MIDDLETOWN PUBLIC SCHOOLS MATHEMATICS CURRICULUM Grade 1 · PDF fileGrade 1 Elementary School ... he Middletown Public Schools Mathematics Curriculum for grades K‐12 was revised

MIDDLETOWN PUBLIC SCHOOLS�

MATHEMATICSCURRICULUMGrade1�Elementary School�

Curriculum Writers: Alison Belcher, Anne Coogan, Camille Guerin, Laura Pasyanos, Christa Robinson REVISED June 2014

MATHEMATICS CURRICULUM Grade 1 Curriculum Writers: Alison Belcher, Anne Coogan, Camille Guerin, Laura Pasyanos, Christa Robinson

8/20/2014 Middletown Public Schools 1



he Middletown Public Schools Mathematics Curriculum for grades K‐12 was revised June2014 by a K‐12 team of teachers. The team, identified as the Mathematics Task Force and Mathematics Curriculum Writers referenced extensive resources to design the document that included:

o Common Core State Standards for Mathematics o Common Core State Standards for Mathematics, Appendix A o Understanding Common Core State Standards, Kendall o PARCC Model Content Frameworks o Numerous state curriculum Common Core frameworks, e.g. e.g. Ohio , Arizona, North Carolina, and New Jersey o High School Traditional Plus Model Course Sequence, Achieve, Inc. o Grade Level and Grade Span Expectations (GLEs/GSEs) for Mathematics o Third International Mathematics and Science Test (TIMSS) o Best Practice, New Standards for Teaching and Learning in America’s Schools; o Differentiated Instructional Strategies o Instructional Strategies That Work, Marzano o Goals for the district

The Middletown Public Schools Mathematics Curriculum identifies what students should know and be able to do in mathematics. Each grade or course includes Common Core State Standards (CCSS), Grade Level Expectations (GLEs), Grade Span Expectations (GSEs), grade level supportive tasks, teacher notes, best practice instructional strategies, resources, a map (or suggested timeline), rubrics, checklists, and common formative and summative assessments. The Common Core State Standards (CCSS):

o Are fewer, higher, deeper, and clearer. o Are aligned with college and workforce expectations. o Include rigorous content and applications of knowledge through high‐order skills. o Build upon strengths and lessons of current state standards (GLEs and GSEs). o Are internationally benchmarked, so that all students are prepared for succeeding in our global economy and society. o Are research and evidence‐based.

Common Core State Standards components include:

o Standards for Mathematical Practice (K‐12) o Standards for Mathematical Content:

o Categories (high school only): e.g. numbers, algebra, functions, data o Domains: larger groups of related standards o Clusters: groups of related standards o Standards: define what students should understand and are able to do

The Middletown Public Schools Common Core Mathematics Curriculum provides all students with a sequential comprehensive education in mathematics through the study of:

o Standards for Mathematical Practice (K‐12) o Make sense of problems and persevere in solving them o Reason abstractly and quantitatively o Construct viable arguments and critique the reasoning of others o Model with mathematics* o Use appropriate tools strategically o Attend to precision o Look for and make use of structure o Look for and express regularity in repeated reasoning

TMission Staement

Our mission is to provide a sequential and comprehensive K‐12 mathematics curriculum in a collaborative student

centered learning environment that develops critical thinkers, skillful problem solvers, and

effective communicators of mathematics.

COMMON CORE STATE STANDARDS



o Standards for Mathematical Content: o K – 5 Grade Level Domains of

Counting and Cardinality Operations and Algebraic Thinking Number and Operations in Base Ten Number and Operations – Fractions Measurement and Data Geometry

o 6‐8 Grade Level Domains of

Ratios and Proportional Relationships The Number System Expressions and Equations Functions Geometry

o 9‐12 Grade Level Conceptual Categories of

Number and Quantity Algebra Functions Modeling Geometry Statistics and Probability

The Middletown Public Schools Common Core Mathematics Curriculum provides a list of research‐based best practice instructional strategies that the teacher may model and/or facilitate. It is suggested the teacher:

o Use formative assessment to guide instruction o Provide opportunities for independent, partner and collaborative group work o Differentiate instruction by varying the content, process, and product and providing opportunities for:

o anchoring o cubing o jig‐sawing o pre/post assessments o tiered assignments

o Address multiple intelligences instructional strategies, e.g. visual, bodily kinesthetic, interpersonal o Provide opportunities for higher level thinking: Webb’s Depth of Knowledge, 2,3,4, skill/conceptual understanding, strategic reasoning, extended reasoning o Facilitate the integration of Mathematical Practices in all content areas of mathematics o Facilitate integration of the Applied Learning Standards (SCANS):

o communication o critical thinking o problem solving o reflection/evaluation o research

o Employ strategies of “best practice” (student‐centered, experiential, holistic, authentic, expressive, reflective, social, collaborative, democratic, cognitive, developmental, constructivist/heuristic, and

challenging) o Provide rubrics and models

RESEARCH‐BASED INSTRUCTIONAL STRATEGIES



o Address multiple intelligences and brain dominance (spatial, bodily kinesthetic, musical, linguistic, intrapersonal, interpersonal, mathematical/logical, and naturalist) o Employ mathematics best practice strategies e.g.

o using manipulatives o facilitating cooperative group work o discussing mathematics o questioning and making conjectures o justifying of thinking o writing about mathematics o facilitating problem solving approach to instruction o integrating content o using calculators and computers o facilitating learning o using assessment to modify instruction

The Middletown Public Schools Common Core Mathematics Curriculum includes common assessments. Required (red ink) indicates the assessment is required of all students e.g. common tasks/performance‐based tasks, standardized mid‐term exam, standardized final exam.

Required Assessments o Common Unit Assessment o Common Tasks o NWEA Test

Common Instructional Assessments (I) ‐ used by teachers and students during the instruction of CCSS. Common Formative Assessments (F) ‐ used to measure how well students are mastering the content standards before taking state assessments

o teacher and student use to make decisions about what actions to take to promote further learning o on‐going, dynamic process that involves far more frequent testing o serves as a practice for students o Common Summative Assessment (S) ‐ used to measure the level of student, school, or program success o make some sort of judgment, e.g. what grade o program effectiveness o e.g. state assessments (AYP), mid‐year and final exams

o Additional assessments include: o Anecdotal records o Conferencing o Exhibits o Interviews o Graphic organizers o Journals o Mathematical Practices o Modeling o Multiple Intelligences assessments, e.g.

Role playing ‐ bodily kinesthetic Graphic organizing ‐ visual Collaboration ‐ interpersonal

o Oral presentationso Problem/Performance based/common tasks o Rubrics/checklists (mathematical practice, modeling) o Tests and quizzes o Technology o Think‐alouds

COMMON ASSESSMENTS



Textbooks

Supplementary Exemplar EnVisions Problem Solver Technology Calculator Computers ELMO Interactive boards LCD projectors MIMIO Overhead calculator Smart board™

Websites AIMS Web http://illuminations.nctm.org/ http://ww.center.k12.mo.us/edtech/everydaymath.htm http://www.achieve.org/http://my.hrw.com http://www.discoveryeducation.com/ http://www.ode.state.oh.us/GD/Templates/Pages/ODE/ODEDefaultPage.aspx?page=1 http://www.parcconline.org/parcc‐content‐frameworks http://www.parcconline.org/sites/parcc/files/PARCC_Draft_ModelContentFrameworksForMathematics0.pdf www.commoncore.org/maps www.corestandards.org www.cosmeo.com www.explorelearning.com (Gizmo™) www.fasttmath.com www.glencoe.com www.khanacademy.com www.mathforum.org www.phschool.com www.ride.ri.gov www.studyIsland www.successnet.com Materials Attribute blocks 2‐sided colored chips 3_d shapes Base 10 blocks Clock Colored chips Dice Expo markers Fact triangle Number grid Number line Pattern blocks Rulers Sticks Student white boards Templates Ten‐frames Unifix cubes

RESOURCES FOR MATHEMATICS CURRICULUM Grade 1



DOMAINS UNIT

CLUSTERS AND STANDARDSMiddletown Public Schools

INSTRUCTIONALSTRATEGIES

RESOURCES ASSESSMENTS

OPERATIONS AND ALGEBRAIC THINKING

(1.OA) Use Mathematical Practices to 1. Make sense of problems and

persevere in solving them 2. Reason abstractly and

quantitatively 3. Construct viable arguments

and critique the reasoning of others

4. Model with mathematics 5. Use appropriate tools

strategically 6. Attend to precision 7. Look for and make use of

structure 8. Look for and express

regularity in repeated reasoning

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Students represent and solve problems involving addition and subtraction. Major content

1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Major content

TEACHER NOTESSee instructional strategies in the introduction Collaborate in small groups to develop problem‐solving strategies using a variety of models such as drawings, words, and equations with symbols for the unknown numbers to find the solutions. Additionally students need the opportunity to explain, write and reflect on their problem‐solving strategies. The situations for the addition and subtraction story problems should involve sums and differences less than or equal to 20 using the numbers 0 to 20. They need to align with the 12 situations found in Table 1 of the Common Core State Standards (CCSS) for Mathematics. Students need the opportunity of writing and solving story problems involving three addends with a sum that is less than or equal to 20. For example, each student writes or draws a problem in which three whole things are being combined. The students exchange their problems with other students, solving them individually and then discussing their models and solution strategies. Now both students work together to solve each problem using a different strategy. Literature is a wonderful way to incorporate problem‐solving in a context that young students can understand. Many literature books that include mathematical ideas and concepts have been written in recent years. For Grade 1, the incorporation of books that contain a problem situation involving addition and subtraction with numbers 0 to 20 should be included in

RESOURCE NOTES See resources in the introduction Refer to Algebra I PBA/MYA @ Live Binder http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements and clarification ORC # 2807 From the International Reading Association, National Council of Teachers of English and Verizon Thinkfinity: Giant story problems: Reading comprehension through math problem‐solving

Using drawings, equations, and written responses, students work cooperatively in two class sessions to solve Giant Story Problems while they gain practice in reading for information.

ASSESSMENT NOTES See assessments in the introduction REQUIRED ASSESSMENTS Pre and Post Test Common Unit Assessment

Common Tasks NWEA Test



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Essential Knowledge and skills Addition finds the missing whole when given the parts,

and subtraction finds a missing part when given the whole and a part.

Word problem situations can be: add‐to, take‐from, put‐together/take‐apart, and compare (see Table below).

Examples Take From example: Abel has 9 balls. He gave 3 to

Susan. How many balls does Abel have now?

Nine bunnies were sitting on the grass. Some more

bunnies hopped there. Now, there are 13 bunnies on the grass. How many bunnies hopped over there? Student: Niiinnneee…. holding a finger for each next number counted 10, 11, 12, Holding up her four fingers, 4! 4 bunnies hopped over there.” (Counting On Method)

8 red apples and 6 green apples are on the tree. How many apples are on the tree? Student: I broke up 6 into 2 and 4. Then, I took the 2 and added it to the 8. That’s 10. Then I add the 4 to the 10. That’s 14. So there are 14 apples on the tree. (Making Tens Method)

13 apples are on the table. 6 of them are red and the rest are green. How many apples are green? Student: I know that 6 and 6 is 12. So, 6 and 7 is 13. There are 7 green apples. (Doubles +/‐ 1 or 2)

Academic vocabulary Adding Adding to Addition Comparing

Describe (+) symbol) Equal to, minus Less than Putting together Subtract Subtraction Sum Taking apart Taking from The same amount as Ten‐frame Unknown(s) Mathematical Practices 1. Make sense of

problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Look for and express regularity in repeated reasoning

Assessment Problems:

1.OA.2 Solve word problems that call for addition of three whole numbers

whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Major content

the curriculum. Use the situations found in Table 1 of the CCSS for guidance in selecting appropriate books. As the teacher reads the story, students use a variety of manipulatives, drawings, or equations to model and find the solution to problems from the story. ODE Common misconception A second misconception that many students have is that it is valid to assume that a key word or phrase in a problem suggests the same operation will be used every time. For example, they might assume that the word left always means that subtraction must be used to find a solution. Providing problems in which key words like this are used to represent different operations is essential. For example, the use of the word left in this problem does not indicate subtraction as a solution method: Seth took the 8 stickers he no longer wanted and gave them to Anna. Now Seth has 11 stickers left. How many stickers did Seth have to begin with? Students need to analyze word problems and avoid using key words to solve them. ODE



Essential Knowledge and skills Students solve multi‐step word problems by adding

(joining) three numbers whose sum is less than or equal to 20, using objects, drawings, equations with symbols.

Examples Mrs. Smith has 4 oatmeal raisin cookies, 5 chocolate chip

cookies, and 6 gingerbread cookies. How many cookies does Mrs. Smith have? Student A: I put 4 counters on the Ten Frame for the oatmeal raisin cookies. Then, I put 5 different color counters on the ten‐frame for the chocolate chip cookies. Then, I put another 6 color counters out for the gingerbread cookies. Only one of the gingerbread cookies fit, so I had 5 leftover. Ten and five more makes 15 cookies. Mrs. Smith has 15

Student B: I used a number line. First I jumped to 4, and then I jumped 5 more. That’s 9. I broke up 6 into 1 and 5 so I could jump 1 to make 10. Then, I jumped 5 more and got 15. Mrs. Smith has 15 cookies.

Student C: I wrote: 4 + 5 + 6 = �. I know that 4 and 6 equals 10, so the oatmeal raisin and gingerbread equals 10 cookies. Then I added the 5 chocolate chip cookies. 10 and 5 is 15. So, Mrs. Smith has 15 cookies.

Academic vocabulary Drawings Equal to Equations How many

altogether How many are left How many in all Less than Missing number Objects Objects Sum Symbol Unknown number Word problem Mathematical Practices 1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically






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Students understand and apply properties of operations and the relationshipbetween addition and subtraction. Major content 1. OA. 3 Apply properties of operations as strategies to add and subtract.. Major content

Essential Knowledge and skills Addition and subtraction are connected. Addition names the whole in terms of the parts, and subtraction names a missing part.

Academic vocabulary First Order Second

TEACHER NOTES See instructional strategies in the introduction Students need not use formal terms for these properties. One focus in this cluster is for students to discover and apply the commutative and associative properties as strategies

RESOURCE NOTES See resources in the introduction Refer to Algebra I PBA/MYA @ Live Binder









Subtraction can be done using addition, through finding the missing addend.

Examples If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.)

To add 2 + 6 + 4, the second two numbers can be added to make

a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

Cubes A student uses 2 colors of cubes to make as many different combinations of 8 as possible. When recording the combinations, the student records that 3 green cubes and 5 blue cubes equals 8 cubes in all. In addition, the student notices that 5 green cubes and 3 blue cubes also equals 8 cubes.

Number Balance A student uses a number balance to investigate the commutative property. “If 8 and 2 equals 10, then I think that if I put a weight on 2 first this time and then on 8, it’ll also be 10.”

Associative Property Examples: Number Line: � = 5 + 4 + 5 Student A: First I jumped to 5. Then, I jumped 4 more, so I landed on 9. Then I jumped 5 more and landed on 14.

Student B: I got 14, too, but I did it a different way. First I jumped to 5. Then, I jumped 5 again. That’s 10. Then, I jumped 4 more. See, 14!

Mental Math: There are 9 red jelly beans, 7 green jelly

Strategies Mathematical Practices 2. Reason abstractly and quantitatively 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

for solving addition problems. Students do not need to learn the names for these properties. It is important for students to share, discuss and compare their strategies as a class. The second focus is using the relationship between addition and subtraction as a strategy to solve unknown‐addend problems. Students naturally connect counting on to solving subtraction problems. For the problem “15 – 7 = ?” they think about the number they have to add to 7 to get to 15. First graders should be working with sums and differences less than or equal to 20 using the numbers 0 to 20. Provide investigations that require students to identify and then apply a pattern or structure in mathematics. For example, pose a string of addition and subtraction problems involving the same three numbers chosen from the numbers 0 to 20, like 4 + 13 = 17 and 13 + 4 = 17. Students analyze number patterns and create conjectures or guesses. Have students choose other combinations of three numbers and explore to see if the patterns work for all numbers 0 to 20. Students then share and discuss their reasoning. Be sure to highlight students’ uses of the commutative and associative properties and the relationship between addition and subtraction. ODE

http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements and clarification Dot Card and Ten Frame Activities (pp. 9‐11, 21‐24, 26‐30, 32‐37) Numeracy Project, Winnipeg School Division, 2005‐2006

ORC # 3992 From the National Council of Teachers of Mathematics: Balancing Equations In this lesson, students imitate the action of a pan balance and record the modeled subtraction facts in equation form.

ORC # 3978 From the National Council of Teachers of Mathematics: How Many Left? This lesson encourages the students to explore unknown‐addend problems using the set model and the game Guess




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beans, and 3 black jelly beans. How many jelly beans are there in all? Student: “I know that 7 + 3 is 10. And 10 and 9 is 19. There are 19 jelly beans.”


1. OA. 4 Understand subtraction as an unknown‐addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. Major content

Essential Knowledge and skills Subtracting a whole from a part is not the same as

subtracting a part from a whole. (7 – 3 is not the same as 3 – 7.)

By understanding the relationship between addition and subtraction, First Graders are able to use various strategies described below to solve subtraction problems.

Examples For Sums to 10 Think‐Addition:

Think‐Addition uses known addition facts to solve for the unknown part or quantity within a problem. When students use this strategy, they think, “What goes with this part to make the total?” The think‐addition strategy is particularly helpful for subtraction facts with sums of 10 or less and can be used for sixty‐four of the 100 subtraction facts. Therefore, in order for think‐addition to be an effective strategy, students must have mastered addition facts first. For example, when working with the problem 9 ‐ 5 = �, First Graders think “Five and what makes nine?”, rather than relying on a counting approach in which the student counts 9, counts off 5, and then counts what’s left. When subtraction is presented in a way that encourages students to think using addition, they use known addition facts to solve a problem. o Example: 10 – 2 = � o Student: “2 and what make 10? I know that 8 and 2

make 10. So, 10 – 2 = 8.” For Sums Greater than 10 The 36 facts that have sums greater than 10 are often

considered the most difficult for students to master. Many students will solve these particular facts with Think‐Addition (described above), while other students may use other strategies described below, depending on the fact.

Academic vocabulary Addend Unknown Mathematical Practices 2. Reason abstractly and quantitatively 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning



Regardless of the strategy used, all strategies focus on therelationship between addition and subtraction and often use 10 as a benchmark number.

Build Up Through 10: This strategy is particularly helpful when one of the numbers to be subtracted is 8 or 9. Using 10 as a bridge, either 1 or 2 are added to make 10, and then the remaining amount is added for the final sum. o Example: 15 – 9 = � o Student A: “I’ll start with 9. I need one more to make

10. Then, I need 5 more to make 15. That’s 1 and 5‐ so it’s 6. 15 – 9 = 6.”

o Student B: “I put 9 counters on the 10 frame. Just looking at it I can tell that I need 1 more to get to 10. Then I need 5 more to get to 15. So, I need 6 counters.”

Back Down Through 10 This strategy uses take‐away and 10 as a bridge. Students take away an amount to make 10, and then take away the rest. It is helpful for facts where the ones digit of the two‐digit number is close to the number being subtracted. o Example: 16 – 7 = � o Student A: “I’ll start with 16 and take off 6. That makes

10. I’ll take one more off and that makes 9. 16 – 7 = 9.” o Student B: “I used 16 counters to fill one ten frame

completely and most of the other one. Then, I can take these 6 off from the 2nd ten frame. Then, I’ll take one more from the first ten frame. That leaves 9 on the ten frame.”



(1.OA)

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Students add and subtract within 20. Major content 1. OA. 5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

Major content

TEACHER NOTES See instructional strategies in the introduction

RESOURCE NOTES See resources in the introduction

ASSESSMENT NOTES See assessments in the introduction



Use Mathematical Practices to 1. Make sense of problems and








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Essential Knowledge and skills When solving addition and subtraction problems to 20:

o use counting strategies, such as counting all, counting on, and counting back, before fully developing the essential strategy of using 10 as a benchmark number.

o move students toward strategies that focus on composing and decomposing number using ten

Examples Counting All: Students count all objects to determine the

total amount. Counting On & Counting Back: Students hold a “start

number” in their head and count on/back from that number.

Example: 15 + 2 = � o Counting All: The student counts out fifteen counters.

The student adds two more counters. The student then counts all of the counters starting at 1 (1, 2, 3, 4,…14, 15, 16, 17) to find the total amount.

o Counting On: Holding 15 in her head, the student holds up one finger and says 16, then holds up another finger and says 17. The student knows that 15 + 2 is 17, since she counted on 2 using her fingers.

Example: 12 – 3 = � o Counting All: The student counts out twelve counters.

The student then removes 3 of them. To determine the final amount, the student counts each one (1,2, 3, 4, 5, 6, 7, 8, 9) to find out the final amount.

o Counting Back: Keeping 12 in his head, the student

counts backwards, “11” as he holds up one finger; says “10” as he holds up a second finger; says “9” as he holds up a third finger. Seeing that he has counted back 3 since he is holding up 3 fingers, the student states that 12 – 3 = 9.

Academic vocabulary Addition Composing Counting all Counting back Counting on Decomposing Subtraction Mathematical Practices 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning


1. OA. 6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as:

counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10

– 1 = 9); using the relationship between addition and subtraction (e.g.,

knowing that 8 + 4 = 12, one knows 12 – 8= 4); creating equivalent but easier or known sums (e.g., adding 6 +7 by

Provide many experiences for students to construct strategies to solve the different problem types illustrated in Table 1 in the Common Core State Standards on page 88. These experiences should help students combine their procedural and conceptual understandings. Have students invent and refine their strategies for solving problems involving sums and differences less than or equal to 20 using the numbers 0 to 20. Ask them to explain and compare their strategies as a class. Provide multiple and varied experiences that will help students develop a strong sense of numbers based on comprehension – not rules and procedures. Number sense is a blend of comprehension of numbers and operations and fluency with numbers and operations. Students gain computational fluency (using efficient and accurate methods for computing) as they come to understand the role and meaning of arithmetic operations in number systems. Primary students come to understand addition and subtraction as they connect counting and number sequence to these operations. Addition and subtraction also involve part to whole relationships. Students’ understanding that the whole is made up of parts is connected to decomposing and composing numbers. ODE Provide numerous opportunities for students to use the counting on strategy for solving addition and subtraction problems. For example, provide a ten frame showing 5 colored dots in one row. Students add 3 dots of a different color to the next row and write 5 + 3. Ask students to count on from 5 to find the total number of dots. Then have them add an equal sign and the number eight to 5 + 3 to form the

Refer to Algebra I PBA/MYA @ Live Binder http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements and clarification Five‐frame and Ten‐

frame

A variety of objects for counting

A variety of objects for modeling and solving addition and subtraction problems

Mentoring Minds 1.OA.5 and 1.OA.6

REQUIRED ASSESSMENTS Pre and Post Test Common Unit

Assessment Common Tasks NWEA Test



creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). Major content Essential Knowledge and skills To solve addition and subtraction problem, students must

study patterns and relationships learning the skill and performing flexibility, accurately and efficiently, in addition to the relationship between addition and subtraction.

Examples Developing Fluency for Addition & Subtraction within 10 Example: Two frogs were sitting on a log. 6 more frogs

hopped there. How many frogs are sitting on the log now? o Counting‐ On: I started with 6 frogs and then counted

up, Sixxxx…. 7, 8. So there are 8 frogs on the log. 6 + 2 = 8

o Internalized Fact: There are 8 frogs on the log. I know this because 6 plus 2 equals 8.

6 + 2 = 8 Add and Subtract within 20 Example: Sam has 8 red marbles and 7 green marbles. How

many marbles does Sam have in all? o Making 10 and Decomposing a Number: I know that 8

plus 2 is 10, so I broke up (decomposed) the 7 up into a 2 and a 5. First I added 8 and 2 to get 10, and then added the 5 to get 15.

7 = 2 + 5 8 + 2 = 10 10 + 5 = 15

o Creating an Easier Problem with Known Sums I broke up (decomposed) 8 into 7 and 1. I know that 7 and 7 is 14. I added 1 more to get 15.

8 = 7 + 1 7 + 7 = 14 14 + 1 = 15

Example: There were 14 birds in the tree. 6 flew away. How many birds are in the tree now? o Back Down Through Ten I know that 14 minus 4 is 10.

So, I broke the 6 up into a 4 and a 2. 14 minus 4 is 10. Then I took away 2 more to get 8.

6 = 4 + 2 14 – 4 = 10 10 – 2 = 8

Relationship between Addition & Subtraction: I thought, ‘6 and what makes 14?’. I know that 6 plus 6 is 12 and two more is 14. That’s 8 altogether. So, that means that 14 minus 6 is 8.

6 + 8 = 14 14 – 6 = 8

Academic vocabulary Addition Equivalent Fluency Patterns in addition

and subtraction Subtraction Mathematical Practices 2. Reason abstractly and quantitatively 6. Look for and make

use of structure 7. Look for and

express regularity in repeated reasoning


equation 5 + 3 = 8. Ask students to verbally explain how counting on helps to add one part to another part to find a sum. Discourage students from inventing a counting back strategy for subtraction because it is difficult and leads to errors. ODE












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Students work with addition and subtraction equations. Major content 1. OA. 7 Understand the meaning of the equal sign, and determine if equations

involving addition and subtraction are true or false. Major content

Essential Knowledge and skills Determine whether an equation is true or false Understand the meaning of the equal sign Examples For example which of the following equations are true and

which are false? o 6 = 6 o 7 = 8 – 1 o 5 + 2 = 2 + 5 o 4 + 1 = 5 + 2

When students understand that an equation needs to “balance”, with equal quantities on both sides of the equal sign, they understand various representations of equations, such as: o an operation on the left side of the equal sign and

the answer on the right side (5 + 8 = 13) o an operation on the right side of the equal sign and

the answer on the left side (13 = 5 + 8) o numbers on both sides of the equal sign (6 = 6) o operations on both sides of the equal sign (5 + 2 = 4 +

3). Once students understand the meaning of the equal sign, they are able to determine if an equation is true (9 = 9) or false (9 = 8).

Academic vocabulary Equal Balance equal sign Equation Equivalent false Joining Quantity Relationship Same amount Separating Symbols (+, ‐, =) The same

amount/quantity as True Mathematical Practices 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 6. Attend to precision 7. Look for and make use of structure


1. OA. 8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. Major content

Essential Knowledge and skills They should be exposed to problems with the unknown in

different positions. Examples For example, determine the unknown number that makes

the equation true in each of the equations

Academic vocabulary Mathematical

TEACHER NOTES See instructional strategies in the introduction Provide opportunities for students use objects of equal weight and a number balance to model equations for sums and differences less than or equal to 20 using the numbers 0 to 20. Give students equations in a variety of forms that are true and false. Include equations that show the identity property, commutative property of addition, and associative property of addition. Students need not use formal terms for these properties. 13 = 13 Identity Property 8 + 5 = 5 + 8 Commutative Property

for Addition 3 + 7 + 4 = 10 + 4 Associative

Property for Addition Ask students to determine whether the equations are true or false and to record their work with drawings. Students then compare their answers as a class and discuss their reasoning. Present equations recorded in a nontraditional way, like 13 = 16 – 3 and 9 + 4 = 18 – 5, then ask, “Is this true?”. Have students decide if the equation is true or false. Then as a class, students discuss their thinking that supports their answers. ODE Provide situations relevant to first graders for these problem types illustrated in Table 1 of the Common Core State Standards (CCSS): Add to / Result Unknown, Take from / Start Unknown, and Add to / Result Unknown. Demonstrate how students can use graphic organizers such as the Math Mountain to help them think about problems. The Math Mountain shows a sum with diagonal lines going down to connect with the two addends, forming a triangular shape. It shows two known

RESOURCE NOTES See resources in the introduction Refer to Algebra I PBA/MYA @ Live Binder http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements and clarification ORC # 4321 From the National Council of Teachers of Mathematics: Finding the Balance This lesson encourages students to explore another model of subtraction, the balance. Students will use real and virtual balances. Students also explore recording the modeled subtraction facts in equation form. Click on Pan Balance – Shapes to get to the online tool Pan Balance – Numbers. This virtual tool can be used to strengthen students’ understanding and computation of numerical expressions and equality.

ASSESSMENT NOTES See assessments in the introduction REQUIRED ASSESSMENTS Pre and Post Test Common Unit




o 8 +? = 11 o 5 = – 3 o 6 + 6 =

Examples Five cookies were on the table. I ate some cookies. Then

there were 3 cookies. How many cookies did I eat? o Student A: What goes with 3 to make 5? 3 and 2 is 5.

So, 2 cookies were eaten. o Student B: Fiiivee, four, three (holding up 1 finger for

each count). 2 cookies were eaten (showing 2 fingers).

o Student C: We ended with 3 cookies. Threeeee, four, five (holding up 1 finger for each count). 2 cookies were eaten (showing 2 fingers).

Example: Determine the unknown number that makes the equation true. 5 ‐ � = 2 o Student: 5 minus something is the same amount as 2.

Hmmm. 2 and what makes 5? 3! So, 5 minus 3 equals 2. Now it’s true!

Practices2. Reason abstractly and quantitatively 6. Attend to precision 7. Look for and make use of structure


quantities and one unknown quantity. Use various symbols, such as a square, to represent an unknown sum or addend in a horizontal equation. For example, here is a Take from / Start Unknown problem situation such as: Some markers were in a box. Matt took 3 markers to use. There are now 6 markers in the box. How many markers were in the box before? The teacher draws a square to represent the unknown sum and diagonal lines to the numbers 3 and 6.

Have students practice using the Math Mountain to organize their solutions to problems involving sums and differences less than or equal to 20 with the numbers 0 to 20. Then ask them to share their reactions to using the Math Mountain. ODE

NUMBER AND OPERATIONS IN BASE

TEN (1.NBT)









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Students extend the counting sequence. Major content 1.NBT.1 Count to 120, starting at any number less than 120. In this range, read

and write numerals and represent a number of objects with a written numeral. Major content

Essential Knowledge and skills Quantities can be represented by a written numeral. Counting can begin at any number and go forward or

backward. There are patterns in numbers. Some patterns of the count sequence make counting

predictable. Examples Students use objects, words, and/or symbols to express

their understanding of numbers. They extend their counting beyond 100 to count up to 120 by counting by 1s.

Some students may begin to count in groups of 10 (while other students may use groups of 2s or 5s to count). Counting in groups of 10, as well as grouping objects into 10 groups of 10, will develop students’ understanding of place value concepts.

Students extend reading and writing numerals beyond 20

Academic vocabulary Number words Numerals Represent Sequence Mathematical Practices 2. Reason abstractly and quantitatively 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

TEACHER NOTES See instructional strategies in the introduction

In this grade, students build on their counting to 100 by ones and tens beginning with numbers other than 1 as they learned in Kindergarten. Students can start counting at any number less than 120 and continue to 120. It is important for students to connect different representations for the same quantity or number. Students use materials to count by ones and tens to a build models that represent a number, then they connect this model to the number word and its representation as a written numeral. Students learn to use numerals to represent numbers by relating their place‐value notation to their models. They build on their experiences with

RESOURCE NOTES See resources in the introduction Refer to Algebra I PBA/MYA @ Live Binder http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements and clarification Mentoring Minds 1.NBT.1

Place value arrow cards

Quick draw multiples

Pregrouped materials





to 120. After counting objects, students write the numeral or use numeral cards to represent the number. Given a numeral, students read the numeral, identify the quantity that each digit represents using numeral cards, and count out the given number of objects.

Students should experience counting from different

starting points (e.g., start at 83; count to 120). To extend students’ understanding of counting, they

should be given opportunities to count backwards by ones and tens.

They should also investigate patterns in the base 10 system.


numbers 0 to 20 in Kindergarten to create models for 21 to 120 with groupable and pregroupable materials (see Resources/Tools). Students represent the quantities shown in the models by placing numerals in labeled hundreds, tens and ones columns. They eventually move to representing the numbers in standard form, where the group of hundreds, tens, then singles shown in the model matches the left‐to‐right order of digits in numbers.

Base‐ten blocks Ten‐frame Hundreds chart and Blank hundreds chart

Groupable models Unifex cubes SUGGESTED LITERATURE Wake Up City, Alvin Tresselt

A Fair Bear Share, Stuart Murphy

Shark Samantha, Stuart J. Murphy

One Hundred Hungry Ants, Elinor Pinczes

Mike’s Kite, Elizabeth MacDonald


TEN (1.NBT)









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Students understand place value. Major content 1.NBT.2 Understand that the two digits of a two‐digit number represent amounts

of tens and ones. Understand the following as special cases: a. 10 can be thought of as a bundle of ten ones — called a “ten.” 1.NBT.2a

b. The numbers from 11 to 19 are composed of a ten and one, two,

three, four, five, six, seven, eight, or nine ones. 1.NBT.2b

c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones). 1.NBT.2c

Major content Essential Knowledge and skills The position of digits in numbers determines the value

they represent (which size group they count). Two‐digit numbers can be decomposed into a unit of ten

ones and some more ones. Examples

First Grade students are introduced to the idea that a bundle of ten ones is called “a ten”. This is known as unitizing. When First Grade students unitize a group of ten ones as a whole unit (“a ten”), they are able to count groups as though they were individual objects. For example, 4 trains of ten cubes each have a value of 10 and

Academic vocabulary Bundle Digit Equal to Greater/less than Groups Left‐overs Ones Singles Tens Value

TEACHER NOTES See instructional strategies in the introduction Essential skills for students to develop include making tens (composing) and breaking a number into tens and ones (decomposing). Composing numbers by tens is foundational for representing numbers with numerals by writing the number of tens and the number of leftover ones. Decomposing numbers by tens builds number sense and the awareness that the order of the digits is important. Composing and decomposing numbers involves number relationships and promotes flexibility with mental computation. The beginning concepts of place value are developed in Grade 1 with the understanding of ones and tens. The major concept is that putting ten ones together makes a ten and that there is a way to write that down so the same

RESOURCE NOTES See resources in the introduction Refer to Algebra I PBA/MYA @ Live Binder http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements and clarification National Library of Virtual Manipulatives Base Block (Adjust the application to only deal with ones and tens)





would be counted as 40 rather than as 4. This is a monumental shift in thinking, and can often be challenging for young children to consider a group of something as “one” when all previous experiences have been counting single objects. This is the foundation of the place value system and requires time and rich experiences with concrete manipulatives to develop.

A student’s ability to conserve number is an important aspect of this standard. It is not obvious to young children that 42 cubes is the same amount as 4 tens and 2 left‐overs. It is also not obvious that 42 could also be composed of 2 groups of 10 and 22 leftovers. Therefore, first graders require ample time grouping proportional objects (e.g., cubes, beans, beads, ten‐frames) to make groups of ten, rather than using pre‐grouped materials (e.g., base ten blocks, pre‐made bean sticks) that have to be “traded” or are non‐proportional (e.g., money).

Example: 42 cubes can be grouped many different ways and still remain a total of 42 cubes.

“We want children to construct the idea that all of these are the same and that the sameness is clearly evident by virtue of the groupings of ten. Groupings by tens is not just a rule that is followed but that any grouping by tens, including all or some of the singles, can help tell how many.” (Van de Walle & Lovin, p. 124)

As children build this understanding of grouping, they move through several stages: Counting By Ones; Counting by Groups & Singles; and Counting by Tens and Ones. Counting By Ones: At first, even though First Graders will have grouped objects into tens and left‐overs, they rely on counting all of the individual cubes by ones to determine the final amount. It is seen as the only way to determine how many. Example:

Mathematical Practices 2. Reason abstractly and quantitatively 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

number is always understood. Students move from counting by ones, to creating groups and ones, to tens and ones. It is essential at this grade for students to see and use multiple representations of making tens using base‐ten blocks, bundles of tens and ones, and ten‐frames. Making the connections among the representations, the numerals and the words are very important. Students need to connect these different representations for the numbers 0 to 99.



Counting By Groups and Singles: While students are able to group objects into collections of ten and now tell how many groups of tens and left‐overs there are, they still rely on counting by ones to determine the final amount. They are unable to use the groups and left‐overs to determine how many. Example:

Counting by Tens & Ones: Students are able to group objects into ten and ones, tell how many groups and leftovers there are, and now use that information to tell how many. Ex: “I have 3 groups of ten and 4 left‐overs. That means that there are 34 cubes in all.” Occasionally, as this stage is becoming fully developed, first graders rely on counting by ones to “really” know that there are 34, even though they may have just counted the total by groups and left‐over Example:



First Grade students extend their work from Kindergarten when they composed and decomposed numbers from 11 to 19 into ten ones and some further ones. In Kindergarten, everything was thought of as individual units: “ones”. In First Grade, students are asked to unitize those ten individual ones as a whole unit: “one ten”. Students in first grade explore the idea that the teen numbers (11 to 19) can be expressed as one ten and some leftover ones. Ample experiences with a variety of groupable materials that are proportional (e.g., cubes, links, beans, beads) and ten frames help students develop this concept. Example: Here is a pile of 12 cubes. Do you have enough to

make a ten? Would you have any leftover? If so, how many leftovers would you have? Student A o I filled a ten frame to make one ten and had two

counters left over. o I had enough to make a ten with some leftover. o The number 12 has 1 ten and 2 ones

Student B o I counted out 12 cubes. I had enough to make 10. I

now have 1 ten o and 2 cubes left over. So the number 12 has 1 ten

and 2 ones. o In addition, when learning about forming groups of

10, First Grade students learn that a numeral can stand for

o many different amounts, depending on its position or place in a number. This is an important realization as young

o children begin to work through reversals of digits, particularly in the teen numbers.

Example: Comparing 19 to 91

First Grade students apply their understanding of groups of ten as stated in 1.NBT.2b to decade numbers (e.g. 10, 20, 30, 40). As they work with groupable objects, first grade students understand that 10, 20, 30…80, 90 are comprised of a certain



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amount of groups of tens with none left‐over.Assessment Problems:

1.NBT.3 Compare two two‐digit numbers based on meanings of the tens and ones

digits, recording the results of comparisons with the symbols >, =, and <. Major content

Essential Knowledge and skills Use understanding of groups and order of digits to

compare two numbers by examining the amount of tens and ones in each number. Connect the vocabulary to the symbols: greater than (>), less than (<), equal to (=).

Examples Compare these two numbers. 42 __ 45

o Student A: 42 has 4 tens and 2 ones. 45 has 4 tens and 5 ones. They have the same number of tens, but 45 has more ones than 42. So, 42 is less than 45.

42 < 45 o Student B: 42 is less than 45. I know this because

when I count up I say 42 before I say 45. 42 < 45 This says 42 is less than 45.

Academic vocabulary Compare Digit Greater than Less than Ones Place value Symbols Tens <, >, = Mathematical Practices 2. Reason abstractly and quantitatively 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning



TEN (1.NBT)






strategically 6. Attend to precision

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Students use place value understanding and properties of operations to addand subtract. Major content 1.NBT.4 Add 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 concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two‐digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. Major content

Essential Knowledge and skills Academic vocabulary

TEACHER NOTES See instructional strategies in the introduction Students should solve problems using concrete models and drawings to support and record their solutions. It is important for them to share the reasoning that supports their solution strategies with their classmates. Students will usually move to using base‐ten concepts, properties of operations, and the relationship between addition and subtraction to

RESOURCE NOTES See resources in the introduction Refer to Algebra I PBA/MYA @ Live Binder http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements and clarification





7. Look for and make use of structure


Students extend their number fact and place value strategies to add within 100. They represent a problem situation using any combination of words, numbers, pictures, physical objects, or symbols.

Also, students should be able to apply their place value skills to decompose numbers. o For example, 17 + 12 can be thought of 1 ten and 7

ones plus 1 ten and 2 ones. Numeral cards may help students decompose the numbers into 10s and 1s.

Students should be exposed to problems both in and out of context and presented in horizontal and vertical forms.

As students are solving problems, it is important that they use language associated with proper place value (see example). They should always explain and justify their mathematical thinking both verbally and in a written format.

Examples Example: 24 red apples and 8 green apples are on the

table. How many apples are on the table? o Student A: I used ten frames. I put 24 chips on 3 ten

frames. Then, I counted out 8 more chips. 6 of them filled up the third ten frame. That meant I had 2 left over. 3 tens and 2 left over. That’s 32.

o So, there are 32 apples on the table.

o Student B: I used an open number line. I started at

24. I knew that I needed 6 more jumps to get to 30. So, I broke apart 8 into 6 and 2. I took 6 jumps to land on 30 and then 2 more. I landed on 32. So, there are 32 apples on the table.

o Student C: I turned 8 into 10 by adding 2 because it’s

easier to add. So, 24 and ten more is 34. But, since I added 2 extra, I had to take them off again. 34 minus 2 is 32. There are 32 apples on the table.

Example: 63 apples are in the basket. Mary put 20 more

apples in the basket. How many apples are in the basket? o Student A: I used ten frames. I picked out 6 filled ten

frames. That’s 60. I got the ten frame with 3 on it.

100 chart Add subtract Number line Ones Operations Place value Relationship Tens Mathematical Practices 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

invent mental and written strategies for addition and subtraction. Help students share, explore, and record their invented strategies. Recording the expressions and equations in the strategies horizontally encourages students to think about the numbers and the quantities they represent. Encourage students to try the mental and written strategies created by their classmates. Students eventually need to choose efficient strategies to use to find accurate solutions. ODE Students should use and connect different representations when they solve a problem. They should start by building a concrete model to represent a problem. This will help them form a mental picture of the model. Now students move to using pictures and drawings to represent and solve the problem. If students skip the first step, building the concrete model, they might use finger counting to solve the problem. Finger counting is an inefficient strategy for adding within 100 and subtracting within multiples of 10 between10 and 90. Have students connect a 0‐99 chart or a 1‐100 chart to their invented strategy for finding 10 more and 10 less than a given number. Ask them to record their strategy and explain their reasoning. Provide multiple and varied experiences that will help students develop a strong sense of numbers based on comprehension – not rules and procedures. Number sense is a blend of comprehension of numbers and operations and fluency with numbers and operations. Students gain computational fluency (using efficient and accurate methods for computing) as they come to understand the role and meaning of arithmetic operations in number systems. ODE



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That’s 63. Then, I picked one more filled ten frame for part of the 20 that Mary put in. That made 73. Then, I got one more filled ten frame to make the rest of the 20 apples from Mary. That’s 83. So, there are 83 apples in the basket.

o Student B: I used a hundreds chart. I started at 63

and jumped down one row to 73. That means I moved 10 spaces. Then, I jumped down one more row (that’s another 10 spaces) and landed on 83. So, there are 83 apples in the basket.

o Student C: I knew that 10 more than 63 is 73. And

10 more than 73 is 83. So, there are 83 apples in the basket.


1.NBT.5 Given a two‐digit number, mentally find 10 more or 10 less than the

number, without having to count; explain the reasoning used. Major content

Essential Knowledge and skills Academic vocabulary



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This standard requires students to understand and apply the concept of 10 which leads to future place value concepts.

It is critical for students to do this without counting. Prior use of models such as base ten blocks, number lines, and 100s charts helps facilitate this understanding.

It also helps students see the pattern involved when adding or subtracting

Examples There are 74 birds in the park. 10 birds fly away. How

many birds are in the park now? o Student A: I thought about a number line. I started

at 74. Then, because 10 birds flew away, I took a leap of 10. I landed on 64. So, there are 64 birds left in the park.

o Student B: I pictured 7 ten frames and 4 left over in

my head. Since 10 birds flew away, I took one of the ten frames away. That left 6 ten frames and 4 left over. So, there are 64 birds left in the park.

o Student C: I know that 10 less than 74 is 64. So there

are 64 birds in the park.

10 less 10 more Mentally Mathematical Practices 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning


1.NBT.6 Subtract multiples of 10 in the range 10‐90 from multiples of 10 in the

range 10‐90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Major content

Essential Knowledge and skills Use concrete models, drawings and place value strategies

to subtract multiples of 10 from decade numbers (e.g., 30, 40, 50). They often use similar strategies as discussed in 1.OA.4.

Examples Example: There are 60 students in the gym. 30 students

leave. How many students are still in the gym?

Academic vocabulary Concrete models Place value Relationship Mathematical Practices 2. Reason abstractly and quantitatively



o Student A: I used a number line. I started at 60 and moved back 3 jumps of 10 and landed on 30. There are 30 students left.

o Student B: I used ten frames. I had 6 ten frames‐

that’s 60. I removed three ten frames because 30 students left the gym. There are 30 students left in the gym.

o Student C: I thought, “30 and what makes 60?”. I

know 3 and 3 is 6. So, I thought that 30 and 30 makes 60. There are 30 students still in the gym.

3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning


MEASUREMENT AND DATA (1.MD)









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Students measure lengths indirectly and by iterating length units. Major content 1.MD.1 Order three objects by length; compare the lengths of two objects

indirectly by using a third object. Major content

Essential Knowledge and skills First Grade students continue to use direct comparison to

compare lengths. Direct comparison means that students compare the amount of an attribute in two objects without measurement.

Examples Example: Who is taller?

o Student: Let’s stand back to back and compare our heights. Look! I’m taller!

Example: Find at least 3 objects in the classroom that are the same length as, longer than, and shorter than your forearm.

Academic vocabulary A little less than A little more than About First Gap Height Length Less Longer than Measure More

TEACHER NOTES See instructional strategies in the introduction Have students use reasoning to compare measurements indirectly, for example to order the lengths of Objects A, B and C, examine then compare the lengths of Object A and Object B and the lengths of Object B and Object C. The results of these two comparisons allow students to use reasoning to determine how the length of Object A compares to the length of Object C. For example, to order three objects by their lengths, reason that if Object A is smaller than Object B and Object B is

RESOURCE NOTES See resources in the introduction Refer to Algebra I PBA/MYA @ Live Binder http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements and clarification ORC # 4329 From the National Council of





Sometimes, a third object can be used as an intermediary, allowing indirect comparison. For example, if we know that Aleisha is taller than Barbara and that Barbara is taller than Callie, then we know (due to the transitivity of “taller than’) that Aleisha is taller than Callie, even if Aleisha and Callie never stand back to back. This concept is referred to as the transitivity principle for indirect measurement.

Example: The snake handler is trying to put the snakes in order‐ from shortest to longest. She knows that the red snake is longer than the green snake. She also knows that the green snake is longer than the blue snake. What order should she put the snakes?

Student: Ok. I know that the red snake is longer than the green snake and the blue snake because, since it’s longer than the green, that means that it’s also longer than the blue snake. So the longest snake is the red snake. I also know that the green snake and red snake are both longer than the blue snake. So, the blue snake is the shortest snake. That means that the green snake is the medium sized snake. NOTE: The Transitivity Principle (“transitivity”)1: If the length of object A is greater than the length of object B, and the length of object B is greater than the length of object C, then the length of object A is greater than the length of object C. This principle applies to measurement of other quantities as well.

Example: Which is longer: the height of the bookshelf or

the height of a desk? o Student A: I used a pencil to measure the height of

the bookshelf and it was 6 pencils long. I used the same pencil to measure the height of the desk and the desk was 4 pencils long. Therefore, the bookshelf is taller than the desk.

o Student B: I used a book to measure the bookshelf and it was 3 books long. I used the same book to measure the height of the desk and it was a little less than 2 books long. Therefore, the bookshelf is taller than the desk. Another important set of skills and understandings is ordering a set of objects by length. Such sequencing requires multiple comparisons (no more than 6 objects). Students need to understand that each object in a seriation is larger than those that come before it, and shorter than those that come after.

Example: The snake handler is trying to put the snakes in

Order Overlap Second Shorter than Third Mathematical Practices 6. Attend to precision 7. Look for and make use of structure

smaller than Object C, then Object A has to be smaller than Object C. The order of objects by their length from smallest to largest would be Object A ‐ Object B ‐ Object C. Measurement units share the attribute being measured. Students need to use as many copies of the length unit as necessary to match the length being measured. For instance, use large footprints with the same size as length units. Place the footprints end to end, without gaps or overlaps, to measure the length of a room to the nearest whole footprint. Use language that reflects the approximate nature of measurement, such as the length of the room is about 19 footprints. Students need to also measure the lengths of curves and other distances that are not straight lines. Students need to make their own measuring tools. For instance, they can place paper clips end to end along a piece of cardboard, make marks at the endpoints of the clips and color in the spaces. Students can now see that the spaces represent the unit of measure, not the marks or numbers on a ruler. Eventually they write numbers in the center of the spaces. Encourage students not to use the end of the ruler as a starting point. Compare and discuss two measurements of the same distance, one found by using a ruler and one found by aligning the actual units end to end, as in a chain of paper clips. Students should also measure lengths that are longer than a ruler. ODE

Teachers of Mathematics, Illuminations: The Length of My Feet This lesson focuses students’ attention on the attributes of length and develops their knowledge of and skill in using nonstandard units of measurement. ORC # 1485 From the American Association for the Advancement of Science: Estimation and Measurement In this lesson students will use nonstandard units to estimate and measure distances.



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order‐ from shortest to longest. Here are the three snakes(3 strings of different length and color). What order should she put the snakes? o Student: Ok. I will lay the snakes next to each other. I

need to make sure to be careful and line them up so they all start at the same place. So, the blue snake is the shortest. The green snake is the longest. And the red snake is medium‐sized. So, I’ll put them in order from shortest to longest: blue, red, green. (Progressions for CCSSM: Geometric Measurement, The CCSS Writing Team, June 2012.


1.MD.2 Express the length of an object as a whole number of length units, by

laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same‐size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps. Major content

Essential Knowledge and skills First Grade students use multiple copies of one object to

measure the length larger object. They learn to lay physical units such as centimeter or inch manipulatives end‐to‐end and count them to measure a length. Through numerous experiences and careful questioning by the teacher, students will recognize the importance of careful measuring so that there are not any gaps or overlaps in order to get an accurate measurement. This concept is a foundational building block for the concept of area in 3rd Grade.

Examples How long is the pencil, using paper clips to measure?

o Student: I carefully placed paper clips end to end.

The pencil is 5 paper clips long. I thought it would take about 6 paperclips. When students use different sized units to measure the same object, they learn that the sizes of the units must be considered, rather than relying solely on the amount of objects counted.

Example: Which row is longer?

Academic vocabulary Mathematical Practices 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure



o Student Incorrect Response: The row with 6 sticks is

longer. Row B is longer. o Student Correct Response: They are both the same

length. See, they match up end to end. In addition, understanding that the results of measurement and direct comparison have the same results encourages children to use measurement strategies.

Example: Which string is longer? Justify your reasoning. o Student: I placed the two strings side by side. The red

string is longer than the blue string. But, to make sure, I used color tiles to measure both strings. The red string measured 8 color tiles. The blue string measure 6 color tiles. So, I was right. The red string is longer.

NOTE: The instructional progression for teaching measurement begins by ensuring that students can perform direct comparisons. Then, children should engage in experiences that allow them to connect number to length, using manipulative units that have a standard unit of length, such as centimeter cubes. These can be labeled “length‐units” with the students. Students learn to lay such physical units end‐to‐end and count them to measure a length. They compare the results of measuring to direct and indirect comparisons. (Progressions for CCSSM: Geometric Measurement, The CCSS Writing Team, June 2012.)










regularity in repeated

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Students tell and write time. Additional content 1.MD.3 Tell and write time in hours and half‐hours using analog and digital

clocks. Additional content

Essential Knowledge and skills Time can be measured in units of time. Units of time include hours (60 minutes), half‐hours (30

minutes), and minutes. The hour hand moves sequentially around the clock. It

moves from one number to a new number in one hour (sixty minutes). It moves one half the distance between numbers in one‐half hour (30 minutes).

Clocks can be analog or digital. Examples Ideas to support telling time:

Academic vocabulary Half‐hour about Hour O’clock Past “six”‐thirty Time Mathematical Practices

TEACHER NOTES See instructional strategies in the introduction Students are likely to experience some difficulties learning about time. On an analog clock, the little hand indicates approximate time to the nearest hour and the focus is on where it is pointing. The big hand shows minutes before and after an hour and the focus is on distance that it has gone around the clock or the distance yet to go for the hand to get back to the top. It is easier for students to read times on digital clocks, but these do not relate times

RESOURCE NOTES See resources in the introduction Refer to Algebra I PBA/MYA @ Live Binder http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements and clarification ORC # 4328 From the





reasoning

within a day, the hour hand goes around a clock twice (the hand moves only in one direction)

when the hour hand points exactly to a number, the time is exactly on the hour

time on the hour is written in the same manner as it appears on a digital clock

the hour hand moves as time passes, so when it is half way between two numbers it is at the half hour

there are 60 minutes in one hour; so halfway between an hour, 30 minutes have passed

half hour is written with “30” after the colon “It is 4 o’clock

“It is halfway between 8 o’clock and 9 o’clock. It is 8:30.”

The idea of 30 being “halfway” is difficult for students to

grasp. Students can write the numbers from 0 ‐ 60 counting by tens on a sentence strip. Fold the paper in half and determine that halfway between 0 and 60 is 30. A number line on an interactive whiteboard may also be used to demonstrate this.

5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure


very well. Students need to experience a progression of activities for learning how to tell time. Begin by using a one‐handed clock to tell times in hour and half‐hour intervals, then discuss what is happening to the unseen big hand. Next use two real clocks, one with the minute hand removed, and compare the hands on the clocks. Students can predict the position of the missing big hand to the nearest hour or half‐hour and check their prediction using the two‐handed clock. They can also predict the display on a digital clock given a time on a one‐ or two‐handed analog clock and vice‐versa. Have students tell the time for events in their everyday lives to the nearest hour or half hour. Make a variety of models for analog clocks. One model uses a strip of paper marked in half hours. Connect the ends with tape to form the strip into a circle. ODE

National Council of Teachers of Mathematics, Illuminations: Grouchy Lessons of Time This lesson provides an introduction to and practice with the concept of time and hours. Judy clocks








structure

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Students represent and interpret data. Supporting content 1.MD.4 Organize, represent, and interpret data with up to three categories; ask

and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. Supporting content

Essential Knowledge and skills Data are gathered to answer questions about the

population from which the data was gathered. Data can be organized, represented, and interpreted in

different ways (e.g. chart or table). The number of data points can be compared in different

ways (most, least, differences, similarities).

Academic vocabulary Category Collect Data Different Interpret Least

TEACHER NOTES See instructional strategies in the introduction Ask students to sort a collection of items in up to three categories. Then ask questions about the number of items in each category and the total number of items. Also ask students to compare the number of items in each category. The total number of items to be sorted should be less than or equal to 100 to allow for sums and differences less than or equal to 100 using the numbers 0 to 100.

RESOURCE NOTES See resources in the introduction Refer to Algebra I PBA/MYA @ Live Binder http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements and clarification






Examples Survey Station

During Literacy Block, a group of students work at the Survey Station. Each student writes a question, creates up to 3 possible answers, and walks around the room collecting data from classmates. Each student then interprets the data and writes 2‐4 sentences describing the results. When all of the students in the Survey Station have completed their own data collection, they each share with one another what they discovered. They ask clarifying questions of one another regarding the data, and make revisions as needed. They later share their results with the whole class.

o Student: The question, “What is your favorite flavor of

ice cream?” is posed and recorded. The categories chocolate, vanilla and strawberry are determined as anticipated responses and written down on the recording sheet. When asking each classmate about their favorite flavor, the student’s name is written in the appropriate category. Once the data are collected, the student counts up the amounts for each category and records the amount. The student then analyzes the data by carefully looking at the data and writes 4 sentences about the data.

Less More Most Organize Question Represent Same Mathematical Practices 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision

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Instructional StrategiesConnect to the geometry content studied in Grade 1. Provide categories and have students sort identical collections of different geometric shapes. After the shapes have been sorted, ask these questions: How many triangles are in the collection? How many rectangles are there? How many triangles and rectangles are there? Which category has the most items? How many more? Which category has the least? How many less? Students can create real or cluster graphs after they have had multiple experiences with sorting objects according to given categories. The teacher should model a cluster graph several times before students make their own. A cluster graph in Grade 1 has two or three labeled loops or regions (categories). Students place items inside the regions that represent a category that they chose. Items that do not fit in a category are placed outside of the loops or regions. Students can place items in a region that overlaps the categories if they see a connection between categories. Ask questions that compare the number of items in each category and the total number of items inside and outside of the regions. ODE

Yarn or large paper for loops A variety of objects to sort Geometric shapes ORC # 5777 From the Charles A. Dana Center, University of Texas at Austin: Buttons, Buttons, Everywhere! In this lesson students use attributes such as shape, color, size, etc. to describe, compare, and sort button

GEOMETRY 1.G) Use Mathematical Practices to 1. Make sense of problems and






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Students reason with shapes and their attributes. Additional content 1.G.1 Distinguish between defining attributes (e.g., triangles are closed and

three‐sided) versus non‐defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes. Additional content

Essential Knowledge and skills Shapes have attributes that can be described in words. Some attributes are defining attributes while others are

non‐defining. Examples All triangles must be closed figures and have 3 sides. These

Academic vocabulary Attributes Circle Closed Cone Cube

TEACHER NOTES See instructional strategies in the introduction Students do not need to learn formal names such as “right rectangular prism.” Students can easily form shapes on geoboards using colored rubber bands to represent the sides of a shape. Ask students to create a shape with four sides on their geoboard, then copy the

RESOURCE NOTES See resources in the introduction Refer to Algebra I PBA/MYA @ Live Binder http://www.livebinders.com/play/play/1171650 for mid‐year evidence statements







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are defining attributes. Triangles can be different colors, sizes and be turned in different directions. These are non‐defining attributes. o Student: I know that this shape is a triangle because it

has 3 sides. It’s also closed, not open.

o Student: I used toothpicks to build a square. I know

it’s a square because it has 4 sides. And, all 4 sides are the same size.

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Cylinder Feature Features Open Pyramid Rectangle Rectangular prism Shape Side Sphere Square Trapezoid Triangle Two‐dimensional Mathematical Practices 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 7. Look for and make use of structure


1.G.2 Compose two‐dimensional shapes (rectangles, squares, trapezoids, triangles, half‐circles, and quarter‐circles) or three‐dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. Additional content

Essential Knowledge and skills Shapes can be composed and decomposed. Examples What shapes can you create with triangles?

Academic vocabulary 2‐dimensional 3‐dimensional Composite Half Quarter

shape on dot paper. Students can share and describe their shapes as a class while the teacher records the different defining attributes mentioned by the students. Pattern block pieces can be used to model defining attributes for shapes. Ask students to create their own rule for sorting pattern blocks. Students take turns sharing their sorting rules with their classmates and showing examples that support their rule. The classmates then draw a new shape that fits this same rule after it is shared. ODE Students can use a variety of manipulatives and real‐world objects to build larger shapes. The manipulatives can include paper shapes, pattern blocks, color tiles, triangles cut from squares (isosceles right triangles), tangrams, canned food (right circular cylinders) and gift boxes (cubes or right rectangular prisms). ODE Folding shapes made from paper enables students to physically feel the shape and form the equal shares. Ask students to fold circles and rectangles first into halves and then into fourths. They should observe and then discuss the change in the size of the parts. ODE

and clarification



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First graders learn to perceive a combination of shapes as a single new shape (e.g., recognizing that two isosceles triangles can be combined to make a rhombus, and simultaneously seeing the rhombus and the two triangles). Thus, they develop competencies that include: o Solving shape puzzles o Constructing designs with shapes o Creating and maintaining a shape as a unit

As students combine shapes, they continue to develop their sophistication in describing geometric attributes and properties and determining how shapes are alike and different, building foundations for measurement and initial understandings of properties such as congruence and symmetry. (Progressions for the CCSS in Mathematics: Geometry, The Common Core Standards Writing Team, June 2012)

Mathematical Practices 1. Make sense of problems and persevere in solving them 4. Model with mathematics 7. Look for and make use of structure


1.G.3 Partition circles and rectangles into two and four equal shares, describe

the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. Additional content

Essential Knowledge and skills Shapes can be divided into equal size parts called shares. Shares are fractional parts of a whole that can be named. The more equal shares one creates from a whole the

smaller the parts become. The size of the share depends on the size of the whole. Examples Example: How can you and a friend share equally

(partition) this piece of paper so that you both have the

Academic vocabulary Decomposing Equal shares Fourths Half Partition Quarters Mathematical Practices



same amount of paper to paint a picture?

Example: Let’s take a look at this pizza.

o Teacher: There is pizza for dinner. What do you notice about the slices on the pizza?

o Student: There are two slices on the pizza. Each

slice is the same size. Those are big slices! o Teacher: If we cut the same pizza into four slices

(fourths), do you think the slices would be the same size, larger, or smaller as the slices on this pizza?

o Student: When you cut the pizza into fourths, the

slices are smaller than the other pizza. More slices mean that the slices get smaller and smaller. I want a slice from that first pizza!

2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 6. Attend to precision 7. Look for and make use of structure


Documents

MIDDLETOWN PUBLIC SCHOOLS MATHEMATICS CURRICULUM Grade 1 · PDF fileGrade 1 Elementary School ... he Middletown Public Schools Mathematics Curriculum for grades K‐12 was revised