. I can write it withnumbers: 82 2 10 5 72.

• Write other +10 and –10 number sentences on the board. Prompt students to: Show methis (written problem) on the number line.


Demonstrate with a student how to play the game Plus 10, Minus 10. Play 1 to 3 rounds; then have students play with a partner, alternating roles.

Rules for Plus 10, Minus 10

1. Students place the Number Cards 10–90 in a facedown pile.

2. Student A selects a card and writes a +10 and a –10 numbersentence.

3. Student B checks the number sentences using a 10-Stack on thenumber line.

4. Student A gets a point for each correct number sentence.

5. The game ends when a student reaches 8 points.

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LESSON 7 Plus and Minus 1—10


OBJECTIVES

Connect physical quantities 1–10 to the number line and to the corresponding addition and subtraction sentences.

SUMMARY

Students place any of the stacks 1–10 at any starting point on the number line to determine the resulting quantity.

MATERIALS

For each student:

• 1 set of cube stacks 1–10• Number Line 0–100• Number Cards 1–100


• number line• more, less, the same as• plus, minus, equals• number sentence

Model & ExploreMap to the Number Line (5 minutes)

• Model placing one of the stacks 1–10 along the Number Line 0–100and naming the quantities and ending numbers. Choose a cube stackand place it at 0. I place the green 2-Stack at 0 on the number line.It takes up 2 spaces. It ends at 2. Point to each space on the numberline and verbalize what you are mapping.

• Slide the stack to a different starting number. I place the 2-Stack at 15on the number line; it ends at 17. So, 17 is 2 more than 15. Continuesliding the 2-Stack along the number line talking about 2 more.

• Invite students to choose a stack between 1–10 and place it at anumber along the number line. Have students verbalize the increaseusing these sentence frames.

When I place the -Stack at on the number line,it ends at .

So, is more than .

11 1210 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

• Model mapping 1- through 10-Stacks to the number line at differentstarting numbers to show “one less.” Place a 2-Stack between 78 and76. I can show 2 less on the number line. I place a 2-Stack so that itbegins at 78 and ends at 76. I can say that 76 is 2 less than 78.

• Continue mapping 1- through 10-Stacks to the number line to show“[number] less.” To help students verbalize the decreases, ask themquestions such as these:

When you place the 2-Stack at 51, where does it end? (49)

How does 49 compare to 51? (49 is 2 less than 51.)

Where does the 2-Stack end when you slide it to 49? (47)

• Have students map other 1- through 10-Stacks to the number line toshow “[number] less.” Encourage them to discuss decreases using thephrase “less˛than.”

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175Lesson 7: Plus and Minus 1—10


• Model mapping to the number line, saying, and writing +1 through +10 numbersentences. Begin by selecting a Number Card, naming it, and pointing to the number onthe number line. Here’s a 32 card, and here is 32 on the number line.

• Connect addition to the corresponding number expression. I want to add 4 more. I canshow it and tell it in words: I place a 14-Stack starting at 32. That is 32 plus 4. I canwrite it with numbers: 32 1 4.

• Make the connection between the physical quantities and the corresponding writtennotation. I see that the +4-Stack ends at 36. So 36 is 4 more than 32. I say and write:32 1 4 5 36.

• Repeat, modeling how to show, say, and write –1 through –10 number sentences.Here is a 45 card and here is 45 on the number line. I want to show 7 less. I place a27-Stack starting at 45. That is 45 2 7. I see that the 27-Stack ends at 38. I say andwrite: 45 2 7 5 38.

• Write other 1 and – number sentences on the board. Prompt students to: Show me this(written problem) on the number line.


Demonstrate with a student how to play the game Plus and Minus. Play 1 to 3 rounds; then have students play with a partner, alternating roles.

Rules for Plus and Minus

1. Students place the Number Cards 1–10 face-down in a pileand Number Cards 10–90 face-down in a second pile.

2. Student A selects a card from each pile and writes a + and a –number sentence.

3. Student B checks the number sentences using a cube-stackon the number line.

4. Student A gets a point for each correct number sentence.

5. The game ends when a student reaches 8 points.

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Counting System Intervention

Developing Skills for Navigating the Number SystemThe Counting System intervention lessons develop robust and fl exible number sense by connecting counting to quantities and relationships among quantities. They are designed to build ease and assurance in navigating our base-ten number system.

The lessons promote a variety of counting skills—counting up; counting on from any given number; counting by 10s, 5s, and 2s; and counting back, to name a few. They connect counting to numbers on a number line and number grid, and they show the relationship between counting and the expressions that show relationships between numbers: counting forward by 10 can be represented as expressions with +10. Well-developed counting skills, both oral counting and counting in conjunction with number lines and number grids, comprise a repertoire of skilled moves on the mental number line—and are crucial to developing students’ number sense.

Using the Counting InterventionFrequency of InterventionThe lessons in the intervention have been carefully crafted to support progress and success. They are designed for

• small-group usage (can also be used one-on-one, if necessary).

• 20–30 minute sessions.

• 4 days per week over 2 weeks.

Elongating lessons doesn’t make them more effective; tightening your use of time will.

• Practice lesson routines, such as Round Robins and count-alouds, so that studentsknow what to expect and how to respond during each routine.

• When an activity drags, move on. Be aware of time and push forward to the nextlesson routine.

• If students are struggling, repeat the lesson another day instead of extending the time.

Although it sounds simple and is usually mastered easily, . . . counting is actually a complex skill with several underlying components. . . . Counting opens the door to simple arithmetic.

—Krasa and Shunkwiler, 2009, pages 55–60

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Counting System Intervention

Progression of ConceptsThe lessons are sequenced from basic to more diffi cult concepts. Before starting the intervention, fi rst determine where students’ counting is breaking down. Be sure students practice each counting skill until they show a secure, consistent, and nimble mastery, which may take more than one session.

Counting RoutinesChoral Count-Aloud: The teacher leads this count-aloud with students, setting the pace and recounting at any wobble spots. This is to help the students “catch” by ear the repeating sequences of the number system. Some physical accompaniment such as fi st-pumping or hand-clapping can help to set a steady rhythm.

Round Robin: In a Round Robin, students count one after the other in sequence. The teacher interrupts periodically with a HALT gesture and provides new counting instructions such as, start with , reverse counting direction, count backward from there. Round Robins are helped by a steady rhythm.

Teacher PreparationThe materials and reproducibles used throughout the intervention lessons can be gathered, constructed, and copied ahead of time. These materials are used in most lessons in the Counting System Intervention. Print them prior to Lesson 1 and keep them on hand throughout the intervention. For your convenience, we have included templates (available as PDFs) for making your own number lines, grids, and number cards.

Print and Assemble• 100 Chart (5 for each student and

1 for the class)

• Blank 10×10 Grid (9 for each student)

• Number Line 0–300 (3 for the class)Print PDF Number Line 0–60, 61–120,121–180, 181–240, 241–300 andassemble to create your own.

• 5-Stick (3 for the class)

• 10-Stick (3 for the class)

Note: There are additional reproduciblesfor each lesson. Please review thematerials list for each lesson to makesure you have all necessary materials.

Accessing the PDF filesThe PDF fi les for the Counting System Intervention are accessible through SAM. (See Getting Started in the Program Overview section for more details.)

Gather• Folders (1 for each student)

These are to store the number gridsand worksheets that each studentcompletes at the end of the lessons.

• 25¢ plastic or cardboard coins(20 for each student)

MATERIALS

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LESSON 1 Counting by 1s


OBJECTIVE

Establish the connection between counting forward and backward by 1s and adding and subtracting 1.

SUMMARY

Students count aloud forward and backward by 1s. They use 100 Charts and number sentences to read and record adding 1 and subtracting 1 from different numbers.

MATERIALS

For the teacher:

• 100 Chart

For each student:

• 100 Chart• Blank 10310 Grid• One-Minute Worksheet 1


• plus, more, add, adding, addition• minus, less, take away, subtract,

subtracting, subtraction

Model & ExploreWARM-UP (1 minute)

Lead a brief choral counting warm-up, having students count in unison and at a pace that keeps everyone together. Start out counting from 1 to 10 forward and then count backward from 10 to 1. Accompany the counting with physical movements to set the beat.

Count Together (2 minutes)

• Distribute 100 Charts to students. Model moving forward by 1s fromany number on the 100 Chart, pointing as you count aloud. Let’s counttogether from 20 to 30. As you point to each row in the 100 Chart, putstress on the decade word. Thirty-one, thirty-two, thirty-three, . . .

• Then model counting backward from any number, again pointing to thechart. Use physical movements to set a steady rhythm.

• Conduct a Round Robin, having students count by 1s from any numberusing their 100 Charts. Interrupt periodically to call out redirections:Count backward from there. Start over at 50. Reverse the RoundRobin direction. Continue until students can count in a steady rhythm.

Connect Counting to Adding & Subtracting (2 minutes)

• Explain that counting by 1 is the same as adding 1. Have studentscount with you, using the 100 Chart. Let’s count forward from 10 to20. You know that 1 more than 10 is 11.And 1 more than 11 is 12.And 1 more than 12 is .Continue to 20, pointing to each number on the 100 Chart as you count.

• Connect counting to addition using the phrase “plus 1.”You also know that 10 plus 1 is 11.And 11 plus 1 is 12.And 12 plus 1 is .Have students verbally fi nish the number sentences through 20.

• Ask students to share more examples of adding 1 to other numbersusing their 100 Charts.

• Repeat, connecting counting back to subtraction, using the phrases “1less” and “minus 1.”

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179Lesson 1: Counting by 1s

• Discuss different words for addition (plus 1, 1 more, add 1, adding 1) and subtraction(minus 1, 1 less, take away 1, subtract 1, subtracting 1). Then have students use thesewords to describe more challenging counting examples (85 to 95, 65 to 55, 120 to 130).

Connect Counting to Number Sentences (5 minutes)

• Connect counting forward by 1 to “+1” expressions. You can use words to show thatcounting by 1 is the same as adding 1: 10 plus 1 is 11, 11 plus 1 is 12. You can alsoshow it with numbers.

• Write the number 12 on the board. Below it write the +1 expression. 12 + 1. Use the 100Chart to connect counting forward by 1 to the expression. I know “12 1 1” is anotherway to say “1 more than 12.” I count forward 1 from 12 and get to 13. So, 12 1 1 5 13.Complete the number sentence on the board and read it aloud.

• Repeat with other examples. Write the 11 expression and have students connect it tocounting forward and addition.

• Connect counting back by 1 to “–1” expressions. Write the expression 12 – 1 on the board,demonstrate moving back 1 on the 100 Chart, and complete the number sentence. Twelveminus 1 is another way to say one less than 12. One less than 12 is 11, so 12 21 5 11.

• Write incomplete +1 and –1 number sentences on the board using other numbers through100 (e.g., 38, 45, 57, 60, 71, 83, 99). Have students read and verbally solve the numbersentences demonstrating +1 and 21 on the 100 Charts. Encourage them to use thedifferent addition and subtraction vocabulary for each number sentence.

Explore: 11, 21 (5 minutes)

• Distribute Blank 10×10 Grids to pairs of students. Let’s explore “1 more” and “1 less” onour number grids—but, without seeing the numbers.

• Guide student pairs to write the number 3 in the correct location on a Blank 10310 Grid.Then have students determine the numbers that are 1 more and 1 less than 3 and writethese on the grid.

• Continue using other starting numbers on the Blank 10310 Grid: 13, 23, 33, 44, 54, 64,75, 85, 95. Have student pairs check to make sure the numbers are in the correct locationon the 100 Chart. If students are having diffi culty determining where to write a number orwhat number is 1 more or 1 less, suggest they refer to their 100 Charts as a guide.


• Give students 60 seconds to complete One-Minute Worksheet 1.

• Have students continue to fi ll in the numbers (1–100) on their Blank 10310 Grid.

• Add students’ completed work to their Counting Folders.

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OBJECTIVE

Establish the connection between counting forward and backward by 10s and adding and subtracting 10 from multiples of 10.

SUMMARY

Students count aloud forward and backward by 10s starting at multiples of 10. They use 100 Charts, the number line, and number sentences to read and record adding and subtracting 10 from multiples of 10.

MATERIALS

For the teacher:

• Number Line 0–300• 100 Chart• 10-Stick

For each student:

• 100 Chart• 10-Stick• Blank 10310 Grid• One-Minute Worksheet 2





Lead several choral count-alouds by 1s from various starting numbers. Focus on short stretches of numbers that students need to strengthen. For example, if students struggle counting from the 90s to the 100s, count from 95 to 105 several times rather than counting once from 75 to 125. Interrupt periodically to have students count backward. Use physical movements to set a steady pace.


• Display the Number Line 0–300. Lead several choral count-alouds by10s from 0 to 100. Then lead a count-aloud in reverse from 100 to 0,which will necessarily be slower.

• Conduct a Round Robin, having students count one after another insequence. Have students count by 10s from 0 to 300. Encourage them touse the number line as they count. Interrupt the Round Robin periodicallycalling out redirections such as to count backward by 10 from there.Count forward by 10 again. Start over at 130. Reverse the RoundRobin direction. Continue until students count in a steady rhythm.


• Distribute 100 Charts. Have students count together by 10s, from 0 to100, on the chart. Explain that counting by 10 is the same as adding 10.

• You know that 10 more than 10 is 20. And 10 more than 20 is 30. And10 more than 30 is . Continue on through 50, pointing to thenumbers on the chart.

• Connect counting to addition using the phrase “plus 10.” You alsoknow that 10 plus 10 is 20. And 20 plus 10 is 30. And 30 plus 10is . Continue through 50, pointing to the numbers on the chart and having students verbally complete the number sentences.

• Ask students to share more examples of adding 10 on the 100 Chart.

• Repeat, connecting counting back by 10 to subtracting 10, using thephrases “10 less” and “minus 10.”

• Lead a discussion about different addition words ( plus 10, 10 more, add10, adding 10 ) and subtraction words (minus 10, 10 less, take away10, subtract 10, subtracting 10 ). ©

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• Repeat, connecting counting back by 10 to subtracting 10, using the phrases “10 less”and “minus 10.”

• Lead a discussion about different addition words ( plus 10, 10 more, add 10, adding 10 )and subtraction words (minus 10, 10 less, take away 10, subtract 10, subtracting 10 ).


• Connect counting forward by 10 and “110” expressions. Write the number 50 on theboard. Below it write the 110 expression and read it aloud.

• Using the 100 Chart, explain and model how to move forward 10 to represent theexpression. I know “50 1 10” is another way to say “10 more than 50.” I countforward 10 from 50 and get to 60. So, 50 1 10 5 60. Complete the number sentenceon the board.

• Then connect counting back by 10 to “210” expressions. Write the expression 50 – 10,demonstrate moving back 10 on the 100 Chart, and complete the number sentence.

• Write incomplete 110 and 210 number sentences on the board using multiples of 10through 300 (for example, 10, 60, 90, 120, 150, 180, 210, 230, 270, 300). Have studentsread and verbally solve the number sentences demonstrating 110 and 210 on the 100Chart or Number Line 0–300. Prompt them to use the different addition and subtractionvocabulary to describe the number sentences.


• Have students explore the Number Line 0–300 using a 10-Stick to show 110, then 210.Place a 10-Stick on 30 and model how to describe the relationship in words and numbers:30 plus 10 more is 40, so 30 1 10 5 40.

• Have student pairs practice with other multiples of 10 from 0 to 200 (for example, 30, 70,80, 110, 140, 190). Students name the starting number, place the 10-Stick, and describethe relationship with words and numbers.

26 2725 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45



• Have students fi ll in a Blank 10310 Grid with multiples of 10 to 1,000.


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OBJECTIVE


SUMMARY

Students count aloud forward and backward by 5s starting at multiples of 5. They use 100 Charts, the number line, and number sentences to read and record adding 5 to and subtracting 5 from multiples of 5.

MATERIALS

For the teacher:

• Number Line 0–300• 100 Chart• 5-Stick

For each student:

• 100 Chart• 5-Stick• Blank 10310 Grid• One-Minute Worksheet 3




Model & Explore

WARM-UP (1 minute)

Lead several short choral count-alouds by 10s in the range 0 to 300; vary the starting number (always a multiple of 10).

Then lead a count-aloud by 10s in reverse from 300 to 0, which will necessarily be slower. Use movement to set a steady pace.


• Show the Number Line 0–300. Lead several choral count-alouds by 5sfrom 0 to 300. Set a pace using physical motions such as hand-wavingor fi nger-tapping.

• Then lead choral count-alouds by 5s—for short stretches ofnumbers—from different starting numbers (always a multiple of 5).

• Conduct a Round Robin, counting by 5s from 0 to 300. Interrupt theRound Robin periodically calling out redirections: Count backward by5 from there. Count forward by 5 again. Start over at 50. Reverse theRound Robin direction. Encourage students to count using the numberline. Continue until students count without hesitation.


• Distribute 100 Charts. Invite students to count with you by 5s to 100.As you count, model how to move forward by 5s on the 100 Chart. Thenhave students circle all the multiples of 5 on the chart.

• Explain that counting forward by 5s is the same as adding 5.Count with me by 5s from 0 to 100 using our marked 100 Charts.You know that 5 more than 5 is 10.And 5 more than 10 is 15.And 5 more than 15 is .Continue on through 50, pointing to the multiples of 5 on the 100 Chart.

• Then connect counting to addition using the phrase “plus 5.”You also know that 5 plus 5 is 10.And 10 plus 5 is 15.And 15 plus 5 is .Continue on to 50, pointing to the numbers and having students verballycomplete each number sentence.

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• Have students give other examples of adding 5 using the 100 Chart.

• Repeat, connecting counting backward by 5s to subtraction using the phrases “5 less” and“minus 5.”

• Talk together about different addition words (plus 5, 5 more, add 5, adding 5, addition) andsubtraction words (minus 5, 5 less, take away 5, subtract 5, subtracting 5, subtraction).


• Write the number 35 on the board. Below it, write, 35 1 5 , and read it aloud.

• Connect counting forward by 5 to “1 5” number sentences using the 100 Chart. I know“35 1 5” is another way to say “5 more than 35.” I count forward 5 from 35 to get to40. So, 35 1 5 5 40. Complete and read aloud the number sentence on the board.

• Then connect counting back by 5 to “25” sentences. Write the expression 35 – 5 on theboard, demonstrate moving back 5 on the 100 Chart or Number Line 0 – 300, and completethe number sentence.

• Write several incomplete 1 5 and 2 5 number sentences on the board using multiplesof 5 (for example 5, 20, 45, 70, 95, 150, 185, 205, 230, 300). Have students read andverbally solve the number sentences demonstrating 1 5 and 2 5 on the 100 Chart orNumber Line 0 – 300. Prompt them to use the different addition and subtraction vocabulary.


• Have students explore the Number Line 0–300 using a 5-Stick to show 15, then 25.Place a 10-Stick on 35 and model describing the relationship in words and numbers:35 plus 5 more is 40, so 35 1 5 5 40.

• Have students practice with other multiples of 5 from 0 to 300 (for example, 30, 55, 75,100, 115, 195, 225, 250). Students name the starting number, place the 5-Stick, anddescribe the relationship with words and numbers.

26 2725 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45



• Have students fi ll in a Blank 10310 Grid from 101–200.


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LESSON 4 Counting by 10s From Any Number


OBJECTIVE

Establish the connection between counting forward and backward by 10s and adding and subtracting 10 from any number.

SUMMARY

Students count aloud forward and backward by 10s starting at any number. They use 100 Charts, the number line, and number sentences to read and record adding and subtracting 10 from any number.

MATERIALS

For the teacher:

• Number Line 0–300• 100 Chart

For each student:

• 100 Chart• 2 Blank 10310 Grids• One-Minute Worksheet 4




Model & Explore

WARM-UP (1 minute)

Lead several short choral count-alouds by 5s and 10s˛in the range 0 to 300; vary the starting numbers (always multiples of 5 or 10).


• Display Number Line 0–300. Lead a choral count-aloud. Call out anumber (0 – 300) and ask students to count by 10 from that number.Point to the number line as you count. Repeat the process two moretimes choosing different starting numbers.

• Then lead the class in three backward count-alouds by 10 fromdifferent starting numbers. These will necessarily be slower.

• Conduct a Round Robin, counting by 10s starting at any number from0 to 300. Interrupt the Round Robin periodically calling out redirectionssuch as Count backward by 10s from there. Restart at 23 and countforward by 10. Continue until students count in a steady rhythm.

• Discuss the similarities between counting by 10s from a multiple of 10(Lesson 2) and counting by 10s from any number.


• Distribute 100 Charts. Reinforce the connection between countingforward by 10s and adding 10. Let’s count together by 10s from 4.Count with students by 10s from 4 to 54 pointing to the 100 Chart.You know that 10 more than 4 is 14.And 10 more than 14 is 24.And 10 more than 24 is .Continue on through 54, pointing to the numbers on the grid.

• Then connect counting to addition using the phrase “plus 10.”You also know that 4 plus 10 is 14.And 14 plus 10 is 24.And 24 plus 10 is . Continue on.

• Repeat, connecting counting backward to subtraction using the phrases“10 less” and “minus 10.”

• Invite students to share other examples of adding 10 and subtracting10 using the 100 Chart.

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185Lesson 4: Counting by 10s From Any Number


• Write the number 17 on the board. Then write the expression 17 1 10 and say it aloud.

• Using the 100 Chart, connect counting forward by 10 to “110” number sentences.I know “17 1 10” is another way to say “10 more than 17.” I count forward 10from 17 to get to 27. So, 17 1 10 5 27. Complete and read aloud the number sentenceon the board.

• Then connect counting back by 10 to “210” sentences. Write the expression 17 – 10,demonstrate moving back 10 on the 100 Chart, and complete the number sentence.

• Write other incomplete 110 and 210 number sentences on the board using numbersthrough 300 (for example, 24, 58, 99, 137, 160, 181, 203, 242, 275, 290). Have studentsread and verbally solve the number sentences demonstrating 110 and 210 on the100 Chart or Number Line 0–300. Prompt them to use the different addition andsubtraction vocabulary to describe the number sentences.


• Demonstrate exploring numbers that are –10 and +10 using a 100 Chart.Here is 24 on my 100 Chart. I want to fi nd the number that is 210, or 10 less than 24.I count back 10 to 14. I see that 14 is directly above 24 on my 100 Chart.

• Provide student pairs with Blank 10310 Grids. Guide students to write the followingnumbers on the correct spots—11, 22, 33, 44, 55, 66, 77, 88.

• Ask students to fi nd and write the 110 and 210 neighbors of those numbers on the grid.



• Have students fi ll in a Blank 10×10 Grid with the numbers 201–300.


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LESSON 5 Counting by 2s (Even Numbers)


OBJECTIVES

• Identify numbers in the even-number count sequence as evennumbers.

• Connect counting by 2 from aneven number to adding 2 andsubtracting 2, focusing on theevenness of the resulting answer.

SUMMARY

Students count aloud forward and backward by 2s starting at any even number. They use 100 Charts and number sentences to identify even numbers and to read and record adding 2 and subtracting 2 from any even number. Note: In Lesson 6, students will identify odd numbers and count by 2s from any odd number.

MATERIALS

For the teacher:

• 100 Chart

For each student:

• 100 Chart• One-Minute Worksheet 5• Blank 10310 Grid


• even, even number, skip count• plus, more, add, adding, addition• minus, less, take away, subtract,



Lead choral count-alouds by 1s starting at different numbers. Interrupt periodically to have students count backward. Focus on short stretches of numbers that students need to strengthen. Use physical movements such as hand waving or toe tapping to set a steady pace.


• Explain that counting by 2s is like counting by 1s, but skipping everyother number (skip-counting). Display the 100 Chart. Lead studentscounting by 2s from 0 to 50. Point to each number as you say it.

• Lead count-alouds by 2s for short stretches, starting at any evennumber from 2 to 100. Continue until students count rhythmically.

• Conduct a Round Robin, counting by 2s (even numbers). Interruptperiodically to start at a different even number or reverse directions.

• Explain that when you count by 2s, starting with 2, all the numbersare called even numbers. When you count by 2s starting with 1, all thenumbers are called odd numbers.

• Explain that students will focus on the even numbers. Ask questions sostudents use the terms “even” or “even number”:

What kind of numbers are you counting? (even)

What kind of number was the last number you counted? (even)


• Distribute 100 Charts. Discuss that counting forward by 2s is the sameas adding 2.You know that 2 more than 50 is 52.And 2 more than 52 is 54.And 2 more than 54 is . Continue on through 60.

• Then connect counting to addition using the phrase “plus 2.”You also know that 50 plus 2 is 52.And 52 plus 2 is 54.And 54 plus 2 is . Continue on, pointing to the 100 Chart and having students verbally complete the number sentences.

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187Lesson 5: Counting by 2s (Even Numbers)

• Then ask students to share their own examples of adding 2 to an even number.

• Repeat, connecting counting backward by 2s to subtracting 2s using the phrases “2 less”and “minus 2.”


• Connect counting forward by 2s to “12” expressions. Write 14 on the board. Below it,write the expression 14 1 2 and read it aloud. Using the 100 Chart, model how to add 2from an even number. I know “14 1 2” is another way to say “2 more than 14.” I countforward 2 from 14 and get to 16. So, 14 1 2 5 16. Complete the number sentence.

• Then connect counting back by 2 to “22” sentences. Write the expression 14 – 2, on theboard, demonstrate moving back 2 on the 100 Chart, and complete the number sentence.

• Guide students to understand that when you add or subtract 2 from an even number, thesolution is also even.

What kind of number is 14? (even)

When I add 2 to 14, I get 16. What kind of number is 16? (even)

When I subtract 2 from 14, I get 12. What kind of number is 12? (even)

• Write other incomplete 12 and 22 number sentences using even numbers through 100(for example, 2, 16, 28, 34, 46, 50, 68, 72, 98, 100). Have students read and verbally solvethe number sentences demonstrating 12 and 22 on the 100 Chart. Prompt them to usethe different addition and subtraction vocabulary to describe the number sentences.


• Have students count by 2s on their 100 Charts. As they count each number, have themcross out numbers they are not saying (the odd numbers). Have them describe thepatterns they see. Stress that the numbers showing are even˛numbers.

• Ask questions to help students recognize patterns with even numbers.

What numbers repeat on your chart? (0, 2, 4, 6, 8)

What patterns do you notice? (numbers repeat, vertical rows)

What is the greatest even number on your chart? (100)



• Have students fi ll in a Blank 10×10 Grid with the numbers 301–400.


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LESSON 6 Counting by 2s (Odd Numbers)


OBJECTIVES

• Identify numbers in the odd-number count sequence as oddnumbers.

• Connect skip counting by 2 toadding 2 and subtracting 2,focusing on the oddness of theresulting answer.

SUMMARY

Students count aloud forward and backward by 2s starting at any odd number. They use 100 Charts and number sentences to identify odd numbers and to read and record adding 2 and subtracting 2 from any odd number.

MATERIALS

For the teacher:

• 100 Chart

For each student:

• 100 Chart• One-Minute Worksheet 6• Blank 10310 Grid


• odd, odd number, skip count• plus, more, add, adding, addition• minus, less, take away, subtract,



Lead students in short count-alouds by 2s from any even number from 2 to 100. If students struggle counting on a particular stretch, repeat it several times until students can count at a steady pace.


• Review that when you count by 2s, starting with 2, all the numbers arecalled even numbers and when you count by 2s starting with 1, all thenumbers are called odd numbers.

• Display the 100 Chart. Lead students in counting by 2s from 1 to 51,pointing to each number on the chart.

• Lead several count-alouds by 2s for short stretches from any oddnumber from 1 to 99. Set a steady pace so students count rhythmically.

• Conduct a Round Robin. Interrupt the Round Robin periodically toreverse direction or start at a different odd number.

• Demonstrate, using drawings, that even numbers can be representedwith pairs and that odd numbers have “one extra.”

2 4 6 8

3 5 7 9


• Remind students that counting forward by 2s is the same as adding 2,and that counting backward by 2s is the same as subtracting 2. Invitestudents to skip count by 2s from 31 to 41 using the 100 Chart.You know that 2 more than 31 is 33.And 2 more than 33 is 35.And 2 more than 35 is . Continue on through 41.

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189Lesson 6: Counting by 2s (Odd Numbers)

• Then connect counting to addition using the phrase “plus 2.”So you also know that 31 plus 2 is 33.And 33 plus 2 is 35.And 35 plus 2 is . Have students verbally complete the number sentences.

• Ask students to share other examples of adding 2 to odd numbers.

• Repeat, connecting counting backward by 2s from an odd number to subtracting 2s usingthe phrases “2 less” and “minus 2.”


• Write the number 63 on the board. Below it, write the expressions 63 1 2 and 63 2 2,and read each aloud. Model adding 2 and subtracting 2 from an odd number using the 100Chart. I know “63 2 2” is another way to say “2 less than 63.” I count back 2 from 63and get to 61. So, 63 2 2 5 61. Complete each number sentence.

• Guide students to understand that when you add or subtract 2 from an odd number, thesolution is also odd.

What kind of number is 63? (odd)

When I add 2 to 63, what kind of number is the solution? (odd)

When I subtract 2 from 63, what kind of number is the solution? (odd)

• Write several incomplete 12 and 22 number sentences on the board using numberssuch as 1, 17, 23, 49, 85, 99. Have students read and verbally solve the number sentencesusing their 100 Charts. Then have students complete number sentences with and withoutusing their 100 Charts.


• Have students count by 2s on their 100 Charts, crossing out numbers they are not saying(even numbers). Stress that the numbers showing are odd numbers.

• Have students discuss odd numbers and describe the patterns they see.

What kind of numbers are you counting? (odd)

What numbers repeat? (1, 3, 5, 7, 9)

What patterns do you notice? (numbers repeat, vertical rows)



• Have students fi ll in a Blank 10310 Grid with the numbers 401–500.


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OBJECTIVE

Establish the connection between counting forward and backward by 100s and adding 100 and subtracting 100.

SUMMARY

Students count aloud forward and backward by 100 starting from any number. They use number lines and number sentences to read and record adding and subtracting 100 from any number.

MATERIALS

For the teacher:

• 3 Number Lines 0–300

For each student:

• One-Minute Worksheet 7• Blank 10310 Grid


• plus, more, add, adding, addition,• minus, less, take away, subtract,



Lead several choral count-alouds by 10 in the range 0 to 300; vary the starting numbers. Periodically have students count backward by 10.


• Lead several choral count-alouds by 100 from 100 to 2,000 untilstudents count in a steady rhythm.

• Then lead choral count-alouds by 100 starting from different numbers(5, 22, 35, 41) until students grasp the concept of counting by 100.

• Conduct a Round Robin, counting forward by 100s. Interrupt periodicallyto reverse direction or start at a different number. Continue until studentscan count fl uently in a steady rhythm.


• Make the connection between counting forward by 100s and adding 100.Let’s count together by 100: 100, 200, 300 . . .You know that 100 more than 100 is 200.And 100 more than 200 is 300.And 100 more than 300 is . Continue on through 1,000.

• Connect counting to addition using the phrase “plus 100.”Counting by 100 is like adding by 100.So, you also know that 100 plus 100 is 200.And 200 plus 100 is 300.And 300 plus 100 is .

• Have students continue to verbally complete the number sentences, untilstudents answer 1100 questions with understanding.

• Reinforce the connection between counting on by 100s from any numberand adding 100 to any number. Follow the structure above, giving severalverbal examples, so that students catch the pattern. For example, counton 100 from 163. Then add 100 to 163.

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INTENSIVE SUPPORT


• ekam stneduts pleH .s001 yb ,srebmun rehto neht dna ,000,1 morf drawkcab tnuoc ,txeNthe connection between counting backward by 100 and subtracting 100. Use the phrases“100 less” and “minus 100.”

• Use different addition words (plus 100, 100 more, add 100 ) and subtraction words (minus100, 100 less, take away 100, subtract 100 ).


• Write the expressions 140 1 100 and 140 2 100 on the board and read them aloud.Using the Number Line 0–300, demonstrate how to move forward or back 100 torepresent each expression. I know “140 1 100” is another way to say “100 more than140.” I can count forward 100 from 140 to get to 240. So, 140 1 100 5 240. Completethe number sentences.

• Write several incomplete 1100 and 2100 number sentences on the board using numberssuch as 103, 158, 200, 245, 370, 498, 561, 687, 814, 900. Have students read andverbally solve the number sentences 1100 and 2100. Prompt them to use the differentaddition and subtraction vocabulary to describe the number sentences. If students arestruggling, have them solve simpler problems using the Number Line 0–300.


• Explore the quantity represented by numbers 100 and higher. Line up and connect threeor more Number Lines 0–300. Invite students to visualize how many 100s fi t on thecombined number line.

• Then have students take turns showing differing 1100 and 2100 combinations on theNumber Line 0–300 (for example, 125 1 100, 125 2 100; 150 1 100, 150 2 100;193 1 100, 193 2 100).


• Have students complete One-Minute Worksheet 7.

• Have students fi ll in a Blank 10310 Grid with multiples of 100 to 10,000.


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OBJECTIVE


SUMMARY

Students count aloud forward and backward by 25s. They use number lines and number sentences to read˛and record adding and subtracting 25.

MATERIALS

For the teacher:

• Number Line 0–300 (one for class)

For each student:

• 20 25¢ plastic or cardboard coins• paper bag• Blank 10310 Grid• One-Minute Worksheet 8


• plus, more, add, adding, addition,• minus, less, take away, subtract,


Model & Explore

WARM-UP (1 minute)

Lead several choral count-alouds by 5s in the range 0 to 500; vary the starting number (a multiple of 5). Occasionally count backward by 5.


• Lead several choral count-alouds by 25s from 0 to 525; vary thestarting number (a multiple of 25). Continue until students count at aconsistent pace.

• Display the Number Line 0–300. Conduct a Round Robin, counting by25s from 0 to 300, pointing to each number as you say it. Interruptthe Round Robin periodically calling out redirections such as Countbackward by 25 from there. Reverse the Round Robin direction.Continue until the Round Robin progresses smoothly and rhythmically.


• Explain that counting forward by 25s is the same as adding 25. Displaythe Number Line 0–300. Invite students to count forward with you by25s to 200.You know that 25 more than 25 is 50.And 25 more than 50 is 75.And 25 more than 75 is .Continue on through 300, pointing to the numbers on the number line.

• Then connect counting to addition using the phrase “plus 25.”So, you also know that 25 plus 25 is 50.And 50 plus 25 is 75.And 75 plus 25 is .Continue on, pointing to the numbers on the number line and havingstudents verbally complete the number sentences.

• Repeat, connecting counting backward to subtraction using the phrases“25 less” and “minus 25.”

• Use different addition words (plus 25, 25 more, add 25, adding 25)and subtraction words (minus 25, 25 less, take away 25, subtract 25,subtracting 25).

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Technical SupportFor questions or other support, visit the HMH Education Support website at www.hmhco.com/fasttmath/productsupport.

At the site, users will fi nd program documentation, manuals, and guides, as well as Frequently Asked Questions and live chat support.

For specifi c questions about the HMH program suite, contact Customer Service at 800-283-5974 to speak to a representative.

For specifi c questions about using SAM with your HMH programs, click Help in the Quick Links along the top of any screen in SAM.

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