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This work is protected by United States copyright laws and is provided solely for the use of teachers and administrators in teaching courses and assessing student learning in their classes and schools. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. Copyright 2012 Pearson Education, Inc., or its affiliate(s). All Rights Reserved. Printed in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. The publisher hereby grants permission to reproduce these pages, in part or in whole, for classroom use only, the number not to exceed the number of students in each class. Notice of copyright must appear on all copies. For information regarding permissions, write to Pearson Curriculum Group Rights & Permissions, One Lake Street, Upper Saddle River, New Jersey 07458. America s Choice, the America s Choice A logo, Math Navigator, the Pearson logo, and the Pearson Always Learning logo are trademarks, in the U.S. and/or other countries, of Pearson Education, Inc. or its affiliate(s). ISBN: 978-0-66364-305-9 1 2 3 4 5 6 7 8 9 10 16 15 14 13 12

Contents Lesson 1 Letter to Parents 1 Lesson 2 Ten Frame Workspace 3 Lesson 5 Double Ten Frames Workspace 4 Lesson 8 Tens and Ones Place-Value Table Workspace 5 Lessons 10, 11, 16 Hundred Chart 6 Lesson 16 Little Ten Frames Workspace 7 Lesson 3 Place Value Cards 8 Misconceptions 16 Class Profile 23 A Complete Solution to a Math Story 27 What to Do If You Get Stuck 28 Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. iii

Letter to Parents Dear Parent/Guardian, Introduction to Math Navigator has been selected to participate in Math Navigator! Math Navigator is one of the ways that our school is working to help all students succeed in mathematics. The program gives students the additional time and instruction they need to improve their performance in this important subject. Your child will be participating in the Place Value and Computational Strategies to 100 module. The main goals of this module are to help students extend their knowledge of the counting sequence to 120, to lay a strong foundation for understanding place value, and to help students use place value and properties of operations to add and subtract. Students will develop number sense for 10 and for the teens numbers, then develop the understandings that 10 ones make a ten and that two-digit numbers represent amounts of tens and ones. Students will explore patterns in spoken number words and written numerals to 120 as they read, write, compare, and order two-digit numbers. They will explore strategies for adding two-digit numbers such as making ten, counting on, and adding on. Students will also add and subtract multiples of 10. At the end of the module, students will explore the general method of adding the tens first, then the ones. Throughout the module, students will record their thinking about computations in writing, and explain their thinking to others. There are a variety of materials students will use with this module: one of them is a set of Study Cards. These cards include mathematical ideas for students to master, game cards, and blank cards that students can customize with concepts that they need to work on. Students are encouraged to use these cards during the lessons, as well as during free time and at home. Please encourage your child to share them with you. The more enthusiastic you can be about Math Navigator, the more it will help your child. Ask questions each day about what your child learned and how the Math Navigator class was different from your child s regular math class. It is important for you to acknowledge what your child has accomplished both on a day-to-day basis and after completing the Math Navigator module. We are excited about using Math Navigator with students. Learn more about this special program and how it works by reading the short description that follows. If you have any questions about the program, please do not hesitate to contact us here at school. Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1

Letter to Parents How Math Navigator Works Structure of a Module Each module contains 20 days of 30- or 45-minute lessons, including a pre-test and post-test. During the 20 days, students have two or three checkpoint lessons that assess their understanding of the concepts in the module. Frequent Skills Practice Most lessons include a Show Me session in which students practice and reinforce skills. It is also a time for students to learn strategies and techniques that make computation easier. Emphasis on Understanding The lessons are carefully designed to uncover mistakes that result from students misunderstanding something. We call such mistakes misconceptions. Misconceptions need to be corrected because they can interfere with new learning. Math Navigator modules do not attempt to reteach everything that students have learned about a topic. Instead, they help students understand the mathematics of the procedures and concepts that they have already learned so that they can correct the misconceptions that are getting in the way of their progress. Learning to Think Mathematically Lessons are structured to teach students to think like mathematicians. Students will learn how to ask themselves questions before beginning a problem; to use diagrams, tables, and other methods of representing problems; and to estimate as a way of determining whether their answers are reasonable. Most importantly, they will come to see that mistakes are opportunities for learning, rather than something to hide. Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 2

Lessons 10, 11, 16 Hundred Chart 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 6

Lesson 3 Place Value Cards Preparing and Using the Place Value Cards For this module, the Place Value Cards are a set of ones cards and tens cards. When folded into a tent, each card shows a one- or two-digit numeral on one side, and a base-10 drawing of the number on the other side. The drawings use dots to represent ones and sticks to represent tens. To prepare the cards, print the sheets onto stiff paper, cut out the cards, and fold them into tents. print cut fold flip The cards can now be used to show two-digit numbers. Below, the 6 card is placed on top of the 10 card to show 16. stack When turning the tents around to see the drawings, the cards must be rearranged so the drawings show the tens on the left and the ones on the right. Students can rearrange the cards as shown below. flip slide The place value cards are used with students first to develop the understanding that each teen number consists of two groups: 1 group of ten things and a group of the extras (the leftovers that are counted by 1s). The cards then help students see that 10 ones equal one ten: the 1 in 16, for, means 1 group of ten. Lifting the ones card (the 6) reveals the 0 from the 10: it is hiding behind the ones number. The place value cards are then used to help students see that for all two-digit numbers, the tens digit refers to groups of tens, and the ones digit refers to some ones. By again lifting the ones card, students see that the 0 from the tens number is hiding behind the ones number. Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 8

Misconceptions Misconceptions and Errors Although this module is designed as initial teaching, it is likely that student misconceptions will become evident as they work through the lessons. Possible misconceptions include: AT1 AT2 F21 O16 O22 O23 PV1 PV2 PV3 PV4 PV5 PV6 PV7 PV8 PV9 PV10 PV11 Does not recognize a pattern including a number pattern Always counts by 1s Does not understand the concept of equivalence Does not recognize or misapplies the associative property Fails to link addition and subtraction as inverse operations Is unable to transfer between different representations of the same operation Does not understand or misinterprets the role of 0 as a placeholder Does not represent a number correctly in different forms: expanded, place value, and numeric Ignores place value and treats each number as a separate number Misapplies the counting sequence when counting Confuses place value in whole numbers Does not understand that the value of any digit in a number is a combination of the face value of the digit and the place Orders numbers based on the value of the digits, instead of place value Reads and writes the numbers between 10 and 20 incorrectly Confuses decade numbers with teens numbers (such as sixty with sixteen) Places the symbols for greater than and less than incorrectly when recording the results of comparisons Thinks about any multi-digit number only as the number reached after counting by 1s AT1 Does not recognize a pattern including a number pattern What is the missing number? 7,, 27, 37, 47, 57, 26 Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 16

Misconceptions AT2 Always counts by 1s Skip-count by 6s. 6, 12, 18, 19, 20, 21, 22, 23, 24 F21 Does not understand the concept of equivalence The student thinks the equals sign always means the result is or here comes the answer (rather than is the same as ). Or, the student thinks a missing number in an equation is supplied by writing the total of the numbers given. Write the missing number. 4 + 3 = 1 + 8 10 + 7 = 17 + 8 O16 Does not recognize or misapplies the associative property The student fails to find an easier way to add because he fails to recognize that the order in which numbers are added can be changed without affecting the result. The student labors to find the total. 38 + 7 =? 38 + 7 = 38 + 5 + 2 I counted on: 38 39, 40, 41, 42, 43. Then 43 + 2 = 45 Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 17

Misconceptions O22 Fails to link addition and subtraction as inverse operations The student does not use the adding up strategy ( think addition ) to make subtraction problems easier to solve. She does not view subtraction problems as missing addend problems. She may count down, making the common error of counting the starting number as the first subtraction. 80 60 =? 80, 70, 60, 50, 40, 30. The answer is 30. O23 Is unable to transfer between different representations of the same operation The student has problems reading or creating different representations of addition or subtraction. Explain how the recording shows the make ten strategy. After 28, you count on 5: 29, 30, 31, 32, 33, and then you count on 2: 34, 35 Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 18

Misconceptions PV1 Does not understand or misinterprets the role of 0 as a placeholder Write the number for the one hundred six. 16 PV2 Does not represent a number correctly in different forms: expanded, place value, and numeric The student has limited his understanding of numbers to one or two representations, or applies the alternate conception Write the numbers you hear when writing numbers in standard form. Write the number I say: thirty-seven. 307 Write the missing number. 36 = 3 + 6 PV3 Ignores place value and treats each number as a separate number The student adds or subtracts each place without grouping or ungrouping to make a unit in an adjacent place. The student writes the total or difference of the parts all as one number. 12 + 9 111 Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 19

Misconceptions PV4 Misapplies the counting sequence when counting Student counts: twenty-nine, twenty-ten, twenty-eleven, PV5 Confuses place value in whole numbers Read 306. Thirty-six Write the numeral for four hundred eight. 4008 PV6 Does not understand that the value of any digit in a number is a combination of the face value of the digit and the place What is the value of the digit 2 in 126? 2 Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 20

Misconceptions PV7 Orders numbers based on the value of the digits, instead of place value 69 > 102, because 6 and 9 are bigger than 1 and 2. PV8 Reads and writes the numbers between 10 and 20 incorrectly The student has difficulty because the teen numbers do not make their base-10 meanings evident. How much is twelve? Ten and nine, ten, eleven, twelve What does this number mean (17)? one seven PV9 Confuses decade numbers with teens numbers (such as sixty with sixteen) The student has difficulty because decade number names sound similar to teens number names, as in forty and fourteen. Write the number I say: Forty 14 Sixteen 60 Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 21

Misconceptions PV10 Places the symbols for greater than and less than incorrectly when recording the results of comparisons The student finds challenging placing the > < = symbols correctly. Write the symbol to make the number sentence true. 35 > 53 PV11 Thinks about any multi-digit number only as the number reached after counting by 1s For, the student thinks about 50 not as 5 tens but as the number reached after 49 counts of 1. How much is 25? 20, 21, 22, 23, 24, 25 Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 22

Class Profile Class Profile Instructions About the Class Profile Completing an analysis of student work gives you a clear picture of the strategies an individual student is applying to a particular problem or topic in mathematics. Such an analysis is even more powerful when it is applied to the Math Navigator class as a whole. The Class Profile gives you both. By reading the Class Profile across a row, you can see where each student stands at any point in time. Reading down the columns allows you to see the strengths and needs of the entire class at a glance. By reviewing the Class Profile, you will be able to make decisions that target appropriate instruction to individuals, small groups, and the whole Math Navigator class. The first pages of the Class Profile provide assessment items related to the content of the module. The last page is based on the mathematical practices from the Common Core State Standards for Mathematics. 1 On this page, record evidence of students using these practices. Recording Data on the Class Profile When you see either through discussion, analysis of student work, or direct observation that a student understands a concept, still has a misconception, or engages in a mathematical practice, make a note on your Class Profile. As the student s understanding increases, update the Class Profile. Using the Class Profile Review the Class Profile periodically during the lesson to help you decide which topics would be most beneficial for your students to focus on during the class discussion. Address topics that most of the students in the Math Navigator group need to learn during the show me, work time, or probing for understanding parts of the lesson. Address topics that only some students are struggling with during partner work or in conferences. If only one or two students need help with a topic, address the topic in an individual conference. Give a copy of the completed Class Profile to each student s classroom teacher at the end of the module. 1 Common Core State Standards Initiative. 2010. Common Core State Standards for Mathematics : 6 8. Accessed July 1, 2011. http://www.corestandards.org/assets/ccssi_math%20standards.pdf. Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 23

Class Profile (1 of 3) 10 9 8 7 6 5 4 3 2 1 Concepts Student Name Observed Errors C1: Understands that the two digits of a two digit number represent amounts of tens and ones C2: Understands that 10 can be thought of as a bundle of 10 ones called a ten C3: Understands that the numbers from 11 to 19 are composed of a ten and 1, 2, 3, 4, 5, 6, 7, 8, or 9 ones C4: Understands that the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to 1, 2, 3, 4, 5, 6, 7, 8, or 9 tens (and 0 ones) C5: Compares two-digit numbers based on meanings of the tens and ones digits C6: Understands that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten C7: Uses visual representations to represent addition and subtraction C8: Explains the reasoning used in computation strategies, including for mental computation Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 24

Class Profile (2 of 3) 10 9 8 7 6 5 4 3 2 1 Procedures Student Name P1: For any number from 1 to 9, finds the number thata makes 10 when added to the given number, e.g., by using objects or drawings, and records the answer with a drawing or equation P2: Uses a drawing or equation to record the composition or decomposition of numbers from 11 to 19 into 10 ones and some further ones P3: Counts to 120, starting at any number less than 120 P4: Reads and writes numerals to 120 and represents a number of objects with a written numeral P5: Uses the symbols >, =, and < to record the results of comparisons of two-digit numbers P6: Adds within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction P7: Given a two-digit number, mentally finds 10 more or 10 less than the number, without having to count; explain the reasoning used. P8: Subtracts multiples of 10 from multiples of 10 in the range 10-90 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction P9: Uses a written method to represent a computation strategy Observed Errors Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 25

Class Profile (3 of 3) 10 9 8 7 6 5 4 3 2 1 Student Name Observations Mathematical Practice Standards MP4: Model with mathematics. MP8: Look for and express regularity in repeated reasoning. MP3: Construct viable arguments and critique the reasoning of others. MP7: Look for and make use of structure. MP2: Reason abstractly and quantitatively. MP6: Attend to precision. MP1: Make sense of problems and persevere in solving them. MP5: Use appropriate tools strategically. Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 26

A Complete Solution to a Math Story Show your thinking. 1 + 2 = 3 Equation Math drawing or diagram Answer the question. There are 3 apples. Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 27

What to Do If You Get Stuck Look at posted charts. Use counters or other materials. Make a math drawing or diagram. Use easier numbers. Ask a friend. Write what you do know. Think of questions to ask later. Place Value and CompuTATIONAl STRATeGIes to 100 Copyright 2012 Pearson Education, Inc. or its affiliate(s). All rights reserved. 28